INTRODUCTION

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Emphysema usually causes dysfunctional interaction between the chest wall and the lungs. The hyperinflated lungs keep the chest wall greatly expanded, limiting the capacity of inspiratory muscles to generate adequate inspiratory flow rates and tidal volumes (VT). At the same time, the limited ability of the chest wall to expand keeps the emphysematous lungs from being inflated to the supernormal volumes at which greater recoil pressures and higher expiratory flow rates could be achieved. This mutually disadvantageous interaction between lungs and chest wall can be seen as a problem of relative size, the lungs being too large for optimal function of the chest wall and the chest wall being too small for optimal function of the lungs. Stated simply, the lungs are functionally restricted by the chest wall.

Two new surgical therapies for end-stage emphysema acutely alter the relationship between the sizes of lungs and chest wall. Lung transplantation replaces one hyperinflated emphysematous lung with a relatively normal lung; lung reduction surgery reduces the size of one or both emphysematous lungs. Both therapies can increase lung elastic recoil and maximal expiratory flow rates, and reduce chest wall hyperinflation. To better understand interactions between the lungs and chest wall in emphysema, and to quantitatively predict the effects of lung transplantation and lung reduction surgery, we developed a computational model that incorporates respiratory muscle function, lung elastic recoil characteristics, and airway mechanics, including flow limitation. The model's parameters can be fitted to characteristics measured in individual patients, and the model then used to predict those patients' responses to surgical interventions.

	MODEL DESCRIPTION

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The model is designed to serve several purposes: (1) to show how lungs and chest wall interact under normal and pathological conditions, and after surgical interventions; (2) to simulate various respiratory maneuvers and their effects on spirometric variables; and (3) to predict postoperative function in individual patients. For the third purpose, the model incorporates parameters describing details of individual patients' mechanics, parameters that would not ordinarily be included in a heuristic model.

The model comprises two lungs enclosed in a chest wall and separated by a compliant mediastinum (Figure 1). The alveolar region of each lung is connected to the upper airway and mouth by a collapsible intrapulmonary airway segment. To simulate breathing, initial conditions are chosen, including initial values of dependent variables such as lung volumes (VL), and of independent variables such as respiratory muscle activation, and pressure or flow imposed by any external respiratory equipment. These values are used to calculate pressures (pleural pressure [Ppl], alveolar pressure [Palv], pressure at the airway opening [Pao], and pressure at the tracheal carina [Pcar]) according to empirical and physical equations describing respiratory system components. Pressure at the body surface is taken to be zero, and gas is considered incompressible. Flow rates into both lungs are then calculated from the appropriate pressure differences, and new VL are found by integration of flow. New values of independent variables are then specified, and the sequence of calculations is repeated to simulate pressure, volume, and flow as a function of time.

View larger version (26K): [in this window] [in a new window]   Figure 1.   Schematic diagram of the model's essential components. Pressures at specific locations are found by computation: Ppl, pleural pressure; Palv, alveolar pressure; Pcar, pressure at the tracheal carina; Pao, pressure at the airway opening (i.e., mouth pressure during spontaneous breathing or pressure at the external end of the endotracheal tube during mechanical ventilation).

	COMPONENT CHRACTERISTICS IN THE MODEL

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Lung Parenchyma

The lungs are characterized by elastic recoil pressure (PelL = Palv  Ppl), usually equal to the static transpulmonary pressure (PstL = Pao  Ppl), which varies at high lung volume (VL) according to the exponential description of Salazar and Knowles (1).

(1)

where parameters k, Vmax, and Vmin specify curvature, asymptotic maximal volume, and volume at which PstL = 0, respectively. At low VL, where this single exponential curve is nearly straight, patients' PstL-VL data sometimes show curvature attributed to small airway closure. To reproduce such curvature, thereby making simulations consistent with patients' respiratory function, another exponential term is added, specified by a volume of divergence from the single exponential (maximal closing volume, Vclmax) and a volume at which PstL is zero (closing volume, Vcl), as shown in Figure 2. In patients exhibiting such curvature at low VL, inclusion of this feature prevents simulations of a prolonged exhalation from reducing VL below the observed residual volume (RV).

View larger version (8K): [in this window] [in a new window]   Figure 2.   PstL and VL of a patient with emphysema, showing measured values (symbols) and the curve fit to the data. At the higher volumes, the data are approximated by the single exponential (1). Below Vclmax (maximal closing volume), the exponential curve is modified empirically to account for loss of PL, presumably owing to airway closure. Vcl is defined as the volume at which PstL is zero.

Pulmonary Airways

In the model, each pulmonary airway extending from alveolar space to the tracheal carina is characterized by a conductance and by the cross-sectional area at the end near the tracheal carina. Intrapulmonary airway conductance (the inverse of resistance, R) is an increasing function of VL. Furthermore, this functional dependence of conductance on VL is itself a function of age, older people having a reduced conductance over a broader range of volumes. Figure 3 shows the relationship in the model between intrapulmonary conductance and VL in subjects age 20 and 67, based on data published by Mead and coworkers (2).

View larger version (11K): [in this window] [in a new window]   Figure 3.   Relationship of pulmonary airway conductance to lung volume in normal subjects age 20 and 67, approximating data from Mead and coworkers (2). TLC = total lung capacity.

In each lung, inspiratory flow () is driven by the difference between pressure at the carina and alveolar pressure (Pcar  Palv). This driving pressure is equal to the sum of flow-resistive pressure loss caused by friction, and convective accelerative pressure recovery caused by deceleration of gas flowing from carina to alveoli. (Convective acceleration is most significant in normal subjects, in whom it modifies the shape of simulated maximal flow-volume curves.) Expiratory flow is governed by the same relationship except during forceful expiration. Then, whenever PelL is less than Palv  Pcar, the central ends of the intrapulmonary airways partially collapse, becoming flow-limiting segments or "choke points." Under these conditions, the effective driving pressure for expiratory flow is limited to PelL = Palv  Ppl, in agreement with the "equal pressure point" concept* of Mead and coworkers (2). Thus, for each lung in general,

(2)

where Ppl, Palv, R, and are variables specific to the right or left lung, and  is gas density. Brackets { } indicate that Pcar is used as the central pressure when Ppl < Pcar; otherwise, Ppl is the central pressure. "A" is the cross-sectional area of the uncollapsed flow-limiting regions in both pulmonary airways, equal to one-half the area of the trachea at the carina (Acar). In the model, the choke points are fixed in location.

Upper Airway

Flow in the trachea and upper airway is driven by the pressure difference between airway opening and carina, and is the sum of resistive, convective accelerative, and temporal accelerative (inertial) terms,

(3)

where R is upper airway resistance, I is the inertance of airway gas, and . is temporal volume acceleration. (Inertance is included to prevent abrupt changes in flow when effort changes abruptly, as during maximal voluntary ventilation [MVV] maneuvers.) The cross-sectional area is assumed to be very large at the airway opening, so the convective accelerative pressure depends on area at the carina (Acar) only.

Chest Wall

Pressure drop across the chest wall is the sum of passive elastic pressure and pressure resulting from respiratory muscle contraction (Pmus). The passive (relaxed) chest wall is characterized by an assumed pressure-volume relationship. Another empirical function relates Pmus to muscle activation (specified by its independent variable), thoracic gas volume [which affects maximal static pressure-volume characteristics of the chest wall (3)], and rate of change of thoracic volume [which affects maximal pressure-flow characteristics (4, 5)]. The inspiratory or expiratory Pmus is maximal at zero flow and decreases to zero when flow rates reach three vital capacities (VC) per second. To more closely simulate muscle action in abrupt maneuvers, the delay between a change in voluntary muscle activation (as would be inferred from the electromyogram) and the resulting change in Pmus is characterized by an exponential function with a time constant of 0.215 s, based on published data (5).

Mediastinum

Asymmetrical inflation of the two lungs causes a pressure difference between the two pleural spaces. In the model, this pressure difference is generated by a compliant mediastinum characterized by a linear pressure-displacement characteristic. Displacement is taken to be zero when left and right lungs are inflated equally in proportion to their size.

	PARAMETER ESTIMATION

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Patient-specific parameters in the equations above characterize the model components. Parameters can be chosen to simulate normal characteristics, derived from preoperative measurements in individual patients, or chosen to test a hypothesis. For example, to characterize elastic recoil of an individual's lungs, parameters are derived from static deflation pressure- volume curves. Where possible, the volume of each lung at total lung capacity (TLC) is determined separately by analysis of thoracic computed tomograms; otherwise, left and right lungs are assigned 47% and 53% of TLC, respectively. For transplanted lungs, normal characteristics are chosen, based on donor height, sex, and age.

Passive pressure-volume characteristics of the chest wall, which are usually too small to affect respiratory function significantly and are rarely measured in patients, are assumed to be normal as scaled to the measured TLC. Maximal static inspiratory muscle pressure-volume characteristics (inspiratory Pmus) are measured approximately in patients during maximal static inspiratory efforts at various VL. Maximal expiratory muscle characteristics (expiratory Pmus) are rarely so abnormal as to limit expiratory flow in emphysema, so they are assumed to have a normal dependence on VL and are scaled to reflect respiratory muscle strength, as inferred from the measured values of maximal static inspiratory pressure. The normal compliance of the mediastinum is not known. We estimate mediastinal compliance to be 1 L/cm H₂O, which would give a transmediastinal pressure of about 4 cm H₂O during total collapse of one lung.

The resistance of the pulmonary airways (upstream resistance in the equal pressure point formulation) can be determined from maximal expiratory flow and elastic recoil pressure by rearrangement of Equation 2, and is set so that simulated maximal expiratory flows at high VL are similar to measured ones. (In the measured maximal expiratory flow-volume curve [MEFVC], we ignore any brief high flow transients at the beginning of exhalation, which are usually attributed to collapse of central airways or to small, rapidly emptying parts of the lung. We set upstream resistance so that the simulated MEFVC matches the subsequent, more linear part of the patient's MEFVC, which characterizes the lungs' expiratory flow limitation.) Alternatively, resistance of the pulmonary airways, which accounts for most of the total airway resistance in patients, can be inferred from pulmonary resistance measured during inspiration. The cross-sectional areas of the intrapulmonary airways and of the trachea near the carina are set to 1 cm² and 2 cm², respectively, and the resistance and inertance of the upper airway are set to normal values of 2 cm H₂O · s/L and 0.01 cm H₂O · s²/L, respectively.

	SIMULATIONS

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Simulations in the model are specified via independent variables for inspiratory and expiratory muscle activation and/or pressure or flow at the mouth. Having derived parameters of the model from measurements in subjects performing particular respiratory maneuvers, we tested the model by simulating those same maneuvers, and comparing the data recovered from those simulated maneuvers with subjects' data. For example, we simulated a slow deflation from TLC to RV with periodic occlusion of the external airway (Figure 4) from which we recovered the static recoil pressure-volume relationship of the lungs, transpulmonary pressure (PL) being taken after it stopped changing during the occlusions. Another simulated maneuver generated maximal static inspiratory pressures at various lung volumes. We then compared the simulated characteristics with those measured in the patients. Figure 5 shows maximal static inspiratory pleural pressures and static recoil pressures of the lungs (PstL = Ppl) in a normal subject and a subject with emphysema. Data from simulated maneuvers agree closely with data from actual maneuvers in patients, indicating that the model functions as designed.

View larger version (18K): [in this window] [in a new window]   Figure 4.   Simulated VL and PL in a normal subject during a deep inhalation, followed by relaxed exhalation with periodic airway occlusions.

View larger version (21K): [in this window] [in a new window]   Figure 5.   Relationships between pressure and VL in a normal subject and a patient with emphysema, measurements and simulations. Pleural pressure (Ppl, which equals transthoracic pressure) during maximal static inspiratory efforts with the airway occluded is used for the maximal inspiratory pressure-volume characteristic of the chest wall. PL during periodic occlusions in a slow exhalation (as in Figure 4) is used for the static pressure-volume characteristic of the lung. The lines are fitted to model simulations for these subjects (not to the subjects' data).

Furthermore, the model can simulate relationships and phenomena not easily observed in patients. For example, the plot of driving pressure (Pao  Palv) versus flow () in the airways from a simulated panting maneuver at low VL (Figure 6A) reveals a loop in which expiratory flow initially increases, but then diminishes even as alveolar pressure continues to rise (between labels 2 and 3 in Figure 6A). In this simulation, the shape of the pressure-flow curve results from the volume- dependence of maximal expiratory flow, the progressive decrease in VL during expiration, and the temporal rise in Palv after the onset of flow limitation. In Figure 6A, the simulation resembles plots of plethysmograph ("box") pressure versus flow measured during the panting maneuvers performed for plethysmographic determination of airway resistance. In the plethysmograph, if the volume of gas exhaled and inhaled during the panting is small relative to thoracic gas volume, the "box" pressure can be used as an index of Palv analogous to that on the abscissa in Figure 6A. Figure 6B shows a plethysmographic pressure-flow loop from a normal subject panting at a low VL, where expiratory flow limitation produces distortion of the pressure-flow loop similar to that in Figure 6A. Such loops are characteristically seen in patients with obstructive airways disease.

View larger version (16K): [in this window] [in a new window]   Figure 6.   (A) Simulated pressure-expiratory flow relationship of the airways (Pao  Palv versus ) during a panting maneuver at low V L, demonstrating effects of flow limitation. After inhalation (1), exhalation begins along a linear pressure-flow characteristic until flow limitation begins (at 2). Expiratory flow progressively decreases as VL diminishes despite an increasing Palv, until (at 3) expiratory effort starts to diminish. Expiratory flow ceases to be maximal at 4. (B) Plethysmographic data showing effects of flow limitation in a normal subject during a panting maneuver at low VL. Plethysmographic ("box") pressure, which is approximately proportional to driving pressure (Pao  Palv) for flow analogous to that in A, is plotted against expiratory flow.

	PREDICTIONS FROM THE MODEL

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The model can be used to predict the effects of surgical interventions, pathological changes, or other alterations of lung or chest wall mechanical properties. Insofar as predictions are later proven to be accurate, the model can be accepted as valid. Differences between model predictions and postoperative function may indicate that important variables were omitted from the model or that some assumptions were not valid.

Unilateral Lung Transplantation

We used the model to predict the effects of unilateral lung transplantation in chronic obstructive pulmonary disease (COPD) by replacing parameters describing one of the patient's lungs with those describing a normal lung of the donor's size and age. In a simulated prolonged maximal inspiratory effort in a patient after transplantation (Figure 7), the transplanted lung initially inflates more rapidly than the remaining native lung, reaching its peak volume (at the dashed vertical line) and then actually emptying a bit into the native lung before TLC is achieved (at the continuous vertical line), exhibiting pendelluft. During the subsequent simulated forced exhalation, the transplant empties rapidly and remains at its minimal air volume as the native lung slowly deflates. Finally, during simulated MVV, there is progressive hyperinflation of the native lung and chest while peak volumes of the transplanted lung actually decrease slightly as the peak inspiratory pleural pressures become less negative. Despite the greater static compliance of the native lung, the transplanted lung has much greater VT because it has greater conductance, even at lower than normal recoil pressures and volumes.

View larger version (20K): [in this window] [in a new window]   Figure 7.   Simulated traces of individual and total VL in a patient after unilateral lung transplantation for emphysema during a maximal sustained inspiration, a maximal expiration, and an MVV maneuver.

When simulated data from the forced expiratory VC maneuver are replotted, the maximal expiratory flow-volume curve shows a change in slope near mid-VC as does the actual measurement (vertical dashed line, Figure 8). The initial period of high flow reflects the transplanted lung's normal characteristics, and the subsequent period of low flow reflects the characteristics of the native emphysematous lung. This pattern of two slopes is characteristic of some patients after transplantation.

View larger version (17K): [in this window] [in a new window]   Figure 8.   Measured (thin lines) and model-predicted (heavy lines) maximal expiratory flow-volume curves of the patient in Figure 7, before (pre-op) and after (post-op) unilateral lung transplantation. The vertical dashed line indicates the region of transition from the high expiratory flows of the transplant to the low flows of the native lung. Note that the measured postoperative TLC is lower than predicted.

Predictions of postoperative function depend on chest wall size as well as on the size of the transplanted lung. For example, for the patient shown in Figures 7 and 8, increasing the chest wall size in the postoperative simulation by 15% increased VC by 38% and increased VT during the MVV maneuver by 37%. By contrast, increasing the size (volume and airway conductance) of the transplanted lung by 30% (thus increasing total lung size by about 15%) increased simulated postoperative VC by only 1.4% and increased VT during the MVV maneuver by just 7%.

In this simulation, postoperative TLC is substantially greater than TLC actually measured in the patient postoperatively. Such a discrepancy could be explained by changes in chest wall function not initially recognized. Indeed, several patients showed postoperative changes in the maximal inspiratory pressure-volume characteristics of the chest wall, discussed subsequently, that could explain a reduction in TLC.

Lung Volume Reduction Surgery (LVRS)

To predict the effects of bilateral LVRS in COPD, we reduced volumetric values describing both lungs of emphysematous patients, and we reduced the conductance of intrapulmonary airways in direct proportion to amount of lung remaining to simulate a reduced number of airways in parallel. Two postoperative simulations for each subject were made. The first simulation ("prediction" in Figure 9) was based on the naïve assumptions that VL were reduced by exactly 20% and that chest wall function was not changed by the surgery. In retrospect, these assumptions were probably not valid. The other simulation ("postoperative simulation" in Figure 9) was made after changing the parameters in the model that characterized the fraction of VL reduction and inspiratory chest wall function, based on postoperative static pressure-volume measurements.

View larger version (27K): [in this window] [in a new window]   Figure 9.   (A) Pressure-volume curves for three patients with emphysema showing measured static elastic recoil pressures of the lungs (circles) and maximal static inspiratory pressures (triangles) versus VL before (open symbols) and after (solid symbols) LVRS. Simulations of preoperative characteristics (thin lines) fit the data well. Predictions of characteristics after surgery (dashed line), based on a standard 20% volume reduction and unchanged inspiratory chest wall function, can be made to fit postoperative data more closely by adjusting the volume of lung reduced and maximal inspiratory pressure-volume characteristics of the chest wall according to postoperative measurements (simulation post-op, bold lines). (B) Maximal expiratory flow-volume curves in the patients before and after surgical lung reduction. Predicted flow rates (dashed lines) fit the postoperative data reasonably well for Patient J but not for the others. Postoperative simulations (bold lines) could be made to reflect postoperative flow rates more by closely adjusting parameters for volume of lung removed and maximal inspiratory chest wall characteristics according to postoperative measurements, as above.

Figure 9 shows preoperative and postoperative data (symbols) and the corresponding model simulations (lines) for three patients, plotted as static pressure-volume plots (Figure 9A) and as maximal expiratory flow-volume plots (Figure 9B). The three model simulations shown on each plot are: a "preoperative simulation," a "prediction" based on preoperative measurements and a 20% surgical lung reduction, and a "postoperative simulation" based on postoperative measurements of TLC and maximal static inspiratory chest wall characteristics. Figure 9A shows that LVRS caused a reduction in TLC and an increase in pulmonary elastic recoil pressure at any given VL. These results were consistent with predictions (dashed line). In Patients D and P, chest wall characteristics also changed, with a downward shift in the maximal static inspiratory pressure-volume curve, reducing TLC in an apparent new chest wall restriction. Postoperative simulations in all patients could be made to fit data more closely by adjusting the values of input parameters of the model, including the volume of surgical lung reduction and the size and strength of the chest wall, to account for changes in measured characteristics (bold lines, Figure 9A).

Surgical lung reduction had varying effects on maximal expiratory flow rates (Figure 9B). The predictions (dashed lines), which assumed no change in chest wall characteristics, all showed increased flow rates throughout the VC (e.g., forced expiratory volume in one second, peak expiratory flow rate). However, when the postoperative changes in chest wall characteristics of Patients D and P were incorporated in the model, postoperative simulations in these patients showed little change in expiratory flow rates within the VC, consistent with the actual results. The improved accuracy of predicted expiratory flow rates brought about by refining the underlying assumptions of static lung and chest wall characteristics suggests that these variables may be important in determining postoperative function in such patients.

To explore the effects that surgical lung reduction would have in a normal individual, we simulated reducing the size of normal lungs in a normal chest. Here, a 30% surgical lung reduction reduced the simulated VC by 22% and reduced the simulated MVV by 21%, in contrast to predictions for emphysematous patients in whom lung reduction increases VC and VT. To explore the ventilatory effects of larger than normal lungs, as might be introduced into a small recipient by double lung transplantation from a larger donor, we increased the volume and conductance of normal lungs in a normal chest. Interestingly, a 30% lung augmentation actually reduced the VC by 21% and reduced the MVV by 27%, illustrating the importance of relative size of lung and chest wall to the predictions.

Lung Transplantation Followed by Surgical Lung Reduction

Some patients with emphysema who undergo single lung transplantation experience persistent underexpansion of the transplanted lung (6). We used model simulations to predict the effects of subsequent surgical reduction of the native emphysematous lung in such patients. Figure 10 shows the predicted change in the flow-volume curve of the patient shown in Figures 7 and 8 after a 30% reduction of the native lung. According to the model, greater expansion of the transplanted lung at full inspiration (not shown) resulted in greater initial expiratory flow (Figure 10, interrupted trace) which was contributed mostly by the transplanted lung.

View larger version (17K): [in this window] [in a new window]   Figure 10.   Predicted effects of a 30% surgical lung volume reduction (LVRS) of the native lung (interrupted line) in the patient with emphysema shown in Figure 8 after single lung transplantation. Parameters for the post-transplantation simulation (post-XP) were adjusted to reflect the maximal inspiratory pressure-volume characteristics of the chest wall measured after transplantation. Subsequent LVRS (dashed line) is predicted to increase expiratory flow rates markedly.

	DISCUSSION

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This model illustrates interactions of the lungs and chest wall under a variety of conditions, including dissimilarity of the two lungs. The model achieves a minimal criterion of validity in that it simulates certain features of the behavior of the individual respiratory systems on which its characteristics were based, reproducing patients' pressure-volume and flow-volume data in simulated maneuvers. To be useful and instructive, such a model should also (1) test the assumptions and physiological equations on which it is based, (2) reveal functional relationships that are difficult to measure in vivo, (3) suggest mechanisms to explain observed phenomena, or (4) predict the effects of experimental or therapeutic interventions in individual patients. The model shows promise in these areas.

Several of the simulations described illustrate the effects of functional restriction of the lungs, here defined as chronic underinflation of lung tissue caused by an inability to lower pleural pressure normally at full inspiration. Functional restriction can affect pairs of lungs, individual lungs, or portions of lungs. Its effects include lower maximal expiratory flow rates, greater airways resistance, and a lower VC than should occur with greater expansion of the lung. Functional restriction may be due to inspiratory muscle weakness or to chest wall conditions such as deformity or stiffening which reduce maximal chest wall size, but may also occur with a normal chest wall, for example, when normal but large lungs are transplanted into a normal but smaller chest. In emphysema, in which the primary defect is airway obstruction, dynamic hyperinflation creates a situation in which the chest wall is unable to lower pleural pressure sufficiently to expand the oversized lungs to supernormal volumes at which expiratory flow rates (and ventilatory capacities) would be more normal. In this sense, emphysema is a disease in which functional restriction of oversized lungs in a normal or abnormally large chest wall contributes to ventilatory dysfunction. After single lung transplantation for emphysema, functional restriction occurs when hyperinflation of the native lung overfills a chest wall that may be supernormal in its ability to expand (6). The model predicts that surgical reduction of the lungs in patients with emphysema, or of the native (emphysematous) lung in patients after transplantation, can reduce functional restriction and underexpansion of both lungs, improving ventilatory function.

The predicted effects of LVRS could have been anticipated from the assumptions of the model, whose maximal expiratory flow rates depend on elastic recoil pressure of the lungs. These predictions largely agree with those of two analytical models of LVRS published recently. Hoppin (12) explored effects of LVRS with a theoretical model of expiratory flow limitation in which maximal expiratory flow, as in our model, is assumed to depend on lung elastic recoil pressure. His model does not include interaction between lung and chest wall, so for purposes of computation, the lungs are inflated to the same PstL after LVRS. Consequently, his analysis shows maximal expiratory flow rates at a given fraction of the VC to be diminished after surgery that removes ventilated lung, and to be unchanged after surgery that removes nonventilated lung. Although Hoppin notes that flows would actually be increased after LVRS, in his model the principal salutary effect of lung reduction is caused by the decrease in absolute VL at which a given maximal expiratory flow can be generated, thus improving the function of the inspiratory muscles.

Fessler and Permutt (13) published a mathematical model of expiratory airflow limitation based on linearized pressure- volume and pressure-flow relationships of both lung and chest wall. Their model, which is based on the same assumptions and principles as our own, predicts an increase in maximal expiratory flow rates with surgical lung reduction in emphysematous lungs and a decrease in maximal expiratory flow rates after surgical reduction of normal lungs, as does our model (Figure 9). These investigators conclude, "It is the relative size of the lung to the size of the thorax that is of greater significance than the specific pathologic features of the lung, per se" (13).

The increased maximal expiratory flow rates predicted after LVRS in our model result from the greater elastic recoil pressure achieved by an unchanged chest wall operating on a smaller lung. In fact, however, the inspiratory function of the chest wall appears to be impaired in many patients after LVRS, and as a result, maximal VL, elastic recoil pressures, and expiratory flow rates achieved after LVRS are often less than those predicted by the model. In Patients D and P, who had apparent postoperative chest wall restriction, expiratory flow rates failed to improve after LVRS, a result that was reproduced when our postoperative simulations included postoperative changes in chest wall characteristics. Hoppin has suggested that for many such patients, benefit from LVRS may result from improved inspiratory muscle performance associated with lower lung and chest wall volumes. However, if new chest wall restriction results from LVRS, this mechanism of improvement may need reevaluation.

To simulate expiratory flow limitation, we chose the equal pressure point formulation of Mead, which is conceptually simple, and relies almost entirely on variables that can be measured in patients. For example, it is not necessary to know the branching structure of the airways or the location or mechanical properties of the choke points, as in more rigorous models of flow limitation (14, 15). The equal pressure point approach effectively precludes expiratory flow in a lung whose static elastic recoil pressure has fallen to zero. This constraint is problematic because some subjects appear to exhibit substantially negative PstL at VL above RV where, by definition, expiratory flow can still occur. To simulate breathing in such patients, our model requires the lung's pressure-volume relationship to be shifted to more positive values. Such a shift seems arbitrary, but may be excusable in light of the fact that we use a single value of esophageal pressure, which is subject to numerous artifacts, to calculate the elastic recoil pressure everywhere in an inhomogeneous lung exposed to regional differences in pleural pressure. Furthermore, there are other behaviors resulting from inhomogeneity of diseased lungs that are not reproduced in our simulations. Examples include the transiently high flow rates seen early in the maximal expiratory flow-volume curve attributed to central airway collapse (16), and the greater expiratory flow rates at a given volume during partial as compared with full forced expiratory maneuvers, attributed to nonuniform emptying (17). Many of these manifestations of inhomogeneity could be simulated in our model by assigning different mechanical properties and volumes to left and right lungs and making the mediastinal compliance infinite, thus creating a two-compartment inhomogeneous "lung."