Mathematical models of some physiological system

Abstract

This thesis presents some theoretical investigations on certain problems concerning the respiratory, excretory, and circulatory systems in human physiology, from a fluid mechanics point of view. It consists of an introductory chapter and four main chapters dealing with specific problems of current interest. Chapter I gives a general description of the aforesaid physiological systems with special emphasis on the organs considered for modelling, and the basic equations utilized in the thesis along with the data pertaining to the relevant physiological background. Chapter II considers a model study of the pulsatile blood flow in the lung alveolar sheets by idealizing each of them as a channel covered by a porous medium. Analytical and numerical results for the velocity and pressure distributions in the porous medium are presented and discussed in detail. In the first section of Chapter III, a mathematical analysis of a model for substrate (oxygen) concentration in tissue is presented for the case of first-order consumption by the tissue. The results are obtained numerically for capillary and tissue concentrations. Capillary permeability, which has no influence on the amount of substrate consumed in a zeroth-order rate of tissue consumption, is found to alter it in this study. In the second section, time-dependent transport and consumption of oxygen is analysed using the above model and asymptotic expansions are developed for the capillary and tissue oxygen concentrations. Chapter IV deals with two specific problems in the excretory system. In the first section, a hydrodynamical model for nephrons is presented. The possible values of the wall permeability are calculated for the case when the mean pressure at the entrance is prescribed. Further, an attempt has been made to describe the concentration distribution patterns of certain substances like urea, glucose and potassium by considering mass transport equations with the appropriate boundary conditions. In the second section, a model of the urometrogram is developed and the pressure distribution in the ureter with an inserted catheter is studied. Using long wavelength approximation, analytical and numerical solutions are obtained. It is found that the presence of a catheter does influence the pressure distribution within the ureter. The final chapter consists of three sections dealing with flow of blood treating it as a suspension. The first section presents an exact similar solution of the Navier-Stokes equations for unsteady flow of a suspension in a semi-infinite contracting or expanding circular pipe. The presence of the solid particles has been observed to influence the flow behaviour significantly. The second section deals with the suspension flow in a curved tube, whose curvature varies spatially. The governing equations are solved numerically and the results show that the delay in adaptation of the flow to the axial changes in curvature is substantially influenced by the particulate phase. The final section, apart from its relevance to bio-fluid mechanics, is of general interest in engineering and is concerned with the suspension flow in a corrugated tube. It is found that the frictional loss in a corrugated tube is larger than in a smooth one for a dilute suspension and the extent of deviation in frictional loss from the case of flow in a straight tube is larger in an axisymmetric tube than in an asymmetric one. For a fairly concentrated suspension, this loss is observed to be influenced by the other parameters involved in the problem. The books and original papers referred to in the text of the thesis are enlisted at the end of each chapter. Figures and tables relevant to each chapter are presented at the end of the chapter. The thesis is partly based on the following papers: (i) Pulsatile blood flow in a rigid pulmonary alveolar sheet with porous walls. Bull. Math. Biol., No. 5, pp. 563�7, 1981. (ii) Unsteady flow of a dilute suspension in a semi-infinite contracting or expanding pipe. Int. J. Engng. Sci., 21, No. 4, pp. 327�4, 1983. Papers based on the remaining work reported in the thesis have either been communicated or will be communicated shortly.