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CO, Torr
0.85 0.80 0.75 0.70 0.65 0.60 0.55
0
0.1
0.2
0.3
0.4
0.5
0.6 C 0.7
Figure 20: The effect of foreign particle concentration on the average, partial pressure of CO inside the representative elementary volume of the alveolar region after ten seconds, for a single-breath test. DL
18 17 16 15 14 13 12
0
0.1
0.2
0.3
0.4
C
0.5
Figure 21: Lung-diffusing capacity versus concentration of single particles. Figure 20 shows the time evolution of CO partial pressure in the course of the single-breath transient simulation for the case of different values of the particle concentration (single particles only; there are no particle clusters in the domain). As seen from the figure, the value of the partial pressure increases nonlinearly with the increase of the particle concentration. The corresponding values of the lung-diffusing capacity are shown in Fig. 21. An interesting observation can be made by considering Fig. 21. It is clearly seen that the behavior of the curve changes at the value of concentration C = 0.12. The same effect was observed when different values of the red blood cells concentration were considered (see the preceding section). What is even more interesting is the prediction made by Wijayanto et al [20] (see also Chap. 5 of this book). It has been shown that the value C = 0.12 corresponds to the maximum value of
60 Human Respiration
L
D , ml of CO/(min mmHg)
16.5 16 15.5 1 ppc 2 ppc 3 ppc 4 ppc 5 ppc 6 ppc 7 ppc
15 14.5 14 13.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 C 0.8
Figure 22: The effect of particle clustering on the lung-diffusing capacity [the numbers in the legend show the average number of particles per cluster (ppc)]. concentration to be achieved in the alveolar region during a single respiration cycle, if the value of concentration at the entrance to the trachea is equal to unity. The effect of particle clustering on the lung-diffusing capacity is shown in Fig. 22. From Fig. 22, it is possible to see that the value of the lung-diffusing capacity decreases monotonically with increased particle concentration. The effect of the particle clustering can be clearly observed from the figure: the value of the lungdiffusing capacity steadily decreases as the average number of particles per cluster increases. One can see, however, that, as the average number of particles per cluster increases, the effect on the lung-diffusing capacity becomes less pronounced. This can be explained by the fact that the larger the number of particles per cluster the more particles are shielded from participating in the diffusion process. This, in turn, leads to the lung-diffusing capacity value becoming less sensitive to the clustering effect. Nevertheless, it is necessary to bear in mind that, as the average number of particles per cluster increases, more red blood cells become screened from the gasexchange process. Hence, the absolute amount of CO remains large at the end of the single-breath test (see Fig. 20). Therefore, although values of the lung-diffusing capacity may remain large enough, the dose of CO obtained during a single respiration cycle becomes larger as the average number of particles per cluster increases. Thus, the effect of intoxication by CO may take place (this effect is quantified in Chap. 6 of this book). Exact values of the lung-diffusing capacity corresponding to different values of particle concentration and particle clustering are given in Table 2. In an attempt to quantify the effect of particles on the lung-diffusing capacity, it is possible to propose the following empirical relationship that fairly well describes the effect observed in the course of numerical simulations: DL (C; k) =
16.6 − 0.39k , 1 + 0.075C
(44)
Fundamentals of Alveolar Gas Diffusion
61
Table 2: Values of the lung-diffusing capacity corresponding to different values of particle concentration and particle clustering. C
1 ppc
2 ppc
3 ppc
4 ppc
5 ppc
6 ppc
7 ppc
0.050000 0.10000 0.20000 0.30000 0.40000 0.50000
16.215 16.159 15.979 15.854 15.572 15.334
16.190 16.121 15.884 15.703 15.380 15.022
15.897 15.783 15.583 15.392 15.182 14.956
15.769 15.446 15.139 14.867 14.605 14.478
15.185 14.930 14.491 14.200 14.071 13.946
14.514 14.127 13.876 13.753 13.743 13.733
13.874 13.721 13.638 13.638 13.638 13.638
where C is the particle concentration and k is the average number of particles per cluster. Thus, it has been shown how the presence of microscopic solid particles in the alveolar region affects the lung-diffusing capacity and, consequently, the overall lung performance. The effect of the particles clustering on the lung-diffusing capacity has also been investigated. It has been found that particle clustering moderately affects the value of the lung-diffusing capacity. However, due to the particles’ presence, more red blood cells become shielded from the gas-exchange process that leads to a larger amount of CO accumulated in the alveolar region. This may lead to a more pronounced intoxicating effect. In addition, an attempt was made to quantify the effect of the particles’ presence on the lung-diffusing capacity. An empirical relationship for doing so has been proposed (eqn. (44)).
3 Visualization of the results Bearing in mind the extreme geometrical complexity of the domain in question and the complexity of the solution itself, it becomes crucial to be able to visualize the results. Such an attempt is presented in this section. Volume visualization has been used for rendering when simulating the respiration process. The visualization of the model has been done with the free-ware software VolPack (The VolPack Volume Rendering Library). VolPack is a portable software library for volume-rendering based on the fast volume-rendering algorithms (Lacroute and Levoy [11]). Based on this library, we developed the software, which makes computer animation from the key-frame volume data calculated during the simulation. The software lets us interpolate the in-between frames, manipulate the volume on the screen with time-dependent affine transformations (rotation and zooming), and control the visual appearance of the volume (color and opacity). While the simulation software runs on a personal computer, the volume visualization subsystems runs on an SGI workstation remotely so that the resulting movie file
62 Human Respiration
Figure 23: Snapshots of the simulated video for four different RBC distributions. can be downloaded back to the PC for further post-production processing. The user may either control the process of rendering visually, or run it in the batch mode. Several video files for different RBC distributions have been created for this project (see Fig. 23). The streaming video versions of them are available on-line at http://www.ntu.edu.sg/home/assourin/Scivi.htm. The first row of snapshots in Fig. 23 represents the results obtained in the case of the random distribution of the red blood cells within the domain. The second and third rows represent the center- and corner-clustering cases, respectively; whereas the fourth row exemplifies the results obtained for the chain distribution. By comparing the four cases, one can clearly see that, in terms of the diffusion efficiency, the first and last cases are much more effective than the clustering cases. This can be explained by the presence of the so-called shielding effect. Indeed, when the red blood cells form clusters, those red blood cells that are nearer the center of the cluster become partly or totally excluded from the gas-exchange process. Hence, this process goes slower or, in other words, is less effective. It can be seen from the figure that the normal distribution is the most efficient one. This is not surprising because almost no screening effect is present in this case. It is noteworthy to point out here that a simple visual analysis of the results leads to the same conclusions that were drawn in the first section of this chapter after analyzing the results analytically. This proves that visualization is a very powerful analysis tool. Figure 24 presents a detailed view of the snapshots obtained upon visualizing the results corresponding to the case of the normal (random) distribution of the red blood cells within the alveolar region. From Fig. 24, one can clearly see that the diffusion process in the case of the randomly distributed red blood cells goes very fast: almost no carbon monoxide remains in the representative elementary volume at the middle of the test.
Fundamentals of Alveolar Gas Diffusion
63
Figure 24: Snapshots of the simulated video for the random distribution of RBC. Looking at the first row in the figure, one can see the distinct granular structure on the snapshots. This may serve as a measure of how many separate red blood cells participate in the gas-exchange process. Since almost all red blood cells in this case stay far from each other, all of them suck CO effectively, so that no shielding effect takes place. Note here that the snapshots corresponding to the uniform distribution of the red blood cells are not presented: the picture obtained is indistinguishable from the picture corresponding to the random (normal) distribution (Fig. 24). Both distributions have about the same effectiveness and their effect looks similar. This was explained in detail in the first section of this chapter. Figure 25 presents a detailed view of the snapshots obtained upon visualizing the results corresponding to the case of the central clustering of the red blood cells within the alveolar region. The effectiveness of the diffusion process in this case is very low. This can be seen by comparing the second and fourth rows in the figure: almost no difference in the snapshots is observed. The diffusing strength of the center cluster is not enough even to remove the test gas (carbon monoxide) from the corners of the representative elementary volume. The figure clearly shows that the gas exchange takes place mainly in the near vicinity of the cluster, so that the space freed from CO (the white spot on the snapshots) does not noticeably expand in the course of the test. Figure 26 presents a detailed view of the snapshots obtained upon visualizing the results corresponding to the case of the corner clustering of the red blood cells within the alveolar region.
64 Human Respiration
Figure 25: Snapshots of the simulated video for the center clustering of RBC.
Figure 26: Snapshots of the simulated video for the corner clustering of RBC. The result is similar to that observed in Fig. 25, except that, in this case, the red blood cells form the cluster in the front-upper corner of the representative elementary volume. This circumstance leads to an even lower effectiveness of diffusion that characterized the case of the center clustering. Indeed, fifty per cent
Fundamentals of Alveolar Gas Diffusion
65
Figure 27: Snapshots of the simulated video for the chain distribution of RBC.
of the red blood cells that form the boundary of the cluster are now excluded from the diffusion process (unlike in the previous case, where all the boundary cells participated in the gas exchange) (see the first section of this chapter for details). Figure 27 presents a detailed view of the snapshots obtained upon visualizing the results corresponding to the case of the chain distribution of the red blood cells within the alveolar region. The same granular structure that was observed in Fig. 24 can be seen on the two first snapshots of Fig. 27. Yet the linear patterns become pronounced starting from the second row of the snapshots. It can be clearly seen from the figure that the chain distribution is much more effective than the two clustering distributions considered previously. However, the effectiveness of the chain distribution is lower than the effectiveness of the normal (random) distribution of the red blood cells. This is because each red blood cell in the chain distribution has at least two neighbors that screen it partly from access to CO, whereas the probability to have neighbors in the case of the random (normal) distribution is much lower (see the first section of this chapter for details). Volume visualization proves to be useful for presenting the results obtained from direct single-breath test measurements or their numerical simulations. By the use of volume visualization, one can have a clearer picture of the lung performance in general and of the abnormalities of the diffusion process in particular. Volume visualization allows one to have an overall picture of the process that is impossible to have otherwise.
66 Human Respiration There is a hope that volume visualization of the microscopic gas exchange will become a powerful tool for pronouncing preliminary judgment on respiratory abnormalities. It is believed that further developments in this area may become useful for early diagnostics of obstructive respiratory diseases in different classes of patients.
Appendix A: Porous medium alveolar model Consider a volume in space representative, from a statistical point of view, of the alveolar region of the lung. The main structures (constituents) of this porous region are (see Fig. 4): the alveolus or gas region (a), the alveolus membrane (m), the interstitial fluid region (f ), the capillary membrane (c), the plasma region (p), the red blood cell membrane (r), and red blood cell interior (h). In treating the gas transport within the alveolar region, it is usual to lump regions m, f , c, and p into one single representative region, denominated here the tissue region (t), mainly because these constituents have comparable molecular gas diffusivities (Hsia et al, [8]). Moreover, the intricacies of the chemo-biological process within the red blood cell are avoided by considering a certain gas concentration (or partial pressure) uniformly distributed within h. The gas-diffusion transport is then driven, in principle, by a concentration imbalance between the gas concentration along the outer boundary of the alveolar region and the gas concentration within each red blood cell. The microscopic equation modeling the gas-diffusion transport within each alveolar constituent, can be written as: ∂c = −∇ · j + S, ∂t
(A.1)
where c is the representative concentration of solute (gas), j is the flux of solute into the medium (e.g., air mixture), and S is a source of solute (negative if sink). Considering a Fikian-type diffusion, the flux is related to the local gradient of gas concentration as j = −D∇c,
(A.2)
where D is the molecular diffusivity of solute into the solvent (D accounts also for the solubility of the gas in a nongaseous medium, e.g., within the tissue region). Equation (A.2) can be written in terms of the gas partial pressure, P (instead of gas concentration), by using the ideal gas law: P = c(RT /M ), where R is the universal gas constant, T is the temperature of the gas, and M is the molecular mass of the gas. The partial-pressure-based diffusion equation equivalent to eqn. (A.2) is P M j = −D M ∇ (T ∇P − P∇T ) . = −D R T RT 2
(A.3)
Fundamentals of Alveolar Gas Diffusion
67
Substitution of eqn. (A.3) into eqn. (A.1), with (DM/R) considered invariant in space and using the ideal gas law, leads to M R
1 ∂P P ∂T − 2 T ∂t T ∂t
2P DM 1 2 2 2 (∇T ) −2∇T · ∇P + T ∇ + S. P − P∇ T + = R T2 T
(A.4)
In the case of isothermal diffusion, as within the alveolar region of homeotherms, eqn. (A.4) simplifies to M M ∂P = D ∇ 2 P + S. TR ∂t TR
(A.5)
For a region with no source or sink, S = 0, and the groups M/TR cancel out. Considering now the three-constituent model of the alveolar region (i.e., alveolus (a), tissue (t), and red blood cell membrane (r)), one can write the equation and boundary conditions valid for each constituent in terms of the gas partial pressure. For the alveolus a-region: ∂Pa = Da ∇ 2 Pa , ∂t
(A.6)
where Pa and Da are the partial pressure and the molecular diffusivity of the gas within the alveolus region. This equation can be solved with an appropriate initial condition and boundary conditions Pa = P0 along A0a , where P0 is a representative partial-pressure value along the outer boundary surface area A0a of the alveolus, and Pa = Pt along the interface Aat between the alveolus and the tissue regions. For the tissue region: ∂Pt = Dt ∇ 2 P t , ∂t
(A.7)
where Pt and Dt are the partial pressure and the equivalent diffusivity of the gas within the tissue region, respectively. Boundary conditions in this case are Pt = Pr along the interface area Atr between the tissue and the red blood cell membrane regions, and nat · Dt ∇Pt = nat · Da ∇Pa .
(A.8)
Equation (A.8) represents a balance of gas flux across the alveolus/tissue interface surface area Aat , with normal unit vector nat . For the red blood cell membrane region: ∂Pr = Dr ∇ 2 P r , ∂t
(A.9)
68 Human Respiration where Pr is the gas partial pressure within the red blood cell membrane region, and Dr is the molecular diffusivity of the gas in the red blood cell membrane. Boundary conditions in this case are ntr · Dr ∇Pr = ntr · Dt ∇Pt ,
(A.10)
representing a balance of gas flux across the tissue/red blood cell membrane interface surface area Atr with normal unit vector ntr . The second and final boundary condition is Pr = Ph along the interface area Arh between the red blood cell membrane and the red blood cell interior region. In general, Ph is not known. However, when simulating the diffusion of CO within the lung (CO is the gas used in clinical tests to determine the lung-diffusing capacity), the partial pressure of CO within a red blood cell can be assumed zero, because the red blood cells behave effectively as an infinite sink for CO (Comroe et al [4]). Equations (A.6)–(A.10) with boundary conditions and appropriate initial conditions, can be used to simulate the diffusion process within the alveolar region. The difficulty is the need to describe (locate) precisely the region occupied by each constituent and the related interface surfaces. To circumvent this difficulty, one might consider a certain volume V of the alveolar region with representative dimension d , where d is much smaller than L (the characteristic length of the alveolar region within the lung), much larger than da (the characteristic alveolus dimension inside V ) and tc (the characteristic capillary bed thickness inside V ), and much larger than tr (the characteristic red blood cell membrane dimension). As defined, V is a representative elementary volume (REV) of the alveolar region. The alveolar region can be built by placing several regions like V next to each other. The fact that V is a REV of the alveolar region means that it contains all the necessary information for describing the physical properties of the alveolar region. Therefore, all constituents of the alveolar region must be represented in (or contained by) this REV. Inside this REV, there can be many alveoli and a long capillary bed with several red blood cells. Because the pore-level (alveolus-erythrocyte level, or microscopic level) solution of the diffusion equation within such a medium seems impractical at the moment, a model that predicts the diffusion process inside the alveolar region at the REV level is now considered. The model in question is developed by applying the method of volume averaging (Carbonell & Whitaker [3], Nozad et al [13], Whitaker [19]) to transform the effects of the internal geometry of the alveolar region from a microscopic level to a macroscopic representative elementary volume level (REV-level). This transformation does not come free: the mass-transfer interaction at every interface surface area between constituents within a REV must be accounted for and modelled. Let us initially define an intrinsic average of the partial pressure Pi as
P i =
1 Vi
Pi dV , Vi
(A.11)
Fundamentals of Alveolar Gas Diffusion
69
where Vi is the volume within V occupied by constituent ‘i’ (‘i’ can be the alveolus a-region, the tissue t-region or the red blood cell membrane r-region). Now write Pi = P i + P˜ i , where P˜ i is the space (within V ) variation of Pi from the volumeaveraged value P i . After integrating eqns. (A.6), (A.7) and (A.9) within V , one obtains, respectively,
Va 2 1 Va ∂ P a ˜ = Da ∇ P a + ∇· nat Pa dA + nat · ∇ P a dA V ∂t V V Aat Aat
Vt ∂ P t = Dt V ∂t
Vt 2 1 ∇ P t + V V
Vr ∂ P r Vr 2 1 = Dr ∇ P r + V ∂t V V
(A.12)
∇ · Ata nta P˜ t dA + Atr ntr P˜ t dA + Ata nta · ∇ P t dA + Atr ntr · ∇ P t dA
∇· Art
nrt P˜ r dA +
(A.13)
nrt · ∇ P r dA
.
Art
(A.14)
Equations (A.12), (A.13) and (A.14) are the REV-averaged diffusion equations for alveolus, tissue, and red blood cell membrane regions, respectively. Now, all three equations are added term by term keeping in mind that: Va + Vt + Vr = V , and, nij = −nji . The resulting equation, simplified further by invoking the equilibrium assumption, i.e., P a = P t = P r = P (where P is the representative – effective – partial pressure within V ), and P˜ i = P˜ j (along Aij , where ‘i’ and ‘j’ can be a, t, or r), is ∂ P = ∂t
Vt Vr Va Da + Dt + Dr ∇ 2 P V V V
1 (Da − Dt ) ∇ · + n at P˜ a dA + (Dt − Dr ) ∇ · n tr P˜ t dA V Aat Atr (A.15)
The difficulty in applying eqn. (55) for modeling the diffusion process still lies on describing precisely the location of the interface surface areas Aat and Atr . We ˜ namely proceed with a closure constitutive equation for the perturbation term P, nij P˜ i dA = di ∇ P , (A.16) Aij
where di is a pseudodiffusion coefficient (a tensor, in general), considered here to be uniform and isotropic. Note that the form of the closure, eqn. (16), proposed by Nozad et al (1985) when modeling heat conduction in porous media, is similar in scope to the eddy-diffusivity concept used in modeling turbulence (i.e., a pseudodiffusivity can be defined as the ratio between the fluctuating quantity and the
70 Human Respiration gradient of the related average quantity). Substituting eqn. (A.16) into eqn. (A.15) and after rearranging the terms: ∂ P = Deff ∇ 2 P , ∂t
(A.17)
where Deff is the effective mass diffusivity coefficient of the REV, formally defined as 1 Va Vt Vr Da + Dt + Dr + [(Da − Dt )da + (Dt − Dr )dt ] . (A.18) Deff = V V V V Observe that once Deff is known, eqn. (A.17) is easily solved by applying a suitable initial condition (in terms of P ) and boundary conditions, for instance,
P = P 0 along the outer surface of the alveolar region, and P = P h within the region occupied by the red blood cells inside the alveolar region of the lung. One does not need to know the effective diffusivity of the alveolar region when considering a steady diffusion process. In this case, eqn. (A.17) reduces to the Laplace equation in P . This detail is explored in section 1.6.1, where the numerical simulation of the steady diffusion process in an alveolar region is discussed. Appendix B: Relationship between the lung-diffusing capacity and the effective diffusion coefficient of the alveolar region: unifying theory The lung-diffusing capacity DL , measured in the laboratory (using CO) and available in the literature (Comroe et al [4], pp. 117–133), is defined as ◦
DL =
V , ( P 0 − P h )
(B.1)
◦
where V is the volumetric gas flow rate crossing the outer surface of the alveolar region (gas uptake) and P 0 is the mean alveolar partial pressure of CO. It is easy to verify that the volumetric flow rate is related to the absolute value of the mass flux js crossing the boundary (surface) of the domain, via ◦
V = js As
RT M,
P b
(B.2) ◦
where As is the total surface area through which js flows. Eliminating V between eqns. (B.1) and (B.2), setting P b = P 0 , and using js = (Ai /As )Deff (M /RT ) (|∇ P s |), Ai 1 DL = Deff ∇ P s , (B.3)
P b − P h P b where Ai is the node surface area. The group [As ( P b − P h )/(Ai ∇ P s )] represents the characteristic diffusion length of the domain.
Fundamentals of Alveolar Gas Diffusion
71
For a certain domain geometry, RBC distribution, red blood cell density ρ and known (measured) DL value, one can then use eqn. (B.3) to determine numerically an estimate of the effective lung diffusivity of the medium. This effectivediffusivity estimate can be used with the macroscopic model, eqn. (A.17), to simulate the transient diffusion process within the cubic alveolar region. Equation (B.3) provides only estimates of Deff because the equation depends on the surface gradient, a quantity obtainable only from a steady-state model (because Deff is not known). Recall that DL is obtained from transient experiments. (Note: even though one of the experimental schemes to measure DL is termed the steadystate technique – Comroe et al [4], p. 121 – the diffusion process during measurement is still inherently unsteady because of the periodicity of the respiration process.) On the other hand, if Deff is known, one can, for instance, change the form of the domain or the RBC distribution (keeping the same red blood cell density), find the ∇ P s value for the new configuration, and then use eqn. (B.3) to determine the corresponding DL of the new domain. In this way, one can now anticipate (predict) the effect on DL of a certain change in the morphology of the alveolar region.
Appendix C: Lumped-capacity approach to the alveolar diffusion The Krogh equation can be derived from a lumped-capacity approach to the diffusion process inside the alveolar region (Comroe et al [4], p. 351). (Observe that the hint for exploring a lumped-capacity approach emanates from the time evolution of P v being exponential, as illustrated in Fig. 15.) To derive the Krogh equation from the lumped-capacity approach, consider the alveolar region as a closed impermeable system of volume VA , filled with a gas having a certain concentration of CO. The volumetric rate of CO being depleted (sink) from inside this system, expressed in terms of the lung-diffusing capacity, is ◦
V CO = DL ( P 0 − P h ) .
(C.1)
Observe that P 0 in eqn. (C.1), in the case of a transient configuration, is in fact
P v (t). Also recall that P h for the case of CO diffusion can be considered equal to zero. Now, the species concentration conservation law, assuming the gas concentration within the domain as uniform, requires m ˙ = VA
d c v (t) , dt
(C.2)
where m ˙ CO is the CO depleting mass flow rate and c v is the volume average of the macroscopic gas concentration within the domain.
72 Human Respiration Equations (C.1) and (C.2) can be combined by setting m ˙ CO t V˙ CO (t) = , cref
(C.3)
where cref is a constant reference-gas concentration value. Using eqns. (C.1) and (C.2), eqn. (C.3) becomes DL P v (t) = −
VA d c v (t) . cref dt
(C.4)
The last term of eqn. (C.3) can be rewritten in terms of the partial pressure of CO, using the ideal gas law (c = MP/RT ), then DL P v (t) = −
VA d P v (t) . Pref dt
(C.5)
Solving eqn. (C.5),
P v (t) = P v (0)e
−
DL Pref VA
t
,
(C.6)
where P v (0) is the initial value of the volume-averaged partial pressure. When the reference pressure Pref is chosen as the total pressure of dry gases (equal to 9.51 × 104 Pa, or 713 mmHg), eqn. (C.6) becomes identical to the Krogh equation (Comroe et al [4], p. 351). The characteristic time constant for the diffusion process, according to eqn. (38), is VA /(DL Pref ), typically equal to 24.4 s (using VA = 4930 ml, DL = 17 mlCO /mmHg min, and Pref = 713 mmHg). This means that it takes approximately 25 seconds for the volume-averaged partial pressure of CO (or concentration) in the lungs to be reduced to 37.3 per cent of the initial value. Note that using the diffusion and solubility laws (Comroe et al [4], p. 350), it is possible to show that the lung-diffusing capacity of oxygen is about 1.23 times the diffusing capacity of carbon monoxide. Therefore, the characteristic time for diffusion of oxygen is approximately equal to 20 seconds. Observe how the solution of eqn. (C.6) leads to an exponential decay of P v with time. The lung-diffusing capacity can be obtained easily from the numerical results (Fig. 10), via
VA
P v (0) ln . (C.7) DL = Pref t
P v (t) During clinical single-breath tests, eqn. (C.7) is used for determining the lungdiffusing capacity considering the alveolar volume VA as equal to the inspired volume plus the residual lung volume. It is easy to determine a condition for the validity of the lumped-capacity approach by considering the definition of the lung-diffusing capacity as equivalent to the
Fundamentals of Alveolar Gas Diffusion
73
definition of the mass-transfer coefficient from Newton’s law of cooling. In this regard, one has m ˙ CO = cref DL ( P v − P h ) =
( P v − P h ) . RD L
(C.8)
Therefore, the equivalent resistance RDL is simply 1/(cref DL ). The representative resistance of the internal diffusion process is m ˙ CO ∼ ADeff
M ( P v − P h ) ( P v − P h ) , = RT
RDeff
(C.9)
where is a representative (scale) diffusion length for the diffusion in the global (Krogh) domain and A is the representative area of the alveolar region (a good candidate for this area would be the surface area of all RBCs inside the domain). Therefore, the equivalent resistance RDeff is RT /(ADeff M ). For the lumped-capacity approach to be valid, one would expect RDL to be larger than RDeff . Using eqns. (C.8) and (C.9), the criterion becomes cref RT DL
DL ∼ Pref < 1, M A Deff A Deff
(C.10)
where estimates for parameters are Pref ∼ 105 Pa, /A ∼ 10−1 /[0.034(10−1 )3 (6/12 × 10−6 )] ∼ 10−3 m−1 , DL ∼10−9 m3 /s Pa, Deff ∼ 10−7 m2 /s, and the left side of the inequality shown in eqn. (C.10) becomes approximately equal to 1.0. This result indicates that the lumped-capacity model is close to not being valid. Observe, from Fig. 10, that the curve γ versus t does not follow the perfectly straight dashed line. Keep in mind that because the time duration of the clinical single-breath test is generally short (about 10 seconds), the deviation from the lumped-capacity model might not be easily observed.
References [1] American Thoracic Society, Single breath carbon monoxide diffusing capacity (transfer factor): recommendations for a standard technique, American Review of Respiratory Disease, 136, pp. 1299–1307, 1987. [2] Burri, P.H., Gil, J. & Weibel, E.R., Ultrastructure and Morphometry of Human Lung, General Thoracic Surgery, Vol. 1, 4th edn, Williams & Wilkins, 1996. [3] Carbonell, R.G. & Whitaker, S., Heat and mass transfer in porous media. Fundamentals of Transport Phenomena in Porous Media, eds. J. Bear & M.Y. Corapcioglu, Martinus Nijhoff, Dordrecht, The Netherlands, pp. 123– 198, 1984. [4] Comroe, J.H., Forster, R.E., Dubuis, A.B., Briscoe, W.A. & Carlsen, E., The Lung Clinical Physiology and Pulmonary Functions Tests, Chicago, Year Book Medical Publisher, pp. 117–121, 1962.
74 Human Respiration [5]
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[16] [17] [18]
[19]
Crapo, J.D. & Crapo, R.O., Comparison of total lung diffusion capacity and the membrane component of diffusion capacity as determined by physiologic and morphometric techniques, Respiratory Physiology, 51, pp. 183–194, 1983. Crapo, J.D., Crapo, R.O., Jensen, R.L., Mercer, R.R. & Weibel, E.R., Evaluation of lung diffusing capacity by physiological and morphometric techniques, Journal of Applied Physiology, 64, pp. 2083–2091, 1988. Fedrespiel, W.J., Pulmonary diffusing capacity: implications of two-phase blood flow in capillaries, Respiratory Physiology, 77, pp. 119–134, 1989. Hsia, C.C.W., Chuong, C.J.C. & Johnson, R.L. Jr., Critique of conceptual basis of diffusing capacity estimates: a finite element analysis, Journal of Applied Physiology, 79, pp. 1039–1047, 1995. Johnson, R.L. Jr., Spicer, W.S., Bishop, J.M. & Forster, R.E., Pulmonary capillary blood volume, flow, and diffusing capacity during exercise, Journal of Applied Physiology, 15, pp. 893–902, 1960. Koulich (Kulish), V.V., Lage, J.L., Hsia, C.C.W. & Johnson, R.L. Jr., A Porous Medium Model of Alveolar Gas Diffusion, Journal of Porous Medium, 2, pp. 263–275, 1999. Lacroute, P. & Levoy, M., Fast Volume Rendering Using a Shear-Warp Factorization of the Viewing Transformation, Proceedings SIGGRAPH ’94, pp. 451–458, 1994. Newth, C.J.L., Cotton, D.J. & Nadel, J.A., Pulmonary diffusing capacity measured at multiple intervals during a single exhalation in man, Journal of Applied Physiology, 43, pp. 617–625, 1977. Nozad, I., Carbonell, R.G. & Whitaker, S., Heat conduction in multiphase systems 1: Theory and experiments in two-phase system. Chemical Engineering Science, 40, pp. 843–855, 1985. Philips Compact Encyclopedia, 1999, 2nd edn, Octopus Publishing Group Ltd: London. Roughton, F.J.W. & Foster, R.E., Relative importance of diffusion and chemical reaction rates in determining the rate of exchange of gases in the human lung, with special reference to true diffusing capacity of the pulmonary membrane and volume of blood in lung capillaries, Journal of Applied Physiology, 11, pp. 290–302, 1957. Shields, T.W., General Thoracic Surgery, Williams & Wilkins: Baltimore, 1994. Weibel, E.R., Morphometric estimation of pulmonary diffusion capacity. I. Model and method, Respiratory Physiology, 11, pp. 54–75, 1970. Weibel, E.R., Federspiel, W.J., Fryder-Doffey, F., Hsia, C.C.W., Konig, M., Stalder-Navarro, V. & Vock, R., Morphometric model or pulmonary diffusing capacity. I. Membrane diffusing capacity, Respiratory Physiology, 93, pp. 125–149, 1993. Whitaker, S., Improved constraints for the principle of local thermal equilibrium. Industrial Engineering Chemistry Research, 30, pp. 983–987, 1991.
Fundamentals of Alveolar Gas Diffusion
75
[20] Wijayanto, B., Kulish, V.V. & Lim Chu Sing, Modeling of Two-Phase Flow in the Human Tracheo-Bronchial System. Singapore General Hospital Proceedings, 11(1), pp. 40–46, 2002. [21] The VolPack Volume Rendering Library, http://graphics.stanford.edu/software/volpack/ [22] Streaming video files for different RBC distributions, http://www.ntu.edu.sg/ home/assourin/Scivi.htm
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CHAPTER 3 Lung-gas composition and transfer analysis: O2 and CO2 diffusion coefficients and metabolic rates D.N. Ghista1 , K.M. Loh2 & D. Ng3 1 School
of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798; email:mdnghista@ntu,edu.sg 2 School of Engineering (Electronics), Nanyang Polytechnic, Singapore. 3 Department of Nuclear Medicine and PET, Singapore General Hospital, Singapore.
Abstract The primary function of the lung is to (i) oxygenate the blood and thereby provide oxygen to the cells for metabolization purposes, and (ii) to remove the collected CO2 from the pulmonary blood. Herein, we will analyze the compositions of the inspired and expired air per breath, and from there compute the O2 consumption and CO2 production rates. Next, we derive expressions for diffusion coefficients DO2 and DCO2 in terms of the evaluated cardiac output, O2 and CO2 concentrations in arterial and venous blood, alveolar and blood O2 and CO2 partial pressures. We then take up a typical case study, and demonstrate the computation of DO2 and DCO2 , to represent the lung-performance capability to oxygenate the blood.
1 Introduction The lung-functional performance is characterized by (i) its ventilatory capacity, to bring air (and hence O2 ) into the alveoli, and (ii) its capacity to transfer O2 and CO2 into and from the pulmonary capillary bed. Hence, the O2 and CO2 diffusion coefficients as well as the O2 consumption rate and the CO2 production rate represent the lung-performance indices.
78 Human Respiration
2 Lung-air composition analysis (and O2 consumption and CO2 production rates) We carry out a mass-balance analysis, involving: (i) (ii)
compositions of air breathed in and out consumption or losses of O2 , CO2 and H2 O.
Table 1 provides clinical data on partial pressures and volumes of N2 , O2 , CO2 and H2 O of atmospheric air breathed in and expired out, one breath cycle. The monitored breathing rate (BR) = 12 breaths/min, and we assume PH2 O at 37◦ C = 47 mmHg. It can be noted that the expired air volume exceeds the inspired air volume for this particular breath cycle. The H2 O loss of 30.1 ml (=32.6–2.5 ml) contributes the major portion of this difference. 2.1 Calculation of O2 consumption rate and CO2 production rate We now determine the O2 consumption rate and CO2 production rates from the inspired and expired gases. Assuming the patient breathes at 12 times per minute we have O2 Consumption Rate = (Inspired O2 − Expired O2 ) × 12 = (104.2 − 80.6) × 12 = 283.2 ml/min
Table 1: Inspired air composition and partial pressures. Atmospheric air Respiratory gases
mmHg
N2
597
O2
159
CO2
0.3
H2 O
3.7
Total
760
Expired air
ml/%
mmHg
ml/%
393.1 78.55% 104.2 20.84% 0.2 0.04% 2.5 0.49% 500 100%
566
393.1 74.5% 80.6 15.7% 19.1 3.6% 32.6 6.2% 525.3 100%
120 27 47 760
Lung-Gas Composition and Transfer Analysis
79
Dead space (150 ml)
Figure 1: Dead-space volume CO2 Production Rate = (Expired CO2 − Inspired CO2 ) × 12 = (19.1 − 0.2) × 12 = 226.8 ml/min The amount of water vapor in the humidified expired air amounts to 6.2% of the expired air (compared to 0.49% of the dry inspired air) corresponding to the partial-pressure ratio of water vapor in the expired air (= 47/760). The volume of the dry expired air = (525.3 − 32.6) ml = 492.7 ml. Now, assume that out of 500 ml of inspired air, the dead-space air volume (not taking part in the gas-transfer process) is 150 ml and the alveolar air volume is 350 ml. We next compute the dead-space air volume composition. 2.2 Dead-space air composition The clinical data of expired air composition is: N2 = 393.1 ml O2 = 83.36 ml CO2 = 16.87 ml H2 O = 34.15 ml Total = 527.49 ml Now, the dead-space air will be made up of (i) a dry air portion from the inspired air (assumed to be = 141 ml), plus (ii) the water vapor taken up by the dry air
80 Human Respiration (estimated to be = 9 ml) since the expired air portion of 141 ml will not have under gone O2 and CO2 transfer, its composition is the same as that of the inspired air: N2 = 111 ml (78.55%), O2 = 29.40 ml (20.84%), CO2 = 0.06 ml (0.04%), H2 O = 0.69 ml (0.49%). When this inspired air (in the dead space) of 141 ml is fully humidified, it will take up a further X ml of H2 O vapor, in the ratio of the partial-pressures, as: X 47 = = 0.0659 141 713 ∴ X = 0.0659 × 141 = 9.29 ml of H2 O vapor (which is close to our estimate). So, by adding 9.29 ml of H2 O vapor to 0.69 ml of water vapor in the inspired air volume of 141 ml, the total water vapor in the dead-space air is 9.98 ml. The humified dead-space air composition will be: N2 = 111.00 ml O2 = 29.40 ml
(= 73.78%) (= 19.55%)
CO2 = 0.06 ml
(= 0.04%)
H2 O = 9.98 ml
(= 6.63%)
Total = 150.44 ml
2.3 Alveolar-air composition and partial pressures We can now compute the alveolar air composition, by subtracting the dead-space air from the expired air. These values are tabulated in column 4 of the table below. Finally, we compute the partial pressure of O2 and CO2 (as well as of N2 and H2 O), so that we can determine next the diffusion coefficients of O2 and CO2 based on the monitoring of arterial and venous blood concentrations. These values are tabulated in column 5 of the below table.
N2 O2 CO2 H2 O Total
Expired air (ml)
Dead-space air (ml)
Alveolar air (ml)
Alveolar-air partial pressure (mmHg)
393.1 80.53 19.12 34.21 526.96
111.00 29.40 0.06 9.98 150.44
282.1 51.13 19.06 24.23 376.52
569.41 103.21 38.47 48.91 760
Lung-Gas Composition and Transfer Analysis
81
3 Lung gas-exchange model and parametric analysis 3.1 Expressions for DO2 and DCO2 The gas exchange between the alveolar air and pulmonary capillary blood is represented by the following O2 and CO2 conservation equations (Fig. 2): ◦
Q VE COVE = QAE COAE + V O2 (from the alveolar air to capillary blood) 2 2 O cap AE 2 D + Pav PO2 = PO , (1) = QAE COAE O2 ; 2 2 cap
PRB (O concentration of the preoxygenated blood) in which PO2 = PO 2 2 ◦
VE AE QVE CCO = QAE CCO − V CO2 2 2 CO AE 2 D = QAE CCO − Pav CO2 ; 2
cap
VE PCO2 = PCO , 2
(2)
cap
PRB (CO concentration of the preoxygenated blood), in which PCO2 = PCO 2 2 wherein
QAB and QVB are arterial and venous blood flow-rates; QAB = QVE (at venous end), QVB = QAE (at arterial end) al and P cap are the alveolar and capillary O partial pressures (ii) PO 2 O2 2 al and P cap are the alveolar and capillary CO partial pressure (iii) PCO 2 CO2 2 (iv) DO2 and DCO2 are the O2 and CO2 diffusion coefficients O2 al − P cap ) over the capillary length; = average of (PO (v) Pav O2 2 (i)
cap
CO2 al − P = average of (PCO Pav O2 ) over the capillary length. 2 Now we can equate the arterial and venous blood flow rates, as
Q AB = QVB = Q = (SV)/(EP) CO/60, SV, EP and CO being the stroke volume, ejection period and cardiac output, respectively. Hence the above equations can be rewritten as: ◦ (vi) V O2 is the O2 transfer rate from alveolar air to capillary blood (= O2 con◦ sumption rate), V CO2 is the CO2 transfer rate from capillary blood to alveolar air.
Figure 2: Schematic of blood-gas concentration in the pulmonary capillary.
82 Human Respiration From eqn. (1): O cap AE 2 D QVE COVE = QAB COAB + Pav O2 ; PO2 = PO2 = PO2 2 2 O 2 D QCOVB = QCOAB + Pav O2 2 2 DO 2 =
Q(COVE Q(COAB − COAE ) − COVB ) 2 2 2 2 = . O2 O2 Pav Pav
(3)
From eqn. (2): CO cap VE AE AE VB 2 D = QAE CCO − Pav QVE CCO CO2 ; PCO2 = PCO2 = PCO2 2 2 AE CO2 QCOVE = QCCO − Pav DCO2 2 2 DCO2 =
VE − C AB ) Q(CCO CO2 2 . CO2 Pav
(4)
wherein VE and C AE can be monitored because Q, COVE and COAE , CCO CO2 2 2 2 VE = C AB and C AB and C AE and C AE = C VB and C VB COVE and C CO O CO O2 CO2 O2 CO2 2 2 2 2 (ii) DO2 and DCO2 (eqns. (3) and (4)) represent the lung gas-exchange parameters.
(i)
Now from eqns. (3) and (4), if we want to evaluate the diffusion coefficients DO2 and al , P cap and P al , P cap in terms of monitorable DCO2 , we need to also express PO O2 CO2 CO2 2 quantities. In this regard, ◦
◦
al can be expressed in terms of V (the ventilation rate) and V (i) Alveolar PO O2 2 (the O2 consumption rate) as Fig. 3:
◦ ◦ ◦ −k2 (V /V m )/V O2
al = k1 1 − e PO 2
,
◦
(5)
◦
where V m is the maximum ventilation rate and V O2 (the O2 consump− COVB ). Equation tion rate or absorption rate from the alveoli) = Q(COAB 2 2 ◦
◦
(5) implies that as (V /V m ) increases, (the exponential term decreases, and) al increases (as in Fig. 3), and as V◦ al PO O2 increases PO2 decreases (as in Fig. 3). 2 ◦
◦
al can be expressed in terms of V and V (ii) Alveolar PCO O2 as in Fig. 4. 2
◦ ◦ ◦ −k4 (V /V m )/V CO2
al PCO = k3 e 2
,
(6)
◦
where V CO2 (the CO2 production rate or excretion rate from the blood) = VB − C AB ). This equation implies that as V◦ /V◦ al Q(CCO m increases, PCO2 CO2 2 ◦
decreases; also, as V CO2 increases (the exponential term decreases, and al increases. hence) PCO 2
Lung-Gas Composition and Transfer Analysis
83
Figure 3: Effect on alveolar PO2 of (i) alveolar ventilation, and (ii) rate of oxygen absorption from alveolar PO2 or O2 consumption rate [from Guyton (1971), p. 476].
Figure 4: Effect on alveolar PCO2 of alveolar ventilation and rate of carbon dioxide excretion from the blood or CO2 production rate [from Guyton (1971), p. 476]. (iii)
Blood PO2 can be obtained in terms of blood CO2 , from the O2 disassociation curve (providing concentrations in arterial and venous blood), is represented in Fig. 5 as:
PO
−k5 P m2
CO2 = COm2 1 − e
O2
, or
∗ −k5 PO
CO∗ 2 = 1 − e
2
,
(7)
84 Human Respiration
Figure 5: O2 dissociation curves, showing the total oxygen in each 100 ml of normal blood, the portion dissolved in the water of the blood [from Guyton [2], p. 485].
Figure 6: The carbon dioxide dissociation curve [from Guyton [2], p. 491].
where m are the maximum values of blood O partial pressure • COm2 and PO 2 2 ∗ • CO2 = CO2 /COm 2 ∗ = P /P m . • PO O2 O2 2
(iv)
Blood PCO2 can be obtained in terms of CCO2 , from the CO2 disassociation curve or CO2 concentration in arterial and venous blood can be represented
85
Lung-Gas Composition and Transfer Analysis
as per Fig. 6 as: −k P /P m CCO2 = COm2 1 − e 6 CO2 CO2
or,
∗ CCO 2
=1−e
m −k6 PCO2 /PCO 2
∗ −k6 PCO
=1−e
2
(8)
.
3.2 Alveolar O2 and CO2 partial-pressure expressions al partial pressure curve (Fig. 3), represented by Now, let us refer eqn. (4) for the PO 2 the equation:
◦ ◦ ( ◦ V O2 −k2 V /V m al = k1 1 − e PO 2
◦ (◦ −k2 V ∗ V O2
◦
◦ , where V ∗ = ◦V Vm
= k1 1 − e ◦
(9)
◦
where V is the alveolar ventilation rate (in liters/min), V m is the maximum venti◦ lation rate (= 50 l/min) and V O2 is the O2 consumption rate (in liters/min). Herein, the coefficients k1 and k2 can be determined by having this equation match the ◦ al = 0 from the equation, which Fig. 3 data. Note, in this equation, when V = 0, PO 2 satisfies the data. ◦ ◦
◦
Now for V O2 = 0.25 l/min, when V ∗ =
140 = k1 ◦
−k2
1−e
V
◦
Vm
0.5 0.25
al = 140 mmHg. Hence, = 0.5, PO 2
= k1 (1 − e−2k2 ).
(10)
◦
al = 100 mmHg. Hence Also, when V O2 = 1 l/min, V ∗ = 0.3 l/min, PO 2
−k 0.3 100 = k1 1 − e 2 1 = k1 (1 − e−0.3k2 ).
(11)
From eqns. (10) and (11), we get: k1 (1 − e−2k2 ) 1 − e−2k2 140 = = −0.3k 2) 100 k1 (1 − e 1 − e−0.3k2 ∴ 140 − 140e−0.3k2 = 100 − 100e−2k2 or, 40 = 100e−2k2 + 140e−0.3k2 , so that k2 = 4.18 min/l. Upon substituting k2 = 4.18 min/l into eqn. (10) we obtain: 140 = k1 1 − e−(2×4.18) , so that k1 ≈ 140 mmHg.
(12)
(13)
86 Human Respiration al curve can be represented by: Hence, the PO 2
◦ (◦ −4.18 V ∗ V O2
,
al PO = 140 1 − e 2
◦
◦
(14)
◦
where, V O2 = Q(COAB − COVB ) and V ∗ = V /50 l/min. 2 2 al Now, let us look at the PCO2 expression: −k4
al PCO = k3 e 2
◦ ◦ ( ◦ V CO2 V /V m
= k3 e
◦
◦ (◦ −k4 V ∗ V CO2
.
◦
al = 12. Hence, We note from Fig. 4 that for V CO2 = 0.2 l/min and V m = 0.2, PCO 2 from the above equation, we get:
12 = k3 e−k4 ◦
(15)
◦
al = 62 mmHg. Hence Also, for V O2 = 0.8 l/min and V m = 0.2, PCO 2 −k4
62 = k3 e
0.20 0.80
= k3 e
−k4 4
(16)
.
From eqns. (15) and (16), we get: 2 12 e−k4 = −k = e− 3 k4 4 62 e 4 12 2 ln = − k4 , so that k4 = 2.46. 62 3
(17)
Substituting k4 = 2.46 into eqn. (16), we obtain: 62 = k3 e
−2.46 4
,
∴ k3 = 114.68.
(18)
al curve can be represented as Hence, the PCO 2
al = 114.68e PCO 2
◦ ◦ ( ◦ V CO2 V /V m
−2.46
,
◦ ◦) ◦ VB − C AB ). where V ∗ = V 50 l/min and V CO2 = Q(CCO CO2 2
3.3 Arterial and venous O2 and CO2 partial-pressure expressions AB and P VB in terms of C AB and C VB . We now need to express PO CO2 O2 CO2 2
(19)
Lung-Gas Composition and Transfer Analysis
87
So that let us look at the O2 disassociation curve, as shown in Fig. 5. ) CO2 = CO2 | max 1 − e−k5 PO2 PO2 | max , CO∗ 2 = 1 − e
or,
where CO∗ 2 = ∗ = From Fig. 5, at PO 2
∗ −k5 PO
2
CO2 CO2 | max
,
∗ , PO = 2
PO2 PO2 | max
(20)
.
40 mmHg = 0.29 (for normal venous blood), and 140 mmHg CO∗ 2 =
15 = 0.75. 20
Hence from eqn. (20): 0.75 = 1 − e−0.29k5 ∴ k5 = 4.78. ∗ = Also, PO 2
(21)
95 mmHg = 0.68 (for normal arterial blood), and 140 mmHg CO∗ 2 =
19 = 0.95. 20
Hence from eqn. (20): 0.95 = 1 − e−0.68k5 , or
k5 = 4.4.
(22)
So, we take the average value of k5 : ∴ k5 =
(4.78 + 4.4) = 4.59. 2
Then the O2 disassociation curve is given by: CO2 = and P O2
COB2
−4.59
= 0.21 − e
PO 2 140
(23)
,
(24)
0.2 0.2 140 ln = 30.5 ln . = 4.59 0.2 − COB2 0.2 − COB2
Finally, we look at the CO2 disassociation curve P
CO2 CO2 | max
−k6 P
CCO2 = CCO2 / max 1 − e
or,
∗ CCO 2
=1−e
PCO 2 CO2 | max
−k6 P
(25)
, ∗ −k6 PCO
=1−e
2
.
(26)
88 Human Respiration ∗ Based on Fig. 6, when PCO = 2
20 mmHg 38 ∗ = = 0.14, CCO = 0.475, so that 2 140 mmHg 80
0.475 = 1 − e−0.14k6 , ∗ = when PCO 2
k6 = 4.60,
(27)
70 mmHg 60 ∗ = = 0.5, CCO = 0.75, so that 2 140 mmHg 80 0.75 = 1 − e−0.5k6 ,
k6 = 2.77.
(28)
So, we take the average value of k6 : k6 =
(4.60 + 2.77) = 3.69. 2
(29)
Then the CO2 concentration is given (from eqns. (26–29) by:
B = 0.8 1 − e CCO2 = CCO 2
and PCO2
−4.71
0.8 = 29.72 ln B 0.8 − CCO 2
PCO 2 140
(30)
.
(31)
3.4 Sequential procedure to compute DO2 and DCO2 ◦
(1) We first monitor: V (t), V (t), SV (stroke volume), EP (cardiac ejection period), VB and C AB (O and CO concentrations in pre oxygenated COVB , COAB , CCO 2 2 CO2 2 2 2 and post oxygenated blood). (=COVE ) and COVB (=COAE ) into eqn. (3), and (2) We substitute the values of COAB 2 2 2 2 AB VE VB AE the values of CCO2 (=CCO2 ) and CCO2 (=CCO2 ) into eqn. (4). (3) We next determine: Q = SV/ejection period, ◦ − COVB , V O2 (t) = Q COAB 2 2 VB ◦ AB V CO2 (t) = Q CCO2 − CCO . 2 ◦
◦
(32) (33) (34)
(4) We then substitute the expressions for V O2 (t) and V CO2 (t) into the equations al (eqn. (14)) and P al (eqn. (19)). for PO CO2 2 VB (=C AE ) into (5) We substitute the monitored values of COVB (=COAE ) and CCO CO2 2 2 2 AE and P AE . eqns. (25) and (31), to obtain the values of PO CO 2 2 (6) Now, in order to determine the values of the lung gas-exchange parameters al DO2 and DCO2 , we substitute into eqns. (3) and (4) for Q from eqn. (32), PO 2 al VB VB from eqn. (14), PCO2 from eqn. (19), PO2 from eqn. (26), and PCO2 from eqn. (31).
Lung-Gas Composition and Transfer Analysis
89
3.5 Determining DO2 and DCO2 al − P cap = P al − P AB along the Figure 7 illustrates the variation of P O2 = PO O2 O2 O2 2 length (l) of ) the capillary bed. Let l ∗ = l lm . Then we can express: O2 P O2 = Pmax fO2 (l ∗ ).
Then,
O2 Pav
=
O2 Pmax
1
0
∗
fO2 (l ) dl
∗
(35)
O2 = Pmax FO2 .
O
(36)
O
2 = 65 mmHg, we have Based on data [3], since Pav2 = 12 mmHg for Pmax FO2 = 0.185. CO2 We can similarly determine the average value of Pav from Fig. 8 as: ) Let l ∗ = l lm . Then, we can represent Fig. 8 as:
CO2 P CO2 = Pmax fO2 (l ∗ ).
Then,
CO2 Pav
=
O2 Pmax
0
1
∗
fCO2 (l ) dl
∗
CO2 FCO2 . = Pmax
(37)
(38)
Figure 7: Uptake of oxygen by the pulmonary capillary blood. (The curve in this figure was constructed from data in Mihorn and Pulley: Biophys. J., 8: 337,1968). [from Guyton (1971), p. 434.]
90 Human Respiration
Figure 8: Diffusion of carbon dioxide from the pulmonary blood into the alveolus. (The curve in this figure was constructed from data in Mihorn and Pulley: Biophys. J., 8: 337,1968). [from Guyton (1971), p. 435.] CO
CO
Based on data [3], since Pav 2 = 0.5 mmHg for Pmax2 = 5 mmHg, we have FCO2 = 0.1. O2 CO2 and Pav expressions, we can determine the O2 consumption From the Pav and the CO2 production rates, as follows: ◦ VB − C Q COAB O Total O2 consumed V O2 2 2 = = (39) DO2 = O2 O2 O2 Pav Pav Pav ◦ VB − C AB Q CCO CO2 Total CO2 produced V CO2 2 DCO2 = = = . (40) CO2 CO2 CO2 Pav Pav Pav
4 Case studies (A) We monitor the partial pressures blood concentrations of O2 and CO2 as: AE VB COAE = COVB = 0.13, COVE = COAB = 0.18, CCO = CCO = 0.525, 2 2 2 2 2 2 VE AB = CCO = 0.485. CCO 2 2
From eqn. (26), we obtain:
VB PO 2
0.2 = 30.5 ln 0.2 − COVB 2 = 32.02 mmHg.
= 29.72 ln
0.2 0.2 − 0.13
(41)
Lung-Gas Composition and Transfer Analysis
From eqn. (31), we obtain: VB PCO 2
0.8 = 37.94 ln VB 0.8 − CCO 2
= 37.94 ln
0.8 0.8 − 0.525
91
= 40.51 mmHg.
(42) ◦
◦
We now also monitor Q = 5 l/min, V ∗ = 0.1 and V = 5 l/min. Then, from eqn. (33): ◦ VB , − C V O2 (t) = Q COAB O 2 2 so that from the above data, ◦
V O2 (t) = 5000 × 0.05 = 250 ml O2 /min Consumption rate
(43)
From eqn. (34): ◦ VB AB V CO2 (t) = Q CCO = 5000(0.04) − CCO 2 2 = 200 ml CO2 /min production rate.
(44)
Now, from eqn. (14). ◦ ◦ al : For V ∗ = 0.1 and V O2 = 0.25 l, we obtain PO 2
◦ (◦ −4.18 V ∗ V O2
,
al PO = 140 1 − e 2
= 140 1 − e−4.18[0.1/0.25] = 113.7 mmHg. ◦
(45)
◦
al : From eqn. (19), for V ∗ = 0.1 and V CO2 = 0.20 l, we obtain PCO 2
al = 107.18e PO 2
◦ (◦ −2.19 V ∗ V CO2
= 107.18e−2.19[0.1/0.2]
= 35.86 mmHg.
(46)
Now, we can evaluate the diffusion coefficients: From eqns. (3), (36), (41), and (45): D O2 =
− COVB ) Q(COAB 2 2 O
Pav2 5000(0.18 − 0.13) = = 16.84 ml O2 /min/mmHg. (113.7 − 31.2) × 0.18
(47)
92 Human Respiration From eqn. (4): DCO2 =
VB − C AB ) Q(CCO CO2 2
CO2 Pav 5000(0.04) = 430.11 ml CO2 /min/mmHg. = (40.51 − 35.86) × 0.1
(48)
(B) Alternately, we derive data from: (i) the inspired and expired air analysis (such as that carried out in Section 2.3): O2 consumption rate = 283.2 ml/min, CO2 production rate = 226.8 ml/min, al = 103.03 mmHg and P al = 38.41 mmHg PO CO2 2 and (ii) venous blood gas analysis: VB COVB = 0.13, CCO = 0.548. 2 2
Then, as per eqn. (41), VB = 31.2 mmHg, PO 2
(49)
= 0.13 and, as per eqn. (42): corresponding to COVB 2
0.8 0.8 VB PCO2 = 37.94 ln = 37.94 ln VB 0.8 − 0.548 0.8 − CCO 2 = 43.84 mmHg.
(50) ◦
We obtain, from air-composition analysis, that V O2 (t) = 283.3 ml/min ◦
and V CO2 (t) = 226.8 ml/min.
(51) (52)
Hence, ◦
DO2 = =
V O2 O2 Pav
283.2 = 21.90 mlO2 /min/mmHg, (103.03 − 31.2) × 0.18
(53)
and ◦
DCO2 = =
V CO2 CO2 Pav
226.8 = 417.68 mlCO2 /min/mmHg. (43.84 − 38.41) × 0.1
(54)
The advantage of this method (B) over (A) is that it does not require monitoring of the cardiac output, and is hence simpler to implement clinically.
Lung-Gas Composition and Transfer Analysis
93
References [1]
Guyton, A.C., M.D. Text Book of Medical Physiology, 8th edn, HBJ I.E. Saunders, ISBN 0-7216-3994-1, pp. 422–442, 1991. [2] Comroe, J.H., Physiology of Respiration, 2nd edn, Year Book Medical Publishers Incorporated, ISBN 0-8251-1827-9, pp. 268–269, 1977. [3] Conney, D.O., Biomedical Engineering Principles, Marcel Dekker Inc., ISBN 0-8247-6347-5, pp. 248–251, 342–397, 1976.
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CHAPTER 4 Lung ventilation modeling and assessment D.N. Ghista1 , K.M. Loh2 , & M. Damodaran1 1 School
of Mechanical and Production Engineering, Nanyang Technological University, Singapore. 2 School of Engineering (Electronics), Nanyang Polytechnic, Singapore.
Abstract We have developed a lung-ventilation model by modeling the lung-volume response to mouth minus pleural driving pressure (by means of first- and second-order differential equations) in terms of resistance to airflow (R) and the lung compliance (C). The lung-volume solution of the differential equation is matched with the clinicalvolume data, to evaluate the parameters, R and C. These parameter values can help us to distinguish an obstructive lung and a lung with stiffened parenchyma from a normal lung, and hence diagnose lung diseases such as asthma and emphysema. We have also formulated a nonlinear compliance lung model, and demonstrated deceased lung compliance with filling volume. We then formulated a nondimensional lung-ventilatory index (VTI ), incorporating the parameters R and C as well as the lung-breathing rate. When the VTI is evaluated for various lung diseases, it will conveniently enable us to diagnose lung diseases in terms of just one VTI number. Finally, we have shown how to model a two-lobe lung, and differentiate between normal and diseased lobes.
1 Introduction 1.1 Role of lung ventilation Lung ventilation constitutes inhalation of an appropriate air volume under driving pressure (=mouth pressure − pleural pressure), so as to: (i) provide an adequate alveolar O2 amount at an appropriate partial pressure, (ii) oxygenate the pulmonary blood, and (iii) thereby provide adequate metabolic oxygen to the cells.
96 Human Respiration
T
Diaphragm
Pel R
T
Po
V,Pa
Air
Pp: pleural pressure
Figure 1: Alveolar model.
Hence, ventilatory function and performance assessment entails determining how much air volume is provided to the alveoli, to make available adequate alveolar oxygen for blood oxygenation and cellular respiration. Based on Fig. 1, we get: (i) (ii) (iii) (iv) (v)
(Pa – Pp ) − Pel = 0 Pel = (2ah)/Rr = 2T /r = V /C + Pel0 (Pm – Pa ) = R(dV /dt) PL = Pm – Pp R(dV /dt)+VIC = PL – Pel0 (lung elastic recoil pressure at end of expiration)
2 Lung-ventilation performance using a linear first-order model We first analyze the lung-ventilation function by means of a very simple model represented by a first-order differential equation (Deq ) in lung-volume (V ) dynamics in response to the driving pressure (Pl = atmospheric pressure – pleural pressure), as displayed in Fig. 1. The clinical pressure-volume data is in Fig. 2. The model-governing equation (shown derived in Fig. 1) is as follows: ◦
RV +
V = PL (t) − Pel0 = PN (t), C
(1a)
wherein: (i) the values of pressure are obtained from the given PL (=Pm – Pp ) data (ii) the parameters of this governing Deq are lung compliance (C) and airflow resistance (R); in the equation both R and C are instantaneous values (iii) V = V (t) − Vo (the lung volume at the end of expiration (iv) Pel0 is the lung elastic-recoil pressure at the end of expiration, and Pel0 = Pel −
V . C
(1b)
Lung Ventilation Modeling and Assessment
97
(VT)+0.5
V, Tidal Volume (litres) (FRC) 0
Air-flow resistance is higher during inspiration than expiration.
+0.5 0
V, Air Flow (litres/second)
–0.5 +1
Pa, Alveolar Pressure (cm H2O) 0
The alveolar pressure (Pa) curve is in phase with the air-flow curve. It constitutes the resistance pressure.
–1
Pp, Pleural Pressure (cm H2O) –5
2
1 –7
1
2
–9 0
1
2 3 Seconds
4
5
Figure 2: Lung-ventilatory model and lung-volume and pleural-pressure data. Curve 1 on the curve represents Pel , the pressure required to overcome lung elastance (=V /C). Curve 2 curve represents Pp , the summation of Pel and Pa . The pressure PN (t) in eqn. (1a) equals Pp minus Pel at end of expiration. At the end of expiration when ωt = ωT , PL = Pel0 = PN (t), which is represented by 3 Pi sin (ωi t + ci ), PN (t) = i=1
and the governing eqn. (1a) becomes: V = PN (t) = Pi sin (ωi t + ci ), C 3
◦
RV +
(2a)
i=1
where the right-hand side represents the net driving pressure minus pleural pressure. PN = (Pm −Pp )−Pel0 . This PN is, in fact, the driving pressure (Pm −Pp ) normalized with respect to its value at end of expiration. Eqation (2a) can be rewritten as follows: 1 V = Pi sin (ωi t + ci ), V + RC R ◦
3
i=1
(2b)
98 Human Respiration wherein the P(t) clinical data (displayed in Fig. 2) is assumed to be represented by: P(t) =
3
Pi sin (ωi t + ci ),
(3)
i=1
P1 = 1.581 cmH2 O ω1 = 1.214 rad/s c1 = −0.3132 rad
P2 = −5.534 cmH2 O ω2 = 0.001414 rad/s c2 = 3.297 rad
P3 = 0.5523 cmH2 O ω3 = 2.401 rad/s c3 = −2.381 rad.
The pressure curve (in Fig. 3A) represented by the above eqn. (3) closely matches the pressure data of Fig. 2. If, in eqn. (1), we designate Ra and Ca as the average values (R and C) for the ventilatory cycle, then the solution of eqn. (1) is given by: V (t) =
3 t Pi Ca [sin (ωi t + ci ) − bi Ra Ca cos (ωi t + ci )] − H e− Ra Ca , 2 2 1 + ωi (Ra Ca ) i=1
(4)
wherein the term (Ra Ca ) is denoted by τa . We need to have V = 0 at t = 0. Hence, putting V (at t = 0) = 0, gives us: H=
3 Pi Ca [sin (ωi t + ci ) − bi Ra Ca cos (ωi t + ci )] . 1 + ωi2 (Ra Ca )2 i=1
(5)
Then from eqns. (4) and (5), the overall expressions for V (t) becomes 3 Pi Ca sin (ωi t + ci ) − ωi τa2 cos (ωi t + ci ) V (t) = 1 + ωi2 τa2 i=1 3 Pi Ca sin (ωi t + ci ) − ωi τa2 cos (ωi t + ci ) − t e τa − 2τ2 1 + ω a i i=1
3 t Pi Ca sin (ωi t + ci ) − ωi τa2 cos (ωi t + ci ) = 1 − e− τa . 2 1 + ωi τa2 i=1
(6)
We also want that dV /dt = 0 at t = 0, implying no air-flow at the start of inspiration. So, by differentiating eqn. (6), we get: 3 t Pi Ca ωi cos (ωi t + ci ) + ωi2 τa sin (ωi t + ci ) 1 − e− τa V = 2 1 + ωi τa2 i=1 ◦
3 Pi Ca sin (ωi t + ci ) − ωi τa cos (ωi t + ci ) − t + e τa . 2τ2 τ 1 + ω a i a i=1 ◦
(7)
From eqn. (7), we get V = 0 at t = 0, thereby also satisfying this initial condition.
Lung Ventilation Modeling and Assessment
99
A
B
Figure 3: AThe pressure curve represented by eqn. (3) matched against the pressure data (represented by dots). B The volume curve represented by eqn. (6), for from Ca = 0.2132 (cmH2 O)−1 and Ra = 2.275 cmH2 Osl−1 pp. 3 matched against the volume data represented by dots.
100 Human Respiration Now, by matching the above V (t) expression (6) with the given V (t) data in Fig. 2, and carrying out parameter identification, we can determine the in vivo values of Ra and Ca , to be Ca = 0.2132 (cmH2 O)−1 ,
Ra = 2.275 cmH2 Osl−1
The computed V (t) curve, represented by eqn. (6) for the above values of Ca and Ra , is shown in Fig. 3B. We can however analytically evaluate Ra and Ca by satisfying some conditions. For this purpose, we first note that V is maximum (= tidal volume, t TV) at about t = tV = 2.02 s. At t = tV , the exponential term e− τa in (6) becomes of the order of e−10 , and hence negligible. Then by putting V˙ (t = 2.02) = 0 in eqn. (7), without the exponential term we obtain: 3 Pi Ca ωi cos (ωi × 2.02 + ci ) + ωi2 τa sin (ωi × 2.02 + ci ) ◦ = 0, V |t=2.02 = 1 + ωi2 τa2 i=1 (8) in which the values of Pi , ωi , and ci are given by eqn. (3). Then by solving ◦ eqn. (8), we get τa = 0.522 s. We can also put V = 0 at t ∼ = 1.81/2.87 s and obtain a similar value for τ. Then, we also note that at tv = 2.02 s (at which dV /dt = 0) and V = 0.55 l. Hence upon substituting into eqn. (6), and neglecting the exponential term, we get the following algebraic equation: 3 Pi Ca sin (ωi t + ci ) − ωi τa2 cos (ωi t + ci ) = 2.55Ca , (9) V (t)|t=2.02 = 1 + ωi2 τa2 i=1 by employing the values of Pi , ωi and ci from eqn. (3). Now since V (t = 2.02 s) = 0.55 l, we get 2.55Ca = 0.55 Ca = 0.22 l (cmH2 O)−1 .
(10)
We can substitute, therein, the values of P1 and P2 from eqn. (3), and obtain the value of Ca as: Ca = 0.22 l (cmH2 O)−1 . Since we have computed τa = 0.485 s, therefore Ra = 2.275(cmH2 O)sl−1 . These are the average values of resistance to airflow and lung compliance during the ventilatory cycle shown in Fig. 2. Since lung disease will influence the values of R and C, these parameters can be employed to diagnose lung diseases. For instance in the case of emphysema, the destruction of lung tissue between the alveoli produces a more compliant lung, and hence results in a larger value of C. In asthma, there is increased airway resistance (R) due to contraction of the smooth muscle around the airways. In fibrosis of the lung, the membranes between the alveoli thicken and hence lung compliance (C) decreases. Thus, by determining the normal and diseased ranges of the parameters R and C, we can employ this simple lung-ventilation model for differential diagnosis.
Lung Ventilation Modeling and Assessment
101
3 Ventilatory index Let us, however, formulate just one non dimensional number to serve as a ventilatoryperformance index VTI1 (to characterize ventilatory function), as: VTI1 = [(Ra Ca )(Ventilatory rate in s−1 ) 60]2 = τa2 (BR)2 602 ,
(11)
where BR is the breathing rate. Now, let us obtain its order of magnitude by adopting representative values of Ra and Ca in normal and disease states. Let us take the above computed values of Ra = 2.275 (cmH2 O)sl−1 and Ca = 0.2132 l (cmH2 O)−1 and BR = 12 m−1 or 0.2 s−1 , computed for the data of Fig. 2 and eqn. (3). Then, in a supposed normal situation, the value of VTI1 is of the order of 33.88. In the case of obstructive lung disease, (with increased Ra ), let us take Ra = 5 (cmH2 O)sl−1 , Ca = 0.12 l (cmH2 O)−1 and BR = 0.3 s−1 ; then we get VTI1 = 116.6. For the case of emphysema (with enhanced Ca ), let us take Ra = 2.0 cmH2 Osl−1 , Ca = 0.5 l (cmH2 O)−1 and BR = 0.2 s−1 ; then we obtain VTI1 = 144. In the case of lung fibrosis (with decreased Ca ), we take Ra = 2.0 cmH2 Osl−1 , Ca = 0.08 l (cmH2 O)−1 and BR = 0.2 s−1 ; then we obtain VTI1 = 3.7. We can hence summarize that VTI1 would be in the range of 2–5 in the case of fibrotic lung disease, 5–50 in normal persons, 50–150 in the case of obstructive lung disease and 150–200 for the case of emphysema. This would of course need verification by analyzing a big patient population. Now, all of this analysis requites pleural-pressure data, for which the patient has to be intubated. If now we evaluate the patient in an outpatient clinic, in which we can only monitor lung volume and not the pleural pressure, then can we develop a non invasively obtainable ventilatory index?
3.1 Noninvasively determinable ventilatory index In order to formulate a non-invasively determinable ventilatory index from eqn. (1), we need to recognize that in this case PN (t) (and hence Pi , ω and ci ) will be unknown and we need to redesignate the model parameters and indicate their identification procedure. So we make use of the following features from the volume–time data to facilitate evaluation of the following three parameters: (Pi , Ca ), ωi , ci , and τa . At t = tv = 2.02 s, V is max and dV /dt = 0; hence we rewrite eqn. (9) as: ◦
V |t=2.02
3 (Pi Ca ) ωi cos (2.02 × ωi + ci ) + ωi2 τa sin (2.02 × ωi + ci ) = 0. = 1 + ωi2 τa2 i=1 (12)
102 Human Respiration ◦
Also, at t = tm = 1.82/2.87 s, V = 0. Hence by differentiating eqn. (7), without the exponential term, we obtain: 3 tm (Pi Ca ) − sin (ωi tm + ci )ωi2 + ωi3 τa2 cos (ωi tm + ci ) V (t) = 1 − e− τa 2 1 + ωi τa2 i=1 ◦◦
3 (Pi Ca ) ωi cos (ωi tm + ci ) − ωi2 τa sin (ωi tm + ci ) − tm e τa +2 2τ2 1 + ω τ a i a i=1 3 (Pi Ca ) sin (ωi tm + ci ) − ωi τa2 cos (ωi tm + ci ) − tm e τa = 0. − 2 1 + ω2 τ 2 τ a a i i=1
(13)
Then, at t = 1 s, V1 = 2.02l. From eqn. (6), without the exponential term, this condition yields: 3 (Pi Ca ) − sin (ωi + ci )ωi2 + ωi3 τa2 cos (ωi + ci ) V1 = = 2.02. 1 + ωi2 τa2 i=1 In addition, we can utilize data information concerning Vj at tj ( j = 1 to 8), and put down: 3 (Pi Ca ) − sin (ωi tj + ci )ωi2 + ωi3 τa2 cos (ωi tj + ci ) Vj = ; j=1 1 + ωi2 τa2 i=1
to
8.
(14) From eqns. (12)–(14), we can obtain the values of Pi Ca (but not of P1 , P2 and P3 by themselves), ωi , ci and τa . On the other hand, by also fitting eqn. (6), (without the exponential term) to the V (t) data, we obtain: P1 C = 0.3223
P2 C = 0.3143
P3 C = −0.02269
(15)
ω1 = −1.178
ω2 = 0.5067
ω3 = 1.855
(16)
c1 = 90223
c2 = 0.2242
c3 = −3.961
τa = 0.5535.
(17)
We can now also formulate another noninvasively determinable nondimensional ventilatory index (VTI2 ) in terms of these parameters as follows: VTI2 =
(BR)R[TV ]2 (BR)τ[TV ]2 = . |P1 C||P2 C||P3 C| |P1 P2 P3 C 2 |
(18)
It is seen that VTI2 can in fact be expressed in terms of P1, P2 , P3 and R, C. This VTI2 index can be evaluated by computing the values of (BR) and τ, along with (Pi C), as given by eqn. (17). Then, after evaluating VTI2 for a number of patients, its distribution can enable us to categorize and differentially diagnose patients with various lung disorders and diseases.
Lung Ventilation Modeling and Assessment
103
4 Variations in R and C during a respiratory cycle (towards nonlinear) Thus far, we have adopted the average cyclic values Ca and Ra for our DE q model parameters. However, we expect that C will vary with lung volume (V ), and that ◦ R will perhaps vary with the airflow rate or (V ) or even ω. Hence, for a true representation of the lung properties C and R, let us determine their values for different times during the ventilatory cycle, and compare them with their average values Ca and Ra , so as to make a case for a nonlinear ventilatory-function model. Let us hence compute the instantaneous value of compliance (C) at time (t = tm ), ◦◦ when V = 0. Let us differentiate eqn. (2a), giving: ◦
V RV + Pi Cωi cos (ωi t + ci ). = C ◦◦
3
(19)
i=1
◦
◦◦
Now at about mid-inspiration, when t = tm = 1.18 and V = 0.48 l/s, V = 0 l/s and ◦◦ ◦ V = 0.29 l (based on Fig. 2). By substituting for V , V and V in eqn. (19), we obtain, C = 0.486 l/cmH2 O (compared to its Ca value of 0.21). Now, in order to compute R, we utilize the data information that at tV = 2.02 s we substitute ◦ ◦◦ V = 0 l/s, V = − 0.89 l/s and V = 0.54 l (from the Fig. 2 data) into eqn. (2a), to obtain: ◦◦
RV =
3
Pi ωi cos (ωi t + ci )
i=1 3
Pi ωi cos (ωi t + ci )
i=1
R=
◦◦
.
V
(20)
Substitute C (at tm = 1.18 s) = 0.486 l/cmH2 O in either eqns. (6) or (2b), and obtain R = 1.122 (cmH2 O)sl−1 . This gives us some idea of the order of magnitude of R and C, in comparison to their average values Ca and Ra . We could naturally expect C at t = tm (which is about mid-inspiration) to be higher than its value at the end of inspiration, when the lung is fully inflated. Also, we could expect the flow resistance ◦ to be minimum at the peak of inspiration, when V = 0. ◦ Because C and R are not constant, but a function of V and V , we can hence represent lung compliance (C) and resistance (R) as follows: C = C0 e−kC V
or
E=
1 = E0 eke V C
(21a)
◦
R = R0 ekR V , ◦
(21b)
wherein V can also be varied by having the subjects breathe at different ventilation frequencies (ω).
104 Human Respiration 4.1 Nonlinear compliance We note as per the conventional formulation of compliance, given by eqn. (2) in Fig. 1 as: V Pel = + Pel0 = VE + Pel0 . (22) C In the above formulation, we assume that C and E(=1/C) remains constant throughout the ventilation cycle. However, at the start of inspiration, C = Co at t = 0, and it decreases as the lung volume increases, based on the lung (static) volume vs pressure curve. So let us improve upon this (22) model, by making Pel a nonlinear function of volume, as follows: Pel = Pel0 + VE0 ekV .
(23a)
We can alternatively write eqn. (23) as: Pel = Pel0 + V (E0 + E1 t + E3 t 2 ).
(23b)
Employing the above format of compliance, the governing DEq (1) becomes ◦
RV + VE0 ekV = PL (t) − Pel0 = PN (t) =
3
Pi sin (ωi t + ci ).
(24)
i=1
Again at the end of expiration, Pel0 = intrapulmonary pressure = (P0 + P1 ). Hence eqn. (24) becomes: ◦
RV + VE0 ekV =
3
Pi sin (ωi t + ci )
(25a)
i=1
whose RHS is similar to that of eqn. (2a), and the values of P1 , P2 , and P3 are given by eqn. (3) for the Fig. 2 data. Solving eqn. (25a): ◦
RV + VE0 e
kV
=
3
Pi sin (ωi t + ci ),
i=1
or,
◦
V +
VE0 kV Pi e = sin (ωi t + ci ), R R 3
i=1
or, based on eqn. (23b), ◦
V +
Pi V E0 + E1 t + E2 t 2 = sin (ωi t + ci ). R R 3
i=1
105
Lung Ventilation Modeling and Assessment
This yields: V (t) = e
t(6E0 +3E1 t+2E2 t 2 ) 6R
t
e
u(6E0 +3E1 u+2E2 u2 ) 6R
3 Pi
0
i=1
R
sin (ωi u + ci )du.
(25b)
We could employ this expression for V (t) to fit the clinical V (t) data. However, let us try a simpler approach to evaluate these parameters k and E0 . For this purpose, we again bring to bear the situation that at the end of inspiration, for t = tv = 2.02 s, ◦ we have V = 0 and V = Vmax = TV = 0.55 l. Hence, from Fig. 2 data, and eqns. (3) and (25a), we obtain: (26) 0.55E0 e0.55k = 2.55. ◦◦
Let us now employ the volume data point at which V = 0. For this purpose, we differentiate eqn. (25a), to obtain: Pi Ca ωi E0 kV e (1 + kV ) = cos (ωi t + ci ) R R 3
◦◦
V +
i=1
◦◦
PC ω (1 + kV ) i a i E0 + E1 t + E2 t 2 = cos (ωi t + ci ). R R 3
V +
(27)
i=1
◦◦
From the Fig. 2 data at about mid-inspiration, for which at t = tm = 1.18 s, V = 0, V = 0.29 and P = 2.53, from Fig. 2 data. Substituting these values into eqn. (27), we get: (1 + 0.29k)(E0 + 1.18E1 + 1.39E2 ) = 2.53. (28) Now, in eqns. (26) and (28), we have four unknowns to be identified: k, E0 , E1 , and E2 . Hence we need two more equations, corresponding to two additional time instants. From the values in the following table, ◦
◦◦
t
V
V
V
P
Using eqn.
1.18 2.02 2.87 4.19 4.76
0.29 0.55 0.29 −0.03 −0.02
0.48 0 −0.47 0 0.02
0 −0.89 0 0.16 0
2.53 2.55 0.29 −0.15 −0.06
26 26 28 26 28
we can determine the unknowns: k = −0.13, E0 = 4.98, E1 = −2.24 and E2 = 0.21.
(29)
Hence, by employing the nonlinear formulation, Pel = Pel0 + E0 e−kV ,
(30)
106 Human Respiration we obtain the following expression for nonlinear lung compliance (or elastance): dPel 1 =E= = E0 kekV = 0.65e0.13V . dV C
(31)
Based on this expression, we obtain , for t = tm and V = 0.29 l: E=
1 = 0.67 cmH2 O/l and C = 1.48 l/cm H2 O. C
(32)
Equation (31) can now provide us a more realistic characterization of lung compliance as follows: 1 = 0.65 and C = 1.53 cmH2 O/l At t = 0 and V = 0, we compute E = C 1 At t = tm = 1.18 s and V = 0.29 l, E = = 0.67 and C = 1.48 cmH2 O/l C 1 At t = tv = 2.02 s and V = 0.55 l and E = = 0.70 and C = 1.43 cmH2 O/l C (33) which corresponds to the value of Ca . Our nonlinear formulation of lung compliance, as depicted by eqns. (31) and (33), indicates that compliance decreases from 1.53 cm H2 O/l at the start of inspiration to 1.48 cmH2 O/l at about mid-inspiration, and then to 1.43 cmH2 O/l at the end of inspiration. What this also tells us is that the ventilatory model (1) gives the correct reading of the compliance at Vmax , i.e. at the end of inspiration. At other times of inspiration and expiration, the Ca parameter underestimates the instantaneous value of lung compliance. Now, we could also obtain an analytical solution of eqn. (25) for V (t), and fit the expression for V (t) to the lung-volume data, to evaluate the parameters (i) R, E0 and k for an intubated patient (ii) R, E0 , k and P1 , P2 and P3 for a non-intubated patient in the out-patient clinic. However, this is outside the scope of this chapter.
5 Work of breathing (WOB) This is an important diagnostic index, especially if it can be obtained without intubating the patient and even without using the ventilator. The premise for determining WOB is that the respiratory muscles expand the chest wall during inspiration, thereby lowering the pleural pressure (i.e., making it more negative) below the atmospheric pressure to create a pressure differential from the mouth to the alveoli during inspiration. Then, during expiration, the lung recoils passively. Hence, the work done during a respiratory life cycle, is given by the area of the loop generated by plotting lung volume (V ) versus net driving pressure (Pp ).
Lung Ventilation Modeling and Assessment
107
Figure 4: Plot of pressure versus volume. The area under the curve provides the work done. This plot is shown in Fig. 4. Its area can by obtained graphically, as well as analytically as shown below: T t=T dPp (t) VdPp (t) = V dt (34) WOB = dt 0 0
t=T
= 0
3 Pi Ca sin (ωi t + ci ) − ωi τa2 cos (ωi t + ci ) 1 + ωi2 τa2 i=1 3
Pi ωi cos (ωi t + ci )dt
i=1
=
3 i=1
−Pi Ca cos (ωi T + ci ) + ωi τa sin (ωi T + ci ) − cos ci − ωi τa sin ci . (35) ωi 1 + ωi2 τa2
The above expression for WOB can be evaluated, once the values of Ci and τ (or ωτ) and P1 , P2 and P3 and have been computed (as shown in the previous section). So let us substitute into this equation, the following values associated with eqn. (3). P1 = 1.581 cmH2 O ω1 = 1.214 rad/s c1 = −0.3132 rad
P2 = −5.534 cmH2 O ω2 = 0.001414 rad/s c2 = 3.297 rad
P3 = 0.5523 cmH2 O ω3 = 2.401 rad/s c3 = −2.381 rad.
108 Human Respiration We compute the value of WOB to be 0.9446 (cmH2 O l) in 5 s, or 0.19 cmH2 O l s−1 or 0.14 mmHg l s−1 or 0.02 W, which is equivalent to an oxygen consumption of ◦ about 0.51 ml/min or about 0.18% of the resting V O2 of 28.87 ml/min. This value can be verified by calculating the value of the area of the pressure-volume loop in Fig. 4 which is equal to 0.8 cmH2 O l.
6 Second-order model for single-compartment lung model Let us now consider the dynamic (instead of static) equilibrium of a spherical segment of the lung model in Fig. 1, obtained as (by dividing throughout by the elemental lung area): ◦◦
ms u + (Pp − Pa ) + Pelas = 0,
(36a)
wherein: Pa and Pp are the alveolar and pleural pressures, u is the alveolar-wall displacement, ms = lung mass (M ) per unit surface area = M/4πR2 , (1b) and Pelas =
V 2σh = + Pel0 , R C
(36b)
where: (i) C is in l (cmH2 O)−1 (ii) ms (wall mass per unit surface area or surface density) = ρh, ρ is the density (mass per unit volume) (iii) σ is the wall stress (iv) h and R are the wall thickness and radius of the equivalent-lung model. Now, the displaced alveolar volume, V = 43 π(R + u)3 , from which we get
◦◦
◦◦
V ≈ 4πR2 u .
(37)
Now, from eqn. (1), by putting (i)
Pp − Pa = (Po − Pa ) + (Pp − Po )
and
PL = Po − Pp , ◦
(ii)
so that Pp − Pa = Po − Pa − PL = RV − PL , (38) ◦◦ ◦◦ ◦ MV V M M = ms u¨ = = M ∗V ; M ∗ = 4πR2 4πR2 16π2 R4 16π2 R4 =
ms , 4πR2
(39)
109
Lung Ventilation Modeling and Assessment
we obtain, from eqns. (1), (2) and (3): ◦◦
M ∗ V + (Po − Pa ) +
V = PL − Pel,o ; C
M∗ =
ms M = . 2 4 16π R 4πR2
(40)
◦
Now, putting Po − Pa = RV , we obtain: ◦◦
◦
M ∗ V + RV +
V Pi sin (ωi t + ci ) − Pel0 = PL − Pel0 = C 3
i=1
= PN .
(41)
Since at the end of expiration when ωi t = ωi T for i = 1 to 3 and PL = Pel0 so that Pel0 = 0. In eqn. (6), we have: wherein: M ∗ = ms /4πR2 = ρs h; ρs is the lung volume-density per unit surface area (in Kgm−5 ) and M ∗ is in Kgm−4 ; (ii) the clinical data in Fig. 2 is assumed to be represented by (i)
PN (t) =
3
Pi sin (ωi t + ci )
with
(42)
i=1
P1 = 1.581 cmH2 O ω1 = 1.214 rad/s c1 = −0.3132 rad
P2 = −5.534 cmH2 O ω2 = 0.001414 rad/s c2 = 3.297 rad
P3 = 0.5523 cmH2 O ω3 = 2.401 rad/s c3 = −2.381 rad.
Then we can rewrite eqn. (6) as: ◦◦
V +
R M∗
◦
V +
Pi V = sin (ωi t + ci ), CM ∗ M∗ 3
(43a)
i=1
or as: ◦◦
◦
V + 2nV + p2 V =
3
Qi sin (ωi t + ci ).
(43b)
i=1
In the above equation: (i) the damping coefficient, 2n = R/M ∗ (ii) the natural frequency of the lung-ventilatory cycle, p2 = 1/CM ∗ (iii) Qi = Pi /M ∗ .
(43c)
So the governing eqn. (8) of the lung-ventilatory response to the inhalation pressure has three parameters: M ∗ , R and C (if the lung pressure is also monitored by
110 Human Respiration intubating the patient). The solution of this equation is given by:
V (t) =
3
.
Qi (−2ωi cos (ωi t + ci )n + sin (ωi t + ci )p2 − sin (ωi t + ci )ωi2 ) 4n2 ω2 + p4 − 2p2 ω2 i=1 1 1 − 1/2Qi −(n2 − p2 ) 2 ci sin ωi2 + p2 (n2 − p2 ) 2 sin ci − 2ωi n2 cos ci 1
+ p2 n sin ci − 2ωi n(n2 − p2 ) 2 cos ci − ωi3 cos ci + ωi2 n sin ci 1 2 + ωi p2 cos ci e − n− −(p−n)(p+n) t / 0 2 2 2 2 12 4 2 2 4 (n − p ) 4n ωi + p − 2p ωi + ωi +
3
1/2
1
−p2 (n2 − p2 ) 2 sin ci + np2 sin ci + ωi cos ci p2
i=1 1
+ ωi2 n sin ci − 2ωi n2 cos ci + 2ωi n(n2 − p2 ) 2 cos ci 1 1 2 + ωi2 (n2 − p2 ) 2 sin ci − ωi3 cos ci e −(n−(−(p−n)(p+n) )t 1 1 . (n2 − p2 ) 2 4n2 ωi2 + p4 − 2p2 ωi2 + ωi4
(44)
We will ignore the exponential terms and perform parameters identification by matching the above expression for V (t) to the clinical data, shown in Fig. 2. The matching is illustrated in Fig. 5, wherein the first- and second-order differential equation solutions for V (t) are depicted. The computed values of the model parameters are also shown in the table below the figure. Further, the first- and second-order model values of R and C are compared in the table. Let us compare these values with those obtained by simulating the first-order model to the clinical data.
7 Two-compartmental linear model Now, it is possible that only one of the two lungs (or lung lobes) may be diseased. So, let us develop a procedure to distinguish between the normal lung and the pathological lung? We hence employ the 2-compartment model (based on our firstorder differential equation of lung ventilatory function) to solve the problem of a two-lung model (schematized in Fig. 6). For this purpose we make the subject breath at different values of frequency (ω), and monitor the total lung pressure PiT (t) (i.e., P1i and P2i ) and total lung volume Vi (t). Correspondingly, we have PiL (t), and ViL (t) and PiR (t) and ViR (t) for the left and right lungs, respectively. The governing equations will be as follows
Lung Ventilation Modeling and Assessment
R [cmH2 O l−1 s] C [l/cmH2 O] M∗ [cmH2O l−1 s2 ] R n = ∗ [s−1 ] M 1 2 [s−2 ] p = CM ∗
First order model
Second order model
2.28 0.21
3.44 0.85 3.02
111
1.14 0.39
Figure 5: Results of single compartmental model based on differentiate equation formulation, compared with the first-order differential equation model. (refer to Fig. 3) PT = Pl = PR,
i.e. P1T = P1l = P1R
&
P2T = P2l = P2R
VT = Vl + Vr
(45) (46)
corresponding to ωi ; wherein (i)
V l (t) = f (ω, Rl , C l , P T (t))
(47)
(ii)
V R (t) = f (ω, Rr , C r , P T (t)).
(48)
In these equations (20), (i) the variables ω, PT (t), V T (t) are deemed to be known, i.e. monitored. (ii) the parameters Rl , C l , and Rr , C r are to be evaluated.
112 Human Respiration
Figure 6: Schematic of two-compartment first-order lung-ventilation model.
Using the first-order differential equation model, (presented in sect. 2, as given by eqn. (6) or (14): 3 (Pi Ca ) −sin (ωi t + ci )ωi2 + ωi3 τa2 cos (ωi t + ci ) . V (t) = 1 + ωi2 τ 2 i=1
(49)
We put down the expression for V T (t) = V L (CL , τL ) + V R (CR, τR ), match it with the volume data (using a parameter-identification technique (software), to obtain the values of (CL , τL ) and (CR, τR ) by means of which we can differentially diagnose left and right lung lobes’ ventilatory capacities and assosciated disorders (or diseases). 7.1 Two compartmental model using first-order ventilatory model Using eqn. (6) without the exponential term, we put down the expression for the total lung volume equal to the sum of left and right lung volumes, as follows: 3 Pi CL sin (ωi t + ci ) − ωi τL2 cos (ωi t + ci ) V (t) = 1 + ωi2 τL2 i=1 3 Pi CR sin (ωi t + ci ) − ωi τR2 cos (ωi t + ci ) , + 1 + ωi2 τR2 i=1
(50)
Lung Ventilation Modeling and Assessment
113
Two compartmental model First order model
R [cmH2 O l−1 s] C [l/cmH2 O] VTL1 VTL2
Left lung
Right lung
1.137 0.1066 2.115 0.2198
1.137 0.0533 0.5289 1.0320
Figure 7: Results of the two-compartment model, based on the first-order differential equation model. Based on our assumption of TV L /TV R = 0.92 we have TV L = 0.48 × 0.48 = 0.2304 l and TV R = 0.52 × 0.48 = 0.2496 l. wherein, for the clinical data, we have: P1 = 1.581 cmH2 O ω1 = 1.214 rad/s c1 = −0.3132 rad
P2 = −5.534 cmH2 O ω2 = 0.001414 rad/s c2 = 3.297 rad
P3 = 0.5523 cmH2 O ω3 = 2.401 rad/s c3 = −2.381 rad.
We further assume that the ratio of TV of the left lung to that of the right lung is 0.92. Now, in order to develop a measure of confidence in our analysis, we first generate the total lung-volume data by assuming different values of C and R for left
114 Human Respiration
Two compartmental model First order model
R [cmH2 O l−1 s] C [l/cmH2 O] VTL1 VTL2
Left lung
Right lung
1.138 0.1066 2.1192 0.3553
2.276 0.1066 8.4766 0.8341
Figure 8: Results of the two-compartment model, based on the first-order differential equation model. Based on our assumption of TV L /TV R = 0.92 we have TV L = 0.48 × 0.61 = 0.2928 l and TV R = 0.52 × 0.61 = 0.3172 l.
and right lung lobes. We then use eqn. (50) along with the above data on pressure and frequency, to generate the total lung-volume data. We adopt this generated lung volume data as the clinical-volume data. We now make our volume-solution expression (eqn. (50)) match this generated clinical-volume data, by means of the parameter-identification procedures, to evaluate C and R for the left and right lung-lobes and hence VTL1 and VTL2 (eqns. (11) and (18)) for these lobes. Based on the values of VTL1 and VTL2 , we can differentially diagnose the left and right lung lobes.
Lung Ventilation Modeling and Assessment
115
7.1.1 Stiff right lung (with compliance problems) We now simulate a normal left lung and stiff right lung, represented by: RL = RR = 1.14 cmH2 O l−1 s and C L = 0.11, C R = 0.05 l/cmH2 O.
(51)
Substituting these parametric values into eqn. (50), we generate the total lungvolume data, as illustrated in Fig. 7. Now our clinical data for this two-compartment model comprises of the pressure data of Fig. 2 and the generated volume data of Fig. 6. For this clinical data, we match the volume solution given by eqn. (50) with the generated volume data, illustrated in Fig. 7, and carry our parameter identification. The computed values of R and C, listed in the table of Fig. 7, are in close agreement with the initially assumed parametric values of eqn. (51). This lends credibility to our model and to our use of parameter-identification method. Now for differential diagnosis, we compute the lung-ventilatory indices, as shown in the table in Fig. 7. 7.1.2 Right lung with R problems Now, we simulate a lug with an obstructive right lung, as represented by: RL = 1.14 and RR = 2.28 cmH2 O l−1 s and CL = CR = 0.11 l/cmH2 O.
(52)
As in the case of the stiff right lung, we first generate the lung-volume data for the above values of compliance and resistances. We then match the total lung-volume solution given by eqn. (50) with the generated lung-volume data, and compute the compliance and flow resistance values of the right and left lung. These are tabulated in Fig. 8, and found to have good correspondence with the assumed values of eqn. (52).
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CHAPTER 5 Modeling of two-phase flow in the human respiratory system V.V. Kulish1,2 , B. Wijayanto1 & C.S. Lim1,2 1 School
of Mechanical & Production Engineering, Nanyang Technological University, Singapore. 2 Biomedical Engineering Research Centre, Nanyang Technological University, Singapore.
Abstract The presence of solid particles in air flowing through the human respiratory channels influences the gas-exchange process in our lungs. This can be related to the causative effects of various respiratory diseases, such as asthma. In order to model the effects of such particles on the gas-exchange process, we need to know the initial concentration of particles in the alveolar region. Two-phase flow through air passages of the human respiratory system has been simulated for all the 23 bifurcational generations by means of the computational fluid dynamics (CFD) with application of fractal modelling of the duct geometry and description of the flow as a diffusion-like process in the constructed fractal space. The result of the simulation allows us to trace and describe the behavior of solid particles as they appear in the alveoli, where their presence can influence the gas-exchange process. Results showed that only 12% of particles entering the trachea reach the alveolar region and influence the gas exchange. Therefore, it can be concluded that in the respiratory duct, there are zones in which the solid phase accumulates, thus reducing the availability for air (oxygen) to reach the region where the gas exchange occurs.
1 Introduction It is a common belief among the medical community that the presence of solid (or liquid) particles in the breathing air affects diffusion occurring in the alveolar
118 Human Respiration region of human lungs. It is known that these particles affect the gas-exchange process leading either to a negative effect (respiratory diseases) or to a positive one (medication inhalation). However, no quantitative model exists regarding the influence of the second phase on the breathing process. This work aims to present such a model based on two novel bioengineering concepts – fractals and fractional calculus – and, to go further, to find a numerical solution to the problem in question. If the classical means were used, this solution could hardly be achieved. Fortunately, the new formulation allows us to obtain the numerical solution for the concentration of the second phase particles at the entrance of the alveolar region without complications related to nonlinearity of the Navier–Stokes equations. Moreover, the extreme geometrical complexity of the fractal domain was avoided by introducing the fractal time and keeping the space Euclidean. The achieved result is also compared with the result obtained for the same problem by semiclassical means, i.e., by solving the linearised Navier–Stokes equation in the nonmapped domain.
2 Methodology 2.1 Geometry of the human respiratory duct The respiratory duct of a human (Fig. 1) is a system of 23 bifurcating generations of self-similar air passages, whose diameters follow the scaling law: dm = d0 × 2−m/3 ,
(1)
where, d is the passage diameter and m is the number of generation. The detailed view of a single bifurcation is shown in Fig. 3. Note that the length of the duct pipe also follows a scaling law, namely: lm = 0.1L × 2−m/4 ,
(2)
where, L = 0.281, m is the total length of the respiratory duct. Since the bifurcations of the respiratory duct (the domain of interest) are selfsimilar, i.e., follow the scaling (power) law given by eqns. (1) and (2), the domain can be modelled as a fractal whose structure is similar to that of the Cantor set (Fig. 2). The fractals are self-similar structures in which all processes remain invariant with spatial scaling, provided that an appropriate time scaling is performed. Consider now the geometry of a single bifurcation shown in Fig. 3, the opening angle ϕ is uniquely determined by the domain geometry as: cos ϕ =
dm /2 = 2−2/3 . dm+1
(3)
Therefore, ϕ = arccos (2−2/3 ) ≈ 50.95◦ . δm+1 = dm+1 sin ϕ = d0 sin ϕ × 2−(m+1)/3 = 0.01 × 2−m/3 [m].
(4)
Modeling of Two-Phase Flow in the Human Respiratory System
119
Figure 1: Schematic of the respiratory duct. BR – bronchi; BL – bronchioles; TBL – transitional bronchioles; RBL – respiratory bronchioles; AD – alveolar ducts; AS – alveolar sacks; z – generation number; T – terminal generation number (Shields, 1994).
Figure 2: Six generations of the Cantor dust (set).
The upper estimate for the domain length:
= l0 + cos ϕ
17
lm = (12.00 + 2−2/3 × 15.186) × 10−2 = 0.22 [m],
m=1
which is in an excellent accord with the physiological data.
(5)
120 Human Respiration lm+1= lm × 2–1/4
dm+1 = dm × 2–1/3
lm
dm / 2 j
j d
Figure 3: Schematic of a single bifurcation of the respiratory duct (lm and lm + 1 – lengths of the ducts of generations m and m + 1 respectively; dm and dm + 1 – diameters of the ducts of generations m and m + 1, respectively; ϕ – the bifurcation angle; δ – the bifurcation length).
2.2 Self-similarity of the flow through the human respiratory duct Since the velocity field is uniquely determined by the domain geometry (which is fractal), it must follow a certain power law, i.e., be self-similar at different scales. This is really the case. From continuity, the following expression must hold: N (m)vm Am = v0 A0 = V˙ ,
(6)
where, vm is the average velocity through a single branch of the mth generation, N (m) = 2m is the number of branches depending on the generation number, Am = (πdm2 )/4 = A0 × 2−2m/3 is, according to eqn. (1), the cross-sectional area of a single branch, and V˙ is the volumetric gas uptake by humans. Therefore, the velocity indeed follows the power law: vm = v0 × 2−m/3 =
V˙ × 2−m/3 , A0
(7)
and, thus, reflects the fractionality of the domain. However, in order to answer the question about self-similarity of the velocity field, it is first necessary to answer the question about the level of organisation of the flow from scale to scale. If the flow is chaotic, its pattern might not be conserved from scale to scale, and, hence, the velocity field might not be self-similar. If, however, the flow is well organised, its pattern, after being scaled, must be the same.
Modeling of Two-Phase Flow in the Human Respiratory System
121
The Reynolds number based on diameter is the criterion of flow organisation. For the domain in question, the Reynolds number follows the power law: Rem =
v m dm = Re0 × 2−2m/3 . ν
(8)
Using the physiological data, i.e., V˙ = 1.25×10−4 m3 /s and d0 = 1.8×10−2 m, one can find Re0 = (4V˙ )/(πνd0 ) = 550 < Recr = 2300. Thus, it is shown that the airflow through the human respiratory duct is laminar and, hence, self-similar at all scales. 2.3 Transformation of the Navier–Stokes equation It was shown, in [1] and [2], that the Navier–Stokes equation written for an irrotational incompressible flow of a constant viscosity fluid, i.e., ∂v 1 + (v∇)v = − ∇p + ν∇ 2 v, ∂t ρ
(9)
can be, by substitution, know as the Cole–Hopf transformation, v = −2ν
∇ψ , ψ
(10)
reduced to the reaction-type diffusion equation (the Einstein–Kolmogorov equation) ∂ψ p − ν∇ 2 ψ = ψ, (11) ∂t 2µ where, ψ is the de Broglie wave function associated with the fluid particle, p is the pressure difference between the local and a certain reference pressure, µ and ν are dynamic and kinematic viscosity, respectively. 2.4 The initial and boundary conditions From eqn. (10), one can derive the expression for the wave function when the velocity is given, that is, xi 1 (12) v(ξi )d ξi , ψ(xi ) = K exp − 2ν x0 where, xi and ξi denote an independent variable and the corresponding dummy variable, respectively. Therefore, if the direction of the domain boundary is chosen along a co-ordinate axis, the no-slip condition becomes simply ψ(xi ) = K, where K is an arbitrary constant. Indeed, the de Broglie wave function can be determined to a certain constant only. However, due to the exponential nature of the relation between this function and velocity, this does not affect the velocity value itself. So, the constant K can be chosen arbitrarily. Nevertheless, since the wave function,
122 Human Respiration in fact, determines the probability of locating a moving particle in a certain place, such that this probability must decrease as the particle’s velocity increases, it is quite reasonable to choose K = 1 for a particle whose velocity is zero. Consequently, the initial condition, if started with the fluid at rest, is also ψ(xi ) = K. Here, care must be taken in order for the constant K chosen for the no-slip condition, to be of the same value as that chosen for the initial condition. Finally, in order to transform the entrance condition, one has to directly use this condition in eqn. (12) to determine the corresponding wave function. 2.5 Computation of the velocity field In order to compute the velocity field, eqn. (11) was discretised with the Laplacian written in cylindrical co-ordinates and, taking into account the flow symmetry, such that ∂/∂θ ≡ 0. The pressure was modelled as the time-dependent function of the form p = p0 (z/) sin (πt/τ), where, according to the physiological data, p0 = 200 Pa (the maximum pressure difference during the breathing cycle), τ = 2 s is the duration of the breathing in phase, is determined from eqn. (5), and z is the discretisation step in the z direction (parallel to the symmetry axis of the domain). The no-slip condition was imposed as ψ = 1 on the solid boundaries of the respiratory duct and, consequently, the initial condition was chosen ψ = 1 everywhere in the domain at t = 0. As for the boundary condition at the entrance, the values of the wave function there were determined from eqn. (10) for the parabolic velocity profile vz (0, r) = vz,max (1 − 4(r/d0 )2 ) and then: ψ 0, rj ≈ ψ 0, rj−1
2ν , 2ν + vz (0, rj )
provided that ψ(0, 0) = 1 due to the no-slip condition. The boundary condition at the outlet was then naturally chosen so that it satisfied the mass-conservation principle. The physical properties of air, necessary for computation, were chosen for the standard atmosphere conditions, i.e., ν = 1.5 × 10−5 m2 /s. The nonhomogeneous diffusion equation (11) was then solved numerically within the domain shown in Fig. 3. After the wave-function field has been computed, the velocity field was restored following the simple algorithm: vz (rj , zi ) ≈ −2ν
ψ(rj , zi−1 ) − ψ(rj , zi ) , ψ(rj , zi )
vr (rj , zi ) ≈ −2ν
ψ(rj , zi ) − ψ(rj−1 , zi ) , ψ(rj , zi )
and
that is, according to the discretisation scheme for eqn. (10).
Modeling of Two-Phase Flow in the Human Respiratory System
123
2.6 Computation of the concentration field The main assumption of the present study was that the solid particles constituting the second phase of the air flow through the human respiratory duct are small enough spheres, i.e., their effective diameter is much smaller than the smallest diameter of a single branch of the duct. Besides, these particles have no slip with respect to the flow. Under these conditions, the concentration field can be found by solving ∂C + (v · ∇)C = D∇ 2 C, ∂t
(13)
provided that the velocity field is given and there is no chemical reaction between particles and the air, i.e., these particles behave like a passive constituent of the mixture. In eqn. (13), C is concentration, v the velocity field, and D is the particles diffusivity, which, in general, depends on the nature of the particles. Equation (13) was discretised and solved numerically after imposing the boundary condition C(0, r) = C0 = const for all t, and the initial condition C(z, r) = 0 at t = 0. 2.7 Fractal (fractional) approach to the problem The convection of passive solid particles, which build the second phase of the flow through the human respiratory system, can be described by the generalised diffusion equation derived in [1]: ∂2(1−H ) C = D∇ 2 C, (14) ∂t 2(1−H ) where C is the particle concentration, t denotes time, and D stands for the diffusion coefficient, which equals the kinematic viscosity of the fluid carrier, when convection is treated. The exponent H , called the Hurst–Hölder exponent, is a measure of the persistence of a statistical phenomenon (in the present case – diffusion or Brownian motion). For a certain random process X (t), characterised by the range R(t) of values X over a time interval t and the sample standard deviation S(t), the Hurst–Hölder exponent can be defined by H = log (R/S)/ log (t). If 1/2 < H ≤ 1, the process is persistent, meaning that it continues in its initial direction rather than changes this direction. Therefore, the diffusion is faster than normal. When, however, 0 ≤ H < 1/2 the process is antipersistent, i.e., the process tends to change its direction with each time step rather than continue in the same direction, and therefore the diffusion is slower than normal. Finally, normal diffusion occurs when H = 1/2 and is usually referred to as Brownian motion or white noise. In this case, eqn. (14) reduces to the classical diffusion equation. The classical Brownian motion (or diffusion), B(t), is related to that taking place in a fractal space, BH (t), by means of the fractional differentiation operator [3]: t 1−2H
BH (t) =
d 1/2−H B(t) . dt 1/2−H
(15)
124 Human Respiration The geometry of the bronchial tree is determined by the least possible resistance to airflow in the entire bronchial system. The tree itself is a self-similar fractal structure of the Cantor dust type. The Hurst–Hölder exponent for the respiratory duct of an average human can be determined if the phase shift between the maximum inhaling velocity and the maximum velocity at the alveolar region is known. Then, taking into account that the time variation of the air velocity in the entrance to the respiratory duct and the time variation of the air velocity at the entrance to the alveolar region must be related by the transform expressed by eqn. (15), i.e.,
π sin ωt + 2
1 −H 2
=
d 1/2−H sin [ωt] , dt 1/2−H
(16)
where ω is the inhaling frequency, one can find the value of H for the system in question. Since the value of the Hurst–Hölder exponent has been found, it becomes possible to solve eqn. (14) numerically, keeping in mind that the diffusion coefficient becomes time dependent, i.e., D(t) = Dt 2H −1 , and that the fractional differentiation can be defined by the Riemann–Liouville integral as [4]: t f (τ) d τ 1 d q f (t) = , (17) q dt ( − q) 0 (t − τ)q+1 for any q < 0. Thus, the generalised diffusion equation (14) can be rewritten in the form t 2H −1 D∇ 2 C =
1 ∂2 (H ) ∂t 2
0
t
C dτ , (t − τ)1−H
(18)
which is valid for 0 ≤ H ≤ 1/2. Using the physiological data on breathing of humans, the value of the Hurst– Hölder exponent for the respiratory duct was found to be 0.5383 from eqn. (16). This means that, in spite of the domain fractionality, which always reduces the speed of the diffusion process, this process goes in the respiratory duct faster than normal. This is purely due to the convection effect. In fact, the flow helps the solid phase to penetrate regions far from the entrance in a shorter time. The value of the Hurst exponent, however, is close to 0.5. This shows that in the respiratory duct, there exists a kind of balance between subdiffusion caused by fractionality and superdiffusion caused by convection. Then, eqn. (18) was discretised with the right-hand side operator approximated by the ordinary ∂/∂t in order to obtain the initial guess for the concentration field. The value of the diffusion coefficient, however, changed with time as Dn+1 = Dn (t)0.0766 . The solution was found in a semi-infinite one-dimensional domain with the boundary condition C = const and the initial condition C = 0 for any time t ≤ 0. The solution, found in such a manner, was then compared with the one found by solving eqn. (13), i.e., by the classical approach. The comparison shows an
Modeling of Two-Phase Flow in the Human Respiratory System
125
excellent agreement between the solution obtained in the two different ways. Hence, the result confirms applicability of the fractal (fractional) approach. It is worth mentioning here that the new approach is more effective. First, it does not require the knowledge of the velocity field, which, in fact, is incorporated in the Hurst exponent. The time dependency of the diffusivity also reflects the convection, i.e., the diffusion happens faster than in the case of a motionless medium. Secondly, to solve eqn. (18) numerically is much easier and faster than even a linearised set of the Navier–Stokes equations together with the equation for the concentration field.
3 Results Figure 4 shows the main result of the computation, i.e. time variation of the solid particle concentration at the entrance into the respiratory zone. The time span t = 2 s on Fig. 4, corresponds to the duration of a single inhalation. Then, the concentration value reaches about 12 per cent of the value at the entrance into the respiratory system (duct). Therefore, if one is to simulate the gas-exchange process in the alveolar region in the case of the presence of a solid phase, it has to account for about 12 per cent of the particles totally present at the trachea entrance. Another important conclusion can be drawn from analysis of the nondimensionalised concentration profile in a single branch of the respiratory duct (Fig. 5). One can see that the maximum concentration occurs near the walls, where the velocity tends to zero due to the no-slip condition. Thus, in a single respiratory branch there are zones in which the solid phase accumulates causing the decrease in the area available for the air (oxygen) in order to reach the region where the gas exchange takes place. This situation resembles that happening in the case of stenosis due to the presence of a benign or malignant tumour in the respiratory duct. From Fig. 5, one can see that, as the concentration profile evolves downstream, up to ten per cent of the cross-sectional area of a single respiratory branch becomes “unavailable” for the air. 0.12
Concentration
0.1 0.08 0.06 0.04 0.02 0 0
0.5
1
1.5
time, s
2
Figure 4: Time variation of concentration at the entrance to the respiratory region.
126 Human Respiration
Dimensionless concentration
The typical nondimensionalised velocity profile is shown in Fig. 6. From Fig. 6, one can see that the velocity profile resembles that occurring in the entrance region of a circular pipe, where the flow is not yet fully developed. It is also no surprise that the shape of the concentration profile (see curve A in Fig. 5) resembles the shape of the velocity profile. It can be easily understood, if one recalls the assumptions made regarding solid particles constituting the second phase of the flow. Since there is no slip between these particles and the carrying medium and the particles are very small, they must track the flow pattern. (In this case, the particles behave in a way similar to the light-scattering particles applied in Laser Doppler Anemometry (LDA).)
1 A D K
0.8 0.6 0.4 0.2 0 0
0.2 0.4 0.6 0.8 1 Distance from the duct center / the duct diameter (r/d)
Dimensionless longitudinal velocity
Figure 5: Development of the concentration profile downstream:A – 75%, D – 50%, K – 25% of the distance from the entrance, respectively.
1 0.8 0.6 0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
Distance from the duct center / the duct diameter (r/d)
Figure 6: Nondimensionalised velocity profile (corresponds to 75% of the distance from the entrance).
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127
4 Discussion The task of experimentally determining flow parameters and distribution of passive solid particles in the respiratory duct is nowadays impossible. However, the advanced methods of fractional calculus, combined with the modern means of computational fluid dynamics (CFD), allow us not only to quantitatively predict those parameters but also to visualise the air flow in question. The results obtained by these means can now be used as the initial data necessary for a numerical simulation of the gas-exchange process in alveoli when the presence of solid microparticles is to be taken into account. It is, however, necessary to keep in mind that the proposed model has its limitations. Thus, for instance, the mechanical properties of the respiratory duct (such as its elasticity) are left outside the model. Also, the condition of normal breathing has been considered, that is, the breathing that provides the basic metabolic activity. However, the results may be used to quantitatively estimate the effects of particles on the air flow under the ‘best’ condition of breathing. Thus, if the breathing conditions ‘worsen’, a worse influence of particles must be expected. All this makes our model a limiting case for estimating the influence of the second phase present in the air.
5 Conclusions A novel method for determining the concentration of passive solid particles in the entrance of the alveolar region has been developed. The results of the present work can be used to properly impose the initial conditions necessary to numerically simulate effects of the microscopic particles on the gas-exchange process. It has been found that that the maximum concentration occurs near the walls of a single respiratory branch. Thus, in the respiratory duct, there are zones in which the solid phase accumulates, reducing the availability for air (oxygen) to reach the region where the gas exchange takes place. this reduction of the available area has been found to be ten per cent.
References [1]
Kulish, V.V., Heat and mass diffusion in microscale: fractals, Brownian motion and fractional calculus. Ph.D. Thesis, Mechanical Engineering Department, Southern Methodist University, pp. 126–130, 1999. [2] Kulish V.V. & Lage, J.L., On the relationship between fluid velocity and de Broglie’s wave function and the implications to the Navier-Stokes equation. International Journal of Fluid Dynamics Research, 29(1), pp. 39–54, 2003. [3] Mandelbrot, B.B., The Fractal Geometry of Nature. 3rd edn. Freeman & Co.: New York, pp. 349–357, 1983. [4] Oldham, K.B. & Spanier, J., The Fractional Calculus, Academic Press: New York and London, p. 49, 1974.
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CHAPTER 6 Quantification of human physiological response to toxic substances V. Kulish1 & V. Novozhilov2 1 School
of Mechanical and Production Engineering, Nanyang Technological University, Singapore. 2 Faculty of Engineering and Built Environment, University of Ulster, United Kingdom.
Abstract The work presented here is an attempt to quantify physiological responses of humans to toxic substances that can be present in the breathing air. Based on the Weber–Fechner law and the data available for carbon monoxide (CO), a scale has been proposed that enables quantification of the main symptoms of intoxication. As a result, a number within the range from zero to unity has been assigned to each of such symptoms depending on the dose of CO received by an average healthy human. By this, zero corresponds to the zero capability (death) and unity corresponds to the absence of any intoxication symptoms. Thus, it becomes possible to perform continuous quantification of the intoxication level. Furthermore, it has been demonstrated that the proposed scale may be used for different toxic substances provided that their solubility in water is known.
1 Introduction It is generally agreed that the main toxic products can be divided into two major categories, namely asphyxiants or narcosis-producing products and the irritants, which may be sensory or pulmonary [1, 2]. In many practical cases (e.g., fire safety), only carbon monoxide (CO) and hydrogen cyanide (HCN) – both belong to the first group – have been detected in the combustion products in sufficient concentration to cause significant acute toxic effects [1]. There is little doubt that CO is responsible for a large proportion of the deaths that arise from accidental fires [3]. It combines
130 Human Respiration with hemoglobin in the blood to form carboxyhemoglobin (COHb), which results in a toxic narcosis by reducing the amount of oxygen supplied to the body tissue. Thus,thetoxicityofcarbonmonoxideisprimarilyduetoitsaffinityforhemoglobin in the blood. Even the partial conversion of hemoglobin to carboxyhemoglobin will result in anemic hypoxia. This is particularly marked for the brain tissues and leads to unconsciousness with cerebral depression. The levels of COHb, which produce this incapacitation, are also common in victims surviving exposure to fire. For carbon monoxide, during the early stages as the carboxyhemoglobin concentration builds up in the bloodstream, the effects are minimal. Researchers have found no effect on human psychomotor performance at levels up to 12% COHb. When the significant effects do occur, their onset is sudden, and the degree of incapacitation rapidly becomes severe, so that, by the time the victim is aware that he or she is affected, effective action is probably not possible. Although exposure to HCN ultimately has the same effect as exposure to CO, it acts much more quickly; in addition, while the onset of incapacitation of CO is gradual, that from HCN is rapid [4]. For hydrogen cyanide, although the ultimate effects of exposure, which are unconsciousness with cerebral depression, are similar to those produced by carbon monoxide, the pattern of toxicity during the early stages is very different. HCN is also not held exclusively in the blood, as COHb is, but is instead carried rapidly to the brain. However, the above is not the main reason why the present study concentrates on carbon monoxide. Having an affinity to hemoglobin about 213 times higher than oxygen (infinite saturation of red blood cells), carbon monoxide is used in clinical measurements to characterize the diffusing properties of the alveolar membrane and, by this, to determine the lung-diffusing capacity. This circumstance makes carbon monoxide unique not only for determining diffusion abnormalities in respiratory patients, but it is the only reference gas for quantifying the physiological response to toxic substances that have a finite saturation level with respect to hemoglobin. The main objective of this chapter is to build a scale that enables one to quantify the negative effects caused by toxic materials on human health.
2 Preliminary considerations For carbon monoxide, in order to predict the time to incapacitation the percentage of COHb (%COHb) must be predicted as a function of carbon monoxide in the environment, as well as the activity level, health, and size of the individual. A detailed model of the physiology of COHb resulting from exposure to CO was developed by Coburn, Foster and Cane in 1965, although it has not been applied much by the combustion toxicology community [5]. The two limiting cases of this model are expressed as follows: The initial uptake of CO when the COHb is well below the concentration in the blood. The equilibrium condition.
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In the initial uptake regime, the transport of CO into the blood stream is limited by the rate at which CO is transported to the lungs through breathing and the transport of CO across the lung/capillary interface. Under these conditions, the total CO uptake would be expected to be proportional to the concentration of CO, ppm CO in the air breathed, the volume rate of breathing (RMV), and the exposure time. This linear relationship between dose and concentration time is known as Haber’s rule: D = κCt,
(1)
where D is the dose, t is time, and κ is the so-called Haber’s constant. Human experiments by Stewart et al [5] are represented by the expression: %COHb = 3.317 × 10−5 (ppmCO)1.036 (RMV)t.
(2)
The deviation from Haber’s rule for CO is very small. This expression has been shown to be useful in constant concentration environments that would be fatal in one hour or less.
3 Basic definitions and the scale of capability In order to start building the quantitative scale of capability, one has first to identify numerical parameters on which this scale will depend. One such possible candidate is undoubtedly the dose of a toxic substance. This quantity can be defined in an exact mathematical way and measured. Untenable toxicity is defined as the lethal dose of the combustion products at a certain height. This is calculated as: t C(t)dt, (3) D= 0
where D is the toxic dose in % minutes, C is concentration of toxic gas in %, t is time in minutes. For convenience, the operational criterion considers only CO and the lethal dose selected from the literature is 9.0% minutes at 1.9 m above floor level. The definition above is similar to that of untenable temperature, which uses the concept of thermal dose at a certain height. This is calculated as: t H= (T )2 dt, (4) 0
where H is the thermal dose in K2 s, T is temperature rise in K, t is time in seconds. In summary, untenable conditions that can cause death in fires are defined as: CO dose
9.0% minutes (i.e., 90,000 ppm min)
Thermal dose
4000 K2 s.
132 Human Respiration It is obvious that the untenable toxicity can constitute the lower limit of the desired scale. We now prescribe unity to the level of capability corresponding to the healthy state, i.e., when no symptoms of intoxication are present, and denote this level as L. We can now write: LCO (Dthr ) = 1,
(5a)
where Dthr denotes the threshold value of the toxic dose. The threshold value corresponds to the state in which no intoxication symptoms are present and is equal to the normal content of carbon monoxide in the breathing air, i.e., Dthr = 0.009% min. The level of capability corresponding to the lethal outcome of intoxication must be zero, that is (5b) LCO (Dlethal ) = 0, where Dlethal = 9.0% min for carbon monoxide. We now have the necessary two limits to build the scale between them, namely LCO (0) = 1 and LCO (9.0) = 0. Having these, one should look at different incapability effects corresponding to the range of doses between Dthr and Dlethal (see Table 1). The toxic dose can be viewed as an external stimulus to which the human organism responds according to the Weber–Fechner law [6]. The most famous example of the use of this law for building scales of physiological responses is the decibel scale that defines a quantitative measure of the human physiological response to sound. The Weber–Fechner law is simply the information gain due to an external stimulus. Mathematical expression of this law follows from the famous Shannon formula for the informational entropy that, in its turn, is related to the Boltzmann formula for the thermodynamic entropy. In the case considered here, the Weber–Fechner law becomes D LCO (D) = 1 − q log , (6) Dthr where the logarithm base is two, D is the dose of CO in % min, and q is a nondimensional constant defined so as to satisfy the scale limits discussed previously. The value of the proportionality constant q in eqn. (6) may be determined if the value Table 1: Characteristic symptoms of CO intoxication depending on the dose received [5]. Dose of CO, % min
Symptom
0.6 2.0 3.0–4.5 9.0
dizziness headache collapse/unconsciousness death
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1 L CO 0.8 dizziness 0.6 headache 0.4 unconsciousness 0.2
death
collapse
0 0
2
4
6
8 D, %min 10
Figure 1: Dependence of the capability degree on the received dose of CO (defined by eqn. (6)). of the lethal dose, Dlethal = 9.0% min, is substituted into eqn. (6) and requiring that LCO (Dlethal ) = 0. It follows then that the value of q must be 0.10034, in order to satisfy this requirement. Then, for a dizziness threshold, for instance, LCO = 0.392; LCO = 0.218 for headache; and 0.100 < LCO < 0.159 for the state of unconsciousness and collapse (see Fig. 1). Now, it is important to note that, by the use of the diffusion and solubility laws [7], it is possible to show that DL-X χX = DL-CO χCO
MCO MX
1/2
χX = 0.0185
28 MX
1/2 = 286.03
χX MX1/2
,
(7)
where DL-X denotes the lung-diffusing capacity with respect to the gas X , χ is the solubility in water (at moderate pressure) and M is the molecular mass of the gas. However, the intoxication effect is proportional to the lung ability to diffuse the intoxicating substance into the blood. Therefore, eqn. (7) gives the coefficient that enables one to compute the physiological response to the gas X , given the physiological response to the same dose of carbon monoxide, i.e., LX (D) = 1 −
DL-X [1 − LCO (D)], DL-CO
(8)
where LCO (D) is given by eqn. (6).
4 Conclusions Based on the Weber–Fechner law and the data available for carbon monoxide (CO), a scale has been proposed that enables quantification of the main symptoms of intoxication. As a result, a number within the range from zero to unity has been
134 Human Respiration assigned to each of such symptoms depending on the dose of CO received by an average healthy human. By this, zero corresponds to the zero capability (death) and unity corresponds to the absence of any intoxication symptoms. Thus, it becomes possible to perform continuous quantification of the intoxication level. Then, inverting the scale, i.e., relating the number corresponding to a certain symptom to the dose of a substance other than CO, it appears possible to compare different toxic substances with respect to their effect on human health conditions. Continuous quantification of intoxication symptoms is important in several ways. First, it allows us to obtain a numerical result during a computer simulation of different health conditions not only in the study of toxic substances, but in a much broader sense (to describe a patient’s conditions numerically). Secondly, it allows, from now on, to avoid a subjective (nonexact) judgment of health conditions, be it by medical diagnostics or by bioengineering research. This first step to the exact – mathematically rigorous – evaluation of health conditions paves a new way towards a better modeling of different cases related to human health and safety. As far as the authors know, no similar approach of evaluating the health symptoms has ever been reported.
References [1]
[2] [3] [4] [5]
[6] [7]
Hartzell, G.E., Priest, D.N. & Switzer, W.G., Proceedings of the First International Symposium on Fire Safety Science, Hemisphere Publishing Company, New York, p. 1059, 1986. Purser, D.A., SFPE Handbook of Fire Protection Engineering, Society of Fire Protection Engineers, Boston, Section 1–14, p. 201, 1988. Anderson, R.A., Watson, A.A. & Harland, W. A., Medical Science Law, 21, p. 175, 1981. Purser, D.A., Grimshaw, P. & Berrill, K.R., Archive of Environmental Health, 39, p. 394, 1984. Beyler, C.L., Fire Dynamics and Chemistry: An Engineering Approach, TMs, Center for Firesafety Studies, Worcester Polytechnic Institute, Worcester MA, 1988. Fechner, G.Th., Elemente der Psychophysik, Ed. Benjamin Rand, 1912. Kulish, V.V., Lage, J.L., Hsia, C.W.H. & Johnson, R.L. Jr., A Porous Medium Model of Alveolar Gas Diffusion. Journal of Porous Medium, 2, 1999.
CHAPTER 7 Anatomically based modeling of pulmonary structure M.H. Tawhai Bioengineering Institute, The University of Auckland, New Zealand.
Abstract Detailed and accurate representation of geometry in computational models of anatomical structures is essential to deliver meaningful results from numerical simulations. Advances in imaging technologies – specifically CT (Computed Tomography) for the lung – means that high-resolution data is now readily available upon which the structure of anatomical computational models can be based. Three major considerations are that 1) model geometry relates directly to individual data, 2) the models are integrative, and 3) the models do not become too large. In this chapter we present methods for constructing anatomically based finite-element models of inter-related pulmonary structures. Segmented human lung-lobe data is fit to highorder elements, and we demonstrate how this model is customized to an individual. By employing high-order elements for the lung-volume mesh, detailed surface geometry is represented using relatively few elements. Finite-element models of the conducting airways are constructed within the lung-lobe model, using a combination of segmented data and a bifurcating-distributive algorithm to generate an airway-consistent mesh. We show how the lung parenchyma can be modeled as a space-filling three-dimensional Voronoi mesh, with a generated geometry consistent with the measured alveolated airway structure. The pulmonary capillaries are generated over the alveolar model, as a two-dimensional Voronoi mesh. The capillary model covers the surface of adjacent alveoli in a continuous network, with a single capillary plexus between the adjacent alveoli. All of the models presented here have been compared extensively with morphometric data, to verify that their geometry is representative of the real pulmonary structure. The models are designed to be integrative: they relate multiple structural systems within the same individual, and their use as computational meshes allows spatial distribution of, for example, air pressure or material properties within the lung.
136 Human Respiration
1 Introduction Advances in medical-imaging technology and computing power provide both highquality data for defining the geometry of pulmonary structures and the means to derive anatomically detailed models. This ‘anatomically based modeling’ approach enables the derivation of geometric models that are specific to an individual, rather than seeking to be representative of the mean population geometry. By developing models that are patient specific, functional predictions can be made for the individual by solution of model systems within the customized geometry. For example, predictions of three-dimensional flow distribution through numerical solution of the Navier–Stokes equations in models of the conducting airways have been shown to be very sensitive to the model branching pattern and the surface geometry [1]. Airway models that are both detailed and subject specific are therefore important for accurate numerical predictions and for understanding the influence of intersubject variability on flow, or on predicted particle transport and deposition. The marked variability that exists in whole-lung geometry between individuals may also influence predictions of soft-tissue deformation, and hence the flow distribution to which it is coupled. An anatomically accurate structure is also important for direct comparison of simulation predictions with regional functional measurements from medical imaging. Finite-element analysis solves model equations over a complex domain by dividing the domain into small ‘elements’ [2]. The discretized domain is referred to as a computational mesh. Because the lungs are comprised of several coupled subsystems – including the airways, the pulmonary circulation, the lymphatic system, and the parenchymal tissue in which the airway and vascular trees are embedded – computational meshes of the lung must vary in structure over a range of scales of interest. The model structure is also dependent on the problem to which it is applied: fluid-flow simulation requires three-dimensional meshes [1], whereas inert-gas mixing [3–5] or airway thermodynamics [6] can be modeled satisfactorily using one-dimensional trees with diameter information. By developing computational meshes of the interdependent structures for a specific individual’s lung geometry, the resulting meshes of the airways, circulation, and tissue are consistent and have anatomically based spatial relationships. This makes the process of coupling functional models within the systems far more straightforward than for models that have been developed independently from different data sets. Creation of a computational mesh of an organ requires a data set that describes either the representative or individual organ structure. For a muscular organ such as the heart it is possible to extract data from post-mortem specimens to develop a generic model [7, 8], or to use MR (Magnetic Resonance) imaging to customize the generic model to an individual [9]. However, these techniques are not suitable for the lung: the delicate structure of the parenchyma precludes ex vivo measurements unless the lung is air dried or similarly preserved, but this alters the lung geometry; and MR imaging does not currently provide adequate anatomical detail for the air-filled lung tissue. The Visible Man project [10] provides a data set from which
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it is possible to develop a generic model, however, because the specimen is post mortem the lung geometry is quite deformed. CT imaging provides a high level of anatomical detail for the lung. Current technology provides enough resolution to enable segmentation of the human lung fissures (boundaries between the lobes), a limited number of major airways, and a large number of major blood vessels [11, 12]. A Lung Atlas is currently being collated by an interdisciplinary team of researchers, based upon measures from high-resolution CT imaging [13]. The Lung Atlas aims to describe the structural and functional properties of the healthy male and female lung and its normal range of variation over three decades of human life. CT data from the Lung Atlas project was used for development of the geometric models that are described in this chapter. In Section 2 a geometry-fitting technique is described and applied to fit a highorder volume mesh to the human lobes, then a technique for customizing a generic mesh to low-resolution data is outlined. In Section 3 the same fitting technique is used to fit a high-order surface mesh to CT-segmented major airways. Section 3 also describes the application of a bifurcating-distributive algorithm to generate a mesh that is representative of the branching structure of the nonsegmented conducting airways. In Section 4 a technique is described for modeling the pulmonary parenchymal tissue as a space-filling network of polyhedral cells, and in Section 5 a technique is presented to generate a segmented capillary network over model alveoli.
2 Finite-element models of the human lung lobes The human lungs are typically divided into five lobes: two lobes in the left lung and three lobes in the right lung. The lobes wrap around the heart and fill the pleural cavity. The lobes are wrapped in the visceral membrane, with a small amount of pleural liquid between them. Because the lung tissue is highly deformable, the pleural surface bears ‘imprints’ from the ribs (Fig. 1a) that change in location as the lungs slide over the ribs during ventilation. Simulation of soft-tissue deformation for the whole lung requires development of independent meshes of the lobes, so that the lobes can move freely over each other. Fitting to geometry data that defines the fissures produces meshes of the lobes that sit close to one another along their boundaries. That is, the definition of a single fissure provides data to each lobe at the common interface. 2.1 High-order elements In Fig. 1a the surface geometry of the right-upper and middle lobes from a highresolution CT scan of a human male at 80% TLC (total lung capacity) is shown rendered in 3D from masked lobe images. A coarse linear mesh is shown in the second panel of the figure. To represent the lobe geometry using only a linear mesh would require a vast number of elements, which quickly becomes impractical when applying the mesh in numerical simulations. Therefore to maintain a small mesh
138 Human Respiration (a)
(b)
(c)
Figure 1: A single mesh fit to both the right-upper and right-middle lobes. (a) Rendered surfaces from masked CT images, and data-points generated over the surface; (b) initial linear mesh of both lobes; (c) fitted cubic Hermite mesh showing surface curvature.
size high-order elements can be used to represent the surface curvature far more efficiently. High-order elements use polynomials to introduce boundary curvature. This means that they provide a closer fit to geometry data with fewer elements than the linear mesh, however, they do not guarantee derivative continuity across element boundaries. One element type that exploits both the higher accuracy of polynomials and ensures derivative continuity is the cubic Hermite. The formulation of this element type is detailed in Bradley et al [14]. The meshes described in the current and following section use tri-cubic Hermite elements for the lobes (volume meshes) and bi-cubic Hermite elements for the airway surfaces (surface mesh). 2.2 Geometry-fitting The geometry-fitting technique solves for nodal positions and first-derivative values to minimize the distance between data-points that represent the surface geometry of an object and a finite-element mesh surface [8]. This technique has previously been used to fit models of the heart [8], lungs [15], diaphragm [16], and musculoskeletal system [15]. Geometry-fitting can be applied to surface or volume meshes, but in each case it is only the nodes on the mesh surface that form part of the solution in the method described below. That is, the location of nodes internal to the mesh do not change during the solution. Data-points that are external to the mesh at the start of the fitting process are projected down to the mesh surface, and data-points that are internal are projected outwards.
Anatomically Based Modeling of Pulmonary Structure (a)
(b)
(c)
(d)
139
Figure 2: The geometry-fitting technique. (a) initial linear mesh (3 nodes and 2 line elements) and scatter of data-points; (b) orthogonal projection of datapoints on to mesh surface; (c) intermediate solution for nodal positions and derivatives, and re-projection of data-points; (d) final mesh solution.
The geometry-fitting technique illustrated in Fig. 2 can be summarized in the following steps: 1. Data-points are gathered to describe the geometry of the object’s surface (Fig. 2a). This may necessitate manual image digitization, or imagethresholding techniques to identify the boundary between the structure of interest and the surrounding tissue. Data-points in Fig. 1a were calculated as a random distribution scattered over the rendered 3D surfaces. 2. An initial linear mesh is positioned close to the data-points (Fig. 2a). An appropriate location and structure for the initial mesh is critical for convergence of the fitting solution to a well-constructed final mesh. 3. Each data-point is projected orthogonally onto the closest surface element of the initial mesh (Fig. 2b). The orthogonal projection defines the shortest distance from the point to the surface. The sum of all data-point to surface distances is calculated as the current error in the mesh. 4. Each data-point projection, and therefore the summed error, can be written in terms of nodal coordinates and derivatives. This expression is the objective function that the geometry-fitting solution seeks to minimize. For each element, a linear system of equations is generated by differentiating the objective function with respect to each element parameter, and equating the resulting expression to zero. 5. The system of equations is solved for new nodal coordinates and derivatives, to minimize the objective function and excessive element curvature (Fig. 2c). 6. Steps 3–5 are repeated until the solution converges and the error is small. That is because the mesh location and surface curvature changes during the solution the data-points must be reprojected onto the current mesh surface and an updated objective function formulated for solution. The volume mesh in Fig. 1c results from applying the geometry-fitting technique to the data in Fig. 1a and initial volume mesh in Fig. 1b.
140 Human Respiration 2.3 Mesh customization To create lobe models specific to an individual, the geometry-fitting technique can be reapplied to a new dense data set. For the well-defined CT image boundaries from the Lung Atlas, compilation of a high-resolution data set and geometry-fitting for an individual is straightforward. Although this approach produces a model with generally small error, an alternative technique – the affine transformation of a generic mesh – is more appropriate for sparse or low-resolution data. An affine transformation differs from a rigid-body transformation in that as well as including translation and rotation, it also includes stretching and shearing. Because CT is a high-dose imaging modality the number of scans a patient is subjected to must be limited, hence scanning the entire lung is not always possible. MRI is low dose, but this method generally produces poor anatomical detail for the lung. Sparse data sets from CT and low-resolution data sets from MRI can be used to define target locations for affine transformation of a generic mesh. An affine transformation procedure, also called ‘host-mesh customization’ has been detailed by Fernandez et al [15] for both surface and volume meshes. The method is illustrated in Fig. 3 for a volume mesh, and can be summarized as follows: 1. A generic mesh – with structure representative of the body of interest (the leftupper lobe is shown in Fig. 3a) – is embedded within a host mesh. The host mesh is comprised of one or more volume elements with any specified interpolation function. The example in Fig. 3 uses a single high-order element. For greater accuracy several elements could be used, but the number of landmark and target points required (step 3) would need to increase accordingly. 2. The location of each generic mesh node is calculated with respect to the host mesh. That is, the host element that contains each node is determined and the node’s local (ξ) coordinates with respect to the host element are calculated. 3. Pairs of landmark and target points on the generic mesh and data set, respectively, are identified (Fig. 3b). Target and landmark points are locations that are clearly identifiable on both the image-based data set and the generic mesh. For example the carina, apex or base of each lung, or bifurcation points of airways that are visible inside the lung and have been constructed as part of the generic mesh. 4. An affine transformation of the host mesh is calculated to minimize the sum of the distances between the pairs of target and landmark points (Fig. 3c). 5. The location of the generic mesh is updated by calculating the global position of each node using the ξ coordinates calculated in step 2 and the new global locations of the host nodes and elements (Fig. 3d). The error in an affine transformation of a generic mesh is generally larger than for a full fit of a dense data set, but for sparse or low-resolution data there is little advantage in applying the geometry-fitting technique. The affine transformation can also be used to position and arrange the initial linear mesh before the projection of data-points in the geometry-fitting procedure. The initial mesh structure is very important for geometry-fitting: if a portion of the
Anatomically Based Modeling of Pulmonary Structure
(a)
(c)
141
(b)
(d)
Figure 3: Host-mesh customization technique. (a) An initial generic mesh of the left-upper lobe surrounded by the host mesh (cube); (b) host mesh enclosing landmark points (small crosses) and target locations (black points); (c) transformation of the host mesh; (d) transformation of the generic mesh. initial mesh is positioned far from the data-points, or the initial element sizes are very different, this will persist throughout the fit and the final mesh will have an unsatisfactory structure. By using the affine transformation to position the mesh first, the initial mesh can be forced to sit within the entire data set, and to have elements of similar size.
3 Finite-element models of the conducting airways The conducting airways extend from the trachea to the terminal bronchioles. Beginning with the trachea, each generation bifurcates into two (or occasionally more) child branches of generally unequal diameter, length, and branching angle [17–19].
142 Human Respiration Each terminal bronchiole (the last purely conductive portion of each airway path) supplies a single pulmonary acinus, which is the basic ‘unit’ of gas exchange [20]. The terminal bronchioles are positioned on average at generation 16 [19], but because of branching asymmetry the terminal bronchiole generation ranges from approximately generation 8 to generation 25 [21]. Anatomical studies of airway-branching pattern and dimensions have given rise to two widely used models of the human conducting airways: Weibel’s symmetric model A [19], and Horsfield’s asymmetric model [21]. Both of these models are still in widespread use in computational studies because they summarize detailed anatomical data into idealized constructs that are relatively easy to implement. In Weibel’s symmetric model A the mean length and diameter are presented for an average number of generations from the trachea to the terminal bronchioles, with each generation (N ) having 2N branches (where N = 0 for the trachea). The Weibel symmetric model has been so widely used because it enables the vast conducting airway system to be represented by only 16 branches (one per generation), hence making it feasible to solve equations within a very small anatomically based description of the airway system. With current computer technology we are not limited to solution in such a small system, however, the symmetric model is still preferable for some simulation studies. For example, when investigating inert-gas mixing in the respiratory airways, Paiva and Engel [4] and Dutrieue et al [3] used the symmetric model to introduce an average ‘delay’ in transit of the inspired gas from the mouth to the terminal bronchiole. When it is sufficient to treat the conducting airways as an ‘averaged’ structure, it is preferable to use the symmetric model than the more accurate but computationally expensive asymmetric models. The Horsfield model is based upon the ordering system proposed by Horsfield and Cumming [17]. In Horsfield ordering the terminal airways are order 1 and each parent branch is one order higher than the child branch of highest order. The Horsfield model represents airway-branching asymmetry by , the difference in order of the two child branches. Horsfield et al [21] presented models with constant through the majority of the conducting airway tree (the terminal airways require a slightly different pattern), and models with different values in each lobe or bronchopulmonary segment. Because each terminal branch in the model has a unique connectivity path, the model size cannot be reduced to the extent of the symmetric model. The Horsfield model also does not include the spatial information that is necessary for some studies. Recent computational studies have highlighted the need for accurate anatomically based airway models with spatially distributed branches [6, 22, 23]. Tawhai and Hunter [22] investigated the influence of conducting airway asymmetry on predictions of inert-gas mixing using an anatomically based model. They concluded that while the branching asymmetry did influence the inert-gas dynamics, a more important factor was an accurate description of the flow distribution. However, to accurately predict the distribution of inspired gas requires coupling of the anatomically based model to deformation of the lung during the breathing cycle. To investigate the influence of heterogeneous airway constriction on global measures of lung function, Tgavalekos et al [23] applied constrictions in an anatomically
Anatomically Based Modeling of Pulmonary Structure
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based airway model, with the location of the constricted airways determined from PET (Positron Emission Tomography) imaging. This type of simulation study can only be performed in a model that has both accurate branching structure and spatial distribution of the airways to link the applied model airway constrictions to the regional PET measures. Tawhai and Hunter [6] have demonstrated that the asymmetry of the branching pattern in the human airways has a significant influence on the rate of heating and humidification in different airway paths when gas is inspired at less than core conditions. Because the location of the airways partly determines the flow through them, the local thermodynamic processes are influenced by the rate of air flow. Therefore this type of study also requires the use of accurate airway models with spatial information. 3.1 Modeling the central conducting airways Advanced segmentation algorithms can uniquely identify human airways down to generation 7, and in some airway paths down to generation 9 [11]. Beyond this level it becomes increasingly difficult to identify the individual airways as the diameters decrease. Therefore to develop models from this limited data set we use a combination of surface-mesh fitting to segmented larger airway data, and an airway-generation algorithm to create centerlines of branches down to the terminal bronchioles. The geometry-fitting technique detailed in Section 2 for the lobes can also be applied to fitting finite element meshes to segmented airway data. Fig. 4 compares two models of several major airways (the first 3 generations): in Fig. 4a the airways are modeled as straight elements, starting and ending at the airway branch points,
(a)
(b)
Figure 4: Models of the trachea and several major bronchi (to generation 3). (a) 1D centerline model with average diameter values for each airway (7 × 1D elements); (b) surface mesh fit to CT data (112 × 2D elements).
144 Human Respiration and with associated mean diameters; in Fig. 4b CT data has been fit to a surface mesh (note that the technique can also be used to fit a volume mesh for the airways, but in the current example only a surface mesh is fit). The second mesh has a more accurate representation of the local surface geometry, whereas the first mesh assumes cylindrical airways with constant cross-sectional area along each branch. The first mesh has the form of the ‘traditional’ airway model. Its structure can be described simply by the location of the branch points, and the diameter of each airway. While the model is small and relatively simple to use, it does not incorporate any change in cross section down the airway, and has discontinuities in cross sections and overlap of the airways at the bifurcations. The second mesh requires storing more information, but it provides a more accurate description of the change in geometry along each airway. The surface mesh can be used in a similar way to the simpler structure in finiteelement simulation studies by integrating the governing equations over the airway cross section. The local area is then used in the formulation of the solution system. 3.2 Modeling the conducting airway tree To model the airway tree from the last segmented airways to the terminal bronchioles, a bifurcating-distributive algorithm can be used to fill the available volume with an asymmetric tree [24]. The algorithm relates the model airways directly to the fit lobe geometry described in Section 2: the airways are generated from the last CT-based airways into each of the fit lobar volume meshes so that the airway lengths, orientation, and branching pattern depend directly on the different lobe shapes. The algorithm generates linear centerlines – no curvature is included. The structure is therefore similar to that shown in Fig. 4a, with straight airways and constant circular cross sections. The algorithm produces only bifurcations: each division is from a single parent into two child branches. Each airway in the unrefined generated tree is a single 1D finite element, and each bifurcation point is the location of a node. The airway algorithm is illustrated in a simple 2D geometry in Fig. 5. Figure 5a shows an initial rectangular host mesh and several airway generations. The algorithm proceeds as follows: 1. The host mesh is filled with a uniformly spaced grid of seed points (Fig. 5a). The density of the seed points is equal to the volume of the whole lung divided by the expected number of acini (=the expected number of terminal branches, approximately 30,000 for the human lung). The seed points can also be randomly distributed, but this has not been shown to produce a significantly different tree structure, and the points are more time consuming to generate. 2. The seed points are grouped with the closest segmented branch end point by calculating the minimum distance from each point to a branch end (Fig. 5b). 3. For each initial group of seed points and parent branch a plane is calculated that contains both the parent branch and the center of mass of the seed points. This plane is used to split each set of seed points into two new sets of generally
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Figure 5: The airway-generation algorithm illustrated in 2D. (a) regular seed points inside the rectangular host area; (b) seed points grouped with the closest airway ending; (c) centers of mass of seed point sets are calculated, and used to split each set of points in two; (d) centers of mass of new sets are calculated; (e) new airways generated from end of parent airway towards center of mass; (f) the resulting area-filling tree.
4. 5.
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unequal size (Fig. 5c). The asymmetry in the seed point split is influenced by the shape of the host volume and introduces asymmetry in the branching pattern and branch geometry. The center of mass is calculated for each of the new seed point sets by averaging the xyz-coordinates of the points (Fig. 5d). For each set of seed points, a new branch is created that starts at the end of the parent branch, and points towards the center of mass. The branch terminates 40% of the distance from the parent to the center of mass (Fig. 5e). Each generated branch and seed point set is inspected to determine whether the generated airway is terminal. An airway is terminal if only one seed point remains in the set, or if the length is less than an anatomically defined minimum value. For terminal airways with more than one seed point, all but the seed point closest to the branch end are distributed to the other seed point sets by calculating the closest current branch ending. This ensures that no ‘gaps’appear in the generated tree. Each set of seed points (including any points that have been reassigned to the set from the terminal airways) is split using the plane containing the newly generated branch and its parent branch. Steps 4–7 are repeated until all of the paths have terminated.
146 Human Respiration The airway diameters can be included using a variety of assumptions. Diameters can be assigned based on generation, Horsfield order, Strahler order [25], or lengthto-diameter ratio. The mesh in Fig. 4a has diameters calculated from the crosssectional areas of the masked CT airways. The conducting-airway mesh generated into the CT-based lobe geometry described in Section 2 is shown in Fig. 6. The method started with the location of airways defined down to generations 6–9. Airways were generated into each lobe separately. Figure 6a views the model from the front, and Fig. 6b from the left-hand side. The different grayscale of each lobe highlights the left-upper and left-lower lobes in Fig. 6b. The geometry of the models produced using the generation algorithm has been compared extensively with published morphometric values [17–19, 21, 25] and has been shown to have anatomically consistent numbers of airways in each generation and Horsfield order, accurate length and branching ratios, and consistent branching angles and planes of branching [24]. A full description of the development of the method, sensitivity to model parameters, and comparison of the model airway trees with anatomical measurements is given in Tawhai [26]. Because the generated airway mesh is always fully contained by the host volume, the algorithm can be applied in any shaped host lung and is not constrained by assumptions about the lobe geometry. This makes the method simple to apply in different human subjects, and in volume models of different animal lungs. The inclusion of additional CT-based airway generations to increase the model accuracy is straightforward using this method: additional airways are simply incorporated in the initial airway-model description, and therefore automatically form part of the mesh from which the algorithm starts branching. (a)
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Figure 6: Airway model generated from 6–9 generations of CT-segmented airways, into a CT-defined volume mesh of the human lung lobes. The model is comprised of approximately 60,000 individual airways (a) from front, (b) from left.
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4 Modeling the respiratory airways as a volume-filling mesh Each terminal conducting airway (the terminal or transitional bronchiole) supplies a single pulmonary acinus. The airspaces in the acinus have walls that contain, or are comprised entirely of, alveoli. The alveoli are densely packed, mechanically connected, and have multifaceted shapes [20]. The alveoli and alveolar ducts therefore form a tethered sponge-like structure. For computational studies the respiratory airspaces have been modeled as a 1D multibranching tree for simulating inert-gas mixing [3, 5], a 2D reconstruction from scanning electron micrographs of the parenchyma [27], a 3D polyhedral network for mechanics studies [28], and as a 3D space-filling base structure for modeling the pulmonary microcirculation [29]. In this section we describe the Delaunay– Voronoi technique employed by Burrowes et al [29] to construct a space-filling alveolar mesh. 4.1 3D Voronoi meshing technique Burrowes et al [29] modeled the alveolar geometry using a Voronoi meshing technique, which creates a volume-filling polyhedral mesh of Voronoi ‘cells’. One of the traditional applications of this technique is for constructing computational meshes for fluid-flow simulation [30, 31]. Here the technique is used to fill a host volume with space-filling model alveoli (one Voronoi cell = one alveolus) and central duct spaces to produce a 3D mesh with anatomically consistent alveolar dimensions and surface-to-volume ratio. The meshing procedure is illustrated for a simple 2D host mesh in Fig. 7. 1. Seed points are generated within a host volume and around the host boundary (Fig. 7a). In the method used by Burrowes et al [29], the host boundary is defined by creating sets of seed points orthogonal to the bounding surface, such that a sphere passing through the points has its center located exactly on the boundary [30, 31]. That is, an area bounded by a set of lines is defined by pairs of orthogonal points along the lines and two external and one internal point at any convex corner (one point orthogonal to each joining line). And a volume bounded by planes is defined by pairs of orthogonal points on the bounding surfaces, and sets of three external and one internal point at any convex corner (where three planes meet). 2. The seed points are triangulated (Fig. 7b). This is a ‘Delaunay triangulation’, which means that the circle passing through three points defining any triangle (or sphere passing through four points defining any tetrahedral element in 3D) does not contain any other point. Note that the host boundary passes through the triangulation (tetrahedralization) of the orthogonal boundary points. 3. The circumcenters of the triangles or tetrahedra are calculated. The circumcenters are the center points of the circle or sphere. The Voronoi ‘dual’ of the Delaunay triangulation has vertices at the circumcenters of neighboring
148 Human Respiration (a)
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Figure 7: Space-filling Voronoi meshing technique for modeling the alveolar (respiratory) tissue, illustrated in 2D. (a) Host area filled with regular grid of seed points; (b) Delaunay triangulation of seed points; (c) circumcenters of adjacent triangles are joined; (d) Voronoi cells comprised of joined circumcenters; (e) faces removed to form ‘ducts’ and to aerate the entire structure. triangles or tetrahedra (Figs. 7c and d). The circumcenters of the boundary triangles or tetrahedra lie on the boundary lines or surfaces. 4. Specific Voronoi cells are allocated as airway duct cells. This step starts at one cell close to the boundary, and with a specified initial direction. Every face adjacent to the first duct cell is removed. The algorithm steps to the adjacent cell that is closest to the initial direction and removes all of its faces. Forward stepping continues until 3–4 ‘rings’ of alveoli have been formed, at which point the duct bifurcates. Two new directions are calculated, and the forward stepping, face removal, and bifurcation continues as above until all of the cells have been classified as either duct or alveolar cells (Fig. 7e). 5. Each cell face is converted into triangular finite elements. A node is placed at the center of the face and at each face vertex. The final structure forms a volume-filling honeycomb-like mesh. The Delaunay–Voronoi meshing technique provides a relatively simple way of creating alveolated airway structures within general host-volume shapes. The finiteelement meshes produced by the method are consistent with the alveolar mesh developed for mechanical analysis by Denny and Schroter [28]. Several ducts isolated from a larger 3D mesh are shown in Fig. 8a, and a single alveolar sac
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Figure 8: Model alveolated ducts isolated from an alveolar mesh generated by the 3D Delaunay–Voronoi meshing technique. (a) system of isolated ducts; (b) a single alveolar duct.
is isolated in Fig. 8b. The triangular elements are shown dividing each cell face. The mesh is continuous, space-filling, and has an alveolar surface to volume ratio within 2.7% of the published value [19].
5 Modeling the microcirculation as a segmented network The pulmonary capillaries form a dense sheet-like meshwork composed of short interconnected capillary segments. The capillaries are wrapped over the alveoli, with only a single sheet of capillaries between adjacent alveoli on the same alveolar duct. The classic model of the pulmonary microcirculation treats the capillary bed as a ‘sheet’ of blood bounded by two compliant layers of endothelium [32–34]. The sheet-flow model of alveolar blood flow simplifies the complex capillary geometry into a model that is appropriate for many computational studies of the pulmonary microcirculation. However, the sheet-flow model cannot be used for investigating individual cell transit, segment blocking by neutrophils, or identification of preferential pathways for cell transit [35]. Segmented models of the pulmonary microcirculation have been used by Huang et al [35], Dhadwal et al [36], and Burrowes et al [29] to investigate cellular transit. Dhadwal et al [36] modeled the capillaries on a single alveolar septum as a 6 × 6 square grid of interconnected segments. Huang et al [35] extended this model to generate a random orientation of the capillary segments, and connected six of the septae at opposite corners for simulation of red blood cell and neutrophil transit. Blood exiting a septal face was constrained to enter the adjacent face at the corner junction. Burrowes et al [29] developed an anatomically based capillary model with multiple inlet and outlet vessels, for solution of the Huang et al [35] blood-flow
150 Human Respiration model in different vertical locations in the lung. The model developed by Burrowes et al [29] is detailed in the following section. 5.1 2D Voronoi meshing technique Burrowes et al [29] used a 2D Delaunay–Voronoi meshing method to generate a segmented capillary mesh that wraps over the surface of model alveoli, with a single model capillary sheet between the adjacent alveoli. The method generates a capillary mesh to cover the 2D alveolar septae of the volume-filling alveolar model described in Section 4. The meshing technique can be applied to not only the fairly regular alveolar structure shown in Section 4, but also to any irregular finite-element representation of the alveoli. The meshing technique as described below is illustrated over the surface of a simplified single alveolus in Fig. 9:
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Figure 9: Voronoi-mesh construction over a portion of the surface of a unit sphere, for modeling the pulmonary capillary bed. (a) Regular grid of seed points generated on the surface of a unit sphere, with a simplified mesh of a single alveolus placed at the center; (b) Delaunay triangulation of initial points; (c) circumcenters of triangles joined to form Voronoi cells; (d) Voronoi cells projected onto alveolar mesh, with each cell edge representing a single capillary segment.
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1. Regularly spaced points are generated over the surface of a portion of a unit sphere (Fig. 9a). The perimeter of the spherical portion is defined by pairs of points that are orthogonal to the base and lie on the sphere surface. 2. The points are triangulated (Delaunay triangulation) over the 2D sphere surface (Fig. 9b). 3. The circumcenters of the triangles are calculated, and neighboring circumcenters are joined to form Voronoi cells (Fig. 9c). The cells are 2D polygons in this method, in contrast to the 3D polyhedra generated by the method in Section 4. 4. The circumcenters are projected onto the surface of a single alveolus (Fig. 9d). This is a normal projection from the curved sphere surface to the flat alveolar faces. 5. Each Voronoi cell edge represents a single capillary segment. A finite-element mesh of the capillary network is constructed by transforming the circumcenter coordinates into nodal positions, and each Voronoi cell edge into a 1D element. The ‘rings’ of capillaries that are formed typically have either six or five segments. 6. For the next alveolus, any Delaunay points that are on a shared face (between adjacent alveoli) are replaced with the points from the neighboring alveolus that lie on the common face. The triangulation produces an identical network on the shared face that is continuous with the networks on each of the adjoining faces in each direction. This ensures that the entire generated capillary network is continuous over the alveoli that surround a common duct, with only a single capillary layer between the adjacent alveoli. The capillary mesh generated over the entire alveolar sac model in Fig. 8b is shown in Fig. 10. The mesh has approximately 1,000 capillary segments per
Figure 10: Capillary model generated over the surface of an alveolar sac. Five venules are shown on the external surface of the alveoli. Two of the four supplying arterioles are visible at the bottom of the figure.
152 Human Respiration alveolus, and 85 segments on average that a red blood cell would pass through as it traverses from a supplying arteriole to a draining venule. These values are consistent with published measurements [19, 37, 38]. Capillary diameters were included using published values from the literature [35].
6 Summary In this chapter we have described geometric models of pulmonary structures that are inter-related through development from the same imaging set (lung, lobes, airways), or through generation within the geometry of another model (capillary network). That is, the models have spatial relationships that can be exploited when simulating pulmonary behavior. But to be integrative the models also need to relate to one another functionally. Because the large-scale models presented here were derived from a single lung geometry, spatial positioning of their embedded structures allows accurate distribution of material properties, or interpolation of parameters from one model subsystem for inclusion in another. For example, the soft-tissue deformation of the lobes can be coupled to the embedded conducting airway model to drive air flow through the tree. Conversely, the airways can be treated as ‘fibers’ within the parenchyma, which will influence the solution of tissue deformation. Because the human lungs display marked variability in shape, volume, and airway and vessel orientation, it is important to be able to produce individualized computational meshes. The meshing techniques described in this chapter – geometry-fitting, affine transformation, Voronoi meshing, and algorithms to generate host-shapedependent airway models – allow us to create computational models that relate anatomical information for an individual to function that can be predicted through numerical simulations.
Acknowledgements This work was supported by The Royal Society of New Zealand Marsden Grant 01-UOA-070 MIS. The author would like to acknowledge the contributions of Professor P.J. Hunter, Ms K.S. Burrowes, Dr P. Mithraratne, and Mr J. Fernandez at the University of Auckland to the modeling work described in this chapter. The author also thanks Dr E.A. Hoffman and collaborators on the Lung Atlas project for access to, and assistance with, CT images.
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CHAPTER 8 Applied chest-wall vibration therapy for patients with obstructive lung disease J.H. Ong & D.N. Ghista School of Mechanical and Production Engineering, Nanyang Technological University, Singapore.
Abstract Patients with chronic lung disease (such as emphysema and chronic bronchitis) have ventilatory difficulties (i.e. getting adequate tidal volume), due to lung compliance and airflow resistance caused by mucus build-up in the lung airways. For these patients, high-frequency chest-compression therapy (HFCCT) is gaining acceptance (as an alternative to conventional physical chest physiotherapy). For arresting the decline in pulmonary function of patients with chronic obstructive pulmonary disease (COPD), the method involves the use of an inflatable vest system and is designed for permitting self-administration. However, the commonly used pulsating frequency range of 6–19 Hz employed in the inflatable vest system is probably not optimized for mucus drainage. This is because for maximal effectiveness, the HFCCT needs to be effected at the chest-resonance frequencies. In an effort to enhance the efficacy of the method, chest-resonance frequencies were measured on a sample of 23 volunteers. The average chest-resonance frequencies found for male and female healthy volunteers are 26.7 Hz and 27.8 Hz, respectively. The results suggest that HFCCT should be carried out at about 18–34 Hz. Ideally, however, it is necessary to first obtain the chest-resonance frequency of the patient and then administer HFCCT at that value. This is because patients suffering from COPD will have impaired respiratory muscle and will have a smaller than normal chest-wall cavity, which in-turn will affect the average chest-resonant frequency of patients suffering from COPD.
1 Introduction There is often a build up of mucus within the lungs of patients caused by a variety of diseases such as cystic fibrosis, emphysema, asthma, and chronic bronchitis.
158 Human Respiration For example, cystic fibrosis (CF) is a deadly hereditary disease that affects the mucus-secreting glands of the body so that there is an overproduction of mucus. The mucus is very thick and accumulates in the intestines and lungs, causing increased lung compliance and resistance to airflow, with impaired breathing. Hence, the excess mucus must be removed on a daily basis, so as to avoid build-up and reduce the risk of contracting infections that in turn may lead to severe impairment of lung function and ultimately to patient’s demise. Traditionally, the removal of mucus from lungs is accomplished by the application of chest physiotherapy (CPT). CPT is an airway-clearance technique that combines manual percussion of the chest-wall by a respiratory therapist, strategic positioning of the patient for mucus drainage and the removal of these secretions by effective coughing and breathing techniques. Studies have shown that regular CPT is able to enhance expectoration [1–3] and slow down the deterioration of lung function [4]. The problem with conventional CPT is that it has to be administered by a trained physiotherapist. It is time consuming, and hence contributes to the low compliance rates that are reported in the range of 26 to 47% [5–8]. As an alternative to manual percussion, pneumatic chest-compression devices [9–13] have been developed to produce high-frequency chest-wall oscillation (HFCWO). Essentially, a pneumatic chest-compression device consists of a largevolume variable-frequency air-pulse delivery system and an inflatable vest worn by the patient. The technique is based on the theory that the action of the pulsating air pressure delivered to the chest-wall will transmit shock waves through the chest-wall and enhance airflow velocity through the air passages. This action produces a cough-like shear force and reduction in mucus viscosity that results in an upward motion of the mucus. Clinical trials have shown that these devices are effective in facilitating mucus clearance [14–17] and inducing HFCWO is equal to or more effective than the conventional CPT involving a respiratory therapist pounding on a patient’s chest to loosen mucus for expectoration. It has also been noted that HFCWO or HFCCT is well tolerated by the majority of patients, and that there has been no adverse effects [14–17] observed in this form of chest percussion. As an example of the clinical efficacy of HFCWO, consider the case of a 48-year-old man with cystic fibrosis who had Pseudomonas aeruginosa in his sputum and had experienced a two-year worsening of pulmonary function [9]. After a year of using the HFCWO vest, his pulmonary function had improved to the level of five years earlier. Furthermore, before HFCWO therapy the technetium aerosol scan showed an absence of ventilation in the upper lobes. However, after eight months of use of the HFCWO vest, it showed restoration of ventilation in these areas. This particular HFCWO vest user did not participate in the clinical trials by Hansen and Warwick [9]. Thus, the observed improvement in his pulmonary function could be due to regular self-administered chest therapy. Hence, unlike conventional CPT, a HFCWO system can be used for self-therapy, which may promote increased patient compliance. Now, if the methodology can be further improved by determining the optimal chest-pulsating frequency (of HFCWO) to be equal to the chest-resonance frequency, the results could be even better.
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The inflatable-vest system based on HFCWO works on the principle of rapid compression and relaxation of the chest-wall. The vest inflates and deflates rapidly, gently compressing and releasing the chest-wall 5–20 times per second. The oscillations can be adjusted and typically the oscillation frequency used in an inflatable vest system is set at 6 to 19 Hz. The present study is based on the premise that if the oscillation frequency of the inflatable vest is tuned to the patient’s chestdynamic characteristics, it will effectively maximize the energy transmission to the lung. This will result in optimal vibrations, which will loosen the mucus, increase the airflow velocity, and hence increase the efficiency of clearing the lungs of mucus. Hence, in the present study, experiments were conducted on 21 healthy volunteers to measure the resonance frequencies of the human chest. The volunteers were mainly in their twenties, and comprised 8 males and 13 females. The study investigated the variation of the measured resonance frequency in relation to the effects of wearing thicker clothing, breathing rate, as well as whether the test is conducted before or after a meal. The tests suggested that the results are independent of these variables. The resonance frequencies were also analyzed to test the correlation between the chest-resonant frequency in relation to the subject’s mass, chest size, height and body mass index (BMI). The results indicate that since the chest-resonance frequency is highly correlated to the subject’s chest size, the latter can be used to estimate their chest-resonant frequency. However, as mentioned earlier, the optimal frequency of applied HFCWO is deemed to be the patient’s chest-resonance frequency (CRF). The CRF would also be affected by COPD, involving the mucus build-up, weakening of the respiratory muscle and resulting in changes to a smaller than normal chest-wall cavity. It would be interesting to determine the ranges of CRF of patients with lung diseases, and to custom-tailor the application of HFCWO (or HFCCT) to the CRF of the patient, so as to deliver maximal energy to the chest to loosen and induce drainage of the mucus.
2 Experimental methodology Twenty-one healthy adults volunteered as subjects for this study. All the subjects were students, comprising 8 males and 13 females, whose age was mostly in the twenties. The subjects were measured for height, weight and chest size. Two measurements of chest size were taken for the female subjects; one is the upper chest size, (which is the circumference of the upper body around the breast) and the other is the lower chest size (which is the circumference of the upper body immediately below the breast). The same experimenter made all measurements. From the physical measurements, each volunteer’s body mass index (BMI) was calculated based on the weight (kg) of the person divided by the square of the person’s height (meter). The body-mass index (BMI) is one of the most accurate ways to determine whether a person is too thin, normal, overweight or obese. Alternatively, it can also be seen as a gauge of the overall mass-to-stiffness ratio of the body. For the measurements of chest-resonance frequency, the volunteers were made to sit upright on a stool and to lean with their backs against a light stiff pad that was
160 Human Respiration 0.2 0.18
Inertance (m/s2 N)
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 15
20
25
30 35 Frequency (Hz)
40
45
50
Figure 1: Typical frequency-response function. linked by a flexible stinger to an electrodynamic shaker. A force transducer was placed between the stinger and the pad so as to record the force delivered to the subject; the output was measured using an accelerometer held firmly against the front of the chest at the sternum. The amplitude of the excitation force delivered to the subjects was adjusted manually, starting from a low level to the maximum level felt comfortable by the subjects. The level is then reduced slightly from the maximum level. White noise was used as the input signal. The output from both the force transducer and the accelerometer was fed to a dual-channel spectrum analyzer. The setting on the spectrum analyzer was adjusted to display the frequency-response function for the acceleration response per unit input force. The coherence function [18], an indicator of how well the output of the system under test is related to the input to the system, was also displayed. A typical frequency-response function and its corresponding coherence function are shown in Figs. 1 and 2, respectively. As with all random input signals, it is necessary to perform an averaging process so as to remove spurious random noise from the signals. The frequency-response function (FRF) and the coherence function (COH) as shown in Figs. 1 and 2, are based on 100 averages. Using 100 averages, it was found that the coherence value near the resonance frequency is always greater than 0.8 for all the tests conducted on the 21 volunteers. This implies that the results from the tests are reliable with good input signal-to-noise ratio. It can be seen from Fig. 1 that a clear maximum peak in the FRF enables the volunteer’s chest-resonant frequency of 26.5 Hz to be identified. The equivalent viscous damping ratio (ζ) for the chest-resonant frequency is estimated from the FRF using the half-power points; in the case of the example in Fig. 1, the damping ratio is 0.16. The vibration test was repeated for all volunteers and the chest-resonance frequencies and damping factors recorded are based on an average of two measurements.
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161
1 0.9 0.8
Coherence
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 15
20
25
30
35
40
45
50
Frequency (Hz)
Figure 2: Corresponding coherence function.
3 Theoretical considerations based on guidelines for HFCCT Let us assume that the chest vibration corresponds to the case of vibration with viscous damping, which can be represented by: x¨ + 2ζωn x˙ + ωn2 x = 0, wherein:
(i) The natural frequency of a person’s chest vibration is ωn = k/m, with k representing the chest stiffness and m the chest mass. The damped natural frequency of chest vibration is: ωd = ωn 1 − ζ 2 (ii) ζ is the viscous damping ratio of a person’s chest vibration When the chest is pulsed by the application of an external force (in HFCCT), we can represent the forced vibration of the chest as: x¨ + 2ζωn x˙ + ωn2 x = Q sin ωt, where ω is the applied pulsating frequency (of vest inflation and deflation) and Q is the amplitude of applied force on the chest. In response to the application of Q sin ωt, the chest-wall displacement response can be written as: x = D sin (ωt − θ), where D is the amplitude of the chest displacement and is given as: 1 Q , β= , D=β 2 ωn (1 − (ω/ωn )2 )2 + (2ζ(ω/ωn ))2
162 Human Respiration and β can be termed as the chest-vibration amplitude amplification. The implication of β is that the amplitude of the chest displacement increases as the applied forcing frequency ω approaches the natural frequency (ωn ). However, the value of β in the neighborhood of ωn is very much influenced by the amount of damping in the chest vibration. For varying ζ, the resonance (or maximum β) occurs at the applied forcing frequency of: 2 ω = ωn 1 − 2ζ 2 ,
and the corresponding maximum amplification is given as: β=
1 . 2ζ 1 − ζ 2
4 Results and discussions The experimental data for anthropometric characteristics, average chest natural frequencies and damping ratios for all the male and female subjects are presented in Tables 1 and 2, respectively. Also included in the tables are the means, standard deviations and the Pearson correlation coefficients between the resonant frequencies and the various physical attributes of the volunteers. As can be seen from the tables, the natural frequency for the human chest is specific to the individual, and varies from 24 Hz to 30 Hz. For the individuals participating in the investigation, an average natural frequency for the male participants is 26.7 Hz and for the female volunteers, the average natural frequency is 27.8 Hz. It is important to note the significance of these varying natural frequencies. That is, when the excitation frequencies are almost equal to the human chest frequencies, large vibration amplitudes develop. At approximately 10 Hz away from the chest frequencies, the vibration amplitude drops dramatically as compared to the peak values. Therefore, the premise is that if the HFCWO vest is pulsed at the human chest frequencies, it will provide the optimal setting for the chest-wall oscillator to transport mucus in a cephalad direction. Consider the HFCWO protocol as recommended by a vest’s manufacturer. Six frequencies (6, 8, 14, 18 and 19 Hz) are recommended. Except for the two highest frequencies of 18 Hz and 19 Hz, at all the other frequencies, the vibrations of the chest-wall will be low as compared to the vibrations at resonance. Even at 18 Hz and 19 Hz, the two frequencies are still far away from the average chest frequencies of the volunteers, and the difference in the vibration amplitudes will still be substantial as compared to exciting chest vibration at resonance. Apart from the resonance frequencies, the sharpness of the peak as seen in Fig. 1 is an important factor. It is related to the damping of the chest-walls. In this exercise, we assume viscous damping and the experimental damping ratios (ζ) are given in Tables 1 and 2. The average damping ratios for the male and female volunteers are 0.15 and 0.16, respectively. The sharpness of resonance (Q) is calculated as Q = 1/2ζ. Calculated on the basis of ζ = 0.15, Q = 1/3. Here, it is important to note that the most frequently reported complaint by patients who used the HFCWO vests was
Table 1: Measured parameters of 8 male volunteers.
Subject
23 27 23 23 39 28 27 28 27.3 5.3
Mass (kg) 53 70 64 55 55 57 60 62 59.5 5.7 –0.79
Height (cm) 162 170 165 162 161 168 172 168 166 4.1 –0.21
Chest size (cm) 83 95 96 92 84 86 83 89 88.5 5.3 –0.94
Body-mass index (BMI)
Chest-resonance frequency (Hz)
Damping ratio (ς)
20.20 24.22 23.51 20.10 21.22 20.20 20.28 21.97 21.57 1.55 –0.89
28.35 24.95 24.43 26.50 28.10 27.70 28.15 25.40 26.70 1.59
0.11 0.13 0.17 0.16 0.13 0.19 0.15 0.17 0.15 0.03
Applied Chest-Wall Vibration Therapy
M1 M2 M3 M4 M5 M6 M7 M8 Mean STD Pearson
Age (Years)
163
Subject F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 Mean STD Pearson
Age (Years) 23 24 24 36 24 23 24 24 22 22 23 23 23 24.2 3.6
Mass (kg) 48 51 56 50 40 48 50 54 58 49 46 70 50 51.5 7.1 –0.78
Height (cm) 170 162 155 152 152 163 164 169 164 154 163 167 160 161.2 6.2 –0.04
Lower chest size (cm) 69 74 81 74 68 69 76 68 75 68 73 86 75 73.5 5.4 –0.93
Upper chest size (cm) 79 87 90 86 78 78 87 79 86 78 80 95 84 83.6 5.4 –0.97
Body-mass index (BMI)
Chest-resonance frequency (Hz)
Damping ratio (ς)
16.61 19.43 23.31 21.64 17.31 18.07 18.59 18.91 21.56 20.66 17.31 25.10 19.53 19.85 2.51 –0.81
29.05 27.05 25.55 27.53 28.80 29.15 26.90 28.45 27.55 29.35 29.15 24.85 28.18 27.81 1.43
0.15 0.13 0.19 0.12 0.20 0.17 0.15 0.19 0.13 0.13 0.16 0.19 0.15 0.16 0.03
164 Human Respiration
Table 2: Measured parameters of 13 female volunteers.
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a sense of chest constriction while the vest was inflated. However, patients tolerated this form of chest percussion since they felt it was easier to breathe once therapy was completed [8]. It should be noted that clinical trials have shown that HFCCT devices are effective in facilitating mucus clearance [6–9], and the action of the HFCWO is at least equal to and in fact more effective than the conventional CPT involving a respiratory therapist pounding on a patient’s chest to loosen mucus for expectoration. Hence, the significant of Q = 1/3 is that if the oscillating compressive force to the patient’s chest is reduced by a third of the pressures used in the literature [6–9], and the pulsating frequencies are set at resonance, then it will probably be as effective as the original trials. Reducing the pulsating air pressure will probably help in reducing the most frequently reported complaint of chest constriction by patients. An interesting observation from Tables 1 and 2 is that the values of the coefficientof-correlation between the resonant frequency and chest size for male and female volunteers are –0.94 (male) and –0.93 (female lower chest size) and –0.97 (female upper chest size). Therefore, the implication is that it is possible to estimate the chest natural frequency from the chest size. Taking a linear regression analysis on all of the data for male chest sizes, the chest-resonant frequency (CRF) can be predicted according to the equation: Male CRF = 51.549–0.281 (Chest size). This data is plotted in Fig. 3, and can be used to estimate the male chest-resonant frequency directly from the figure. In the case of female subjects, there are two chest size values. Since, the upper chest size has a higher coefficient of correlation; a backward linear regression analysis is done on the upper chest sizes. The data is plotted in Fig. 4, and the equation for predicting the resonant frequency is: Female CRF = 49.048–0.254 (Chest size).
5 Concluding remarks It is important to note that all the data collected from the experiment relate only to a small sample of healthy volunteers who are young adults. These chest frequencies 29 Frequency = 51.549 – 0.281 (ChestSize) Frequency (Hz)
28 R2 = 0.8806 27 26 25 24 80
85
90
95
Male Chest Size (cm)
Figure 3: Graph of resonant frequency versus male chest size.
100
166 Human Respiration 30 Frequency = 49.048 – 0.254 (ChestSize) Frequency (Hz)
29 R2 = 0.9368 28 27 26 25 24 70
75
80
85
90
95
100
Female Upper Chest Size (cm)
Figure 4: Graph of resonant frequency versus female upper chest size.
may inevitably be different from patients who suffer from cystic fibrosis or any form of chronic obstructive pulmonary disease (COPD). Nevertheless, Figs. 3 and 4 can be used to provide an initial estimate of the chest-resonant frequencies of patients with COPD. To estimate the resonance frequencies of patients with COPD from their chest sizes, it will be more appropriate to obtain the data from volunteers who suffer from COPD. For these volunteers, it is not advisable to do the experiment as described in this chapter on these unhealthy subjects. For these subjects, the HFCWO vest can be used for conducting the experiments, as it has been shown to be safe and there are no detrimental effects to the users. Based on an average chest frequency of 26 Hz, the pulsating frequencies can be set between 18 Hz to 34 Hz. Alternatively, the resonant frequency of a patient with COPD can be estimated using the regression equations or directly from Figs. 3 and 4. The range for the pulsating frequency can then be set at plus or minus 8 Hz from this predicted resonance frequency. Since, at resonance, maximum vibrations occur, it can be construed that the frequency that gives the maximum airflow rate is the chest-resonant frequency.
References [1]
Gallon, A., Evaluation of chest percussion in the treatment of patients with copious production, Respiratory Medicine, 85, pp. 45–51, 1991. [2] Maxwell, M. & Redmond, A., Comparative trial of manual and mechanical percussion technique with gravity assisted bronchial drainage in patients with cystic fibrosis, Archives of Disease in Childhood, 54, pp. 542–544, 1979. [3] Salh, W., Bilton, D. & Dodd, M., Effect of exercise and physiotherapy in aiding sputum expectoration in adults with cystic fibrosis, Thorax, 44, pp. 1006–1008, 1989.
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[4] Warwick, W.J. & Hansen, L.G., The long-term effect of high frequency chest compression therapy on pulmonary complications of cystic fibrosis, Pediatric Pulmonology, 11, pp. 265–271, 1991. [5] Currie, D.C., Munro, C. & Gaskell, D., Practice, problems and compliance with postural drainage: a survey of chronic sputum producers, British Journal of Disease in Chest, 80, pp. 249–253, 1986. [6] Davids, L.M. & Heneley, L.D., Investigation into the compliance with physiotherapy regimens in cystic fibrosis, South African Journal of Physiotherapy, 46, pp. 7–10, 1990. [7] Fong, S., Dales, R. & Tierney, M., Compliance amongst adults with cystic fibrosis, Pediatric Pulmonology, 5(suppl), p. 140, 1988. [8] Muszinsky-Kwan, A.T., Perlman, R. & Rivinton-Law, B., Compliance with and effectiveness of chest physiotherapy in cystic fibrosis: a review, Physiotherapy Canada, 40, pp. 28–32, 1988. [9] Hansen, L.G. & Warwick, W.J., High-frequency chest compression system to aid in clearance of mucus from the lung, Biomedical Instrumentation Technology, 24(4), pp. 289–294, 1990. [10] Bose, A.A., Finkelstein, S.M., Warwick, W.J. & Budd, J.R., Automation in high frequency chest compression. Proceeding of the Annual Conference on Engineering in Medicine and Biology, Publ by IEEE, IEEE Service Center, Piscataway, NJ, USA. Vol 15, pt2, pp. 1008–1009, 1993. [11] Warwick, W.J. & Leland, G.H., Chest Compression Appratus, United States Patent, 4838263, June 13, 1989. [12] Van Brunt, N.P. & Gagne, D.J., Disposable Pneumatic Chest Compression Vest, United States Patent, 6155996, Dec 5, 2000. [13] Van Brunt, N.P., Airway Treatment Appratus with Airflow Enhancement, United States Patent, 6340025 B1, Jan 22, 2002. [14] Arens, R., Gozal, D., Omlin, K., Vega, J., Boyd, K., Keens, T. & Woo, M., Comparison of high-frequency chest compression and conventional chest physiotherapy in hospitalized patients with cystic fibrosis, American Journal of Respiratory and Critical Care Medicine, 150, pp. 1154–1157, 1994. [15] Scherer, T.A., Barandun, J., Martinez, E., Wanner, A. & Rubin, E.M., Effect of high frequency oral airway and chest wall oscillation and conventional chest physiotherapy on expectoration in patients with stable cystic fibrosis, Chest, 113(4), pp. 1019–1027, 1998. [16] Kluft, J., Beker, L., Castagnino, M., Gaiser, J., Chaney, H. & Fink, R., A comparison of bronchial drainage treatments in cystic fibrosis, Pediatric Pulmonology, 22, pp. 271–274, 1996. [17] Braggion, C., Cappelletti, L., Cornacchia, M., Zanolla, L. & Mastella, G., Short-term effects of three chest physiotherapy regimens in patients hospitalized for pulmonary exacerbations of cystic fibrosis: a cross-over randomized study, Pediatric Pulmonology, 19, pp. 16–22, 1995. [18] McConnell, K.G., The Dual Channel Analyzer (Section 5.8). Vibration Testing – Theory and Practice, John Wiley & Sons: New York, pp. 333– 344, 1995.
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CHAPTER 9 Indicator for lung status in a mechanically ventilated COPD patient using lung-ventilation modeling and assessment D.N. Ghista1 , R. Pasam2 , S.B. Vasudev3 , P. Bandi4 & R.V. Kumar5 1 School
of MPE, Nanyang Technological University, Singapore. of Bioengineering, University of Illinois at Chicago, USA. 3 Department of Biomedical Engineering, University of South Florida, Tampa, USA. 4 Department of Population Health Sciences, University of Wisconsin, Madison, USA. 5 Department of Pulmonary Medicine, Mediciti Hospitals, Hyderabad, India. 2 Department
Abstract We have developed a lung-ventilatory index (LVI ), based on a lung model represented by a first-order differential equation in lung-volume dynamics to assess lung function and efficiency in the case of chronic-obstructive-pulmonary-disease (COPD) patients requiring mechanical ventilation because of acute respiratory failure. Herein, we have attempted to evaluate the efficacy of the LVI in identifying improving or deteriorating lung condition in such mechanically ventilated chronicobstructive-pulmonary-disease (COPD) patients, and whether it could provide valuable information regarding their lung status and consequently if LVI can be used as a potential indicator to predict ventilator discontinuation. For this purpose, we undertook a bioengineering study of 13 COPD patients who were mechanically ventilated because of acute respiratory failure. When their LVI was evaluated, it provided a clear separation between patients with improving and deteriorating lung condition. Finally, we formulated a nondimensional lung-improvement index (LII ) representative of the overall lung response to treatment and medication and κ that corresponds to the rate of lung improvement and reflects the stability of lung status with time.
170 Human Respiration
1 Introduction In mechanically ventilated patients with chronic-obstructive-pulmonary-disease (COPD), elevated airway resistance and decreased lung compliance (i.e. stiffer lung) are observed with rapid breathing. Many indices have been developed based on indicators of lung-status improvement to predict the successful discontinuation of mechanical ventilation. The existing indices include breathing pattern, arterial blood gas, frequency of breathing, tidal volume, and rapid shallow-breathing index, all of which are related to respiratory-system output. The need for accurate predictive indicators of lung-status improvement is essential for ventilator discontinuation through stepwise reduction in mechanical support, as and when patients are increasingly able to support their own breathing, followed by trials of unassisted breathing preceding extubation, and ending with extubation. For determining if a patient is ready to be discontinued from mechanical ventilation after the clinician has chosen an appropriate indicator to assess lung status s/he will incorporate this indicator into three general approaches for ventilator discontinuation that are (i) synchronized intermittent mandatory ventilation (SIMV) wherein the number of breaths is supplied by the ventilator and lowering the ventilator breaths will initiate more spontaneous breaths in the patient, (ii) pressuresupport ventilation (PSV) that provides inspiratory pressure assistance based on spontaneous efforts, and (iii) spontaneous breathing trial (SBT). The intent of the ventilatory discontinuation process is to decrease the level of support provided by the ventilator, requiring the patient to assume a greater proportion of the ventilatory workload. For stepwise reduction in mechanical ventilatory support, the most useful clinically employed indicators have been the rapid shallow-breathing index (RSBI) < 65 breaths/min/L (measured using ventilatory settings) and respiratory rate or frequency (RF) < 38 breaths/min. However, these are extrinsic empirical indices, currently there is no known easy-to-use, reliable indicator that incorporates the intrinsic parameters governing the respiratory-system mechanics for indicating lung-status improvement or deterioration, and eventual ventilator discontinuation. For this reason, we have developed an easy-to-employ lung-ventilatory index (LVI ), involving the intrinsic parameters of a lung-ventilatory model, represented by a firstorder differential equation in lung-volume response to ventilator driving pressure. The LVI is then employed for evaluating the lung status of chronic-obstructivepulmonary-disease (COPD) patients requiring mechanical ventilation because of acute respiratory failure. Herein, we have also tested the efficacy of LVI for assessing the lung status in mechanically ventilated COPD patients in acute respiratory failure. In general, the ventilator discontinuation process can be outlined in three stages. In the first stage, the clinician progressively reduces (in a stepwise fashion) the level of support. In the second stage, the patient undergoes a trial of unassisted breathing. In the final stage, the clinician extubates the patient. Any successful indicator for lung-status improvement will help validate the extubation process in the first stage itself. All COPD patients recruited in our study were discontinued from SIMV
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171
(synchronized intermittent mandatory ventilation) mode to a combination of SIMV and PSV (pressure support ventilation), and to a breathing trial.
2 Methodology We recruited 13 mechanically ventilated patients with chronic-obstructivepulmonary-disease (COPD) in acute respiratory failure. All patients met the diagnostic criterion of COPD. The first attempt of discontinuation for every patient was made within a short duration (not exceeding 88 hours). The patients in the study were between the ages of 54–83 years. All the patients were on SIMV mode with mandatory ventilation at initial intubation. Based on the physician’s judgment, the modes were changed for eventual discontinuation of mechanical ventilation. The time period for recording observations was one hour. The indices developed were not involved in any of the decision making during the stages of ventilator discontinuation. For all purposes in this study a successful ventilator discontinuation is defined as the toleration to extubation for 24 hours or longer and a failed ventilator discontinuation is defined as either a distress when ventilator support is withdrawn or the need for reintubation. From hereon, we will be using the term discontinuation for “ventilator discontinuation”.
3 Lung-ventilation model From a ventilatory-mechanics viewpoint, the lungs can be considered analogous to a balloon, which can be inflated and deflated (passively). The gradient between the mouth-pressure (Pm ) and the alveolar pressure (Pal ) causes respiration to occur. During inspiration, Pm > Pal that causes air to enter the lungs. During expiration Pal increases, and is greater than Pm ; this causes the air to be expelled out of the lungs passively. These pressure differentials provide a force driving the gas flow. The pressure difference between the alveolar pressure (Pal ) and pleural pressure (Pp ) counterbalances the elastic recoil. Thus the assessment of respiratory mechanics involves the measurement of flows, volumes (flow integrated over time) and pressure-gradients. The lung-ventilation model (shown in Fig. 1) is based on the following dynamic-equilibrium differential equation (eqn. (1)), expressing lungvolume response to pressure across the lung. ◦
RV +
V = PL (t) − Pe = PN (t), C
(1)
wherein: (i) the driving pressure, PL = Pm − Pp (ii) the parameters of the governing equation (1) are lung compliance (C) and airflow resistance (R), with both R and C being instantaneous values (iii) V = V (t) − Ve (wherein Ve is the end-expiratory lung volume) (iv) Pe is the end-expiratory pressure
172 Human Respiration
T
Diaphragm
Pel T
R
Pm
V.Pal
Air
Pp
Figure 1: Alveolar model. Let B be the amplitude of the net pressure wave form applied by the ventilator, Ca be the averaged dynamic lung compliance, Ra the averaged dynamic resistance to airflow, the driving pressure PL and the net pressure PN be given as PL = Pe + B sin(ωt), PN = B sin(ωt). The governing equation (1) then becomes: ◦
Ra V +
V = PN = B sin (ωt). Ca
(2)
The volume response to PN (the solution to eqn. (2)) is given by: V (t) =
BCa {sin (ωt) − ωka cos (ωt)} + H e−t/ka , 1 + ω2 ka2
(3)
wherein: (i) ka (=Ra Ca ) is the averaged time constant, (ii) the integration constant H is determined from the initial conditions, (iii) the model parameters are Ca and ka (i.e. Ca and Ra ), and (iv) ω is the frequency of the oscillating pressure profile applied by the ventilator. An essential condition is that the flow rate is zero at the beginning of inspiration and end of expiration. Hence, the flow rate dV /dt = 0 at t = 0. Applying this initial condition to our differential equation (3), the constant H is obtained as: H=
BCa ωka . 1 + ω2 ka2
(4)
Then, from eqns. (3) and (4), we obtain: V (t) =
BCa ωka −t/ka BCa {sin (ωt) − ωka cos (ωt)} + e . 2 2 1 + ω ka 1 + ω2 ka2
(5)
◦
Figure 2 illustrates typical data of V , V and PN . For evaluating the parameter ka , we will determine the time at which V (t) is maximum and equal to the tidal volume (TV), Hence, putting dV /dt = 0 in eqn. (5), we obtain: cos (ωt) + ωka sin (ωt) = e−t/ka .
(6)
Indicator for Lung Status TV, Tidal Volume (liters)
173
TV=0.5
(FRC) 0 0.3 V, Air Flow (liters/sec)
0 –0.3 Pk
∆P B
Ptm PN, Net Pressure (cm H2O)
∆t 0
5 secs
tm ◦
Figure 2: Lung-ventilatory model data shows air flow (V ) and volume (V ) and net pressure (PN ). Pause pressure (Ptm ) occurs at tm , at which the volume is maximum (TV = tidal volume). t is the phase difference between the time of maximum volume and peak pressure (Pk ), it also the time lag between the peak and pause pressures. B is the amplitude of the net pressure waveform PN applied by the ventilator. This PN oscillates about Pe with an amplitude of B. For t = tm , V (t) is maximum and equal to TV. Now in a normal person ka is of the order of 0.1, and 0.5 in ventilated patients with respiratory disorders, which is relevant to our study of COPD patients. Hence the term et/ka is of the order of 10t to 2t. At t = tm , at which the lung volume is maximum, we note from Fig. 2 that tm is of the order of 2 s. Hence tm /ka is of the order 20–4, so that e−t/ka is of the order of e−20 to e−4 , which is very small and hence negligible. Hence from eqn. (6), we obtain the following expression for ka : tan (ωt) = −1/ωka ,
(7a)
ka = −1/ω tan (ωtm ),
(7b)
or, wherein: (i)
If tan (ωtm ) = − 1/ωka , then
(8a)
tan (π − ωtm ) = 1/ωka , i.e., π − ωtm = tan−1 (1/ωka )
(8b)
Hence, tm =
π − tan −1 (1/ωka ) , ω
(8c)
174 Human Respiration or,
π−θ (8d) , wherein θ = tan−1 (1/ωka ). ω Hence, the phase difference between t = π/2ω at which pressure PN (= B sin (ωt)) is maximum and tm is, tm =
tm − t = or, t =
π − tan−1 (1/ωka ) π − , ω 2ω
tan−1 (1/ωka ) (π/2) − tan−1 (1/ωka ) π − = . 2ω ω ω
From eqn. (8c),
tan−1 (1/ωka ) = θ = π − ωtm .
(9a)
(9b) (10a)
Hence, sin θ = cos θ =
1
(10b)
1 + ω2 ka2 ωka
1 + ω2 ka2
.
(10c)
Fig. 2 shows the clinical lung-volume response dynamics in response to the now net ventilatory driving pressure (PN ). Referring to Fig. 2, tm denotes the time at which the lung volume is maximum. Since ω (the frequency of the oscillatory pressure profile applied by the ventilator) and tm (the time at which V (t) = TV) are known, we can evaluate the model parameter ka , from eqns. (10a, b, c). Hence, from eqns. (5) and (6) we obtain: V (t = tm ) = TV =
BCa {sin (ωt) − ωka cos (ωt)} BCa ωka −t/ka + e . 1 + ω2 ka2 1 + ω2 ka2
(11a)
Therein, based on eqn. (6), at t = tm the second term, H=
BCa ωka −tm /ka e ≈ 0. 1 + ω2 ka2
(11b)
Hence, eqn. (11a) becomes: V (t = tm ) = TV =
BCa {sin (ωt) − ωka cos (ωt)} . 1 + ω2 ka2
(12)
Now, in eqn. (12) if we put N = sin (ωt) − ωka cos (ωt)
(13a)
then, from eqns. (8d, 10b, c), eqn. (13a) becomes: N = sin (θ) − ωka cos (θ) =
ω2 ka2 + , 1 + ω2 ka2 1 + ω2 ka2 1
(13b)
Indicator for Lung Status
or, N=
2
1 + ω2 ka2 .
175
(13c)
Then, based on eqns. (13a) and (13c), eqn. (12) becomes: V (t = tm ) = TV =
BCa 1 + ω2 ka2
(14)
.
4 Determining lung compliance (Ca ) and air-flow resistance (Ra ) Now, as shown in Fig. 2, the peak pressure (Pk ) is the maximum pressure in the lungs during inspiration. The pause pressure is defined to occur when V (t) is maximum, i.e. (at t = tm ) at the end of inspiration. The peak pressure occurs when the driving pressure is maximum at t = π/2ω, while the pause pressure occurs when the lung volume is maximum, i.e. at the end of inspiration, when tm = (π − θ)/ω. It can be interpreted that there is always a phase lag of t between pause and peak pressures, which is described by eqn. (9). It is known that the driving pressure (PL ) is given as PL = Pe + B sin (ωt), which leads us to: Peak Pressure (Pk ) = PLat t=π/2ω = Pe + B Pause Pressure (Ptm ) = PLat t=(π−θ)/ω
3 4 π−θ = Pe + B sin ω , ω
or, Pause Pressure (Ptm ) = Pe + B sin θ.
(15)
(16) (17)
Based on eqn. (10b), eqn. (17) becomes: Pause Pressure (Ptm ) = Pe +
B 1 + ω2 ka2
.
(18)
Then, from eqns. (15) and (18): (i) Pk − Ptm = P = B − B sin θ = B(1 − sin θ),
(19a)
or, B=
P Pk − Ptm = , (1 − sin θ) (1 − sin θ)
(19b)
Ptm − Pk sin θ , (1 − sin θ)
(20)
(ii) Pe = Pk − B =
wherein, sin θ is given by eqn. (10b).
176 Human Respiration In eqn. (19b), since Pk , and Ptm as well as θ(= π − ωtm ) are measurable, hence B can be evaluated. Then from eqn. (14), we can evaluate the averaged lung compliance (Ca ), because ka has already been evaluated by eqn. (7b). Hence, based on eqns. (14), (10b) and (19b), we obtain: TV 1 + ω2 ka2 TV TV(1 − sin θ) Ca = = = . B B sin θ P sin θ
(21)
Hence, from eqns. (21) and (10a) the average value of airflow resistance (Ra ) can be evaluated as: Ra = ka /Ca =
P sin θ(1/ω tan θ) P cos θ = . TV(1 − sin θ) TVω(1 − sin θ)
(22)
For our patients, the computed values of the parameters are: Ra = 9 − 43 cmH2 O s/L
(23)
Ca = 0.020 − 0.080 L/cmH2 O. Now, that we have determined the expressions for Ra and Ca , the next step is to develop an integrated index incorporating these parameters.
5 Formulating a lung-ventilatory index (LVI) incorporating Ra and Ca We believe that the correlations between averaged airflow resistance (Ra ), averaged lung compliance (Ca ), tidal volume (TV), respiratory rate (RF), and maximum inspiratory pressure or peak pressure (Pk ) could be used as a possible indicator for determining lung status in a mechanically ventilated COPD patient with acute respiratory failure. We hence propose that a composite index (LVI ) incorporating these isolated parameters can have a higher predictive power. For this purpose, we note that patients with COPD have higher Ra , lower Ca , lower TV, higher Pk and higher respiratory rate (or frequency) RF. If we want the lung-ventilatory index (LVI ) to have a high value for a COPD patient and further increasing LVI for deteriorating lung status and decreasing LVI for improving lung status in a mechanically ventilated COPD patient in acute respiratory failure, then the lungventilatory index (LVI ) can be expressed as: LVI =
Ra (RF)Pk . Ca (TV)
(24)
Let us obtain the order-of-magnitude values of LVI for a mechanically ventilated COPD patient in acute respiratory failure (by using representative computed values of the parameters Ra , Ca , RF, TV, and Pk ) in order to verify that the formula for
Indicator for Lung Status
177
LVI (eqn. (22)) can enable distinct separation in mechanically ventilated COPD patients in acute respiratory failure. LVI (Intubated COPD) =
[15 cmH2 O s/L][0.33 s−1 ][20 cmH2 O] . [0.035 L/cmH2 O][0.5 L]
= 5654 (cmH2 O/L)3 ,
(25)
wherein, Ra = 15 cmH2 O s/L TV = 0.5 L
Ca = 0.035 L/cmH2 O
RF = 0.33 s−1
Pk = 20 cmH2 O.
Now, let us obtain the order of magnitude of LVI (by using representative computed values of Ra , Ca , RF, TV , and Pk ) as above for a COPD patient with improving lung status just before successful discontinuation. LVI (Outpatient COPD) =
[10 cmH2 O s/L][0.33 s−1 ][12 cmH2 O] [0.05 L/cmH2 O][0.35 L]
= 2263 (cmH2 O/L)3 ,
(26)
wherein, Ra = 10 cmH2 O s/L TV = 0.35 L
Ca = 0.050 L/cmH2 O Pk = 12 cmH2 O.
RF = 0.33 s−1
Hence, for LVI to reflect lung-status improvement in a mechanically ventilated COPD patient in acute respiratory failure, it has to decrease to the range of LVI for an outpatient COPD patient at the time of discontinuation. Additionally, we now write: (27) LVI (t) = LVI0 e−κt , wherein: (i)
LVI0 represents the value of LVI at the time of admission of the patient to the respiratory care unit. (ii) the coefficient κ, represents the rate of improvement (or deterioration) of the patient’s lung status; κ = 0 implies no change in lung condition. (iii) the coefficient κ (the rate of decrease of LVI or improvement in lung status) will be positive with deteriorating lung condition and negative for improving lung condition. We can also formulate an index (LII ) for overall lung-status improvement or deterioration as: LII (%) =
LVI (at entry or intubation) − LVI (at discharge or extubation) × 100. LVI (at entry) (28)
178 Human Respiration
6 Evaluating lung-ventilatory index (LVI ) 6.1 LVI characteristics Now, that we have formulated LVI, let us verify for selected patient data that LVI is indicative of lung-status improvement or deterioration. For this purpose, we have categorized the intubated patients into two categories, (i) patients who were successfully discontinued and (ii) patients who failed discontinuation. Table 1 provides the range of LVI values for the two categories, wherein for all successful discontinuations the LVI is close to the value for an outpatient COPD patient. In Table 1 all patients who were successfully discontinued have LVI at discontinuation in the range of 1194–4589 (cmH2 O/L)3 . Similarly, patients with failed discontinuation have LVI at discontinuation in the range of 7144–15658 (cmH2 O/L)3 , thus LVI is indicating a clear separation between successful and failed discontinuation. It is also observed that the rate of lung-status improvement was faster after successful discontinuation, due to reduced dead space by mechanical ventilation of CABG patients. In Fig. 3, we have provided the distribution of LVI at outcome for successful and failed discontinuation of mechanically ventilated COPD patients in acute respiratory failure (approximated as a normal distribution). In Fig. 4, we have provided LVI characteristics for four patients, indicating their lung-status. 6.2 Comparing the efficacies of Ra and Ca with LVI Now, let us evaluate the significance of Ra and Ca with lung status. In Table 2, we have provided information on the values of Ra and Ca at discontinuation for the two classes of patients. In Figs. 5 and 8, we have provided Ra and Ca characteristics for the same four patients discussed in Fig. 4. We observe that patients with successful discontinuation had (i) Ra at discontinuation in the range of 9– 14 cmH2 O s/L compared to 17–23 cmH2 O s/L in failed-discontinuation cases, and (ii) Ca for successful discontinuation in the range of 0.03–0.08 L/cmH2 O compared to 0.028–0.042 L/cmH2 O in failed-discontinuation cases. Figure 5 shows the time variation in Ra of successfully extubated patients (SEPs) and the unsuccessfully extubated patients (UEPs). It is noted that Ra decreases
Table 1: Range of LVI values at intubation and outcome. Outcome
Successful Discontinuation Failed Discontinuation
Number
Age (Years)
Sex (M/F)
Time of intubation (Hrs)
LVI at intubation (cmH2 O/L)3
LVI at outcome (cmH2 O/L)3
6
54–74
6/0
11–55
3959–13568
1194–4589
7
64–83
5/2
29–88
3350–21152
7144–15658
Indicator for Lung Status
179
14000 12000
LVI
10000 8000
LVI Successful Discontinuation LVI Failed Discontinuation
6000 4000 2000 0 (6) Successful Discontinuation Cases
(7) Failed Discontinuation Cases
Figure 3: Distribution of LVI at discontinuation for patients with failed and successful discontinuation. For the 6 successfully discontinued cases, the LVI was (2900) ± (567) (cmH2 O/L)3 ; for the 7 failed-discontinuation cases the LVI was (11400) ± (1433) (cmH2 O/L)3 . It is observed that LVI indicates clear separation between failed and successful discontinuation. 35000 30000 25000 LVI (Patient 1) LVI (Patient 2) LVI (Patient 3) LVI (Patient 4)
LVI
20000 15000 10000 5000 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 Hours
Figure 4: LVI showing lung status in four mechanically ventilated COPD patients in acute respiratory failure. Note that Patients 1, 2 and 3 were successfully discontinued and patient 4 failed discontinuation.
steadily in SEPs but continues to be high in the UEPs. The distribution of Ra (in Fig. 6), graphically illustrates the distinct separation of unsuccessfully and successfully extubated patients. Yet in the case of patient 2 in (Fig. 5) the resistance at intubation was 12.38 cmH2 O s/L and decreased only slightly at discontinuation to 10.82 cmH2 O s/L within the period of 15 hours of intubation to a value corresponding to that of an outpatient COPD patient. However, for this patient—2, the LVI changes from 7300 to 4500 (cmH2 O/L)3 , i.e. it is significantly closer to the
180 Human Respiration Table 2: Range of Ra and Ca at intubation and outcome. Outcome
Ra at intubation cmH2 Os/L
Ra at outcome cmH2 Os/L
Ca at intubation L/cmH2 O
Ca at outcome L/cmH2 O
ka at intubation s
ka at outcome s
Successful Discontinuation
14–32
9–14
0.03–0.047
0.030–0.080
0.42–1.50
0.27–1.12
Failed Discontinuation
14–25
17–23
0.03–0.037
0.028–0.042
0.42–0.925
0.47–0.99
Average Airflow Resistance
45 40 35 30
Patient 1
25
Patient 2
20
Patient 3
15
Patient 4
10 5 0 1
4
7 10 13 16 19 22 25 28 31 34 37 40 43 46 Hours
Figure 5: The variations in Ra for four mechanically ventilated patients indicating that Patient 1, 2, and 3 are all discontinued at lower values 8–10 cmH2 O s/L, i.e. closer to outpatient COPD values. It is also observed that the average airflow resistance is higher for deteriorating lung condition.
outpatient COPD patient LVI at successful discontinuation, thus indicating that Ra is less sensitive to the change in lung status compared to LVI . Now, let us observe the dynamics of Ca , which is also deemed to provide separation between normal and COPD patients. Fig. 8, illustrating the Ca dynamics of the same 4 patients, does not show a definitive trend of decreasing Ca , while the Ca for patient—1 even decreased. Also, from Table 2 and Fig. 7, the value for Ca at discontinuation for patients with successful extubation is 0.03–0.08 L/cmH2 O and for failed extubation is 0.028–0.042 L/cmH2 O, indicating that Ca does not provide a clear separation between patients with improving and deteriorating lung status. Hence, we believe Ca by itself cannot be a reliable indicator for assessing lung status in mechanically ventilated COPD patients in acute respiratory failure.
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181
Average Airflow Resistance
25 20 Successful Discontinuation Failed Discontinuation
15
10 5 0 (6) Successful Discontinuation Cases
(7) Failed Discontinuation Cases
Figure 6: Distribution of Ra at discontinuation for successful and failed discontinuation. Ra = (11.5) ± (0.83) (cmH2 O S/L) for the 6 successfuldiscontinuation cases, and (20) ± (1) (cmH2 O S/L) for the 7 failed-discontinuation cases. Hence, Ra indicates a clear separation between failed and successful discontinuations.
Average Lung Compliance
0.07 0.06 0.05 0.04
Successful Discontinuation Failed Discontinuation
0.03 0.02 0.01 0 (6) Successful Discontinuation Cases
(7) Failed Discontinuation Cases
Figure 7: Distribution of Ca at discontinuation for successful and failed discontinuation. Ca = (0.055) ± (0.0083) (L/cmH2 O) for the 6 successfuldiscontinuation cases, and (0.034) ± (0.0027) (L/cmH2 O) for the 7 failed-discontinuation cases. Hence, Ca does not provide a clear separation between failed and successful discontinuations.
6.3 LVI as a reliable predictor of ventilator discontinuation We have noted that LVI can, in fact, be a reliable indicator of lung status in a mechanically ventilated COPD patient with acute respiratory failure. But, can it
182 Human Respiration 0.09
Average Compliance
0.08 0.07 0.06
Patient 1
0.05
Patient 2
0.04
Patient 3
0.03
Patient 4
0.02 0.01 0 1
4
7 10 13 16 19 22 25 28 31 34 37 40 43 46 Hours
Figure 8: The variations in Ca for four mechanically ventilated patients, indicating that the Ca values for Patients 1, 2 and 4 were all in the lower values and did not change significantly from the time of intubation. Though the lung status for Patients 1 and 2 improved and were successfully discontinued and Patient 4 failed discontinuation. be used as a predictor for ventilator discontinuation? Now, an ideal predictor of ventilator discontinuation would safely and easily distinguish patients needing discontinuation and continued ventilatory support, which we have successfully indicated with our patient data. Also, it has been speculated that as much as 42% of the time spent by patients on mechanical ventilation are instances when they could have been extubated, and in many cases unnecessary delays in the discontinuation process is associated with further complications. In this context, we observe that LVI has adequately addressed the issue by identifying instances when the patient could potentially be discontinued. For example, in Fig. 4 there are four instances when the LVI value for patient 4 is lower than 5000 (cmH2 O/L)3 , and closer to the value for an ideal outpatient COPD patient. This leads us to believe that LVI can be a reliable factor in the clinician’s judgment to identify patients needing discontinuation. Thus far, what we have developed leads us to believe that LVI is reasonably representative of a reliable index for discontinuation. However, we will be carrying out a more evidence-based clinical approach for validation.
7 Assessing lung-improvement index (LII) and rate of lung improvement (κ) The rationale behind designating LII eqn. (24) as an index is not as an indicator for lung status per se, because there are instances when the patient’s lung had improved from the time of initial intubation but could not be sustained upon discontinuation. However, LII could be representative of the overall lung response to treatment
Indicator for Lung Status
183
Table 3: Range of LII and κ at intubation and outcome.
Number
Age (Years)
Sex (M/F)
Time of intubation (Hrs)
LII (%) at outcome
κ at outcome
Successful Discontinuation
6
54–74
6/0
11–55
26–86 %
0–0.18
Failed Discontinuation
7
64–83
5/2
29–88
−101–49 %
−0.045–0.015
Outcome
0.14 Rate of Lung Improvement
0.12
Successful Discontinuation Failed Discontinuation
0.1 0.08 0.06 0.04 0.02 0 –0.02 –0.04
(6) Successful Discontinuation Cases (7) Failed Discontinuation Cases
Figure 9: The distribution of κ for patients with successful and failed discontinuation. κ = (0.09) ± (0.03) for the 6 successful-discontinuation cases, and (−0.015) ± (0.01) for the 7 failed-discontinuation cases. Hence, κ indicates appreciable separation between successful and failed discontinuations.
and medication. The coefficient κ corresponds to the rate of improvement or deterioration in lung status. It is to be noted that for a patient with improving lung condition, κ will be positive and vice versa. We have observed that in most patients, κ decreased immediately after the first few hours of mechanical ventilation, and later on varied somewhat and stabilized before successful discontinuation. Hence, we propose that κ reflects the patient stability in lung status with time, and can provide a clear separation between patients with improving and deteriorating lung status, as indicated by Fig. 9. Table 3, provides information on LII and κ for patients with successful and failed discontinuation, while Figs. 9 and 10 illustrate their distributions. From Table 3 and Fig. 10, we note that some patients with positive response to treatment (i.e. +LII ) did not necessarily succeed discontinuation. Hence, it is
184 Human Respiration
Lung Improvement Index (LII)
80
Successful Discontinuation Failed Discontinuation
60 40 20 0 –20 –40 –60
(6) Successful Discontinuation Cases
(7) Failed Discontinuation Cases
Figure 10: The distribution of the lung-improvement index (LII ) for patients with successful and failed discontinuations. LII = (56%) ± (10%) for the 6 successful-discontinuation cases, and (−26%) ± (25%) for the 7 faileddiscontinuation cases.
indicated that LII and κ can show a distinct lung-improvement trend, but may not be regarded as absolute indicators for patient-candidates for successful extubation. The way in which we could employ LVI, LII and κ, in combination is as follows. Starting with evaluation of LVI at the time of intubation, we can employ LII and κ to designate signs of lung improvement. Then, when LVI value persists being less than 3000 (cmH2 O/L)3 for 2–3 hours, we could decide to extubate the patient.
8 Conclusion We have shown LVI, LII, and κ to be reliable indicators for mechanically ventilated COPD patients in acute respiratory failure. We have also indicated how they can be collectively employed to decide on extubating a patient. Now, we need to further testify their efficacies, in assessing COPD therapy in an intensive-care unit, in a larger patient population.
References [1]
Ghista, D.N., Loh, K.M. & Damodharan, M., Lung Ventilation Modeling and Assessment, in “Human Respiration: Anatomy and Physiology, Mathematical Modeling, Numerical Simulation and Application”, ed. by V. Kulish, WIT Press (Southampton, UK), 2005. [2] MacIntyre, N.R., Evidence-based Guidelines for Weaning and Discontinuing Ventilatory support by A Collective task Force of the American College of Chest Physicians: Respiratory Care, 47(1), 69–90, 2002.
Indicator for Lung Status
[3]
[4]
[5]
[6]
[7]
[8]
185
Menzies, R., Gibbons, W. & Goldberg, P., Determinants of weaning and survival among patients with COPD who require mechanical ventilation for acute respiratory failure, Chest., 95(2), 398–405, 1989. Yang, K.L. & Tobin, M.J., A prospective study of indexes predicting the outcome of trials of weaning from mechanical ventilation, N Engl J Med., 324(21), 1445–1450, 1991. Alvisi, R., Volta, C.A., Righini, E.R., Capuzzo, M., Ragazzi, R., Verri, M., Candini, G., Gritti, G. & Milic-Emili, J., Predictors of weaning outcome in chronic obstructive pulmonary disease patients, Eur Respir J., 15(4), 656– 662, 2000. Pauwels, R.A., Buist,A.S., Calverley, P.M., Jenkins, C.R. & Hurd, S.S., Global strategy for the diagnosis, management, and prevention of chronic obstructive pulmonary disease. NHLBI/WHO Global Initiative for Chronic Obstructive Lung Disease (GOLD) Workshop summary, Am J Respir Crit Care Med., 163(5), 1256–1276, 2001. Meade, M., Guyatt, G., Cook, D., Griffith, L., Sinuff, T., Kergl, C., Mancebo, J., Esteban, A. & Epstein, S., Predicting success in weaning from mechanical ventilation. Chest., 120(6 Suppl), 400S–424S, 2001. Jubran, A., de Graaff, W.B.V. & Tobin, M.J., Variability of patient-ventilator interaction with pressure support ventilation in patients with chronic obstructive pulmonary disease, Am J Respir Crit Care Med., 152(1), 129–136, 1995.
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CHAPTER 10 Mechanics of proportional-assist ventilation A.C. Lua & K.C. Shi School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50, Nanyang Avenue, Singapore 639798.
Abstract Proportional-assist ventilation (PAV) is an effective and improved mode of ventilation over the conventional pressure-support ventilation due to its attractive features of better patient harmony and comfort. The conventional ventilation systems in hospitals will either deliver a specified tidal volume to the patient or raise the airway pressure to a preset level during the inspiratory phase of each breath. These systems do not take into consideration the patient’s breathing patterns and varying requirements during each inspiration. To assist patients with breathing difficulties, PAV has been used as a mode of synchronized partial ventilatory support, by applying airway pressure in proportion to volume and flow rate to overcome the patient’s higher respiratory elastance and resistance, respectively, as a result of pulmonary disease. A new PAV system using a proportional solenoid valve to control the air supply to patients suffering from respiratory disabilities, was studied. The outlet flow and pressure from the valve at various air-supply pressures were tested and proven to be suitable for pressure and flow control in a PAV system. In vitro tests using a breathing simulator, which has been proven to possess the general characteristics of the human respiratory system in spontaneous breathing tests, were conducted and the results demonstrated the viability of this PAV system in normalizing the breathing patterns of patients with abnormally high resistances and elastances as well as neuromuscular weaknesses. In vivo tests using this PAV system were also conducted on healthy volunteer subjects with artificial resistance and elastance incorporated. The test subjects felt comfortable to breathe, with reduced breathing effort when PAV was active, demonstrating the positive assist effect of PAV.
188 Human Respiration
1 Introduction Respiration consists of two processes – external respiration that involves the intake of oxygen and removal of carbon dioxide from the body as a whole, and internal respiration that is the utilization of oxygen and the production of carbon dioxide by cells and the gaseous exchanges between the cells and their fluid medium. The respiratory system is made up of a gas-exchanging organ (the lung) and a pump that ventilates the lungs. The pump consists of the chest wall and the respiratory muscles that increase and decrease the size of the thoracic cavity. At rest, a normal human breathes 12–15 times a minute. Five hundred milliliters of air per breath or 6–8 l/min are inspired and expired. Inspiration is an active process. The contraction of the inspiratory muscles increases the intrathoracic volume. During quiet breathing, the intrapleural pressure at the base of the lungs is about –2.5 mmHg gauge pressure at the start of inspiration and it finally decreases to about –6 mmHg. The lungs are pulled into a more expanded position. The pressure in the airway and therefore the intrapulmonary pressure become slightly negative and air flows into the lungs (Fig. 1). At the end of inspiration, the lung recoil begins to pull the chest back to the expiratory position where the recoil pressure of the lungs and chest wall balance. The pressure in the airway becomes slightly positive and air flows out of the lungs. Expiration during
Figure 1: Changes in intrapleural and intrapulmonary pressures relative to atmospheric pressure during respiration [1].
Mechanics of Proportional-Assist Ventilation
189
quiet breathing is passive in the sense that the muscles that decrease intrathoracic volume do not contract. Mechanical ventilators are used to artificially ventilate the lungs of patients who are unable to breathe normally. Mechanical ventilation is generally classified into positive-pressure, negative-pressure and high-frequency ventilation. Positive pressure means that the air that is at above atmospheric pressure, is applied to the respiratory tract with the result of pushing the air into the patient’s lungs in order to inflate them. Negative-pressure ventilation is performed with the use of a body ventilator that surrounds a patient with adjustable negative-pressure and creates a vacuum around the chest wall. The negative pressure moves the thoracic walls outward expanding the intrathoracic volume and decreasing the pressure inside the lungs. The pressure difference between the atmosphere and the lungs allows the air to flow into the lungs. High-frequency ventilation delivers the air at a much higher frequency of 50–150 cycles per minute and at a low air volume, providing adequate gas exchange while minimizing the degree that the lungs are actually inflated. Positive-pressure ventilation is the most widely used and this principle is used in the assist-mode ventilation to be described below. In the assist-mode ventilation, the delivery of ventilatory support can be coordinated by the patient’s effort. This mode allows the patient to participate in the breathing process and the initialization of the machine-cycled breath is triggered by the patient’s own effort. This is in contrast to the other mode, i.e., the controlledmode ventilation that is totally predetermined by the ventilator settings and therefore the patient’s spontaneous breathing is totally depressed. Currently, research in ventilation is mainly directed towards the assist mode because it can (i) synchronize patients’ effort and ventilatory response, (ii) reduce the need for sedation, (iii) prevent disused atrophy of the respiratory muscles, (iv) improve haemodynamic tolerance of ventilatory support, and (v) facilitate the weaning process. The assist mode can be subdivided into five modes, namely, assist-control ventilation, synchronized intermittent mandatory ventilation (SIMV), pressure-support ventilation (PSV), continuous positive airway pressure (CPAP) and positive end expiratory pressure (PEEP). Assist-control ventilation provides preset volume or pressure support every time the patient initiates a spontaneous breath. In addition, the ventilator is preset to provide a minimum number of breaths if the patient fails to trigger inspiration. SIMV coincides the mandatory machine breaths to occur with the patient’s inspiratory efforts, but not every breath is supported. SIMV is usually used as a weaning mode because it requires the patient to take on some of the work of breathing. The intermittent frequency can be gradually reduced according to the patient’s recovery condition. PSV is a mode of ventilation that supplements the patient’s spontaneous inspiration with a preset amount of positive airway pressure. During PSV, the patient determines the rate and duration of the breath. It is useful in compensating for high airway resistance and it is generally regarded as comfortable for spontaneous inspiration for patients. CPAP provides a constant flow of air (positive airway pressure) during both inspiratory and expiratory phases. The ventilation supply is mostly controlled by a solenoid valve or a proportional solenoid valve that can be triggered by either pressure or flow signal. PEEP functions to maintain
190 Human Respiration positive pressure at the end of expiration, serving to keep the lungs and the alveoli inflated at the end of expiration and reducing the potential for alveolar collapse, thereby improving oxygenation levels. Therefore, PEEP is useful for patients who have difficulties in maintaining adequate oxygenation and this will eliminate the need for high levels of applied oxygen. However, none of the above modes normalizes the continuous relationship between the patient’s effort and ventilatory consequences. The patient’s spontaneous effort triggers the ventilator and it then runs its own course according to its predetermined settings. All these systems do not take into consideration the patient’s breathing patterns and varying requirements during each inspiration. Therefore, PAV is proposed as a mode of ventilatory support that will normalize the patient–ventilator interaction throughout the respiration process. PAV provides synchronized partial ventilatory support by applying airway pressure in proportion to volume and flow rate to overcome respiratory-system elastance and resistance, respectively. The PAV system makes use of the patient’s spontaneous breathing patterns and amplifies proportionally the instantaneous effort throughout each inspiration cycle. It uses positive-feedback control theory to determine the anticipated requirements from a knowledge of its existing values and make instantaneous adjustments accordingly to achieve the necessary flow requirements for the patient. As the patient pulls his/her chest muscles more to inspire, the machine will generate more airflow to supplement this. The ventilator simply amplifies the patient’s effort without imposing any ventilatory or pressure targets. The role of PAV is to allow the patient to comfortably attain whatever ventilation and breathing pattern his or her control system sees fit. The responsibility for determining the level and pattern of breathing is shifted entirely from the caregiver to the patient. In this respect, PAV differs fundamentally from all other methods of ventilatory support where ventilatory variables are, by and large, determined by machine settings.
2 Development of proportional-assist ventilation The early PAV system was described by Younes et al [2] for an initial clinical trial on some ventilator-dependent patients. Essentially, this prototype consisted of a rolling-seal piston coupled to a motor to generate a force in proportion to the airway pressure. This PAV system is schematically shown in Fig. 2. The current supplied to the motor is proportional to the instantaneous flow rate V˙ and its volume of inspiration V . The control gain G determines the proportionality. At the early stage of inspiration, as the patient pulls, air moves from the cylinder to the patient. The flowmeter measures the flow rate and the volume can be calculated by integrating the flow rate with respect to time. As flow rate and volume signals are generated, the motor applies pressure according to the flow rate and volume as well as the control gain G. If the patient pulls harder, more flow rate and volume are generated; the assisted ventilation will increase accordingly. The relationship between the instantaneous effort and ventilatory consequences in healthy persons and in patients with diseases is shown in Fig. 3. The effort is defined as a fraction of maximal inspiratory muscle activation. The application
Mechanics of Proportional-Assist Ventilation
191
FLOWMETER Pairway
Pmus
PATIENT
V
G
dV/dt G
Normal Y-axis
FLOW AND/OR VOLUME
Figure 2: A simple proportional-assist ventilation system.
Disease PAV
X-axis
INSTANTANEOUS EFFORT
Figure 3: The relation between instantaneous effort and ventilatory consequences in healthy persons and in patients with diseases. of PAV compensates for the deficiencies in the diseased case to attain the actual instantaneous requirements of a normal person for a given effort. Another PAV system that used a linear actuator to push a bellow to provide ventilation to the patients according to the calculated airway pressure was reported by Chua et al [3] and Li et al [4]. This system is shown in Fig. 4 in which a mechanical-driven unit is used to generate a pressurized-air supply. A summary of the research progress on PAV is as follows: (i) Clinical trials of PAV on patients with respiratory diseases and evaluation of the short- and long-term effects on the respiratory system, breathing patterns as well as the whole physiological system [2, 5–7]. (ii) Comparison of PAV with other conventional ventilation methods [8]. (iii) Evaluation of the effect of varying degree of assistance on patients and the appropriate determination of proportionalities [3, 4, 9]. (iv) Evaluation of the effect of PAV in some special circumstances such as exercise tolerance and infant ventilatory support [10, 11].
192 Human Respiration
Controller
PENTIUM
P1
Linear motor
F : Flow transducer P : Pressure transducer
P2
F Exhalation valve
Bellow
Oneway valve Oxygen supplier Proportional soleniod valve
Figure 4: Proportional-assist ventilation system [3].
3 Theory of proportional-assist ventilation During tidal breathing, the elastic and resistive properties of the total respiratory system determine the pressure required to inflate the lungs. This pressure (P) can be divided into its elastic component (pressure required to change volume, V ) and resistive component (pressure required to generate flow, V˙ ) by the simplified equation of motion of the lungs [12]: P = E × V + R × V˙ ,
(1)
where E is the elastance (the reciprocal of the commonly used compliance, C) and R is the resistance. The relevant elastic properties can be divided into two components, those of the lungs and those of the chest wall, which are arranged in series. Similarly, the resistive pressure can be defined as the total pressure drop in phase with the flow through the separate segments of the tracheobronchial tree and across the lung tissue and chest wall, which are arranged in series. The elastance and resistance of the respiratory system are defined as: E=
P V
(2)
R=
P , V˙
(3)
where V , P and V˙ are the changes in lung volume, airway pressure and flow rate, respectively. The typical normal adult values for elastance and resistance are 10 cmH2 O/l and 2.5 cmH2 O/l/s, respectively. In the operation of a mechanical ventilator, parameter settings are based on a prior knowledge of the values of elastance and resistance of the respiratory system. Therefore, the assessment of elastance and resistance is important in lung-function tests.
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The static pressure–volume curve of the lungs, constructed by measuring the difference between the alveolar and pleural pressures (transpulmonary pressure) at several lung volumes, can be used for the estimation of the static compliance of the respiratory system. Normally, it is obtained during breath holding with a balloon-catheter system or a small pressure transducer in the lower esophagus to approximate pleural pressure. This pressure is compared to the pressure at the airway opening that during breath holding, reflects the alveolar pressure. Measurements of transpulmonary pressure and volume can also be recorded continuously during tidal breathing; the so-called ‘dynamic compliance’ can then be obtained. In normal subjects, dynamic compliance is only slightly less than the static compliance [13, 14]. Generally, spontaneous breathing is entirely accomplished by the respiratory muscles via the generating muscle pressure (Pmus ) while the ventilator provides ventilatory support by changing the airway pressure (Paw ). Therefore, for patients receiving synchronized ventilatory support, breathing is accomplished through the combined actions of respiratory muscles and ventilator. Thus, at any instant the total applied pressure (Papp ) is given by: Papp = Pmus + Paw .
(4)
This total applied pressure is dissipated primarily against two forces, the elastic recoil of the respiratory system (Pel ) and the resistance to gas flow offered by the airways (Pres ), i.e., Pmus + Paw = Pel + Pres . (5) During the inflation phase, the pressure difference between the airway pressure (Paw ) and the alveolar pressure (Palv ) is the pressure dissipated against the resistance of the airways (Pres ). This difference is related to the flow rate and the resistive properties of the airways (R). Thus, Pres = V˙ × R.
(6)
The elastic recoil of the respiratory system (Pel ) is related to the extent the lung volume is above the passive functional residual capacity (FRC) and the elastance (E) of the respiratory system. If a linear relationship between the pressure and volume in the tidal volume range is assumed, then Pel = V × E,
(7)
where V is the volume above passive FRC. Substituting eqns. (6) and (7) into eqn. (5) gives (8) Pmus + Paw = V × E + V˙ × R. In the PAV system, Paw is a function of V and V˙ that are generated by the patient’s effort. Thus (9) Paw = f1 (V ) + f2 (V˙ ).
194 Human Respiration In the case of linear functions: Paw = K1 × V + K2 × V˙ ,
(10)
where K1 is the proportionality between the airway pressure and inspired volume (in elastance units, cmH2 O/l) and K2 is the proportionality between the airway pressure and inspired flow rate (in resistance units, cmH2 O/l/s). K1 and K2 values are determined according to both abnormal neuromuscular and abnormal mechanical disabilities of the patient. Substituting eqn. (10) into eqn. (8) yields (11) Pmus + K1 × V + K2 × V˙ = V × E + V˙ × R. In this method, if the patient increases or decreases his effort, V and V˙ will be altered accordingly. Also, as Paw changes in relation to V and V˙ , machine assist will increase or decrease in concert with Pmus . The net effect is to amplify the patient’s effort while, at the same time, allow the patient to have total control over all aspects of breathing. In the PAV system, the appropriate ventilatory support, to a great extent, is dependent on the determination of K1 and K2 . Younes [15] proposed the following 4 ways to determine K1 and K2 . (i) Support in proportion to estimated Pmus . Pmus is amplified by the same factor regardless of how it is partitioned between elastic and resistive components and regardless of the time during the inspiration. This approach would be ideal in cases where the problem is primarily in the inability to generate pressure while the mechanical properties are reasonably normal. (ii) Adjustment according to the nature of mechanical abnormality. In this mode, the machine-generated pressure will compensate for the extra work resulting from disease, leaving the patient’s muscles to cope with the normal load. (iii) Combination of (i) and (ii) above. When the disorder involves both neuromuscular weakness and abnormal mechanics, K1 and K2 can each be the sum of two components, one specifically for the mechanical abnormality and the other for neuromuscular weakness. (iv) Adjustment of gain factors according to the patient’s comfort. Implementation of the above three approaches requires a knowledge of the patient’s mechanical properties (i.e., E and R). This is not always possible and the data may not be reliable because of incomplete relaxation. It would seem reasonable to utilize the patient’s own sensory mechanisms to adjust the type and magnitude of assist. The K1 and K2 settings may be adjusted by trial and error to the values that result in the greatest comfort for the patient.
4 Proportional solenoid valve PAV system The PAV system to be described here uses a proportional solenoid valve (PSV) to regulate airflow from a compressed air source. The schematic diagram and
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photograph of this system are shown in Figs. 5 and 6, respectively. The proportional solenoid valve (MYPE-5-M5-010B, FESTO) controls the flow continuously and proportionally to the analogue input. When the input analogue voltage increases from 5 to 10 VDC, the flow will increase from 0 to a maximum value.
PT : Pressure transducer PNT : Pneumotachograph PENTIUM
PT 1
PT 2
: One way valve
Free airway Regulator
Proportional solenoid valve
PNT Breathing simulator
Exhalation valve Compressed air source
Figure 5: Schematic diagram of the proportional solenoid valve PAV system.
Figure 6: Photograph of the proportional solenoid valve PAV system.
196 Human Respiration The outlet flow and pressure of the PSV are related to its inlet air pressure. Connecting the PSV to a compressed-air source via a pressure regulator and adjusting its analogue input by varying the output signal from a computer, the outlet flow from the PSV can be varied. The flows at different analogue input values are measured by a pneumotachograph (4700A, Hans Rudolph). The pressure drop across the pneumotachograph is measured using a pressure transducer (DP103-14, Validyne) and carrier demodulator (CD15, Validyne). The outlet pressure of the PSV is measured by another pressure transducer (DP15-30, Validyne). Calibrating the PSV at a fixed inlet supply air pressure, the outlet pressure and flow from the PSV for a range of analogue inputs from 5.75 to 9 VDC for various inlet supply pressures are shown in Figs. 7 and 8, respectively. In Fig. 7, the relationship between the outlet pressure and the analogue input can be approximated by a second-order polynomial except for an inlet supply pressure of 0.5 bar. The outlet flow is almost linearly proportional to the analogue input for any fixed inlet supply pressure as shown in Fig. 8. Hence, the use of this solenoid valve to control the desired flow during inspiration can be easily attained. The PSV has a very short response time of 20 ms to an analogue input and this is crucial for short breathing cycles, typically about 3 seconds. If the assisted airway pressure based on the inspired flow and integrated volume, as expressed in eqn. (10), is amplified and taken as the analogue input signal to the proportional solenoid valve, then a proportional-assisted air flow can be delivered to the patient. A commercial breathing simulator (Series 1101, Hans Rudolph Inc.) was used to mimic human spontaneous breathing for the in vitro tests. The spontaneous breathing pattern of the breathing simulator is determined by the lung-model 40
PSV outlet pressure (cm H2O)
PSV supply pressure 0.5 bar 1.0 bar 1.5 bar 2.0 bar
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Figure 7: Pressure output from the proportional solenoid valve at various inlet air source pressures.
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parameters and the patient effort. The two main lung-model parameters are airway resistance (R) and total elastance (E) of the patient. Normal airway-resistance values for adults range from 5 to 20 cmH2 O/l/s, whilst normal values of elastance for adults are from 20 to 50 cmH2 O/l. The patient effort embodies the breathing rate, effort amplitude, effort slope and inhale percentage (% inhale). The effortamplitude parameter controls the maximum patient effort or peak negative plural pressure generated. By adjusting this parameter, the tidal volume and flow waveform can be changed. The normal amplitude value ranges from 5 to 30 cmH2 O. The effort slope parameter is used to adjust the shape of the patient-effort waveform by controlling its rise and fall rate. The PSV-controlled PAV support system incorporating the breathing simulator (Fig. 5) and the control program (Fig. 9) function as follows. As the breathing simulator generates spontaneous effort, air is inhaled through the free airway via a oneway valve and its flow is measured by a pneumotachograph (4700A, Hans Rudolph). This flow signal is sent to a computer via a data acquisition card (PCI-20428W, Intelligent Instrumentation Inc.). The flow signal is then integrated to obtain the volume. The airway pressure is measured by a pressure transducer (DP15-30, Validyne). A closed-loop control program (Visual Designer, Intelligent Instrumentation Inc.) is used in the PAV system. The theoretical airway pressure calculated according to eqn. (8) is used as the setpoint for the PID (proportional, integral and derivative) process value whilst the actual airway pressure will be treated as the current input data. The pressure difference between the calculated and the actual airway pressure is the error input of PID. The output of the PID is multiplied by an amplification factor (Kc ) to transfer the pressure signal to an analogue voltage signal. As the operating input signal of the proportional solenoid valve ranges from 5 to 10 VDC, a 5-VDC offset value is added to the analogue voltage signal. This total voltage is
PSV outlet flow rate (l/s)
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PSV supply pressure 0.5 bar 1.0 bar 1.5 bar 2.0 bar
0.3
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0.0 5
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7 8 PSV analog input (volt)
9
Figure 8: Flow output from the proportional solenoid valve at various inlet air source pressures.
198 Human Respiration Free airway
Exhalation valve
Breathing simulator
F
P
PSV Compressed air source
V Paw
KC
K2
V K1
Paw'
e = Paw'–Paw
PID
Figure 9: Schematic diagram of the control program.
then used as the analogue input to the proportional solenoid valve to control the flow. The PAV-assisted air joins the atmospheric air intake through the free airway via a ‘Y’ type connector (Fig. 5). The exhalation valve (ADK12-15A, CKD Corporation) shown in Fig. 5 is a normally open-mode solenoid valve. It operates as a switch that will open during exhalation and close during inhalation. To ensure that PAV support is cut off at the end of each inspiration, the parameter Kc is set to zero when the flow declines and falls below a certain preset value that is 0.05 l/s in the tests reported here. Immediately, the exhalation valve is triggered to open so that the exhaled air can exit into the atmosphere. The exhalation valve will be triggered to close at the beginning of inspiration when the inspired flow rises above zero. Flow rate, volume and airway pressure are continuously monitored and displayed on a strip-chart recorder. These signals are also stored as data files for subsequent analyses. In a PSV-controlled PAV system, pressure ‘run-away’ is a common phenomenon when the proportionality settings are inappropriately used and an excessive-pressure limit mechanism is not incorporated. Pressure ‘run-away’ occurs when the pressure provided by the ventilator at the end of inspiration exceeds the opposing elastic and resistive pressures. Flow delivery continues during patient’s neural expiration and airway pressure continues to increase during expiration [16]. A peak inspiratory pressure greater than 30–40 cmH2 O increases the risk of barotrauma. Therefore, a pressure-limit mechanism is designed to prevent the occurrence of pressure ‘runaway’ by switching off the proportional solenoid valve when it occurrs. Basically, the control program has the following safety features. A timer is used to monitor the opening duration of the proportional solenoid valve. The PSV will be triggered to switch off when the opening duration reaches a preset value. Simultaneously, the inspired flow, volume and airway pressure are monitored and compared with clinically determined maximum thresholds. If any of these values exceeds the
Mechanics of Proportional-Assist Ventilation
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: Pressure transducer
PNT
: Pneumotachograph
199
: One way valve
PENTIUM
PT 2
PT 4 PT 1
PT 3
Free airway
Regulator
PNT 1
Proportional solenoid valve PNT 2
Exhalation valve Compressed air source
Figure 10: Schematic diagram of the proportional solenoid valve PAV system for human subject. corresponding threshold value, the proportional solenoid valve will be switched off and the exhalation valve triggers to open. In this way, pressure ‘run-away’ can be effectively curbed. For the in vivo tests, the human subject replaces the breathing simulator in the PAV system in Fig. 5 and the modified system is shown in Fig. 10. The subject breathes spontaneously from a two-way nonrebreathing facemask (Hans Rudolph). The proportional solenoid valve used in this system is an air/O2 servoid metering valve (South Bend Controls) that is medically safe for patients. As the activating signal of this valve is based on current instead of voltage, a 1-A output transmitter is used to convert a 0–10-VDC to 50–350 mADC. An additional pneumotachograph is installed at the expiration line to monitor the expired flow. Also, the pressure in the facemask is measured by a pressure transducer (Type 7261, Kistler) and charge amplifier (Type 5011, Kistler) to monitor the subject’s effort.
5 Experimental results and discussion Experimental results on the PAV system are presented for the in vitro tests based on the breathing simulator and in vivo tests for the healthy subjects. 5.1 Spontaneous breathing tests on breathing simulator To verify whether the breathing simulator does mimic the human respiratory system, a group of spontaneous tests was conducted to determine the response of tidal
200 Human Respiration 0.8 E = 50 cm H2O/l; Effort = 10 cm H2O E = 20 cm H2O/l; Effort = 10 cm H2O E = 50 cm H2O/l; Effort = 20 cm H2O E = 30 cm H2O/l; Effort = 20 cm H2O
Tidal volume (l)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10
20
30 40 50 60 Resistance (cm H2O/l/s)
70
80
Figure 11: Tidal-volume response to different resistances under various elastance and effort combinations. volume to different values of resistance, elastance and effort. These tidal volumes will be used as the normal breathing consequences in support of subsequent PAVperformance tests. To determine the influence of the airway resistance to spontaneous breathing, the response of the tidal volume to different resistances was tested under a given elastance and patient-effort combination. In these tests, various elastance and patienteffort combinations were used. The resistance was varied from normal (low) to abnormal (high); these values were 10, 20, 30, 50 and 80 cmH2 O/l/s. Figure 11 shows the tidal-volume response to different resistances under various elastance and effort combinations. Increasing the resistance from 10 to 80 cmH2 O/l/s results in reducing the tidal-volume progressively for every elastance and effort combination. Thus this breathing simulator can be used to mimic patients with high airway resistance. Similarly, the resistance and the patient effort were fixed whilst the lung elastance was varied to determine its effect on spontaneous breathing. The elastance was varied from 20 to 100 cmH2 O/l for various resistance and effort combinations; the tidal volume response can be observed in Fig. 12. Increasing the elastance progressively decreases the tidal volume, which is to be expected in the human respiratory system. Normally, a patient with neuromuscular weakness cannot generate enough negative pleural pressure to pull the lungs to a more expanded position. Therefore, the normal required tidal-volume for the patient cannot be met. This neuromuscular weakness can be simulated by having a small effort amplitude in the breathing simulator. In the following tests, the effort amplitude was set at 5, 10, 15 or 20 cmH2 O in each test. These simulated values would mimic from a very weak effort or
Mechanics of Proportional-Assist Ventilation
0.8
R = 20 cm H2O/l/s; Effort = 10 cm H2O R = 10 cm H2O/l/s; Effort = 10 cm H2O R = 20 cm H2O/l/s; Effort = 20 cm H2O R = 10 cm H2O/l/s; Effort = 20 cm H2O
0.7 Tidal volume (l)
201
0.6 0.5 0.4 0.3 0.2 0.1 0.0
20
30
40
50 60 70 80 Elastance (cm H2O/l)
90
100
Figure 12: Tidal-volume response to different elastances under various resistance and effort combinations.
0.8
R = 20 cm H2O/l/s; E = 50 cm H2O/l R = 10 cm H2O/l/s; E = 50 cm H2O/l R = 20 cm H2O/l/s; E = 20 cm H2O/l R = 10 cm H2O/l/s; E = 20 cm H2O/l
Tidal volume (l)
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
5
7
9 11 13 15 17 Patient effort amplitude (cm H2O)
19
21
Figure 13: Tidal-volume response to different effort amplitudes under various elastance and resistance combinations. breathing at rest to heavy breathing or a person carrying out strenuous exercise. Figure 13 shows the tidal-volume response to different effort amplitudes under various resistance and elastance combinations. At each constant resistance and elastance setting, the tidal-volume increases almost linearly with increasing patienteffort amplitude. The combination with lower resistance and elastance values corresponds to a larger gradient; and consequently a larger lung expansion with a greater
202 Human Respiration tidal-volume. In all the combinations, lung expansion is directly proportional to the patient effort. Therefore, this breathing simulator can simulate both normal subjects and patients with neuromuscular weaknesses. In summary, the breathing simulator can mimic a wide range of patient conditions by setting different lung-mechanics parameters and patient-effort shapes. Therefore, it is suitable to be used in the evaluation of the PAV system. 5.2 Performance tests on PAV system using breathing simulator Tests were conducted on the breathing simulator to mimic patients with either high resistance, high elastance or weak effort. In this study, the normal resistance, elastance and patient effort values were assumed to be 20 cmH2 O/l/s, 20 cmH2 O/l and 10 cmH2 O, respectively, and the corresponding normal tidal volume given by the breathing simulator was about 0.269 l as shown in Fig. 11. The abnormal resistance, elastance and patient-effort values were set at 30 cmH2 O/l/s, 50 cmH2 O/l and 5 cmH2 O, respectively. In all the tests, the spontaneous breathing effort was assumed to be constant for simplicity. The breath rate, effort slope and % inhale were set at 20 breaths per minute, 10 and 40%, respectively. 5.2.1 PAV tests on breathing simulator with high resistance In this study, the elastance and patient effort of the breathing simulator were set at normal values of 20 cmH2 O/l and 10 cmH2 O, respectively, but the resistance was set at a relatively higher value of 30 cmH2 O/l/s. Hence, the PAV should mainly overcome the extra resistance and the proportionalities K1 and K2 were set accordingly. The breathing patterns in terms of flow, volume and airway pressure with and without the implementation of PAV are compared in Fig. 14. During the first six breaths, the ‘patient’(breathing simulator) breathed without assistance, and from the seventh breath onwards, the PAV commenced to supply air to assist the ‘patient’. It can be observed that, with the implementation of PAV, the flow, volume and airway pressure increase significantly. During spontaneous breathing without assistance (Fig. 14(c)), the airway pressure was observed to be slightly negative during inspiration. However, when the PAV was initiated, the airway pressure increased sharply. Subsequently, different K1 and K2 combinations were tested. The breathing patterns in a typical breathing cycle before and after the implementation of PAV are illustrated in Fig. 15. The solid line represents the breathing pattern without the implementation of PAV, whilst the rest represent the breathing patterns with the implementation of PAV under different K1 and K2 combinations. It can be observed that with the implementation of PAV, the peak inspiration and expiration flow, tidal-volume, airway pressure and inspiratory time increase for each K1 and K2 combination as compared to those without PAV. However, the flow only increases marginally at the beginning of the inspiration period, i.e. during the first 0.75 second (Fig. 15(a)). This may be due to two reasons. First, at the beginning of the inspiration, both flow and inspired volume are relatively small and therefore the calculated theoretical airway pressure is also small.
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Flow rate (l/s)
(a) 0.6 0.5 0.4 0.3 0.2 0.1 0.0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6
PAV initiated
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10
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30
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30
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(b)
0.6 PAV initiated
Volume (l)
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Airway pressure (cm H2O)
(c) PAV initiated
16 14 12 10 8 6 4 2 0 –2 –4 0
10
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Time (s)
Figure 14: Tracings of breathing patterns before and after the implementation of PAV (R = 30 cmH2 O/l/s; E = 20 cmH2 O/l; Effort = 10 cmH2 O; K1 = 0; K2 = 30 cmH2 O/l/s).
204 Human Respiration (a) 0.4
Flow rate (l/s)
0.2 0.0 –0.2 –0.4 –0.6
No PAV K1 = 0; K2 = 20 cm H2O/l/s K1 = 0; K2 = 30 cm H2O/l/s K1 = 10 cm H2O/l, K2 = 20 cm H2O/l/s
–0.8 –1.0 0.0
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0.6
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0.5 0.4 0.3 0.2 0.1 0.0
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(c)
17
No PAV K1 = 0; K2 = 20 cm H2O/l/s K1 = 0; K2 = 30 cm H2O/l/s K1 = 10 cm H2O/l, K2 = 20 cm H2O/l/s
15 13 11 9 7 5 3 1 –1 0.0
0.5
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Figure 15: Illustrations of the effects of PAV in overcoming high resistance with different K1 and K2 combinations. Hence, the computed signal for the proportional solenoid valve, which is proportional to the difference between the theoretical and the actual airway pressures, is also small, resulting in a small opening of the valve. Secondly, there are time delays in (i) the opening of the proportional solenoid valve (this can be possibly a very
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small time delay as the valve response time is 20 ms) and (ii) gas transmission from PSV to the patient-connection port. One possible way to reduce the delay time is to decrease the tube length between the PSV and the patient-connection port. The proportional solenoid valve was not triggered to close at the end of spontaneous inspiration but only after the ‘patient’ had generated sufficient positive pressure to balance the airway pressure. The inspired flow will persist a little longer after the commencement of the exhalation phase of spontaneous breathing. This accounts for the PAV inspiration time to be longer than that of spontaneous breathing. The sharp ‘spike’ in the flow (Fig. 15(a)) during exhalation is due to the sudden opening of the exhalation valve at the end of inspiration, resulting in the venting of the airway pressure at the patient connection port to the atmosphere. In Fig. 15(b), it can be observed that one of the benefits of PAV is that the tidalvolume increases. However, the minimum volume at the end of each breathing cycle increases slightly when PAV is initiated. This is due to the initial higher resistance set and shorter exhalation time such that the lung cannot be emptied sufficiently, resulting in an increase in the residual volume. A comparison between two groups of proportionality settings K1 = 0, K2 = 30 cmH2 O/l/s and K1 = 10 cmH2 O/l, K2 = 20 cmH2 O/l/s shows that the tidalvolume in both cases has exceeded the original normal value (0.269 l in this case), demonstrating a positive effect of PAV in overcoming high resistance. However, the tidal volume and airway pressure increase more for the former settings than those of the latter. Since the abnormality is resistance in this case, the effect of increasing flow-assist proportionality (K2 ) is better than that of increasing volumeassist proportionality (K1 ). This result clearly shows that proper proportionality settings should be based on a good knowledge of the patient’s resistance and elastance. 5.2.2 PAV tests on breathing simulator with high elastance In these tests, the resistance and patient effort of the breathing simulator were set at normal values of 20 cmH2 O/l/s and 10 cmH2 O, respectively, but the elastance was set at a relatively higher value of 50 cmH2 O/l. Hence, the function of the PAV is to overcome the higher elastance and thus the proportionalities K1 and K2 were set accordingly. The breathing patterns in a typical breathing cycle before and after the implementation of PAV for different K1 and K2 combinations are illustrated in Fig. 16. The solid line represents the breathing pattern without the assistance of PAV, whilst the rest represent the breathing patterns with the implementation of PAV under different K1 and K2 combinations. From Fig. 16, it can be observed that the flow, tidal-volume and airway pressure increase significantly with the implementation of PAV as compared to those without PAV. In Fig. 16(a), with the implementation of PAV, the reasons for the marginal increase in flow at the beginning of the inspiration period, the longer inspiration time and the ‘spike’ in flow at the beginning of exhalation are similar to those for Fig. 15(a). The tidal volumes attained for these different K1 and K2 combinations have exceeded the normal tidal volume, demonstrating a positive effect of PAV in overcoming high elastance.
206 Human Respiration (a) 0.4
Flow rate (l/s)
0.2 0.0 – 0.2 – 0.4 – 0.6 – 0.8 – 1.0 – 1.2 – 1.4
No PAV K1 = 30 cm H2O/l; K2 = 0 K1 = 40 cm H2O/l; K2 = 0 K1 = 30 cm H2O/l, K2 = 10 cm H2O/l/s
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18 No PAV K1 = 30 cm H2O/l; K2 = 0 K1 = 40 cm H2O/l; K2 = 0 K1 = 30 cm H2O/l, K2 = 10 cm H2O/l/s
Airway pressure (cm H2O)
16 14 12 10 8 6 4 2 0
–2 0.0
0.5
1.0
1.5
Time (s)
Figure 16: Illustrations of the effects of PAV in overcoming high elastance under different K1 and K2 combinations. Comparing the proportionality settings of K1 = 40 cmH2 O/l, K2 = 0 and K1 = 30 cmH2 O/l, K2 = 10 cmH2 O/l/s in Fig. 16(b) and (c) show that increasing volume-assist proportionality K1 is more effective than flow-assist proportionality K2 to overcome the patient’s high elastance abnormality.
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5.2.3 PAV tests on breathing simulator with weak effort Tests were carried out for a weak patient-effort amplitude of 5 cmH2 O whilst the resistance and elastance were set at normal values of 20 cmH2 O/l/s and 20 cmH2 O/l, respectively. The K1 and K2 values were increased gradually at interval of 10, with the starting values at 10 cmH2 O/l and 10 cmH2 O/l/s for K1 and K2 , respectively. The PAV system performed well for the first two values of K1 and K2 . However, when K1 and K2 were increased to 30 cmH2 O/l and 30 cmH2 O/l/s, respectively, pressure ‘run-away’ occurred. The stable breathing patterns in one breath before and after the implementation of PAV using the first two K1 and K2 combinations are shown in Fig. 17, whilst Fig. 18 shows the breathing patterns when pressure ‘run-away’ occurred after PAV was initiated. It can be seen in Fig. 17 that although there are some increases in the flow, tidal-volume and airway pressure, the small PAV enhanced effect is not capable of providing sufficient tidal-volume to ‘pull’ the patient to his/her normal breathing patterns. This implies that the PAV system is not quite suitable for patients with very weak muscle effort and therefore some kind of forced mechanical ventilation should be provided for them. To avoid damage to the breathing simulator, the proportional solenoid valve is automatically switched off when pressure ‘run-away’ occurs. Otherwise, the flow will continue to increase with increases in the valve opening until it is completely open. The flow will then remain at that maximum value and therefore the volume will increase unboundedly. Then pressure ‘run-away’ ensues. The upper limit of the opening duration for the proportional solenoid valve was set at a large value of 5 s. If the inspiration period is 5 s or more, the solenoid valve will automatically cut off the supply flow to the breathing simulator. The 5-s limit is longer than the normal tolerant inspiration time. The cutoff of supply flow is only momentary; in the next inspiration cycle, the proportional solenoid valve will open and continue to control the supply flow until the next pressure ‘run-away’ occurs. In Fig. 18(a), the flow patterns and breathing rates before and after the implementation of PAV differ significantly although the spontaneous breathing effort remains unchanged. In the pressure ‘run-away’ situation, the breathing patterns are quite irregular after PAV is initiated. During the spontaneous expiration when PAV is on, the proportional solenoid valve is still supplying inflow to the breathing simulator as reflected by the increasing inspired volume (Fig. 18(b)) and the increasing airway pressure (Fig. 18(c)). However, when the opening duration of the solenoid valve has reached the pre-set 5 s, it suddenly shuts off and stops any further inflow, as shown in Fig. 18(a). This demonstrates that the pressure ‘run-away’ limit mechanism functions properly. As the proportional solenoid valve suddenly cuts off the inflow completely, the air continues to flow into the breathing simulator and momentarily the peak alveolar pressure is even greater than the airway pressure. Therefore, the exhalation valve is triggered to open and the airway pressure drops sharply (Fig. 18(c)), accompanied by a high peak expiratory flow as shown in Fig. 18(a). In practice, the maximum opening duration of the solenoid valve will not be set for more than 2 s in the program for safety consideration. The reason for setting
208 Human Respiration (a)
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Volume (l)
(b) 0.23 0.21 0.19 0.17 0.15 0.13 0.11 0.09 0.07 0.05 0.03 0.01
No PAV K1 = 10 cm H2O/l; K2 = 10 cm H2O/l/s K1 = 20 cm H2O/l; K2 = 20 cm H2O/l/s
0.0
0.5
1.0
1.5
Time (s)
Airway pressure (cm H2O)
(c) No PAV K1 = 10 cm H2O/l; K2 = 10 cm H2O/l/s K1 = 20 cm H2O/l; K2 = 20 cm H2O/l/s
3 2 1 0
–1 0.0
0.5
1.0
1.5
Time (s)
Figure 17: Illustrations of the effects of PAV in overcoming weak effort under different K1 and K2 combinations.
Mechanics of Proportional-Assist Ventilation (a)
2
Flow rate (l/s)
1
209
Assisted airflow was terminated by switching off the solenoid valve PAV initiated
0
–1 –2 –3 0
10
20
30
40
30
40
30
40
30
40
Time (s)
(b) 2.4
Volume (l)
1.9 1.4 0.9 0.4
–0.1 0
10
20
Time (s)
(c) Airway pressure (cm H2O)
60 50 40 30 20 10 0
–10
0
10
20
Time (s)
Alveolar pressure (cm H2O)
(d) 80 70 60 50 40 30 20 10 0
–10 0
10
20
Time (s)
Figure 18: Illustrations of the breathing patterns when pressure ‘run-away’ occurs and the triggering of the pressure-limit mechanism.
210 Human Respiration 5 s in the above instance is to demonstrate clearly that without a pressure-limit mechanism, the airway and alveolar pressures will continue to increase unabatedly and lead to a serious pressure ‘run-away’ situation. 5.3 Simulated tests on healthy human subjects Breathing tests were carried out on six normal subjects with artificial resistance and elastance incorporated. Four males and two females, with ages ranging from 23 to 32, volunteered for this study. In order to mimic patients with high resistance, a series of orifice resistors were incorporated in the airway upstream of the pneumotachograph (PNT 1) in Fig. 10. The resistance of the resistor was varied by changing the diameter of the center hole from 5 to 10 mm at a step interval of 1 mm. The resistances of the various orifice resistors are given in Table 1. For simplicity, the orifice resistors are referred to here as resistor 5 to 10 where the number specifies the diameter of the orifice. Before the in vivo PAV tests, it is necessary to determine the influence of the orifice resistors on the normal subjects’ breathing conditions. Each person was instructed to breathe through a facemask that was connected to an orifice resistor and a pneumotachograph. Figure 19 shows the typical flow patterns when a subject breathed through resistors 10 to 5. Increasing resistance (or decreasing orifice diameter) decreases the flow rate and hence the tidal volume also decreases. The subjects were able to maintain a stable breathing effort when breathing through resistors 10 to 8 and they could sustain this for a longer time without assist. Their flow patterns were relatively stable, unlike those for resistors 7 to 5 as shown in Fig. 19. For resistors 7 to 5, the subjects
Table 1: Resistance of orifice resistors. Orifice Diameter (mm)
5
6
7
8
9
10
Resistance (cmH2 O/l/s)
100.88
54.80
37.08
15.81
10.71
8.68
Resistor 10 0.8
Flow rate (l/s)
0.6 0.4 0.2 0.0
–0.2 –0.4 –0.6 0
10
20
30
40
50
60
70
Time (s)
Figure 19: Typical flow patterns of a normal subject breathing through different orifice resistors.
Mechanics of Proportional-Assist Ventilation Resistor 9
Flow rate (l/s)
0.8 0.6 0.4 0.2 0.0 –0.2
–0.4 –0.6 –0.8
0
10
20
30
40
50
60
70
40
50
60
70
40
50
60
70
40
50
60
70
Time (s)
Resistor 8
Flow rate (l/s)
0.8 0.6 0.4 0.2 0.0 –0.2
–0.4 –0.6 –0.8
0
10
20
30
Time (s)
Flow rate (l/s)
Resistor 7 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8
0
10
20
30
Time (s)
Flow rate (l/s)
Resistor 6 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8
0
10
20
30
Time (s)
Figure 19: Continued.
211
212 Human Respiration
Flow rate (l/s)
Resistor 5 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8
0
10
20
30
40
50
60
70
Time (s)
Figure 19: Continued.
opined that it was very difficult to breathe with a normal breathing effort. Therefore, incorporating orifice resistors could, to some extent, mimic patients with high resistance. To mimic patients with high elastance, a slim belt was strapped to the chest of the six normal subjects at varying degrees of tightness. The muscular movement of the subject was restricted in some extent by the slim belt and the subject would exert much more breathing effort to obtain the desired ventilation. This is similar to a patient with a higher elastance whose breathing effort is mainly dissipated towards expanding the stiff lungs. The typical flow-rate patterns of a healthy subject breathing with different belt tightness exerted on the chest are shown in Fig. 20. Increasing belt tightness that corresponds to increasing elastance of the respiratory system, decreases the inspiratory flow rate. Therefore, to some extent, it is possible to mimic a patient with higher elastance by a slim belt strapped on the subject’s chest. 5.3.1 In vivo PAV tests to overcome high resistance In vivo PAV tests were carried out on the six normal subjects to overcome the high resistance induced by the orifice resistors. The subjects were requested to remain as relaxed as possible and to maintain a relatively steady breathing effort throughout the entire test cycles. Figure 21 shows the breathing patterns before and after the implementation of PAV for a subject when breathing through resistor 7 with proportionalities K1 and K2 set at 5 cmH2 O/l and 10 cmH2 O/l/s, respectively. Before PAV was initiated, the subject breathed unassisted for several cycles. Figure 21 shows that the flow rate, inspired volume and airway pressure of the subject increase after the activation of PAV. The most significant change with the implementation of PAV is the airway pressure that shows a sharp increase when PAV is activated. It also shows that the assist in each cycle is not consistent due to varying spontaneous efforts. Increasing effort increases the assist, which is one of the advantages of PAV. In some of the tests, the flow rate and inspired tidal-volume decreased after the activation of PAV as illustrated in Fig. 22. In these cases, the subjects felt
Mechanics of Proportional-Assist Ventilation
213
Flow rate (l/s)
Medium tight 0.8 0.6 0.4 0.2 0.0 – 0.2 – 0.4 – 0.6 – 0.8
0
10
20
30
40
50
60
70
Time (s)
Flow rate (l/s)
Tight 0.8 0.6 0.4 0.2 0.0 – 0.2 – 0.4 – 0.6 – 0.8 0
10
20
30
40
Time (s)
Flow rate (l/s)
Very tight 0.8 0.6 0.4 0.2 0.0 – 0.2 – 0.4 – 0.6 – 0.8
0
10
20
30
40
50
Time (s)
Figure 20: Typical flow-rate patterns of a normal subject with different belt tightness exerted on the chest.
comfortable and therefore they relaxed their efforts when breathing with the help of PAV. This is reflected in the positive airway pressures at the commencement of each breathing cycle after the activation of PAV, indicating that the subject had not maintained the same breathing efforts as compared to those before the activation of PAV. If the subject pulls less, PAV will also reduce its assist. Therefore, it is reasonable to observe in Fig. 22 that the tidal-volumes after the activation of PAV are less than those before its activation. This phenomenon demonstrates the positive effect of PAV as it relieves the subject’s effort to obtain a comfortable tidalvolume.
214 Human Respiration (a) Flow rate (l/s)
1.5
PAV initiated
1.0 0.5 0.0 –0.5 –1.0 0
10
20
30
40
50
40
50
40
50
Time (s)
Volume (l)
(b) 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
PAV initiated
0
10
20
30
Time (s)
(c)
Paw (cm H2O)
20
PAV initiated
15 10 5 0 –5
0
10
20
30
Time (s)
Figure 21: Representative breathing patterns before and after the implementation of PAV for a healthy subject when breathing through resistor 7 with proportionalities K1 and K2 set at 5 cmH2 O/l and 10 cmH2 O/l/s, respectively.
5.3.2 In vivo PAV tests to overcome high elastance In vivo PAV tests were carried out on the six normal subjects to overcome the high elastance induced by the slim belt. To overcome high elastance, PAV is mainly employed as a volume-assist mode. Therefore, K1 was varied according to different belt tightness. Flow rate, volume and airway pressure patterns in a typical breathing cycle before and after the activation of PAV at different K1 levels are shown in Fig. 23. Increasing K1 increases the ventilatory consequences, demonstrating the positive effect in assisting patients with higher elastance than normally should be.
Mechanics of Proportional-Assist Ventilation
215
Flow rate (l/s)
(a) 0.5 0.4 0.3 0.2 0.1 0.0 –0.1 –0.2 –0.3 –0.4 –0.5
PAV initiated
0
10
20
30
40
Time (s)
(b)
0.6
PAV initiated
Volume (l)
0.5 0.4 0.3 0.2 0.1 0.0
0
10
20
30
40
Time (s)
Paw (cm H2O)
(c)
9 8 7 6 5 4 3 2 1 0 –1 –2
PAV initiated
0
10
20
30
40
Time (s)
Figure 22: Breathing consequences of PAV in relieving a healthy subject’s breathing effort. The subjects opined that they felt much more comfortable to breathe when PAV was activated.
6 Conclusions A proportional-assist ventilation system based on a proportional solenoid valve to control air delivery from a pressurized source to a patient, was evaluated. Respiratory tests using this proposed system were conducted on a breathing simulator that is capable of mimicking the spontaneous breathing responses of patients suffering from pulmonary diseases and respiratory disabilities. The results demonstrated the viability of the PAV system in normalizing the breathing patterns of patients with abnormal resistance, elastance and weak effort after appropriate selection of K1 and K2 proportionalities. Pressure ‘run-away’ can be prevented
216 Human Respiration
Flow rate (l/s)
(a) No PAV With PAV, K1 = 5; K2 = 5 With PAV, K1 = 10; K2 = 5 With PAV, K1 = 15; K2 = 5
1.2 1.0 0.8 0.6 0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0 0
1
2
3
4
5
Time (s)
Volume (l)
(b) No PAV With PAV, K1 = 5; K2 = 5 With PAV, K1 = 10; K2 = 5 With PAV, K1 = 15; K2 = 5
1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
1
2
3
4
5
Time (s)
Airway pressure (cm H2O)
(c) No PAV With PAV, K1 = 5; K2 = 5 With PAV, K1 = 10; K2 = 5 With PAV, K1 = 15; K2 = 5
14 12 10 8 6 4 2 0 –2
0
1
2
3
4
5
Time (s)
Figure 23: Representative breathing consequences with and without PAV for a healthy subject strapped with a slim belt. by incorporating a back-up safety mechanism in the control program to ensure the safe operation of the PAV system. In vivo tests using the proposed PAV system were carried out on volunteer healthy persons with incorporated artificial resistance and elastance. Significant increases in inspired flow, tidal-volume
Mechanics of Proportional-Assist Ventilation
217
and airway pressure after the implementation of PAV were obtained. The test subjects felt comfortable to breathe with reduced breathing effort when PAVassist ventilation was initiated, thereby demonstrating the positive assist effect of PAV.
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[3]
[4]
[5]
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[7]
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Ganong, W.F., Review of Medical Physiology (16th edn.), Appleton & Lange: Connecticut, 1993. Younes, M., Puddy, A., Roberts, D., Light, R.B., Quesada, A., Taylor, K., Oppenheimer, L. & Cramp, H., Proportional assist ventilation: Results of an initial clinical trial. American Review of Respiratory Disease, 145(1), pp. 121–129, 1992. Chua, L.P., Li, N., Lua, A.C. & Lim, T.K., In-vivo tests on ventilatory responses to control parameter variations for a proportional assist ventilation system. Proc. of the 9th Int. Conf. on Biomedical Engineering, eds. J.C.H. Goh & A. Nather, Singapore, pp. 577–579, 1997. Li, N., Chua, L.P., Lua, A.C., Liu, C.Y. & Lim, T.K., Ventilatory response of a proportional assist ventilation system to control parameter variations. Proc. of ASME Fluids Engineering Division, Summer Meeting, FEDSM973032, Canada, pp. 1–8, 1997. Patrick, W., Webster, K., Ludwig, L., Roberts, D., Wiebe, P. & Younes, M., Noninvasive positive-pressure ventilation in acute respiratory distress without prior chronic respiratory failure. American Journal of Respiratory and Critical Care Medicine, 153, pp. 1005–1011, 1996. Navalesi, P., Hernandez, P., Wongsa, A., Laporta, D., Goldberg, P. & Gottfried, S.B., Proportional assist ventilation in acute respiratory failure: Effects on breathing pattern and inspiratory effort. American Journal of Respiratory and Critical Care Medicine, 154(5), pp. 1330–1338, 1996. Ranieri, V.M., Grasso, S., Mascia, S., Mascia, L., Martino, S., Fiore, T., Brienza, A. & Giuliani, R., Effects of proportional assist ventilation on inspiratory muscles effort in patients with chronic obstructive pulmonary disease and acute respiratory failure. Anaesthesiology, 86(1), pp. 79–91, 1997. Ranieri, V.M., Giuliani, R., Mascia, L., Grasso, S., Petruzzelli, V., Puntillo, N., Perchiazzi, G., Fiore, T. & Brienza, A., Patient-ventilator interaction during acute hypercapnia: Pressure support vs proportional assist ventilation. Journal of Applied Physiology, 81(1), pp. 426–436, 1996. Marantz, S., Patrick, W., Webster, K., Roberts, D., Oppenheimer, L. & Younes, M., Response of ventilator-dependent patients to different levels of proportional assist. Journal of Applied Physiology, 80(2), pp. 397–403, 1996. Meza, S. & Younes, M., Ventilatory stability during sleep studied with proportional assist ventilation (PAV). Sleep, 19(10), pp. S164–S166, 1996. Dolmage, T.E. & Goldstein, R.S., Proportional assist ventilation and exercise tolerance in subjects with COPD. Chest, 111, pp. 948–954, 1997.
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Pride, N.B., Airflow resistance. Lung Function Tests: Physiological Principles and Clinical Applications, eds. J.M.B. Hughes & N.B. Pride, W.B. Saunders: Philadelphia, pp. 27–43, 1999. Gibson, G.J., Lung volumes and elasticity. Lung Function Tests: Physiological Principles and Clinical Applications, eds. J.M.B. Hughes & N.B. Pride, W.B. Saunders: Philadelphia, pp. 44–56, 1999. Taylor, A.E., Rehder, K., Hyatt, R.E. & Paker, J.C., Clinical Respiratory Physiology, W.B. Saunders: Philadelphia, 1989. Younes, M., Proportional assist ventilation, a new approach to ventilatory support: Theory. American Review of Respiratory Disease, 145(1), pp. 114–120, 1992. Younes, M., Proportional assist ventilation. Principles and Practice of Mechanical Ventilation, ed. M.J. Tobin, McGraw-Hill: New York, pp. 349–369, 1994.
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