Mathematical modeling of Cardiovascular System
Zdeněk Brož, František Maršík, Světlana Převorovská
e-mail : broz@km1.fjfi.cvut.cz
Abstract
Cardiovascular system distributes blood with oxygen and many others of vital
concernment substances and therefore is the most important part of human body. Knowing
how cardiovascular system works allows us to treat some heart diseases, but it also helps us
to learn about its disfunctionality. Numerical simulation of cardiovascular system has also
become a useful tool of surgeon who diagnoses cardiovascular diseases and recommends the
way of their medical treatment
1. Introduction
The numerical model of the cardiovascular system of pulsating type imitating the
electrochemical and mechanical activity of heart is good for non-invasive diagnosis. In the
course of design of the mathematical-physical model we laid stress on “real-time” responses
of numerical simulation on computer and in consideration of this requirement we developed
software that can do such simulation. With this software you can directly see haemodynamic
responses of cardiovascular system in dependence of changes in its parameters (e.g.
hydrodynamical resistance of aorta or sodium channels conductance in heart tissue) and watch
specific haemodynamic values like ejection fraction or mean pressure. This model is also
suitable to simulate things like heart support pump, dialysis, valve oscillation or so-called
Korotkoff’s sound which is experimentally heard in system arteries and is probably caused of
self-excited oscillation of arterial system.
2. Scheme of Cardiovascular System
A real cardiovascular system has been dived into several compartments for the modeling
and simulation purposes (see fig. 1). The circulatory system is represented by four
compartments of a pulsating heart (left and right atria and left and right ventricle) and by
several vascular segments of the pulmonary and systemic circuits connected with heart
chambers in series. The pulmonary circuit consists of pulmonary artery, arteries, capillaries
and veins and analogically systemic circuit was compartmentalized into aorta, arteries,
capillaries, veins and head arteries and veins. The latest version of the model was enriched by
three segments that allow simulating dialysis process.
The above-mentioned model provides one-dimensional flow of incompressible blood
through the network of elastic blood vessels. Heart compartments are supposed to be made of
anisotropic and viscoelastic incompressible material.
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Fig. 1. Scheme of the cardiovascular system
The behavior of the cardiovascular system is described by its haemodynamic variables,
i.e. the blood pressure, volume, and by the cardiovascular parameters such as compliances and
resistances in corresponding compartments.
3. Chemical and Mechanical Performance of Heart
The heart is pressure-volume pump, where the pressure pulsations are generated by
changes in concentration of calcium ions that regulates the force of contractility of the
cardiomyocytes. The Beeler-Reuter equations where used to describe the membrane potentials
during cardiac cycle that affects the concentration of calcium ions transported through the
cardiomyocytes membrane. The potentials of atrial and ventricular action Vm[mV] are
determined by following four major ionic currents :
-
INa – fast inward sodium current
IS – slow inward calcium current
IK – time independent potassium current
IX – time dependent outward potassium current
dVmi (t )
dt


1
I Nai (t )  I Si (t )  I Ki (t )  I X i (t )  I Sti (t )
Cm

(1)
2
where Cm[F/cm2] is the capacitance of myocardial tissue and Ist(i)[A/cm2] is the stimulating
current. The ionic currents are formulated in this way :


I Nai (t )  g Nai mi3 (t )hi (t ) ji (t )  g NaCi (Vmi (t )  E Na )
(2)
I Si (t )  g Si d i (t ) f i (t )(Vmi (t )  E Si (t ))
(3)
0.04Vmi ( t ) 85 
Vmi (t )  23 

e
1


I K i (t )  0.35 g K i 4 0.08Vm (t ) 53

0
.
2
0.04Vmi ( t )  53
 0.04Vmi ( t )  23 
 e
i

e
1

e


0.04Vmi ( t )  77 
e
1
I X i (t )  0.8 g X i xi (t ) 0.04Vm (t )35
i
e
(4)
(5)
The pressure in the heart is described by the equation derived from the general energy and
entropy balance :
2.hi Ei
ri
pi (t ) 
 Vi (t )  2.hi k chem,i

 1 
cCai (t )
ri
 V0i

(6)
where the intracellular concentration of calcium ions cCa is calculated from the following
differential equation :
dcCai (t )
dt

 10 7 I Si  0.07 10 7  cCai (t )

(7)
For more information and better explanation of above-mentioned equations see [1],[2] or [3]
4. Haemodynamic Performance of Cardiovascular System
Following equation describes pressure changes in the pulmonary artery and aorta
Pi (t ) 
(Vi (t )  VU i   i
Ci
dVi (t )
)
dt
i=2,8
(8)
For other segments in pulmonary and systemic circuit we can omit the last item in the
equation and use it in following form :
Pi (t ) 
(Vi (t )  VU i )
Ci
i=3,4,5,9,10,11,12,13,14,15
(9)
where C[m3/Pa] denotes the compliance, VU[m3] is the residual volume and [1] represents
the wall viscosity. Pressure in dialysis pump is described by following equation :
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
P16 (t )  Pampl sin 2 ( t ) ; t  0,T
T
(10)
Blood flow between compartments is determined by the balance of momentum in the
following differential and ordinary equations :

between ventricles and output arteries, and between dialysis pump and systemic veins:
dF j ,k (t )
dt

dt

j=1,7,16; k=2,8,11
(11)
1
 p j (t )  pk (t )  R j ,k F j ,k (t )
Lj
j=2,8,8; k=3,9,12
(12)
j=3,4,9,9,9,9,10,13,14,15;
k=4,5,10,14,15,16,11,11,11,11
(13)
j=0,5,6,11,12 ; k=1,6,7,0,13
(14)
flow between other segments are described in following form :
F j ,k (t ) 

2

F j ,k (t ) 2 

 p j (t )  p k (t )  R j ,k F j ,k (t ) 


2 

2
A
(
t
)
j


in flow that comes from pulmonary artery or aorta we can omit the last item that
represents blood inertia :
dF j ,k (t )

1

Lj
2
1
 p j (t )  pk (t ). Vi (t2)
R j ,k
VU i
or
F j ,k (t ) 
1
 p j (t )  pk (t )
R j ,k
where L[Pa.s2/m3] characterizes the blood inertia, R[Pa.s/m3] is the hydrodynamical
resistance, [1] is the coefficient of blood inertia and A[m2] is the flow area. The volume
changes in all segments of the cardiovascular system are determined by the balance of mass.
d
V j (t )  Fi , j (t )  F j ,k (t )
dt
i, j, k=0,1,2,…,16
(15)
5. Results of Numerical Simulation
In order to show the capabilities of the prescribed model, a haemodynamic behavior of the
human cardiovascular system in normal state was simulated. Our developing program gives
very good results compared with SimBioSys SW (analogical software used in medical
training [4]) and it gives you full access to all implemented cardiovascular parameters that
describe the whole system. The software uses Runge-Kutt method for solving differential
equations numerically and it gives the results practically in real-time, so you can use this
simulator in very interactive way.
The following figure shows pressure – volume loop of healthy patient.
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Fig. 2. Pressure-Volume loop
With the simulator you can depict various dependencies of all haemodynamic variables in
the cardiovascular system of human body and quest for still unknown principles,
Fig. 3. Snapshot of the program
(The figure on its toper part shows the dependency between flow and volume of atrial parts of heart.)
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but the software is also suitable to simulate things like Korotkoff‘s sound.
Fig. 4. Korotkoff’s sound
In clinical usage there are some
commonly used parameters that
describe haemodynamic of the whole
cardiovascular system. The aim of
every medical investigation is to
evaluate these parameters from
pressure and volume characteristics
during cardiac cycle.
This
simulating
program
automatically evaluates these variables
and displays them for possible later
analysis.
Fig. 5. Sheet with haemodynamic variables
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6. Conclusion
Mathematical models of the cardiovascular system are useful for a deeper understanding
of complex processes occurring in the heart and blood vessels in the normal and
pathophysiological state. All the findings that were done from pressure waves, volume
changes or evaluated haemodynamic variables can be used for diagnosing a patient illness and
therefore this simulator can be helpful for recommending of the way of medical treatment.
The simulation results are compatible with the published clinical data.
Bibliography
[1] Beeler G.W., Reuter H., Reconstruction of the action potential of the ventricular
myocardial fibers, J. Physiol. (London), 1977, 268, pp. 177-210.
[2] Převorovská S., Maršík F., Musil J. Human cardiovascular system with heart failure
under baroreflex control (numerical model), Acta of Bioengineering and Biomechanics
Vol. 3, No. 1, 2001
[3] Brož. Z Diploma Thesis, Faculty of Nuclear Sciences and Physical Engineering,
CVUT, 2001
[4] Simbiosys SW – www.simbiosys.ca
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