Ranjan K. Dash

Casson fluid flow in a pipe filled with a homogeneous porous medium

Abstract The flow characteristics of a Casson fluid in a tube filled with a homogeneous porous medium is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. This analysis can model the pathological situation of blood flow when fatty plaques of cholesterol and artery-clogging blood clots are formed in the lumen of the coronary artery. Two cases of permeability of the porous medium are considered, namely (i) permeability has a constant value K o , and (ii) permeability varies in radial direction according to K ( r ) = K o (1− r )/ r . The generalized equation of motion, which is an integral equation for shear stress, is solved iteratively and is coupled with the Casson constitutive equation to find the velocity distribution. For the case of constant permeability, the analytical solution is found for the shear stress distribution in terms of modified Bessel functions of order 0 and 1. Finally, the effect of permeability factor K o and yield stress θ of the fluid on shear stress distribution, wall shear stress, plug flow radius, flow rates and frictional resistance are examined.

Flow in a catheterized curved artery with stenosis

The fluid mechanics of blood flow in a catheterized curved artery with stenosis is studied through a mathematical analysis. Blood is modelled as an incompressible Newtonian fluid and the flow is assumed to be steady and laminar. An approximate analytic solution to the problem is obtained through a double series perturbation analysis for the case of small curvature and mild stenosis. The effect of catheterization on various physiologically important flow characteristics (i.e. the pressure drop, impedance and the wall shear stress) is studied for different values of the catheter size and Reynolds number of the flow. It is found that all these flow characteristics vary markedly across a stenotic lesion. Also, increase in the catheter size leads to a considerable increase in their magnitudes. These results are used to obtain the estimates of increased pressure drop across an arterial stenosis when a catheter is inserted into it. Our calculations, based on the geometry and flow conditions existing in coronary arteries, suggest that, in the presence of curvature and stenosis, and depending on the value of k (ratio of catheter size to vessel size) ranging from 0.1 to 0.4, the pressure drop increases by a factor ranging from 1.60 to 5.16. But, in the absence of curvature and stenosis, with the same range of catheter size, this increased factor is about 1.74-4.89. These estimates for the increased pressure drop can be used to correct the error involved in the measured pressure gradients using catheters. The combined effects of stenosis and curvature on flow characteristics are also studied in detail. It is found that the effect of stenosis is more dominant than that of the curvature. Due to the combined effect of stenosis, curvature and catheterization, the secondary streamlines are modified in a cross-sectional plane. The insertion of a catheter into the artery leads to the formation of increased number of secondary vortices.

ESTIMATION OF INCREASED FLOW RESISTANCE IN A NARROW CATHETERIZED ARTERY -- A THEORETICAL MODEL

The changed flow pattern in a narrow catheterized artery is studied and an estimate of the increased flow resistance is made. The anomalous behaviour of blood in small blood vessels has been taken into account by modelling blood as a Casson fluid possessing some finite yield stress. Both the cases of steady and pulsatile flow situations are studied. The pulsatile flow is analysed by considering the pressure gradient as a periodic function of time with small inertial effects. The resulting quasi-steady non-linear coupled implicit system of differential equations governing the flow are solved using a perturbation analysis, where it is assumed that the Womersley frequently parameter is small (alpha < 1) which is reasonable for physiological situations in small blood vessels as well as in coronary arteries. The effect of pulsatility, catheter radius and yield stress of the fluid on the yield plane locations, velocity distribution, flow rate, shear stress and frictional resistance are investigated. Because of the yield stress theta, two yield surfaces are found to be located in the flow field. Depending on the ration kappa (catheter size/vessel size) ranging from 0.3 to 0.7 (which is widely used in coronary angioplasty procedures), the frictional resistance to flow in large blood vessels, where the effect of yield stress can be neglected (i.e. theta = 0), increases by a factory ranging from 3 to 33. In small blood vessels with the same range of catheter size and an unit pressure gradient, frictional resistance increase was by a factor of 7-21 when theta = 0.05 and 11-294 when theta = 0.1. For small values of kappa and theta, the frictional resistance increased to several hundred times thus implying that the combined effect of increased catheter radius and yield stress is to obstruct the fluid movement considerably.

Analysis of cardiac mitochondrial Na+–Ca2+ exchanger kinetics with a biophysical model of mitochondrial Ca2+ handing suggests a 3: 1 stoichiometry

Calcium is a key ion and is known to mediate signalling pathways between cytosol and mitochondria and modulate mitochondrial energy metabolism. To gain a quantitative, biophysical understanding of mitochondrial Ca2+ regulation, we developed a thermodynamically balanced model of mitochondrial Ca2+ handling and bioenergetics by integrating kinetic models of mitochondrial Ca2+ uniporter (CU), Na+–Ca2+ exchanger (NCE), and Na+–H+ exchanger (NHE) into an existing computational model of mitochondrial oxidative phosphorylation. Kinetic flux expressions for the CU, NCE and NHE were developed and individually parameterized based on independent data sets on flux rates measured in purified mitochondria. While available data support a wide range of possible values for the overall activity of the CU in cardiac and liver mitochondria, even at the highest estimated values, the Ca2+ current through the CU does not have a significant effect on mitochondrial membrane potential. This integrated model was then used to analyse additional data on the dynamics and steady‐states of mitochondrial Ca2+ governed by mitochondrial CU and NCE. Our analysis of the data on the time course of matrix free [Ca2+] in respiring mitochondria purified from rabbit heart with addition of different levels of Na+ to the external buffer medium (with the CU blocked) with two separate models – one with a 2: 1 stoichiometry and the other with a 3: 1 stoichiometry for the NCE – supports the hypothesis that the NCE is electrogenic with a stoichiometry of 3: 1. This hypothesis was further tested by simulating an additional independent data set on the steady‐state variations of matrix free [Ca2+] with respect to the variations in external free [Ca2+] in purified respiring mitochondria from rat heart to show that only the 3: 1 stoichiometry model predictions are consistent with the data. Based on these analyses, it is concluded that the mitochondrial NCE is electrogenic with a stoichiometry of 3: 1.

Blood HbO2 and HbCO2 Dissociation Curves at Varied O2, CO2, pH, 2,3-DPG and Temperature Levels

New mathematical model equations for O2 and CO2 saturations of hemoglobin (SHbO2 and SHbCO2) are developed here from the equilibrium binding of O2 and CO2 with hemoglobin inside RBCs. They are in the form of an invertible Hill-type equation with the apparent Hill coefficients KHbO2 and KHbCO2 in the expressions for SHbO2 and SHbCO2 dependent on the levels of O2 and CO2 partial pressures (PO2 and PCO2), pH, 2,3-DPG concentration, and temperature in blood. The invertibility of these new equations allows PO2 and PCO2 to be computed efficiently from SHbO2 and SHbCO2 and vice-versa. The oxyhemoglobin (HbO2) and carbamino-hemoglobin (HbCO2) dissociation curves computed from these equations are in good agreement with the published experimental and theoretical curves in the literature. The model solutions describe that, at standard physiological conditions, the hemoglobin is about 97.2% saturated by O2 and the amino group of hemoglobin is about 13.1% saturated by CO2. The O2 and CO2 content in whole blood are also calculated here from the gas solubilities, hematocrits, and the new formulas for SHbO2 and SHbCO2. Because of the mathematical simplicity and invertibility, these new formulas can be conveniently used in the modeling of simultaneous transport and exchange of O2 and CO2 in the alveoli-blood and blood-tissue exchange systems.

A biophysically based mathematical model for the kinetics of mitochondrial calcium uniporter

Ca2+ transport through mitochondrial Ca2+ uniporter is the primary Ca2+ uptake mechanism in respiring mitochondria. Thus, the uniporter plays a key role in regulating mitochondrial Ca2+. Despite the importance of mitochondrial Ca2+ to metabolic regulation and mitochondrial function, and to cell physiology and pathophysiology, the structure and composition of the uniporter functional unit and kinetic mechanisms associated with Ca2+ transport into mitochondria are still not well understood. In this study, based on available experimental data on the kinetics of Ca2+ transport via the uniporter, a mechanistic kinetic model of the uniporter is introduced. The model is thermodynamically balanced and satisfactorily describes a large number of independent data sets in the literature on initial or pseudo-steady-state influx rates of Ca2+ via the uniporter measured under a wide range of experimental conditions. The model is derived assuming a multi-state catalytic binding and Eyrings free-energy barrier theory-based transformation mechanisms associated with the carrier-mediated facilitated transport and electrodiffusion. The model is a great improvement over the previous theoretical models of mitochondrial Ca2+ uniporter in the literature in that it is thermodynamically balanced and matches a large number of independently published data sets on mitochondrial Ca2+ uptake. This theoretical model will be critical in developing mechanistic, integrated models of mitochondrial Ca2+ handling and bioenergetics which can be helpful in understanding the mechanisms by which Ca2+ plays a role in mediating signaling pathways and modulating mitochondrial energy metabolism.

Modeling Cellular Metabolism and Energetics in Skeletal Muscle: Large-Scale Parameter Estimation and Sensitivity Analysis

Skeletal muscle plays a major role in the regulation of whole-body energy metabolism during physiological stresses such as ischemia, hypoxia, and exercise. Current experimental techniques provide relatively little in vivo data on dynamic responses of metabolite concentrations and metabolic fluxes in skeletal muscle to such physiological stimuli. As a complementary approach to experimental measurements and as a framework for quantitatively analyzing available in vivo data, a physiologically based model of skeletal muscle cellular metabolism and energetics is developed. This model, which incorporates key transport and reaction processes, is based on dynamic mass balances of 30 chemical species in capillary (blood) and tissue (cell) domains. The reaction fluxes in the cellular domain are expressed in terms of a generalized Michaelis-Menten equation involving energy controller ratios ATP/ADP and NADH/NAD+. This formalism introduces a large number of unknown parameters (~90). Estimating these parameters from in vivo sparse data and evaluating dynamic sensitivities of the model outputs with respect to these parameters is a challenging problem. Parameter estimation is accomplished using an efficient, nonlinear, constraint-based, optimization algorithm that minimizes differences between available experimental data and corresponding model outputs by explicitly utilizing equality constraints on resting fluxes and concentrations. With the estimated parameter values, the model is able to simulate dynamic responses to reduced blood flow (ischemia) of key metabolite concentrations and metabolic fluxes, both measured and nonmeasured. A general parameter sensitivity analysis is carried out to determine and characterize the parameters having the most and least effects on the measured outputs.

A class of model equations for bi-directional propagation of capillary-gravity waves

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a ¼ a=h0 and wavelength parameter b ¼ð h0=lÞ 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h0 is the height of the undisturbed water surface from the flat bottom topography.These equations are also characterized by the surface tension parameter, namely the Bond number s ¼ C=qgh 2 , where C is the surface tension coefficient, q is the density of water, and g is the acceleration due to gravity. The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq

Erratum to: Blood HbO2 and HbCO2 Dissociation Curves at Varied O2, CO2, pH, 2,3-DPG and Temperature Levels

New mathematical model equations for O2 and CO2 saturations of hemoglobin (