Airway mechanics, gas exchange, and blood flow in a nonlinear model of the normal human lung

C. H. Liu, S. C. Niranjan, J. W. Clark Jr., K. Y. San, J. B. Zwischenberger, A. Bidani,


A model integrating airway/lung mechanics, pulmonary blood flow, and gas exchange for a normal human subject executing the forced vital capacity (FVC) maneuver is presented. It requires as input the intrapleural pressure measured during the maneuver. Selected model-generated output variables are compared against measured data (flow at the mouth, change in lung volume, and expired O2 and CO2concentrations at the mouth). A nonlinear parameter-estimation algorithm is employed to vary selected sensitive model parameters to obtain reasonable least squares fits to the data. This study indicates that 1) all three components of the respiratory model are necessary to characterize the FVC maneuver; 2) changes in pulmonary blood flow rate are associated with changes in alveolar and intrapleural pressures and affect gas exchange and the time course of expired gas concentrations; and 3) a collapsible midairway segment must be included to match airflow during a forced expiration. Model simulations suggest that the resistances to airflow offered by the collapsible segment and the small airways are significant throughout forced expiration; their combined effect is needed to adequately match the inspiratory and expiratory flow-volume loops. Despite the limitations of this lumped single-compartment model, a remarkable agreement with airflow and expired gas concentration measurements is obtained for normal subjects. Furthermore, the model provides insight into the important dynamic interactions between ventilation and perfusion during the FVC maneuver.

  • ventilation
  • perfusion
  • convective-diffusion transfer
  • parameter estimation
  • pulmonary function testing

human external respiration is a complex process consisting of at least three component parts: 1) ventilation via airways and lung mechanics; 2) perfusion of lung via the pulmonary circulation; and 3) gas exchange based on the transport of species across the alveolar-capillary barrier and the O2-CO2 binding properties of blood. Mathematical modeling to date has focused largely on the component parts, i.e., either exclusively on airway mechanics (14, 23, 25, 54), lung mechanics (18, 49, 50-52), gas exchange (22, 27, 34, 35), pulmonary circulation (8, 9, 26), occasionally on the linkage of any two components (17, 29, 43, 46, 56), but never on a treatment involving all elements collectively. This study attempts to describe the three constituent components concurrently, including the inherent coupling between them.

In an effort to characterize the dynamics of the forced vital capacity (FVC) maneuver in normal human subjects, a nonlinear one-compartment mathematical model of respiration combining airway/lung mechanics, pulmonary blood flow, and gas exchange is presented. Measured intrapleural pressure waveforms generated during the execution of the FVC maneuver were used as model input. The FVC maneuver was chosen as the appropriate driving function, since it involves the generation of full muscular effort covering the full range of admissible lung volumes. A nominal set of model parameter values is derived by using information from a variety of sources, including 1) our previous studies of airway mechanics (20, 40), 2) the pulmonary circulation report of Milnor (37), and 3) the pulmonary gas-transport model of Flummerfelt and Crandall (17). A nonlinear least squares estimation algorithm (Marquardt) was employed to adjust a sensitive subset of model parameters to achieve an acceptable fit to measured data. The ventilation and perfusion models are naturally coupled within the gas-transport model. Additional interactions between intrapleural and alveolar pressures and pulmonary blood volume occur during the FVC maneuver. Specifically, this affects the time course of the observed expired gas (O2 and CO2) concentration (see results).

This study aims to 1) describe a methodology for characterizing data collected during the performance of the FVC maneuver, and2) provide biophysically based explanations of the interactions between ventilation and perfusion and the concomitant effects on gas exchange. A theoretical basis for physiological interpretation of events occurring during the execution of an FVC maneuver is provided. A subset of output variables predicted by the model and compared against data includes changes in lung volume, airflow at the mouth, and the partial pressures of O2 and CO2 in the expired gas. The model also yields predictions of quantities not measured routinely, such as 1) alveolar pressure, 2) excursions in airway resistance and lung compliance, 3) gas composition in the airways, 4) blood perfusion rates, and 5) capillary blood volume variation. Direct measurement of these latter quantities cannot be obtained clinically without invasive procedures. The crucial role of component dynamics during the FVC maneuver is analyzed and discussed. Model-based sensitivity analysis reveals that parameters associated with all three of the forenamed respiratory components affect and influence the data. Feasibility and predictive capability are established by characterizing the data collected from four normal subjects.


Model Development

The choice of the specific model structure adopted was motivated by the requirements that the model 1) satisfactorily describe the dynamics of airway/lung mechanics over the full range of lung volumes from residual volume (RV) to total lung capacity (TLC) (therefore, a nonlinear description); 2) emulate flow-limiting behavior during forced expiration (hence, use of a resistive-compliant collapsible midairway segment); 3) simulate temporal profiles of expired gas concentration in normal subjects during the FVC maneuver; and 4) describe changes in gas exchange and perfusion rates. A schematic diagram of the complete model incorporating airway mechanics, gas exchange, and pulmonary circulation is depicted in Fig.1 A, along with an equivalent representation of the corresponding pneumatic and hydraulic subsystems in Fig. 1, B and C, respectively. The readers are referred to appendix for a complete description of the dynamic equations comprising the model.

Fig. 1.

Schematic representation of airway/lung mechanics, gas exchange, and pulmonary circulation system. Symbols are explained inGlossary. A: components of airway mechanics, pulmonary circulation, and gas exchange model. B: pneumatic representation of airway/lung mechanics and gas exchange. C: hydraulic representation of pulmonary circulation.


Compliance of alveolar compartment, l/cmH2O
Concentration of species i in the kthairway compartment, gram-mol/l
Compliance of collapsible airway segment, l/cmH2O
Compliance of lumped pulmonary capillary region, l/mmHg
Total content of gaseous species i in blood in compartmentj, ml i/ml blood
Mean pulmonary capillary compliance during passive breathing, l/mmHg
Lung diffusing capacity of species i, ml [stpd] ⋅ min−1 ⋅ mmHg−1
Linear resistance of upper airways, cmH2O ⋅ l−1 ⋅ s
Flow-dependent resistance of upper airways, cmH2O ⋅ l−2 ⋅ s2
Magnitude of Rc at Vc = Vcmax, cmH2O ⋅ l−1 ⋅ s
Length of pulmonary capillary, cm
Discretized number of segments in the capillary, 35
Total number of gaseous species considered in this study, 3
Partial pressure of species i in the alveolar space, Torr
Partial pressure of species i in the collapsible airway, Torr
Partial pressure of species i in the upper airway (dead space), Torr
Saturated partial pressure of species i in the ambient, Torr
Total pressure in alveolar region, cmH2O
Total pressure in collapsible region, cmH2O
Total pressure in rigid dead space region, cmH2O
Total pressure in the external ambient, cmH2O
Lung elastic recoil, cmH2O
Outer envelope of Pel during expiratory phase, cmH2O
Outer envelope of Pel during inspiratory phase, cmH2O
Pel at Va = V*, cmH2O
Partial pressure of water vapor, Torr
Intrapleural pressure, cmH2O
Equilibrium intrapleural pressure during tidal breathing, cmH2O
Minimum intrapleural pressure achieved during FVC maneuver, cmH2O
Maximum intrapleural pressure achieved during FVC maneuver, cmH2O
Constant characterizing arterial and venous resistance relation to effort (typically, greater than −Pplmin), cmH2O
Spatially averaged intrapleural pressure relative to reference, cmH2O
Standard pressure, 760 mmHg
Transmural pressure across collapsible airway, cmH2O
Transmural pressure across pulmonary capillary, cmH2O
Ptm at Vc = Vcmax, cmH2O
Total pressure in pulmonary artery (and arterioles), Torr
Partial pressure of species i in blood in segment j, Torr
Total pressure in pulmonary capillary, Torr
Total pressure in pulmonary veins (and venules), Torr
Transmural pulmonary arterial pressure relative to Ppl¯, Torr
Transmural pulmonary venous pressure relative to Ppl¯, Torr
Reference pressure at Tbody, Torr
Partial pressure of O2 in expired gas at mouth, Torr
Partial pressure of CO2 in expired gas at mouth, Torr
Airflow rate between the collapsible airway and alveolar space, l/s
Airflow rate between the dead space and collapsible airways, l/s
Airflow rate in the upper airways, l/s
Blood flow rate into pulmonary capillary, l/s
Blood flow rate out of pulmonary capillary, l/s
Collapsible airway resistance, cmH2O ⋅ l−1 ⋅ s
Rl, ti
Pulmonary tissue resistance, cmH2O ⋅ l−1 ⋅ s
Small airway resistance, cmH2O ⋅ l−1 ⋅ s
Parameter characterizing curvature of Rs
Rs at V*, cmH2O ⋅ l−1 ⋅ s
Rsc max
Rsc at the instant of Ppl = Pplmax(>0), cmH2O ⋅ l−1 ⋅ s
Magnitude of (Rs − Rsc) at minimal alveolar volume, cmH2O ⋅ l−1 ⋅ s
Upper airway resistance, cmH2O ⋅ l−1 ⋅ s
Pulmonary arterial (and arteriolar) resistance, mmHg ⋅ l−1 ⋅ s
Approximately mean Ra during passive breathing, mmHg ⋅ l−1 ⋅ s
Pulmonary capillary resistance, mmHg ⋅ l−1 ⋅ s
Magnitude of Rpc at Vpc = Vpcmax, mmHg ⋅ l−1 ⋅ s
Pulmonary venous (and venuolar) resistance, mmHg ⋅ l−1 ⋅ s
Approximately mean Rv during passive breathing, mmHg⋅ l−1 ⋅ s
Body temperature, 310 K
Ambient temperature, 298 K
Standard temperature, 273 K
Alveolar volume at end inspiration, assuming that Ppl¯ = Pplmin at all times during forced inspiration, liters
Airflow rate at the mouth detected by pneumotachometry, l/s
Alveolar volume, liters
Pulmonary capillary volume, liters
Maximum pulmonary capillary volume, liters
Collapsible airway volume, liters
Maximum collapsible airway volume, liters
Anatomic dead space volume, liters
Total lung volume (=Va + Vc + Vd), liters
Lung volume at which Rs increases abruptly during forced expiration, liters
Parameter characterizing volume dependence of Ra and Rv, cmH2O ⋅ l−5 ⋅ s
Mean molar-averaged axial velocity of blood flow in capillary segmentj, l/s
Rate of transfer of species i between blood and alveolar region, ml (stpd) i/min
Total rate of transfer of all species, ml (stpd)/min
Overall density of air in alveolar region, g/l
Overall density of air in collapsible airway region, g/l
Overall density of air in dead space region, g/l
Overall density of air in ambient under reference conditions, g/l
Scale factor used to create inspiratory Pel for each subject

Airway/lung mechanics model.

The general form is similar to that previously reported (20, 40). A brief review of the model with incorporated modifications is provided below.


The lung and airways were assumed to be enclosed within a rigid-walled thoracic cage, with the airways open to the atmosphere. The intrapleural space was assumed to be subject to a time-varying, spatially averaged driving intrapleural pressure [Ppl¯(t)], which was assumed to be equivalent to the average pressure in the pleural space acting on the lungs and produced by the muscles of respiration. Excursion in Ppl¯ was dictated by the effort generated by the subject.


Alveolar region (of volume Va) was assumed to exhibit nonlinear, time-varying viscoelastic behavior (18, 24, 51, 52). Static elastic behavior of the lung (Pel vs. Va) was described by a hysteretic pressure-volume (P-V) relationship (see appendix for details). The extent of hysteresis in Pel was presumed to be a function of breathing effort, which, in turn, was assumed to be proportional to Ppl¯ (reflecting muscular effort). Hence, the dependence of Pel on Ppl¯ served to define the well-known hysteretic path (31). Viscous dissipative characteristics exhibited by lung tissue (1, 18) were characterized by using a constant lung tissue resistance (Rl, ti).


Peripheral airways were characterized by a resistance (Rs) that was inversely proportional to Va (20, 40). Airway closure during forced expiration causes occlusion of these airways at low alveolar volumes (4, 6, 13, 39, 42). Because of the effect of large intrathoracic pressures generated during the effort-dependent portion of forced expiration, Rs was modified to be a function of both Va and Ppl¯.


Collapsible airway region (of volume Vc) has been characterized before in terms of a volume-dependent resistance and a volume-pressure relationship (Vc-Ptm) (20, 40). The functional importance of this collapsible segment has since been confirmed by Barbini et al. (2), who analyzed the input impedance spectrum vs. frequency and demonstrated that adequate reconstruction of pressure-flow data could not be achieved with a conventional single-compartment resistive-compliant model. Previous studies have demonstrated that in lumped models expiratory flow limitation during the FVC maneuver cannot be simulated without the presence of this collapsible segment (2, 40). Verbraak et al. (55) modeled the elastic properties of the compressible segment as a family of curves dependent on the lung elastic recoil. This more complex approach proved to be of little benefit in achieving good fits to subject data, and, hence, the original formulation was utilized in this work.


Upper airway region (of volume Vd) was assumed to be rigid, with its resistance to airflow characterized by a nonlinear, flow-dependent Rohrer resistor (23), as in Refs. 20 and 40.

Pulmonary circulation model.

The pulmonary capillaries were considered as a single tubular compartment of constant length of 0.05 cm (17) and a variable volume. The lumped pulmonary circulation model developed (Fig. 1 C ) was based on the following assumptions. 1) Upstream pulmonary arterial pressure (Ppa) and downstream pulmonary venous pressure (Ppv) were assumed to be constant at 15 and 5 Torr, respectively, referenced to intrapleural pressure (26, 58). 2) Pulmonary vascular resistance was partitioned into three components: a proximal, precapillary arteriolar resistance (Ra); a pulmonary capillary resistance (Rpc); and a distal, postcapillary venous resistance (Rv). Perivascular pressure was assumed to be intrapleural pressure for the proximal and distal (extra-alveolar vessels) but alveolar pressure for the capillary (intra-alveolar vessel). The proximal and distal resistances were assumed to be inversely proportional to Vabut proportional to the pleural pressure (15, 21), whereas the capillary resistance was presumed to be affected solely by alveolar pressure (37). Blood flow rate into and out of the capillary (Q˙bin and Q˙bout, respectively) was then governed by the nodal pressure drops (Pa, Ppc and Pv) developed across the corresponding vascular resistances. Consequently, capillary blood volume (Vpc) was modulated by the inequality between blood inflow and outflow and the transmural pressure across the lumped capillary wall.

Gas exchange model.

Gas exchange occurring in the constant-volume dead space and variable-volume collapsible and alveolar compartments was described by using species-conservation laws. On the air side of the exchanger, it was assumed that inspired air was instantaneously warmed to body temperature and fully saturated with water vapor. The gaseous mixture was presumed to obey the ideal gas law. On the blood side, the discrete constituents (plasma and erythrocytes) were lumped together and assumed to statistically behave as a uniform, homogeneous phase (3). Within a control volume, the instantaneous specific reactions were then considered to be at equilibrium; relationship between species content and their corresponding equilibrium partial pressures was consequently represented by empirical dissociation curves (12, 28, 48). One-dimensional axial convection provided the sole means for bulk transport of blood and movement of species along the pulmonary circulation; diffusion in the radial and axial directions was ignored. Two-phase flow created due to blood heterogeneity was further disregarded. Transport of gaseous species across the alveolar-capillary membrane, assumed to be solely by diffusion, was characterized by a lumped species lung diffusing capacity (Dl i), which accounted for the total diffusion-resistive path taken by species i(i = O2, CO2, N2) as it diffused across the alveolar-capillary barrier. O2 was taken up by the blood, and CO2 was excreted, whereas N2 (a relatively inert gas) diffused in either direction, depending on the instantaneous overall ventilation-perfusion ratio (39). The contribution of the physiological shunt (35) was neglected. The model used here was directly adapted from Flummerfelt and Crandall (17), with the provision that alveolar pressure was not held atmospheric but, rather, was calculated via the airway mechanics model.

Experimental Pulmonary Measurements

Measurements of airflow at the mouth, expired Pco 2 and Po 2 at the mouth, and esophageal pressure were made in four volunteer human male subjects in the Pulmonary Function Laboratory at John Sealy Hospital, Galveston, TX. A System 2800 Autobox Body Plethysmograph with associated pneumotachometer from SensorMedics (Dayton, OH) was used to perform the tests as well as to collect the data. A latex balloon was inserted through the subject’s nose and positioned in the esophagus (nasogastric), at a location where the largest pressure deflection could be observed. The balloon was then connected to a pressure transducer in the body box. Expired gas was sampled continuously at the mouthpiece and analyzed by a Datex Capnomac Ultima System to yield continuous measurements of CO2 and O2concentrations in the expirate. The CO2 and O2data exhibited time delay; their traces were manually synchronized to the recordings of the pressure and flow data to accommodate the resulting transportation lag. The esophageal pressure signal [assumed equivalent to intrapleural pressure (36)] was sampled at 50 Hz (i.e., sampling interval = 0.02 s), which was more than adequate to ensure the reproduction of the pressure signal from its samples (the maximum Nyquist sampling rate was calculated to be 40 Hz, based on the Fourier transforms of the flow data that had the highest frequency content of all the recorded waveforms). The functional residual capacity (FRC) was obtained by having the subject pant against a closed shutter. Analog recordings were digitally sampled by using a National Instruments NB_MIO-16x DAQ board and an AMUX-64T multiplexer board, controlled by using LabVIEW 4.0 software, all of which were connected to a Macintosh Quadra 800. LabVIEW virtual instruments were developed to 1) acquire continuous waveform data from multiple analog channels; 2) integrate airflow data to obtain instantaneous thoracic gas volume data; 3) continually display flow-volume plots; 4) calibrate (direct or volume referenced) input transducers; 5) apply a Butterworth filter to lightly smooth the data; and 6) accummulate data records in separate ASCII files as needed. For the FVC maneuver, the subject deflated the lung to close to RV and, without pausing, inflated fully to TLC. Again without pausing, the subject exhaled forcefully to RV until no airflow was detected at the mouth. The maneuver was completed with another forceful inspiration to TLC.

Each experimental episode was recorded after the subject rested adequately (for ∼5 min) and followed by several cycles of tidal breathing to ensure full recovery. The end-tidal gas composition was monitored to ensure that the CO2 level reached 39–40 Torr. When this level was achieved, it was assumed that a steady-state condition had been reached and that the mixed venous blood tension achieved constant nominal values consistent with those commonly reported (59). The duration of the recording episode was <1 min; hence, it was presumed that the mixed venous composition did not change significantly during this time. This seemingly reasonable modeling assumption does require experimental verification, however. Within the noninvasive constraints observed in the pulmonary function laboratory (except for the use of a nasogastric esophageal balloon), it is unlikely that such a measurement could be adopted easily. Four volunteer human subjects with normal lung function (i.e., no respiratory abnormalities) were recruited for this study. Their particulars are listed in Table 1.

View this table:
Table 1.

Physical parameters for volunteer subjects

Computational Aspects

A block diagram depicting the overall implementation is shown in Fig.2. Measured Ppl¯ associated with the FVC maneuver [first filtered by using a zero-phase shift, third-order Butterworth digital filter (41) to reduce cardiogenic artifacts] was used as the input to the model. Other information necessary to initialize the model included 1) analytic descriptions of P-V relationships associated with the collapsible airway segment, alveolar region, and the lumped pulmonary capillary; 2) the pressure-flow relationships that characterize resistances of the upper, collapsible, and small airways; pulmonary arterial; and capillary and venous resistances; 3) gas composition of inspired air; and 4) mixed venous blood-gas composition (assumed constant for reason explained in Experimental Pulmonary Measurements). Model implementation of the ensuing system of ordinary differential equations was done in the C programming language. Numerical integration of the differential equations was performed by using Epsode (5), with a tolerance of 10−4 s and a maximum time step size of 5 × 10−3 s. A subset of model output (lung volume variation, flow at the mouth, and expired gas concentration) was compared against the data obtained in the pulmonary function laboratory. A parameter-estimation algorithm was applied to adjust a selected set of sensitive parameters so as to achieve acceptable fits to data for a particular normal subject during the FVC maneuver.

Fig. 2.

Block diagram of simulation implementation. Intrapleural pressure is input to both the actual system and the mathematical model. Measured output variables are compared off-line against corresponding prediction. A nonlinear least square parameter-estimation algorithm is utilized to modify and estimate model parameter values to minimize discrepancy between measurements and the corresponding model predictions during forced vital capacity (FVC) maneuver. λ denotes the Levenburg adjustment parameter. See Glossary for other definitions.

Parameter Estimation

Values for the adjustable parameters were obtained by using an iterative nonlinear least-squares parameter-identification method, viz., Marquardt (30). A sequential process was adopted for parameter estimation. In the first stage, only flow at the mouth and lung volume were used as data to estimate parameters describing airway mechanics. The estimation was performed separately for the inspiratory and expiratory phases by using subsets of parameters in each phase. In the second stage, O2 and CO2 concentrations at the mouth were used as data to obtain estimates on parameters related to gas exchange and pulmonary circulation. During this time, the parameter estimates obtained from the first stage were held constant. This adjustment strategy was justified based on the observation that changes in pulmonary circulation model parameters did not affect the results achieved in tuning the airway/lung mechanics model. Further details on this aspect are furnished in appendix .

For practical reasons, it was necessary to have good nominal values for parameters to ensure convergence of the estimation algorithm. Initial simulations employing parameter values from previous studies (see introductory section) provided initial fits. Further manual adjustment yielded even better fits to the data, ultimately leading to a nominal set of model parameters that was used to initialize the Marquardt scheme (30). The adjustable parameters were chosen based on their known influence on portions of the maximum flow-volume curve associated with the FVC maneuver, as well as parameter variation checks performed in a separate study (not presented here), by using relative sensitivity coefficients to assess the sensitivity of flow and volume to these variations. The estimation algorithm was terminated when the maximum relative change in the adjustable parameters did not exceed 1% on subsequent iterative cycles.


Model predictions compared against data for a human subject performing an FVC maneuver are shown in Fig. 3. The last cycle of tidal breathing before the subject exhaled to RV prior to the onset of the FVC maneuver is also shown for reference. Note that the major features of the loop predicted by the model (depicted by solid lines in Fig. 3), such as peak inspiratory flow, initial expiratory upstroke slope, peak expiratory flow, and final expiratory slope, all agree reasonably well with the experimental data.

Fig. 3.

Vital capacity maneuver. Model predictions are denoted by solid lines, and the measured data are represented as dots. InBF, vertical dashed lines (from left toright, marked as e, i, e*, respectively) mark the transition to residual volume (RV), inspiration from RV to total lung capacity (TLC) with full effort, and forced expiration from TLC to RV during FVC. A: plot of maximal flow-volume loop for a subject. B: intrapleural pressure generated by the subject during FVC maneuver. C: alveolar pressure developed. D: flow at mouth. E: lung volume variation from RV. F: collapsible segment volume. SeeGlossary for other definitions.

Airway Mechanics

A phase-plane plot called the “maximal flow-volume” loop is constructed in Fig. 3 A. The dynamic description is restricted solely to the FVC portion of the maneuver. During the early inspiratory phase, Ppl¯(t) drops considerably lower than baseline values (Fig. 3 B ) and is transmitted across the alveolar wall creating subatmospheric alveolar pressure (Pa), as indicated in Fig. 3 C. The ensuing elevation in transairway pressure gradient (Pe − Pa) favors airflow into the lungs (Fig. 3 D ), causing their subsequent inflation (Fig.3 E ). As inspiration proceeds, however, Pareverts to equilibrium because of continued air filling (Fig.3 C ), thereby lowering the transairway pressure gradient and leading to a reduction in the flow at the mouth (Fig. 3 D ). During the early portion of the forced expiration, both Ppl¯ and Pa rise sharply to positive levels much greater than the normal baseline values (Fig. 3,B and C ). The reversal in direction and elevated magnitude of the transairway pressure gradient now causes maximal or peak expiratory airflow at the mouth (Fig. 3 D ), resulting in a rapid drop in lung volume from TLC (Fig. 3 E ). As expiratory effort continues, Ppl¯ and Pa remain positive, andV˙a o gradually approaches zero while lung volume declines to RV (Fig. 3, D and E ). The model was constrained to limit lung volumes to never fall below RV. The corresponding excursion in the volume of the collapsible segment Vc during FVC is shown in Fig. 3 F. It rises steeply during the inspiratory phase and falls rapidly to very low values as it experiences the full effect of positive transmural pressure during the prolonged forced expiratory period. At low alveolar volumes, high Rs causes the collapsible volume to inflate rapidly. Subsequent increase in Va increases peripheral airway patency, thereby lowering Rs. This facilitates outflow from the collapsible segment into the alveolar region, causing the momentary dip in Vc (Fig.3 F ) just after the onset of inspiration. This is termed as “serial pendelluft.”

Pulmonary Circulation

Nodal driving pressure drops (Pa − Ppc and Ppc − Pv) and the corresponding transnodal resistances dictate blood flow rates and capillary blood volume changes. The dynamics of circulation are easily explained by considering nodal pressures referenced to intrapleural pressure, namely, Ppc referenced to intrapleural pressure, i.e., Ppc′ ≡ Ppc − Ppl¯ (= Ptmb + Pel + Rl, tiV˙a o), whereas the new arterial and venous pressures referenced to intrapleural pressure (designated by Ppa and Ppv, respectively, and depicted as dotted lines in Fig. 4 A) are arbitrarily set at 15 and 5 Torr, respectively, for these calculations. Figure 4 Adepicts these modified nodal pressures referenced to Ppl¯ as well as the transmural pressure across the capillary wall, Ptmb. As the subject inspires from RV (i.e., i → e*), reduction in Ra and Rv due to alveolar inflation (thin lines, Fig. 4 B ) creates an increase in both inlet and outlet blood flow rates at the capillary (Fig.4 C ). The difference in inlet and outlet blood flow rates (Q˙bin and Q˙bout, respectively), caused by the disparity in (Ppa − Ppc′) and (Ppc′ − Ppv), respectively, results in a slight decrease in capillary blood volume Vpc. As inspiration proceeds, the rise in Pel and positive Rl, tiV˙a o (despite lower Ptmb) causes a net increase in Ppc′. The outflow flow rate exceeds the inlet flow rate, which causes a sharp drop in capillary blood volume (Fig. 4 D ) and a concomitant increase in capillary resistance Rpc (thick line, Fig. 4 B ). At this point, and as V˙a o → 0, the effect of Ptmb on Ppc′ dominates, and Ppc′ falls well into the early part of forced expiration (thin line, Fig. 4 A ). The minimum in Vpc actually occurs past the end of inspiration (Fig.4 D ).

Fig. 4.

Pulmonary circulation description during vital capacity maneuver. Vertical dashed lines in all panels are as defined in Fig. 3.A: nodal pressures (referenced to Formula and transmural pressure (Ptmb) across lumped capillary. B: pulmonary arterial (Ra), capillary (Rpc), and venous (Rv) resistances. C: inlet Formula and outlet Formula blood flow rates through capillary. D: capillary blood volume excursion. E: lumped capillary compliance. SeeGlossary for other symbol definitions.

In the early part of the expiratory phase (t ≥ e*), Ppc′ is low, which causes a greater inlet blood flow compared with outflow; the capillary refills quickly to recover its blood volume lost earlier. As expiration proceeds, however, decreasing Va increases Ra and Rv, which (despite lowered Rpc) lowers the blood flow rates. Inlet and outlet blood flow rates closely match one another, thereby minimizing variation in Vpc toward the end of the FVC maneuver. Capillary blood volume is also constrained to not exceed Vpcmax. The variation in the instantaneous capillary compliance (Cpc) resulting from the nonlinear (sigmoidal-like) shape of the Ptmb vs. Vpc curve is shown in Fig. 4 E. Also note the slight backflow in Q˙bin and Q˙bout in the brief instances when Ppc − Ppl¯ either exceeds Ppa (zone-1-like behavior) or is lower than Ppv (zone-3-like behavior), respectively, during the transition from inspiration to expiration.

Resistive and Compliant Properties

Figure 5 presents model-generated compliant and resistive properties of the lung and airways for subject 1during the FVC maneuver. Figure 5 A shows the hysteretic behavior associated with Pel, where the lower curve is traversed during inspiration and the upper curve during expiration. The subject’s collapsible airway compliance curve is shown in Fig. 5 B. As Ptm becomes negative during forced expiration, expiratory flow limitation occurs. Figure 5 C shows the lumped pulmonary capillary exhibiting similar qualitative compliant characteristics.

Fig. 5.

Description of lung and airway characteristics obtained from parameter estimation for subject 1. Dashed-dotted lines corresponds to the episode when subject expired to RV before executing FVC maneuver. Solid line corresponds to when subject executed the FVC maneuver from RV to TLC back to RV. A: static lung elastic recoil characteristic. B: compliance characteristic of the collapsible airway. C: pulmonary capillary compliance characteristic. Note the tapering at higher positive transmural pressures. D: excursion in small airway resistance. Note the difference in behavior during positive (forced expiration) and negative (inspiration) Formula efforts. E: resistance variation in collapsible airways. F: excursion in pulmonary capillary resistance. G: pulmonary arterial and venous resistances. Resistance offered by the capillary region is much greater than that offered by the extra-alveolar arterial and venous resistances. Note the hysteretic behavior exhibited by all the resistances. See Glossary for symbol definitions.

The effect of Va on Rs is shown in Fig. 5 D, where the lower curve is traced during inspiration and the upper curve during active expiration (with positive Ppl¯). The transition point in the Rs curve during expiration where the slope changes corresponds to a critical volume (Vcrit; assumed to be 70–80% of FVC), below which the caliber of the peripheral airways is considered to be sensitive to the surrounding positive intrapleural pressure during forced expiration (Fig. 5 D ). Incorporation of this property is purely a modeling construct, necessary to produce the strong concavity observed in the flow-volume loop following peak expiratory flow (e.g., see Figs. 3 A, 8 AC, and 10A ). Rc is similarly described by two curves, traversed differently on inspiration and expiration (Fig.5 E ). The model-predicted excursions in Rpc, Ra, and Rv shown in Fig. 5, F and G, agree qualitatively with trends reported in Ref. 37. Clearly, pulmonary vasculature is dominated by transmural effects due to changes in alveolar pressure and the capillary resistance during the FVC maneuver.

Isovolume Pressure-Flow (IVPF) Description

An IVPF curve can be constructed from flow-volume loop data corresponding to various levels of effort (4) and is often used to illustrate expiratory flow limitation. Figure6 depicts model-generated IVPF curve forsubject 1. Here, the subject’s maximum inspiratory input Ppl¯ (Fig. 3 B ) was scaled to achieve graded lung inflations from RV. Each inflation was followed by forceful expiration with full effort. In addition, with maximal lung inflation from full inspiratory effort, submaximal and supramaximal expiratory efforts were simulated by scaling the positive Ppl¯ record accordingly. Data pairs consisting of predicted airflow rate at the mouth and the corresponding Ppl¯ were separated based on lung volume. The cluster of doublets so obtained then referred to a fixed lung volume (within 1%). Figure 6 shows the results for four lung volumes (1, 2, 2.5, and 3 liters measured from RV; or 27, 54, 68, and 82% of vital capacity). At high lung volumes, a steady increase in expiratory airflow with increasing pleural pressure simulates the effort-dependent expiration characterized by high alveolar elastic recoil. At lower lung volumes, the curve flattens, suggesting a limitation of expiratory flow, regardless of the magnitude of the positive pleural pressure encountered (effort-independent region). Increased dynamic compression of the airways at higher pleural pressures increases peripheral airway resistance contributing to expiratory flow limitation.

Fig. 6.

Isovolume pressure-flow relationship evaluated for subject 1.Various levels of expiratory effort were simulated by scaling expiratory intrapleural pressure waveform. Symbols denote simulation results, whereas dashed line was manually traced. VC, vital capacity.

Effect of Perfusion on Gas Exchange

Figure 7 compares the temporal profile of expired Po 2 and Pco 2observed at the mouth (Pe O2 and Pe CO2, respectively) forsubject 1 against model predictions for the nominal case (solid line) and for the cases in which the blood flow rate is assumed constant (dashed lines) throughout the maneuver. To provide acceptable fits to the dynamic profiles, it was necessary to have higher blood flow rates during the early part of expiration and lower blood flow rates thereafter. Simulation results assuming fixed blood flow rates (of 1 and 5.4 l/min) are also shown in Fig. 7, A and B. Clearly, a better fit is obtained with a variable blood flow rate, particularly in the case of the expired CO2 profile. The relative sensitivity of the CO2 profile to changes in blood flow rates suggests that CO2 exchange is more perfusion dependent than is O2 exchange. Because a single alveolar compartment was employed herein, a change in blood flow rate in effect created a variation in ventilation-perfusion ratio during the course of the FVC maneuver.

Fig. 7.

Effect of changing perfusion rate on expired Po 2 and Pco 2 in expired gas at mouth during FVC maneuver for subject 1. The same Formula corresponding to the reference case (see Fig. 3 B ) was used for all cases presented here. A similar qualitative effect is observed for other subjects (results not shown). A: effect on expired O2. B: effect on expired CO2. See Glossary for symbol definitions.

Intersubject Variability

Figure 8 shows model-generated fits to the vital capacity maneuver performed by three additional subjects. The same value of Rl, ti (0.2 cmH2O ⋅ l−1 ⋅ s) assumed earlier for subject 1 was utilized for these calculations. Physical input parameters for all four subjects are provided in Table 1, with model parameters obtained from the parameter-estimation algorithm shown in Table2. There is some difference among the subjects in the actual parameter values obtained. Differences in vital capacity can be attributed in part to differences in the size of the subjects (38); hence, in Fig. 8, lung volumes are shown normalized to body surface area (BSA) instead (assumed to be proportional to the available surface area for gas exchange). Peak expiratory flow rates are comparable for all cases, and the normalized lung volumes lie in the range of 0.33–0.43 ml/cm2 BSA. Model predictions of the temporal profiles for O2 and CO2concentrations obtained in the expirate show good agreement with experimental data (second and third rows of Fig. 8). The final end-expiratory Po 2 and Pco 2 values obtained are comparable despite differences in the individual time histories.

Fig. 8.

Comparison of model prediction against data for the VC maneuver performed by 3 other volunteer subjects, provided for reference. Note that volumes are normalized with respect to body surface area (BSA).A: flow-volume loops. B: time course of expired Po 2. C: time course of expired Pco 2. See Glossary for symbol definitions.

View this table:
Table 2.

Model parameters for volunteer subjects

Model-generated spirogram indexes for all four subjects compared against data are depicted in Table 3, again indicating a reasonable agreement for all the subjects and further demonstrating the good fits achieved for the flow-volume loops in general.

View this table:
Table 3.

Comparison of spirogram indexes for the subjects

Component Resistances

The contributions of component resistances during forced expiration for each of the aforementioned subjects are depicted in Fig. 9,AD. In this case, the expired lung volumes are normalized with respect to vital capacity rather than BSA. In addition, input Ppl¯ traces are superimposed on the same plots to indicate the maximum expiratory effort generated by each subject. The general trend in these records indicates that Ppl¯ increased linearly with Vl during the initial 10–30% of lung volume during expiration; thereafter, it remains approximately constant, declining only during the last 20% of expiration. The maximum Ppl¯ maintained ranged between 90 and 150 cmH2O. In all cases, over the majority of the volume range, both Rc and Rs far exceed Ruaw (which lies close to the abscissa). At high lung volumes, Rc and Rs are small for all subjects and have comparable effects. The relative contribution of Rs diminishes at lower lung volumes as Rcbecomes much greater. Both increase, however, as lung volume decreases. Clearly, the behavior of the maximum expiratory flow-volume (MEFV) loop toward the end of the FVC maneuver is dominated by the resistances describing the peripheral and midairways (Rs and Rc, respectively). At high lung volumes, the Ruaw limits the peak expiratory flow rates.

Fig. 9.

Contribution of component resistances for the FVC maneuver for all volunteer subjects (AD ). Differences in type of effort generated are reflected in variation in Formula in the 4 subjects. Note that Rc dominates beyond peak expiratory flow, whereas effects of Rs are increasingly apparent only at low lung volumes. Upper airway resistance contributes only to peak expiratory flow. Vcritoccurs toward early part of the forced expiratory phase in all subjects. See Glossary for symbol definitions.

Sensitivity Analysis

The effects of variations in a sampling of the parameters (related to airway/lung mechanics and the pulmonary circulation) on a subset of the model predictions are discussed in this section. Additional calculations (not shown here) were performed to determine the sensitive parameters to be used as candidates in the parameter-estimation algorithm. The intrapleural pressure used as model input corresponds to that generated by subject 1 during the FVC maneuver (solid line in Fig. 3 B ) and is maintained the same for the simulation study described in this section.

In general, an increase in airway resistance tends to lower peak flow rates as well as impede airflow into and out of the alveolar compartment. The effects are more pronounced during expiration because of the greater magnitude of resistance encountered and are reflected in the expiratory portion of the flow-volume loop. Slower deflation of the lung assists in maintaining lower vascular resistances and increases perfusion, albeit to a very small extent. Because expiration is forceful in this maneuver, the contents of the alveolar compartment are quickly emptied out. Alveolar composition is not significantly affected, thereby resulting in no marked differences in O2and CO2 composition observed at the end of the FVC maneuver. This is unlike during tidal breathing when decrease in the upstroke slope leads to lower end-tidal CO2 composition. During forced expiration, these small differences are, in general, attenuated, and insignificant effects on expired-gas tracings are observed.

In contrast, alterations in alveolar compliance result in marked variation in resulting lung volume changes for the same intrapleural pressure variation. This causes marked changes in alveolar composition and is reflected in the final levels of gas composition observed in the expired gas. Alveolar composition is dictated by the extent of gas exchange occurring across the alveolar-capillary membrane and is mainly governed by the ratio of perfusion to ventilation. Changes in parameters describing pulmonary circulation cause alterations in perfusion rates which, in turn, modify gas-exchange rates, alter alveolar composition, and significantly affect the time course of the expired-gas composition.

A more detailed sensitivity analysis is conducted by altering the functional descriptors for the resistances and compliances. A summary of the qualitative effects of the individual component parameters and the resulting correlation between model parameters and property attributes of the physiological variables is provided in Table4. This is useful in eliciting mechanistic insight into resulting system behavior. Detailed illustrations for a sample subset of the parameters listed in Table 4 are depicted in Figs.10 and 11.

View this table:
Table 4.

Qualitative description of the effects of individual model parameters on functional dependencies for resistances and compliances

Fig. 10.

Effect of airway mechanics parameters during FVC. Parameters describingsubject 1 were used as baseline for all cases. For each scenario, only 1 of parameters was modified while all others were unchanged. Same driving intrapleural pressure (shown in Fig.3 B ) was used in all cases. A and C: effect on flow-volume loop. B and D: effect on Pco 2 at the mouth during forced expiration. SeeGlossary for symbol definitions.

Fig. 11.

Effect of modifying vascular parameters during FVC. Parameters describing subject 1 were used as baseline for all cases. For each scenario, only 1 of parameters was modified while all others were maintained unchanged. Same driving intrapleural pressure (shown in Fig.3 B ) was used in all cases. A and D: effect on inlet blood flow rate. B and E: effect on outlet blood flow rate. C and F: effect on Pco 2 at the mouth during forced expiration. SeeGlossary for symbol definitions.

Effect of airway mechanics parameters.


Obstructed small airways exaggerate the concavity of the MEFV curve. This is illustrated by adjusting a couple of model construct parameters describing Rs.

1) Effect of Rsc max. Concavity of the effort-dependent portion of the expiratory flow-volume loop can be reduced by increasing the patency of the small airways. This is equivalent to decreasing Rsc max in Eq.6A in appendix . The effect of reducing Rsc max to 50% of control is shown in Fig.10 A, where the resulting flow during expiration is less influenced by positive pleural pressure; hence, less concavity is exhibited in the expiratory flow loop. For reasons explained earlier, the expired CO2 profile is unaffected (Fig.10 B ).

2) Effect of Vcrit. The abrupt increase in small airway resistance due to reduction in its caliber during the effort-dependent portion of the FVC maneuver below which pleural pressure effects become evident was analyzed by using the nominal parameter, Vcrit (Eqs. 5A,a and 6A describing Rsc in appendix ). The value of Vcrit is assumed to vary among subjects. Delaying the onset of this switching (simulated by decreasing Vcritto 50% of control and shown by dashed-dotted line in Fig.10 A ) produces larger expiratory flow rates for the same lung volume until such time that airway closure becomes dominant. Profile of CO2 in the expirate is marginally affected.


An increase in alveolar compliance (simulated by lowering Pelmax; resulting Pel is only 50% of the control value) causes overinflation, thereby increasing vital capacity (dotted line in Fig. 10 C ). The maximum expiratory flow rate achieved is greater than that for the control case. Resulting dilution of alveolar contents consequently results in a lowered value for Pco 2 in the airways and is correspondingly reflected in the expired gas at the mouth (dotted line in Fig.10 D ). Buildup of CO2 in the expirate is lowered and approaches the final value at a different slope.


The influence of Rc extends throughout the FVC maneuver, as indicated in Fig. 9. An increase in Rc (simulated by doubling K3 during inspiration and expiration) tends to lower both the inspiratory and expiratory peak flows (dashed line in Fig. 10 C ). Once again, the time course of CO2concentration in the expired gas at the mouth is unaffected (dashed line in Fig. 10 D ).


An increase in Ruaw produces significant effects in both lung volumes and airflows. Because the Ruaw is also dependent on flow, the effects of increasing Ruaw (200% of control) are more pronounced, yielding much lower vital capacities and peak inspiratory and expiratory values (dashed-dotted line in Fig. 10 C ).

Effect of vascular parameters.

The effect of modifying selected parameters describing the pulmonary vasculature during the FVC is shown in Fig. 11. The flow-volume loop was not affected by the perturbation of the vascular parameters. The control case is depicted by the solid line in Fig. 11. A decrease in the vascular resistances [simulated by setting either Ra0(in Eq. 15A,c ), Rv0 (in Eq. 15A,d ), or Rpc0 (in Eq. 15A,e ) to 50% of baseline] increases the inlet (Fig. 11, A and D ) and outlet (Figs. 11,B and E ) blood flow rates. This causes a higher CO2 transfer to the alveolar space and results in higher values of end-expiratory Pe CO2(Fig. 11, C and F ). Because Rpc dominates vascular resistance, reduction of this resistance greatly affects blood flow rates. The coupling between the alveolar volume and extra-alveolar resistances at lower lung volume was investigated through variation of the nominal parameter Vφ (Eqs. 15A,c and 15A,d ). A reduction in Vφ to one-half of its nominal value effectively reduces Ra and Rv, thereby resulting in increased blood flow rates.

Regional parameter sensitivity.

The comparative effects of the sensitive model parameters can be localized to specific regions in the flow-volume loop and expired-gas concentration temporal profiles and are schematically depicted in Fig.12. Regions of the flow-volume loop influenced by the particular parameter during the inspiratory and expiratory phases are shown in Fig. 12 A. Rc has a dominant effect during most of the FVC maneuver, whereas Rs effects (via model parameters Rsm and Rsa) are more evident at lower lung volumes. The drop in airflow following expiration is mainly dictated by Vcrit and Rsc max (parameters that affect Rs). Ruaw strongly influences peak inspiratory and expiratory flow rates as well as the initial upstroke in forced expiration. Parameters describing compliance of the collapsible segment (Ptmmax and Vc max) and the alveolar region (ξ and V*) affect the inspiratory phase. Effects of parameteric changes on the flow-volume loop are also reflected in the expired-gas composition profile, as shown in Fig. 10, B and D.

Fig. 12.

Schematic representation of localization of contribution of model parameters to flow-volume loop and FVC capnogram (expired Pco 2 in expired gas at mouth during FVC maneuver). A: effect on flow-volume loop. B: effect on expired CO2 waveform at mouth. See Glossary for symbol definitions.

Figure 12 B shows the model parameters that significantly affect the FVC capnogram (Pe CO2). The initial upstroke in CO2 tension in the expirate remains unaffected. The initial peak attained is affected by pulmonary capillary compliance (Vpcmax,C̅pc) and resistance (Rpc0). The ramplike increase in the temporal profile is influenced by the arterial and venous resistive parameters (Ra0, Rv0, Vφ, and Pφ). Note, however, that end-expiratory compositions so obtained depend on the cumulative effect of all the parameters.


To develop a mathematical model that emulates the functional behavior of the respiratory system, it is essential to characterize the airways and lung and alveolar-capillary gas transport. The lumped model presented consists of nonlinear resistive-compliant airway and alveolar compartments interacting with pulmonary vascular compartments. Measured pleural pressure was used to drive the model, and a nonlinear parameter-estimation scheme was employed to identify model parameters that yielded good agreement between model predictions and experimental data. The FVC maneuver was chosen to illustrate the excursion over the full range of permissible lung volumes. After the system under the vital capacity maneuver has been identified, it should be possible to predict its behavior during other breathing maneuvers, i.e., tidal breathing and panting, holding the parameter set unchanged (not shown here).

Airway/Lung Mechanics

The Ppl¯ waveform measured during the FVC maneuver differed among subjects but was characterized by a sharp transition between initial maximal inspiratory and expiratory efforts, followed by a prolonged positive offset beyond the point when peak expiratory flow was achieved. The curve showing Ppl¯ as a function of lung volume (Fig.9, AD ) clearly demonstrates that the flow work (Ppl¯dVL) developed by individual subjects differed during the forced expiratory period. To simulate zero flow at the mouth toward the end of the FVC maneuver, it was necessary to assume high airway resistance at low lung volumes (Figs. 5, D and E and9 AD ), in effect simulating progressive airway closure toward end expiration (32, 33, 39). An increase in airway resistance at low lung volumes via elevation of Rs alone can also produce this effect on the flow-volume loop (13, 14), but it creates significant discrepancies between model predictions and experimentally measured data corresponding to the time course of Po 2 and Pco 2 values observed in the expired gas. Large values of Rs did not permit efficient transport of gases from the alveolar region to the mouth, and underpredicted Pco 2 and overpredicted Po 2 values in the expired gas (at end expiration). It was, therefore, necessary to allocate the resistance changes at low volumes to both Rs and Rc, to achieve reasonable fits to all aspects of the data. Figure 9 again demonstrates the importance of the collapsible segment during the FVC maneuver. Simulation results presented here suggest that the contribution of Rc during the latter part of forced expiration is greater than any of the other component airway resistances.

To obtain concavity of the flow-volume loop past peak expiratory flow, it was necessary to incorporate two parameters, Vcrit and Rsc max, which describe the abrupt increase in Rs due to the effects of positive Ppl¯ (Fig. 5 D ). Vcrit corresponds to a critical lung volume below which this effect is apparent. The increase in magnitude in Rs when lung volume equals Vcrit is then characterized by Rsc max (Eq. 6A ).

Pulmonary Circulation

In this work, it was found necessary to include a time-varying description of the pulmonary blood flow rate to obtain gas-exchange predictions that were consistent with experimental observations on the time course of Pe O2 and Pe CO2. The model presented includes the externally imposed effects of alveolar volume and alveolar and intrapleural pressures. The effects on the extra-alveolar and intra-alveolar volumes are dissimilar. The model is capable of simulating capillary recruitment-derecruitment in terms of changing capillary blood volume for the breathing maneuver considered (Fig.4 D ). The prediction is qualitatively consistent with observations reported in the literature during tidal breathing in terms of the direction of changes in blood volume (37). The dynamics of changes in pulmonary blood flow rate determine O2 influx and CO2 excretion.

Gas Exchange

An adaptive pulmonary circulation model has also been included in addition to an air-side model to describe gas exchange at the alveolar-capillary membrane. Dynamic interaction of air-side variables with the pulmonary circulation variables clearly affects capillary gas exchange. For instance, blood flow rate through the pulmonary capillary is governed by the pulmonary vasculature resistances, which, in turn, depend on alveolar volume and pleural pressure. Hence, they exhibit variations throughout the phase of the respiratory cycle that affects the residence time of blood in the capillary bed. Changes in forward and retrograde blood flow produced during forced inspiration are responsible for the nature of the early profile of Pco 2 in the expired gas. Use of constant pulmonary capillary blood flow rate did not match the early phase of the expired CO2 time course (Fig. 7). This coupled aspect of pulmonary airway/lung mechanics, circulation, and gas exchange has not been included in previously reported mathematical models.

Model Limitations

Although the model presented in this work is quite detailed, like all mathematical models, it must be considered in the context of known limitations.

1. The performance of the model in tracking expiratory behavior at submaximal inspiratory effort as well as at low lung volumes (near RV) is less than satisfactory (Fig. 3, A andD ). Improvements in better emulation of airway closure are warranted.

2. Distributed-parameter models based on morphometric representations of the airways (57) have been employed to simulate expiratory flow limitation by using wave-speed mechanisms (10, 25, 53). Wave phenomena cannot be addressed by using lumped models. Herein flow limitation is attributed solely to viscous dissipation.

3. In general, subjects with respiratory system abnormalities often tend to have regional dysfunction (39). These regional differences can hinder gas-transport efficiency in a nonuniform, heterogeneous fashion, which suggests that the one-compartment approach applied here to normal subjects would not be adequate for characterizing patients with airway disease. The model has obvious limitations in characterizing diseased subjects. To simulate airway disease, it would be necessary to partition the alveolar compartment into multiple subcompartments with variable regional ventilation and perfusion. This would significantly increase model complexity, and validation of the expanded model would require additional measurements on regional ventilation and perfusion. Because conventional clinical pulmonary function laboratories are often limited to minimally invasive procedures, a trade-off exists between model complexity and the number and types of pulmonary variables that can be monitored with conventional measurement techniques. It has been suggested in the literature that ventilation inhomogeneity associated with lung disease can be examined by incorporating an insoluble gas, e.g., Ar, to help distinguish the effects of nonuniform ventilation and capillary gas exchange (45). This approach, which reportedly does not require the use of additional compartments, needs to be further explored. A point worth investigating, however, is the degree to which reduced, lower order models (including this single-lung-compartment model with altered characteristics) are capable of capturing the essential dynamic features of the complete system. These lower order models, although less accurate, require fewer measurements for model validation and yet provide useful insights into the potential interaction between ventilation and pulmonary blood flow.

4. The sloping capnogram (shown in Fig. 7) was characterized by incorporating time-varying blood flow rate. The effect of including multiple compartments with varied ventilation and perfusion levels (as mentioned above) to generate the desired time-course description of the expired gases was beyond the scope of the present study.

5. Airway inertance is assumed to be negligible at normal respiratory frequencies (39). Ignoring inertial effects may not be leading to accurate results during the early part of forced expiration, which is characterized by high airflows. Depending on the specific application, an inertance might need to be incorporated between the upper airway resistance and the collapsible segment in Fig. 1 B. Inertance would be mandatory for analysis of high-frequency ventilation (11).

6. Lung tissue resistance, Rl, ti, assumed to be constant for all the subjects during the FVC maneuver, ignores stress relaxation (24, 51) in lung tissue.

7. The static lung relaxation curve is dependent on breathing frequency and history (4, 39). As breathing frequency increases, one would expect the relaxation curve to flatten, resulting in a stiffer (lower compliance) characterization of the lung. This aspect is not considered in the present model formulation.

8. The lumped description of the pulmonary circulation presented here is oversimplified. Differences in regional perfusion due to the gravitationally induced changes in hydrostatic pressure [zones 1, 2, and 3 (39, 59)] are not considered. The use of constant-pressure sources to describe the pulmonary arterial and pulmonary venous pressures ignores the dynamic aspect of the pulmonary circulation (the result of right ventricular and left atrial pumping) and the compliant properties of the extra-alveolar vessels. These aspects were neglected because of our present limitation regarding measurement of pulmonary hemodynamic data via indirect means. Obtaining measurements of blood flow during respiratory maneuvers (e.g., utilizing ultrasonic methods) would greatly enhance the ability to identify the appropriate structure for an adequate pulmonary circulation model and the values of associated parameters. Modulation of the pulmonary vascular resistance by the O2 tension in blood (37, 39) has also been neglected.

9. The open-loop model as formulated is not driven by the metabolic demands of the tissues. The metabolic requirements of O2 and CO2 by the tissues cause a corresponding change in the mixed venous gas tensions entering the pulmonary circulation, especially when long breathing episodes are considered as part of any breathing maneuver. The ramifications of artificial panting could be quite different physiologically from those caused by demand panting.

10. During the experimental protocol, the volunteer subjects were allowed to rest between episodes while the capnogram recovered to nominal values. Nevertheless, it is quite possible that during the experiment mixed venous tensions varied from the constant values assumed here. Additional noninvasive measurements of blood gases would yield important new information, and prove useful in providing better model-based assessment of blood-gas concentrations in the pulmonary capillary and gas transport across the alveolar capillary membrane.


Despite these stated limitations, the lumped nonlinear one-compartment model of airway/lung mechanics, pulmonary circulation, and gas exchange presented in this study satisfactorily describes the dynamics of the FVC maneuver in normal human subjects. The study also demonstrated the feasibility of employing parameter-identification techniques to match experimental data obtained noninvasively in the pulmonary function laboratory. The model serves as a template for future development of other single- and multiple-compartment models that describe abnormalities in pulmonary function. It also serves as a framework to investigate cardiopulmonary interactions. Results indicate that an accurate characterization of the interaction between ventilation and perfusion is essential to achieve satisfactory match to expired O2 and CO2 concentration waveforms when using a single-compartment model. As a result of this interaction, the model predicts a steep decline in transient blood flow rates during the execution of the FVC maneuver; this could have clinical consequences under pathophysiological abnormalities. However, such putative changes in pulmonary blood flow rate during the FVC maneuver need to be verified experimentally.

To characterize the inspiratory and expiratory flow-volume loops, it was necessary to incorporate a hysteretic description of airway resistances. Airway resistance values encountered during expiration exceeded those calculated during inspiration for the same compartmental volume. Specifically, during expiration, significant elevation in both the collapsible midairway and small airway resistances was necessary to adequately characterize the shape of the expiratory flow-volume loop and time course of the FVC capnogram.

Although it provides a quantitative and theoretical basis for physiological interpretation, the model, in its present form, is not expected to be utilized clinically. Even though the model has limitations, nevertheless, it provides useful insight in assessing the inherent coupling between airway mechanics and pulmonary blood flow and the resulting effects on the output variables characterizing gas exchange.


The support and encouragement of Dr. L. C. Sheppard is highly appreciated. The authors also thank Dr. S. T. Kuna, Pulmonary Division, Dept. of Internal Medicine at the University of Texas Medical Branch, Galveston, TX, for the use of his laboratory facilities and Dr. F. Ghorbel and Athanasios Athanasiades, Dept. of Mechanical Engineering, Rice University, Houston, TX, for assistance and useful discussions.


  • Address for reprint requests: J. W. Clark, Jr., Dept. of Electrical and Computer Engineering, Rice University, Houston, TX 77251.

  • Financial support for this work, provided by the Biomedical Engineering Center at the University of Texas Medical Branch, Galveston, TX, is gratefully acknowledged. Additional financial support, provided to A. Bidani by the Moody Foundation, Galveston, TX (no. 94-48) and to J. B. Zwischenberger and A. Bidani by The Shriner Hospital for Crippled Children, Galveston, TX (no. 15859) is also deeply appreciated.


Modeling equations were developed by employing macroscopic balances on the overall mass, overall linear momentum, and the species mass by using a control volume approach. Air is assumed to behave ideally with its overall mass density (ρ) being defined by a constitutive equation of state corresponding to the ideal gas law (P = ρRT/M), where T is absolute temperature, M is the molecular weight of air, andR is the universal gas constant. The molecular weight of air is assumed to be invariant to changes in composition in the ensuing formulation. Blood is considered to be incompressible and behave as a homogeneous one-phase mixture.


Air side.


The total pressures in the alveolar and collapsible regions at any time, denoted by Pa and Pc, respectively, in Fig. 1, are described by the balance of applied and developed pressures across their respective wall boundaries [D’Alembert principle, (31)] and are written asPA=Pel+RL,tidVAdt+Ppl¯+Pref Equation 1A,a PC=Ptm+Ppl¯+Pref Equation 1A,bwhere Pref is the reference pressure at 37°C.

The minimum and maximum values of Ppl¯ achieved during the vital capacity maneuver with full effort are denoted by Pplmin and Pplmax, respectively. The static P-V relationship for the alveolar compartment, Pel(Va, Ppl¯) , is characterized by two curves, PelE and PelI, which delineate the outer boundaries of the Pel envelope. The PelI and PelE curves illustrated in Fig.13 A are generated by assuming that Ppl¯ was held constant at Pplmin (marked X # → Xin Fig. 13 B ) and Pplmax (markedY → Y # in Fig. 13 B ) throughout the inspiratory and expiratory phases, respectively. The mean of these two curves refers to an equilibrium curve corresponding to the case where Ppl¯ = Pplmean (= −3 to −5 cmH2O typically obtained at end-tidal conditions). The recoil pressure Pel is assumed to be graded according to effort; hence Pel is computed via linear interpolation based on the actual Ppl¯ recorded. Interpolation is performed by grading between the equilibrium curve and the PelI curve for the inspiratory phase, whereas the equilibrium curve and the PelE curve are used for the expiratoryphase. The resulting trace is a hysteretic loop contained within the envelope defined by PelI and PelE. This, in effect, defines the actual P-V relationship generated for a given subject specific to the Ppl¯ recorded while executing the FVC maneuver. The resulting set of equations is then given byna=V*RV+0.1V*RV+0.0010.99 Equation 2A,a nb=V*RV+0.1VARV+0.0010.99 Equation 2A,b nc=V*RV+0.10.0010.99 Equation 2A,c nd=Pelmax+25lnncna Equation 2A,d PelE=PelmaxVARV+0.001V*RV+0.0013 Equation 2A,e PelI=(ξPelE)Pelmax+ndlnnanbξ+1 Equation 2A,f ne=0.5PplmeanPplmin [PplmeanPpl¯(t)]for inspiration Equation 2A,g ne=0.5PplmeanPplmax [PplmeanPpl¯(t)]for expiration Equation 2A,h Pel=(0.5+ne)PelE +(0.5ne)PelIfor expiration Equation 2A,i Pel=(0.5ne)PelE+(0.5+ne) PelIfor inspiration Equation 2A,jwhere Pelmax (>0) is the magnitude of the minimum Ppl¯ obtained during maximal inspiratory effort; and V* and ξ are parameters describing Pel and are determined for each subject through the parameter-estimation algorithm.

Fig. 13.

Generation of static recoil pressure tracing for a given subject.A: reference curves by using constant Pplmin and constant Pplmax defined the outer Pel envelope (solid lines in A and dashed-dotted lineX #XYY # in B ). Dotted lines correspond to recoil pressure at graded levels (illustrated in B ) of Formula Actual Pel tracing corresponding to subject 1 during FVC is denoted by ⋄ (PelI and PelE). B: actual intrapleural pressure generated by subject 1 during FVC maneuver shown by solid line. Dotted lines are isobars and depict graded levels of set Formula used during inspiratory and expiratory phases to generate corresponding traces within the Pel envelope shown in A.

The transmural pressure across the collapsible compartment, Ptm, is expressed as a function of Vc (40) and is given aslbptm=(Ptmmax5.6)6.908 Equation 3A,a Ptm=saptmsbptmVCVCmax0.72 ifVCVCmax0.5 Equation 3A,b Ptm=5.6lbptmlnVCmaxVC0.999 ifVCVCmax>0.5 Equation 3A,cwhere Vc max again is a parameter determined for each subject using the estimation algorithm; andsaptm and sbptm are constants determined by forcing continuity and differentiability of Eq. 3A,b and 3A,c at Vc/Vc max = 0.5.


All flows are evaluated at body temperature (Tbody). Equations describing the flows beteen the collapsible and alveolar regions (Q˙ca), between the dead space and the collapsible airway (Q˙dc), and in the upper airway (Q˙ed), are derived from the balance on overall linear momentum in each region. Ignoring the inertial contributions, the expressions for flow are given asQ˙CA=PCPARs Equation 4A,a Q˙DC=PDPCRC Equation 4A,b Q˙ED=PrefPDRuaw Equation 4A,c

The small airway resistance Rs has different values for the two respiratory phases. The value for Rs during expiration is greater than that during inspiration for the same lung volume because of the effects of airway closure during forced expiration. The expressions for Rc and Ruaw are similar to those listed elsewhere (40, 44). The various airway resistances are thereby given asRs=RsmexpRsaVARVV*RV+Rsc Equation 5A,a RC=K3VCmaxVC2 Equation 5A,b Ruaw=K1+K2Q˙ED Equation 5A,cNote that Rsc is held constant at 0.02 when Ppl¯ is negative. However, when Ppl¯ exceeds zero (e.g., during forced expiration) and when Vl is below a critical volume [Vcrit, ∼70–80% of vital capacity (VC)], the resistance to airflow offered by the small airways is assumed to be a function of effort resulting in increased Rs. This aspect is modeled by defining Rsc to be effort dependent according toRsc=Rscmax0.02Pplmax(Ppl¯Pplmax)+Rscmax Equation 6AAirflow detected by the flow transducer located within the mouthpiece is designated asV˙a o. Model-generated flow at the mouth is calculated according toV˙Ao=Q˙EDTamTbody(PrefPH2OTbody)Preffor inspiration V˙Ao=Q˙EDfor expiration Equation 7A


Invoking overall continuity (conservation of total mass) in the two compliant compartments, the dynamic expressions for the rate of change of volume in the compartments may be written as

During inspirationdVCdt=ρDρCQ˙DCQ˙CAVCρCdρCdt Equation 8A dVAdt=ρCρAQ˙CAVAρAdρAdtPSTbodyTSPAΦtot* Equation 9A

During expirationdVCdt=Q˙DCρAρCQ˙CAVCρCdρCdt Equation 10A dVAdt=Q˙CAVAρAdρAdtPSTbodyTSPAΦtot* Equation 11Awhere Φtot* is the total gas exchange of all species considered from the alveolar region to the blood in the capillary across the alveolar-capillary barrier, and is given asΦtot*=i=1Ntotj=1NsegDLi(PAiPbi(j))ΔVpc(j)Vpc=i=1Ntotφi Equation 12AThe inner summation represents the total transfer rate (stpd) of species i across the capillary wall, φi, and is evaluated by summing the transfer across each of the discretized capillary segments j. ΔVpc( j ) denotes the volume of thejth capillary segment. Note that Eqs. 9A and 11A when expanded result in a second-order ordinary differential equation and can be alternately expressed as two first-order ordinary differential equations using state companion form (19).

Application of overall continuity in the rigid dead-space region results in a dynamic expression for the total density in the dead space and is given bydρDdt=ρDVDρrefρDQ˙EDQ˙DCfor inspiration Equation 13A dρDdt=ρDVDQ˙EDρCρDQ˙DCfor expiration Equation 14A

In this paper, the variation of the total air density throughout the pulmonary pathways is assumed to be negligible. Hence, the time derivatives of the density terms are set to zero in Eqs.8-14, and the ratio of density terms are set to one inEqs. 8-11 and 14A. For ease of implementation, the density terms are replaced by the corresponding pressure terms by substituting for the constitutive equation of state (ideal gas law).

Blood side.

The pulmonary capillary is modeled as a lumped compliant element the volume of which varies in time and is governed by pleural and alveolar pressure changes. Resistance to blood flow through the pulmonary vasculature is partitioned into 1) extra-alveolar resistances of the pulmonary arteries and veins modulated by the pleural pressure and 2) intra-alveolar resistance of the capillaries modulated by the alveolar pressure. The mean pulmonary arterial and venous hydrostatic pressures (Ppa and Ppv) are assumed to be constant and are denoted by constant-pressure sources of 15 and 5 Torr relative to the intrathoracic pleural pressure, respectively. Pulsatile effects resulting from the pumping of the right ventricle and left atrium during the cardiac cycle are hereby ignored. The corresponding dynamic equations are given asdVpcdt=(PaPpc)Ra+Rpc/2(PpcPv)Rv+Rpc/2 Equation 15A,a Ppc=Ptmb+PA Equation 15A,b Ra=[Vφ(VAV*)4+Ra0]1+Ppl¯Pφ Equation 15A,c Rv=[Vφ(VAV*)4+Rv0]1+Ppl¯Pφ Equation 15A,d Rpc=Rpc0VpcmaxVpc2 Equation 15A,ePtmb exhibits qualitative characteristics similar to those described for the collapsible airways and is given byma×Vpcmax0.001Vpcmax13.6C¯pc0.001 Equation 16A,a mb=13.66.908+ln(ma0.999) Equation 16A,b mc=20.46.908mb Equation 16A,c Ptmb=mcmblnVpcmax0.001Vpc0.0010.999 Equation 16A,dThe forms for Eqs. 15A,c through 15A,e were dictated by the observation that the extra-alveolar resistances decrease while the intra-alveolar resistance increases during lung inflation (39). The large excursion in the perivascular intrapleural pressure modulates the extra-alveolar resistances and is described by an empirical expression given in Eqs. 15A,c and 15A,d. The parameters Ra0, Rv0, Rpc0, Vφ, Vpcmax, C¯pc, and Pφ are determined through the parameter-estimation scheme.

Gas Exchange

Air side.

Each of the airway compartments is assumed to be well mixed (hence, no spatial variation within). A generation term representing gas exchange across the alveolar-capillary barrier is present in the alveolar compartment only; no generation terms are present in the descriptions for the rigid and collapsible airway compartments. Inspiratory and expiratory phases are treated separately in the ventilated airways. Species molar balance equations describing the change in partial pressure of the alveolar gaseous species i can be written for each of the compartments as (adapted and modified from Ref. 17)

InspirationdPDidt=1VDQ˙EDPamisatTbodyTamQ˙DCPDi Equation 17A dPCidt=1VCQ˙DCPDiQ˙CAPCiPCidVCdt Equation 18A dPAidt=1VAQ˙CAPCiPAidVAdt PSTbodyTSj=1NsegDLi(PAiPbi(j))ΔVpc(j)Vpc Equation 19A

ExpirationdPDidt=1VD[Q˙EDPDiQ˙DCPCi] Equation 20A dPCidt=1VCQ˙DCPCiQ˙CAPAiPCidVCdt Equation 21A dPAidt=1VAQ˙CAPAiPAidVAdt PSTbodyTSj=1NsegDLi(PAiPbi(j))ΔVpc(j)Vpc Equation 22A

Blood side.

The lumped pulmonary capillary is spatially discretized into equal compartments (N seg = 35), and species molar balance is employed to describe the dynamics of the species concentration in each of the discretized compartments. The corresponding equation for species i in compartmentj is given byCbi(j)t=(Vzb(j)Cbi(j))z+DLi(PAiPbi(j))Vpc Equation 23AThe first-order spatial derivative is approximated by using a four-point upwind biased quadratic interpolation formula (47). Fictitious points created at the capillary entrance and exit are eliminated by using the physical and numerical boundary conditions at the entrance and exit, respectively. The effective diffusing capacities for each species (in units of mlstpd ⋅ min−1 ⋅ mmHg−1) are designed to account for changing capillary blood volume by scaling the nominal values (17, 22, 35) as shown underDLO2=VpcVpcmax(23.86+0.5119PO2 0.007983PO22+2.306×105PO23) Equation 24A,a DLCO2=VpcVpcmax×400.0 Equation 24A,b DLN2=VpcVpcmax×15.0 Equation 24A,c


A nonlinear least squares parameter-identification algorithm [Marquardt-Levenburg (30)] is applied to achieve fits to the flow, lung volume, and expired gas concentrations (Po 2 and Pco 2) during an FVC maneuver. The objective function employed is the square of weighted residuals in flow, total lung volume, and expired O2 and CO2 concentrations at the mouth and is expressed as Ω = We̲We̲. One may define an alternate concatenated error vector, e̲*=We̲, in terms of the weighted residuals as followse̲*=ωv1ev1,,ωvmevm,ωf1ef1,, ωfmefm,ωO21eO21,,ωO2meO2m, ωCO21eCO21,,ωCO2meCO2m Equation 1BIn Eq. 1B, the elements of the positive definite diagonal matrix W, ωrs, are the individual weights assigned to each residual r at timet s (r = flow at the mouth, total lung volume, expired Pco 2, and expired Po 2). The individual weights are determined by imposing penalty functions (16) associated with specific portions of the time record, as discussed below. The estimation problem now translates to the determination of the elements of the parameter vector α that minimizes a scalar functional Ω = e̲ * e̲*. This is accomplished by using the Levenburg-Marquardt procedure with the modified error term e̲*.

In the first stage of adjustment, and during the inspiratory phase with full muscular effort, the parameters that describe the compliant and resistive properties of the alveolar space (ξ, V*), small airways (Rsm, Rsa), collapsible airways (inspiratory K3, Vc max, Ptmmax), and upper airways (K2 for inspiration) were estimated (refer to appendix ). In the following expiratory phase, the estimation method was used to determine other parameters related to the small airways (Rsa, Rsm, Vcrit, Rsc max), collapsible airways (expiratory K3), and upper airways (K2 for expiration). The combined parameter estimate then provided a good fit to both the inspiratory and expiratory portions of the flow-volume loop. In the second stage of adjustment, previously identified parameters were now held unchanged, and the estimation procedure was invoked (for both inspiratory and expiratory phases) to estimate the parameters describing the pulmonary circulation and gas exchange (Ra0, Rv0, Rpc0, Vφ, Vpcmax, C¯pc, and Pφ). At this time, the expired Po 2 and Pco 2 values at the mouth were utilized as the data for the estimation algorithm.

Assignment of Weights

Certain regions of the flow-volume loop are emphasized in identifying the model parameters: 1) total lung volume; 2) peak inspiratory flow; 3) peak expiratory flow; and 4) the slope of the flow-volume relationship (effort independent) at low lung volumes near RV. A penalty function is imposed by increasing the weighting elements ωrs in these regions to indicate their importance in achieving good fits to the data. These regions are believed to have clinical significance in diagnosis of pulmonary mechanics abnormalities. Initial values of ωrs are uniformly set as ωvs = 1, ωfs = 5, ωO2 s = 1 and ωco2s = 1. The penalty functions imposed at specific time points modify the weighting factors at those points and are as described below.

1. TLC fit. This is enforced only during the inspiratory phase. For the data points in the volume range TLC − 0.5 liters ≤ (Va + Vc + Vd) ≤ TLC, a penalty is enforced on e vs in Eq.1B and the weighting factor is given as ωvs = 1 + (100 × ‖TLCmodel − TLCdata‖ ).

2. Peak inspiratory flow fit. The weighting factor for the inspiratory flow is given as ωfs = 1 + (1,000 × ‖peak  V˙AomodelpeakV˙Aodata). This is used only for those time points when flow at the mouth is within 0.5 l/s of the peakV˙a o data values (mid one-thirdof VC).

3. Peak expiratory flow fit. During expiration, ωvs = 1 + (1,000×peakV˙AomodelpeakV˙Aodata) and ωfs = 5 + (1,000×peakV˙AomodelpeakV˙Aodata) for those time points when the flow at the mouth is within 1 l/s below the peak V˙Aodata (highest two-third of VC). The data sampling in the vicinity of the expiratory peak is very sparse, thus requiring a larger weighting factor compared with the other weights.

4. Expired CO2 and O2 at the mouth fit. Penalty is imposed to achieve good fits on the phase III plateau of the profile of CO2 and O2concentration in the expirate. The weighting factors are given as ωCO2 = 100.0, ωO2 = 10.0.

Penalty and barrier functions constitute a global approach to nonlinear programming in which weighting factors are incorporated into objective functionals to enforce a specific search direction, thereby accelerating rate of convergence.


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