A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries

Theoretical analysis of the determinants of lung oxygen diffusing capacity

The process of pulmonary oxygen uptake is analyzed to obtain an explicit equation for lung oxygen diffusing capacity in terms of hematocrit and pulmonary capillary diameter. An axisymmetric model with discrete cylindrical erythrocytes is used to represent radial diffusion of oxygen from alveoli through the alveolar-capillary membrane into pulmonary capillaries, through the plasma, and into erythrocytes. Analysis of unsteady diffusion due to the passage of the erythrocytes shows that transport of oxygen through the alveolar-capillary membrane occurs mainly in the regions adjacent to erythrocytes, and that oxygen transport through regions adjacent to plasma gaps can be neglected. The model leads to an explicit formula for diffusing capacity as a function of geometric and oxygen transport parameters. For normal hematocrit and a capillary diameter of 6.75 μm, the predicted diffusing capacity is 102 ml O₂ min⁻¹ mmHg⁻¹. This value is 30-40% lower than values estimated previously by the morphometric method, which considers the total membrane area and the specific uptake rate of erythrocytes. Diffusing capacity is shown to increase with increasing hematocrit and decrease with increasing capillary diameter and increasing thickness of the membrane. Simulations of pulmonary oxygen uptake in humans under conditions of exercise or hypoxia based show closer agreement with experimental data than previous models, but still overestimate oxygen uptake. The remaining discrepancy may reflect effects of heterogeneity of perfusion and ventilation in the lung.

Diffusing capacity reexamined: relative roles of diffusion and chemical reaction in red cell uptake of O2, CO, CO2, and NO

This paper presents an analytical expression for the diffusing capacity (Θt) of the red blood cell (RBC) for any reactive gas in terms of size and shape of the RBC, thickness of the unstirred plasma layer surrounding the RBC, diffusivities and solubilities of the gas in RBC and boundary layer, hematocrit, and the slope of the dissociation curve. The expression for Θthas been derived by spatial averaging of the fundamental convection-diffusion-reaction equation for O2in the RBC and has been generalized to all cell shapes and for other reactive gases such as CO, NO, and CO2. The effects of size and shape of the RBC, thickness of the unstirred plasma layer, hemoglobin concentration, and hematocrit on Θthave been analyzed, and the analytically obtained expression for Θthas been validated by comparison with different sets of existing experimental data for O2and CO2. Our results indicate that the discoidal shape of the human RBC with average dimensions of 1.6-μm thickness and 8-μm diameter is close to optimal design for O2uptake and that the true reaction velocity in the RBC is suppressed significantly by the mass transfer resistance in the surrounding unstirred layer. In vitro measurements using rapid-mixing technique, which measures Θtin the presence of artificially created large boundary layers, substantially underpredicts the in vivo diffusing capacity of the RBC in the diffusion-controlled regime. Depending on the conditions in the RBC, uptake of less reactive gases (such as CO) undergoes transition from reaction-limited to diffusion-limited regime. For a constant set of morphological parameters, the theoretical expression for Θtpredicts that Θt,NO> Θt,CO2> Θt,O2> Θt,CO.

A numerical model for the oxygenation of blood in lung capillaries--effect of nth order one-step kinetics of oxygen uptake by haemoglobin

A numerical model is described for the oxygenation of blood in lung capillaries by considering the transport mechanisms of molecular diffusion, convection and the facilitated diffusion due to the presence of haemoglobin. In order to represent the oxygen dissociation curve accurately in the model, the nth order one-step kinetics of oxygen uptake by haemoglobin has been used. The resulting system of coupled, non-linear partial differential equations is solved numerically. It is shown that the blood is required to traverse a larger distance in the capillary before becoming fully oxygenated with nth order one-step kinetics in comparison to first-order one-step kinetics.

Pulmonary diffusing capacity: implications of two-phase blood flow in capillaries

The classical view of oxygen (O2) uptake in pulmonary capillaries assumes implicitly that capillary blood can be regarded as a continuous homogeneous hemoglobin solution. In this study a theoretical model was used to examine the role played by the particulate (two-phase) nature of blood on pulmonary oxygen exchange. Red cells were modelled as discrete hemoglobin (Hb) containing spheres flowing in single file suspension through a cylindrical capillary surrounded by a uniform annulus of alveolar tissue. The model accounted for the free diffusion of O2 from alveolar air space through tissue and plasma, free and Hb facilitated diffusion of O2 inside red cells, and the intracellular kinetics of O2-Hb binding. Oxygen uptake was driven by a specified O2 tension at the alveolar surface. The computed pulmonary diffusing capacity (DLO2) decreased with increasing spacing (Ls) between red cells. The reduction in DLO2 with increasing Ls was marshalled more by a reduction in membrane diffusing capacity (DMO2), than by the reduction in erythrocyte diffusing capacity (DeO2). The dependence of DMO2 on cell spacing stemmed from the manner in which O2 flowed across the alveolar surface into the discrete sinks (red cells) within the capillaries. The degree to which Ls influenced DMO2 was dependent on tissue and plasma layer thickness relative to red cell dimensions. The results indicate that the functional area of the alveolo-capillary membrane for O2 exchange depends on the red cell content of capillaries. Thus, DMO2 is not dictated solely by the morphology of the exchange apparatus (and physical parameters), but has functional determinants as well.

A mathematical model for the computation of the oxygen dissociation curve in human blood

The mathematical relations developed by various researchers for the oxygen dissociation curve are reviewed. Using well-known mechanisms of chemical kinetics of various species in the blood, we have developed a mathematical formula to compute the oxygen dissociation curve in the blood showing its dependence on the pH and PCO2. The functional form, proposed here, is much simpler in comparison to those available in the literature for use in the mathematical modelling of O2 transport in the pulmonary and systemic circulations. In the process, the well-known Hill's equation has been generalized showing an explicit dependence on PCO2 and pH. It is shown that the oxygen dissociation curve computed from our comparatively simpler equation, fits in fairly well with the documented data and shows realistic shift with PCO2 and pH.

A numerical study of the nonsteady transport of gases in the pulmonary capillaries

A mathematical model is formulated for simulating the unsteady transport of gases in the blood flowing through the pulmonary capillaries. The formulation takes into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion of the species due to haemoglobin. A time dependent situation is created by allowing to vary suddenly the partial pressures of the gases either in the venous blood or in the alveolar air. A numerical technique is described to solve the resulting time-dependent system of nonlinear coupled partial differential equations with the physiologically relevant boundary, entrance and initial conditions. The time required by the gases to achieve equilibrium is computed. It is shown that the dissolved oxygen takes longest in reaching equilibration whereas the carbon dioxide is the fastest. The various physiologically relevant unsteady situations have been examined.

The process of gas exchange in the pulmonary circulation incorporating the contribution of axial diffusion

A mathematical model is made to describe the process of gas exchange in the pulmonary circulation incorporating the contribution of axial diffusion. The model takes into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion due to the presence of haemoglobin as a carrier of the gases. The mathematical formulation leads to a coupled system of non-linear elliptic partial differential equations. A numerical scheme is described to solve such a system. It is found that the axial diffusion does not have an appreciable effect on the transport of the species in the blood.

A numerical model for blood oxygenation in the pulmonary capillaries--effect of pulmonary membrane resistance

A study of the blood oxygenation in pulmonary capillaries is made by considering the transport mechanisms of molecular diffusion, convection and the facilitated diffusion due to the presence of haemoglobin. The resistance offered by the pulmonary membrane on the transport of gases has been incorporated. The resulting system of coupled, non-linear partial differential equations is solved numerically. It is found that, in the immediate neighbourhood of the entry, the amount of dissolved O2 decreases. This decreases further as the resistance offered by the pulmonary membrane increases. The rate of oxygenation of blood increases as the permeability coefficient for O2 (PO) increases. It is shown that the ideally permeable case for both O2 and CO2 can be approximated by taking PO approximately 10 cm/s. Further, it is shown that the oxygen takes longest and CO2 is the fastest to attain equilibration. The equilibration length increases as the resistance offered by the membrane increases. Finally, some of the pulmonary diseases such as pulmonary oedema and fibrosis have been analyzed.

Numerical simulation of pulmonary O2 and CO2 exchange

The process of gas exchange leading to the oxygenation of blood in pulmonary capillaries is simulated numerically, taking into account the main transport mechanisms of molecular diffusion, convection and the facilitated diffusion due to the presence of haemoglobin, as well as physiologically relevant boundary conditions and variable initial data. An algorithmic program to solve the relevant equations is run on a computer. It is found that, in the immediate neighbourhood of the entry, the amount of dissolved oxygen decreases, whereas the amount of carbaminohaemoglobin increases and the facilitated diffusion is more dominant over the molecular diffusion. Further, it is shown that (i) O2 takes longest and CO2 is the fastest to attain equilibration, (ii) the blood is completely oxygenated with one fifth part of its transit. Finally, the effect of various physiological parameters on equilibration length is examined.

Equivalence between one step kinetics and Hill's equation

One step kinetics between oxygen and haemoglobin are shown to be equivalent to the well-known Hill's equation. We have modified the one step kinetics while dealing with the mathematical modelling of simultaneous transport of oxygen and carbon-dioxide in the blood flowing through the pulmonary and the systemic capillaries. In the process, the Hill's equation has been modified showing an explicit dependence on PCO2. We have proposed comparatively simpler equations to represent Hb saturation with O2 and CO2 allowing for the interaction between the gases. It is shown that the oxygen dissociation curve, obtained from modified equations, fits in fairly well with the experimental data and shows realistic shift with PCO2. The results computed from our comparatively simpler equations based on physical laws, are in good agreement with those obtained from Kelman's empirical relations that are accepted in anaesthesia and respiratory physiology as providing very good matches to reality.

Transport of gases in pulmonary capillaries- a computation of equilibrium constants arising in the oxygen dissociation curve

A study of the simultaneous diffusion of O2 and CO2 in the presence of haemoglobin is made by considering the intermediate compound hypothesis of Adair. Our analysis exhibits explicit dependence of the fractional saturation (of haemoglobin with O2) on pCO2. It has been emphasized that discrepancies in the various numerically computed values of equilibrium constants arising in the oxygen dissociation curve may be due to inadequacy of the methods used to solve the normal equations obtained in the process of least-square curve-fitting, which are highly non-linear in character in the present context. It has been shown, by using the method suggested by Marquardt that the values of these constants lead to a saturation function which fits very well with the one based on documented data. The analysis further shows that under normal physiological conditions, the main bulk of oxygenated haemoglobin exists in the fully saturated form as HbO8 which also seems obvious from our computed values of equilibrium constants.