Mathematical modelling of oxygen transport to tissue

 The equations governing oxygen transport from blood to tissue are presented for a cylindrical tissue compartment, with blood flowing along a co–axial cylindrical capillary inside the tissue. These governing equations take account of: (i) the non–linear reactions between oxygen and haemoglobin in blood and between oxygen and myoglobin in tissue; (ii) diffusion of oxygen in both the axial and radial directions; and (iii) convection of haemoglobin and plasma in the capillary. A non–dimensional analysis is carried out to assess some assumptions made in previous studies. It is predicted that: (i) there is a boundary layer for oxygen partial pressure but not for haemoglobin or myoglobin oxygen saturation close to the inflow boundary in the capillary; (ii) axial diffusion may not be neglected everywhere in the model; (iii) the reaction between oxygen and both haemoglobin and myoglobin may be assumed to be instantaneous in nearly all cases; and (iv) the effect of myoglobin is only significant for tissue with a low oxygen partial pressure. These predictions are validated by solving the full equations numerically and are then interpreted physically. Received: 13 October 2000 / Revised version: 12 June 2001 / Published online: 17 May 2002     

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.     

The effect of carbon monoxide on haem-facilitated oxygen diffusion

The aim of this paper is to quantify the effect of small quantities of carbon monoxide on the facilitated diffusion of oxygen by haemoglobin in the steady state. It is the first phase in the study of a mathematical model for carbon monoxide poisoning. Here we extend the Wyman model for facilitated diffusion to the case in which there are two ligands. The equations are solved using an asymptotic technique developed by Murray. We obtain accurate analytic approximations for the biologically important quantities of the problem for various percentages of carbon monoxide. These are the concentrations of free oxygen, haemoglobin, oxyhaemoglobin and carboxyhaemoglobin, and hence the saturation of the protein and the facilitated oxygen flux. The major effect of very small quantities of carbon monoxide on the oxygen flux is shown.     

Ligand binding kinetics of des arginine haemoglobin

Des arginine 141 a haemoglobin (the haemoglobin in which the C-terminal arginine of the a chain has been removed) has a high affinity for oxygen and a reduced co-operativity in its oxygen equilibrium binding. The kinetic consequences of this modification are investigated in this paper. Deoxy des Arg haemoglobin binds carbon monoxide faster than does haemoglobin A, whilst oxy des Arg haemoglobin loses oxygen more slowly. These results are correlated with the oxygen equilibrium binding properties of des Arg haemoglobin. The carbon monoxide binding kinetics have been interpreted as implying a change in the parameter c (of the allosteric model), as well as L, when this arginine is removed from haemoglobin.     

A numerical study of the nonsteady transport of gases in the pulmonary capillaries

A mathematical model is formulated for simulating the unsteady transport of gases in the blood flowing through the pulmonary capillaries. The formulation takes into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion of the species due to haemoglobin. A time dependent situation is created by allowing to vary suddenly the partial pressures of the gases either in the venous blood or in the alveolar air. A numerical technique is described to solve the resulting time-dependent system of nonlinear coupled partial differential equations with the physiologically relevant boundary, entrance and initial conditions. The time required by the gases to achieve equilibrium is computed. It is shown that the dissolved oxygen takes longest in reaching equilibration whereas the carbon dioxide is the fastest. The various physiologically relevant unsteady situations have been examined.     

Kinetics of the reaction of intraerythrocytic haemoglobin by single cell microspectroscopy: effect of shape and osmolarity

The kinetics of the reaction of CO with intraerythrocytic haemoglobin has been studied in single red blood cells (RBC) using a scanning microspectrophotometer and a photochemical perturbation method. Measurements have been carried out using red blood cells from man and camel (Camelus dromedarius), the latter at different osmotic pressures. Camel RBC, which are smaller and different in shape compared to human RBC, are known to remain intact even at an osmolarity 6-times lower than physiological (280–290 mosm/l), swelling up to twice their normal volume. The results show that the recombination time course is affected by diffusion of CO through a stagnant layer of solvent around the cell membrane, but that it is also influenced by other parameters such as intracellular diffusion of ligand and haemoglobin.

Microspectroscopy Camel hemoglobin Red blood cell Diffusion Osmolarity Cell volume     

The effect of temperature on the ability of Tilapia mossambica Peters to enter deep water

The ability of adult Tilapia mossambica Peters to enter deep water was determined at 15, 22 and 30°C. At 30°C adults compensate to about 20m depth but at 15°C to only 7 m. Compensation is more rapid at high than at low temperatures. T. mossambica haemoglobin has a marked Root effect which is the same at 22 and 30°C. The oxygen affinity of the haemoglobin is higher at 15°C than at 30°C. There was no measurable difference in the rate of passive oxygen diffusion across the swimbladder wall in the temperature range 15–30°C. It is concluded that the ability to enter deeper water at higher temperatures is related to decreased oxygen affinity of the haemoglobin and higher rates of oxygen secretion and blood circulation.     

Brain oxygen transport related to levels of fetal haemoglobin in stable preterm infants.

The relative amount of regional cerebral oxygen transport was compared between different preterm infants by performing measurements of cerebral blood flow velocity, mean arterial blood pressure, whole blood viscosity and haemoglobin content for each individual. In addition the percentage of fetal haemoglobin was determined. On 25 occasions measurements of fetal haemoglobin and cerebral oxygen transport have been performed prior to and following a blood transfusion with adult red blood cells. Comparison of the data for cerebral oxygen transport suggests that the actual amount of cerebral oxygen transport is lowest at fetal haemoglobin levels below 30% and will increase progressively as soon as the percentage of fetal haemoglobin rises about 30%. Thus, at increasing fetal haemoglobin levels, cerebral haemodynamic mechanisms in the human neonate cause elevations of regional cerebral blood flow and oxygen transport. The found increase of cerebral blood flow and oxygen transport at high fetal haemoglobin levels will minimize the impeded dissociation and delivery of oxygen to brain tissues.     

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.     

Functional arterio-venous anastomoses between the testicular artery and the pampiniform plexus in the spermatic cord of rams

Blood flow to the testis, haemoglobin oxygen saturation and testosterone concentration in arterial and venous testicular blood vessels were studied in Texel rams in the breeding and non-breeding season. Blood flow in the proximal and distal testicular artery was measured electromagnetically. The mean flow in the proximal testicular artery was 18.5 ml/min and in the distal testicular artery 7.5 ml/min, and there was no detectable seasonal influence. Haemoglobin oxygen saturation and testosterone concentration were measured in the saphenous artery and vein, the distal testicular artery and vein, and in the proximal testicular vein. The haemoglobin oxygen saturation in the proximal testicular vein was significantly higher than in the distal testicular vein in both seasons. The mean testosterone concentration was significantly lower in the proximal testicular vein than in the distal testicular vein in both seasons. Based on haemoglobin oxygen saturation and testosterone data, it was calculated that between 28 and 46% of the testicular arterial blood was bypassing the testis and was directly flowing through arterio-venous anastomoses towards the pampiniform plexus in the spermatic cord of conscious rams. In anaesthetized rams 55 and 64% of the blood was flowing directly from the testicular artery to the pampiniform plexus based on blood flow data. Transfer of testosterone and oxygen by passive diffusion from the testicular artery to the pampiniform plexus and vice versa in the spermatic cord was not detected.     

Kinetics and Mechanism of Bacterial Disinfection by Chlorine Dioxide

Survival data are presented for a fecal strain of Escherichia coli exposed to three concentrations of chlorine dioxide at four temperatures. Chick's first-order reaction equation is generalized to a pseudo nth-order model. Nonlinear least squares curve-fitting of the survival data to the nth order model was performed on an analogue computer. The data were observed to follow fractional order kinetics with respect to survival concentration, with an apparent activation energy of 12,000 cal/mole. Initial experiments support the thesis that the mechanism of chlorine dioxide kill occurs via disruption of protein synthesis.     

Haemoglobin Rahere (beta Lys-Thr): A new high affinity haemoglobin associated with decreased 2, 3-diphosphoglycerate binding and relative polycythaemia.

A new haemoglobin with increased oxygen affinity, beta82 (EF6) lysine leads to threonine (Hb Rahere), was found during the investigation of a patient who was found to have a raised haemoglobin concentration after a routine blood count. The substitution affects one of the 2, 3-diphosphoglycerate binding sites, resulting in an increased affinity for oxygen, but both the haem-haem interaction and the alkaline Bohr effect are normal in the haemolysate. This variant had the same mobility as haemoglobin A on electrophoresis at alkaline pH but was detected by measuring the whole blood oxygen affinity; it could be separated from haemoglobin A, however, by electrophoresis in agar at acid pH. The raised haemoglobin concentration was mainly due to a reduction in plasma volume (a relative polycythaemia) and was associated with a persistently raised white blood count. This case emphasises the need to measure the oxygen affinity of haemoglobin in all patients with absolute or relative polycythaemia when some obvious cause is not evident.     

A re-examination of oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics

Oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is re-examined and the results of a recent paper by Lin corrected. An extension of this model to include external diffusion is made and its effect is shown to be significant. A model which attempts to model the nucleus as a central sphere which does not consume any oxygen is also investigated. Finally, a perturbation solution for a small Michaelis constant is developed.     

2,3-Diphosphoglycerate Content of Human Arterial and Venous Blood

THE glycolytic intermediate, 2,3-diphosphoglycerate, is an intracellular regulator of the oxygen affinity of haemoglobin^1,2. At high altitudes there is a direct relationship between the decreased oxygen affinity of haemoglobin and the increased concentration of diphosphoglycerate in the blood³. This was explained by Benesch et al.⁴ and Chanutin et al.⁵, who found that the binding of diphosphoglycerate to haemoglobin reduces the oxygen affinity and by our finding that the concentration of diphosphoglycerate increases when the red cells are incubated under low oxygen tension^6,7, thereby releasing oxygen from haemoglobin. For the same reason, the oxygen tension is reduced during the circulation of blood from the pulmonary alveoli to the tissues; the decreased level of the diphosphoglycerate facilitates the binding of oxygen to haemoglobin in the pulmonary alveoli and the increased level of the diphosphoglycerate in the blood of the capillaries decreases the affinity of haemoglobin for oxygen. We have measured the amount of 2,3-diphosphoglycerate and other glycolytic intermediates in arterial and venous blood to test this supposition.     

New artificial oxygen carriers made of pegulated polymerised pyridoxylated porcine haemoglobin (P(4)Hb)

Oxygen-carrying plasma expanders are designed for use as iso-oncotic 'blood substitutes' to combat oxygen deficiencies caused by blood loss. In contrast, a hypo-oncotic artificial oxygen carrier can be added to existing blood - as a 'blood additive'. It has potential therapeutic use for deficiencies of oxygen which are not entailed by blood (volume) lack, and can therefore not be treated by a 'blood substitute', e.g. anaemias, local ischaemias and their complications such as stroke or myocardial infarction, or lack of oxygen in tumours, reducing the effectiveness of anti-cancer treatments by irradiation or chemotherapy. For such a novel approach haemoglobin-based oxygen-carrying additive, the haemoglobin must be highly polymerised in order to decrease the oncotic pressure, which can be received many times lower compared with smaller molecular size haemoglobins. Our aim is to produce haemoglobin polymers with narrow distributions of molecular weights of approximately 1,000,000 g/mol, preferably produced in high yield and at low cost. But polymerising haemoglobin by cross-linking normally results in a so-called percolation distribution of molecular weights, with a large amount of insoluble material, and with only poor yields of the desired polymers. A newly developed one-vessel synthesis procedure, which includes a controlled marked dilution of the synthesis medium during the cross-linking reaction, enables yields of polymerised haemoglobin (P(4)Hb) of over 80 %. Those preparations are easy and cheap to perform at large scales. P(4)Hb hyperpolymers (the high molecular moiety of P(4)Hb) are suitable for an oxygen-carrying blood additive: their oxygen-binding properties are sufficient, they are fully compatible with human blood plasma, and at the intended therapeutic concentration of approximately 30 g/l oncotic pressures are very low, and the impact on blood viscosity is tolerable.     

A theoretical model for studying the rate of oxygenation of blood in pulmonary capillaries

A mathematical analysis of the process of gas exchange in the lung is presented taking into account the transport mechanisms of molecular diffusion, convection and facilitated diffusion of the species due to haemoglobin. Since the rate at which blood gets oxygenated in the pulmonary capillaries is very fast, it is difficult to set up an experimental study to determine the effects of various parameters on equilibration rate. The proposed study is aimed at determining the effects of various physiological parameters on equilibration rate in pathological conditions.Among the significant results are that 1. dissolved oxygen takes longer to achieve equilibration across the pulmonary membrane and carbon dioxide attains equilibration faster, 2. the equilibration length increases with increase in blood velocity, haemoglobin concentration, calibre of pulmonary capillaries and fall in alveolar PO₂, 3. the alveolar PCO₂ and forward and backward reaction rates of haemoglobin with CO₂ do not materially affect the equilibration rate or length. 4. At complete equilibration, by the end of the pulmonary capillary 92% of the total haemoglobin has combined with oxygen and 8% free pigment is left which is present as carbamino haemoglobin, met haemoglobin, carboxy haemoglobin etc.These results are of some importance for anaemic conditions, muscular exercise, meditation, altitude physiology, hypo-ventilation, hyperventilation, etc.Symbols H⁺ hydrogen ion - O₂ oxygen - CO₂ carbondioxide - HbO₂ oxyhaemoglobin - HbCO₂ carbaminohaemoglobin - PO₂ partial pressure of O₂ - PCO₂ partial pressure of CO₂ - PaO₂ O₂ tension in arterial blood - PaCO₂ CO₂ tension in arterial blood - k₁ forward rate constant for Eq. (1) - k₂ backward rate constant for Eq. (1) - m₁ forward rate constant for Eq. (2) - m₂ backward rate constant for Eq. (2) - k equilibration rate - a radius of the capillary - Q velocity of blood - L length of the capillary - D₀ diffusion coefficient of O₂ - D_c diffusion coefficient of CO₂ - D_H diffusion coefficient of Hb - H total haemoglobin concentration - A matrix - c₁ concentration of dissolved O₂ in blood - c₂ concentration of HbO₂ in blood - c₃ concentration of dissolved CO₂ in blood - c₄ concentration of HbCO₂ in blood - c₅ concentration of haemoglobin - c1alv concentration of O₂ in the alveolar region - c_3alv concentration of CO₂ in the alveolar region - c_iven concentration of the ith species in venous blood - c_iart concentrations of the ith species in arterial blood - F _is concentrations of the species in dimensionless form - J₀, I₀ Bessel's functions - P_alvO₂ tension of O₂ in alveolar region - P_alvCO₂ tension of CO₂ in alveolar region.     

Oxygen uptake,the circulatory system,and haemoglobin function in the intertidal polychaete Terebella haplochaeta (Ehlers)

The intertidal polychaete Terebella haplochaeta (Ehlers) shows a high degree of oxyregulation in declining pO₂ when confined to its burrow at low tide. This response is achieved by a number of adaptations to the respiratory system. The worm ventilates its burrow in a headward direction by rhythmical conractions of the body. The rate of these pulsations increases at low pO₂ and assists the circulation of the coelomic and vascular fluids. Haemoglobin in the vessels has a high affinity for oxygen and a sigmoidal equilibrium curve. Both the shape and position of the oxygen-binding curve are sensitive to changes in pH, pCO₂, and temperature in a way that suggests augmentation of oxygen delivery at low tide. The concentration of haemoglobin in the vessels is high and is further raised following warm acclimation, presumably to meet an increase in oxygen demand. The ultrastructure of the gills and blood vessels indicates a design for function at low oxygen tensions where diffusion distances must be short and surface areas large in order to enhance the rate of diffusion of oxygen from the near environment.     

Role of nutrient supply on cell growth in bioreactor design for tissue engineering of hematopoietic cells

In the present study, a dynamic mathematical model for the growth of granulocyte progenitor cells in the hematopoietic process is developed based on the principles of diffusion and chemical reaction. This model simulates granulocyte progenitor cell growth and oxygen consumption in a three-dimensional (3-D) perfusion bioreactor. Material balances on cells are coupled to the nutrient balances in 3-D matrices to determine the effects of transport limitations on cell growth. The method of volume averaging is used to formulate the material balances for the cells and the nutrients in the porous matrix containing the cells. All model parameters are obtained from the literature. The maximum cell volume fraction reached when oxygen is depleted in the cell layer at 15 days and is nearly 0.63, corresponding to a cell density of 2.25 x 10(8) cells/mL. The substrate inhibition kinetics for cell growth lead to complex effects with respect to the roles of oxygen concentration and supply by convection and diffusion on cell growth. Variation in the height of the liquid layer above the cell matrix where nutrient supply is introduced affected the relative and absolute amounts of oxygen supply by hydrodynamic flow and by diffusion across a gas permeable FEP membrane. Mass transfer restrictions of the FEP membrane are considerable, and the supply of oxygen by convection is essential to achieve higher levels of cell growth. A maximum growth rate occurs at a specific flow rate. For flow rates higher than this optimal, the high oxygen concentration led to growth inhibition and for lower flow rates growth limitations occur due to insufficient oxygen supply. Because of the nonlinear effects of the autocatalytic substrate inhibition growth kinetics coupled to the convective transport, the rate of growth at this optimal flow rate is higher than that in a corresponding well-mixed reactor where oxygen concentration is set at the maximum indicated by the inhibitory kinetics.     

Oxygen-related processes in red blood cells exposed to tert-butyl hydroperoxide

The correlation between the oxidative processes in tert-butyl hydroperoxide (tBHP)-exposed red blood cells and the reactions of oxygen consumption and release were investigated. Red blood cell exposure to tBHP resulted in transient oxygen release followed by oxygen consumption. The oxygen release in red blood cells was associated with intracellular oxyhaemoglobin oxidation. The oxygen consumption proceeded in parallel with free radical generation, as registered by chemiluminescence, but not to membrane lipid peroxidation. The oxygen consumption was also observed in membrane-free haemolyzates. The order of the organic hydroperoxide-induced reaction of oxygen release with respect to the oxidant (tBHP) was estimated to be 0.9 +/- 0.1 and that of the oxygen consumption reaction was determined to be 2.4 +/- 0.2. The apparent activation energy values of the oxygen release and oxygen consumption were found to be 107.5 +/- 18.5 kJ/mol and 71.0 +/- 12.5 kJ/mol, respectively. The apparent pKa value for the functional group(s) regulating the cellular oxyHb interaction with the oxidant in tBHP-treated red blood cells was estimated to be 6.7 +/- 0.2 and corresponded to that of distal histidine protonation in haemoprotein. A strong dependence of tBHP-induced lipid peroxidation on the oxygen concentration in a red blood cell suspension was observed (P50 = 32 +/- 3 mmHg). This dependence correlated with the oxygen dissociation curve of cellular haemoglobin. The order of the membrane lipid peroxidation reaction with respect to oxygen was found to be 0.5 +/- 0.1. We can conclude that the intensity of the biochemical process of membrane lipid peroxidation in tBHP-exposed erythrocytes is controlled by small changes in such physiological parameters as the oxygen pressure and oxygen affinity of cellular haemoglobin. Neither GSH nor oxyhaemoglobin oxidation depended on oxygen pressure.