A mathematical model of the functional state of the oxygen transport system

Based on the principle of minimum power, a mathematical model of the functional state of the oxygen transport system is presented. The optimization model minimizes the power expenditure of the heart, bone marrow, lung and other tissues. The model is used to determine the functional parameters of the oxygen transport system in man under both normal and varying barometric pressures. Theoretical results are compared with experimental data.     

A mathematical model of the exercise functional state of the oxygen transport system

Based on the principle of minimum power, a mathematical model of the exercise functional state of the oxygen transport system is presented. Aerobic and anaerobic muscular efficiencies are considered. The energetically optimal arteriovenous oxygen content difference, cardiac output and ventilation during exercise in man are determined depending on mechanical power. Theoretical results are compared with experimental data.     

Optimal arterial blood pressure

Based on the principle of minimum power, a mathematical model of the functional state of the circulatory system is presented. The optimization model minimizes the power expenditure of the heart, bone marrow and the power expended in carrying blood. Mean arterial blood pressure is determined depending on oxygen transport parameters and locomotor activity. Theoretical results are compared with experimental data for man.     

Hypoxic pulmonary vasoconstriction as a regulator of alveolar-capillary oxygen flux: A computational model of ventilation-perfusion matching

The relationship between regional variabilities in airflow (ventilation) and blood flow (perfusion) is a critical determinant of gas exchange efficiency in the lungs. Hypoxic pulmonary vasoconstriction is understood to be the primary active regulator of ventilation-perfusion matching, where upstream arterioles constrict to direct blood flow away from areas that have low oxygen supply. However, it is not understood how the integrated action of hypoxic pulmonary vasoconstriction affects oxygen transport at the system level. In this study we develop, and make functional predictions with a multi-scale multi-physics model of ventilation-perfusion matching governed by the mechanism of hypoxic pulmonary vasoconstriction. Our model consists of (a) morphometrically realistic 2D pulmonary vascular networks to the level of large arterioles and venules; (b) a tileable lumped-parameter model of vascular fluid and wall mechanics that accounts for the influence of alveolar pressure; (c) oxygen transport accounting for oxygen bound to hemoglobin and dissolved in plasma; and (d) a novel empirical model of hypoxic pulmonary vasoconstriction. Our model simulations predict that under the artificial test condition of a uniform ventilation distribution (1) hypoxic pulmonary vasoconstriction matches perfusion to ventilation; (2) hypoxic pulmonary vasoconstriction homogenizes regional alveolar-capillary oxygen flux; and (3) hypoxic pulmonary vasoconstriction increases whole-lobe oxygen uptake by improving ventilation-perfusion matching.     

Molecular aspects of embryonic hemoglobin function

In order to provide the appropriate level of oxygen transport to respiring tissues, we need to produce a molecular oxygen transporting system to supplement oxygen diffusion and solubility. This supplementation is provided by hemoglobin. The role of hemoglobin in providing oxygen transport from lung to tissues in the adult is well-documented and functional characteristics of the fetal hemoglobin, which provide placental oxygen exchange, are also well understood. However the characteristics of the three embryonic hemoglobins, which provide oxygen transport during the first three months of gestation, are not well recognized. This review seeks to describe the state of our understanding of the temporal control of the expression of these proteins and the oxygen binding characteristics of the individual protein molecules. The modulation of the oxygen binding properties of these proteins, by the various allosteric effectors, is described and the structural origins of these characteristics are probed.     

Nonequilibrium Thermodynamic Analysis of the Coupling between Active Sodium Transport and Oxygen Consumption

Sodium transport and oxygen consumption have been simultaneously studied in the short-circuited toad skin. A constant stoichiometric ratio was observed in each skin under control condition (NaCl-Ringer's solution bathing both sides of the skin) and after block of sodium transport by ouabain. During alterations of sodium transport by removal and addition of K to the internal solution the stoichiometric ratio is constant although having a value higher than that observed in other untreated skins. The coupling between active sodium transport and oxygen consumption was studied after a theoretical nonequilibrium thermodynamic model. Studies were made of the influence of Na chemical potential difference across the skin on the rates of Na transport and oxygen consumption. A linear relationship was observed between the rates of Na transport and oxygen consumption and the Na chemical potential difference. Assuming the Onsager relationship to be valid, the three phenomenological coefficients which describe the system were evaluated. Transient increases in the rate of sodium transport and oxygen consumption were observed after a transitory block of sodium transport by removal of Na from the external solution. Cyanide blocks completely the rate of oxygen consumption in less than 2 min and the short-circuit current measured after that time decays exponentially with time, suggesting a depletion of ATP from a single compartment.     

Investigations of the validity of e.a. müller's "leistungspulsindex" (author's transl)]

In order to obtain a critical review of E.A. Müller's "Leistungspulsindex", we worked out a functional representation of that index, using a simple physiologic-mathematic model of the oxygen transport system. We found that the LPI is influenced by the economy of the oxygen transport system and that it is inversely proportional to stroke volume, caloric coefficient, and to the efficiency of muscular work. Based on the theoretic analysis we tried to interpret the results of an investigation carried out with pupils of an coeducative intermediate school (years of birth 1964, 1965, 1966). We obtained statistically significant correlations (p less than 0.1%) between LPI and parameters characterizing performance capacity (PWC170, heart rate at 100 W work load) and between LPI and body weight. The results of both the theoretic and experimental investigation indicate a dependence of the LPI on factors influenced by physical training processes, but no evidence was found, however, on an interrelationship between LPI and congenital factors of physical performance.     

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.     

Role of myoglobin in the oxygen supply to red skeletal muscle.

The contribution of nyoglobin to the oxygen uptake of red skeletal muscle was estimated from the difference in oxygen uptake with and without functional myoglobin. The oxygen uptake of bundles (25 mm long, 0.5 mm mean diameter) of muscle fibers teased from pigeon breast muscle was measured in families of steady states of oxygen pressure from 0 to 250 mm Hg. The oxygen-binding function of myoglobin, in situ in muscle fiber bundles, was abolished by treatment with nitrite of hydroxylamine, which convert oxymyoglobin in situ to high spin ferric myoglobin, or with phenylhydrazine, which converts oxymyoglobin to denatured products, or with 2-hydroxyethylhydrazine, which appears to remove myoglobin from the muslce. The oxygen uptake was again measured. At higher oxygen pressure, where oxygen availability does not limit the respiration of the fiber bundles, oxygen uptake is not affected by any of the four reagents, which is evidence that mitochondrial oxygen uptake is not impaired. At lower oxygen pressure, where oxygen uptake is one-half maximal, the steady state oxygen consumption is roughly halved by abolishing functional myoglobin. Under the steady state conditions studied, the storage function of myoglobin, being static, vanishes and the transport function stands revealed. We conclude from these experiments that myoglobin may transport a significant fraction of the oxygen consumed by muscle mitochondria.     

The influence of hepatic venous oxygen saturation on the liver's synthetic response to metabolic stress.

Hepatic oxygen consumption (HVO2) and hepatic venous oxygen saturation (ShvO2) were assessed in the isolated perfused rat liver under conditions that mimic critical illness in an effort to assess their utility in predicting the functional status of the liver. Flow rates were adjusted over the physiologic range of oxygen transport as indicated by the hepatic venous O2 saturation range of 10%-75%. HVO2 was found to be transport (HDO2) dependent only when perfusate conditions contained an increased counterregulatory hormone (glucagon, epinephrine, dexamethasone) stimulus or a high lactate concentration. In the absence of a metabolic load, (substrate and hormone-free perfusate), HVO2 was transport independent even at an ShvO2 as low as 10%. Although transport dependency of HVO2 is frequently used to infer tissue ischemia, hepatic oxygen consumption was poorly correlated with synthetic function under all conditions. In contrast, hepatic albumin production was directly related to ShvO2 at all levels of HDO2 and under all perfusion conditions indicating that this metabolic process is particularly sensitive to reductions in oxygen availability, which is more reliably predicted by venous saturation measurements. However, glucose and urea synthesis were almost independent of ShvO2. These findings indicate that various hepatic processes are affected differentially by stress conditions and flow alterations that may exist during critical illness, and protein synthesis is particularly sensitive to oxygen deprivation. Additionally, hepatic venous oxygen saturation measurement, but not HVO2, serves as a useful surrogate marker for hepatic albumin production suggesting that regional venous oximetry may play an important role in the detection of hepatic functional impairment.     

Microbial biosensor array with transport mutants of Escherichia coli K12 for the simultaneous determination of mono-and disaccharides

An automated flow-injection system with an integrated biosensor array using bacterial cells for the selective and simultaneous determination various mono- and disaccharides is described. The selectivity of the individually addressable sensors of the array was achieved by the combination of the metabolic response, measured as the O(2) consumption, of bacterial mutants of Escherichia coli K12 lacking different transport systems for individual carbohydrates. Kappa-carrageenan was used as immobilization matrix for entrapment of the bacterial cells in front of 6 individually addressable working electrodes of a screen-printed sensor array. The local consumption of molecular oxygen caused by the metabolic activity of the immobilized cells was amperometrically determined at the underlying screen-printed gold electrodes at a working potential of -600 mV vs. Ag/AgCl. Addition of mono- or disaccharides for which functional transport systems exist in the used transport mutant strains of E. coli K12 leads to an enhanced metabolic activity of the immobilized bacterial cells and to a concomitant depletion of oxygen at the electrode. Parallel determination of fructose, glucose, and sucrose was performed demonstrating the high selectivity of the proposed analytical system.     

The effect of sustained compression on oxygen metabolic transport in the intervertebral disc decreases with degenerative changes

Intervertebral disc metabolic transport is essential to the functional spine and provides the cells with the nutrients necessary to tissue maintenance. Disc degenerative changes alter the tissue mechanics, but interactions between mechanical loading and disc transport are still an open issue. A poromechanical finite element model of the human disc was coupled with oxygen and lactate transport models. Deformations and fluid flow were linked to transport predictions by including strain-dependent diffusion and advection. The two solute transport models were also coupled to account for cell metabolism. With this approach, the relevance of metabolic and mechano-transport couplings were assessed in the healthy disc under loading-recovery daily compression. Disc height, cell density and material degenerative changes were parametrically simulated to study their influence on the calculated solute concentrations. The effects of load frequency and amplitude were also studied in the healthy disc by considering short periods of cyclic compression. Results indicate that external loads influence the oxygen and lactate regional distributions within the disc when large volume changes modify diffusion distances and diffusivities, especially when healthy disc properties are simulated. Advection was negligible under both sustained and cyclic compression. Simulating degeneration, mechanical changes inhibited the mechanical effect on transport while disc height, fluid content, nucleus pressure and overall cell density reductions affected significantly transport predictions. For the healthy disc, nutrient concentration patterns depended mostly on the time of sustained compression and recovery. The relevant effect of cell density on the metabolic transport indicates the disturbance of cell number as a possible onset for disc degeneration via alteration of the metabolic balance. Results also suggest that healthy disc properties have a positive effect of loading on metabolic transport. Such relation, relevant to the maintenance of the tissue functional composition, would therefore link disc function with disc nutrition.     

Structural requirements for,and the effects of chemicals on,the rat pulmonary inactivation of prostaglandins

We have investigated the removal of prostaglandins (PGs) from the pulmonary circulation using the isolated perfused rat lung in order to determine which parts of the PG molecule were essential for transport into the pulmonary tissue. From these studies we propose that three functional groups of the PG molecule are necessary for transport into the lung tissue: the carboxylic acid group at carbon 1, hydroxyl group at carbon 15, and an oxygen group at carbon 11. The geometrical relationship between these groups is important for transport since reduction of the 13,14-double bond reduced transport, and changing the C-15 hydroxyl from an S to R configuration abolished transport. Various chemicals, drugs, and PG antagonists were tested for their ability to inhibit the transport system responsible for PG removal from the circulation. Diphloretin phosphate and polyphloretin phosphate were effective inhibitors, whereas dexamethasone, bromocresol green, N-ethyl maleimide and imipramine were moderately effective inhibitors. The PG antagonist, SC-19220, 7-oxo-13-prostynoic acid, and hydrocortisone were ineffective.     

A model of the development of stable competing relations in the self-organization of biosystems

A model is proposed that allows one to reveal specific features of the biosystem dynamics based on competitive processes of binary interaction. The model assumes a two-level hierarchy of system organization. Objects at the lower level of a system being investigated interact in a random way. The upper level determines the modulation of the development of a random process at the lower level. The application of the results of modeling to processes such as oxygen transport by proteins, adaptation of living organisms to changing environmental conditions, muscle contraction, and adaptability to stresses is discussed.     

Environmental influences on the development of the cardiac system in fish and amphibians

In poikilothermic animals body temperature varies with environmental temperature, and this results in a change in metabolic activity (Q10 of enzymatic reactions typically is around 2-3). Temperature changes also modify gas transport in body fluids. While the diffusion coefficient increases with increasing temperatures, physical solubility and also hemoglobin oxygen affinity decrease. Therefore, an increase in temperature typically requires adjustments in cardiac activity because ventilatory and convectional transport of respiratory gases usually are tightly coupled in adults in order to meet the oxygen demand of body tissues. Hypoxic conditions also provoke adaptations in the central circulatory system, like the hypoxic bradycardia, which has been described for many adult lower vertebrates, combined with an increase in stroke volume and peripheral resistance. In embryos and larvae the situation is much more complicated, because nervous control of the heart is established only late during development, and because the site of gas exchange changes from mainly cutaneous gas exchange during early development to mainly pulmonary or branchial gas exchange in late stages. In addition, recent studies in amphibian and fish embryos and larvae reveal, that at least in very early stages convectional gas transport of the hemoglobin is not essential, which means that in these early stages ventilatory and convectional gas transport are not yet coupled. Accordingly, in early stages of fish and amphibians the central cardiac system often does not respond to hypoxia, although in some species behavioral adaptations indicate that oxygen sensors are functional. If a depression of cardiac activity is observed, it most likely is a direct effect of oxygen deficiency on the cardiac myocytes. Regulated cardiovascular responses to hypoxia appear only in late stages and are similar to those found in adult species.     

A mathematical model of transmural transport of oxygen to the retina

A mathematical model of transmural transport of oxygen to a metabolizing retina is presented based on the equations of fluid dynamics. The equations of oxygen transfer are derived and then solved subject to the condition that the capillaries begin to transport oxygen at an initial time. The resulting transient analysis gives us insight into how diffusive and filtrative processes lead to the oxygen distributions both inside and outside capillaries. On the other hand, the steady state solution allows us to predict the cutoff intraocular pressure above which no oxygen is transferred to retinal tissue. It also gives quantitative relationships which allow us to postulate how intracapillary hypertension counterbalances elevated intraocular pressures and how low pressure glaucoma may arise from ineffective diffusive and filtrative processes of oxygen transport.     

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.     

Effect of different oxygen pressure levels on the transport of alpha-aminoisobutyric acid in myocardial cell cultures

The activity of aminoacid transport, as measured by alpha-aminoisobutyrate uptake, has been studied in confluent myocardial cell cultures exposed to different oxygen tensions. The results obtained indicate that the rate of cellular uptake and accumulation of the inert aminoacid increase with time as the fraction of oxygen is reduced. When alpha-aminoisobutyrate was added in presence of all other aminoacids of the medium, the effect of oxygen was also evident, suggesting a mechanism which overcomes the competitive action of the other aminoacids assigned to the same transport system of alpha-aminoisobutyrate (A system). The modulation of aminoacid transport activity may represent one of the possible mechanisms by which environmental oxygen affect the rate of cellular protein synthesis.     

A mathematical model of diffusional shunting of oxygen from arteries to veins in the kidney

To understand how arterial-to-venous (AV) oxygen shunting influences kidney oxygenation, a mathematical model of oxygen transport in the renal cortex was created. The model consists of a multiscale hierarchy of 11 countercurrent systems representing the various branch levels of the cortical vasculature. At each level, equations describing the reactive-advection-diffusion of oxygen are solved. Factors critical in renal oxygen transport incorporated into the model include the parallel geometry of arteries and veins and their respective sizes, variation in blood velocity in each vessel, oxygen transport (along the vessels, between the vessels and between vessel and parenchyma), nonlinear binding of oxygen to hemoglobin, and the consumption of oxygen by renal tissue. The model is calibrated using published measurements of cortical vascular geometry and microvascular Po(2). The model predicts that AV oxygen shunting is quantitatively significant and estimates how much kidney Vo(2) must change, in the face of altered renal blood flow, to maintain cortical tissue Po(2) at a stable level. It is demonstrated that oxygen shunting increases as renal Vo(2) or arterial Po(2) increases. Oxygen shunting also increases as renal blood flow is reduced within the physiological range or during mild hemodilution. In severe ischemia or anemia, or when kidney Vo(2) increases, AV oxygen shunting in proximal vascular elements may reduce the oxygen content of blood destined for the medullary circulation, thereby exacerbating the development of tissue hypoxia. That is, cortical ischemia could cause medullary hypoxia even when medullary perfusion is maintained. Cortical AV oxygen shunting limits the change in oxygen delivery to cortical tissue and stabilizes tissue Po(2) when arterial Po(2) changes, but renders the cortex and perhaps also the medulla susceptible to hypoxia when oxygen delivery falls or consumption increases.