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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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 :
Q∗ =
P1 − P2
,
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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Periodic functions for LV and RV compliance
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
S=
(
1
0
ifP1
> P2
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
Q∗ =
(P1 − P2 )S
,
R
where R will typically be very low for a healthy valve.
Inserting for S, we have
Q∗ = max((P1 − P2 )/R, 0).
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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
= Qin
i − Qi ,
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 = max((Psv − Plv )/Rmi , 0),
(1)
Qout = max((Plv − Psa )/Rao , 0).
(2)
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We get
d(C(t)Plv )
= max((Psv − Plv )/Rmi , 0)
dt
− max((Plv − Psa )/Rao , 0)
Note: this is an ODE with discontinuous right hand side (if-tests),
which is not ideal from a numerical viewpoint. Implicit solvers
must be used, and the order of convergence is limited to 1.
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Similar calculations for the two other compliance vessels gives the
system
d(C(t)Plv )
= max((Psv − Plv )/Rmi , 0)
dt
− max((Plv − Psa )/Rao , 0),
Csa
dPsa
Psa − Psv
,
= max((Plv − Psa )/Rao , 0) −
dt
Rsys
Csv
dPsv Psa − Psv
− max((Psv − Plv )/Rmi , 0).
=
dt
Rsys
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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
tricuspid valve (systemic veins to right ventricle); Rtri ,
pulmonary valve (right ventricle to pulmonary arteries); Rpuv ,
mitral valve (pulmonary veins to left ventricle); Rmi ,
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This gives the ODE system
d(Clv (t)Plv )
= max((Ppv − Plv )/Rmi , 0) − max((Plv − Psa )/Rao , 0)
dt
Psa − Psv
dCsa Psa
,
= max((Plv − Psa )/Rao , 0) −
dt
Rsys
dCsv Psv Psa − Psv
− max((Psv − Prv )/Rtri , 0),
=
dt
Rsys
d(Crv (t)Prv )
= max((Psv − Prv )/Rtri , 0) − max((Prv − Ppa )/Rpuv , 0),
dt
Ppa − Ppv
d(Cpa Ppa )
,
= max((Prv − Ppa )/Rpuv , 0) −
dt
Rpu
dCpv Ppv Ppa − Ppv
− max((Ppv − Plv )/Rmi , 0).
=
dt
Rpu
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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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