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(P_o) increases. It is shown that the ideally permeable case for both O₂ and CO₂ can be approximated by taking P_o ˜ 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.     

The effects of chemical kinetics on oxygen delivery to tissue.

A mathematical model has been formulated to analyze the effect of nonequilibrium kinetics on oxygen delivery to tissue. The model takes into account molecular diffusion, facilitated diffusion in the capillary blood, convection, chemical kinetics of O2 with hemoglobin, and the rate of metabolic consumption. A line iterative technique is described to solve numerically the resulting coupled system of nonlinear partial differential equations with physiologically relevant boundary and entrance conditions. With nonequilibrium kinetics the end-capillary PO2 is found to be lower than that in the venous blood. The effect is more pronounced during hypoxia and anemia. It is found that the tissue PO2 at the lethal corner decreases with the decrease in blood velocity, arterial PO2, hemoglobin concentration, P50, and increase in COHb concentration or metabolic rate, while the difference between end-capillary PO2 and venous PO2 increases, which reflects the effect of nonequilibrium kinetics on the delivery of O2 to tissue. Thus, the consideration of venous PO2 as an indicator of tissue PO2 in clinical and experimental studies may be questionable.     

A mathematical model for the computation of carboxyhaemoglobin in human blood as a function of exposure time.

A mathematical model is developed for the carbon monoxide (CO) uptake by the blood by taking into account the molecular diffusion, convection, facilitated diffusion and the non-equilibrium kinetics of CO with haemoglobin. The overall rate for the combination of CO with haemoglobin is derived by including the dissociation of CO from carboxyhaemoglobin (COHb). The resulting coupled system of nonlinear partial differential equation with physiologically relevant initial, entrance and boundary conditions is solved numerically. A fixed point iterative technique is used to deal with nonlinearities. The concentration of COHb in the blood is computed as a function of exposure time and ambient CO concentration. The COHb levels computed from our model are in good agreement with those measured experimentally. Also, results computed from our model give better approximation to the experimental values compared with the results from other models. The time taken by the blood COHb to attain 95% of its equilibrium value is computed. The COHb concentration in the blood increases with the increase in ventilation rate, association rate coefficient of CO with haemoglobin and total haemoglobin content in the blood, and with the decrease in dissociation rate coefficient of CO with haemoglobin and mean capillary blood PO2. It is found that the COHb level in the blood is not affected significantly because of endogenous production of CO in the body under normal condition. However, the effect may be significant in the patients with haemolytic anaemia.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

Factors determining temporal pattern of isobaric supersaturation

It is possible to produce a transient supersaturation or undersaturation in tissues and blood by sequentially breathing gases with different equilibration rates. If the ambient gas pressure is sufficiently high, the induced supersaturation can produce vascular bubbles. By means of the classical perfusion-dependent model of inert gas elimination, which assumes that the effects of diffusion are minimal, the magnitude of the total inert gas pressure can be predicted. If, however, the effects of diffusion cannot be ignored, the supersaturation could be substantially larger. This paper estimates the effects of diffusion in a Krogh cylinder on the supersaturation produced by suddenly changing the inert gas partial pressure in the blood. The results of these estimates indicate that diffusion plays a role in this transient supersaturation only in long Krogh cylinders with high blood flows. The effects of diffusion are further reduced by the finite time necessary to switch the inert gases in arterial blood. The conclusions are supported by experiments that measure vascular bubble production after a switch of the inert portion of the inspired gas. These experiments further show that the formation of vascular bubbles after such a switch cannot be entirely explained by the different diffusion constants of the gases used.     

The variance of the distribution of traversal times in a capillary bed

A random walk model of capillary tracer transit times is developed that treats simulataneously: plug flow in the capillary, radial and axial diffusion in the capillary cylinder and tissue annulus, and endothelial barriers to solute transport. The mean transit time is simply the volume of distribution divided by blood flow. Variance of transit times has additive terms for radial, axial, and barrier influences that are reduceable to variances of simpler models of capillary exchange. The dependence of variance on the solute diffusion coefficient is not monotonic, but has a minimum near 0·5 × 10?6 cm²/s for reasonable parameters and no barrier, Small molecules like inert gases are expected to have larger variances with higher diffusion coefficients, while larger molecules and barrier limited solutes will have the reverse dependence. Available literature data indicates that capillary heterogeneity will have a major influence on whole-body variance of transit times.     

A Mathematical Model of the Controlled Plant of the Réspiratory System

Ability to predict the dynamic response of oxygen, carbon dioxide tensions, and pH in blood and tissues to abrupt changes in ventilation is important in the mathematical modeling of the respiratory system. In this study, the controlled plant (the amount and distribution of O₂ and CO₂) of the respiratory system is modeled. Although the body tissues are divided into a finite number of “compartments” (three tissue groups), in contrast to earlier models, the blood and tissue gas tensions within each compartment are considered to be continuously distributed in time and in one spatial coordinate. The mass conservation equations for oxygen and carbon dioxide involved in the blood-tissue gas exchange are described by a set of partial differential equations which take into account convection of O₂ and CO₂ caused by the flow of blood as well as diffusion due to local tension gradients. Nonlinear algebraic equations for the dissociation curves, which take into account the Haldane and Bohr effects in blood, are used to obtain the relationships between concentrations and partial pressures. Time-variable delays caused by the arterial and venous transport of the respiratory gases are also included. The model so constructed successfully reproduced actual O₂ and CO₂ tensions in arterial blood, and in muscle venous and mixed venous blood when ventilation was abruptly changed.     

The diffusive transport of gibberellins and abscisic acid through the aleurone layer of germinating barley grain: a mathematical model

A mathematical model of the diffusive transport of abscisic acid (ABA) and gibberellins (GAs) through the aleurone layer of barley (Hordeum vulgare L.) grain is presented. The model consists of two partial differential equations describing the accumulation of phytohormone in the apoplastic and symplasmic compartments of the aleurone layer, both spatially and temporally. The mathematical model contains the morphology of the barley grain and the physicochemical properties of the two phytohormones. A mathematical derivation of the accumulation ratios for the two phytohormones between the symplast and apoplast under equilibrium conditions resulted in different distribution mechanisms for GAs and ABA. A sensitivity analysis of the accumulation ratio for GAs indicated high sensitivity to the apoplastic pH and the membrane potential, whereas the accumulation ratio for ABA proved to be most sensitive to the pH difference between the apoplast and symplast. The diffusive transport time for GAs to the basal site of the aleurone layer as calculated with the mathematical model is within a physiologically plausible timescale according to experimental data from the literature. Abscisic acid cannot be transported by diffusion to the end of the aleurone layer as quickly as GAs, according to model simulations. Therefore, the functional role of ABA in germination is likely to be in the vicinity of the embryo.     

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.     

An electrochemical model of the transport of charged molecules through the capillary glycocalyx

An electrochemical theory of the glycocalyx surface layer on capillary endothelial cells is developed as a model to study the electrochemical dynamics of anionic molecular transport within capillaries. Combining a constitutive relationship for electrochemical transport, derived from Fick's and Ohm's laws, with the conservation of mass and Gauss's law from electrostatics, a system of three nonlinear, coupled, second-order, partial, integro-differential equations is obtained for the concentrations of the diffusing anionic molecules and the cations and anions in the blood. With the exception of small departures from electroneutrality that arise locally near the apical region of the glycocalyx, the model assumes that cations in the blood counterbalance the fixed negative charges bound to the macromolecular matrix of the glycocalyx in equilibrium. In the presence of anionic molecular tracers injected into the capillary lumen, the model predicts the size- and charge-dependent electrophoretic mobility of ions and tracers within the layer. In particular, the model predicts that anionic molecules are excluded from the glycocalyx at equilibrium and that the extent of this exclusion, which increases with increasing tracer and/or glycocalyx electronegativity, is a fundamental determinant of anionic molecular transport through the layer. The model equations were integrated numerically using a Crank-Nicolson finite-difference scheme and Newton-Raphson iteration. When the concentration of the anionic molecular tracer is small compared with the concentration of ions in the blood, a linearized version of the model can be obtained and solved as an eigenvalue problem. The results of the linear and nonlinear models were found to be in good agreement for this physiologically important case. Furthermore, if the fixed-charge density of the glycocalyx is of the order of the concentration of ions in the blood, or larger, or if the magnitude of the anionic molecular valence is large, a closed-form asymptotic solution for the diffusion time can be obtained from the eigenvalue problem that compares favorably with the numerical solution. In either case, if leakage of anionic molecules out of the capillary occurs, diffusion time is seen to vary exponentially with anionic valence and in inverse proportion to the steady-state anionic tracer concentration in the layer relative to the lumen. These findings suggest several methods for obtaining an estimate of the glycocalyx fixed-charge density in vivo.     

Coupled water transport in standing gradient models of the lateral intercellular space.

A standing gradient model of the lateral intercellular space is presented which includes a basement membrane of finite solute permeability. The solution to the model equations is estimated analytically using the "isotonic convection approximation" of Segel. In the case of solute pumps uniformly distributed along the length of the channel, the achievement of isotonic transport depends only on the water permeability of the cell membranes. The ability of the model to transport water against an adverse osmotic gradient is the sum of two terms: The first term is simply that for a well-stirred compartment model and reflects basement membrane solute permeability. The second term measures the added strength due to diffusion limitation within the interspace. It is observed, however, that the ability for uphill water transport due to diffusion limitation is diminished by high cell membrane water permeability. For physiologically relevant parameters, it appears that the high water permeability required for isotonic transport renders the contribution of the standing gradient relatively ineffective in transport against an osmotic gradient. Finally, when the model transports both isotonically and against a gradient, it is shown that substantial intraepithelial solute polarization effects are unavoidable. Thus, the measured epithelial water permeability will grossly underestimate the water permeability of the cell membranes. The accuracy of the analytic approximation is demonstrated by numerical solution of the complete model equations.     

Simulations of steady quaternary gas diffusion between alveolar and blood compartments

Fick's law is usually employed to analyze diffusion phenomena in the lung. However, this law is only strictly applicable to two component mixtures. When there are more than two gases in the mixture, Stefan's equations are more appropriate. Under physiological conditions during respiration, at least three gases are involved (O₂, CO₂, N₂). Furthermore, helium is often added in studies on the effect of density on pulmonary mixing. Features of quaternary gas diffusion (O₂, CO₂, N₂, He), and computer simulations of physiologically relevant conditions are presented. The results of the simulations were found to be in agreement with results from studies on normal and disease lungs.     

Three-dimensional,unsteady simulation of alveolar respiration

A novel macroscopic gas transport model, derived from fundamental engineering principles, is used to simulate the three-dimensional, unsteady respiration process within the alveolar region of the lungs. The simulations, mimicking the single-breath technique for measuring the lung diffusing capacity for carbon-monoxide (CO), allow the prediction of the red blood cell (RBC) distribution effects on the lung diffusing capacity. Results, obtained through numerical simulations, unveil a strong relationship between the type of distribution and the lung diffusing capacity. Several RBC distributions are considered, namely: normal (random), uniform, center-cluster, and corner-cluster red cell distributions. A nondimensional correlation is obtained in terms of a geometric parameter characterizing the RBC distribution, and presented as a useful tool for predicting the RBC distribution effect on the lung diffusing capacity. The effect of red cell movement is not considered in the present study because CO does not equilibrate with capillary blood within the time spent by blood in the capillary. Hence, blood flow effect on CO diffusion is expected to be only marginal.     

Fluid flow and convective transport of solutes within the intervertebral disc

Previous experimental and analytical studies of solute transport in the intervertebral disc have demonstrated that for small molecules diffusive transport alone fulfils the nutritional needs of disc cells. It has been often suggested that fluid flow into and within the disc may enhance the transport of larger molecules. The goal of the study was to predict the influence of load-induced interstitial fluid flow on mass transport in the intervertebral disc.An iterative procedure was used to predict the convective transport of physiologically relevant molecules within the disc. An axisymmetric, poroelastic finite-element structural model of the disc was developed. The diurnal loading was divided into discrete time steps. At each time step, the fluid flow within the disc due to compression or swelling was calculated. A sequentially coupled diffusion/convection model was then employed to calculate solute transport, with a constant concentration of solute being provided at the vascularised endplates and outer annulus. Loading was simulated for a complete diurnal cycle, and the relative convective and diffusive transport was compared for solutes with molecular weights ranging from 400 Da to 40 kDa.Consistent with previous studies, fluid flow did not enhance the transport of low-weight solutes. During swelling, interstitial fluid flow increased the unidirectional penetration of large solutes by approximately 100%. Due to the bi-directional temporal nature of disc loading, however, the net effect of convective transport over a full diurnal cycle was more limited (30% increase). Further study is required to determine the significance of large solutes and the timing of their delivery for disc physiology.     

Diffusion of peroxynitrite in the presence of carbon dioxide.

Peroxynitrite, the reactive species formed in vivo by the reaction of nitric oxide with superoxide anion, is capable of diffusing across erythrocyte membranes via anion channels and passive diffusion (A. Denicola, J. M. Souza, and R. Radi, Proc. Natl. Acad. Sci. USA 95, 3566-3571, 1998). However, peroxynitrite diffusion could be limited by extracellular targets, with the reaction with CO(2) (k(2) = 4.6 x 10(4) at 37 degrees C and pH 7.4) the most relevant. Herein, we studied the influence of physiological concentrations of CO(2) on peroxynitrite diffusion across intact red blood cells. The presence of CO(2) inhibited the oxidation of intracellular oxyhemoglobin by externally added peroxynitrite. However, the inhibition by CO(2) decreased at increasing red blood cell densities. At 45% hematocrit, 1.3 mM CO(2) (in equilibrium with 24 mM bicarbonate, at pH 7.4 and 25 degrees C) only inhibited 30% of intracellular oxyhemoglobin oxidation. This partial inhibition was also observed in red blood cells pretreated with the anion exchanger inhibitor 4,4'-diisothiocyanatostilbene-2,2'-disulfonic acid, ruling out a competition between peroxynitrite and bicarbonate for the transport through the anion channel. A theoretical model was developed to estimate the diffusion distance and half-life of extracellular peroxynitrite before reacting with intracellular oxyhemoglobin, at different red blood cell densities, and in the presence or absence of CO(2). The theoretical model correlated well with the experimental data. Our results indicate that, even in the presence of CO(2), peroxynitrite is able to diffuse and reach the inside of the erythrocyte.     

Development of a novel intravascular oxygenator catheter: Oxygen mass transfer properties across nonporous hollow fiber membranes

Despite hypoxic respiratory failure representing a large portion of total hospitalizations and healthcare spending worldwide, therapeutic options beyond mechanical ventilation are limited. We demonstrate the technical feasibility of providing oxygen to a bulk medium, such as blood, via diffusion across nonporous hollow fiber membranes (HFMs) using hyperbaric oxygen. The oxygen transfer across Teflon® membranes was characterized at oxygen pressures up to 2 bars in both a stirred tank vessel (CSTR) and a tubular device mimicking intravenous application. Fluxes over 550 ml min^?1 m^?2 were observed in well‐mixed systems, and just over 350 ml min^?1 m^?2 in flow through tubular systems. Oxygen flux was proportional to the oxygen partial pressure inside the HFM over the tested range and increased with mixing of the bulk liquid. Some bubbles were observed at the higher pressures (1.9 bar) and when bulk liquid dissolved oxygen concentrations were high. High‐frequency ultrasound was applied to detect and count individual bubbles, but no increase from background levels was detected during lower pressure operation. A conceptual model of the oxygen transport was developed and validated. Model parametric sensitivity studies demonstrated that diffusion through the thin fiber walls was a significant resistance to mass transfer, and that promoting convection around the fibers should enable physiologically relevant oxygen supply. This study indicates that a device is within reach that is capable of delivering greater than 10% of a patient's basal oxygen needs in a configuration that readily fits intravascularly.     

Facilitated transport of oxygen in the presence of membranes in the diffusion path.

Most of the experimental observations on facilitated transport have been done with millipore filters, and all the theoretical studies have assumed homogeneous spatial properties. In striated muscle there exist membranes that may impede the diffusion of the carrier myoglobin. In this paper a theoretical study is undertaken to analyze the transport in the presence of membranes in the diffusion path. For the numerical computations physiologically relevant values of the parameters were chosen. The numerical results indicate that the presence of membranes tends to decrease the facilitation. For the nonlinear chemical kinetics of the reaction of oxygen with the carrier, this decrement also depends on the location of the membranes. At the higher oxygen concentration side of each membrane the flow of combined oxygen is transferred to the flow of dissolved oxygen. The reverse process occurs at the lower concentration side. Jump discontinuities of the concentration of the oxygen-carrier compound at each membrane are associated with these transfers. The decrement of facilitation is due to the cumulative effect of these jump discontinuities.     

Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels

The effect of blood velocity pulsations on bioheat transfer is studied. A simple model of a straight rigid blood vessel with unsteady periodic flow is considered. A numerical solution that considers the fully coupled Navier-Stokes and energy equations is used for the simulations. The influence of the pulsation rate on the temperature distribution and energy transport is studied for four typical vessel sizes: aorta, large arteries, terminal arterial branches, and arterioles. The results show that: the pulsating axial velocity produces a pulsating temperature distribution; reversal of flow occurs in the aorta and in large vessels, which produces significant time variation in the temperature profile. Change of the pulsation rate yields a change of the energy transport between the vessel wall and fluid for the large vessels. For the thermally important terminal arteries (0.04-1 mm), velocity pulsations have a small influence on temperature distribution and on the energy transport out of the vessels (8 percent for the Womersley number corresponding to a normal heart rate). Given that there is a small difference between the time-averaged unsteady heat flux due to a pulsating blood velocity and an assumed nonpulsating blood velocity, it is reasonable to assume a nonpulsating blood velocity for the purposes of estimating bioheat transfer.     

Growth of a single cell

A theory of growth of a cell which takes up nutrients by diffusion or active transport is discussed. The main conclusion is that the volume should grow at least as fast as the third power of the time. Existing experimental evidence is not a conclusive test of the theory, and further experiments to test it are proposed.