Intrapartum oxytocin use for labor acceleration in rural India.

OBJECTIVE: To examine factors associated with the use of oxytocin for acceleration of labor in women delivered at home in rural India. METHOD: Quantitative data were collected from 527 women who were delivered at home and qualitative interviews were carried out with 21 mothers and 9 birth attendants. RESULTS: Oxytocin use was associated with higher education and socioeconomic status, primigravidity, and delivery by a traditional birth attendant. CONCLUSION: Labor acceleration with oxytocin occurs indiscriminately In India. Oxytocin use should be regulated, and training for birth attendants should be provided as well as health education for pregnant women.

A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall.

A two-phase model for the flow of blood in narrow tubes is described. The model consists of a central core of suspended erythrocytes and a cell-free layer surrounding the core. It is assumed that the viscosity in the cell-free layer differs from that of plasma as a result of additional dissipation of energy near the wall caused by the red blood cell motion near the cell-free layer. A consistent system of nonlinear equations is solved numerically to estimate: (i) the effective dimensionless viscosity in the cell-free layer (beta), (ii) thickness of the cell-free layer (1-lambda) and (iii) core hematocrit (H(c)). We have taken the variation of apparent viscosity (mu(app)) and tube hematocrit with the tube diameter (D) and the discharge hematocrit (H(D)) from in vitro experimental studies [16]. The thickness of the cell-free layer computed from the model is found to be in agreement with the observations [3,21]. Sensitivity analysis has been carried out to study the behavior of the parameters 1-lambda, beta, H(c), B (bluntness of the velocity profile) and mu(app) with the variation of D and H(D).

Parametric analysis of the relationship between end-capillary and mean tissue PO2 as predicted by a mathematical model.

An increase of the partial pressure of oxygen in venules towards larger vessels has been observed experimentally, but the mechanism of this phenomenon has not been established. The present study considers a simple mathematical model of oxygen transport from a capillary to the surrounding tissue cylinder and analyses the conditions under which the end-capillary partial pressure of oxygen is lower than the mean tissue pressure. Under these conditions oxygen would diffuse into the venules since they are surrounded by the tissue with a higher partial pressure of oxygen. Cerebral circulation is chosen for these calculations and conditions of normoxia, hypoxic hypoxia, carbon monoxide hypoxia, anemia, and polycythemia are simulated. The tissue metabolic rate is also varied. It is found that under most conditions the relationship between the end-capillary and mean tissue partial pressures of oxygen can be reversed when one of the parameters is varied within its physiological range, i.e. the difference between these variables could be either positive or negative depending on the value of the parameters. Therefore, under many realistic conditions this mechanism would contribute to an increase of the partial pressure of oxygen in the venules. This conclusion should hold for a more realistic geometrical model of capillary network, but the relationships between the end-capillary and mean tissue partial pressures of oxygen, in addition to their dependence on the parameters considered in this study, would likely be dependent on the spatial location within the network.

An analysis of hypoxia in sheep brain using a mathematical model.

Cerebral blood flow (CBF) increases as arterial oxygen content falls with hypoxic (low PO2), anemic (low hemoglobin) and carbon monoxide (CO) (high carboxyhemoglobin) hypoxia. Despite a higher arterial PO2, CO hypoxia provokes a greater increase in CBF than hypoxic hypoxia. We analyzed published data using a compartmental mathematical model to test the hypothesis that differences in PO2 in tissue, or a closely related vascular compartment, account for the greater response to CO hypoxia. Calculations showed that tissue, but not arteriolar, PO2 was lower in CO hypoxia because of the increased oxyhemoglobin affinity with CO hypoxia. Analysis of studies in which oxyhemoglobin affinity was changed independently of CO supports the conclusion that changes in tissue PO2 (or closely related capillary or venular PO2) are predictive of alterations in CBF. We then sought to determine the role of tissue PO2 in anemic hypoxia, with no change in arterial and little, if any, change in venous PO2. Calculations predict a small fall in tissue PO2 as hematocrit decreases from 55% to 20%. However, calculations show that changes in blood viscosity can account for the increase in CBF in anemic hypoxia over this range of hematocrits.

A mathematical model for the elimination of carbon monoxide in humans.

Carboxyhaemoglobin (COHb) build-up in the blood as a result of exposures to carbon monoxide (CO) affects human beings. It is important to eliminate CO from the blood for treatment and health safety. A mathematical model is proposed to compute COHb level in the blood as a function of post-exposure time as CO is eliminated. The model takes into account molecular diffusion, facilitated diffusion, convection, non-equilibrium kinetics of CO with haemoglobin and the important physiological parameters, such as ventilation rate, blood flow rate and the total volume of blood in the body. Endogenous production of CO in the body is included in the formulation. The resulting coupled system of non-linear partial differential equations with physiologically relevant initial, entrance and boundary conditions is solved numerically. The COHb levels computed from our model agree with those measured experimentally (Pace et al., 1950; Peterson & Stewart, 1970). The half-life of COHb, i.e. the time required for the blood COHb to decrease from the initial level to its half-value is computed. The half-life values of COHb computed from our model are in good agreement with those based on experimental data collected under different physiological conditions (Pace et al., 1950). Also, the results predicted from our model give better approximation to the experimental values than the CFK equation (Coburn et al., 1965). It is found that the rate of elimination of CO increases with the increase of inspired PO2 and ventilation rate.

Mathematical model for the exchange of gases in the lungs with special reference to carbon monoxide.

A mathematical model has been formulated for the simultaneous exchange of gases O2, CO2, CO and N2 in the lungs. The model takes into account the physiological parameters, such as ventilation rate, diffusing capacity of the lungs, cardiac output, total volume of blood in the body and the interaction of gases in the blood. The nonlinear functions for representing O2, CO2 and CO dissociation curves have been used. The results predicted from the model are in good agreement with those based on the ventilation/perfusion relationships. The COHb build-up in the blood, computed from the model as a function of exposure time, is in good agreement with the experimental values. The consideration of capillary blood pO2 as a constant value, instead of an independent variable, is shown to introduce a maximum error of 0.25 per cent in the blood COHb. The model is applied to analyse the COHb levels at high altitude.

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.

A note on Gutierrez's kinetics model for oxygen delivery to tissue.

The problems associated with Gutierrez model (1986, Resp. Physiol. 63, 79-96) on O2 delivery to tissue are discussed. He has used a dimensionally incorrect function for the oxygen dissociation curve (ODC). The dimensionally correct function for the ODC has been used in the analysis and the correct results are given for normoxic, hypoxic and anaemic conditions.

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.

Modelling of oxygen transport in systemic circulation in a hyperbaric environment.

A mathematical model is developed for analysing the transport of oxygen in the systemic capillaries and surrounding tissue in a hyperbaric environment. The governing equation in the capillary describes the transport due to molecular diffusion (radial as well as axial) and the convective effect of the blood. The non-linear oxygen dissociation curve is represented by a linear function simulating the conditions of a hyperbaric environment. The corresponding equation in the tissue region is based on the transport of oxygen due to radial as well as axial diffusion and the zero order metabolic consumption rate. The equations in both the regions are connected through the interface conditions. An analytical solution of the resulting system of elliptic partial differential equations with the physiologically relevant boundary conditions is obtained by the method of eigenfunction expansion. It is found that the amount of oxygen decreases from the core of the capillary to the periphery of the tissue. It is shown that significant radial diffusion of oxygen takes place in the initial part of the tissue close to the arterial end. The accumulation of oxygen in the tissue has been examined in terms of various non-dimensional parameters. The physiological relevance of these parameters in determining the degree of accumulation of O2 in the tissue in a hyperbaric environment is discussed in the light of previous experimental studies.

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.

A mathematical model for the rate of oxygenation of blood in pulmonary capillaries using nth-order one-step kinetics of oxygen uptake by haemoglobin.

A mathematical model is described for the process of gas exchange in pulmonary capillaries by taking into account the transport mechanisms of molecular diffusion, convection, and the facilitated diffusion due to haemoglobin. The nth-order one-step kinetics of oxygen uptake by haemoglobin has been incorporated. The rate k at which blood becomes oxygenated is determined by setting up an appropriate eigenvalue problem. This method eventually leads to a transcendental equation in k. A multiprecision technique due to Verma and Sharan (1980) is employed to obtain a physically acceptable solution. It is shown that, at equilibrium, the saturation of haemoglobin with oxygen computed from the analysis is fairly close to the data of Severinghaus (1966). It was found that 97.15% of the total haemoglobin combined with oxygen. The blood is oxygenated well before it leaves the pulmonary capillary. The dissolved oxygen takes longer to achieve equilibration whereas the carbon dioxide traverses a comparatively smaller distance in the capillary.

A numerical model for studying the effect of plasma layer on the process of blood oxygenation in the pulmonary capillaries.

A two layer model for the blood oxygenation in pulmonary capillaries is proposed. The model consists of a core of erythrocytes surrounded by a symmetrically placed plasma layer. The governing equations in the core describe the free molecular diffusion, convection, and facilitated diffusion due to the presence of haemoglobin. The corresponding equations in the plasma layer are based on the free molecular diffusion and the convective effect of the blood. According to the axial train model for the blood flow proposed by Whitmore (1967), the core will move with a uniform velocity whereas flow in the plasma layer will be fully developed. The resulting system of nonlinear partial differential equations is solved numerically. A fixed point iterative technique is used to deal with the nonlinearities. The distance traversed by the blood before getting fully oxygenated is computed. It is shown that the concentration of O2 increases continuously along the length of the capillary for a given ratio of core radius to capillary radius. It is found that the rate of oxygenation increases as the core to capillary ratio decreases. The equilibration length increases with a heterogeneous model in comparison to that in a homogeneous model. The effect of capillary diameters and core radii on the rate of oxygenation has also been examined.

Mathematical model for the computation of alveolar partial pressure of carbon monoxide.

A mathematical model is formulated for computing alveolar partial pressure of carbon monoxide (PACO) from that in the atmospheric air. The model takes into account parameters like inspired/expired air flow rates, diffusion capacity of the lung, concentration of CO in the atmospheric air, blood flow rate and the non-linear CO dissociation curve. The effect due to the presence of O2 in the blood on CO dissociation curve is also incorporated. It is shown that for a given atmospheric CO concentration, PACO increases exponentialy with time and attains asymptotic value. Alveolar PCO increases further with the increase in the atmospheric CO concentration. The model can also be used to compute carboxyhaemoglobin levels in the blood as a function of exposure time and the results are comparable with the CFK equation and the values measured experimentally.

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.

Numerical simulation of systemic O2 and CO2 exchange in a hyperbaric environment.

The process of gas exchange in systemic capillaries and its surrounding tissue is simulated numerically in a hyperbaric environment, taking into account the molecular diffusion, convection, saturation of haemoglobin with O2 and CO2, and the metabolic activity in the tissue. Krogh tissue-cylinder is used as a geometrical representation of the capillary-tissue system. The resulting system of non-linear governing equations together with the physiologically relevant boundary conditions is solved numerically. It is found that the concentration of oxygen decreases from the axis of the capillary to the tissue periphery whereas the concentration of carbon dioxide increases. It is shown that very little CO2 is transported radially. The location of the vulnerable region from the point of view of CO2 accumulation is found to be the rim (r = R2, z = L) situated at the periphery of the tissue near the venous end of the capillary. It is also found that accumulation of O2 decreases whereas that of CO2 increases in a hyperbaric environment. Finally, it is surmised that one of the reasons in causing discomfort among divers could be excessive accumulation of CO2 in the tissue.

Algorithm for computing oxygen dissociation curve with pH, PCO2, and CO in sheep blood.

By extending the study of Samaja and Gattinoni, an algorithm is described for computing the oxygen dissociation curve with variations in pH, PCO2, and CO in homozygous HbB sheep blood. The difference in the values of O2 pressure at 50% saturation in presence of CO computed from the present algorithm and Hill's equation does not exceed 0.5%. It is shown that O2 affinity increases as the concentration of CO or pH increases or PCO2 decreases. The algorithm is convenient for representing the oxygen dissociation curve with variation in pH, PCO2 and the concentration of CO in modelling oxygen transport in sheep blood even under hypoxic conditions.

A compartmental model for oxygen transport in brain microcirculation.

A compartmental model is formulated for oxygen transport in the cerebrovascular bed of the brain. The model considers the arteriolar, capillary and venular vessels. The vascular bed is represented as a series of compartments on the basis of blood vessel diameter. The formulation takes into account such parameters as hematocrit, vascular diameter, blood viscosity, blood flow, metabolic rate, the nonlinear oxygen dissociation curve, arterial PO2, P50 (oxygen tension at 50% hemoglobin saturation with O2) and carbon monoxide concentration. The countercurrent diffusional exchange between paired arterioles and venules is incorporated into the model. The model predicts significant longitudinal PO2 gradients in the precapillary vessels. However, gradients of hemoglobin saturation with oxygen remain fairly small. The longitudinal PO2 gradients in the postcapillary vessels are found to be very small. The effect of the following variables on tissue PO2 is studied: blood flow, PO2 in the arterial blood, hematocrit, P50, concentration of carbon monoxide, metabolic rate, arterial diameter, and the number of perfused capillaries. The qualitative features of PO2 distribution in the vascular network are not altered with moderate variation of these parameters. Finally, the various types of hypoxia, namely hypoxic, anemic and carbon monoxide hypoxia, are discussed in light of the above sensitivity analysis.

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.

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.