MATHEMATICAL MODELLING OF FLUID EXCHANGE THROUGH BLOOD CAPILLARIES
MATHEMATICAL MODELLING OF FLUID EXCHANGE THROUGH BLOOD CAPILLARIES
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Date
1986-10
Authors
SALEH, AISHATU
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Abstract
Flow properties of blood play an increasing role in
the diagnostic and therapy of many diseases as well as in
different artificial organs. The vessels of human
circulatory system range in internal diameter from
approximately 2.5 cm to 3μ (μ=10-4 cm). The vessels with
an internal dimension less than about 500μ are
commonly reffered to as the microvessels. Microcirculatory
vessels are small artries (70-500μ), arterioless (l0-70μ);
capillaries (4-10μ), venules (10-100μ) and small viens
(110-500μ).
The first interest in mathematical modelling of the
microcirculation developed from experimental investigations
to determine the arrangement of blood vessels in tissue.
These experiments were being conducted as a result of
interest in the mechanism by which oxygen was transported
from blood to tissue and how this transfer could be
controlled. The first theoretical study on the exchange
of oxygen across the capillary walls and the tissue
surrounding them was made by Krogh (1919) for which he
won his Nobel prize. He observed in detail the distribution
of capillaries in muscle, and proposed a functional
unit of capillary beds. Krogh's model in composed of a
long straight cylindrical capillary and a concentric
cylinder of homogeneous tissue surrounding it. The
present thesis, which is concerned with the mathematical
modelling of fluid exchange through blood capillaries,
comprises of four chapters.
The first chapter introduces the subject
'Biomechanics' and outlines the historical background
of the subject and its applications to clinical problems
in the cardiovascular system, quantitative physiology,
orthopedics, artifical limb and etc. The second
chapter describes the various aspects of the microcirculation.
The factors, Starting hypothesis, capillary
filtration coefficient, capillary and tissue hydrostatic
pressures, plasma and tissue osmotic pressures are
discussed. Finally this chapter discusses a variety of
approaches and formulations for the mathematical modelling
of the microcirculation. In the third chapter a mathematical
model is developed to see the effects of the
variable filtration coefficient, lymph flow and reflection
coefficient on the fluid exchange between blood capillary
and the tissue space. It is assumed that the exchange
of fluid across the capillary wall obey Starling's law
and that the filtration coefficient and the reflection
coefficient of the capillary wall are not uniform along
the length of capillary but is a linear function of the
distance from the arterial end. It is further assumed
that the flow of fluid through the tissue region is governed
by a generalized Darey's equation. Under these assumptions
the velocity and pressure distributions in the capillary
and tissue regions are obtained. We also obtain the average
interstitial fluid pressure and the average effective
pressure difference between the blood and interstitial
fluid. While chapter three is concerned about the
uniform capillary radius, chapter four is devoted to
fluid exchange between blood capillary of slowly but
arbitrary varying cross section and the tissue space.
An analytical solution in the general case is obtained
by perturbation technique.
Description
A THESIS SUBMITTED TO THE POSTGRADUATE SCHOOL
AHMAOU BELLO UNIVERSITY IN PARTIAL FULFILMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF
SCIENCE (MATHEMATICS)
Department of Mathematics,
Faculty of Science,
Ahmadu Bello University,
Zaria - Nigeria,
Keywords
MATHEMATICAL, MODELLING, FLUID, EXCHANGE, THROUGH, BLOOD, CAPILLARIES