Abstract
Frank, Andreas O., C. J. Charles Chuong, and Robert L. Johnson. A finiteelement model of oxygen diffusion in the pulmonary capillaries. J. Appl. Physiol. 82(6): 2036–2044, 1997.—We determined the overall pulmonary diffusing capacity (Dl) and the diffusing capacities of the alveolar membrane (Dm) and the red blood cell (RBC) segments (De) of the diffusional pathway for O_{2} by using a twodimensional finiteelement model developed to represent the sheetflow characteristics of pulmonary capillaries. An axisymmetric model was also considered to assess the effect of geometric configuration. Results showed the membrane segment contributing the major resistance, with the RBC segment resistance increasing as O_{2} saturation (
 finiteelement method modeling
 hematocrit
 plasma protein concentration
oxygen uptake within the lungs, from the alveolar air space to a hemebinding site on a hemoglobin molecule inside of a red blood cell (RBC), is dictated by the diffusion characteristics of this pathway and the chemical reactions within the RBC. Conceptually, the total resistance for the oxygen uptake in this pathway (1/Dl, where Dl is lung diffusing capacity) can be expressed as the algebraic sum of that, due to a membrane segment (1/Dm, where Dm is membrane diffusing capacity) and an RBC segment [1/(θ ⋅ Vc), where θ is the specific rate of gas uptake by RBCs and Vc is the pulmonary capillary blood volume]. This was originally defined by the Roughton and Forster equation (18) written as
The lumpedparameter representation of Eq.1 was a conceptual milestone, which allowed Dm and Vc to be quantified from experimental determination of Dl for CO at different oxygen tensions (18). Such data could be translated into oxygen diffusing capacities. Equation 1, however, does not address the spatially distributed nature of the oxygen gas transport process. Federspiel (8) applied a finitedifference numerical method to calculate the Dl by assuming spherical RBCs uniformly spaced inside a cylindrical capillary tube. However, capillaries in the lung are sandwiched between sheets of alveolar membrane as described by sheet flow (9); thus the capillary tube configuration may not accurately reflect the airtissue barrier across which gas transport must occur in the lungs and may lead to an overestimation of diffusing capacities. Wang and Popel (22) studied the oxygen release in the systemic microcirculation using a finiteelement model considering various RBC shapes. Both studies considered the transient diffusion of oxygen and oxyhemoglobin within the RBC coupled with the chemical reaction between oxygen and hemoglobin.
In this paper, we present a model based on the finiteelement method (FEM) that describes the transient oxygen transport in the pulmonary capillaries from the alveolar air space to the RBCs. We considered a modified twodimensional (2D) geometry with parachuteshaped RBCs in a parallelsided channel, which incorporates the available surface area at the airtissue barrier and the RBC wall. This modified 2D geometry was used to represent the sheetflow configuration of the pulmonary capillaries (9). To assess the effects of the capillary geometry, an axisymmetric model was also considered. The contributions to the diffusional resistance imposed by the membrane and RBC segments were determined and expressed as their respective diffusing capacities; from these, the total diffusing capacity was calculated. Effects of varying Hct and varying plasma protein concentrations on diffusive transport were also examined.
METHODS
Geometric model of a typical capillary segment. A modified 2D model with parachuteshaped RBCs in a parallelsided channel was used to represent the sheetflow characteristics (9) of pulmonary capillary blood flow. The model incorporated the available surface area for gas transport at the airtissue barrier and the RBC wall. The model geometry consists of a cross section through the longitudinal axis of a typical pulmonary capillary segment (100 μm length) containing a variable number of equally spaced RBCs depending on the segmental Hct (Fig.1 A).
The parachute shape of the RBC was digitized from a photograph of Skalak and Branemark (20), fitted with cubic splines, and then used as the cross section for all the RBCs in the capillary segment (Fig.1 A). Human RBCs are known to have a mean volume of ∼97 μm^{3} and a mean surface area of ∼137 μm^{2}(10), and it is important to incorporate these values into the model, since they can greatly affect the amount of transient gas transport. The 2D planar model was thus modified to have an effective depth of 4.92 μm, resulting in an RBC volume of 103 μm^{3} and an effective surface area of 125 μm^{2}. Note that the effective surface area considers only that which is used for gas transport at the RBC perimeter along the peripheral surface and does not include the area of the front or back faces of the RBC.
Assuming that the RBCs are equally distributed within a pulmonary capillary bed, we can study the oxygen diffusion characteristics by examining a typical unit segment, i.e., one RBC in its surrounding tissueplasma barrier and the alveolar air. Additionally, we have taken advantage of the symmetry with respect to thexaxis and only modeled onehalf of such a typical unit segment (Fig.1 A). Therefore, the model consists of three different regions (Fig.1 B):1) the bloodgas tissue barrier,2) the plasma fluid within the capillary, and 3) the RBC, each with respective gasdiffusive properties. The effect of varying Hct was studied by adjusting the “unit length” of the capillary segment, which effectively changes the volume of plasma fluid but does not alter the volume of the RBC within the capillary segment (Fig.1 B). In this study, the Hct was varied from 10 to 50% by increments of 5%.
Axisymmetric representation. By using the same shape as the 2D model (Fig.1 B), an axisymmetric model was also constructed so that the effects of capillary geometry could be assessed. Specifically, the differences in diffusingcapacity parameters could be evaluated, resulting from differences in the surface area available for gas transport and geometric shape factors. For the axisymmetric model, the yaxis is now interpreted as the radial direction (Fig. 1,A andB). As with the 2D planar configuration, the unit length of the diffusion space was adjusted to simulate the effect of varying Hct. All of the radial dimensions were maintained to be the same as those in the 2D model to allow for a direct comparison of the geometric configurations. The resulting volume and the surface area of the RBCs for the axisymmetric case were 103 μm^{3} and 133 μm^{2}, respectively.
Passive diffusion in the membrane segment. Due to the low Peclet number in pulmonary capillary blood flow, we have neglected the convective transport (1, 8,13). When a reference frame moving with the RBC is used and the differences in the plasma fluid and RBC traveling speed are neglected, the passive diffusive transport in the membrane segment (bloodgas barrier and plasma fluid) can be described by a simple diffusion process written as
Facilitated diffusion within the RBC segment. The diffusive transport inside the RBC is described by a more complicated facilitated diffusion due to the oxygen and hemoglobin interaction. Under typical conditions in the microcirculation, the facilitated diffusion can increase the rate of oxygen transport almost twofold (13). Thus the passive diffusion of oxygen, the passive diffusion of oxyhemoglobin, and the reaction between oxygen and hemoglobin must be considered, which can be written as
The assumption of instantaneous equilibrium within the RBC is valid as long as the speed of the chemical reaction greatly exceeds that for the diffusive transport. This assumption, however, must break down just inside of the RBC membrane, since the hemoglobin molecule is impermeable to the membrane, whereas oxygen is permeable. The region where deviation from equilibrium occurs has been shown to be only a thin layer inside of the RBC membrane, with the layer thickness varying depending on both spatial position along the membrane and temporal position during the RBC transit (22); i.e., Wang and Popel reported extreme cases for deviations in
Initial and boundary conditions. The initial conditions require that the oxygen distribution (
Discretized FEM model and solution method. The diffusive transport for the membrane segment (bloodgas barrier and plasma regions) is governed byEq.
2, whereas for the RBC segment it is governed by Eq.
6. The physical properties used for each different region of the model are summarized in Table1. With isoparametric formulation, a typical discretized FEM model consists of ∼500 bilinear elements with nodal
Governing Eqs.
2 and6 were solved by using the NewtonRaphson iteration technique, with the time transient portion solved by applying the general form of the trapezoid rule (2). The analysis is to find, at each time step, the
Calculation of Dl, Dm, and De.
Through the transient, at each time step, the distribution of oxygen flux was calculated from
Effects of plasma protein concentration changes. We examined the effect of plasma protein concentration change on Dl, Dm, and De. Three levels of plasma protein concentration (5, 6.9, and 10 gm/100 ml), all within physiological range, were considered. Their respective gas diffusion coefficients (24) are listed in Table 1. At each level of protein concentration, we determined the effect of Hct for a range of 10–50% by increments of 5%.
RESULTS
Transient values in Dl, Dm, and De.
Values for De decreased during the RBC transit along the capillary because of the progressive fall in the number of reduced hemoglobin binding sites available for oxygen binding. Nevertheless, De remains relatively large throughout most of the capillary transit as compared with Dm. The transient variations for all three diffusing capacity parameters are shown as a function of increasing hemoglobin saturation (
Effects of protein concentration and Hct changes. An increase in protein concentration causes a fairly uniform reduction in both Dm and Dl throughout the RBC transit, whereas a decrease in protein concentration leads to a uniform elevation (Fig. 3,A andB, for Hct = 45%). The percent changes seen in Dm and Dl due to protein concentration changes, however, were small (∼2–4%) relative to the actual percent changes in protein concentration (28–45%). Similar results were obtained at all other Hct values. Results of mean diffusing capacities during the RBC transit, i.e.,D̅ l andD̅m per RBC diffusion space and per 100 μm of capillary, are given in Fig. 4,A–D. For the average plasma protein concentration,D̅ l per RBC diffusion space was 17.5 × 10^{−11}ml ⋅ min^{−1} ⋅ mmHg^{−1}at a Hct of 10% but it decreased to 12.2 × 10^{−11}ml ⋅ min^{−1} ⋅ mmHg^{−1}at a Hct of 50% (Fig. 4 A). The decrease at higher Hct is due to the competition among cells for the oxygen influx. TotalD̅ l for the entire 100 μm capillary blood volume was seen to increase from 58.5 × 10^{−11} to 204 × 10^{−11}ml ⋅ min^{−1} ⋅ mmHg^{−1}as the Hct increased from 10 to 50%. A progressively decreasing slope is seen after 35% Hct, indicating a gradual approach toward a plateau at higher Hct values (Fig. 4 B). Similar trends were seen for D̅m per RBC: it decreases from 25.8 × 10^{−11} to 16 × 10^{−11}ml ⋅ min^{−1} ⋅ mmHg^{−1}as the Hct varies from 10 to 50% because of the competition among cells (Fig. 4 C). TotalD̅m for the entire 100 μm capillary blood volume was seen to increase from 86.2 × 10^{−11} to 268 × 10^{−11}ml ⋅ min^{−1} ⋅ mmHg^{−1}as the Hct increased from 10 to 50%. A progressively decreasing slope is seen beyond 35% Hct, indicating a gradual approach toward a plateau (Fig. 4 D).
Axisymmetric vs. 2D configurations.For the axisymmetric cases, the mean diffusing capacities (D̅ land D̅m) were obtained, ranging from 2.0 to 2.4 times those of their 2D planar counterparts. This was due primarily to the increased surface area available for gas transport at the airtissue barrier with the former geometry. Further discussion on the differences between the axisymmetric and 2D cases is presented indiscussion.
DISCUSSION
Variability in transient Dm and De. The progressive importance of the RBC segment in total Dl can be seen in the distribution of oxygen flux through the RBC wall membrane (Fig.5). In an early stage of RBC transit (
Thus there is a gradual progression during the RBC transit toward a more uniform distribution in the oxygen flux across the RBC wall membrane and in the utilization of the plasma fluid with respect to oxygen transport. This results in the gradual decrease in Dm (Fig. 2), since, effectively, the diffusion of oxygen from the airtissue barrier into the RBC takes a longer path because of the increased regional resistance of the RBC. Therefore, the regional changes in De occurring throughout the transient cause the utilization of the RBC wall membrane and plasma fluid to be altered, which affects Dm.
Comparison with Dm measurements. To compare with available experimental measurements, we calculated the mean Dm at the total lung volume by using the following extrapolation
Our 2D FEM model has slightly overpredicted Dm,o _{2}because it assumed an ideal condition of uniform RBC distribution and spacing within the capillaries, which offers the optimal use of the membrane segment, i.e., highest values in Dm (5). It is known that there is spatial and temporal fluctuation in both RBC distribution and spacing in the pulmonary capillary bed under physiological conditions. For the resting state, the corresponding Hct was in the range of 18–30% (Fig. 6), somewhat lower than the physiological values of 28∼37% (3, 7) because of the overprediction in Dm,o _{2}. For the exercise state, this source of error is further exaggerated, and the deviation in the estimated Hct is probably even larger.
2D vs. axisymmetric model. For all cases with axisymmetric configuration, mean diffusing capacities of 2.0–2.4 times their 2D counterparts were obtained. Factors contributing to the difference in diffusing capacities between geometric configurations can be separated into two categories:1) the available surface area for gas transport, and 2) the geometric shape of the diffusion space. For the axisymmetric case, the available surface area at the airtissue barrier was 2.29 times the 2D case considering the same Hct; also, there was a slightly higher RBC surface area (1.06 times). Note that the unit length (Fig. 1,A andB) for the axisymmetric case was necessarily different from the 2D so that we could match their Hct values. The increased surface area, primarily that at the airtissue barrier, was responsible for most of the exaggerated diffusing capacity parameters. In contrast, for the axisymmetric configuration, the converging effect of the radially inward flux of oxygen would counteract the transport because of the progressively reducing area associated with the geometric shape. The effects due to geometric shape between the two configurations can be quantified in terms of Nusselt number (see below).
The calculated mean Dm values for the total lung derived from the axisymmetric cases were found to be significantly higher than the measured values, whereas the 2D cases more closely corresponded to the measurements (Fig. 6). This suggests that the 2D model is more representative of the oxygen transport in the pulmonary capillary bed. Our comparisons demonstrate that the geometric representation of the diffusion space needs to be carefully incorporated for accurate assessment of the diffusing capacity parameters.
Comparison of Nusselt numbers with other models. To compare with other mathematical models, we calculated the Nusselt number (Nu), a measure of mass transport conductance, defined as
The Nusselt number, written in terms of flux (flow/area), represents the ratio between the rates of actual mass transfer and the diffusive transfer. For the axisymmetric case, the Nusselt number was found to be 20% lower than its 2D counterpart at a Hct of 25%, suggesting a higher efficiency with the latter configuration, which is to mimic the sheetflow configuration. Note that the greater diffusing capacity parameters (∼2.2 times) for the axisymmetric configuration are primarily due to its exaggerated surface area at the airtissue barrier. Effect of surface area is removed in the calculation of Nusselt number, since it is based on flux and not flow (flux × area). Thus the differences in Nusselt number between the axisymmetric and 2D cases are due to the converging effect with the former, i.e., oxygen molecules are competing for the progressively reducing surface area available.
As with the diffusing capacity parameters, the Nusselt number from the present work is seen to be strongly dependent on Hct. Calculated mean Nusselt numbers in terms of Nu − Nu_{Hct=25} were plotted as a function of Hct with data from previous works (12, 16) in Fig.7 B. They reveal the dependency of the Nusselt number on Hct for varioussized capillaries. The data from Groebe and Thews (12), based on oxygen delivery with axisymmetric capillaries of a 5.5μm diameter, show a stronger dependency on Hct as compared with the present models (capillary diameter of 8 μm). On the other hand, data from Nair (16) represent larger vessel diameters (20–100 μm), showing a much smaller dependency on Hct. These comparisons suggest that the Nusselt number (oxygen mass transfer conductance) becomes more strongly dependent on Hct as the capillary vessel size decreases, and for small capillary vessels the Hct can have a significant effect on oxygen transport.
Comparison with RBC conductance θ. Using the meanD̅e per RBC throughout the entire transient, we calculated an equivalent θ value of 5.9 ml ⋅ min^{−1} ⋅ mmHg^{−1} ⋅ ml whole blood^{−1} for an Hct of 45%. This was obtained on the assumption of infinite reaction velocity between oxygen and hemoglobin. To calculate it, we have considered a transient analysis from 0 to 80%
Neglect of convective diffusion. Our model neglected convective diffusion of oxygen within the pulmonary capillary. This assumption was made with the consideration that the effect of convective diffusion is small compared with that of conductive diffusion. Two possible mechanisms could contribute to the convective diffusion within the capillary bed:1) relative motion between the RBC and the blood plasma, and 2) relative motion between the RBC membrane and its interior contents.
Convection due to the former mechanism would lead to increased oxygen conductance of the membrane segment, which could reduce the role of the membrane segment as the limiting resistance of the pulmonary diffusion process (13). Flow velocity at capillary is relatively low (∼0.2 cm/s), with a low Reynolds number (∼0.001) (4). The relative motion of plasma fluid with respect to the RBC, particularly that at the cellendothelial gap, is known to be nonuniform with a complex circulating pattern associated with the propulsion of the RBCs (4). The plasma fluid moves at a speed lower than that of the RBC as a consequence of the noslip velocity boundary condition at the capillary wall. With an effective Peclet number of <1 (8), the enhanced mixing due to the stirring motion in plasma should not appreciably affect the diffusive transport of dissolved gasses such as oxygen (1), especially for higher Hct values (13).
The second mechanism considers the relative motion between the RBC membrane and the cytoplasm within, known as “tank treading” (11). It occurs within the RBC segment of the diffusion space and would thus tend to increase the RBC conductance. An increase of the RBC conductance would tend to decrease its effect on either Dl or Dm, due to its already relatively high value.
Summary. We determined diffusing capacities (Dl, Dm, and De) for oxygen using a 2D FEM, developed to represent the sheetflow configuration of pulmonary capillaries. Results showed the membrane segment contributing the major resistance, with the RBC segment resistance increasing as So _{2} rises during the RBC transit. Both Dm and Dlincreased as the Hct was increased but gradually approached a plateau as the Hct exceeded 35%. Both Dm and Dl were found to be relatively insensitive to changes in plasma protein concentration. Axisymmetric results showed similar trends to the 2D model for all Hct and plasma protein concentrations but consistently overestimated the diffusing capacities by ∼2.2 times, primarily due to the exaggerated airtissue barrier surface area. The 2D model correlated reasonably well with experimental data and can better represent the oxygen uptake at the pulmonary capillary bed.
Acknowledgments
The authors thank the reviewers for very constructive comments and suggestions.
Footnotes

Address for reprint requests: C. J. C. Chuong, Biomedical Engineering Program, PO Box 19138, 501 W 1st St., EL220, Univ. of Texas at Arlington, Arlington TX 76019.

This work was supported in part by National Heart, Lung, and Blood Institute Grant T32HL07362 and by National Science Foundation Grant BCS9008455.
 Copyright © 1997 the American Physiological Society