Finite-element analysis of oxygen transport in the systemic capillaries.

A mathematical model is formulated for the transport of O2 in the systemic capillaries and surrounding tissue. The model takes into account molecular diffusion, the convective effect of the blood, the nonlinear effects of oxyhaemoglobin, and the consumption of O2 in the metabolic process. A finite-element formulation for solving the equations governing the mass transfer in the capillary is described. A fixed-point iterative technique is used to deal with the nonlinearities in the model. The concentration of O2 is found to decrease from the axis of the capillary to the periphery of the tissue. It is shown that, owing to the nonlinear effects of oxyhaemoglobin, the partial pressure of oxygen (PO2) in the capillary and tissue increases. It is also shown that the tissue PO2 increases as (i) the arterial blood PO2 increases, (ii) the Péclet number increases, and (iii) the diffusive flux of O2 from the capillary decreases.     

Theoretical predictions of maximal oxygen consumption in hypoxia: effects of transport limitations

A Krogh-type model for oxygen transport is used to predict maximal oxygen consumption (V(.-) O(2max)) of human skeletal muscle under hypoxic conditions. Assumed values of capillary density, blood flow, and hemoglobin concentration are based on measurements under normoxic and hypoxic exercise conditions. Arterial partial pressure of oxygen is assumed to decrease with reductions in inspired partial pressure of oxygen (P(I)O(2)), as observed experimentally. As a result of limitations of convective and diffusive oxygen delivery, predicted V(.-) O(2max) values decline gradually as P(I)O(2) is reduced from 150 mmHg to about 80 mmHg, and more rapidly as P(I)O(2) is further reduced. At very low levels of P(I)O(2), V(.-) O(2max) is limited primarily by convective oxygen supply. Experimentally observed values of V(.-) O(2max) in hypoxia show significant dispersion, with some values close to predicted levels and others substantially lower. These results suggest that maximal oxygen consumption rates in hypoxia are not necessarily determined by oxygen transport limitations and may instead reflect reduced muscle oxygen demand.     

Simultaneous measurements of capillary diffusion and filtration exchange during shifts in filtration-absorption and at graded alterations in the capillary permeability surface area product (PS)

The diffusion exchange of Cr-EDTA, using the single injection indicator diffusion method, was followed simultaneously with estimations of the capillary filtration capacity (CFC) in an “isogravimetric” rat hindquarter preparation during artificial perfusion and maximal dilatation. Measurements were performed at constant flow and during 1) shifts in filtration-absorbtion, 2) alterations of perfused capillary wall area (graded rarification of capillary network by microsphere injection) and 3) during alterations of permeability (i.a. infusion of histamine). At maximal vasodilatation CFC was 0.037 ± 0.001 ml/min × mmHg × 100 g and PS for Cr-EDTA 5.67 ± 0.13 ml/min × 100 g. During filtration or absorbtion, Cr-EDTA transfer from vessels to interstitium changed only slightly but the situation may well be different for solute transfer from interstitium to vessels. Alterations in capillary wall area resulted in proportional changes in PS for Cr-EDTA while the CFC changes were always relatively smaller. Histamine increased CFC some threefold with a marked increase in protein transfer, while PS for Cr-EDTA increased only marginally. This hista mine effect could be ascribed mainly to an increase in the number of large pores which, because of their relative paucity, are of little importance for small molecular diffusion exchange but highly important for convective and macromolecular exchange.     

The effect of myoglobin on the oxygen concentration in skeletal muscle subjected to ischemia

A mathematical model of oxygen transport in skeletal muscle is utilized to investigate myoglobin's effect on the space-time distribution of oxygen in skeletal muscle subjected to ischemia. The model is formulated using the Krogh cylinder as the geometrical representation of the functional unit of transport, i.e., a capillary and the surrounding tissue it supplies. The model includes the convective and diffusive transport of oxygen within the capillary-tissue system, a nonlinear oxygen consumption rate, and the reversible reaction of oxygen with hemoglobin in capillary blood and with myoglobin in the muscle tissue. Myoglobin's role in the maintenance of an adequate oxygen supply for use by the muscle is demonstrated by observing the transient capillary tissue system's response to a 60 sec capillary occlusion. Solutions are obtained for the model with and without myoglobin's presence. A comparison of solutions is presented for two discrete points in the tissue region; namely, the so-called lethal corner which occurs at the venous end at the outer radius of the tissue cylinder and an average point found in the center of the tissue region. Three-dimensional oxygen distributions as a function of time are also presented.     

Finite-Element Analysis of Oxygen Transport in the Systemic Capillaries

A mathematical model is formulated for the transport of O₂ inthe systemic capillaries and surrounding tissue. The model takesinto account molecular diffusion, the convective effect of theblood, the nonlinear effects of oxyhaemoglobin, and the consumptionof O₂ in the metabolic process. A finite-element formulationfor solving the equations governing the mass transfer in thecapillary is described. A fixed-point iterative technique isused to deal with the nonlinearities in the model. The concentrationof O₂ is found to decrease from the axis of the capillary tothe periphery of the tissue. It is shown that, owing to thenonlinear effects of oxyhaemoglobin, the partial pressure ofoxygen (PO₂) in the capillary and tissue increases. It is alsoshown that the tissue PO₂ increases as (i) the arterial bloodPO₂ increases, (ii) the Péclet number increases, and(iii) the diffusive flux of O₂ from the capillary decreases.     

Oxygen supply to contracting skeletal muscle at the microcirculatory level: diffusion vs. convection

An adequate supply of oxygen is essential for the normal function of all cells. Because skeletal muscle cells have the ability to vary their oxygen demand by over an order of magnitude on going from rest to vigorous contraction, it is important that mechanisms be in place to ensure that the supply of oxygen is maintained at sufficient levels. Microcirculation plays a critical role in this process, as the terminal branches of this intricate network of blood vessels determine the distribution of perfusion, as well as the structural framework for diffusion. The oxygen supply depends on proper functioning of both the convective and diffusive components of the transport system. Convection is responsible for the long-range, rapid transport of oxygen by bulk flow of the blood and diffusion is the efficient mechanism for transport over the short distances between capillaries and muscle cells. Convective transport is dominated by the movement of red blood cells, as virtually all the oxygen at normal haematocrit is carried inside them, reversibly bound to haemoglobin. Over the years, specialized techniques, many of them video-based, have been developed for use in intravital microscopy to measure the parameters needed to quantify convection and diffusion in both capillaries and the larger microvessels, arterioles and venules. Most of our knowledge of oxygen transport in the microcirculation of muscle pertains to the resting condition, because one must be able to visualize the structures of interest, such as microvessels and muscle cells, and the large tissue movements that occur during contraction preclude measurements during that time. In resting muscle it has been found that the arterioles are the primary site of the diffusion of oxygen from the circulation, where the oxygen is utilized by nearby muscle cells or diffuses directly to nearby venules or capillaries. Diffusive interactions among neighbouring capillaries have also been observed. In contracting muscles, microvessels observed immediately following the period of stimulation exhibit enhancements of both convective (increased flow of red blood cells) and diffusive (increased perfused capillary surface area) transport. The use of computational models in the interpretation of experimental studies is leading to an increased understanding of the processes that underlie the oxygen transport system in skeletal muscle.     

Simultaneous measurements of capillary diffusion and filtration exchange during shifts in filtration-absorption and at graded alterations in the capillary permeability surface area products (PS).

The diffusion exchange of Cr-EDTA, using the single injection indicator diffusion method, was followed simultaneously with estimations of the capillary filtration capacity (CFC) in an "isogravimetric" rat hindquarter preparation during artificial perfusion and maximal dilatation. Measurements were performed at constant flow and during 1) shifts in filtration-absorbtion, 2) alterations of perfused capillary wall area (graded rarification of capillary network by microsphere injection) and 3) during alterations of permeability (i.a. infusion of histamine). At maximal vasodilatation CFC was 0.037 +/- 0.001 ml/min X mmHg X 100 g and PS for Cr-EDTA 5.67 +/- 0.13 ml/min X 100 g. During filtration or absorbtion, Cr-EDTA transfer from vessels to interstitium changed only slightly but the situation may well be different for solute transfer from interstitium to vessels. Alterations in capillary wall area resulted in proportional changes in PS for Cr-EDTA while the CFC changes were always relatively smaller. Histamine increased CFC some threefold with a marked increase in protein transfer, while PS for Cr-EDTA increased only marginally. This histamine effect could be ascribed mainly to an increase in the number of large pores which, because of their relative paucity, are of little importance for small molecular diffusion exchange but highly important for convective and macromolecular exchange.     

Accessory factors in the quantitative homeostasis of erythropoiesis

The participation of erythropoietin in the quantitative regulation of erythropoiesis is evident only when the oxygen supply to the tissues is clearly removed from normality. The role of the hormone as the sole regulator of the remarkable constancy of the hemoglobin mass becomes controversial as oxygen exchange approachs its normal range. There is at present a lack of positive evidence for the involvement of oxygen in erythrocytic homeostasis in the steady state. A new hypothesis concerning the existence of a complementary mechanism is presented here. Its action would be to correct changes in the hemoglobin mass not large enough to perturb tissue oxygenation and would provide an explanation for the fine regulation of red cell production related to erythrocyte daily renewal. The proposed mechanism involves an oxygen-independent factor (s). Acting as a regulator of the number of divisions in the normoblastic compartment, its activity would be turned on and off by changes in the concentration of circulating erythrocytes. These changes would be detected through the electrokinetic interactions between the charges on the erythrocyte membranes and those on the membrane of some fine nerve endings situated under the endothelium in a specialized capillary circuit.     

The process of gas exchange in systemic circulation in a hyperbaric environment: an analytical approach

A mathematical model is proposed to deal with the simultaneous transport of oxygen (O2) and carbon dioxide (CO2) in systemic capillaries and the surrounding tissue in a hyperbaric environment. The transport in the capillary region depends on molecular diffusion (radial as well as axial), the convective effect of the blood, and the saturation of haemoglobin with O2 and CO2. The corresponding equation in the tissue region describes the transport of the species due to radial and axial diffusion in the tissue and consumption of O2 in the metabolic process. The production of CO2 inside the tissue is incorporated through the respiratory quotient. The saturation of blood with O2 and CO2 have been approximated by linear functions to simulate the conditions of the hyperbaric environment. The resulting system of governing equations with the physiologically relevant boundary conditions is solved analytically. The concentration of O2 is shown to decrease from the core of the capillary to the tissue periphery, whereas the concentration of CO2 increases. It is shown that very little of the CO2 is transported radially. The location of the vulnerable point from the point of view of CO2 accumulation is found to be the corner (x = R2, z = L) situated at the periphery of the tissue near the venous end of the capillary. The accumulation of O2 and CO2 in the tissue is discussed in terms of various dimensionless parameters. It is found that the accumulation of CO2 increases whereas that of O2 decreases in the hyperbaric environment. Finally, it is surmised that one of the major causes of discomfort among divers could be excessive accumulation of CO2 in the tissue.     

Oxygen and lactic acid transport in skeletal muscle

A mathematical model of oxygen and lactic acid transport in skeletal muscle is used to test the effects of reactive hyperemia on oxygen and lactic acid concentrations following a period of ischemia. The model is based on the Krogh cylinder as the geometrical representation of the functional unit of transport, i.e., a capillary and the tissue it supplies. Included in the mathematical development of the model are the convective and diffusive transport of the chemical species, the nonlinear aspects of oxygen and lactic acid kinetics, and the reversible reaction of oxygen with hemoglobin in capillary blood and myoglobin in the tissue. The steady-state solution to the model is obtained first as the baseline for the study. Ischemia is then simulated by the cessation of capillary blood flow. This is followed by a reactive hyperemic response that is a function of the occlusion duration. The general effect of reactive hyperemia is to shorten the time intervals for initial return of tissue oxygen levels and the washout of accumulated lactic acid and to maintain tissue oxygen levels above steady-state values.     

Oxygen gradients in the microcirculation

As arterialized blood transits from the central circulation to the periphery, oxygen exits through the vessel walls driven by radial oxygen gradients that extend from the red blood cell column, through the plasma, the vessel wall, and the parenchymal tissue. This exit determines a longitudinal gradient of blood oxygen saturation whose extent is inversely related to the level of metabolic activity of the tissue, being small for the brain and considerable for skeletal muscle at rest where hemoglobin is only half-saturated with oxygen when blood arrives to the capillaries. Data obtained by a variety of methods show that the oxygen loss is too great to be explained by diffusion alone, and oxygen gradients measured in the arteriolar wall provide evidence that this structure in vivo is a very large oxygen sink, and suggests a rate of oxygen consumption two orders of magnitude greater than seen in in vitro studies. Longitudinal gradients in the capillary network and radial gradients in surrounding tissue also show a dependence on the metabolic rate of the tissue, being more pronounced in brain than in resting skeletal muscle and mesentery. Mean PO2 values increase from the postcapillary venules to the distal vessels of this network while radial gradients indicate additional oxygen loss. This circumstance may be due to pathways with higher flow having higher oxygen content than low flow pathways as well as possible oxygen uptake from adjacent arterioles. Taken together, these newer findings on oxygen gradients in the microcirculation require a reexamination of existing concepts of oxygen delivery to tissue and the role of the capillaries in this process.     

A Mechanobiological Model for Tissue Differentiation that Includes Angiogenesis: A Lattice-Based Modeling Approach

Mechanobiological models have previously been used to predict the time course of the tissue differentiation process, with the local mechanical environment as the regulator of cell activity. However, since the supply of oxygen and nutrients to cells is also a regulator of cell differentiation and oxygen diffusion is limited to few hundred micrometers from capillaries, the morphology of the new vascular network may also play a critical role in the process. In this paper, a computational model for tissue differentiation based on the local mechanical environment and the local vascularity is presented. A regular lattice is used to simulate cell activity (migration, proliferation, differentiation, apoptosis, and angiogenesis). The algorithm for capillary network formation includes mechanoregulation of vessel growth. A simulation of tissue differentiation in a bone/implant gap under shear was performed. The model predicts capillary networks similar to those found in experimental studies and heterogeneous patterns of tissue differentiation, which are influenced by the morphology of the capillary network. Higher mechanical loads caused slower vascular development and delayed bone tissue formations.     

Oxygen gradients in the microcirculation

Early in the last century August Krogh embarked on a series of seminal studies to understand the connection between tissue metabolism and mechanisms by which the cardiovascular system supplied oxygen to meet those needs. Krogh recognized that oxygen was supplied from blood to the tissues by passive diffusion and that the most likely site for oxygen exchange was the capillary network. Studies of tissue oxygen consumption and diffusion coefficient, coupled with anatomical studies of capillarity in various tissues, led him to formulate a model of oxygen diffusion from a single capillary. Fifty years after the publication of this work, new methods were developed which allowed the direct measurement of oxygen in and around microvessels. These direct measurements have confirmed the predictions by Krogh and have led to extensions of his ideas resulting in our current understanding of oxygenation within the microcirculation. Developments during the last 40 years are reviewed, including studies of oxygen gradients in arterioles, capillaries, venules, microvessel wall and surrounding tissue. These measurements were made possible by the development and use of new methods to investigate oxygen in the microcirculation, so mention is made of oxygen microelectrodes, microspectrophotometry of haemoglobin and phosphorescence quenching microscopy. Our understanding of oxygen transport from the perspective of the microcirculation has gone from a consideration of oxygen gradients in capillaries and tissue to the realization that oxygen has the ability to diffuse from any microvessel to another location under the conditions that there exists a large enough PO(2) gradient and that the permeability for oxygen along the intervening pathway is sufficient.     

Effect of histamine on microvasculature of isolated dog gracilis muscle.

Isogravimetric capillary pressure (Pci), capillary filtration coefficient (CFC), and plasma protein concentration were measured before and during adminisistration of histamine in an isolated, independently perfused canine gracilis muscle. Histamine produced an average decrease in Pci of 14.1 mmHg, an increase in CFC of 36-fold, and an increased rate of plasma protein escape of at least 24-fold. These results suggest that histamine reduces the reflection coefficient for protein at the capillary wall and are consistent with predictions of the theory of restricted diffusion assuming that 1-2.5% of available pores increase in radious from 40 to 240 A.     

The thickness of the alveolar capillary wall in guinea-pigs at high and low altitude

The thickness of the alveolar-capillary wall was measured in the lungs of seven guinea-pigs born and living at La Raya, in the Peruvian Andes, at an altitude of 4200 m and in seven sea-level representatives of the same species. This was achieved by carrying out morphometric studies on electron micrographs to obtain the so-called arithmetic and harmonic mean thicknesses. The arithmetic mean thickness was always the larger, this being due to the greater emphasis which the technique employed places on the copious amounts of connective tissue in the interstitial space of the alveolar capillary wall in this species. These thicker portions of the alveolar wall are not concerned with gaseous diffusion. The harmonic mean thickness probably gives a more physiologically realistic estimate of the magnitude of the diffusion barrier to oxygen. This proved to be smaller in the animals from high altitude and may facilitate diffusion of oxygen from alveolar spaces to blood, thus making less steep the "oxygen cascade" from inspired air to mitochondria.     

A theoretical study of restricted convection-diffusion as applied to blood-tissue barrier exchange.

This paper handles a model of the capillary function in the exchange of uncharged molecules between the blood and the tissue. The capillary system is subdivided into a filtering and a reabsorbing part. The exchange is assumed to occur through channels which are described in operational terms as pores. Through these pores there is a transport of solutes by concomitantly acting convection and diffusion influenced by a steric hinderance (restricted convection-diffusion). The outflux of glucose and raffinose is enhanced in the filtering pores, raffinose relatively more than glucose. In the reabsorbing pores the outward diffusion is hindered to some extent, raffinose relatively more than glucose. It is shown that the net effect of filtration and reabsorption is to increase the outflux of raffinose as compared to that of glucose. This mechanism may explain why glucose and raffinose and other small molecules appear to pass across capillary walls in proportion to their free diffusion coefficients and not in proportion to their restricted diffusion coefficients.     

A Theoretical Study of Restricted Convention-Diffusion as Applied to Blood-Tissue Barrier Exchange

This paper handles a model of the capillary function in the exchange of uncharged molecules between the blood and the tissue. The capillary system is subdivided into a filtering and a reabsorbing part. The exchange is assumed to occur through channels which are described in operational terms as pores. Through these pores there is a transport of solutes by concomitantly acting convection and diffusion influenced by a steric hinderance (restricted convection-diffusion). The outflux of glucose and raffinose is enhanced in the filtering pores, raffinose relatively more than glucose. In the reabsorbing pores the outward diffusion is hindered to some extent, raffinose relatively more than glucose. It is shown that the net effect of filtration and reabsorption is to increase the outflux of raffinose as compared to that of glucose. This mechanism may explain why glucose and raffinose and other small molecules appear to pass across capillary walls in proportion to their free diffusion coefficients and not in proportion to their restricted diffusion coefficients.     

基于MATLAB的微循环数值分析－毛细血管几何特性对氧输送特性的影响

目的　考察毛细血管弹性和血管扭曲度对氧分布的影响,从而进一步考察肿瘤组织内形成乏氧区的机理,为肿瘤的放射线疗法提供理论帮助。方法　通过一维毛细血管模型与氧传输模型相结合,考察了Krogh模型和扭曲血管模型内毛细血管与组织内的氧分压分布。血管截面积尺寸由一维模型获得并传输给组织模型。通过有限元分析计算氧分布。结果　获得了在一定血流压力和不同毛细血管初始半径下,血管沿流动方向的尺寸变化,并计算了Krogh模型和扭曲血管模型中,毛细血管与组织内的氧分压分布。结论　当毛细血管较小时,血管弹性对氧分压分布的影响很小;随着管径增大,血管弹性的影响也加大。另外,随着毛细血管扭曲度的增加,组织内氧分压分布的不均一性也增加,计算结果与相关文献的结果一致。该工作将有助于进一步考察肿瘤组织内的氧传输。     

Evaluation of the resistance of membrane and erythrocytes to oxygen transport in pulmonary capillaries

The conductance of the diffusion pathway for oxygen in pulmonary capillaries is usually described in terms of two lumped components; the membrane and erythrocyte diffusing capacities. In this study the relative importance of these two components was investigated theoretically over a wide range of alveolar oxygen partial pressures. It was found that the membrane diffusing capacity is largely independent of the alveolar oxygen partial pressure and the erythrocyte diffusing capacity increases steeply with a decrease in alveolar oxygen partial pressure. The ratio of the resistance of the membrane to that of the erythrocytes to oxygen transport is a strong function of the alveolar oxygen partial pressure [1.5 (PAO2 = 30 mmHg); 1.0 (PAO2 = 100 mmHg); 0.5 (PAO2 = 700 mmHg)].     

Diffusion limits of an in vitro thick prevascularized tissue

Although tissue engineering promises to replace or restore lost function to nearly every tissue in the body, successful applications are currently limited to tissue less than 2 mm in thickness. in vivo capillary networks deliver oxygen and nutrients to thicker (> 2 mm) tissues, suggesting that introduction of a preformed in vitro vascular network may be a useful strategy for engineered tissues. This article describes a system for generating capillary-like networks within a thick fibrin matrix. Human umbilical vein endothelial cells, growing on the surface of microcarrier beads, were embedded in fibrin gels a known distance (Delta = 1.8-4.5 mm) from a monolayer of human dermal fibroblasts. The distance of the growth medium, which contained vascular endothelial growth factor and basic fibroblast growth factor, from the beads, C, was varied from 2.7 to 7.2 mm. Capillaries with visible lumens sprouted in 2-3 days, reaching lengths that exceeded 500 microm within 6-8 days. On day 7, capillary network formation was largely independent of C; however, a strong inverse correlation with Delta was observed, with the maximum network formation at Delta = 1.8 mm. Surprisingly, the thickness of the gel was not a limiting factor for oxygen diffusion as these tissue constructs retained a relatively high oxygen tension of > 125 mmHg. We conclude that diffusion of oxygen in vitro is not limiting, allowing the development of tissue constructs on the order of centimeters in thickness. In addition, diffusion of fibroblast-derived soluble mediators is necessary for stable capillary formation, but is significantly impeded relative to that of nutrients present in the medium.