Journal of Applied Physiology
Vol. 82, No. 6, pp. 1717-1718, June 1997

Invited Editorial on "A finite-element model of oxygen diffusion in the pulmonary capillaries"

Aleksander S. Popel

Department of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore, Maryland 21205

ARTICLE

REFERENCES

ARTICLE

THE TOTAL AMOUNT of oxygen transferred in the lung from the gas phase in the alveoli to the blood is of critical importance to the function of the organism as a whole as well as to its every organ and cell. This amount is proportional to the difference in the oxygen tension in the alveoli and the mean oxygen tension in the pulmonary capillary; the coefficient of proportionality is called the overall pulmonary diffusing capacity (DL). An analogous relationship is considered for other organs where oxygen is transferred from the blood to the tissue. The value of the organ diffusing capacity is a function of the blood-gas barrier morphology (e.g., thickness) in the lung, capillary-parenchymal cell morphology in other organs, and blood hematocrit in all organs. Experiments to establish the determinants of organ diffusing capacity for lung and other organs and the corresponding mathematical modeling have been carried out in parallel, with fruitful interchange of concepts and ideas developed for the pulmonary and systemic circulations (13, 16). Mathematical modeling of oxygen transport has become an important and powerful tool in investigating biophysical and physiological mechanisms of organ function (8, 10). It is instructive to follow the progress in the modeling developments leading to the work of Frank et al. (5) published in this issue of the Journal of Applied Physiology (p. 2036).

The inverse of the diffusing capacity characterizes the resistance to oxygen transport. In a landmark paper, Roughton and Forster (11) expressed the total transport pulmonary resistance to oxygen as the sum of in-series resistances of the membrane element (Dm) and of the red blood cells (RBCs) element. Therefore, they recognized that a fraction of the resistance to oxygen transport may be intracapillary. Two decades later, Hellums (7) formulated a mathematical model of oxygen transport from systemic capillaries with discrete RBCs represented in the model as cylindrical slugs; he showed that at a capillary hematocrit of 50% about one-half of the total resistance is intracapillary. Contribution of the plasma gaps between the RBCs was neglected. An elegant analysis of Clark et al. (2) provided a simple analytic expression for the intra-RBC portion of the transport resistance in terms of hemoglobin concentration, hemoglobin-oxygen reaction rate, PO₂ at 50% oxyhemoglobin saturation, and other pertinent parameters. Federspiel and Popel (4) extended the analysis of intracapillary transport by considering spherical RBCs and taking into account diffusion in the plasma gaps. They calculated the capillary mass transfer coefficient (a characteristic similar to the organ diffusing capacity but applied to a single capillary) as a function of hematocrit and predicted significant variation of the mass transfer coefficient with hematocrit. Wang and Popel (14) carried this analysis further by considering parachute-shaped RBCs. Groebe and Thews (6) found that taking into account the effect of RBC velocity leads to a mild increase of the predicted mass transfer coefficient. A comparative theoretical study of transport resistances along the pathway from the RBC to the mitochondrion in a group of skeletal muscles, diaphragms, and myocardia in different animals (adaptive species) based on a large body of physiological data confirms earlier predictions that a very large fraction of the total transport resistance, as much as 50%, is intracapillary (12).

The first mathematical model of pulmonary capillary oxygen transport with discrete RBCs was formulated by Federspiel (3), who considered spherical RBCs in a cylindrical capillary. He predicted that a significant portion of the membrane transport resistance, Dm¹, resides in the blood plasma gaps between the cells and, thus, is affected by inter-RBC distance or capillary hematocrit. This theoretical analysis prompted a morphological study in which Dm was estimated from micrographs using a measure of distance between RBC surface and alveolar surface, a morphometric method (15). Hsia et al. (9) considered a two-dimensional geometry with cylindrical RBCs positioned symmetrically between two planes representing the membrane and performed detailed calculations for DL and Dm as functions of capillary hematocrit; they also compared the predictions for Dm with the results of the morphometric method. The results were in good agreement for hematocrits >30% but diverged at smaller hematocrits.

The mathematical modeling study of Frank et al. (5) combines most of the features of the previous models; it considers equally spaced parachute-shaped RBCs in two-dimensional and axisymmetric geometries, and it also includes the effect of plasma protein concentration. The results are presented for diffusing capacities (calculated for a single pulmonary capillary) DL and Dm both per RBC and per 100 µm of capillary as functions of hematocrit (these representations are related by simple expressions, but their variation with hematocrit is qualitatively different). Increases by a factor of three or four are predicted for DL and Dm per 100 µm of capillary when hematocrit is increased from 10 to 50%. The effect of plasma protein concentration is minor. The predictions are then rescaled for the total lung, assuming that all pulmonary capillary units are identical, and are compared with experimental measurements of the membrane diffusing capacity at rest and moderate and peak exercise reported earlier. The results for the two-dimensional geometry show a reasonable agreement for the physiological range of hematocrits, considerably better than the results for the axisymmetric geometry. Interestingly, Federspiel (3) compared results of his calculations with spherical cells and axisymmetric capillary geometry with another set of experimental data and found them in good agreement.

Theoretical studies (3, 5, 9) make important predictions of diffusing capacities for a single pulmonary capillary. All of these studies predict that a major fraction of the membrane resistance is intracapillary and is related to the pathway of oxygen diffusion through the plasma gaps between adjacent RBCs and that this resistance varies significantly with hematocrit. The difficulty in comparing the predictions of these models to in vivo experimental data lies in the fact that data for individual pulmonary capillaries are not available, and data for whole lung are likely to be affected by multitude of factors not accounted for in a single capillary model, such as heterogeneities in capillary-tissue geometry, alveolar PO₂, capillary hematocrit, and flow distribution. Similar difficulty is evident in the analyses applied to the systemic circulation. The next stage in mathematical modeling should include these heterogeneity factors, thereby linking the microscopic single-capillary and the whole organ behavior. The emerging experimental data that include not only the effects of hematocrit but also of other factors, such as RBC shape (1, 17), and perhaps future measurements in the pulmonary microcirculation should provide the basis for substantive validation of the theory.

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