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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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120 Chapter 6
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Figure 6.4
Volume of the left ventricle and auricle (A: dynamical model of the valves, B: instant valves
opening/closing).
According to the physiological literature [239,240], one of the well-known effects of the
dynamical closing of the valves is the backward flow from the aorta to the left ventricle
during the early diastole (aortic regurgitation). This backward flow phenomenon results from
the rapid decrease of pressure in the ventricle, which occurs faster than the aortic valve
closing completes. Both models A and B predict backward flow showing negative flow rates
at some instance (see Fig. 6.3). We note that backward flow in model B has no
physiological sense because the instantly closed valve should immediately terminate the
retrograde flow. The observed negative flow in model B may be explained either by the
instability of the numerical solution due to the discontinuous change of parameters or by
inappropriate setting of the predefined periods of opening and closing in Eq. (6.24).Model
A conforms to physiological data and produces realistic dynamics for basic parameters of
the cardiac cycle. Model B may be adjusted to known physiological values, but it requires
manual fitting in every case. It makes model B less useful for patient-specific applications.
6.5.2 Stenosis of the mitral valve
Mitral valve stenosis is a pathology caused by the narrowing of the atrioventricular lumen
between left auricle and left ventricle. The coalescence of mitral valve leaflets is the main
reason of such narrowing. As a result, the blood flow from the auricle to the ventricle
decreases and so it reduces the stroke volume and the cardiac output. In paper [212], the
mitral valve stenosis is modeled by 30% decrease of the maximum opening angle q
This decrease leads to the decrease of the lumen by 25%. The results of the simulations
are shown in Figs. 6.5 and 6.6.
max
mi
.
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Figure 6.5
Flow through the aortic and mitral valve (A: normal conditions, B: mitral valve stenosis).
Volume of the left ventricle and auricle (A: normal conditions, B: mitral valve stenosis).
The change in dynamics of the mitral valve has impact on the dynamics of the aortic
valve. The moment of aortic valve opening is delayed by 0.05 s compared with the normal
reference conditions. The changes in peak pressure in the left auricle from 115 to 110 mm
Hg and in the left ventricle from 8 to 10 mm Hg are not significant. A pronounced effect
is observed for both the flow rate and the volume of the heart chambers. Fig. 6.5
demonstrates the decrease of the systolic blood flow through the aortic valve from 900 to
560 mL/s. The peak systole is delayed by 0.05 s compared with the reference conditions,
Figure 6.6
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which conforms to late aortic valve opening. The shape of the mitral blood flow rate curve
changes significantly. The first maximum at the early diastole and the negative flow at the
end of the cardiac cycle disappear (see Fig. 6.5). Substantial decrease of the left ventricle
volume from 120 to 85 mL is shown in Fig. 6.6. The volume of the left auricle increases
from 40 up to 50 mL.
Thus, the model is able to reproduce the well-known facts that the mitral valve stenosis
increases the load to the left auricle and hence may produce subsequent hypertrophy. The
mitral valve stenosis also causes substantial decrease of the cardiac output, which can be
evaluated on the basis of model personalization.
6.5.3 Aortic regurgitation
Aortic regurgitation (aortic insufficiency) is the incompetence of aortic valve that causes
reverse flow from aorta into the left ventricle during diastole. The reasons of aortic
insufficiency include intrinsic features, aortic root dilation, valvular degeneration, aortic
root dissection, and some others. In paper [212], aortic valve insufficiency is modeled by
the increase of minimum opening angle q
shown in Figs. 6.7 and 6.8. The change of aortic valve dynamics affects the mitral valve
dynamics. The moment of mitral valve opening keeps ahead by 0.05 s compared with
normal reference conditions. The pressure in the left auricle remains the same. The change
of peak pressure in the left auricle from 120 to 125 mm Hg and in the left ventricle from 8
to 10 mm Hg is not significant. However, it is enough for an early opening of the aortic
valve and for resulting substantial changes in cardiac output. In Fig. 6.7, one may see the
min
from 0to 25. Results of simulations are
ao
Flow through the aortic and mitral valve (A: normal conditions, B: aortic regurgitation).
Figure 6.7
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Figure 6.8
Volume of the left ventricle and auricle (A: normal conditions, B: aortic regurgitation).
substantial increase of the systolic blood flow rate through the aortic valve from 900 to
1300 mL/s. The peak systole keeps ahead by 0.05 s compared with the reference
conditions, which conforms to early aortic valve opening. The increase of systolic flow is
formally compensated by the reverse flow during diastole, but such regime causes
substantial overload of the left ventricle and aorta. The straightforward effect of the aortic
insufficiency is the negative flow of 70 mL/s during diastole (see Fig. 6.7). Systolic
volumes of the chambers remain the same. Substantial increase of the left ventricle
volume from 115 to 150 mL is shown in Fig. 6.8.
Conclusions
The 0D model of the heart outflow, which accounts for heart chambers and valves
dynamics, leads to physiologically correct simulations of certain heart diseases. It also
provides a tool for constructing closed hemodynamic model of a vascular network. The
basic feature of the cardiac cycle is the periodic alternation of sharp and smooth solution
variations, which causes stability problems in numerical simulations. Higher-order implicit
A- and L-stable methods are suitable for the numerical integration of the models of cardiac
dynamics. Patient-specific evaluation of basic model parameters (heart rate, angles of valve
opening, the volume of chambers) can be performed in a hospital through echocardiography
or other conventional methods. Other model parameters (inertia coefficient, hydraulic
resistance, variable elastance function) are hard to evaluate directly. They can be identified
by fitting known measured and computed variables (e.g., volume dynamics, cardiac output,
and ejection fraction). The obvious advantage of the lumped parameters approach is its low
computational cost compared with higher-dimension models.
CHAPTER 7
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1D vascular hemodynamics
7.1 Introduction
The development of reduced order models that describe physiological and
pathophysiological processes in the cardiovascular system is an active research area. The
spatial order reduced models were proved to suit well for a number of clinical problems
since they enable fast personalized simulations for planning pharmacological or surgical
treatment.
Section 7.2 of this chapter introduces 1D hemodynamic equations in a single vessel and
their extension to a network of vessels. This extension is done through boundary and
coupling conditions of different types. Advanced approaches use 1D hemodynamic models
as a part of multiscale modeling. Section 7.3 describes 0De1D, 3De1D, and 3De0De1D
multiscale models. Numerical methods are discussed in Section 7.4. Modifications of
reduced order models, which account for physiological conditions and some pathologies,
are presented in Sections 7.5 and 7.6.
7.1.1 Model reduction in hemodynamics
Detailed numerical 3D modeling of blood flow in a realistic vascular network is a
complicated technological and practical task. A typical network includes from dozens to
hundreds of vessels. A mathematical model should account the interaction of flow with a
moving viscoelastic vessel wall (fluidestructure interaction models, FSI). Furthermore, it
is necessary to specify a sufficiently accurate 3D geometry of the vascular bed, elastic
properties of the vessel wall material and boundary conditions, and finally to apply
advanced numerical schemes (see Chapter 5). On the other end of the model complexity
range, we have lumped parameter (0D) models capable to address the mean dynamics of
the system but unable to describe finer effects, in particular, those related to wave
propagation (see Chapter 6). Within this range, spatial order reduction (SOR) approaches
provide mathematical models of various complexities.
Averaging procedures or other methods are applied to eliminate the dependency of flow
variables on one or two spatial coordinates under certain assumptions regarding the flow.
The limitations of these assumptions determine the scope of a given SOR model.
Reduction procedures should preserve important physiological features of the problem so
Personalized Computational Hemodynamics. https://doi.org/10.1016/B978-0-12-815653-7.00007-5
Copyright © 2020 Elsevier Inc. All rights reserved.
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that the resulting model is able to reproduce the flow and vascular dynamics with
sufficient details. Since the original problem is highly nonlinear and the reduction process
builds on a combination of several simplifying assumptions, the resulting models should
be thoroughly validated. The SOR approach applied to the 3D FSI problem leads to a 1D
reduced flow model, a compromise between the detailed 3D FSI and the rough 0D models.
Mechanical interpretation of the resulting 1D flow model is the laminar flow of viscous
incompressible fluid flow through a network of elastic tubes. 1D modeling is an attractive
alternative to fully resolving 3D simulations thanks to much lower computational costs of
studying regional, systemic, and closed (global) circulation. Primary applications of these
models are the transport of blood gases, nutrients, and drugs through the whole organism,
the analysis of the blood flow variations under external or internal mechanical action such
as physical exercise or stimulation by medical devices, and the analysis of the effect of
intravascular surgery, such as stenting or shunting. For description of the full-body
hemodynamics, these models can be combined with models of the heart (see Chapter 6)
and a model of microcirculation (see Chapter 8).
The computational domain of 1D models is the human vascular 1D network or its parts.
A realistic network can be generated using anatomical atlases and physiological data
found, for instance, in Refs. [120,239,240]. Alternative approaches use the reduction of a
detailed 3D anatomical model [241] and segmentation methods of medical images
obtained from MRI/CT [100,242,243]. Algorithms of processing patient MRI/CT data
include 3D segmentation, centerlines identification, and construction of a graph with 1D
straight segments (we refer to Chapter 3 for details). The construction of 1D models may
be also based on aggregated laboratory data and experiments [244e246].
The analysis of blood flows by 1D hemodynamic models in vascular networks
composed of hundreds of vascular segments can be found, for instance, in Refs. [213,247].
Extensive reviews of different aspects of 1D blood flow modeling can be found in
Refs. [43,224,248e250].
7.1.2 Applicability and limitations
In the SOR approach, 1D hemodynamic equations are derived by averaging the full 3D
NaviereStokes equations in a single vessel [248,251]. 1D hemodynamic models are
usually based on the following assumptions (some of them can be alleviated):
1. The ratio of vessel diameter to its length is relatively small.
2. The blood rheology corresponds to Newtonian viscous incompressible fluid.
3. The blood viscosity is constant.
4. The blood flow profile in any cross section orthogonal to the vessel centerline is
radially symmetric.
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5. The shape of the velocity profile in any such cross section does not vary along the
vessel (e.g., it stays flat or parabolic).
6. The pressure is constant in each cross section.
7. Longitudinal stretching of the vessels is negligibly small.
8. The thickness of vessel walls is sufficiently small and constant.
9. The forces act on the wall in normal directions.
10. All cross sections of the vessel are circular.
11. Wall displacements occur only in the radial direction.
12. The deformation gradient of the vessel wall changes along the centerline continuously.
13. The material of the vessel wall is incompressible, and the deformation is linear.
Models based on (a subset of) the above assumptions are often well suited for simulating
blood flow in large and medium arteries and superficial veins. In practice, they are also
used for the hemodynamic simulation in deep veins, since the latter in the standing
position (subject to the gravity field) have a circular cross section. The elliptic shape of the
vessel cross section may be included in 1D hemodynamic models by modifying the
equation describing the elasticity of the wall [252e254] or by different averaging
procedure for the NaviereStokes equations.
We note that the existence of a smooth solution for 1D hemodynamic models in a network
with junctions is not known yet. For single vessel, the existence of a smooth solution to
the 1D hemodynamic equations has been proven with the zero right-hand side and under
certain additional assumptions in Ref. [251], whereas in Ref. [255], the existence is proven
for particular boundary and initial conditions given as linear combinations of trigonometric
functions. Surprisingly, smooth pulsating boundary conditions at the entrance of a
semi-infinite compliant vessel typically lead to the formation of shock waves outside
(2.8 m from the entrance) of the physiologically interesting domain [251].
For realistic cardiovascular simulations, the above assumptions may be considered
somewhat restrictive and arguable. In particular, the shape of the velocity profile is not
constant, and the Coriolis parameter (Boussinesq coefficient) and the friction coefficient
from Eq. (7.12) are not constant either [256]. The model of thin-walled elastic cylinder is
only a rough approximation of the realistic vascular material. Therefore, only extensive
validation can support the practical use of the 1D hemodynamic models. Such validation
includes comparisons with experimental and laboratory data, with the results of modeling
using different methods, as well as with the solutions to the full 3D FSI problems. For
example, studies [209,257e259] report reasonably good agreement of numerical solutions
with physiological data, 3D FSI simulations, and physical experiments with the network of
silicone tubes, while testing on clinical data is presented in Ref. [112]. Comparison of
several 1D models on benchmark problems is given in Refs. [258,260]. Comparison of
different boundary conditions for a 1D model using the Windkessel model and the
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structured tree method as the truncated conditions was done in Ref. [261]. It is interesting
that, despite significant differences in mathematical formulations and numerical methods
used for 1D simulations, in most cases, it is possible to achieve satisfactory agreement.
Moreover, 1D models may successfully incorporate important physiological conditions
such as gravity force, muscles contraction force, regulation, and autoregulation (see
Section 7.5). Therefore, the 1D blood flow modeling is currently accepted as the adequate
approach for regional and global blood flow simulations and as a useful tool for a number
of physiological and clinical problems.
7.2 Derivation of equations
7.2.1 Averaged NaviereStokes equations in a vessel
Based on assumptions outlined in the previous section, one can derive a 1D hemodynamic
model in several ways. For example, one may perform averaging and asymptotic analysis
of the NaviereStokes equations assuming small ratio r/L, where r is the radius and L is
the length of a vessel [262,263], or one may integrate the 3D NaviereStokes equations
over a cross-sectional slice [248].
Let x denote the coordinate along the centerline of a vessel with circular cross section G
S
orthogonal to the centerline, r be the radial coordinate in the cross section, t be the time,
R(t, x) be the radius of the cross section, hðt; xÞ be a given displacement of the
vascular wall in the radial direction, and let the cross-sectional velocity profile be given by
gþ2
xðyÞ¼
(Poiseuille) profile, and g ¼ 9 corresponds to almost flat profile.
ð1 ygÞ; 0 y 1. Note that g ¼ 2 corresponds to the parabolic
g
1
We assume that vxand vrcomponents of the velocity satisfy
ðt; r; xÞ¼vðt; xÞx
v
x
Here
v is the average velocity in cross section GS, v ¼ S
r
Rðt; xÞ
; v
ðt; R; xÞ¼
r
R
1
G
S
vh
ðt; xÞ: (7.1)
vt
vxds. The component v
x
attains its maximum at the centerline (r ¼ 0) and vanishes at the vascular wall (r ¼ R(t, x)).
Since v
(t, R, x) ¼ 0, the condition on vr(t, R, x) represents the nonslip condition on the
x
vessel wall.
To derive 1D hemodynamic equations, we define a vessel slice V as a neighborhood of a
cross section G
the external normal is n with components n
x
Dx=2; xþ Dx=2 are denoted by G1and G2, respectively; GWis a part of the vessel
with coordinate x. The boundary of V is vV ¼ G1WG2WGW, and
S
; nr. The inlet and outlet cross sections at
x
wall, as shown in Fig. 7.1.
1
Alternative options such as the power law profile and the Stokes layer are analyzed in [250].
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Figure 7.1
Integration domain of a 3D model.
The integration of the incompressibility Eq. (4.16) over V yields
0 ¼
Z
div v dx ¼
V
Z
v$n ds ¼
vV
Z
G
vxds
2
Z
G
1
vxds þ
Z
vrds: (7.2)
G
W
For the last term, we get from Eq. (7.1) the equality
Z
vrds ¼
G
W
Denoting the average flow Q
Z
vh
ds ¼
vt
G
W
R
¼
j
G
vxds through the cross section G, substituting
G
vS
vt
x¼x
Dx þ O
2
: (7.3)
jDxj
Eq. (7.3) to Eq. (7.2), dividing by Dx, and taking the limit Dx/0, we obtain
Dx
2
Qx
Dx
0 ¼ lim
Dx/0
R
G
vxds
2
Dx
R
G
1
vxds
Qxþ
Dx/0
þ
vS
vt
¼ lim
which leads to the 1D mass conservation equation:
Dx
2
vS
þ
;
vt
Next, we rewrite the x-component of the NaviereStokes equations for the incompressible
Newtonian fluid
vS
vt
(Eq. (4.18)) as follows:
vv
x
vt
þdivðv
x
þ
vÞþ
vQ
¼ 0: (7.4)
vx
1rvp
vx
nDv
¼ 0: (7.5)
x
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Integrating Eq. (7.5) over the slice V leads to
dx þ
Z
divðvxvÞdx þ
V
1
r
Z
vv
x
vt
V
We transform each term in Eq. (7.6) as follows:
Z
Z
V
Z
Here a ¼ S
R
we used
vv
x
vt
Z
divðvxvÞdx ¼
¼
Z
V
vxv$n ds ¼
vV
2
aS
v
G
2
aSv
Dx
vp
dx ¼S
vx
V
Z
Dvxdx ¼
V
Vvx$n ds ¼
vV
Z
¼
nxVvx$exds þ
vV
Z
v
0
¼
R
1
2
v
i
vV
G
nxVvx$exds ¼ OððDxÞÞ2since
x
ð1Þnrds þ O
R
G
W
2
v
ds, and for Eq. (7.8), we used vx¼ 0onGW. In derivation of (7.10)
x
i
dx ¼
Z
G
Z
G
vQ
Dx þ O
vt
Z
v
G
1
2
vp
vx
vv
x
vx
1
nrVvx$erds
W
jDxj
R
vV
Z
vp
vx
V
Z
2
ds þ
x
G
1
Dx ¼
Dx þ OðDxÞ
x¼x
Z
ds þ
G
2
¼ 2p
nxds ¼ 0.
Z
dx n
jDxj
v
G
2
v
vx
vv
vx
2
Dvxdx ¼ 0: (7.6)
V
2
; (7.7)
2
ds þ
x
2
aQ
S
Z
x
ds þ
vx0ð1ÞDx þ O
Z
vxv$n ds
G
W
Dx þ O
2
; (7.9)
jDxj
2
;
Vvx$n ds
G
W
2
:
jDxj
(7.8)
(7.10)
Substituting Eqs. (7.7)e(7.10) into Eq. (7.6) and passing to the limit for Dx/0 gives the
1D momentum conservation equation:
where
Note that K
is negative since x
r
1D model in (S, v, p(S)) variables. It can be reformulated in (S, Q, p(S)) variables. Indeed,
substituting Q ¼ Sv in Eqs. (7.4) and (7.11), we arrive at
vQ
v
þ
vt
vx
0
ð1Þ
< 0. Eqs. (7.4) and (7.11) constitute the hemodynamic
2
Q
a
S
¼2pnx0ð1Þ: (7.12)
K
r
vS
þ
vt
Srvp
¼ K
þ
vx
vðSvÞ
¼ 0; (7.13)
vx
v; (7.11)
r
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