A simplified model of the cardiovascular system. The left ventricle
is modelled by an elastic compartment (capacitance) which is filled directly from the
venous side of the circulatory system via the mitral valve (diode). When the left
ventricle pressure pLV exceed the constant venous pressure pv, the mitral valve closes
and the heart starts to contract. The contraction is modelled by a time-dependent
pressure source (pump) in parallel with the elastic compartment. When the
contraction is sufficiently strong, the left ventricle pressure exceeds the aortic
pressure p and the aortic valve (diode) opens and the ventricle blood volume is
expelled into the aorta. The systemic arteries are modelled using a simple twoelement Windkessel. The mitral valve dynamics is given by pv & pLV
qin =
pv-pLV
RMV
(pv \u2212 pLV),(1)
where H denote the Heaviside function. For the aortic valve we have in complete
analogy
qout =
pLv-p
RAV
(pLV \u2212 p).(2)
The left ventricle pressure is given by
pLV = ELV(VLV & VLV,0)2 + f(t)g(VLV), (3)
where VLV and VLV,0 are the blood volume and non-inflated (dead) volume of the left
ventricle, respectively. The first term on the right side in Eq. (3) is represents the
passive response of the ventricle during filling. The second term is the time-varying
Figure 1: A simple model of the cardiovascular system.
Parameter Value Parameter Value
RMV 0.0010
mmHg-s/cm3
VLV,0 8.0000 cm3
RAV 0.0600
mmHg-s/cm3
T 1.00 s
R 1.0000
mmHg-s/cm3
tp 0.35 s
ELV 0.0007 mmHg/cm6 ts 0.80 s
E 0.3636 mmHg- cm3 1.5000
mmHg/cm3
pV 10 mmHg 0.6000 cm3
2
Table 1: Data for the cardiovascular model.
elastance where
and g(VLV) = (VLV). The function f is T-periodic and f(t
T) = f(t).
Questions (10p)
(a) Write a program and compute the left ventricle pressure-volume loop using
themodel above. Plot a graph for a single steady-state loop and indicate the
following points: aortic valve opening (AVO), aortic valve closing (AVC), the
mitral valve opening (MVO), mitral valve closing (MVC), end diastolic volume
(EDV ), and end systolic volume (ESV ). Parameters for the model are given in
Table 1. Set the initial conditions for the aortic pressure and the left ventricle
volume to p(0) = 50 mmHg and VLV(0) = 50 cm3, respectively. Note that you
may need to simulate several heart cycles before a stable pressure-volume loop
is obtained.
(b) The Frank-Starling law states that the stroke volume (SV ) increases with the
end diastolic volume of the ventricle. If the stroke volume is defined by SV =
EDV ESV , verify that your model satisfy the Frank-Starling law by computing
the stroke volume and plotting the left ventricle pressure-volume response for
four different end diastolic volumes. Hint: the end diastolic volume can be
changed by altering the ventricle stiffness. Use small changes.
(c) The end systolic pressure volume relationship (ESPVR) is an important
characteristic of the heart. It is obtained by connecting the non-inflated
ventricle volume to the point of aortic valve closing by a straight line. Show
graphically that the pressure-volume loops in (b) roughly fall along the same
ESPVR line.
Reporting
Inthis work is to be presented in terms of a short-written report. Write the report
using a word processor (e.g., LATEX, Mircosoft Word, Open Office Writer, or Google
Docs) with a standard font and size 11-12 pt. Page margins should be normal. The
report must not exceed 4 pages including derivations, figures, results, tables,
references, etcetera. Furthermore, attach any code used to solve in this work to
the same document. This attachment is not subject to the page limit. Submit the
report by sending it as PDF documents.
About the recuiterMember since Mar 14, 2020 Yudhi Fernando
from Kagawa, Japan