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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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Computational hemodynamics in vascular surgery 193
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Circumferential Young’s modulus of left and right coronary arteries for different ages and
genders. Age groups: I 0e1 years; II 1e7 years; III 8e19 years; IV 20e39 years; V 40e59 years;
VI 60e80 years. Image from I. Ozolanta, et al., Changes in the mechanical properties, biochemical contents
and wall structure of the human coronary arteries with age and sex, Med. Eng. Phys. 20 (7) (1998)
a
“Modest smoker” means that a person smokes less than 10 years and in average consumes less than
10 cigarettes per day. “Chronical smoker” means that a person smokes more than 10 years and in
average consumes more than 20 cigarettes per day.
smoke
Figure 9.3
523e533.
Table 9.1: Factor a
Nonsmoker Modest smoker Chronical smoker
1.0 1.2 1.5
for smoking status.
smoke
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Factor a
Table 9.2: Factor a
Sport Nonsportsmen Long-distance running Weightlifting Swimming
a
sport
accounts changes of vessel elasticity for sportsmen [413,416]. Sport activity
sport
1.0 0.6 1.5 0.9
dependence on athletic status.
sport
may both increase and decrease the pulse wave velocity (Table 9.2).
Various diseases have a significant impact on the elastic properties of blood vessels. They
can be taken into account by introducing additional factors in Eq. (9.2). We discuss briefly
some factors that can be determined through noninvasive examination of the patient.
Although we are not aware of any studies that address explicitly the dependence of the
elastic properties of arteries on body mass index or overweight, a relationship between the
volume of pericardial fat and calcification of coronary vessels is statistically proved.
Moreover, it is shown in Ref. [419] that the pericardial fat volume correlates with the body
mass index. Thus, the body mass index and the volume of pericardial fat volume may be
accounted in Eq. (9.2).
The study of the impact of diabetes on the stiffness of the carotid artery and the abdominal
aorta [420] showed that in this case the stiffness of the arteries strongly depends on
gender. Gender correlates with the elastic properties of all arteries in patients with
diabetes. For men, there is no statistical difference between the stiffness of the artery walls
for healthy and diabetics. For women with diabetes, the arterial walls are stiffer than for
healthy women.
Hyperglycemic patients have stiffer coronary vessels than people without hyperglycemia
[421]. In particular, the compliance for people without hyperglycemia is (10.6 4.4)$
3
10
kPa1(40e52 years) and (7.0 2.5)$103kPa1(52e75 years); for hyperglycemic
people, it is (7.9 5.8)$10
years). The large variation (over 70%) in the results for hyperglycemic patients may be
explained by nondistinguishing between men and women in the hyperglycemic group.
Identification of the functional parameters of the model is performed in several stages. At
the first stage, we select the cardiac flow rate profile and the venous pressure, we set the
default values for the resistances of microcirculation and define the pulse wave velocities
by Eqs. (9.1) and (9.2). At the next stage, the resistances are corrected to minimize
deviations of the measured and calculated blood flow rates. At the final stage, the pulse
wave velocity (vascular stiffness) is adjusted to achieve a better fit between the
calculated and measured blood flow rates. This procedure can be made in the black-box
manner if a method identifying arteries is given (for the example of such, we refer to
Chapter 3).
3
kPa1(40e49 years) and (7.2 3.4)$103kPa1(49e75
Computational hemodynamics in vascular surgery 195
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The described three-stage procedure of parameters identification allows us to achieve an
average deviation of 4% of the calculated flow rates from the measured ones. The alternative
method of parameters identification [209,422] is based on the Kalman filter, a recursive
algorithm that optimizes the system state vector using a number of noisy measurements. In
theory, the Kalman filter provides a fully automated selection of parameters. However, in
practice, its use is associated with certain difficulties. In case of nonlinear models, the
numerical implementation of the algorithm is rather cumbersome, and iterations of parameter
sets may not converge. If many model parameters are to be determined, extra computational
resources are required, which makes it difficult to run the model on personal computers.
Finally, the Kalman filter does not solve the problem of arteries identification, since it
requires explicit localization of velocity measurements.
The hemodynamic model is written in terms of time-dependent differential equations,
which describe a quasi-periodic solution. Suitable initial conditions can be chosen from a
sufficiently large range of physiologically correct values (positivity of the cross section S,
continuity of the velocity v, etc.). For instance, one can set S
0
where S
corresponds to the unstressed vessel k as it can be directly determined from
k
ð0; xÞ¼S
k
0
; vkð0; xÞ¼0;
k
anatomical atlases, morphometric data, MRI data, etc. Any choice of initial conditions
requires computation of several cardiac cycles to attain the quasi-periodic state. The
number of cycles depends on the number of vessels in the considered network, their size,
network topology, and proximity of the initial conditions to the quasi-stationary state. In
case of networks consisting of arteries only, one needs five to six cardiac cycles, and
addition of venous system increases the number of cycles to 10e12.
9.3 Stenting of leg artery
We start with occlusion treatment in the femoral artery, which was addressed in detail in
Ref. [242]. The objective of the application is to predict presurgically changes of regional
hemodynamics in thigh vasculature and compare these changes with postsurgical ones.
Segmentation and skeletonization of patient-specific MRI data result in the arterial
network shown in Fig. 9.4.
The region of the thigh vasculature is distanced from the heart in the following sense: the
entry point of the regional network coincides with the beginning of common iliac artery
(vessel 1 in Fig. 9.4). The blood flow inlet profile is obtained by the scaling of the heart
ejection profile Q(t) from Fig. 9.1 with coefficient a ¼ 0.21, which provides the best fit
between calculated and measured presurgical velocity maximum in the femoral artery.
This value is in a good agreement with the value computed in the model of the global
systemic circulation. Moreover, according to Ref. [423], for heart rate 60 beats/min and
stroke volume 60 mL, the blood flow through the femoral artery is 635 mL/min; therefore,
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Figure 9.4
The large arteries of two legs and the arterial network of the left thigh. From T. Gamilov, Y. Ivanov,
P. Kopylov, S. Simakov, Y. Vassilevski, Patient specific haemodynamic modeling after occlusion treatment in
leg, Math. Model. Nat. Phenom. 9 (6) (2014) 85e97.
Table 9.3: Comparison of calculated and measured systolic blood velocities.
Maximum velocity (cm/s)
Measurement at
Artery Index Patient Model Patient Model
Common femoral 3 148 149 150 155
Superficial femoral proximal 4 48 54 65 70
Occlusion 5 >300 340 e 71
Superficial femoral distal 7 e 67 98 86
Popliteal 9 52 56 72 72
Deep femoral 12 103 93 69 83
Index denotes the vessel index from Fig. 9.4.
From T. Gamilov, Y. Ivanov, P. Kopylov, S. Simakov, Y. Vassilevski, Patient specific haemodynamic modeling after occlusion treatment in
leg, Math. Model. Nat. Phenom. 9 (6) (2014) 85e97.
635 mL=min
a >
60 min
Presurgical Postsurgical
1
; (9.4)
1
$60 mL
z
6
which is in a good agreement with a ¼ 0.21 as well. Since the surgery is applied far
downstream from the beginning of common iliac artery, we apply the same inlet velocity
profile for both presurgical and postsurgical states.
Other parameters are fitted to match the available presurgical Doppler ultrasound
measurements at several points of the vasculature located proximal and distal to the
occlusion (see the column presurgical in Table 9.3). These parameters are assumed to
Computational hemodynamics in vascular surgery 197
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Figure 9.5
Model of stenosis in a 1D vessel.
remain intact after the occlusion treatment. The only exception is vessel 5, which is a
fragment of the femoral artery where the occlusion is located. Thus, the diseased artery is
split into the stenosed part and healthy proximal and distal parts (Fig. 9.5).
Presurgical parameters of vessel 5 are the modifications of parameters of healthy vessel 4:
the hydraulic resistance is increased by factor of 3, the diameter and the artery lumen are
decreased by 60% and 84%, respectively (see Section 7.6.1 for this and other options of
the stenosis models). Postsurgical parameters of vessel 5 are assumed to coincide with the
parameters for the healthy vessel 4. Although vessels with atherosclerotic occlusion are
characterized by the increased stiffness, variations of the pulse wave velocity in vessel 5
within a physiologically reasonable range do not influence the computational results
essentially.
The column postsurgical in Table 9.3 presents the simulated and measured postsurgical
velocities. Locations of measurements refer to Fig. 9.4. The flow rates are different before
and after the treatment. Both show a reasonably good coincidence between modeled and
measured values (the maximum relative error does not exceed 20%). The 20% error in the
distal part of the superficial femoral artery is caused by the insufficient MRI resolution.
According to general anatomy, the deep femoral artery branch is connected with the
popliteal artery providing an alternative pass in the case of femoral artery occlusion. The
segmentation of the patient MRI data failed to detect this connection. Introducing the
alternative pass decreases the systolic velocity and the error in the deep femoral artery due
to the collateral flow. This example demonstrates importance of the correct segmentation
of the regional vasculature.
9.4 Stenting of coronary arteries and fractional flow reserve
9.4.1 Model of coronary hemodynamics
The model of coronary hemodynamics inherits some basic features of other models for
regional circulation. However, certain features distinguish it from the others. One difference is
the compression of some coronary arteries, veins, and microvasculature by myocardium
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during systole (see Section 7.5.3 for details). Furthermore, one may adopt Windkessel models
at terminal arteries or generate an artificial venous vasculature to mimic the performance of
the coronary veins. In the latter case, in addition to the venous vasculature that mirrors the
arterial counterpart, virtual vessels with Poiseuille pressure drop (Eq. (7.18)) are introduced
between terminal arteries and terminal veins. The total pressure is assumed to be continuous
(Eq. (7.17)) at bifurcations of the arteries and veins, to mimic the microvasculature resistance.
Following Ref. [264], we take into account compression of vessels by myocardial tissues
during the systole phase by the threefold increase of the virtual vessel resistance R and the
increase of the external pressure P
cor
ðtÞ (see Section 7.5.3).
ext
Personalization of the model implies fitting the virtual vessel resistances R, the pulse wave
velocities c, and boundary conditions at the heart. The coronary vessels lie deep inside the
human body, so it is not possible to measure the velocity or pressure using inexpensive
noninvasive methods. Fitting functional parameters is based on available information about
the patient. The patient information is represented by CT scans of his chest area and
general characteristics: height, weight, lifestyle, clinical records, etc.
If the patient’s cardiac output is known, the averaged flow rate profile at the aorta entrance
is corrected. The cardiac output is given to us in the form of the heart rate and, more
seldom, in the form of the stroke volume. These two characteristics allow us to choose an
appropriate boundary condition by scaling the function Q
ðtÞ (see Section 9.2). If no
heart
data on cardiac output is available, the default heart rate 60 beats per minute and a stroke
volume 62 mL are fitted to patient’s age and clinical record. The pressure at the outlet of
the vein network by default is taken to be 8 mm Hg. Of course, these values can be
changed to known values.
Selecting parameters c and R is based on the separation of arteries in two parts: the
branches of the right coronary artery (RCA) and the branches of the left coronary artery
(LCA). For the aortic arch passing into the thoracic aorta, R ¼ 20 dyn s/cm
branches of the right coronary artery, R ¼ 7200 dyn s/cm
coronary artery, R ¼ 720 dyn s/cm
vessels, as they are strongly affected by the myocardium. The base value of the stiffness
parameter c is taken to be 1200 cm/s for RCA branches and 950 cm/s for LCA branches in
accordance with Ref. [414]. If additional patient information is available, the base value is
modified following Eq. (9.2). The artificial network of coronary veins has the identical
structure, but the veins have doubled diameters d and stiffness parameter c reduced by
20%.
The virtual vessels connecting terminal arteries and veins have the following parameters:
length 20 cm, diameter 3 cm, c ¼ 300 cm/s, and resistance R ¼ 6 kdyn s/cm
parameters are chosen to yield the known pressure drop and flow deceleration between
arteries and veins.
Typical personalized coronary vasculature shown in Fig. 9.6 has parameters l, d, c, and R
presented in Table 9.4.
5
5
5
. The coronary arteries have more rigid walls than other
; and for the branches of the left
; for the
5
. These
Computational hemodynamics in vascular surgery 199
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8
9
4
7
6
5
12
11
13
15
2
3
1
10
Heart
14
16
17
18
(3)(2)(1)
Figure 9.6
Typical coronary vasculature. (1) segmentation of CT imag e, (2) skeletonization, and (3) 1D
network. From S.S. Simakov, T.M. Gamilov, F.Y. Kopylov, Y.V. Vasilevskii, Evaluation of hemodynamic sign if-
icance of stenosis in multiple involvement of the coronary vessels by mathematical simulation, Bull. Exp. Biol.
Med. 162 (1) (2016) 111e114.
Table 9.4: Default parameters for coronary arteries: k is the vessel index according to Fig. 9.6,
(cm) is the length, dk(mm) is the diameter, ck(cm/s) is the pulse wave velocity, and R
l
k
k
(dyn s/cm5) is the resistance.
klkd
k
1 5.28 21.7 1050 20 10 0.59 3.6 950 720
2 60.0 25.1 840 20 11 6.1 3.0 950 720
3 2.72 3.1 1200 7200 12 2.05 1.17 950 720
4 1.44 1.31 1200 7200 13 1.75 1.21 950 720
5 1.40 2.73 1200 7200 14 1.39 3.8 950 720
6 6.75 1.52 1200 7200 15 12.1 2.05 950 720
7 5.01 2.50 1200 7200 16 5.4 1.91 950 720
8 1.27 1.19 1200 7200 17 0.38 1.01 950 720
9 5.65 0.157 1200 7200 18 2.62 1.19 950 720
From Y. Vassilevski, T. Gamilov, P. Kopylov, Personalized computation of fractional flow reserve in case of two consecutive stenoses, in: Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016, Crete,
Greece, 5e10 June, vol. 1, 2016, pp. 90e97.
c
k
R
k
klkd
k
c
k
R
k
9.4.2 Fractional flow reserve
One important clinical task is to estimate the hemodynamic significance of the stenosis as
a decision indicator for a surgical treatment. The vascular occlusion factor (VOF, the
relative lesion cross-sectional area l ¼1 S
¼1 dstd
l
d
1
narrowest place) has been used as the conventional significance index for decades. The
VOF cross sectionebased value l > 75% and the VOF diameterebased value l
were assumed to be hemodynamically significant [425]. Hemodynamic significance of a
stenosis, however, is also affected by collateral flows development, hemostasis,
autoregulation, etc. Recently, the fractional flow reserve (FFR) has become the golden
100%, where S
1
S
st
, dstare the vascular lumen area and diameter at the
st
100% or relative diameter decrease
> 50%
d
200 Chapter 9
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standard for the evaluation of stenosed coronary arteries [426,427]. Clinical decisions
based on FFR have reduced the number of expensive high-risk surgical interventions and
improved the statistics of incidences caused disability or death [428].
FFR is the ratio between the maximum achievable blood flow in a diseased coronary
artery and the theoretical maximum flow in a normal coronary artery. In practice, FFR is
computed as the ratio of the mean distal to the stenosis pressure (P
aortic pressure (P
) under maximal hyperemia conditions (see Fig. 2.9):
mean,a
) to the mean
mean,d
FFR ¼P
mean;d
=P
: (9.5)
mean;a
The hyperemic conditions are achieved using vasodilator (e.g., adenosine or papaverine)
administration [428,429]. FFR close to 1.0 is considered as the normal value. FFR values
less than 0.75e0.80 are associated with myocardial ischemia and possible need for
surgical intervention.
Initially, FFR was measured invasively in clinics through coronary catheterization by a
single-use transducer [427]. The procedure is rather expensive. The recent trend is the
development of noninvasive methods for personalized estimate FFR
on the basis of
CT
personalized mathematical models of the blood flow in atherosclerotic coronary
vasculature. Usually, the input of these noninvasive methods consists of CT scans, blood
pressure, and heart rate.
Evaluation of FFR using three-dimensional models of coronary blood flow is the most
popular approach [430e433] although it raises doubts and criticism of experts [434].
Among the problems of FFR
, we mention the impossibility of setting patient-specific
CT
boundary conditions, rigid wall assumption, restrictions to the computational domain,
difficulty in selecting and fitting model parameters, and high demands on computing
resources, which limit the applicability of models in clinical practice. A remedy for the
latter problem can be the use of machine learning tools that compute FFR
CT
within
seconds instead of lengthy 3D simulations [435].
1D hemodynamic models offer a feasible alternative for computation of FFR
[17,260,264,312,436]. The associated low computational cost is appealing for clinical use,
where the 1D simulations can be performed on-site and faster than the full 3D analog. The
1D approach proposed in Ref. [264] also features user independence since it is fully
automated provided a sufficient quality of CT images, as discussed in Ref. [100].
In Fig. 9.7, we compare invasively measured FFR and FFR
software [433] and the 1D hemodynamic model for several dozens of anonymized patientspecific data. Both techniques demonstrate similar quality of FFR estimation. Importantly,
false indications for surgery are rare that makes both methodologies appealing in clinics.
CT
computed by HeartFlow
CT
Computational hemodynamics in vascular surgery 201
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Figure 9.7
Comparison of invasively measured FFR and FFR
hemodynamic model.
computed by HeartFlow software and 1D
CT
The importance of transition from the use of VOF to estimating FFR can be demonstrated
by the following example. Consider the coronary vasculature shown in Fig. 9.6 and
assume that the left anterior descending (LAD) artery has a stenosis with l ranging from
50% to 99%. The diameter d of the LAD artery may be 2 or 3 mm; both values are
anatomically acceptable.
Fig. 9.8 shows FFR
may give different FFR
for varying l and two cases of d. We observe that the same VOF
CT
for different diameters d. If one assesses the hemodynamic
CT
significance of the stenosis with the help of VOF, stenting should be performed for l >
70%. If one assesses the hemodynamic significance of the stenosis basing on FFR
stenting should be performed for FFR
no clear indication for surgical intervention. These thresholds for l and FFR
< 0.75, and the case 0.75 < FFRCT< 0.8has
CT
CT
are
CT
,
marked by horizontal and vertical lines in Fig. 9.8. The lines define domains with
conflicting indications for stenting. In the case of narrow LAD artery (d ¼ 2mm),
surgical intervention is necessary or possible (FFR
< 0.8) even if the degree of
CT
stenosis is within 57% < l < 70% (curve 2 from a to b). Vice versa, in case of thick
LAD artery (d ¼ 3 mm ), surgical intervention is not required or is likely not required
(FFR
> 0.75) even if the degree of stenosis is in the range of 70% < l < 83% (curve 1
CT
from b to c).
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Figure 9.8
Comparison of FFR
1) or 3 mm (curve 2). FFR, fractional flow reserve; LAD, left anterior descending; VOF, vascular
occlusion factor. From S.S. Simakov, T.M. Gamilov, F.Y. Kopylov, Y.V. Vasilevskii, Evaluation of hemody-
namic significance of stenosis in multiple involvement of the coronary vessels by mathematical simulation, Bull.
for stenoses with the same VOF of LAD artery with diameter 2 mm (curve
CT
Exp. Biol. Med. 162 (1) (2016) 111e114.
Therefore, VOF is not a reliable indicator for assessing the hemodynamic significance of
coronary artery stenosis. The use of this indicator in some cases may lead to both
substantial underestimation and substantial overestimation of the hemodynamic
significance of stenosis.
The hemodynamic significance of stenoses may vary depending on the physiological state.
The common reason of changes in the state is physical activity. Conditions of physical
activity are impossible or difficult to reproduce in a clinical examination of a particular
patient. However, to some extent, they can be taken into account in numerical simulation.
Physical exercise causes intensification of cardiac activity, which is modeled by scaling
the heart outflow condition as shown in Ref. [17] (see also Fig. 9.1).
To study the impact of the cardiac output on FFR
, we consider two anonymized patient
CT
cases with multiple stenoses of coronary arteries. Patient 1 has stenoses in the proximal
part (one-third) of the left main coronary artery (LCA-1, l ¼ 55%), the middle one-third
of the left circumflex artery (LCX-1, l ¼ 80%), and the middle one-third of the left
anterior descending artery (LAD-1, l ¼ 50%). Patient 2 has stenoses in the proximal part
(2 mm length) of the right main coronary artery (RCA-2, l ¼ 55%) and the middle onethird (2 cm length) of the left circumflex artery (LAD-2, l ¼ 80%). The stenoses are
marked by letters A, B, C, D, and E in Fig. 9.9.
The value of FFR
was measured in every case for every stenosis. The computational 1D
CT
network of coronary vessels and functional parameters (stiffness and resistance) was
personalized using the techniques from Sections 3.4.2 and 9.2, respectively, and FFR
CT
was computed for every stenosis with acceptable errors. For details, we refer to Ref. [264].
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