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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3779_Библиотеки_им_академика_М_И_Перельмана
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110 Chapter 6
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ventricles (afterload conditions). These models operate with such parameterized statistics
(called further parameters) as the volume of the heart chambers, the pressure, and
the flow rate through chamberechamber or chamberevessel connections. The
conservation laws and semi-empirical hydrodynamic laws help to relate these parameters
together.
Another common approach relates the lumped parameters of the heart and vasculature
using electromechanical analogy between the elements of electrical circuit (voltage,
current, resistance, capacity, and inductance) and the mechanical variables (pressure, flow
rate, hydraulic resistance due to the blood viscosity, elasticity, and blood inertia). The
resulting system of ODEs has the same mathematical properties for both lumped
hydrodynamic and electric circuit approaches. The number of equations depends on the
compartment decomposition, which may include heart chambers, vascular regions,
microcirculatory regions, aneurysms, etc. The reviews of the underlining principles and
mathematical models for both the heart chambers and the vascular regions can be found in
Refs. [223e226].
6.1.2 The scope and limitations of lumped models
The output of lumped models describes the dynamics of averaged flow statistics for
compartments such as flow rate, pressure, and volume, without providing any spatial details.
Such models may be used to describe blood flows in a segment of a large vessel [227], to
simulate hemodynamics in a part of a network of large vessels [212], to account for a large
region of microcirculation [215], to analyze the heart ejection dynamics [213], to set
boundary conditions for terminal vessels in higher dimensional models [221,228], to
perform the closure of arterial and venous network models [218,221,228] (required for
drug transport simulations), and to facilitate smooth coupling for models of different
dimensions [229] (we refer to Section 7.3 for details). Various aspects of the lumped
parameter approach for blood flow modeling can be found in Refs. [223e225,230]. The
work [231] defines heart outflow as a function of time, which correlates with the auricle
pressure and provides construction of a conservative closed circulation model. Lumped
models may account for several physiological effects such as vascular bed nervous control
[218,225] and viscoelasticity of vascular walls [223].
Among the drawbacks of 0D models, we note the lack of spatial details and the absence
of the heart rate variability accounting for dependence of the systole to diastole ratio
under preload and after-load conditions and the heart rate. The heart rate variability is
an important phenomenon for some clinical applications such as coronary circulation
during tachycardia (Section 7.6.3) and assessment of the fractional flow reserve
(Section 9.4.2).
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6.2 Electromechanical analogy
Although the electric circuit approach is difficult to apply to the detailed compartment
representation of the cardiovascular system due to the indirect analogy between electric
circuits and vascular networks (so-called inexact consistency [224]), one may compare
formulations of two basic 0D models of a lumped compartment with one input and one
output channel. The first model uses mechanical principles and operates with averaged
pressure (P) and flow rate (Q) through an elastic reservoir with volume (V). The second
model is the electric circuitebased analog.
The mass conservation states the balance between the change of volume and flow rate:
dV
¼Q
where Q
in
and Q
are the instantaneous inflow and outflow rates. The Poiseuille’s law
out
relates pressure drop DP
Q
dt
and flow rate over the input or the output through
in;out
in
; (6.1)
out
where R
DP
in¼RinQin
is the hydraulic resistance of the input or output channel, which may be
in,out
; DP
out
¼ R
outQout
(6.2)
constant or may vary depending on time and flow parameters (regulation, autoregulation,
non-Newtonian rheology, etc.).
For the linear elastic material of the vessel wall, we have V ¼ PC, were C is the elasticity
coefficient and P is the pressure inside the compartment. Finally, dynamics of the volume
is subject to second Newton’s law:
d2V
I
dt
where I is the inertia parameter, R is the hydraulic resistance of the compartment, and P
2
þR
dV
dt
þ
V
ext
; (6.3)
¼ P
C
ext
is the pressure exerted by external forces. System (6.1)e(6.3) can be used alone or can be
coupled with other models via input and output pressure drops DP
rates Q
and Q
in
out
.
and DP
in
and flow
out
The standard analysis of ODEs shows that system (6.1)e(6.3) describes free or forced
oscillations.
According to second Kirchhoff’s law, a similar system describes oscillations of an
alternating current in electric circuit that includes resistor, inductance, and capacity:
dI
e
L
þR
e
dt
þ Ue¼ Ee; (6.4)
eIe
112 Chapter 6
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where Ieis the current, Leis the inductance, Reis the electric resistance, Ueis the voltage,
and E
is the electromotance. Recalling that
e
where q
dq
e
I
e
is the electric charge and Ceis the capacity, one can rewrite Eq. (6.4) as
e
d2q
L
e
dt
; U
¼
dt
dq
e
þR
e
2
dt
q
e
¼
; (6.5)
e
C
e
q
e
e
¼ Ee: (6.6)
þ
C
e
The obvious equivalence of Eqs. (6.3) and (6.6), as well as the equivalence of Eq. (6.1)
and the first Kirchhoff law and the equivalence of Eq. (6.2) and Ohm’s law for a part of
circuit, allows us to postulate an analogy between the blood flow in a vascular region and
a heart chamber and the electric current in an electrical circuit with parallel and sequential
elements. Within this framework, the electric potential (U
pressure (P), the electric charge (q
current (I
) corresponds to the flow rate (Q), the electric resistance (Re) corresponds to the
e
hydraulic resistance (R), the capacity (C
the inductance (L
) corresponds to the inertia coefficient (Ie). Methods for electric circuit
e
) corresponds to the volume of blood (V), the electric
e
) corresponds to the elasticity coefficient (C), and
e
) corresponds to the hydraulic
e
modeling are well developed; the initial value problem for the system of ODEs (6.4)e(6.6)
is computationally simple.
It is proved [227] that the solution to Eqs. (6.4)e(6.6) for a single compartment is a
first-order (both in space and time) approximation of solution to a linearized 1D
hemodynamic system. However, inexact consistency of the electric circuit models
complicates the development of complex compartment models [224].
6.3 Dynamical lumped model of the heart
In absence of interchamber defects, the left and right hearts can be considered separately
using similar models. We consider a model for the left heart. It includes the left auricle,
which receives blood from the pulmonary vein, and the left ventricle, which ejects blood
to the aorta. One fundamental concept of the heart function is time-varying elastance.
Periodic heart function can be analyzed by the pressureevolume trajectory (PV diagram)
that represents a closed contour in (P, V) coordinates. The lumped elasticity of the heart
chamber is defined as a slope of PV diagram, which is the instant ratio E ¼
is volume change due to pressure change DP. Details about the cardiac cycle within the
scope of variable elasticity model are given in Refs. [232e235]. The cardiac cycle is
driven by a periodic change of the elasticity E due to electrical stimulation of myocardium
by the sinoatrial node activation. During systole, the myocardium becomes stiffer so
that tension increases to its maximum value and ejection happens. During diastole,
DP
: Here, DV
DV
Lumped parameter models 113
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elasticity decreases to its minimum, which promotes a faster filling of the chambers at
low pressures. A detailed review in Ref. [224] covers lumped models of chamber
interactions, various regulatory effects, and interaction with vascular and respiratory
systems.
For the sake of brevity, we denote by d the diastolic phase, fr is the friction force, max is
the maximum value, min is the minimum value, p is the pressure force, pb is the
beginning of the P wave, pw is the duration of the P wave, r is the resistance force, s is
the systole, s1 is the peak systole, and s2 is the end systole, whereas mi refers to the mitral
valve, ao to the aortic valve, lpv to the input from left pulmonary veins to the left auricle,
and sas to the entrance to aorta (aortic sinus). For notations, see also Fig. 6.1.
Using the concept of variable elastance, one can adopt a basic model of lumped
compartment (Eqs. 6.1e6.3) for the two-chamber dynamical model of the left heart:
V
0
lv
0
þ P
lv
0
la
þ P
0
la
d2V
lv
I
lv
2
dt
d2V
la
I
la
2
dt
þ R
þ R
dV
lv
lv
dt
dV
la
la
dt
þ E
þ E
ðtÞVlv V
lv
ðtÞ
V
la
la
where lv refers to the left ventricle, la refers to the left auricle, V
of the chamber, and P
0
is the reference pressure in the chamber,
d
eðtÞ: (6.8)
2
EðtÞ¼E
d
Es E
þ
For the left ventricle, we set
8
>
>
>
0:51 cos
>
>
>
>
<
ðtÞ
e
¼
lv
>
0:51 þ cos
>
>
>
>
>
>
:
0; T
s2
t
T
s1
t T
Ts2 T
t T:
p; 0 t Ts1;
s1
p; Ts1< t < Ts2;
s1
¼ Plv;
¼ Pla;
0
is the reference volume
(6.7)
(6.9)
Figure 6.1
Scheme of the lumped heart model.
114 Chapter 6
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whereas for the left auricle
8
e
la
ðtÞ¼
>
0; 0 t T
>
<
>
0:51 cos
>
:
pb
t T
T
pw
;
pb
2p; Tpb< t < T:
Flow rate through the left ventricle and auricle is given by
dV
dt
dV
dt
lv
la
¼ Q
¼ Q
mi
lpv
Qao;
Qmi:
The Poiseuille pressure drop condition (6.2) for every connecting channel is
(6.10)
(6.11)
Q
¼ gaoðqaoÞ
ao
¼ gmiðqmiÞ
Q
mi
P
lpv
¼
Q
lpv
where gðqÞ¼q
min
q q
max
; 0 gðqÞ1is a monotone function, which models
valve opening. For the closed valve, it holds gq
max
have gðq
Þ¼1. For the simplest model with the instant valves closing, it is natural to
Plv P
Pla P
P
R
lpv
min
sas
;
R
ao
lv
;
R
mi
la
;
¼ 0; while for the opened valve, we
(6.12)
define
(
P
g
aoðqao
g
miðqmi
Þ¼
Þ¼
(
1; P
0; P
1; P
0; P
lv
lv
lpv
lpv
< P
Plv;
< plv:
sas
sas
;
;
(6.13)
Parameters and coefficients of the model are given in Table 6.1 according to
the study in [212].
Essential elements of the heart are the valves between the auricles and ventricles and
between ventricles and aorta or pulmonary artery. Valves help to maintain the
unidirectional flow from venous to arterial parts, especially during the diastole phase when
the heart chambers are filled with a new portion of venous blood.
The periods of valves rapid opening and closing are rather small, but the dynamics of
valves makes significant impact on the blood flow through the heart during these periods.
The authors of Ref. [213] describe the valve function as an instant opening/closing of
appropriate flow channels between the chambers and between the chambers and outgoing
Lumped parameter models 115
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Table 6.1: Parameters of the model in Eqs. (6.7)e(6.13).
Parameter Value
E
lv;s
E
lv;d
I
lv
R
lv
E
la;s
E
la;d
I
la
R
la
T
pw
p
K
b
K
min
q
ao
max
q
ao
min
q
mi
max
q
mi
P
sas
P
lpv
T
s1
T
s2
T
pb
f
K
v
K
ao
v
K
mi
2:0mmHg=mL
0:05 mm Hg=mL
105mm HgmL
4$105mm Hg s=mL
0:25 mm Hg=mL
0:15 mm Hg=mL
105mm HgmL
4$104mm Hg smL
0.08 s
5:5$103rads2$mmHg
2 rad=s=m
0
75
0
75
100 mm Hg
37 mm Hg
0.3 s
0.44 s
0.92 s
1
50 s
7 rad=s=m
3:5 rad=s=m
arteries at prescribed time moments. Also the work [222] accounts for the valve function
by the sign of pressure drop (Eq. 6.13): negative sign means the closed state. Electrical
analog of the heart valve is a diode combined with a resistor. In reality, valve motion
depends on many flow phenomena such as the pressure gradient across the valve, vorticity
generation, the shear forces acting on the valve leaflets as discussed in Ref. [224]. A
mechanical lumped parameter model accounting for these phenomena is proposed in
Refs. [212,236]. The model uses the angle of valve opening, which is, of course, a lumped
parameter describing the valve status and motion. For a model that takes valve dynamics
into account, one may consider the following valve function:
g
aoðqao
g
miðqmi
ð1 cos qaoÞ
Þ¼
1 cos q
ð1 cos qmiÞ
Þ¼
1 cos q
gðqÞ¼
max
ao
max
mi
8
<
0; q < q
:
1; q > q
2
2
; q
2
; q
2
min
qao q
ao
min
qmi q
mi
min
;
max
;
max
ao
max
mi
;
;
(6.14)
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where qaoand qmiare governed by second Newton’s law:
d2q
dt
d2q
dt
ao
2
mi
2
¼K
¼K
ao
mi
dq
ao
f
f
dq
dt
dt
mi
þðP
þðP
P
ly
PlyÞK
la
sas
p
ÞK
cos qaoþ K
ao
p
cos qmiþ K
mi
b
Qaocos qao K
ao
b
Qmicos qmi K
mi
y
Qaosin 2q
ao
y
Qmisin 2q
mi
mi
ao
e
f
;
ao
e
f
:
mi
(6.15)
Here, we introduced functions:
1
e
f
¼
¼
2
1
2
1 þ tanheA
1 þ tanheA
ao
e
f
mi
ðPlv P
ao
ðPla PlvÞ;eAmi¼ 10;
mi
Þ;eAao¼ 10;
sas
(6.16)
which produce smooth switching of the corresponding terms in Eq. (6.15).
6.4 Numerical methods
6.4.1 Heart model
With parameters chosen in physiological range, the system of ODEs (6.7)e(6.16) is stiff.
It means that the solution includes both sharp and gradual variations, especially in the case
of instant valve opening and closing. Numerical discretization of such ODEs requires
special stability control, which may substantially limit the integration step. An implicit
one-step A- and L-stable third-order accurate method described below provides a
reasonable basic choice of a numerical solver for system (6.7)e(6.16). The idea of the
method can be found in Ref. [237].
Adopting vector notations
y ¼
we rewrite system (6.7)e(6.16) as
where w ¼
implicit RungeeKutta method for the numerical solution of Eq. (6.18) has the form of a
dy
; f is the right-hand side of Eqs. (6.7)e(6.16). The general single-step
dt
system of nonlinear equations:
nþ1
Ry
nþk
dq
;y¼y
T
mi
; (6.17)
dt
¼0 ; (6.19)
nþk
dV
ly
X
k¼0
ly
dt
dy
¼fðt; yÞ;
dt
1
y sbkf s2c
a
k
V
¼
dV
V
la
la
dt
dw
dt
dq
¼
vf
vt
vf
vt
þ
ao
q
dt
þ
mi
vf
f; (6.18)
vy
vf
f
vy
t¼t
q
ao
k
which can be solved by Newton’s method:
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y
nþ1
sþ1
¼y
nþ1
s
B
1
nþ1
nþ1
; y
t
where
B ¼
vy
vR
nþ1
¼ E sb
vf
vy
and
(
C ¼
vy
v
vf
vy
j
The set of parameters
R
s
y
2
c
!f)
nþ1
; s ¼ 1; 2; .; y
s
þ
v
vf
vy
vt
vf
;
vy
i;j
Lumped parameter models 117
nþ1
¼ yn; (6.20)
0
2
vf
þC!; (6.21)
vy
vf
i
¼
: (6.22)
vy
j
a
¼1; a0¼1; b0¼
1
1
2
þ c
þ c1; b1¼
0
1
c
c1; c1¼ c0
0
2
1
; c
¼ 0 (6.23)
0
6
defines a third-order numerical method, which is both A- and L-stable as shown in
Ref. [238].
6.4.2 Coupling with the vascular network
The model in Eqs. (6.7)e(6.16) assumes constant pressure at lpv and ao. One may use
these values for coupling the model with the 1D network model of vascular
hemodynamics discussed in Chapter 7. The values of pressure P
as the values of pressure P(S) at the corresponding points of 1D vessels. Thus, the
complete discrete system of equations includes the mass conservation Eq. (7.16), the
discretized compatibility conditions (7.32), the constitutive relationship (7.44) at lpv and
ao, and the numerical implementation of the model in Eqs. (6.7)e(6.16). The above
system is solved after completing one time step for the 1D model in internal nodes of the
vessels presented in detail in Section 7.4.2. The following iterative algorithm can be
applied to solve the coupled 1De0D system:
1. Set the initial values of pressure P
and Paoequal to the values from the previous time
lpv
step or previous iteration.
2. Perform one time step using the numerical discretization of the 0D heart model according to Section 6.4.1. This step invokes embedded iterations and computes flow rates
Q
and Qao.
lpv
3. Compatibility condition (7.32) and constitutive relationship (7.44) provide the updated
1D linear velocity (v), cross section (S), and pressure values at lpv and at ao (refer to
Section 7.4.2).
and Paoare the same
lpv
118 Chapter 6
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4. Compute pressure difference between the current iteration and previous time step at lpv
and at ao, and check the relative error.
6.5 Accounting for valve pathologies
The heart valves function causes a significant impact on the cardiac output. In this section,
we compare the difference between the model with instant aortic and mitral valve opening
and closing and the model with dynamical valves opening and closing (Eqs. (6.14e6.16)).
The narrowing of the atrioventricular lumen may cause the stenosis of mitral valve and
decrease the cardiac output. It is modeled by a decrease of maximum opening angle of the
mitral valve q
dilatation is known as aortic regurgitation or aortic insufficiency. It is modeled by an
increase of minimum opening angle q
6.5.1 Comparison of valve closing models
Here, we compare the difference between the model with dynamical valves opening and
closing (Eqs. 6.14e6.16) (model A) and the model with the instant aortic and mitral valve
closing (model B). Instead of the instant valves closing controlled by the pressure drop
condition across the valve (Eq. 6.13), we shall use another model with the prescribed valve
function g(q) for the predefined time periods from Ref. [213].
where T
simulations are shown in Figs. 6.2e6.4. In all simulations, periodic solutions are observed
starting from the third cardiac cycle. The time of the valves opening and closing in the
dynamical model of the valves is rather short (0.05 s), but it changes the flow parameters
substantially.
ao
¼ 0:15 s; T
open
max
. The backward flow through the aortic valve during diastole due to aortic
mi
min
.
ao
8
;
open
;
close
mi
¼ 1s: The results of the
close
ao
close
>
<
gðqÞ¼
>
:
¼ 0:33 s; T
0; 0 t < T
1; T
0; T
mi
open
t T
open
< t T;
close
¼ 0:44 s; T
(6.24)
The results shown in Figs. 6.2e6.4 for model A are in a good agreement with the
well-known physiological data presented in Refs. [239,240]. The systolic pressure in the
left ventricle equals to the standard value 120 mm Hg. The maximum flow through the
aortic valve equals to 900 mL/s, which is a typical value for aorta with diameter of 3 cm.
The maximum flow through the mitral valve is 300 mL/s. The volume of the left ventricle
changes from 50 to 120 mL, and the volume of left auricle changes from 40 to 60 mL.
Thus, model A will serve as a reference model for all comparisons here and in Sections
6.5.2 and 6.5.3.
Lumped parameter models 119
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Figure 6.2
Pressure in the left ventricle and auricle (A: dynamical model of the valves, B: instant valves
opening/closing).
Flow through the aortic and mitral valve (A: dynamical model of the valves, B: instant valves
Comparison of models A and B shows substantial differences in all computed parameters.
The most pronounced is the difference of systolic pressures in the left ventricle
A
(P
¼ 120 mm Hg; P
lv;syst
rate through aortic valve is achieved 0.05 s later in model A than in model B. This causes
substantial changes in the dynamics of the heart chambers volumes (see Fig. 6.4).
Figure 6.3
opening/closing).
B
¼ 130 mm Hg; see Fig. 6.2). The peak value of the flow
lv;syst
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