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Computational hemodynamics in vascular surgery 213
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Figure 9.20
Structures of Circle of Willis: full Circle of Willis (1), ACoA is missing (2), ACoA and RPCoA are
missing (3), ACoA, RPCoA, and LPCoA are missing (4), RPCoA is missing (5), RPCoA and
LPCoA are missing (6). ACoA, anterior communicating artery; LPCoA, left posterior
communicating artery; RPCoA, right posterior communicating artery. From S. Simakov, T. Gamilov,
Computational study of the cerebral circulation accounting for the patient -specific anatomical features, in:
Smart Innovation, Systems and Technologies, vol. 133, 2019, pp. 309e 330.
The absence of the anterior and posterior communicating arteries in the Circle of Willis is
a common case, whereas underdevelopment or absence of ACoA is observed in 20%e40%
of cases [438].
For each structure, the impact on blood flow through the middle and posterior brain
arteries from one to four stenoses located in the right and left internal carotid arteries
and right and left vertebral arteries was examined. The results of the numerical study can
be summarized as follows. The full Circle of Willis provides sufficient blood supply to
the cerebral arteries even with significant stenoses of the carotid or vertebral arteries.
However, less than 50% of population has the full Circle of Willis. In other cases, even a
single stenosis may lead to a significant reduction in blood flow in one of the cerebral
arteries, whereas the elimination of stenosis may improve blood flow in one region but
worsen it in another region causing stealing phenomena in certain part of the vascular
bed. Therefore, the accurate identification of the Willis Circle structure is crucial for the
patient-specific simulation of cerebral hemodynamics. If such identification is not
possible, the worst-case scenario should be considered for the sake of reliability of the
prediction. The simulation results suggest the following rules for choosing a treatment
strategy for stenotic lesion of the brachiocephalic arteries: in the presence of the full
214 Chapter 9
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Circle of Willis clearly visible on CT images, the elimination of all stenoses in the
collateral arteries (left internal carotid artery, right internal carotid artery, left vertebral
artery, right vertebral artery) is not necessary; in the absence of both posterior
communicating arteries, compression of the vertebral arteries (for example, in case of
cervical osteochondrosis) is a life threatening condition, and elimination of a stenosis
may lead to stealing blood supply in a remote region.
9.6 Decision support software
Proprietary software MultiVox is a compelling picture archiving and communication
software system developed at Moscow State University [439]. We integrated software
complex implementing algorithms and methods from Chapters 3, 7 and 9 into MultiVox to
create a useful tool for FFR
images and extract the 3D and 1D structure of the aorta and coronary vessels. Several
parameters can be adjusted by the user, such as a vesselness threshold and a stenosis
position. The vesselness threshold is a parameter that restricts the probability of vessel
growing in the Frangi filter discussed in Section 3.4.2. Large values lead to a lot of noise.
Small values reduce the noise but cut off small vessels as well. The default value is
estimated automatically, but the user can change it to obtain a clear structure of coronary
vessels (see Fig. 9.21).
assessment. Algorithms from Chapter 3 process DICOM
CT
Various types of postprocessing, including shaded surface display, maximum intensity
projection, and multiplanar reconstruction, help to identify the presence of stenoses and
occlusions in the coronary arteries. Multiplanar reconstruction involves the process of
converting data from an imaging modality acquired in an axial plane into another plane.
Figure 9.21
3D voxel structure of aorta. Vesselness threshold.
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Figure 9.22
Extracted coronary vessels. Multiplanar reconstruction of a coronary artery.
It is commonly performed with the thin-slice data from volumetric CT scanning in the
axial plane. It also can be accomplished with scanning in any plane and any modality,
which is capable of cross-sectional imaging.
The acquired data from the axial plane can be converted to nonaxial planes such as
coronal, sagittal, or oblique. The MultiVox package provides additional functionality to
work with the data. One of these additional tools, curved planar reformation, involves
tracing a blood vessel and generating a 2D planar image, which transects the structure
along its short axis. Multiplanar reconstruction is a built-in feature of the MultiVox
package. We utilize it to visualize the result of segmentation and simplify stenosis
detection (see Fig. 9.22).
Position and length of the stenosis can be manually adjusted by moving bars along the
vessel as shown in Fig. 9.22. An automated procedure provides black-box calculation of
the degree of the stenosis, although the user can reset the value. It is possible to insert a
stent or additional stenoses to the virtual model as well as remove them. This helps to
compare different treatment strategies or estimate possible risks.
After positioning of the stenoses, the simulations with the 1D hemodynamic model are
performed. The results of the computational processing include the set of FFR
values for
CT
the designated stenoses. It takes 10e25 min to perform segmentation, arterial structure
extraction, and 1D simulations on a laptop. The actual time depends on the complexity of
the arterial structure. In any case, a conventional PC is enough to perform all
preprocessing procedures and simulations that are appealing in clinical practice.
CHAPTER 10
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Applications in antitumor therapy
10.1 Introduction
Following cardiovascular diseases, cancer is the second leading cause of death in
developed countries. Despite the fact that the risk of cancer increases with age,
oncological diseases are not uncommon in all age groups. It is believed that cancer, or
malignant tumor, is not a single disease, but rather a number of interrelated pathological
processes triggered by the growth of cells undergoing malignant transformation. In 2000,
Douglas Hanahan and Robert Weinberg [333] identified six hallmarks, i.e., common
alterations in cell physiology that collectively dictate malignant growth: (1) selfsufficiency in growth signals; (2) insensitivity to antigrowth signals; (3) evasion of
programmed cell death (apoptosis); (4) limitless replicative potential; (5) sustained
angiogenesis; (6) tissue invasion and metastasis. In their later work of 2011, they added
four more typical traits of cancer: (7) deregulated metabolism; (8) evading the immune
system; (9) genome instability; (10) induction of inflammation.
Although the discussion concerning the significance of this way of cancer description is
ongoing [440,441], the approach is generally accepted. As an example of criticism, we
may point to Ref. [441], where the author suggests that cancer is rather a tissue-level
disease and its description via cell-level hallmarks is misleading.
From mathematical point of view, the first four hallmarks can be combined into one
concept that can be formulated as follows: cancer cells proliferate unlimitedly under
sufficient level of nutrients. This concept is clearly supported by in vitro experiments
with cultures of tumor cells, where the law of exponential growth of cell number is
manifested at the initial stage [442]. Subsequently, as nutrients become depleted, growth
rate slows d own, since some cells stop proliferating and pass to a quiescent state, which
requires less nutrients [443]. The dependence of tumor proliferation rate on the
availability of nutrients can be seen even more clearly from results of in vitro
experiments with multicellular tumor spheroids (MCTS), i.e., three-dimensional
aggregates of malignant cells growing in nutrient solution. After a short initial growth
phase, the radius of MCTS increases linearly with time, whereas the spheroid acquires rigid
structure consisting of central necrotic zone and a thin outer rim of living cells, which
Personalized Computational Hemodynamics. https://doi.org/10.1016/B978-0-12-815653-7.00010-5
Copyright © 2020 Elsevier Inc. All rights reserved.
217
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thickness remains constant over time. The thickness of this layer is determined by diffusion
of nutrients from the solution surrounding MCTS [444]. Of course, the viable rim does not
consist of only proliferating cellsda significant fraction of it consists of quiescent cells.
With increase in concentrations of key metabolites, oxygen, and glucose, in the medium, the
living rim enlarges and the MCTS growth rate increases proportionally.
The structure of solid tumors in tissue can both match the structure of MCTS and
significantly differ from it (see Fig. 10.1). Solid tumors, which have different structures,
experience different patterns of nutrient inflow. Encapsulated tumors and minimally
invasive tumors, transplanted under the skin of an animal (usually rat or mouse), grow as
separate compact objects. The inflow of nutrients to such tumors occurs via capillaries
located in the peritumoral region, i.e., the tissue that surrounds the tumor, so their structure
is similar to the one of MCTS [445]. On contrary, tumors of invasive type grow by
infiltrating surrounding tissues, so there may be no clear border between a tumor and a
normal tissue [446]. Inflow of nutrients to these tumors is not that much compromised,
since there are microvessels inside the tumor. However, in this case gradual destruction of
capillaries takes place due to various factors (as discussed in detail in Chapter 8). The
absence of microvessels inside low-invasive tumors and the destruction of capillaries
inside high-invasive ones both contribute to nutrient deficiency and necrosis formation in
internal tumor regions. Note that, even under substantial microvasculature density, tumor
growth in tissue is limited since proliferating tumor cells consume much more nutrients
than normal cells. Lowering of nutrient levels inside the tumor leads to the phenomenon
that a sufficiently large fraction of tumor cells experiences lack of nutrients even if their
supply is adequate for surrounding healthy tissues. In the state of metabolic stress, tumor
cells produce various signaling molecules that stimulate the growth of new capillaries, i.e.,
we have the process of angiogenesis [447]. The most universal and widespread mediator
Figure 10.1
Types of tumor growth.
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of angiogenesis is the vascular endothelial growth factor, or VEGF, which is also known as
vascular permeability factor (VPF), since it causes the increase in permeability of capillary
walls.
Angiogenesis has been marked by Douglas Hanahan and Robert Weinberg as one of the
hallmarks of cancer. Via stimulating of new microvessels formation, tumors obtain more
nutrients and speed up their growth. In 1971, Judah Folkman suggested a new type of
anticancer therapy, i.e., antiangiogenic therapy (AAT), aimed at cessation of angiogenesis,
which should lead to a complete stop or at least to a significant slowdown in tumor growth
[448]. The first antiangiogenic drug bevacizumab, which irreversibly binds to VEGF,
rendering it inactive, was approved for medical use in 2004 and is widely used nowadays.
AAT by bevacizumab not only prevents the formation of new capillaries but also leads to
the normalization of tumor capillaries, i.e., brings them to a more physiologically normal
state with normalized permeability of the walls, both actions limiting the supply of
nutrients to the tumor. Compared with traditional radio- and chemotherapy (CT), AAT has
moderate side effects but, as it was repeatedly shown in clinics, by itself cannot
completely destroy the tumor and therefore has limited effectiveness [449]. Currently,
almost all approved protocols that include bevacizumab combine it with various
chemotherapeutic agents [450], which aim to destroy actively proliferating cells via
various mechanisms. However, since they are not tumor specific and affect all proliferating
tissues, their action is associated with significant side effects, such as anemia, hair loss,
immunosuppression, rheumatoid arthritis, and others, which limit the use of these drugs.
Two different types of drugs administered simultaneously inevitably interact with each
other. Of a special importance is the fact that AAT often ultimately leads to the decrease in
the inflow of chemotherapeutic drug into tumor, which has been observed experimentally
[451,452]. Therefore, a major challenge associated with the use of CT in combination with
AAT is the optimal scheduling of drug administration to maximize antitumor effect and
minimize side effects. Currently, this is a topic of an active research, and mathematical
modeling aims to facilitate progress in this direction.
Historically, first mathematical models in oncology tried to address only tumor growth.
The scope of such studies was to adequately describe the experimentally observed change
of the number of cells in experiments with malignant cultures in vitro or to model the
change in tumor volume in experiments with MCTS or transplanted tumors in vivo
[453,454]. A common trait of such models is small number of parameters, usually two or
three, which allows the authors to successfully fit the experimental data. Although the
progress in this direction continues [455], such models do not take into account the
structure of the tumor and basic processes that determine its development in tissue. For the
first time, the importance of nutrient inflow into the tumor as a key factor for its
development was demonstrated in papers by Burton [456] and Greenspan [457], where the
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MCTS structure was considered. Later, as the models became more and more complicated
in addressing the spatial dynamics of tumor growth, it became feasible to predict the
changes in the tumor structure due to cellular motility [458,459], convective flows in the
tissue [460,461] and angiogenesis [462,463]. Recently, mathematical modeling in
oncology has been increasingly focusing on the description of anticancer therapy.
However, most of the work in this area is devoted to classical antitumor radio- or
chemotherapy alone [464e466]. In contrast to clinical practice, mathematical models that
combine CT and AAT were introduced only recently [467,468]. The existing works on this
topic are still built on purely phenomenological description of the effect of antiangiogenic
drugs, which makes it impossible to account for the interaction of different types of
therapies. Moreover, all works of this type do not consider the spatial structure of tumor,
which indicates that they may be suitable only for modeling of postoperative (adjuvant)
therapy. At the same time, there are many reliable studies that simulate classical
monochemotherapy [469e471] and AAT by bevacizumab, considering its
pharmacokinetics in blood and tissue [472,473].
Further in this chapter, using a spatially distributed model of the tumor growth and
combined therapy, we compare the efficiencies of different schemes of palliative and
neoadjuvant combined chemotherapy with bevacizumab, as well as monochemotherapy.
Palliative therapy is administered when surgical intervention is not possible, and the drug
treatment continues until tumor regrowth or lethal outcome. Neoadjuvant therapy is held
before surgery to reduce the tumor volume. For the modeling of neoadjuvant therapy, it is
necessary to take into account the formation of necrosis during the therapy and the
mechanisms of its elimination from tissue. In the presented approach, this problem is
solved by explicit consideration of dynamics of interstitial fluid (IF), which is the main
constituent of necrosis.
10.2 Mathematical model of antitumor combined chemo- and
antiangiogenic therapy
10.2.1 Governing equations
To visualize the role of different factors during the chemo- and antiangiogenic therapy, we
start with Fig. 10.2, which shows a block scheme of the model we are interested in. We
consider a monoclonal tumor and rely on the accepted principle of migration/proliferation
dichotomy of tumor cells [474]. Thus, the model tumor comprises a heterogeneous colony
of malignant cells, described in terms of the normalized densities of proliferating cells,
ðr; tÞ, where r and t are space and time coordinates, and quiescent cells, n2ðr; tÞ, which
n
1
are able to migrate into surrounding tissue. Cells can change their state depending on the
concentration of glucose Gðr; tÞ, which is chosen as a key metabolite, since it is both a
crucial precursor for organic synthesis and the most important energetic substrate for
tumor cells. Under insufficient levels of glucose, proliferating cells pass to the quiescent
Applications in antitumor therapy 221
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Figure 10.2
Block scheme of the model of tumor growth and combined chemotherapy with bevacizumab: n
and n2are proliferating and migrating tumor cells, respectively, h is normal cells, m is cell
remnants, IF is interstitial fluid, G is glucose, VEGF stands for vascular endothelial growth factor,
NC and AC are normal and angiogenic part of microcirculatory network, respectively, A is
bevacizumab, T
(gray arrows in print version) indicate stimulating links, red lines (black lines in print version) denote
f
and Tbare free and protein-boun d forms of cisplatin, respectively. Green arrows
inhibiting actions, and whit e arrows correspond to cell transitions.
1
state. If the level of glucose is sufficient, then the reverse process takes place. The
surrounding tissue consists of normal cells characterized by density hðr; tÞ and IF. We
denote by IFðr; tÞ the fraction of the IF. Tumor cells and normal cells can die due to lack
of glucose, while tumor cells may also die from action of chemotherapeutic agent, for
which only cycle-specific action is considered. Dead cells form necrosis, which consists of
nonutilized cell remnants and IF. The fraction of cell remnants in the tissue is denoted by
mðr; tÞ. The healthy tissue contains a normal capillary network; the bulk density of its
surface is NCðr; tÞ. As the result of the tumor angiogenesis, the normal capillary network
expands by angiogenic capillaries. The bulk density of angiogenic capillaries surface is
denoted by ACðr; tÞ. The angiogenic capillaries network is introduced in the model to
account for the enhanced permeability (for various substances) of tumor angiogenic
microvasculature. Besides IF formed in result of cell death, more IF enters the tissue from
the microvasculature. Its influx similarly depends on the type of microvessels. The model
assumes that active outflow of IF occurs only through the lymphatic system. We assume
that the amount of lymphatic capillaries is proportional to the density of normal cells.
Thus, the rate of outflow of IF depends on the fraction of normal cells. The model also
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takes into account the concentrations of VEGF, bevacizumab, and cytotoxic drug. We
denote by Vðr; tÞ and Aðr; tÞ the concentrations of VEGF and bevacizumab, respectively.
To account for the role of the cytotoxic drug, we use the parameters of cisplatin, which is
one of the chemotherapeutic agents used in the clinical practice in combination with
bevacizumab [450]. Since it has a high affinity for plasma proteins, we take into account
two forms of cisplatin, using the variables of concentrations of its free and protein-bound
forms, T
f
ðr; tÞ and Tbðr; tÞ, respectively.
Summarizing, the dynamics of cells and necrosis is governed by the following system of
equations:
vn
vt
vn
vt
vh
vt
vIF
vt
proliferation
z}|{
1
¼ Bn
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
2
ðGÞn1 P2ðGÞn
¼ P
1
death
zfflfflfflfflffl}|fflfflfflfflffl{
¼d
h
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
¼ bðd
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
P1ðGÞn1þ P2ðGÞn
1
cell transitions
convection
zfflfflfflfflffl}|fflfflfflfflffl{
ðGÞh
VðvhÞ;
ðGÞ
n
n
2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
where P
þP
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
vm
¼½1 bðd
vt
n
þ n2þ IF þ m þ h ¼ 1;
1
ðGÞ¼k1expðk2GÞ; P2ðGÞ¼
1
IF;NC
NC þ P
n
ð1 þtanh½εdðGdGÞÞ; i ¼ n; h:
cell transitions
cell death
ðGÞ
þ d
h
cell death
death
zfflfflfflfflfflffl}|fflfflfflfflfflffl{
dnðGÞn
2
h þ k
inflow
ACðIFic IFÞ
IF;AC
TTnTn1
death from CT
zfflfflfflfflfflffl}|fflfflfflfflfflffl{
kTTnTn
2
convection
zfflfflfflfflffl}|fflfflfflfflffl{
Vðvn2Þ
2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
Þ
v
dr
ðGÞn2þ dhðGÞh þ kTTnTn1Þ
1
k3ð1 tanh½εtrðGtrGÞÞ; diðGÞ¼
2
convection
zfflfflfflfflffl}|fflfflfflfflffl{
Vðvn1Þ
1
migration
zfflfflffl}|fflfflffl{
þ DnDn
outflow
ð
IF IF
diffusion
zfflfflfflffl}|fflfflfflffl{
þDIFDIF
convection
zfflfflfflffl}|fflfflfflffl{
VðvmÞ
2
Þ
min
h þ h
convection
zfflfflfflfflffl}|fflfflfflfflffl{
VðvIFÞ
;
;
(10.1)
h
;
1
max
d
i
2
In Section 10.2.2, we set up all parameters and discuss how one can estimate their values.
Since the model is one-dimensional, the gradient in the convection operator is a scalar
function. We remark that the form of the cell transition term from proliferation to rest
P
ðGÞ in the first equation is taken from Ref. [475], where it was successfully used for
1
fitting experimental data. The reverse transition, P
tumor regrowth, but its form does not affect the result qualitatively; therefore, we use a
smoothed step function for it. Diffusion approximation is used to describe the transport of
IF throughout the tissue.
ðGÞ, is important in simulating possible
2
Applications in antitumor therapy 223
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In this model, we consider simultaneously two types of growth of solid tumor discussed in
Section 10.1, i.e., the dense growth as a compact object and infiltrative growth. This is a
rare approach in mathematical modeling. However, it is physiologically grounded, since
the growth of real tumor often represents a combination of these two types. Consideration
of infiltrative growth is realized herein in a traditional way, i.e., through the term of
migration of tumor cells. For description of compact growth, we use the following
approach, which represents a special case of a more general method, utilized, e.g., by
Ref. [476]. That method converges to the one presented herein under the assumption that
values of interstitial pressure, arising due to cell dynamics, are infinitesimal, so they do not
hinder the cell proliferation.
Let us consider an incompressible dense tissue in which spatial distribution of components
is affected by their local kinetics, e.g., proliferating tumor cells push out surrounding
tissues, providing an increase in tumor size. In mathematical form, the condition of
incompressible dense tissue means the constancy of all tissue constituents, which is
included in Eq. (10.1). To account for such phenomena, we introduce a convective velocity
field vðr; tÞ.
Convective velocity field vðr; tÞ can be obtained by summing up the equations for all
tissue constituents. During this procedure, left sides of equations produce the derivative of
a constant value, i.e., zero; the transition terms of cell death are canceled out; and the sum
of convection terms provides the gradient of velocity field, which can be expressed via
remaining terms in the following way:
The final expression for v obtains different forms depending on space dimensions and
boundary conditions.
As already mentioned, the model uses two variables to describe the microcirculatory
network, namely, its normal and angiogenic parts, NC and AC. The angiogenic part has
significantly higher wall permeability. With the growth of tumor, the microcirculatory
network is locally destroyed under the influence of increased local pressure due to
proliferation and migration of tumor cells, as well as various chemical factors. These
factors are implicitly taken into account by setting the rate of microvessel destruction
nonzero inside necrosis and lower than in the presence of tumor cells. Under sufficient
amount of VEGF, new angiogenic capillaries formation takes place along with
denormalization (i.e., transformation to angiogenic) of previously existing normal ones. This
is introduced in the model to reflect the increase in permeability of the capillary walls under
the action of VEGF and is described by the transition from NC to AC.Atlow
Vv ¼ Bn
v
þP
1
ðIF IF
dr
IF;NC
NC þ P
Þ
min
h þ h
ACðIFic IFÞ
IF;AC
h
þ DnDn2:
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