[Digital simulation of a spatial diffusion model of O2 supply in biological tissue]

The blood oxygen transport system. A numerical simulation of capillary-tissue respiratory gas exchange

A theoretical model of the process of respiratory gas exchange between capillary and tissue is described, with special reference to the importance of variations in blood properties. The volume of tissue supplied with oxygen from a single vessel, as a function of the blood flowrate (u), hematocrit (h), and 2,3 diphosphoglycerate concentration (DPG), is calculated from a solution to the set of equations governing species distributions in the blood and tissue. The results, which are presented in the form of crossplots of the three blood parameters (u, h, and DPG) at a constant oxygen supply rate, show the possible significance of in vivo variations in the oxygen affinity of hemoglobin as a compensatory mechanism. Of further physiological interest is the sharp increase in venous erythrocyte pH in response to decreases in hematocrit, once the hematocrit is below a certain level. These results, and those relating DPG to hematocrit at constant O2 supply, are consistent with experimental observations of elevated DPG and pH levels in anemic individualts, and the dependence of erythrocyte DPG concentration upon pH.

Stereological estimates of number and length of capillaries in subdivisions of the human hippocampal region

The hippocampal formation is a neuroanatomically well-defined region of the brain involved in memory processes. In view of the functional importance of the region and its involvement in a number of brain pathologies, including Alzheimer's disease and temporal lobe epilepsy, a quantitative description of its vascular supply represents an important first step in evaluating the involvement of vascular changes in these phenomena. Unbiased estimates of the length and connectivity of the vascular supply of brain regions have not been described previously. The total number, total length, and distribution of the diameters of capillaries were estimated in the five major subdivisions of the hippocampal formation (fascia dentata, hilus, CA3-2, CA1, and subiculum) in 5 normal males, 52-84 years of age. These estimates were used to derive several other structural parameters. Both the primary and the derived parameters were used to make inter- and intra-individual comparisons. For each of the five major subdivisions from each individual, the volume was estimated using the Cavalieri principle. The total capillary length was estimated on 3-microm-thick plastic isotropic uniform random sections. Using a topological definition of a capillary unit and the optical disector, total capillary number was estimated in 40-microm-thick plastic sections. Length-and number-weighted three-dimensional diameter distributions were obtained from the thin and thick plastic sections, respectively. In each subdivision the total length of capillaries was correlated with previously obtained data on the number of neurons in the same subdivisions of the same individuals. Intersubdivisional differences were observed, in that the hilus of the dentate gyrus had fewer capillaries per unit volume than the other four subdivisions. Interindividual comparisons indicate that the interindividual variances are of a magnitude suitable for sensitive group comparisons. The design-based stereological methods that were used in the analyses can provide a basis for a new unbiased approach to the estimation of vascular parameters in well-defined regions of the brain.

Oxygen Diffusion from Capillary Layers with Concurrent Flow

Oxygen transport from capillary layers with concurrent flow is considered for symmetric and asymmetric distributions of oxygen concentration between the layers. The analysis is based on the solution previously obtained by the author [1]. Solutions for the symmetric case are shown to be very close to the corresponding solutions of the Krogh cylinder model. Asymmetry in oxygen distribution is introduced systematically by considering different velocities of blood in the alternate capillary layers, different inlet capillary oxygen tensions, and different capillary hematocrits. It is shown that increase of the degree of asymmetry leads to diminution of the mean oxygen tension.

Distribution of the myocardial tissue PO2 in the rat and the inhomogeneity of the coronary bed

The structural inhomogeneity of the myocardial capillary bed is stimulated by microcirculatory units (MCU's) in a diffusion model. This stimulation is based on MCU's in which the arrangement of the capillary ends (concurrent structure, partial and total countercurrent (structure, helical structure) as well as the structure and supply parameters are varied. The variation of these parameters is based on own measurements of the intracapillary HbO2 saturation as well as on the following parameters from the literature: frequency distribution of capillary distance and capillary radius, mean capillary length or capillary section length respectively, arterial and mean venous PO2, mean coronary blood flow, mean O2 consumption and diffusion conductivity. The analysis of O2 supply of the normoxic rat heart shows that an O2 diffusion shunt is obligatory except of MCU's with an extremely large capillary distance or with a concurrent capillary structure. Therefore the minimal tissue PO2 lies at the level of the capillary venous PO2 of a MCU. The maximum of the total PO2 frequency distribution in the normoxic rat myocardium lies at 25 +/- 5 mm Hg, i.e. above the mean venous PO2 (20 mm Hg). Tissue PO2 values between 0 and 5 mm Hg amount to 0.5% i.e. they are extremely rare. Tissue PO2 values of 0-1 mm Hg represent less than 0.2%.

[A mathematical model for the description of oxygen-diffusion in the intervillous space of the human placenta (author's transl)]

The oxygen consumption has a central place in the complicated interaction between diminished oxygen supply and degenerative trophoblastic tissue change. In the present investigation, that serves as a mathematical model, the problem of the diffusion equation for the constant state is discussed and solved. It is shown that the diminishing of length of the radius of the maternal blood compartment has the same meaning as the decrease in the flow velocity. The influence of the mass transfer coefficient h on the oxygen partial pressure and on the oxygen transfer is investigated. The influence of parameter variations on the placental alterations during toxemia is discussed and their clinical importance is obvious. Our theoretical model elusidates the value of an early therapeutical approach in the case of toxemia.

Regulation of local tissue PO2 of the brain cortex at different arterial O2 pressures

Local tissue oxygen pressure (PO2) was recorded with a platinum multiwire surface electrode at adjacent sites of the cat cortex under steady-state conditions and with different arterial oxygen supply. Simultaneously PO2 in the sinus sagittalis was continuously recorded through the vascular wall in some experiments. Under normoxic and steady-state conditions local PO2 values varied between 0 Torr and almost arterial levels of 90 Torr. This was in accordance with the assumption of a diffusive transport of oxygen in tissue. With increased arterial oxygen supply local tissue PO2 reacted quite differently at adjacent sites. Linear increases in local tissue PO2 as compared to arterial PO2 as well as constant levels, very small increases and even small decreases were recorded. Constancy or small changes, respectively, of local PO2 (= local PO2 regulation) may be caused by changes in microflow, but changes in oxygen consumption cannot be excluded completely. The regulation of local PO2 could be abolished by adding CO2 to the gas mixture or by producing tissue anoxia. With severely reduced arterial oxygen supply local tissue PO2 dropped down to hypoxic or anoxic levels at all sites measured.