A mathematical theory of capillary exchange as a function of tissue structure

A mathematical theory of the process of the exchange of substances between the blood in the capillaries of a homogeneous tissue and the extracellular space, and between the extracellular space and the cells is developed. An ideal geometry of the tissue is assumed, based to some extent on recent anatomical work concerning the functional distinction between two types of capillaries, the arteriolo-venular and the true capillaries. Equations are developed relating the concentration in the arterial blood to the mean capillary concentration, the concentration at the wall of the capillary in the extracellular space, and the average concentration in the extracellular space, and also relating the cellular concentration to the average extracellular concentration. The solutions of the equations are given for certain special cases and numerical results obtained. It is shown that the average extracellular concentration is a sensitive function of the permeability of the capillary wall and also is strongly influenced by the diffusion coefficient of the extracellular space. Furthermore, it is shown that the speed with which the average extracellular concentration approaches the steady state is largely a function of the permeability of the capillary wall.     

Analysis of coupled intra- and extraluminal flows for single and multiple capillaries

Matched asymptotic expansions are used to study a model of the coupled fluid flow in the capillaries and tissue of the microcirculation. These capillaries are long, narrow cylindrical tubes embedded in a uniform tissue space. The capillary, or intraluminal, flow is assumed to be that of an incompressible Navier-Stokes fluid wherein colloids are represented as dilute solute; the extraluminal flow in the tissue is according to Darcy's law. Central to this fluid exchange is the boundary condition on the fluid radial velocity at the semipermeable wall of the capillary. This boundary condition, involving the local hydrostatic and colloidal osmotic pressures in both the capillary and the tissue, together with the radial gradient of the tissue hydrostatic pressure, couples the intra- and extraluminal flow fields. With this model we investigate the relationship between transport properties, hydrostatic pressures, and flow exchange for a single capillary, and describe the fluid transport in the tissue space produced by an array of such capillaries.     

Morphology and physiology of capillary systems in subregions of the subfornical organ and area postrema

From recent morphological and physiological studies of capillaries, I shall review four new or revised concepts about blood-tissue communication in the subfornical organ (SFO) and area postrema (AP). First, the capillary systems of SFO and AP exhibit subregional differentiation correlated topographically with cytoarchitecture, densities of immunoreactivity for several peptides and amines, cellular sensitivity to neuroactive substances, afferent neural terminations, and tissue metabolic activity. Thus, contrary to frequent citations, the angioarchitecture and microcirculatory physiology of these small sensory nuclei are not homogeneous. Second, electron microscopic, morphometric, and topographical studies reveal that SFO contains three different types of capillary and AP has two. The differentiated capillary morphology appears to be well organized for specialized functions particularly in SFO subregions. No other body organ or small tissue region is known to have such capillary diversity, further highlighting the complex functions served by SFO. Third, pools of interstitial fluid (Virchow-Robin spaces) surrounding type I and III capillaries in SFO and AP may participate in the receptive properties of these organs as low-resistance pathways for rapid dispersion of blood-borne hormones inside their organ boundaries. The parenchymal walls of Virchow-Robin spaces appear to harbour metabolic mechanisms for hormones such as angiotensin II, and thus could vastly extend the effective blood-brain surface area of permeable capillaries in SFO and AP. Fourth, SFO and AP bear similar physiological characteristics of high blood volume, yet relatively low rates of blood flow. Accordingly, intracapillary blood velocity must be quite slow in these organs, and the duration of transit by blood and circulating messengers rather protracted. This feature of slow blood transit time likely compounds the sensory capability of SFO and AP, rendering increased contact time for blood-borne hormones to penetrate the permeable capillaries of these structures and interact with their known dense populations of receptors for several homeostatic substances involved in regulation of blood pressure and body fluids.     

Analysis of the pathways of substance transport into the acinar cells]

In the exchange link of the microcirculation system of the exocrine part of the pancreas of Rana temporaria the substances moved from the blood capillary into the pericapillary space, then into the intercellular clefts and into the acinar cells by active transport. This is confirmed by the electron microscope studies of the ATP-ase activity localization in the exchange link: there are numerous lead phosphate granules in the endothelium of blood capillaries, on the fibrillae structures of the pericapillary space interstitium, on the lateral plasmic membrane of the exocrine pancreacytes, and on the cytoplasmic plates forming pinocytotic vacuoles.     

Some mathematical aspects of chemotherapy—II: The distribution of a drug in the body

In a previous paper (Bellman, Jacquez, and Kalaba,Bull. Math. Biophysics,22: 181–198, 1960) a model of the processes occurring in the exchange of a drug between capillary plasma, extracellular space and intracellular space was developed. This included the possibility of a reaction between the drug and a component of the intracellular space. The equations developed thus describe the events within a capillary bed. In the present paper, a simplified model of the body is set up. Each organ is treated as a single capillary bed and is linked to other organs via the circulation, in the parallel and/or series arrangements found in the body. Mixing in the circulation is included at the simplest possible level. The concentration of drug entering any one capillary bed is determined by the concentrations leaving all other capillary beds, the time lags, and mixing involved in the circulation. The equations describing these processes in conjunction with the equations of the processes occurring within each capillary bed lead to a large set of differential-difference equations.     

Circulatory system in chicken skeletal muscle in the second half of embryogenesis: Shape,blood flow,and vascular reactivity

A restructuring of the capillary bed—from the embryonic structure with a three-dimensional network of wide and long protocapillaries to the mature structure with high density of thin and short capillaries along the fibers—has been demonstrated in the chick skeletal muscle on embryonic days 10–19 by morphometric analysis. In this case, the specific blood flow and capillary luminal area per cm³ of the muscle remained unaltered, while the blood volume in it significantly dropped. The response of muscle circulation to nitroprusside (increase) and noradrenaline (decrease) appeared in 19-day-old embryos, but this response could develop only under conditions of initially low or high blood flow, respectively. We propose that the arterial trunk lumen area to the total capillary lumen area remains constant as the intraorganic circulation is formed, which provides for the required linear blood velocity in capillaries.     

Theoretical simulation of K(+)-based mechanisms for regulation of capillary perfusion in skeletal muscle

Muscle fibers release K(+) into the interstitial space upon recruitment. Increased local interstitial K(+) concentration ([K(+)]) can cause dilation of terminal arterioles, leading to perfusion of downstream capillaries. The possibility that capillary perfusion can be regulated by vascular responses to [K(+)] was examined using a theoretical model. The model takes into account the spatial relationship between functional units of muscle fiber recruitment and capillary perfusion. Diffusion of K(+) in the interstitial space was simulated. Two hypothetical mechanisms for vascular sensing of interstitial [K(+)] were considered: direct sensing by arterioles and sensing by capillaries with stimulation of feeding arterioles via conducted responses. Control by arteriolar sensing led to poor tissue oxygenation at high levels of muscle activation. With control by capillary sensing, increases in perfusion matched increases in oxygen demand. The time course of perfusion after sudden muscle activation was considered. Predicted capillary perfusion increased rapidly within the first 5 s of muscle fiber activation. The reuptake of K(+) by muscle fibers had a minor effect on the increase of interstitial [K(+)]. Uptake by perfused capillaries was primarily responsible for limiting the increase in [K(+)] in the interstitial space at the onset of fiber activation. Vascular responses to increasing interstitial [K(+)] may contribute to the rapid increase in blood flow that is observed to occur after the onset of muscle contraction.     

How well mixed is inert gas in tissues?

The washout of inert gas from tissues typically follows multiexponential curves rather than monoexponential curves as would be expected from homogeneous, well-mixed compartment. This implies that the ratio for the square root of the variance of the distribution of transit times to the mean (relative dispersion) must be greater than 1. Among the possible explanations offered for multiexponential curves are heterogeneous capillary flow, uneven capillary spacing, and countercurrent exchange in small veins and arteries. By means of computer simulations of the random walk of gas molecules across capillary beds with parameters of skeletal muscle, we find that heterogeneity involving adjacent capillaries does not suffice to give a relative dispersion greater than one. Neither heterogeneous flow, nor variations in spacing, nor countercurrent exchange between capillaries can account for the multiexponential character of experimental tissue washout curves or the large relative dispersions that have been measured. Simple diffusion calculations are used to show that many gas molecules can wander up to several millimeters away from their entry point during an average transit through a tissue bed. Analytical calculations indicate that an inert gas molecule in an arterial vessel will usually make its first vascular exit from a vessel larger than 20 micron and will wander in and out of tissue and microvessels many times before finally returning to the central circulation. The final exit from tissue will nearly always be into a vessel larger than 20 micron. We propose the hypothesis that the multiexponential character of skeletal muscle tissue inert gas washout curves must be almost entirely due to heterogeneity between tissue regions separated by 3 mm or more, or to countercurrent exchanges in vessels larger than 20 micron diam.     

Transcapillary exchange in relation to capillary circulation

Transcapillary exchange of diffusible solutes depends on capillary blood flow, Q; capillary permeability, P; and capillary surface area, S. In a single capillary, the extent of equilibration of a given solute depends on the ratio of Q, to the product of P and S. In a microvascular bed consisting of many capillaries, equilibration depends on the fraction of them which are open to blood flow at any time and on the distribution of Q/PS ratios in the open capillaries. Both these characteristics are subject to control by vascular smooth muscle, particularly by the precapillary sphincters. Vasomotor mechanisms have been shown experimentally to exert a wide range of effective control over blood-tissue transport. In skeletal muscle, effective PS measured with ⁴²K or ⁸⁶Rb may be increased 8-fold from maximum nervous vasoconstriction to optimum metabolic vasodilatation. Most probably, these changes are due to differences in functional capillary surface area and of blood flow distribution relative to permeability and surface area. The extent to which variations in permeability itself can contribute to control of transcapillary exchange is not known.     

A geometrical model of dermal capillary clearance

A new microscopic model is developed to describe the dermal capillary clearance process of skin permeants. The physiological structure is represented in terms of a doubly periodic array of absorbing capillaries. Convection-dominated transport in the blood flow within the capillaries is coupled with interstitial diffusion, the latter process being quantified via a slender-body-theory approach. Convection across the capillary wall and in the interstitial phase is treated as a perturbation which may be added to the diffusive transport. The model accounts for the finite permeability of the capillary wall as well as for the geometry of the capillary array, based on realistic values of physiological parameters. Calculated dermal concentration profiles for permeants having the size and lipophilicity of salicylic acid and glucose illustrate the power and general applicability of the model. Furthermore, validation of the model with published in vivo experimental results pertaining to human skin permeation of hydrocortisone is presented. The model offers the possibility for in-depth theoretical understanding and prediction of subsurface drug distribution in the human skin following topical application, as well as rates of capillary clearance into the systemic circulation. A simpler approach that treats the capillary bed as a homogeneously absorbing zone is also employed. The latter may be used in conjunction with the capillary exchange model to estimate measurable dermal transport and clearance parameters in a straightforward manner.     

On facilitated oxygen diffusion in muscle tissues.

The role of myoglobin in facilitated diffusion of oxygen in muscle in examined in a tissue model that utilizes a central supplying capillary and a tissue cylinder concentric with the central capillary, and that includes the nonlinear characteristics of the oxygen-hemoglobin dissociation reaction. In contrast to previous work, this model exhibits the effect of blood flow and a realistic, though ideal, tissue-capillary geometry. Solutions of the model equations are obtained by a singular-perturbation technique, and numerical results are discussed for model parameters of physiologic interest. In contrast to the findings of Murray, Rubinow, Taylor, and others, fractional order perturbation terms obtained for the "boundary-layer" regions near the supplying capillaries are quite significant in the overall interpretation of the modeling results. Some closed solutions are found for special cases, and these are contrasted with the full singular-perturbation solution. Interpretations are given for parameters of physiologic interest.     

Total and shunting circulation in human central hemodynamics

The goal of the study was to develop a method for the separate measurement of capillary and metarteriolar circulation. Data on the cardiovascular system of 301 male patients (1–49 years of age) and 344 female patients (1–50 years of age) with a diagnosis of functional murmur were used. In the process of heart and major vessel diagnostic catheterization, the diagnosis of heart defect was excluded. The cardiac output (Q) was estimated. The calculated oxygen consumption was converted using the Hüfner’s coefficient to the equivalent quantity of hemoglobin (Hb) delivering this oxygen into exchange capillaries. The Hb content per milliliter of blood is known; therefore, by dividing the total quantity of Hb that passed through capillaries by its content per milliliter of blood, one can obtain the blood volume (in milliliters) that passed through the capillary bed (Q _cap). A shunt in the microvasculature was found as the difference between Q and Q _cap). Thus, there exist in the microvascular module two parallel bloodstreams: a slow one, which goes through true capillaries, where the exchange happens, and a fast shunting stream through metarterioles, direct channels, and arteriolevenous anastomoses. The latter not only takes part in the tissue thermal exchange, but are also channels that ensure the free transfer of white blood cells through the microcirculatory module, especially of those whose characteristic sizes exceed the diameter of the metabolic capillaries. The contribution of these two parallel streams in the microcirculatory module into Q is different. According to other results of this study, the slow capillary stream makes up approximately 20%, whereas the fast shunting bloodstream, 80% of Q.     

Current views on oxygen transport from blood to tissues]

During the recent 25-30 years, sophisticated experiments and mathematical simulation significantly changed the generally accepted theory of oxygen transport in tissue, which was based on two major postulates, namely: 1) Blood flows in capillaries continuously at uniform velocity, 2) Gas circulation between blood and tissue takes place exclusively in capillaries. As was shown by modern research techniques, blood flow in microvessels has irregular sharp velocity fluctuations in very short time intervals (seconds). In addition, mean velocity of blood flow in microvessels of the same caliber and the same micro-region of tissue may differ several times. Therefore, efficiency of microcirculation reactions may be assessed exclusively witH mean blood velocity in capillaries of the whole micro-region, and with complicated changes of the histogram of mean velocity distribution in capillaries. It was shown that arteriolas and venulas of inactive muscles and brain account for 30 to 50% of gas circulation between blood and tissue. This resulted in fundamental change of the previous postulates in the area of tissue gas circulation physiology, and, in effect, in replacement of oxygen transport paradigm created by A. Krog. This study is an attempt to present a new modern concept of oxygen transport in tissue, to show its research significance, and possible applications.     

Red cell distribution and the recruitment of pulmonary diffusing capacity.

The distribution of red blood cells in alveolar capillaries is typically nonuniform, as shown by intravital microscopy and in alveolar tissue fixed in situ. To determine the effects of red cell distribution on pulmonary diffusive gas transport, we computed the uptake of CO across a two-dimensional geometric capillary model containing a variable number of red blood cells. Red blood cells are spaced uniformly, randomly, or clustered without overlap within the capillary. Total CO diffusing capacity (DLCO) and membrane diffusing capacity (DmCO) are calculated by a finite-element method. Results show that distribution of red blood cells at a fixed hematocrit greatly affects capillary CO uptake. At any given average capillary red cell density, the uniform distribution of red blood cells yields the highest DmCO and DLCO, whereas the clustered distribution yields the lowest values. Random nonuniform distribution of red blood cells within a single capillary segment reduces diffusive CO uptake by up to 30%. Nonuniform distribution of red blood cells among separate capillary segments can reduce diffusive CO uptake by >50%. This analysis demonstrates that pulmonary microvascular recruitment for gas exchange does not depend solely on the number of patent capillaries or the hematocrit; simple redistribution of red blood cells within capillaries can potentially account for 50% of the observed physiological recruitment of DLCO from rest to exercise.     

Der O2-Austausch zwischen Capillaren und Gewebe I. O2-Konzentrationsprofile in Blutcapillaren und deren Umgebung

Summary The differential equations valid for technical heat exchangers can also describe the O₂ exchange in the blood capillaries and the exchange of molecules like THO and acetamid in the renal tubules. Differences in the boundary conditions occur, however. Hence, these differential equations were resolved for the corresponding boundary conditions. The results permit us to conclude that the concentration profiles occurring in the capillaries and renal tubules, as a result of diffusion in the capillary cross-section, can, generally speaking, be disregarded for the following reason: Although the differences in partial pressure between the capillary wall and capillary centre, at the beginning of the capillaries come to 40–60 mm Hg, they descrease rapidly. The calculations have shown, that the time constant for the saturation process of the plasma (10 msec), is small in comparison with the contact time of the blood (100 msec). In the tissue capillaries, the differences in partial pressure between the capillary wall and the centre come to about 4–6 mmHg. This difference remains constant over the total capillary length.

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Bedeutung der Symbole A Atmungsintensität des Gewebes - R _a Radius des Standardzylinders - r _i Capillarradius - K ₁ Kroghscher Diffusionskoeffizient im Capillarinnern Mit Unterstützung durch die Deutsche Forschungsgemeinschaft.Herrn Prof. Dr. Dörr, Herrn Prof. Dr. Passow und Herrn Prof. Dr. Dr. Thews danke ich für wertvolle Anregungen.     

O2 transport in cerebral microregions (mathematical simulation)

The effects of the circulation rate in capillaries, the intensity of O2 consumption by nerve cells and the capillary network density on the O2 tension distribution in the cerebral cortex have been studied, utilizing a mathematical model simulating actual neuron-capillary relationships. The model has been written as a system of equations in partial derivatives, its solution obtained by the net-point method. Regulatory variations of the capillary circulation rate in certain cerebral microregions have been shown to ensure similar changes in oxygen supply throughout the region. A drop of the pO2 level in a cerebral microregion with a rising O2 consumption by nerve cells is shown to be due, by 75 percent, to the increase of O2 consumption and by 25 percent, to the lower pO2 in the capillaries. Conversely, an increase in pO2 in microregions resulting from a lower O2 consumption by neurons is due by 75 percent, to a pO2 rise in capillaries and by 25 percent, at the expense of an O2 consumption decrease. In cerebral regions differing in capillary network density by 20 percent, changes in the conditions for oxygen supply to tissue are due by 1/3 to pO2 variations in the capillaries and by 2/3 to alterations in the diffusion distances.     

Inert gas clearance from tissue by co-currently and counter-currently arranged microvessels

To elucidate the clearance of dissolved inert gas from tissues, we have developed numerical models of gas transport in a cylindrical block of tissue supplied by one or two capillaries. With two capillaries, attention is given to the effects of co-current and counter-current flow on tissue gas clearance. Clearance by counter-current flow is compared with clearance by a single capillary or by two co-currently arranged capillaries. Effects of the blood velocity, solubility, and diffusivity of the gas in the tissue are investigated using parameters with physiological values. It is found that under the conditions investigated, almost identical clearances are achieved by a single capillary as by a co-current pair when the total flow per tissue volume in each unit is the same (i.e., flow velocity in the single capillary is twice that in each co-current vessel). For both co-current and counter-current arrangements, approximate linear relations exist between the tissue gas clearance rate and tissue blood perfusion rate. However, the counter-current arrangement of capillaries results in less-efficient clearance of the inert gas from tissues. Furthermore, this difference in efficiency increases at higher blood flow rates. At a given blood flow, the simple conduction-capacitance model, which has been used to estimate tissue blood perfusion rate from inert gas clearance, underestimates gas clearance rates predicted by the numerical models for single vessel or for two vessels with co-current flow. This difference is accounted for in discussion, which also considers the choice of parameters and possible effects of microvascular architecture on the interpretation of tissue inert gas clearance.     

Circulation changes depending on manifestations in soft tissue tensioning in the leg lengthening process

Surgical intervention was found to intensify circulation in the limb and redistribute the blood flow. Leg lengthening led to arterial inflow limitation due to the magisterial artery strain. The changes were accompanied both by increase of functioning capillaries in number and increase of venous outflow dependence on blood inflow. The decreased after surgery oxygen tension in m. gastrocnemius did not change throughout distraction. The increase of functioning capillaries in number in resting contributed to maintenance of tissue oxygenation in the initial period of distraction, and in case of maximal tissue tensioning hydrostatic pressure increased in the capillaries due to arterial pressure rise. Restoration of the circulation parameters in the fixation period started with an increase of circulation volumetric rate in vessels with preservation of the rest mechanisms of the tissue oxygenation maintenance.