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CHAPTER III
The purpose of this chapter is to discuss the variation of blood pressure in the eight
compartments listed earlier during seven phases of the cardiac cycle and analyse the variation
of blood pressure at each compartment using Lumped Parameter Method. A brief note on the
various approaches by different authors is considered first. Next, the step function has been
redefined for satisfactory explanation.
Richard E Klabunde has modelled the blood flow using Bernoulli principle [1]. However,
Peter W Carpenter et al., have modelled the blood flow using experimental and theoretical
research on rotational flows over compliant walls [18].
D S Sankar and Usik Lee analyse the pulsatile flow of blood through a catheterized artery
assuming the blood flow as a two fluid model with the suspension of all the erythrocytes in the
core region as a Non-Newtonian fluid and the peripheral region of plasma as a Newtonian fluid.
This results in a mathematical model resulting in system of nonlinear implicit system of partial
differential equations which is solved using perturbation method [19].
However, James B. Grotberg and Oliver E. Jensen have made an extensive research for
understanding the fundamental mechanics of the flexible tube flows and the physiological
applications spanning the cardiovascular system [21].
Yunlong Huo and Ghassan Kassab have tried to model the pulsatility of coronary circulation
on the basis of branching pattern, vascular geometry and material properties of the coronary
vasculature using Womersley-type mathematical modelling [23].
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Nilmini Saumya Wijeratne has mathematically formulated the circulatory system considering
the two dimensional fluid structure and structure interactions in the presence of a nonNewtonian fluid and isotropic elastic properties [33].
Guyton et al., [49] have made an attempt to describe physiological facts of circulatory function
by diving the whole circulatory system into 354 blocks where each block represents one or
more mathematical equations which describe the physiological facts of circulatory system.
Many mathematical models have been proposed to study many human physiology using
lumped parameter method. However, much work in the area of mathematical modelling to
study variation of blood pressure in heart chambers during one cardiac cycle at higher altitude
has not been done. Therefore, an attempt is made to mathematically model the blood flow in
heart using lumped parameter electric model as this is most suitable for representing a large
portion of closed loop system.
In recent times, mathematical model of cardiac electrical activity has been recognised as one
of the significant approaches which can diagnose and understand the cardiovascular system.
Lumped parameter method is one such method which is used in the field of bio medical
research.
3.1 Lumped Parameter Method for Cardiovascular System
Lumped parameter model is a very useful type of mathematical modelling where the physical
system is represented by an electrical network. This model is represented graphically by a
circuit diagram in which vertices represent the voltages and the edges the current in the circuit.
Using lumped parameter models, the relationships between flow rate and pressure of the
cardiovascular system can be established on the basis of an analogy with current and voltage
in an electric circuit system.
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The heart consists of four chambers namely right atrium (RA), left atrium (LA), right ventricle
(RV) and left ventricle (LV). There are also four valves namely tricuspid valve (Tr), pulmonary
valve (Pu), mitral valve (Mi) and the aortic valve (Ao). Systemic arteries (sa), systemic veins
(sv), pulmonary arteries (pa) and pulmonary veins (pv) form the systemic and the pulmonary
circulations.
In order to represent the cardiovascular system as a lumped parameter method, it becomes
important to understand the overall structure of the circulation, which can be summarized as
follows.
The aortic valve opens when the left ventricular pressure is more than the systemic aortic
pressure and lets the blood flow through the systemic arteries. Through the systemic arteries
the pure blood flows to all parts of the body. From all parts of the body the impure blood
through the systemic veins return to the right atrium. The tricuspid valve opens when the right
arterial pressure is more than the right ventricle. As a result the right ventricle is filled with
impure blood. Again, when the right ventricular pressure increases, the pulmonary valve opens
and the blood through pulmonary arteries reach the tissues of the lung. The blood gets purified
in lungs by exchange of oxygen and pure blood from the lungs through pulmonary veins return
to the left atrium. When the pressure of the left atrium increases and becomes more that left
ventricle the mitral valve opens and blood fills the left ventricle. Thus the circulation gets
completed.
Now, this has to be represented as a lumped pulsatile model by describing a network of the
compliance and resistance vessels. The lumped parameter electric model proposed by Eunok
Jung and Wanho Lee [51] has been studied extensively and in the present work and the
definition of step function has been modified to study the variation of blood pressure at eight
compartments during one cardiac cycle. This is schematically represented as follows [50, 51,
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52]. This electric analogy is interpreted using simple ordinary differential equations in time
which can be solved either numerically or analytically.
Rp
pa
pv
Rpu
CRV(t)
CRA(t )
Rpv
RV
RV
RTrRV
LA
RA
RR
A
LV
CLA(t)
RMi
Rsv
CLV(t)
RAo
sv
sa
ss
Rs
vs
vs
Figure
3.1: Equivalent Circuit for Lumped Model
vs
vs
v
In the above diagram the symbols and their representation is as follows.
Symbol
Rpu
RTr
Rs
Rsv
CRV
CLV
Representation
Symbol
resistance of pulmonary
valve
resistance of tricuspid
valve
resistance of systemic
artery
resistance of systemic
vein
compliance of Right
Ventricle
compliance of Left
Ventricle
RAo
resistance of Aortic valve
RMi
resistance of Mitral valve
Rp
resistance of
artery
resistance of
vein
compliance
Auricle
compliance
Auricle
Rpv
CRA
CLA
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Representation
pulmonary
pulmonary
of
Right
of
Left