## Abstract

It has been suggested that the human pulmonary acinus operates at submaximal efficiency at rest due to substantial spatial heterogeneity in the oxygen partial pressure (Po_{2}) in alveolar air within the acinus. Indirect measurements of alveolar air Po_{2} could theoretically mask significant heterogeneity if intra-acinar perfusion is well matched to Po_{2}. To investigate the extent of intra-acinar heterogeneity, we developed a computational model with anatomically based structure and biophysically based equations for gas exchange. This model yields a quantitative prediction of the intra-acinar O_{2} distribution that cannot be measured directly. Temporal and spatial variations in Po_{2} in the intra-acinar air and blood are predicted with the model. The model, representative of a single average acinus, has an asymmetric multibranching respiratory airways geometry coupled to a symmetric branching conducting airways geometry. Advective and diffusive O_{2} transport through the airways and gas exchange into the capillary blood are incorporated. The gas exchange component of the model includes diffusion across the alveolar air-blood membrane and O_{2}-hemoglobin binding. Contrary to previous modeling studies, simulations show that the acinus functions extremely effectively at rest, with only a small degree of intra-acinar Po_{2} heterogeneity. All regions of the model acinus, including the peripheral generations, maintain a Po_{2} >100 mmHg. Heterogeneity increases slightly when the acinus is stressed by exercise. However, even during exercise the acinus retains a reasonably homogeneous gas phase.

- oxygen diffusion
- acinar gas mixing
- gas exchange
- alveolar gas
- mathematical modeling

the pulmonary acinus (or a subpart) is generally considered to be the functional unit of gas exchange in the lungs (46). It is defined as comprising all alveolated (respiratory) airways distal to a terminal bronchiole (18). Inefficient acinar function may potentially arise due to heterogeneous alveolar air oxygen partial pressures (P_{a}O_{2}) or significant P_{a}O_{2} gradients from the acinar entrance to the periphery (“stratification”). In a healthy individual, the average P_{a}O_{2} of ∼100–110 mmHg is inferred from whole lung experimental measurements. However, these measurements do not provide information about spatial variations in P_{a}O_{2} within the acinus. It is theoretically possible to maintain efficient gas exchange function in the presence of a substantial decrease of P_{a}O_{2} in the peripheral respiratory airways if intra-acinar perfusion is matched to local P_{a}O_{2}. In this situation, the peripheral generations of the pulmonary acinus would not contribute substantially to gas exchange and the acinus would be performing suboptimally.

The consequence of uniform perfusion in the presence of nonuniform P_{a}O_{2} is a depressed end-capillary blood oxygen partial pressure (P_{c′}O_{2}). If significant P_{a}O_{2} heterogeneity or entrance-to-periphery P_{a}O_{2} gradients exist, then it follows that there must be a mechanism to match perfusion to local P_{a}O_{2} to maintain arterial oxygenation. For example, if P_{a}O_{2} drops well below 100 mmHg in the peripheral generations, then it is important this region of the alveolocapillary bed remains relatively unperfused. An understanding of intra-acinar P_{a}O_{2} distributions would also allow assessment of the validity of the common physiological assumption of the acinus as a well-mixed compartment (2, 5, 13, 16, 23, 43).

Because intra-acinar gas exchange cannot be measured directly, the extent of intra-acinar P_{a}O_{2} gradients and heterogeneity is not known. Mathematical models have previously been used to simulate intra-acinar P_{a}O_{2} distributions and elucidate the mechanisms behind them (8, 27, 29, 30). Several mechanisms have been postulated that may contribute to intra-acinar stratification and heterogeneity. First, time-limited gas-phase diffusion gradients propose that inspired O_{2} may have insufficient time for gas-phase diffusion along the full length of the acinar pathways within the duration of a breath (22). Second, intra-acinar P_{a}O_{2} heterogeneity has been postulated to be caused by diffusion-advection-dependent inhomogeneities, which arise due to the asymmetric branching structure of the acinus (4, 9, 10, 24, 26). Third, the phenomenon of diffusional screening suggests persistent O_{2} gradients may exist along the acinar airways regardless of breath duration due to the distribution of O_{2} uptake by the blood (29). Molecules travelling along the acinar pathways encounter the alveolar gas exchange surface, where they can diffuse into the capillary blood. Hence the rate of gas uptake into the blood limits the distance the molecules diffuse into the acinus.

Previous modeling studies have investigated some of these important mechanisms, but have included simplifications such as a symmetric acinar geometry, nonbiophysical equations for gas exchange, or including either only advective or diffusive transport. Interaction of these mechanisms and their compounding effects have not been incorporated in a single computational model to predict quantitative intra-acinar O_{2} distributions.

In this study we investigate the degree of intra-acinar O_{2} heterogeneity and gradients in a healthy acinus using a theoretical model of intra-acinar O_{2} transport and exchange. The model developed in this study extends previous computational approaches by combining a realistic acinar geometry and description of air-side transport, which is important for incorporating advection-diffusion interactions, and accurate blood biochemistry, which is important for a realistic prediction of gas exchange. We use the model to quantify intra-acinar O_{2} heterogeneity and gradients and address the standard assumption of the acinus as a well-mixed homogeneous compartment.

## THE COMPUTATIONAL MODEL

The following O_{2} transport mechanisms are included in the model acinus, representing an average acinus in the lung: advection and diffusion along the lumen of all conducting and alveolated airways; diffusion across the alveolar air-blood membrane into the blood; and O_{2}-hemoglobin binding. Spatial and temporal variations in air-side and blood-side Po_{2} are calculated throughout the breathing and cardiac cycles. This approach is similar to that previously applied to the alveolocapillary mesh in an alveolar sac (32).

### Geometry

#### Respiratory airways.

The respiratory airways are represented using an asymmetric multibranching geometry (34, 36). This multi-branch-point model is based on “acinus-3” from the morphometric study of acinar structure by Haefeli-Bleuer and Weibel (18).

The airway radii and branch lengths are scaled to a preinspiratory volume [functional residual capacity (FRC)], assuming that the volume during measurement (18) was close to total lung capacity (TLC) and that FRC is one-half of TLC. The model acinus hence has an initial FRC volume of 81 mm^{3}, which gives a whole lung alveolar volume of 2.65 liters. The average acinar path length is 6.6 mm, where path lengths are calculated from the end of the transitional bronchiole. Figure 1 shows results in the respiratory airways mesh. Airway segment lengths, gas exchange surface area, inner (duct) and outer (alveolar sleeve) cross-sectional areas that are used to create the acinus mesh are listed in Table A (appendix).

#### Conducting airways.

Although the model is used to investigate O_{2} dynamics inside the acinus, it is important to include the conducting airways in the full model because they introduce a delay in the arrival of inspired air during inspiration and they allow for a gradient in O_{2} concentration down the conducting airways. If the acinus was modeled in isolation, both flow and concentration would need to be assumed at the entrance of the acinus over the course of a breath. In the absence of in vivo measurements for these values under different conditions, a model that couples conducting and respiratory airway geometries has the advantage that only boundary conditions at the trachea are required.

The geometry of the conducting airways is represented with Model “A” from Weibel (39), as modified by Weibel et al. (42). This results in a total of 32,768 (2^{15}) acini. Coupled to the acinus models, the total lung volume at FRC is 2.73 liters, and anatomical deadspace is 81 ml. The model value for the deadspace is comparable to the measured lower airways volume (∼70 ml; Ref. 14), because the upper airways mouthward of the trachea are excluded in the model.

### Air-Side Advection and Diffusion

The partial differential equations governing the advective and diffusive processes in the acinar airways are solved in a simplified geometry of an alveolated airway following the approach of Paiva (25). Each airway is considered to be a straight duct with cross-sectional area A_{i} surrounded by a sleeve of alveoli. The total cross-sectional area of the duct plus alveolar sleeve is denoted A_{o}. Radial and rotational flows are neglected. Radial diffusion into alveolar spaces is assumed to occur instantaneously, which is an acceptable assumption if we consider that the radial diffusional distance (alveolar diameter of ∼0.2 mm plus duct radius of ∼0.2 mm) is much smaller than the axial distance (average path length from acinus entrance to periphery of 6.6 mm).

Under these assumptions and with the requirement of mass conservation in the airways, the O_{2} concentration (*c*) satisfies the one-dimensional equation for advection-diffusion of a gas in an alveolated airway that was originally derived by Paiva (25) and Scherer et al. (30):
_{2} and nitrogen (22.5 mm^{2}/s), *x* is the axial coordinate, *t* is time, and Q_{s} accounts for the exchange of the gas between the alveolar air and the blood. The form of the gas exchange source term, Q_{s}, is given in subsection *Gas Exchange*. The source term provides the interface between the air-side O_{2} transport model and the gas exchange model. Note that by assuming alveolar air to be fully saturated with a constant air pressure equal to atmospheric pressure, then the O_{2} concentration is related to Po_{2} by: Po_{2} = cV_{m}(P_{atm} − P_{H2o}), where V_{m} is the O_{2} molar volume, P_{atm} is atmospheric pressure, and P_{H2o} is water vapor pressure at body temperature.

The model was assumed to change in length and alveolar cross-sectional area during ventilation; the duct-cross sectional area did not change. Uniform flow distribution was imposed by assuming that the change in volume of an element was in proportion to the ratio of the element volume to the total acinus volume. Element length changed in proportion to the cuberoot of volume change, and the change in area of the alveolar cross-section was calculated to satisfy the volume change. The volume of air entering the distal-most elements is equal to the change in volume of the alveolar sleeve, therefore the flow and velocity at the model periphery is zero.

### Gas Exchange

The uptake of O_{2} by the capillary blood is given by Fick's first law, describing the O_{2} flux across the air-blood barrier. The rate of change of blood O_{2} partial pressure, given by *Eq. 2*, is the same as that given by Ben-Tal (2). The derivation of *Eq. 2* is given in the appendix.
_{a}O_{2} and P_{b}O_{2} are the air-side and blood-side O_{2} partial pressures, respectively; T_{t} is the transfer factor (see below); V_{b} is the capillary blood volume; Hb is the hemoglobin concentration in whole blood; σ is the solubility of O_{2} in blood. The oxyhemoglobin saturation function S_{O2} = f(P_{b}O_{2}) is given by the Kelman equations (21). Here, blood is considered a single-phase homogeneous fluid and is assumed to move into the capillary bed instantaneously and remain stationary for the duration of the transit time of the RBC. After the transit time has elapsed, new blood moves into the capillary bed with a P_{b}O_{2} equal to mixed venous blood, which is ∼40 mmHg under normal conditions (17). The blood carbon dioxide partial pressure (P_{b}CO_{2}) and blood pH are both assumed to be constant in the capillary blood. During the solution of the advection-diffusion equation, the value of P_{b}O_{2} is updated at each time step at each solution point (node).

The source term (Q_{s}) in the advection-diffusion *Eq. 1* is the gas exchange flux per unit length, and is calculated as:
_{b}O_{2} is updated from *Eq. 2* at each time step. The transfer factor (T_{t})—commonly called the “diffusing capacity”—is the conductance of the air-blood barrier and incorporates resistance due to the air-blood barrier properties in addition to the binding of O_{2} to hemoglobin. Following the method of Weibel et al. (41), it is calculated as:
_{h} is the harmonic mean barrier thickness; S_{A} is the gas exchange surface area; φ is a surface correction factor (this accounts for the uneven distribution of RBCs along the capillary and folds in the surface that are smoothed out in vivo by surfactant); and θ is the rate of O_{2} uptake by whole blood. An empirical expression from Staub et al. (31) is used for θ in terms of the reaction velocity of O_{2} combining with hemoglobin. This accounts for the dependence of θ on oxyhemoglobin saturation:
_{c}^{′} is the forward reaction velocity for the combination of O_{2} with deoxyhemoglobin and κ is the O_{2} carrying capacity of hemoglobin. In all simulations the value of K_{c}′ is taken to be 4.4 × 10^{2} mM^{−1}·s^{−1} (40), which corrects for the slowing effect of the unstirred layer of plasma surrounding the RBC during experimental measurements.

### Intra-Acinar Perfusion

The intra-acinar perfusion distribution (the amount of blood flow received by different regions of the acinus) determines the transit time of an RBC through the capillary bed and the local capacity (blood volume) available for O_{2} uptake. The local flow also affects the calculation of the pooled Po_{2} in the blood draining from the whole acinus. In this model, perfusion of the capillary bed surrounding the alveolar sleeve is assumed to be proportional to the gas exchange surface; that is, there is no perfusion stratification or local control mechanisms. Capillary blood volume is also calculated as proportional to the gas exchange surface. All regions of the acinus have the same RBC transit time. Blood transport is modeled as a unit of mixed venous blood that instantaneously enters the capillary bed surrounding a section of alveolated airway, then remains stationary for the duration of the RBC capillary transit [this assumption was used and justified by Ben-Tal (2)].

### Solution Methodology and Parameters

Parameter values for equations in subsections *Air-Side Advection and Diffusion* and *Gas Exchange* are given in Table 1. The breathing cycle is a simple step function with constant flow during inspiration and expiration, with flow equal but reversed. The breath duration is 5 s (35), with equal inspiration and expiration times. The representative acinus geometry has a smaller volume (81 mm^{3}) than the average acinus (94 mm^{3}) studied by Haefeli-Bleuer and Weibel (18), hence the tidal volume is scaled down from the normal value cited in the literature of 0.5 (44) to 0.4 liters. This results in a constant flow rate at the trachea of 0.16 l s^{−1}.

The total alveolar surface area of an acinus is assumed to be 30 cm^{2}. This value is obtained by scaling the TLC value of 143 m^{2} for the whole lung (15) to halfway between start- and end-inspiratory volume and assuming 2^{15} acini in the lung. The lung volume at the start of inspiration (FRC) is ∼50% TLC, and end-inspiratory volume is ∼65% TLC; therefore the scaling factor is 0.575^{2/3}. Although the alveolar surface area changes by ∼7% from end expiration to end inspiration, here it is assumed to be constant over the breathing cycle. The model uses a capillary blood volume of 6.6 × 10^{−3} ml/acinus. This equates to 216 ml for the whole lung compared with 213 ± 31 ml measured by Gehr et al. (15).

For rest conditions, mixed venous blood Po_{2} is set to 40 mmHg (17, 35) and the RBC transit time is set to 1.0 s. Commonly cited RBC transit times in the pulmonary capillaries are 0.75–1.0 s (19, 35).

The computational mesh (respiratory and conducting airways) consists of one-dimensional finite elements with linear interpolation functions, where each airway segment is discretized to contain three finite elements. Convergence testing was performed on the advection-diffusion problem (without gas exchange) and it was found that further spatial discretization did not improve the solution significantly.

The distribution of O_{2} is solved over multiple cardiac and respiratory cycles until the model reaches breath-by-breath steady state, defined as a change in breath-averaged Po_{2} of <1% between sequential breaths. Blood from different portions of the acinus combines at the acinus exit and hence the O_{2} content of the mixed end-capillary blood is calculated as the blood flow-weighted average O_{2} content from all nodes.

For each time step, the advection-diffusion equation (*Eq. 1*) and gas exchange equation (*Eq. 2*) are solved iteratively. The gas exchange equation is only solved for each node in the respiratory mesh to determine the updated capillary blood Po_{2}. The source term is calculated using *Eq. 3* and is used in the subsequent advection-diffusion step. In all simulations, the time step is set to 0.05 s. During each inspiratory and expiratory period, boundary conditions at the trachea are set appropriately to be inspiratory or expiratory flows. Zero concentration flux boundary conditions are set at all terminal nodes. The boundary conditions are hence:
*x* = 0 is the model entrance (proximal trachea node) and *x* = *L* corresponds to the most distal acinar nodes. The inspired concentration, c_{insp}, is that of ambient air. Initial O_{2} concentration values are set to be 5.8 mM for all nodes, which is approximately equal to a Po_{2} of 105 mmHg:

The advection-diffusion equation is solved using a Galerkin finite element method, and the gas exchange equation is solved using a stiff ODE solver.

Exercise is simulated by increasing the cardiac output (decreasing RBC transit time, increasing capillary blood volume), increasing the ventilation rate (increasing flow at the mouth, decreasing breath duration), and increasing metabolic consumption of O_{2} (decreasing the mixed venous blood Po_{2}). The changes in representative parameter values for mild and moderate exercise are taken from Tipton (35) and are summarized in Table 2.

## SIMULATION RESULTS

### Intra-Acinar Gas Exchange During Normal Breathing

Normal quiet breathing was simulated in the model and the results were compared with both physiological data and previous models. The model reached the same steady state for air-side and blood-side Po_{2} regardless of the initial Po_{2} in the airways. The volume-weighted spatial average P_{a}O_{2} for all elements in the acinus was averaged over one breath (inspiration and expiration). The model value of 107 mmHg compares well with the normal physiological range of 100–110 mmHg (7). The temporal average of mixed P_{c′}O_{2} was 0.3 mmHg less than the average P_{a}O_{2}.

Figure 2*A* shows the temporal variation in air-side Po_{2} at the trachea, acinus entrance, and acinus periphery over three steady-state breaths. The spatially averaged acinar P_{a}O_{2} is also shown. The sharp transitions at the trachea at end inspiration and end expiration were due to the square-wave breathing function. At the acinus entrance P_{a}O_{2} increased after a delay following the start of an inspiration as fresh air traverses the deadspace to reach the acinus. Small oscillations with a period of 1 s that are visible in the average and peripheral P_{a}O_{2} were due to blood moving through the capillary bed. Figure 2*B* shows the profile of mean air-side Po_{2} from the end of the conducting airways to the acinus periphery at the end of an inspiration and the end of an expiration. Moving peripherally from the acinus entrance, the mean P_{a}O_{2} decreased from the acinus entrance to the acinus periphery at end inspiration. Note that this figure includes only the last few generations of the conducting airway path (∼4 mm) to demonstrate that Po_{2} starts to decrease below the inspired Po_{2} just before the acinar entrance.

Figure 1*A* shows the spatial distribution of P_{a}O_{2} in the acinar geometry at the end of inspiration. Spatial heterogeneity was evident in the acinus at the end of inspiration: for example, shorter paths have higher P_{a}O_{2} than longer paths. The gradient from entrance to periphery was abolished during expiration and P_{a}O_{2} values were relatively uniform at end expiration (Fig. 1*B*) compared with end inspiration (Fig. 1*A*). However, heterogeneity between different acinar pathways was still evident (error bars in Fig. 2*B*). Figure 1, *A* and *B*, respectively, correspond to the end-inspiration and end-expiration curves in Fig. 2*B*.

As expected, the capillary blood equilibrated with alveolar air (P_{b}O_{2} ≥ 0.99 P_{a}O_{2}) in all regions of the acinus before it reached the end of its transit; that is, there was no diffusional limitation. While the equilibration time of blood with alveolar air is not measured experimentally, theoretical gas exchange models that use lumped whole lung parameters and assume mixed venous blood and alveolar air Po_{2} values of 40 and 100 mmHg, respectively, have generally agreed on an equilibration time for O_{2} of 0.25–0.30 s (7, 38); in comparison, the current model gave a value of 0.26–0.27 s.

An analysis of the sensitivity of the model results to the model parameters was performed. The spatially averaged P_{a}O_{2} is relatively sensitive to the parameter P_{v}O_{2}, where a 10% change in parameter results in ∼10% change in spatially averaged P_{a}O_{2}. However, experimental data for this parameter is readily available and is hence likely to be accurate. The spatially averaged P_{a}O_{2} is relatively insensitive to changes in all other parameters. The coefficient of variation of P_{a}O_{2} is insensitive to changes in all parameters. This gives confidence in the results of the model and in the conclusion that there is low intra-acinar heterogeneity.

### Intra-Acinar Gas Exchange During Exercise Conditions

The average P_{a}O_{2} and P_{c′}O_{2} increased slightly with exercise; however, values were still physiologically realistic. During mild and moderate exercise, the average P_{a}O_{2} was 110 and 113 mmHg, respectively. Both spatial and temporal partial pressure heterogeneity increased during exercise. The greater spatial heterogeneity was seen in the approximate doubling of the coefficient of variation of P_{a}O_{2} from rest (3.0%) to moderate exercise (6.6%). Temporal heterogeneity—the amplitude of P_{a}O_{2} oscillations over a breath—increased significantly during exercise from 2.8 mmHg during rest conditions to 6.7 and 8.7 mmHg during mild and moderate exercise, respectively. The effect of heterogeneity on gas exchange function is evident in the difference between mean alveolar and end-capillary Po_{2} values [P(A-c′)O_{2}]. This increased by 290% (0.8 mmHg) and 540% (1.5 mmHg) from rest to mild and moderate exercise, respectively. A comparison of results from rest and exercise simulations in Fig. 3 illustrates the increase in heterogeneity through the acinus. End-inspiration (Fig. 3*A*) and end-expiration (Fig. 3*B*) P_{a}O_{2} values are shown from the end of the conducting airways to the acinar periphery.

The time required for equilibration between air-side and blood-side Po_{2} increased from ∼0.26 s during rest conditions to 0.32 and 0.35 s during mild and moderate exercise, respectively. Due to the lower RBC transit time during exercise, this equates to a significantly larger proportion of the total RBC transit.

### Modeling Simplifications

The model was used to investigate two separate simplifications and the simplified models were compared with the full baseline model presented in subsection *Intra-Acinar Gas Exchange During Normal Breathing*.

### Using a Symmetric Acinar Mesh

A computationally more efficient acinar geometry that is frequently used in modeling studies is a symmetric branching tree. Here a symmetric model was compared with the multibranching asymmetric acinus to determine the effect of neglecting asymmetry on predictions of O_{2} concentration distribution. A symmetric geometry was generated based on the Weibel et al. (42) morphometric data, with the same total acinar volume, average path length, and total alveolar surface area as the asymmetric geometry, which enables a quantitative comparison. The symmetric acinar geometry had nine generations (excluding the transitional bronchiole), compared with the asymmetric geometry that had alveolar sacs terminating between generations 6 and 11 (average 9.1 generations).

Figure 4*A* shows the P_{a}O_{2} distribution at the end of inspiration. Compared with the multibranching model (gray line), the curve for the symmetric mesh had a similar shape (spatial distribution) and similar temporal distribution. However, the distribution of P_{a}O_{2} in the symmetric acinus model varied quantitatively from the multibranching acinus model. The averaged Po_{2} values for both the alveolar air and capillary blood increased by ∼20% in the symmetric model compared with the baseline model. That is, the symmetric geometry resulted in much higher Po_{2} values in the acinus, given the same input parameters and boundary conditions. This is most likely because diffusive-advective inhomogeneities cannot develop in the symmetric structure and intra-acinar heterogeneities cannot be captured.

### Modeling Blood as an Infinite Oxygen Sink

Including a gas exchange term that incorporates the change in blood partial pressure and the dynamics of O_{2}-hemoglobin binding dynamics requires additional iterations to solve blood-side partial pressures at each time step, as well as the solution of the nonlinear oxyhemoglobin saturation equation. Can we simplify the gas exchange process and produce the same results?

If blood is modeled as an infinite O_{2} sink, such that the blood compartment is effectively assumed to have infinite volume with instantaneous mixing, the computational requirements are reduced substantially. This is a simplification used in past models of intra-acinar O_{2} exchange (27, 29). This simplification was included in the current model by holding the blood Po_{2} constant at the value of mixed venous blood. Therefore the flux of O_{2} was greater than the baseline model because the driving partial pressure difference did not decrease as the blood Po_{2} equilibrates with alveolar air. The spatial heterogeneity in O_{2} concentration was greater than the baseline model, with a large entrance-to-periphery Po_{2} gradient (see Fig. 4*B*). In addition, temporal fluctuations over a single breath, once steady state had been reached, were greater than the baseline model. Therefore, this simplification produced results with greater spatial and temporal heterogeneity and with a lower average P_{a}O_{2}.

## DISCUSSION

The computational model presented in this paper examines O_{2} transport in the conducting and acinar airways and exchange with the capillary blood within an anatomically based acinar structure. The model adds to previous models of intra-acinar gas mixing by including a biophysically accurate description of O_{2} uptake by the capillary blood that is coupled to equations for air-side O_{2} transport. Previous models have either examined inert gases (4, 9, 30, 34, 36) or have assumed blood to be an infinite O_{2} sink (27, 29).

### Acinar Heterogeneity

At rest, there is a small degree of intra-acinar heterogeneity that causes a small reduction in the mixed end-capillary blood Po_{2}. This P(A-c′)O_{2} difference is not due to diffusional limitation, where capillary blood Po_{2} does not equilibrate with air-side Po_{2}, but is analogous to the effect of whole lung ventilation-perfusion mismatch that increases the measured alveolar-arterial Po_{2} gradient [P(A-a)O_{2}]. Therefore, the model value of 0.3 mmHg can be viewed as the acinar contribution to whole lung P(A-a)O_{2}. A healthy lung has a P(A-a)O_{2} of at least 4 mmHg due to ventilation-perfusion mismatch (44). Thus the intra-acinar contribution to whole lung ventilation-perfusion mismatch predicted by this “average” acinus model is very small.

The characteristic Po_{2} profile (Fig. 2*B*), as described by other authors (4, 9, 11, 26), is observed in the inspiratory phase, with an O_{2} gradient decreasing from ∼148 mmHg at the acinus entrance to 103–107 mmHg at the most distal alveolar sacs at end inspiration. During expiration there is almost no concentration gradient down the airways. However, along individual acinar airway paths, the partial pressures can either increase or decrease from acinar entrance to periphery. That is, on some pathways the O_{2} molecules diffuse mouthward, and on other pathways the O_{2} molecules diffuse peripherally. This effect, known as diffusive Pendulluft, arises due to the asymmetry of the multibranching acinar geometry and is not present when our model is solved in a symmetric geometry. This highlights the importance of including a realistic asymmetric acinar geometry when modeling gas mixing in the acinus.

### Acinus Efficiency

Simulation results provide an insight into normal gas exchange function inside the acinus. The primary conclusion of this modeling study is that the acinus is highly efficient under normal physiological conditions. Here, “efficient” means the degree of utilization of the gas exchange surface. In this respect, we contradict the conclusions of Felici et al. (12), whose simulations with blood as an infinite sink showed a large reduction in P_{a}O_{2} along the acinar pathway, such that the acinar gas exchange surface was only 30–40% efficient. The gas exchange surface in the anatomically structured model with blood oxygen dynamics is fully functional, with all regions of the acinus exposed to P_{a}O_{2} values >100 mmHg. Discussing the proportion of the acinus with a P_{a}O_{2} below 100 mmHg is fairly arbitrary; however, we find it a useful indication of the effectiveness of the acinar surface area. Blood that equilibrates to 100 mmHg is almost fully saturated. In addition, a lower P_{a}O_{2} gives rise to a lower driving Po_{2} gradient across the air-blood membrane and hence a lower gas exchange flux. Regions with low P_{a}O_{2} would contribute less to the net gas exchange flux of the acinus, and therefore the acinus would be performing suboptimally. An additional potential reason for the different predictions of the current model and the model of Felici et al. (12) is that advective transport was neglected in the latter study.

In our model, the phenomenon of diffusional screening, as first discussed by Sapoval et al. (29), does exist in the sense that some O_{2} molecules are taken up by the gas exchange surface in proximal acinar regions before they reach the periphery. However, this phenomenon only causes a significant decrease in P_{a}O_{2} if blood is assumed to be an infinite sink for O_{2}. In the current model, diffusional screening did not significantly impair acinar gas exchange function. The steady-state distribution is due to the balance between the inspired flux, expired flux, and gas exchange flux. The inclusion of a description of blood biochemistry, and hence a realistic gas exchange flux, is important in quantifying the effect of gas exchange. Assuming a blood O_{2} sink (holding the blood Po_{2} constant) overestimates the gas exchange flux, resulting in high O_{2} transfer and hence large O_{2} gradients along the acinar airways.

### Acinus Performance During Exercise

Applying exercise stress causes an increase in intra-acinar heterogeneity, as evidenced by the higher P(A-c′)O_{2} under these conditions compared with baseline conditions. Increasing the stress from mild to moderate exercise increases P(A-c′)O_{2} further. Tipton (35) gives a threefold increase in whole lung P(A-a)O_{2} from rest to moderate exercise. The acinar contribution to P(A-a)O_{2} increases by a factor of 5.4 from rest to moderate exercise, indicating that intra-acinar heterogeneity becomes more important during exercise. The time taken for blood Po_{2} to equilibrate with air Po_{2} increases during exercise, in both absolute terms and relative to the RBC transit time. This is most likely due to the greater partial pressure difference between mixed venous blood and alveolar air. At rest, blood takes 26% of its transit time for the partial pressure to equilibrate with air-side Po_{2}, leaving an effective safety margin of 0.74 s, whereas during moderate exercise, blood takes ∼59% of its transit to equilibrate, leaving a safety margin of only 0.24 s.

### Addressing Common Assumptions

Models are simplifications of reality. The computational modeler pursues modeling simplifications because they enable a more intuitive understanding of underlying mechanisms and they present computational savings. However, it is often necessary to first build complexity into a model before stripping back unnecessary components. Using this model, it has been shown that exclusion of blood dynamics leads to a quantitatively incorrect Po_{2} distribution. If a single control volume of blood is considered, it is clear that rapid equilibration with P_{a}O_{2} (in ∼0.26 s) means that O_{2} is not being exchanged for the majority of time that the volume of blood spends in the capillary bed (∼0.74 s).

A common assumption used in physiological models describes the acinus as a single well-mixed gas exchanging compartment. Results from the current model show that there is P_{a}O_{2} heterogeneity in the asymmetric acinar branching structure and there is a significant drop from inspired Po_{2} (∼150 mmHg) to the P_{a}O_{2} at the acinar periphery (∼102–110 mmHg) at the end of inspiration. However, heterogeneity is small and the low coefficient of variation of 3% indicates that most of the acinar volume has a P_{a}O_{2} that is close to the mean value. This is also evidenced by the small P(A-c′)O_{2}: a high P(A-c′)O_{2} indicates poorly matched air-side O_{2} delivery and perfusion. Given that the model assumes perfusion to be proportional to the gas exchange surface, the acinar perfusion (and hence also air-side O_{2} delivery) is relatively uniform. The assumption of an acinus as a well-mixed gas exchanging compartment is therefore reasonable.

### Limitations of the Model

Acinar perfusion is assumed to be proportional to capillary surface area in the model. Heterogeneous acinar perfusion may also affect acinar gas exchange efficiency; however, this area has not received the same level of attention as air-side acinar heterogeneity. Experimental and computational studies provide some evidence of intra-acinar perfusion stratification (6, 28, 37, 45), although others have found that intra-acinar perfusion is uniform (33). Our assumption that blood flow is proportional to gas exchange surface area allows investigation of acinus performance in the absence of perfusion heterogeneity that may aid gas exchange function. Heterogeneous perfusion may occur due to vessel structure (6) or dynamic control mechanisms, such as hypoxic vasoconstriction. The reasonably uniform P_{a}O_{2} distribution predicted by the model suggests that if intra-acinar perfusion heterogeneity exists, it is not likely to have a significant effect on acinar gas exchange in this model of the healthy acinus. It would, however, be an important consideration during disease.

Each computational node in the acinar mesh was assumed to be functionally independent on the blood side. A secondary septum is a shared wall that connects neighboring alveoli that open on to different airway segments. In secondary septa in the alveolated airways, the capillary blood contained in the wall could theoretically be exposed to significantly different alveolar air Po_{2} on either side, which could potentially affect the rate of blood oxygenation or even (to some degree) cause reverse diffusion of O_{2} from capillary blood to alveoli with low air Po_{2}. The likely effect of this, if it exists, is that the distribution of O_{2} in the acinus will become more uniform. Investigation of the effect of secondary septa would be an interesting consideration for future modeling studies.

It has not been established whether collateral ventilation via pores of Kohn, which is neglected in the model, is significant within a human acinus. Morphometric studies of human acini have noted that connections exist between alveoli in terminal sacs and those in the proximal generations in regions of the acinus where successive generations branch back toward the respiratory bronchioles (18). In addition, pores of Kohn may lead to collateral ventilation between adjacent acini and hence the interdependence of acinar units. However, collateral ventilation may be prevented by a thin surface lining film covering these pores in situ (1). If collateral ventilation does occur, it is likely that the distribution of P_{a}O_{2} would be more uniform.

### Conclusions

In summary, we developed an anatomically and biophysically based model of intra-acinar O_{2} exchange to investigate steady-state gas exchange and the efficiency of the human pulmonary acinus. Model results show that equilibration between blood and air-side Po_{2} prevents development of large intra-acinar gradients in the alveolated airways during both normal lung function and physiological stress such as exercise conditions. Effective gas exchange can occur throughout the whole acinus without the need for vasoconstriction in the acinar periphery to match local perfusion to P_{a}O_{2}. This behavior can only be predicted and explained using models that consider the chemical and physical processes involved in the blood-side gas exchange dynamics.

The model highlights the physical mechanisms that play an important role in determining the steady-state intra-acinar P_{a}O_{2} distribution. As discussed by previous authors, advection is an important O_{2} transport mechanism in addition to diffusion within the acinar airways. A flux into the capillary blood must be included in the model, and it is important that this includes realistic blood Po_{2} dynamics: blood Po_{2} increases as it traverses the capillary bed. Under normal conditions, an RBC is only available to uptake O_{2} for approximately one-third of its transit, after which time it is saturated with O_{2}. A realistic asymmetric acinar geometry is also necessary to include the diffusion-advection-dependent inhomogeneity mechanism (26).

The composition of average alveolar gas and capillary blood was found to be very similar to that estimated by lumped models that assume that the acinus acts as a well-mixed unit. This implies that mechanisms to control the perfusion distribution are not necessary at the subacinar scale to maintain blood gases during healthy function. The acinar gas exchange efficiency is indicated by the P(A-c′)O_{2} gradient. In the presence of significant intra-acinar heterogeneity, P(A-c′)O_{2} will be large. However, the small P(A-c′)O_{2} predicted by the current model leads to the conclusion that the acinus is an effective gas exchanger.

## GRANTS

This work was supported by a University of Auckland Doctoral Scholarship and National Heart, Lung, and Blood Institute Grant R01-HL-064368.

## ACKNOWLEDGMENTS

The authors thank Alys Clark for reviewing the manuscript and Kim Prisk for fruitful discussions.

## APPENDIX A: DERIVATION OF BLOOD OXYGEN DYNAMICS EQUATION

Here the oxygen-hemoglobin dynamics equation given by Ben-Tal (2) is derived. O_{2} is primarily transported in the blood as oxyhemoglobin and a small fraction is dissolved in the plasma. The rate of change of P_{b}O_{2} depends on the air-side Po_{2} and the saturation of hemoglobin. Dissolved O_{2} binds to hemoglobin in a complex four-step kinetic process inside the RBC but this reaction can be simplified using a one-step approximation as shown in *reaction 10*, where each hemoglobin molecule is replaced by four independent heme groups.

O_{2} is assumed to be in equilibrium with hemoglobin in this model, which has been shown to be a reasonable assumption under normal conditions (2). The saturation of oxyhemoglobin (S_{O2}) depends on a number of factors including P_{b}O_{2}, blood pH, and P_{b}CO_{2}. The function describing the saturation curve is given by the Kelman equations (21).

The number of molecules of O_{2} (n_{O2}) in a blood volume (V_{b}) are the total of dissolved and bound O_{2}:

According to Henry's Law, dissolved O_{2} is proportional to the partial pressure ([O_{2}] = σP_{b}O_{2}), where the proportionality constant (σ) is the solubility coefficient. The bound O_{2} is the concentration of hemoglobin in whole blood (Hb) multiplied by the saturation (S_{O2}), which gives the proportion of binding sites that are occupied by O_{2}. As there are four heme groups on each hemoglobin molecule, Hb is multiplied by a factor of four.

The rate of change of O_{2} molecules is therefore:
*Eq. 3* gives the flux of O_{2} molecules from the alveolar air. Setting this flux equal to the rate of change in O_{2} molecules:

Substituting *Eq. 14* into *Eq. 13* and rearranging for the rate of change of P_{b}O_{2}:
_{O2}/dP_{b}O_{2} is the slope of the dissociation curve and is calculated using the Kelman equations as a function of P_{b}O_{2} and P_{b}CO_{2}. The total O_{2} transfer factor, T_{t}, is calculated as the series summation of the serial membrane and erythrocyte components. As the erythrocyte component is a function of the saturation it is updated at each time step; therefore T_{t} decreases as blood Po_{2} increases.

## APPENDIX B: ACINUS MESH GEOMETRIC PARAMETERS

Table A lists the parameters used to generate the multibranching acinus geometry. Values are scaled to an FRC volume of 80 mm^{3}. Lengths and cross-sectional areas are from the morphometric data of Haefeli-Bleuer and Weibel (18). The gas exchange surface area is proportional to the ratio of inner to outer cross-sectional areas and hence to the ratio of alveolar volume to duct volume for each airway. Proximal acinar airways have less alveolization and hence less gas exchange surface area per unit length.

- Copyright © 2011 the American Physiological Society