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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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1D vascular hemodynamics 151
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lumped element. If A0, As, l0, and lsdenote cross-sectional areas and lengths of
nonstenotic and stenotic vessels, respectively, then parameters of the stenosed region with
degree a ¼ A
A
should be updated:
=
s
0
R
s¼Re
a2; Cs¼ Cea
3=2
; Ls¼ Lea1:
The physiological impact of the stenosis is the pressure drop Dp across the stenosed
region, resulting in the reduction of flow rate Q. Empirical dependencies of Dp on Q may
be diverse:
288r
Dp ¼
Dp ¼ðR
Dp ¼
dp
¼
dx
where Re is the Reynolds number, R
2
2ReA
s
þ R
1
Kym
Q þ
3
2pR
8rQ2B1B
p2R
Q2; ½310
Þ
Q þðK
2
Ktr
2
2A
0
dR
2
5
1
dx
1
A
A
þ K
0
1
s
60mQB
2
pR
Þ
2
;½312
Q
2
QjQjþ
3
4
Kurl
; ½313
dQ
s
; ½222; 311
dt
A
0
, R2, K1, K2, B1, B2, B3, Kv, Kt, and Kuare constants,
and R is the diameter of the vessel. The pressure drop across a stenosis with irregular
geometric structure requires more detailed two-scale 1De3D models such as models
presented in Refs. [271,311], although asymmetric lumen produces no appreciable
difference in the pressure drop as stated in Refs. [312,313]. If the stenosis is considered as
a node of a 1D vascular graph, these relations for Dp can be used as a part of the
boundary conditions in a 1D hemodynamic model instead of Eqs. (7.17), (7.18), or (7.19).
Stenoses may be taken into account by a 1D hemodynamic model directly in two ways.
First, a stenosis with degree a may be incorporated into a healthy vascular network as a
tubular segment with modified cross-sectional area A
and outlet of this segment, one adds to the boundary conditions the Poiseuille pressure
drop condition Dp ¼ R
Refs. [242,314,315]. Second, the presence of a stenosis may modify the constitutive
relation (7.44) for a healthy vessel by locally increasing the vessel wall stiffness.
A more advanced approach is to recover function f from Eq. (7.44) for the healthy and
stenosed vessels as proposed in Refs. [316,317]. Recovery of function f is based on
representation of the vessel wall by concentric elastic shells. It is assumed that the vessel
is cylindrical and the plaque is axisymmetric, uniform, and lengthy; hence, the strain in
the axial direction is negligible. Inhomogeneous distribution of collagen and elastin fibers
in the healthy vessel wall (see Chapter 2) produces nonlinear (e.g., neo-Hookean) elastic
¼ aA0and length Ls. At the inlet
s
Q with hydraulic resistance Rswa2as was done in
s
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response to blood pressure load. The equilibrium state for such thin wall (with middle
radius R and thickness H ) composed of a neo-Hookean material with a material constant m
and inflated by internal pressure p
stretch l
derived in Ref. [317]:
r
ml
2
r
gives the nonlinear equation on the principal radial
0
l
2
r
¼p
ðl
HÞðR=lrlrH=2Þ:
=
r
0
This equation defines implicitly function f since it relates the pressure and the lumen area.
As an atherosclerotic wall contains three layers, it can be represented by a three-layer
circular cylindrical shell inflated by internal pressure p
. The internal (vessel wall) and
0
external layers (fibrous cap) are represented by thin-walled cylindrical shells. The middle
layer is a thick-walled cylinder that represents the lipid pool. The static equilibrium
conditions allow the authors of [316,317] to compute the lumen area under given blood
pressure p
for isotropic incompressible linear elastic (Hookean) [316] or incompressible
0
nonlinear materials [317] of vessel wall, lipid pool, and fibrous cap. In Fig. 7.7,we
compare functions p
plaque and lumen 10%. Since S
transmural pressure corresponds to S/S
(S/S0) for an atherosclerotic vessel with a lengthy axisymmetric
0
denotes the lumen area of the healthy artery, zero
0
¼ 0.1. Parameters for the static equilibrium
0
problem are set according to the properties of the common carotid artery. The substantial
difference in pressures is observed at transmural pressures over 4 kPa as well as for
negative transmural pressures. The nonlinear materials demonstrate higher sensitivity to
inflating transmural pressure and higher resistivity to deflating transmural pressure.
Comparison of the constitutive relations for Hookean and neo-Hookean materials for an
atherosclerotic common carotid artery. From Y. Vassilevskii, S. Simakov, V. Salamatova, Y. Ivanov, T.
Dobroserdova, Vessel wall models for simulation of atherosclerotic vascular networks, Math. Model. Nat.
Figure 7.7
Phenom. 6 (7) (2011), 82e99.
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Derivation of the constitutive relation in case of asymmetry of the vessel and the plaque is
based on numerical methods. For instance, following Ref. [318], the elastic response of a
tube wall may be computed from its representation by sets of axial fibers, ring fibers, and
helical fibers and computation of the response of the fibers collection to a deformation.
The response of an atherosclerotic vessel to inflation may be computed from
representation of the fibrous cap and the arterial wall by sets of linear elastic rings and the
lipid pool by a set of radial springs. The response of the springs is derived from
analytically known radial displacements of an incompressible linear isotropic cylindrical
shell with exerted external and internal pressures. Two concentric grids of ring fibers
represent the external (artery) and internal (fibrous cap) layers. These layers and balancing
forces applied on these layers at each grid point lead to a system of discrete nonlinear
equations to be solved by the Newton method. For all possible pressures and plaques
observed in the common carotid artery, such numerical model is shown in Ref. [316] to
provide the error for displacements less than 2% compared with the semi-analytic solution
known for axisymmetric geometries as demonstrated.
The case of several sequential stenoses is less studied. The circulation disturbance caused
by sequential stenoses is addressed in Refs. [319e322].
7.6.2 Blood flow in tortuous vessels
PT of the carotid and vertebral arteries (CVAs) is the major pathological factor in the
development of insufficiency in the cerebral circulation. PT takes the second place after
the atherosclerosis disease and often develops without symptoms. PT is observed in 30%
cases of the deaths from stroke. The ancestral features and hypertension are the major
reasons of PT. The possible side effect of the PT is the blood flow decrease in CVA. The
PT under low-flow conditions (e.g., hypotension) produces the most pronounced effect to
the patient. PT has one of four general types: C-like (see Fig. 7.8B), S-like PT (see
Fig. 7.8A), kinking (artery bending) (see Fig. 7.8C), and coiling (see Fig. 7.8D). The less
threatening C and S types may transform to the more significant cases (kinking and
coiling). A variety of PTs and their complicated geometry require patient-specific analysis
and treatment that suggest the use of simulation tools.
The blood flow with high Reynolds numbers (Re > 10
3
) in PT regions has a complex 3D
structure exhibiting turbulence and backward flows. High curvature may produce total
occlusion and blood flow termination. Internal viscous friction in PT produces energy
losses, increase of the hydraulic resistance, and decrease of the downstream pressure and
velocity. The pressure loss takes place at every bend. The hydraulic resistance of every
bend is proportional to the velocity. The latter may increase in the bend and decrease after
the bend. Thus, the total effective resistance of a PT is not equal to the sum of the
resistances of its every bend. Patient-specific numerical simulations are needed to assess
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Figure 7.8
Types of pathological deformation of the internal carotid artery. (A) S-like tortuosity, (B) C-like
tortuosity, (C) kinking. and (D) coiling. Image from I.P. Dudanov, S.V. Ordynets, I.A. Lukinskiy, B.C.
Abuazab, V.V. Akhmetov, A.A. Shabonov, O.P. Verbitskiy, Extracranial nonatherosclerotic pathology of the
carotid artery in the causes of acute ischemic stroke, Res. Pract. Med. J. 4 (4) (2017), 35e49.
Patient-specific 3D and 1D segmentation. AB is a region with pathological tortuosity.
the effective resistance of the whole PT region. 1D hemodynamic models have insufficient
accuracy for the detailed simulations of the blood flow in the PT regions. Nevertheless,
they may help to make fast assessment of the disease stage. For the algorithms of
identification of PT and the generation of 1D vascular structure with PT (see Fig. 7.9), we
refer to Chapter 3. The PT region can be treated in detail using a more complex
mathematical model.
Figure 7.9
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The pressure drop in relatively smooth C-like and S-like regions is proportional to the
friction along the route and to the length of the route. One may account for the PT in such
regions via the modification of the friction force:
PT
f
¼ffrð1 þafrkðxkÞÞ; k ¼
fr
d2rðsÞ
2
ds
; (7.64)
where k is the local curvature of the vessel centerline, r(s) is a 3D curve parameterization
provided by the segmentation of the patient-specific medical images, 0 s 1 is the
parameter along the curve, and a
is a factor that may be set according to lab experiments
fr
presented in Ref. [324]. For reference cases, one may use rðsÞ¼ð1 cosð2pnsÞ; 0; 0Þ and
set n ¼ 1 for a C-like PT and n ¼
3
for an S-like PT.
2
In the complex cases of kinking and coiling, one may introduce a node with a lumped
compartment providing the pressure drop conditions:
2
v
; (7.65)
b
2
where v is the upstream velocity, and k
Dp ¼k
is the energy loss coefficient estimated in rigid
b
tube experiments presented in Ref. [325].
Comparisons of the pressure drop and velocity decrease over PT under normal and
hypotension conditions are shown in Figs. 7.10 and 7.11. Hypotension conditions were
simulated by the decrease of the stroke volume (SV) from 55 to 45 mL. The segment of
the right internal carotid artery AB (see Fig. 7.9) was simulated by different geometrical
models of the PT: straight line, node with lumped parameters with the pressure drop
model (7.65), S-like and C-like shapes with friction (7.64), and spline approximation of
patient-specific centerlines.
Comparison of the pressure drop over pathological tortuosity (PT) region in normal and
hypotensive conditions with the different models of PT geometry.
Figure 7.10
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Figure 7.11
Comparison of the velocity decrease over pathological tortuosity (PT) region in normal and
hypotensive conditions with the different models of PT geometry.
The presence of PT produces almost the same velocity decrease and substantially greater
pressure drop compared with the straight case. The larger pressure drop results in the
blood flow redistribution: flow increases in the collateral path (left internal carotid artery),
whereas the flow through the right internal carotid artery decreases. This effect is more
pronounced under the hypotension condition.
7.6.3 Coronary circulation during cardiac pacing and tachycardia
Sinoatrial node (SA) initiates the myocardium contraction. Activity of the SA is modified
by signals from the sympathetic and the parasympathetic nervous system and humoral
factors. Pathological processes (e.g., tachycardia) and external stimulation of the
myocardium (e.g., cardiac pacing) modify the myocardium contractions and change the
coronary blood flow decreasing nutrients delivery to the myocardium and provoking
ischemic events in the heart. The impact of cardiac pacing and tachycardia to the coronary
circulation may be assessed by patient-specific 1D hemodynamic models based on clinical
data collected by noninvasive procedures (see, e.g., Refs. [17,264]). The remainder of this
section is based on the work [303]; the reader is referred to it for details.
7.6.3.1 Effect of asynchronous myocardial stimulation
Cardiac pacing is performed by a pacemaker, which produces electrical pulses to prompt
the heart to beat at the normal rate. Improper placement or synchronization of the
pacemakers may lead to abnormalities in ventricle contractions and violation of the normal
coronary flow in right (RCA) and left (LCA) coronary arteries and their branches.
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Table 7.2: Simulated systolic blood flow through right cor onary artery (RCA) and
left coronary artery (LCA) in the case of the delayed contractions in left ventricle (LV) and
right ventricle (RV).
Blood flow, mL/s
RCA LCA
No pacing 1.68 2.60
RV delay 1.49 3.04
LV delay 1.75 3.33
From T.M. Gamilov, F. Liang, S.S. Simakov, Mathematical modeling of the coronary circulation during cardiac pacing and tachycardia,
Lobachevskii J. Math. 40 (4) (2019), 448e458.
Modeling of the coronary circulation during pacing is based on modifications of the
external pressure and the terminal hydraulic resistance for coronary vessels according to
their proximity to the pacemaker. We compare two opposite cases. First, we assume that a
pacemaker is placed in the left ventricle (LV) and produces 0.03 s delay in the increase of
pressure and peripheral resistance in RCA branches. Second, we assume that a pacemaker
is placed in the right ventricle (RV) and causes 0.03 s delay in the increase of pressure and
peripheral resistance in LCA branches. Results of simulations are shown in Table 7.2.
Comparative simulation results shown in Table 7.2 demonstrate small variations of the
systolic flow in the RCA compared with the normal case without pacing. However, the
systolic flow in the LCA is increased both for the right delay with LV stimulation and for
the left delay with RV stimulation. Moreover, the delayed activation of the RV in the case
of the LV pacing may cause decreased blood supply of the RCA, the known fact from
common clinical practice.
7.6.3.2 Effect of pacemaker position
Pacemaker position may produce asynchronous contractions in both LV and RV. We
examine three options of the pacemaker position: LV base, RV apex, and RV base. The
impact of the position is accounted by scaling the amplitude of the coronary blood
pressure and peripheral resistance proportional to the myocardial tension, which depends
on the distance from the pacemaker. The work [326] provides necessary clinical data for
these simulations.
The simulation results collected in Table 7.3 show distinctive decrease of the systolic flow
in RCA for RV apex and LV pacing compared with the normal case without pacing. We
also observe considerable increase of the systolic flow in LCA for all positions of the
pacemaker compared with the normal case. The computed blood flows support the
common opinion that the optimal pacemaker position is in the RV base.
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Table 7.3: Blood flow through right coronary artery (RCA) and left coronary artery (LCA) for
the different pacemaker positions.
Blood flow, mL/s
Pacemaker position
No pacing 1.68 2.6
RV apex 1.53 3.16
RV base 1.67 3.05
LV 1.44 3.17
LV, left ventricle; RV, right ventricle.
From T.M. Gamilov, F. Liang, S.S. Simakov, Mathematical modeling of the coronary circulation during cardiac pacing and tachycardia,
Lobachevskii J. Math. 40 (4) (2019), 448e458.
RCA LCA
7.6.3.3 Tachycardia
Stable permanent increase of the heart rate (HR) of an adult at rest above 100 beats per
minute is associated with a pathological type of tachycardia. Tachycardia can be induced
by cardiac pacing or caused by various cardiovascular disorders. For elderly people, the
tachycardia may be a consequence of excessive emotional and/or physical stress or
myocardial infarction. Tachycardia contributes to morbidity and mortality of patients with
the ischemic heart disease.
The impact of tachycardia on the coronary circulation can be simulated by the increase of
the HR and the fraction of systole, decrease of the SV, modifications of the external
pressure, and the terminal hydraulic resistances. In normal (healthy) case, the fraction of
systole is 40%. According to the clinical data [327], for the tachycardia conditions, the
fraction can be set to 56%. The SV decreases from 60 mL in healthy case to 35 mL in
case of tachycardia. The increase of the coronary vessels elasticity by 20% reproduces
conditions in pediatric patients with supraventricular tachycardia.
We compare three models of tachycardia designated as A, B, and C. All models use
increased HR, but only model C accounts the increased systolic fraction. Models B and C
take into account the decreased SV.
The simulation results shown in Table 7.4 prove that the blood flow through the coronary
arteries increases during tachycardia. However, the fraction of the coronary flow in the
cardiac output d
¼ 60$Q
of the systolic fraction (case C) leads to the decrease of relative blood supply d
ðHR $SVÞ is different (* denotes LCA or RCA). The increase
=
in RCA
and LCA.
Simulations of coronary circulation in pediatric patients with supraventricular tachycardia
(Table 7.5) show that for model C the blood flow through RCA and LCA as well as the
fraction of the coronary flow in the cardiac output becomes smaller than for the healthy
heart, which can induce the development of ischemia.
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Table 7.4: Blood flow through right coronary artery and left coronary artery for
models A, B, and C.
Healthy heart Case A Case B Case C
HR, bpm 60 120 120 120
SV, mL 60 60 35 35
Q
, mL/s 0.9 2.1 1.2 1.0
RCA
Q
, mL/s 2.8 6.5 3.8 3.05
LCA
d
,% 1.5 1.7 1.7 1.4
RCA
d
,% 4.7 5.4 5.4 4.3
LCA
HR, heart rate; SV, stroke volume.
From T.M. Gamilov, F. Liang, S.S. Simakov, Mathematical modeling of the coronary circulation during cardiac pacing and tachycardia, Lobachevskii J. Math. 40 (4) (2019), 448e458.
Table 7.5: Blood flow through right coronary artery and left coronary artery for pediatric
patients (healthy heart and tachycardia).
Healthy heart Case B Case C
HR, bpm 98 207 207
SV, mL 55 28 28
Q
, mL/s 1.5 1.8 1.4
RCA
Q
, mL/s 4.7 5.6 4.4
LCA
d
,% 1.7 1.9 1.4
RCA
d
,% 5.2 5.8 4.5
LCA
HR, heart rate; SV, stroke volume.
From T.M. Gamilov, F. Liang, S.S. Simakov, Mathematical modeling of the coronary circulation during cardiac pacing and tachycardia, Lobachevskii J. Math. 40 (4) (2019), 448e458.
We should note that HR increase leads to the increase of the mechanical work performed
by the myocardium and, thus, to the increase of the oxygen and nutrients consumption.
Thus, formally normal blood flow through LCA and RCA in cases B and C in Table 7.4
and in case B in Table 7.5 may be insufficient and can induce the development of
ischemia.
7.6.4 Enhanced external counterpulsation
EECP is a noninvasive method of stretching blood vessels and blood flow recovery. The
EECP procedure initiates reverse blood flow in arteries during aortic valve closing
(diastole) by a three-step sequential cuff compression (see Fig. 7.12). EECP pressurizes
during systole and depressurizes during diastole the inflatable cuffs surrounding the legs
and the lower abdomen of a patient (see Refs. [291,328] for the description of EECP
procedure). EECP increases diastolic pressure and stretches coronary arteries and improves
the blood supply of the myocardium, as significant coronary flow occurs during diastole
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Figure 7.12
External compression scheme.
(see Section 7.5.3). The primary application of EECP is a noninvasive treatment of cardiac
ischemia and angina. Different studies indicate its positive effect on blood pressure,
cardiac index (e.g., see Ref. [328]), pulse waves (e.g., see Ref. [329]), and endothelial
function (e.g., see Ref. [330]).
Possible side effects of the increased pressure in systemic arteries may include aneurysm
formation and rupture, atherosclerotic plaques rupture, etc. Therefore, EECP procedure
should be analyzed to keep patient in safe conditions. The 1D hemodynamic model can be
used for such analysis. Cuffs performance is simulated as external pressure (see Section
7.5.3) applied to the lower abdomen, thigh, and calf arteries according to the schedule
shown in Table 7.6. Here, the cardiac cycle is 1 s, T
initiation, and T
is the time of the compression termination. The values of the standard
f
is the time of the compression
s
times and pressure amplitude are taken from Ref. [291].
Table 7.6: Pressure and time synchronization for enhanced external counterpulsation.
External pressure Standard time Delayed time
, mmHg Ts, s Tf,s T
P
ext
Calves 200 0.24 0.8 0.35 0.95
Thighs 150 0.25 0.8 0.6 0.95
Lower abdomen 100 0.26 0.8 0.75 0.95
; s T
s
; s
f
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