## Abstract

A model of parenchymal mechanics is revisited with the objective of investigating the differences in parenchymal microstructure that underlie the differences in regional compliance that are inferred from gas-mixing studies. The stiffness of the elastic line elements that lie along the free edges of alveoli and form the boundary of the lumen of the alveolar duct is the dominant determinant of parenchymal compliance. Differences in alveolar size cause parallel shifts of the pressure-volume curve, but have little effect on compliance. However, alveolar size also affects the relation between surface tension and pressure during the breathing cycle. Thus regional differences in alveolar size generate regional differences in surface tension, and these drive Marangoni surface flows that equilibrate surface tension between neighboring acini. Surface tension relaxation introduces phase differences in regional volume oscillations and a dependence of expired gas concentration on expired volume. A particular example of different parenchymal properties in two neighboring acini is described, and gas exchange in this model is calculated. The efficiency of mixing and slope of phase III for the model agree well with published data. This model constitutes a new hypothesis concerning the origin of phase III.

- nonuniform ventilation
- Marangoni flow
- mixing efficiency

nonuniform ventilation has physiological and clinical significance, both intrinsically and because the ventilation distribution is an underlying component of the ventilation-perfusion ratio (V̇a/Q̇) distribution. Mixing inefficiency, deviations from simple exponential plots of expired concentration as a function of breath number for multibreath washin or washout of a test gas, and the slope of phase III (*S*_{III}) of expiration of a test gas are taken as signs of nonuniform ventilation. Matching of multicompartment models to data on mixing efficiency or data on expired gas concentration vs. breath number yield the result that logSDv̇, where v̇ is regional specific ventilation, is 0.5–0.6. Thus ventilation is quite nonuniform; the ratio of v̇ for a region that lies 1 SD above the mean to that for a region that lies 1 SD below the mean is ∼3. Both functional and anatomic studies show that the bulk of this variability in v̇ occurs at small scale. This raises the question: what regional differences in parenchymal microstructure could produce this variance in v̇?

To investigate this question, a model of parenchymal mechanics (27) is revisited. The effects of variations in the values of the parameters of the model on specific ventilation are described. Particular examples of parameter values for two regions fed by the daughter branches at a bifurcation are chosen, and the volume oscillations of the two compartments are calculated. The magnitude of the oscillations in surface tension (γ) that accompanies the volume oscillations is different for the two compartments. Differences in γ drive Marangoni flows that equilibrate γ. This relaxation process is then included in the model, the volume oscillations are recalculated, and the concentration of N_{2} in the expired gas after an inspiration of pure oxygen is calculated. Because of the phase difference between the volume oscillations of the compartments that is introduced by γ relaxation, the concentration of nitrogen in mixed expired gas increases during the course of expiration. Thus the modeling generates a new hypothesis about the source of the *S*_{III}.

## MODELING AND ANALYSIS

The conceptual basis of the model of parenchymal mechanics (27) is the following. Total lung recoil pressure is assumed to consist of two components: recoil of the saline-filled lung, and the additional recoil of the air-filled lung. The recoil of the saline-filled lung is provided by the connective tissue framework of the lung, consisting of the pleural membrane, the bronchial tree, and the interconnected network of interlobular membranes that extend between the bronchial tree and the pleural membrane. The parenchyma is pictured as a collection of polyhedral alveoli that open on the lumens of the alveolar ducts. Cables, composed of connective tissue, form the free edges of the alveolar walls. The outward pull of γ on the two faces of the alveolar wall extending outward from a cable is balanced by the combination of tension carried by the cable and the curvature of the cable. It is assumed that tissue tension in the alveolar wall is negligible. This assumption is based on the observation that the alveolar walls in saline-filled lungs are undulating and appear to be flaccid. In air-filled lungs, excess wall material collects in pockets at the alveolar intersections, and the surfaces are flat. This assumption is also consistent with later anatomical measurements reported by Oldmixon and Hoppin (18). They found that 80% of the elastin and collagen in parenchyma are located in features that they call “ends,” cables that form the free edges of alveolar walls, and “bends,” cables from which two alveolar walls extend outwards. Thus, mechanically, the alveolar walls are pictured as merely platforms for γ, and the additional recoil of the air-filled lung is provided by γ and cable tension.

### Basic Equations

#### Pressure.

Total recoil pressure (P) is the sum of the recoil pressure in the saline-filled lung (P_{sal}) and the additional recoil of the air-filled lung (P_{γ}). The tissue component of recoil pressure is taken to be linearly proportional to lung volume (V) with an elastance of 2×10^{3} dyn/cm^{2} per total lung capacity (TLC) (V_{o}).
*S*/3·V, where *S* is surface area (12). For randomly oriented cable tangents, the stress contributed by tension in the cables (τ) is τ·*L*/3·(V + V_{ti}), where *L* is cable length, V_{ti} is tissue volume, and (V + V_{ti}) is total V (27).

#### Surface area.

*S* is assumed to depend on V and *L*. This function, *S*(V,*L*), is governed by two constraints. The first is the condition of internal equilibrium between the outward pull of γ and the inward Laplace stress in the cables. At equilibrium, the stored energy in the system is minimum, and the net work done by a virtual displacement of the cables is zero. This principle yields the following equation governing *S*(V,*L*).
*Eqs. 2*, *3*, and *4* yields the following partial differential equation for *S*(V,*L*).
*Eq. 5* is given by *Eq. 6*, where *F* is an arbitrary function of its argument.
*S*(V,*L*), where *C*_{1} and *C*_{2} are constants.
*Eq. 7* describes the dependence of *S* on V for a foam in which geometry remains similar as air volume changes. The second describes the reduction of *S* due to missing surfaces in the volume occupied by the lumen of the duct with lumen radius proportional to *L*.

#### Cable tension.

The properties of the cable are described by *Eq. 8*. In this equation, *C*_{3} and *C*_{4} are constants that describe the elastance of the cable and the strength of the nonlinearity, respectively, and *L*_{o} is the resting length of the cable.

#### Surfactant dynamics.

For small *S* oscillations, surfactant acts like an elastic material with a specific elastance determined by its physical-chemical properties. The value of specific elastance reported by Schurch et al. (22) is 200 dyn/cm. Thus an incremental change in γ (dγ) is related to an incremental change in *S* (d*S*) by *Eq. 9*.

#### Summary of basic equations.

The variable τ can be eliminated by substituting for τ from *Eq. 3* into *Eqs. 2* and *8*. The remaining equations reduce to the following.
*Eqs. 10–13* constitute a complete set of equations for the variables, V, *L*, *S*, and γ, as functions of time.

The variables V, *L*, and *S* in *Eqs. 10–13* are extensive quantities for the whole lung. The variables are transformed into intensive variables by normalizing V by V_{o}, *L* by *L*_{o}, and replacing *S* by the surface-to-volume ratio. The new variables are defined as v = V/V_{o}, *l* = *L*/*L*_{o}, *s* = *S*/V, and the governing equations, expressed in terms of these variables are the following.
*Eqs. 14–17* are related to the upper case constants by the relations: *c*_{1} = *C*_{1}/V_{o}^{1/3}, *c*_{2} = *C*_{2}·*L*_{o}^{2}/V_{o}^{2/3}, *c*_{3} = *C*_{3}/2·*C*_{1}·*C*_{2}·*L*_{o}, and *c*_{4} = *C*_{4}.

### Parameter Values and Mechanical Behavior

#### Base case.

The values of the four parameters in *Eqs. 14–17* were chosen so that the model matches the anatomic and mechanics data for human lungs. For a foam with spherical cells, the surface-to-volume ratio ≈ 3/*r*, where *r* is the radius of the cell. The value of *c*_{1} was taken as 210 cm^{−1} to match the values of alveolar radius reported by Weibel (25). The value of *c*_{2} was taken to be 0.22 so that the second term in the brackets in *Eq. 15* is 0.4, consistent with the observation that the lumen of the duct occupies 40% of the volume of the acinus (25). The values of *c*_{3} and *c*_{4} were chosen rather arbitrarily so that the specific elastance of the model is ≈ 4P (15). These values are listed in Table 1 as parameter values for the base case. The value of v_{ti} was taken to be 0.12 (26).

*Equations 14–17* were solved numerically for an imposed sinusoidal pressure oscillation with mean value 2.5 ×10^{3} dyn/cm^{2}, an amplitude of 1.0 × 10^{3} dyn/cm^{2}, and with γ = 10 dyn/cm at midoscillation. The resulting v-P and relation is shown in Fig. 1.

#### Effect of varying parameter values.

For a larger value of *c*_{1}, corresponding to a smaller alveolar size, the value of v at midoscillation is smaller, but the compliance is essentially unchanged. Also, the rate of change of γ with P is larger. For larger values of *c*_{2}, corresponding to a larger duct lumen, the changes in v and γ are opposite those for larger *c*_{1}. For larger values *c*_{3} and *c*_{4}, corresponding to stiffer alveolar entrance rings, parenchymal elastance is larger, and the γ-P curve is essentially unchanged.

*Equations 14–17* are rather opaque, but some understanding of the changes in mechanical properties with changes in parameter values can be obtained from the form of these equations. The effect of *c*_{1} can be explained from *Eq. 14*. The right side of *Eq. 14* is a weak function of v. Therefore, a relatively small increase in *c*_{1} requires a relatively large decrease in v to match a given value of P at a given γ. Also, for a given change in pressure, the change in γ must be smaller if *c*_{1} is larger. The effect of greater cable stiffness is determined by *Eqs. 15–17*. From *Eq. 15*, it can be seen that *S* increases with V, and that this increase is buffered by increases in *L* with increasing γ. For a stiffer cable, the increase in *l* with increasing γ is smaller (*Eq. 16*), and thus the increase in *S* for a given change in v is larger. The change in γ is proportional to the change in *S* (*Eq. 17*), and since the change in γ is fixed by the change in P (*Eq. 14*), the change in v must be smaller.

### Two-compartment Model

#### Mechanics.

The two-compartment model follows the familiar scheme of two compartments fed by the daughter branches of a common airway. The parameter values for each compartment are listed in Table 1, and the v-P and γ-P curves for the two compartments are shown in Fig. 1. It can be seen from Fig. 1 that differences in γ are generated during oscillatory pressure changes. These differences would drive Marangoni flow of the liquid lining layer from the compartment with low γ into the compartment with high γ. The remaining surface expands in the compartment with low γ and contracts in the compartment with high γ, thereby tending to equilibrate γs. This effect is described by adding a term to the right side of *Eq. 17* for *region 1*, to obtain *Eq. 18* where the superscripts denote compartment number.
*compartment 2*. The value of *k* was taken as 3 cm^{−1}·s^{−1}, and approximate numerical solutions for *Eqs. 14*, *15*, *16*, and *18* for a breathing frequency of 12 breaths/min were obtained by the methods described in the appendix.

#### Gas exchange.

An N_{2} washout maneuver was simulated. The initial concentration of N_{2} in both compartments was taken as 80%. The concentrations of N_{2} in the two regions after an inspiration of pure O_{2} were calculated, assuming a dead space of 0.14 liter, and the concentration of N_{2} in the mixed expired gas during the subsequent expiration was calculated. For this simulation of a single-breath washout, a line was fit to the curve of N_{2} vs. expired volume (V_{exp}) in the range, 0.4 liter < Vexp < 0.8 liter, to obtain the *S*_{III}, and *S*_{III} was divided by mean expired concentration to obtain the normalized *S*_{III} (*S*_{n}).

Two other single-breath maneuvers were simulated. The first was for a 50% larger pressure oscillation with a breathing frequency of 8 breaths/min, and the second was a breath hold of 4 s at constant pressure at end inspiration followed by expiration. In addition, the values of *S*_{n} were calculated as a function of breath number for a multibreath washout. In this calculation, the concentrations of N_{2} in the compartments were calculated, taking account of penduluft that occurs at the beginning and end of inspiration and the reingestion of gas from the dead space at the beginning of inspiration.

## RESULTS

The calculated v-P curves for the two compartments and the curve of γ^{2}-γ^{1} vs. P are shown in Fig. 2. The volume loops shown in Fig. 2 are not pure sinusoids, but, roughly speaking, v^{1} leads P by 8° and v^{2} lags P by 23°. Thus, the v-P curve for *compartment 1* rotates clockwise, and that for *compartment 2* rotates counterclockwise. For TLC = 7 liters, end-expiratory volume in Fig. 2 corresponds to 3 liters, and the volume excursions correspond to a tidal volume of 1 liter. Compartmental volumes during expiration, normalized by their volume at end inspiration, are shown as functions of total volume, normalized by total end-inspiratory volume, in Fig. 3.

The concentrations of N_{2} in *compartments 1* and *2*, which were initially 80%, are 54 and 68%, respectively, at end inspiration. The curve of mixed expired concentration of N_{2} as a function of V_{exp} is shown in Fig. 4. Also shown in Fig. 4 is the concentration of N_{2} for ideal mixing. The efficiency of mixing is 93%. Because of the phase difference between the volume oscillations, the expired concentration of N_{2} rises during expiration. The slope of the line fit to the curve is 3.9%/liter, and the *S*_{n} = 0.068 liter^{−1}.

For the tidal volume of 1.5 liters, the end-inspiratory concentrations in both regions are lower than for the tidal volume of 1.0 liter, the difference in concentrations is slightly lower, and the phase angles are about the same. Therefore, the increase of expired concentration with phase angle is about the same, and the *S*_{III} is smaller primarily because the volume excursion is larger. For this maneuver *S*_{n} = 0.046 liter^{−1}.

For the breath hold at end inspiration, the difference in γ decreases from 1.1 dyn/cm at end inspiration to 0.3 dyn/cm at the end of the 4-s breath hold. Volume decreases in *region 1* and increases in *region 2* during breath hold. During expiration after breath hold, γ differences and surface flows are reestablished, and the expiratory volume trajectories are curved. The changes in volume and γ difference during breath hold and subsequent expiration are shown by the vertical and curved dashed lines in Fig. 2. The difference between the gas concentrations in the two compartments is reduced by the penduluft that occurs during breath hold, the curvatures of the volume trajectories are smaller than for normal breathing, and the resulting value of *S*_{n}, 0.044 liter^{−1}, is smaller.

The plot of *S*_{n} vs. breath number is shown in Fig. 5. *S*_{n} increases with breath number and reaches an asymptote of ∼0.4 liter^{−1} for breath number > 20.

## DISCUSSION

The study of gas mixing contains two schools. The objective of the first school is to determine the magnitude of the variance of the distribution of regional ventilation. This distribution is physiologically significant because it, together with the distribution of perfusion, forms the basis for the V̇a/Q̇ distribution. Despite its significance, it has received relatively little attention.

The functional studies of this school include studies of efficiency of mixing, fitting of multiexponentials to washin data, and matching predictions of multicompartment models to washin data. This school includes anatomic studies of variable ventilation. These began with the work of Hubmayr et al. (13). They found that parenchymal expansion was variable, that the variability increased with increasing resolution, and that this variability at small scale was largely independent of gravity. Subsequent studies, using new imaging techniques and aerosol deposition, yielded results that are consistent with those of Hubmayr et al. To date, the magnitude of the variability measured anatomically is smaller than that inferred from washin studies, perhaps because of the limits on the resolution of the imaging methods. This school has generated no hypotheses about the source of small-scale ventilation inhomogeneity.

The focus of the second school is the *S*_{III}. Although the *S*_{III} has little intrinsic functional significance, this school has been more prolific than the first. Experimental studies include measurements of the *S*_{III} for both single-breath and multibreath N_{2} washouts and measurements of slopes for different test gases and different breathing maneuvers.

This school has included hypothesizing and modeling from the beginning. In his seminal paper, Fowler stated that the slope of the alveolar plateau was the result of both a spatial and a temporal inhomogeneity of ventilation, with well-ventilated regions emptying early in expiration and poorly ventilated regions emptying later (11). Studies of the washin and washout of radioactive gases showed that ventilation was nonuniform, with ventilation increasing from apex to base in upright subjects. Milic-Emili et al. (17) provided a mechanistic explanation for the observed gravitational gradient in ventilation. They argued that, because of the gravitational gradient in pleural pressure, regions at different heights in the gravitational field were situated at different points on the P-V curve, with transpulmonary pressure higher at the apex than the base. With this model, he obtained plots of regional volume, as a fraction of volume at TLC, vs. total V as a fraction of TLC. The slopes of these lines increased from apex to base. To explain his observations on the effect of posture on the slope of the alveolar plateau of radioactive Xe, Anthonison et al. (1) concluded that these lines must be curved: concave downward for apical regions with lower than average slope and concave upward for basal regions with greater than average slope. This diagram, similar to Fig. 3, with lines curved toward the line of identity, was referred to as the onion-skin diagram.

Observations that were contrary to this argument were reported. Piiper and Scheid (21) pointed out that the magnitude of the gravitational ventilation gradient was too small to explain the inefficiency of mixing. Most telling, sampling of gas concentrations from small airways showed that the bulk of the source of the slope was located in regions that were fed by airways of 3 mm diameter or less and were too small to be significantly distorted by gravity (10). Later, modeling of gravitational lung deformation predicted that regional volume curves have the opposite curvature from those postulated by Anthonisen et al. (1), and that the diagram is trumpet-shaped (26). From these curves, one would predict a negative *S*_{III} (14). Despite this contrary evidence, the onion-skin diagram with its gravitational origin remained the conventional explanation for the *S*_{III} into the early 1980's.

Conventional ideas shifted with the work of Bowes et al. (4) and Paiva and Engel (19, 20), beginning in the late 1970's. They and their colleagues used computational methods to analyze convection and diffusion in either continuous or nodal models of the peripheral airways. In these models, the geometries of the branches extending from a bronchiole are assumed to be asymmetrical, so that the relative magnitudes of convective and diffusive transport are different in different branches. As new morphometric data became available, the models became more complex, and, at this point, it is difficult for the reader to come away with a qualitative understanding of the results. Experimental studies by Crawford and colleagues provide valuable data on the *S*_{III} for multibreath N_{2} washouts and their dependence on tidal volume (6), gas diffusivity (8), and breath hold (7) in humans. Usually, only the *S*_{III} normalized by mean expired concentrations (*S*_{n}) are reported. Other data, such as N_{2} concentration as a function of V_{exp} or mean concentration as a function of breath number are usually not reported or compared with model predictions. Paiva, Engel, and colleagues give particular attention to the plot of *S*_{n} vs. breath number for washouts extending to 25–30 breaths. The curve predicted by the convection-diffusion model does not match the observed curve (8), and an interregional convective spatial-temporal inhomogeneity is postulated to explain the difference (23). The convection-diffusion model is the current conventionally accepted explanation for the *S*_{III} (24).

There has been little interaction between the two schools. The first reports mean or end-expiratory concentrations and ignores the slope. The second usually reports data on the slope, normalized by mean concentration, and ignores the data on concentration.

The key feature in the modeling of the first school is nonuniform specific volume expansion and diffusive transport is neglected, whereas, in the second school, diffusive transport is crucial and specific volume expansion is usually assumed to be uniform.

Here, a new mechanism for the *S*_{III} is proposed. The various aspects of that mechanism will be discussed in the following paragraphs.

The parameter values for the base case are supported by data, as described in the modeling section. The differences in the values of the *c*'s for *compartments 1* and *2* can be interpreted as follows. The differences in the values of *c*_{1} and *c*_{2} from the values for the base case imply modest differences in alveolar radii and lumen radius. The differences in *c*_{3} and *c*_{4} are more extreme. Variations in these parameters are justified by the observations of Oldmixon and Hoppin (18) that cable diameter and the relative proportion of elastin and collagen in the cables is quite variable.

The value of *k* used in the calculation, 3 cm^{−1}·s^{−1}, was chosen because the *S*_{III} was maximum for this value. Thus this choice for the value of *k* is crucial and must be justified, and this justification yields new information about the mechanism. The term *k*(γ^{2} − γ^{1}) in *Eq. 18* describes the surface flux between compartments divided by V_{o}. Surface velocity (*u*) is related to the gradient in γ (dγ/d*x*) by *Eq. 19*, where μ is fluid viscosity, and *t* is the thickness of the fluid layer.
^{2} − γ^{1})/(2*b*), where *b* is the length of the airway, and surface flux is *u* times the circumference of the tubes π·*d*, where *d* is the diameter of the airway. It follows that *k* is given by the following equation.
*d* (28, 5), and the viscosity of the fluid is 0.01 dyn·s·cm^{−2}. The value of *k* is inversely proportional to V_{o} because the change in γ for a given flux of *S* is inversely proportional to indigenous *S*. For TLC = 7 × 10^{3} cm^{3}, V_{o} = 7 × 10^{3}/2^{n} cm^{3}, where *n* is the generation number of the branches. For *generation 19* in the Weibel model of the airways, *b* = 0.1 cm, *d* = 0.05 cm, and for these values of the parameters on the right side of *Eq. 20*, *k* = 3 cm^{−1}·s^{−1}. Thus the branches are identified as terminal bronchioles, and the regions are identified as acini with a value of V_{o} of 1.4 × 10^{−2} cm^{3}. It should be noted that the geometry of the peripheral airways is variable, and that the value of *t* was obtained from one study of the liquid lining of peripheral airways in guinea pigs. For comparison, the value of *k* for generation 18 is 1.2 cm^{−1}·s^{−1}, and the value of *S*_{n} for this value of *k* is 0.04 liter^{−1}. If *t* were twice as big, *k* would be 2.4 cm^{−1}·s^{−1} in *generation 18*, and this would be the crucial generation.

The model described here shares a weakness with the convection-diffusion models in that both describe a particular bifurcation of the peripheral airways; they do not provide comprehensive models for the distribution of parenchymal properties for the lung. In addition, the parameter values that are listed in Table 1 have a particular correlation. For *compartment 1*, alveolar radius is smaller than the base case (*c*_{1} is larger) and the entrance ring is more compliant (*c*_{3} and *c*_{4} are smaller). For *compartment 2*, the relations are the opposite. If the values of *c*_{3} and *c*_{4} were switched between the compartments, and the values of *c*_{1} were unchanged, *compartment 1* would still lead pressure, but its specific compliance would be smaller than that of *compartment 2*. As a result, the concentration of N_{2} in *compartment 1* would be higher than that in *compartment 2*, and the predicted *S*_{III} would be negative. A correlation between alveolar size and entrance ring compliance is required to obtain a positive *S*_{III}.

For the parameter values shown in Table 1, the predictions of this model agree well with a number of observations. The mixing efficiency, 93%, is slightly higher than the 90% reported by Cumming and Guyatt (9). The predicted value of *S*_{n}, 0.068 liter^{−1}, agrees with the observed values (6–8). Because the mechanism is purely convective, equal slopes would be predicted for gases of different diffusivities. Meyer et al. (16) found no significant difference in the slopes for SF_{6} and He washouts in dogs, whereas Crawford et al. (8) found that, in humans, the value of *S*_{n} for SF_{6} is 40% higher than the values for He. The predicted value of *S*_{n} for the larger tidal volume of 1.5 liters, 0.046, agrees with observation (6). The predicted value of *S*_{n} for a 4-s breath hold at end inspiration, 0.044 liter^{−1}, is smaller than the value for no breath hold, but larger than the observed value of 0.025 liter^{−1} (7). The shape of the curve of *S*_{n} vs. breath number is like the observed shape. The values *S*_{n} for large breath number reported by Crawford et al. (8) vary widely among the six subjects. The predicted value of *S*_{n} for the 25th breath, 0.40 liter^{−1}, falls within the range of observed values, but is higher than the average value, 0.28 liter^{−1}.

The quantitative differences between predictions and observation involve gas diffusivity or maneuvers with extended times. This suggests that some diffusional equilibration is occurring. The relaxation time for diffusional equilibration between compartments fed by airways of the 19th generation (with no flow) is 7 s. Thus diffusional equilibration could lower *S*_{n} for He, lower *S*_{n} for breath hold, and decrease the asymptotic value of *S*_{n} for the multibreath washout.

In some ways, the proposed mechanism is similar to the original onion-skin explanation for the *S*_{III}. However, the mechanism responsible for the curvatures of the compartmental volume curves is different, and the scale of the compartments is quite different. It also differs in that compartmental volume trajectories are loops: the expiratory trajectory does not retrace the inspiratory trajectory. In addition, Marangoni flows are dissipative, and this mechanism may contribute to the viscoelastic properties of the lung. Furthermore, because of the phase difference between compartmental volume oscillations, the mechanism may contribute to aerosol dispersion.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: T.A.W. conception and design of research; T.A.W. prepared figures; T.A.W. drafted manuscript; T.A.W. edited and revised manuscript; T.A.W. approved final version of manuscript.

## Appendix

Approximate solutions to *Eqs. 14*, *15*, *16*, and *18* were obtained by the following numerical methods.

For breathing with a tidal volume of 1 liter, the driving pressure was expressed by the following equation, where *t* denotes time and τ denotes the period of a breath, 5 s.
*l*, *s*, and γ, for the two compartments were denoted *x*_{p}^{i}, where *p* denotes a particular variable, and *i* denotes the compartment number. Each variable was expressed by a truncated Fourier series.
*t* = *n*·τ/80, where *n* ranges from 0 to 80. The equations were then written in the form of *Eq. A3*, where *j* denotes a particular equation, LHS_{j}^{i}(*n*) denotes the left side of the equation at *t* = *n*·τ/80, and RHS_{j}^{i}(*n*) denotes the right side at that time.
*a*^{1} for the variable γ was set at 10 dyn/cm. Then the values of the remaining 39 parameters, *a*_{p}^{i}, *b*_{p}^{i}, ϕ_{p}^{i}, *c*_{p}^{i}, and ψ_{p}^{i} were obtained by finding the values that minimized the following objective *Q*.
_{j}^{i}(*n*) was 0.04.

The solution for breathing with a tidal volume of 1.5 liters was obtained by the same method with the values of the parameters describing P(*t*) and *a*^{1} for γ as given in the text.

For breath hold, the initial values of the variables were taken as the values of the variables at *t* = π·τ/2 for the solution for breathing with a tidal volume of 1 liter. Time was discretized to values of *n*/5, where *n* ranges from 1 to 20. The value of P for all times was set equal to the initial value, and the values of the remaining variables at the discrete times which minimized *Q* were determined. In calculating *Q*, the derivative in *Eq. 18* was approximated by using finite differences. For the subsequent expiration, the initial values of the variables were constrained to equal the values at the end of breath hold, and the values during expiration were found by the method used for breathing, but applied to the time interval, τ/4 < *t* < 3τ/4.

- Copyright © 2013 the American Physiological Society