Models for the circulatory system
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Outline
Overview of the circulatory system
Important quantities
Resistance and compliance vessels
Models for the circulatory system
Examples and extensions
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The circulatory system
Figure from Hoppensteadt og Peskin: Modeling and
simulation in medicine and the life sciences.
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Important quantities
Heart rate, measured in beats per minute.
Cardiac output: The rate of blood flow through the
circulatory system, measured in liters/minute.
Stroke volume: the difference between the end-diastolic
volume and the end-systolic volume, i.e. the volume of
blood ejected from the heart during a heart beat,
measured in liters.
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The cardiac ouput Q is given by
Q = F Vstroke
Typical values:
F = 80 beats/minute.
Vstroke = 70cm3 /beat = 0.070 liters/beat.
Q = 5.6 liters/minute.
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Resistance and compliance vessels
Pext = 0
Q1
Q2
V
V = vessel volume,
Pext = external pressure,
P1 = upstream pressure,
P2 = downstream pressure,
Q1 = inflow,
Q2 = outflow.
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Resistance vessels
Assume that the vessel is rigid, so that V is constant. Then
we have
Q1 = Q2 = Q∗ .
The flow through the vessel will depend on the pressure
drop through the vessel. The simplest assumption is that
Q∗ is a linear function of the pressure difference P1 − P2 :
P1 − P2
Q∗ =
,
R
where R is the resistance of the vessel.
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Compliance vessels
Assume that the resistance over the vessel is negligible.
This gives
P1 = P2 = P∗
Assume further that the volume depends on the pressure
P∗ . We assume the simple linear relation
V = Vd + CP∗ ,
where C is the compliance of the vessel and Vd is the “dead
volume”, the volume at P∗ = 0.
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All blood vessels can be viewed as either resistance
vessels or compliance vessels. (This is a reasonable
assumption, although all vessels have both compliance
and resistance.)
Large arteries and veins; negligible resistance,
significant compliance.
Arterioles and capillaries; negligible compliance,
significant resistance.
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Heart
Systemic veins
Systemic arteries
Body
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The heart as a compliance vessel
The heart may be viewed as a pair of compliance vessels,
where the compliance changes with time,
V (t) = Vd + C(t)P.
The function V (t) should be specified so that it takes on a
large value Cdiastole when the heart is relaxed, and a small
value Csystole when the heart contracts.
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0.015
0.01
0.005
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.04
0.03
0.02
0.01
0
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Modeling the heart valves
Characteristic properties of a heart valve:
Low resistance for flow in the “forward” direction.
High resistance for flow in the “backward” direction.
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The operation of the valve can be seen as a switching
function that depends on the pressure difference across the
valve. The switching function can be expressed as
(
1 ifP1 > P2
S=
0 ifP1 < P2
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The flow through the valve can be modeled as flow through
a resistance vessel multiplied by the switching function. We
have
(P1 − P2 )S
Q∗ =
,
R
where R will typically be very low for a healthy valve.
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Dynamics of the arterial pulse
For a compliance vessel that is not in steady state, we have
dV
= Q1 − Q2 .
dt
From the pressure-volume relation for a compliance vessel
we get
d(CP )
= Q1 − Q2 .
dt
When C is constant (which it is for every vessel except for
the heart muscle itself) we have
dP
C
= Q1 − Q2 .
dt
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The circulatory system can be viewed as a set of
compliance vessels connected by valves and resistance
vessels. For each compliance vessel we have
d(Ci Pi )
out
−
Q
= Qin
i ,
i
dt
while the flows in the resistance vessels follow the relation
P in − P out
Qj =
.
Rj
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A simple model for the circulatory system
Consider first a simple model consisting of three
compliance vessels; the left ventricle, the systemic arteries,
and the systemic veins. These are connected by two
valves, and a resistance vessel describing the flow through
the systemic tissues. For the left ventricle we have
d(C(t)Plv
= Qin − Qout ,
dt
with Qin and Qout given by
Qin
Qout
Smi (Psv − Plv )
=
,
Rmi
Sao (Plv − Psa )
=
.
Rao
(1)
(2)
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We get
d(C(t)Plv )
Psv − Plv Plv − Psa
=
−
,
dt
Smi Rmi
Sao Rao
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Similar calculations for the two other compliance vessels
gives the system
d(C(t)Plv )
Smi (Psv − Plv ) Sao (Plv − Psa )
−
,
=
dt
Rmi
Rao
dPsa
Sao (Plv − Psa ) Psa − Psv
Csa
−
,
=
dt
Rao
Rsys
dPsv
Psa − Psv Smi (Psv − Plv )
Csv
−
.
=
dt
Rsys
Rmi
(3)
(4)
(5)
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Heart
Systemic veins
Systemic arteries
Body
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With a specification of the parameters Rmi , Rao , Rsys , Csa , Csv
and the function Clv (t), this is a system of ordinary differential equations that can be solved for the unknown pressures
Plv , Psa , and Psv . When the pressures are determined they
can be used to compute volumes and flows in the system.
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A more realistic model
The model can easily be improved to a more realistic model
describing six compliance vessels:
The left ventricle, Plv , Clv (t),
the right ventricle, Prv , Crv (t),
the systemic arteries, Psa , Csa ,
the systemic veins, Psv , Csv ,
the pulmonary arteries, and Ppv , Cpv ,
the pulmonary veins, Ppv , Cpv .
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The flows are governed by two resistance vessels and four
valves:
Systemic circulation, Rsys ,
pulmonary circulation, Rpu ,
aortic valve (left ventricle to systemic arteries), Rao , Sao ,
tricuspid valve (systemic veins to right ventricle),
Rtri , Stri ,
pulmonary valve (right ventricle to pulmonary arteries),
Rpuv , Spuv ,
mitral valve (pulmonary veins to left ventricle) ,Rmi , Smi .
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This gives the ODE system
d(Clv (t)Plv )
Smi (Psv − Plv ) Sao (Plv − Psa )
−
,
=
dt
Rmi
Rao
dCsa Psa
Sao (Plv − Psa ) Psa − Psv
−
,
=
dt
Rao
Rsys
Psa − Psv Stri (Psv − Prv )
dCsv Psv
=
−
,
dt
Rsys
Rtri
(6)
(7)
(8)
d(Crv (t)Prv )
Stri (Psv − Prv ) Spuv (Prv − Ppa )
=
−
,
dt
Rtri
Rpuv
(9)
Spuv (Prv − Ppa ) Ppa − Ppv
dCpa Ppa
=
−
,
dt
Rpuv
Rpu
(10)
dCpv Ppv
Ppa − Ppv Smi (Ppv − Plv )
=
−
.
dt
Rpu
Rmi
(11)
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LV compliance, pressures and flows
0.015
0.01
0.005
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
150
100
50
0
80
60
40
20
0
Top: Cl v, Middle: Ppv (blue), Plv (green), Psa (red), Bottom: Qmi , Qao .
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RV compliance, pressures and flows
0.04
0.03
0.02
0.01
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
20
15
10
5
0
60
40
20
0
Top: Cr v, Middle: Psv (blue), Prv (green), Ppa (red), Bottom: Qtri , Qpuv .
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Systemic and pulmonary flows
8
7
6
5
4
3
2
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Qsys (blue) and Qpu (green). Note the higher maximum flow in the pulmonary system despite
the lower pressure. This is caused by the low resistance in the pulmonaries.
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Pressure volume loops
120
100
80
60
40
20
0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
20
15
10
5
0
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Top: left ventricle, bottom: right ventricle.
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Mitral valve stenosis
Rmi changes from 0.01 to 0.2.
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LV compliance, pressures and flows
0.015
0.01
0.005
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
150
100
50
0
80
60
40
20
0
Reduced in-flow to the LV causes reduced filling and thereby reduced LV pressure and arterial
pressure.
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RV compliance, pressures and flows
0.04
0.03
0.02
0.01
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
30
20
10
0
60
40
20
0
The RV pressure increases because blood is shifted from the systemic circulation to the pulmonary circulation.
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Systemic and pulmonary flows
8
7
6
5
4
3
2
1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Blood is shifted from the systemic circulation to the pulmonary circulation.
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Pressure volume loops
120
100
80
60
40
20
0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
25
20
15
10
5
0
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Reduced filling of the LV, slightly higher pressure in the RV.
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Reduced systemic resistance
Rsys reduced from 17.5 to 8.5.
This can be the result of for instance physical activity,
when smooth muscle in the circulatory system reduce
the resistance to increase blood flow to certain muscles.
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LV compliance, pressures and flows
0.015
0.01
0.005
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
100
50
0
80
60
40
20
0
The arterial pressure drops dramatically. This is not consistent with what happens during
physical activity.
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Summary
Models for the circulatory system can be constructed
from very simple components.
The models are remarkably realistic, but the simple
model presented here has some important limitations.
The models may be extended to include feedback loops
through the nervous system.
The simple components of the model can be replaced
by more advanced models. For instance, the varying
compliance model for the heart may be replaced by a
bidomain and mechanics solver that relates pressures
and volumes.
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