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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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224 Chapter 10
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concentrations of VEGF, capillary normalization and reverse transition occur. The rate of
angiogenesis decreases in the presence of cisplatin, which also kills dividing endothelial
cells. A special term is introduced to account for microcirculatory network tendency to
return to a constant physiologically reasonable density under the cessation of tumor
angiogenesis. This term includes the Heaviside function q, since it differs from zero only
when the total density of normal and angiogenic capillaries exceeds unity. Capillaries also
move owing to convective flows, but more slowly than cells, due to their connectivity in
network. The ability of angiogenic capillaries to grow into the tissue is accounted for by the
diffusion term.
The equations describing the dynamics of capillary network read as follows:
vNC
vt
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
¼½lðn
degradation
þ n2ÞþlmmNC
1
normalization
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
v
V
mat
þ
V þ V
AC
denormalization
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
v
V
dem
V þ V
NC
vAC
vt
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
density normalization
mðNC þ AC 1ÞNCQðNC þ AC 1Þ
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
A
T
T
V þ V
degradation
V
k
¼ Re
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
½lðn1þ n2ÞþlmmAC
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
angiogenesis
ðNC þACÞ1
normalization
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
V þ V
density normalization
v
V
mat
ðNC þACÞ
C
AC
convection
zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{
VðgvNCÞ
max
denormalization
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
v
V
dem
þ
V þ V
;
(10.2)
NC
mðNC þ AC 1ÞACQðNC þ AC 1Þ
migration
zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{
þDACDAC
convection
zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{
VðgvACÞ
:
Glucose enters the tissue from the capillary network, and its inflow is regulated by diffusion
through the pores in the capillary walls, where various capillaries have different
permeabilities. The level of glucose in the blood is assumed to be constant. All cells
consume glucose, and the level of consumption of proliferating tumor cells is much higher
due to the specific features of the tumor metabolism [477]. Glucose diffuses throughout the
tissue, and its local diffusion coefficient decreases with the concentration of cells [478].
For this model, it can be formulated in terms of the fraction of IF. It is known that the
coefficient of diffusion of all substances increases in necrosis (the diffusion MRI principle is
based on this observation [479]). We introduce the dependence of the diffusion coefficient
on the fraction of IF for all substances in the model using the linear relation:
ðIFÞ¼D
D
i
0
ð1 þaIFÞ; i ¼ G; V ; A; Tf; Tb:
i
Applications in antitumor therapy 225
https://t.me/medicina_free
Thus, the change of glucose concentration is determined by the equation:
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
vG
vt
¼P
G;NC
NC þ P
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
q
n1þ q
n
1
inflow
G;AC
consumption
n2þ qhh
n
2
ACðGbl GÞ
G
G þ G
diffusion
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
þDGðIFÞDG
(10.3)
:
Proangiogenic factor VEGF is produced by malignant cells. Since metabolic stress
significantly increases its production [480], its secretion only by quiescent cells is
considered. We do not consider VEGF interactions with receptors of endothelial cells and
tissue elements in all their complexity. We rather use the simplest form of the term to
describe VEGF internalization by endothelial cells, since it does not affect the simulation
results. We also take into account diffusion of VEGF in tissue, its degradation, and outflow
from tissue, the latter being explicitly included in the model under the assumption that
concentration of free VEGF in the blood is negligible.
Bevacizumab is administered intravenously, which is reflected by the ordinary differential
equation for its blood concentration. It contains a term representing its administration,
which leads to the abrupt increase of the concentration of bevacizumab in blood after
moment of its injection. The equation also contains the reaction term representing the
blood clearance. Bevacizumab comes from the blood into tissue, where it diffuses and
irreversibly binds to VEGF. The difference in permeabilities of walls of various capillaries
is significant for bevacizumab, as its size is comparable with the pore sizes of normal body
capillaries. Since bevacizumab is a macromolecule specifically designed to inhibit VEGF,
its interaction with tissue elements is neglected.
vV
vt
secretion
z}|{
¼ pn
internalization
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{
uVðNC þ AC
2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
Þ
P
V;NC
outflow
NC þ P
V;AC
ACV
vA
vt
vA
vt
degradation
zfflffl}|fflffl{
dVV
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
¼P
A;NC
administration
bl
z}|{
¼ F
zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{
ðkAAnÞAV
NC þ P
iv
A
neutralization
inflow
ACðAbl AÞ
A;AC
blood clearance
zfflfflffl}|fflfflffl{
dAA
bl
diffusion
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
þDVðIFÞDV ;
binding with VEGF
:
zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{
ðkAVnÞAV
diffusion
zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
þDAðIFÞDA;
(10.4)
226 Chapter 10
https://t.me/medicina_free
Cisplatin also enters the circulatory system through intravenous injection. The model
reflects this with a sharp increase in the cisplatin concentration at the moments of injection.
Cisplatin has the molecular weight of only 300 Da. It actively binds to blood proteins and
as a result exists in blood and tissue in two formsdin the form of free small molecules of
active drug and a large inactive protein-bound complex. Their concentrations in tissue are
denoted as T
f
and Tb, respectively. Both forms of the drug come from blood into tissue,
where they diffuse and continue to transit from one form to another. The rate of cisplatin
binding to the protein should depend on the concentration of proteins, which is usually
much higher in blood than in normal tissue. However, since we consider a tumor tissue, by
the time of therapy, blood proteins should accumulate inside it to a level comparable with
the blood level due to increased permeability of angiogenic capillaries and the impaired
drainage of IF through the lymphatic system. Hence, we take the coefficient of cisplatin
binding to blood proteins inside the tissue equal to the corresponding value in blood. We
also neglect the blood clearance of protein-bound drug, as well as its binding to tissue
elements because they are small comparing with these parameters for the free agent.
vT
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
f
vt
¼P
NC;Tf
NC þ P
inflow
AC;Tf
AC
f
T
T
bl
nonspecific binding
f
zfflfflffl}|fflfflffl{
dTT
f
vT
vt
vT
vt
vT
vt
interaction with proteins
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{
konTfþ k
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
b
¼P
interaction with proteins
NC þ P
NC;Tb
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{
þkonTf k
administration
f
bl
b
bl
z}|{
¼ F
¼ k
iv
T
interaction with proteins
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{
f
T
k
on
bl
b
T
off
inflow
AC;Tb
b
T
off
interaction with proteins
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{
konT
b
T
off
bl
diffusion
zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{
þDTfðIFÞDTf;
AC
T
diffusion
zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{
þDTbðIFÞDT
f
þ k
T
off
bl
:
b
b
T
bl
b
;
blood clearance
zfflfflfflfflffl}|fflfflfflfflffl{
b
d
bl
T;bl
(10.5)
f
T
;
bl
Here, we neglect convective transport of all the substances across the capillary walls. As
discussed above, this assumption is clearly justified for small molecules such as glucose.
Inclusion of this type of transport for bevacizumab and VEGF would not affect the tumor
growth qualitatively due to specific features of their action and dynamics in tissue. The
Applications in antitumor therapy 227
https://t.me/medicina_free
justifications of this approach for consideration of cisplatin transport and its impact on
obtained results are discussed in Section 10.4.
10.2.2 Model parameters
The model contains several dozens of parameters. When available, the values are taken
from various experiments. Otherwise, they are evaluated to ensure that the tumor growth
reflects some of its known characteristics. The basic set of parameters is given in
Table 10.1, where the following normalization parameters are used to obtain their model
values: t
concentration. The normalization parameters for VEGF, bevacizumab, and cisplatin are
used in the corresponding terms describing the effects of the therapies and are chosen to
be V
two values are estimated from the concentrations of the corresponding drugs in the blood
Parameter Description Value Model References
¼ 1 h for time, Ln¼ 102cm for length, and Gn¼ 1 mM for glucose
n
11
n
¼ 10
mole/mL, An¼ 1:6$109mole/mL, and Tn¼ 5$108mole/mL. The last
Table 10.1: Model parameters.
Value estimations based on tumor cells
0.4 h
19.8
0.16 h
1
1
mL
mg
1
0.02 [481]; see text
0.4 [475]
3.6 [475]
0.4 See Section 10.2.1
1.8 1.8 See Section 10.2.1
1.7 mM 1.7 See Section 10.2.1
10
3$10
3
9
1
h
3$10
10
9
4
5 5 See text
0.55 See text
12
cm
2
s
1
3:6$10
5 10
5
3
B Proliferation rate 0.02 h
k
1
Maximum transition rate
into rest
k
2
Sensitivity of transition
into rest to glucose
k
3
Maximum rate of
transition into
proliferative state
ε
tr
Sensitivity of transition
into proliferative state
G
tr
Threshold concentration
of glucose for transition
into proliferative state
k
T
max
d
n
ε
d
Sensitivity to cisplatin
Maximum rate of death
Sensitivity of death rate
to glucose
G
d
Threshold concentration
of glucose for death
D
n
Migration coefficient
10
Normal cells
max
d
h
Maximum rate of death 5 103h
Varies, see text
[482]
See text
[482]
Continued
228 Chapter 10
https://t.me/medicina_free
Table 10.1: Model parameters.dcont’d
Parameter Description Value Model References
Interstitial fluid
b Fraction in dead cells 0.8 0.8 See text
P
IF;NC
Coefficient of inflow
through normal
capillaries
P
IF;AC
Coefficient of inflow
through angiogenic
capillaries
IF
ic
IF
min
Inflow parameter 0.5 0.5 See text
Minimum fraction in
tissue
v
dr
h
Rate of drainage 0.01 0.01 See text
Michaelis constant for
outflow
D
IF
Rate of transport in
tissue
a Coefficient of increase in
diffusion rate of
substances in necrosis
Capillaries
l Degradation rate in alive
region
l
m
Degradation rate in
necrosis
v
mat
V
Normalization rate 0.05 h
Michaelis constant for
1:7$10
5:1$10
10
angiogenesis rate
v
dem
m Density normalization
Denormalization rate 0.05 h
5
rate
10
g Network elasticity 0.75 0.75 See text
R Maximum angiogenesis
5 10
rate
C
P
P
k
D
G;NC
G;AC
G
max
A
T
AC
bl
Maximum surface density
Sensitivity of
0:35$10
angiogenesis to cisplatin
Migration coefficient
Permeability of
continuous capillaries
Permeability of
angiogenic capillaries
Glucose blood level 5.56
1:1$10
2:8$10
1:1$10
5:5$10
4
4
1:1$10
5:5$10
4
4
See Section 10.2.4
See Section 10.2.4
0.15 0.15 See text
0.425 0.425 See Section 10.2.1
10
9
0.036 See text
0.3 0.3 See Section 10.2.1
10
10
14
500
10
cm
cm
11
cells
mL
cells
mL
1
molemL
1
1
2
3
1
3
h
2
cm
3
cm
8
molemL
2
cm
s
1
1
s
1
1
s
1
s
0.05 See text
0.15 See text
0.05 See text
3
10
0.05 See text
3
10
3
5 10
See Section 10.2.1
5 See text
10 See text
4
3:6 10
See text
See text
See text
Glucose
cm
5
4 See Section 10.2.3
s
cm
5
mmol
l
s
10 See Section 10.2.3
5.56 [483]
Continued
Table 10.1: Model parameters.dcont’d
https://t.me/medicina_free
Parameter Description Value Model References
q
n
1
Proliferating tumor cells
consumption rate
q
n
2
Quiescent tumor cells
consumption rate
q
h
G
Normal cells
consumption rate
Michaelis constant for
consumption rate
0
D
G
Diffusion coefficient
p Secretion rate
u Internalization rate
P
NC;V
Permeability of
continuous capillaries
P
AC;V
Permeability of
angiogenic capillaries
d
V
k
A
Degradation rate 0.01 h
Constant of binding to
bevacizumab
P
D
NC;A
0
V
Diffusion coefficient
Permeability of
continuous capillaries
P
AC;A
Permeability of
angiogenic capillaries
P
0
D
A
d
A
NC;Tf
Diffusion coefficient
Blood clearance rate 0.035 day
Permeability of
continuous capillaries
P
AC;Tf
Permeability of
angiogenic capillaries
k
on
k
off
d
T
Protein-binding rate 0.46 h
Protein-unbinding rate 0.04 h
Rate of nonspecific
binding with tissue
elements
d
P
0
D
T
T;bl
NC;Tb
Diffusion coefficient
Blood clearance rate 0.13 h
Permeability of
continuous capillaries
P
AC;Tb
Permeability of
angiogenic capillaries
0
D
Tb
Diffusion coefficient
6:5$10
2:3$10
0.49
0.04 mM 0.04 [485]
2:6$10
0.4
2:8$10
6:4$10
7:8$10
5:3$10
5:9$10
Bevacizumab
1:6$10
1:2$10
4$10
Free cisplatin
8$10
2:2$10
2:1$10
Protein-bound form of cisplatin
2:8$10
4:5$10
5:8$10
mol
17
cell$s
mol
18
cell$s
mg
min$100 mL
6cm2
VEGF
fg
h$cells
4s1
cm
8
s
cm
7
s
1
5M1s1
7cm2
cm
9
s
cm
7
s
7cm2
s
1
cm
6
s
cm
5
s
1
1
1
3h
6cm2
1
cm
8
s
cm
7
s
7cm2
70 [481]; see text
2 See text
1.6 [484]
s
94 [486]
0.2 [487]
1 [488]
0.023 See Section 10.2.3
0.28 See Section 10.2.3
0.01 [489]
12
1:9$10
s
21.2 [489]
4
6$10
[490]
See Section 10.2.3
0.044 See Section 10.2.3
14.3 See text
3
1:4$10
[450]
3 See Section 10.2.3
7.9 See Section 10.2.3
0.46 [491]
0.04 [491]
3 Varies, see text
s
77.2 See text
0.13 [492] “obtains different
form”
0.01 See text
0.16 See text
s
20.9 See text
230 Chapter 10
https://t.me/medicina_free
in an average person shortly after the administration. The maximum density of tumor
8
cells is 3$10
NC
¼ 100 cm2/cm3based on its average value for human muscle [49].
n
cells/mL [481]. The normal density of capillary surface is assumed to be
Estimating for the tumor cells proliferation rate and the rate of glucose consumption by
proliferating cells, we rely on the data obtained in experiments studying the growth of
tumor spheroids in suspension. However, we assume that these values proportionally
decrease, when the tumor grows in tissue, due to such factors as mechanical pressure and
increased acidity. The estimate of the glucose consumption rate by quiescent tumor cells is
based on the observation made in Ref. [475] that it should be more than 40 times lower than
the rate of glucose consumption by proliferating cells. The maximum rates of death of
normal and tumor cells are estimated on the basis of experimental data about the behavior
of cells under extreme nutrient deprivation. The parameters of sensitivity of cell death rate
to the concentration of glucose are chosen so that the rate of cell death becomes significant
only if glucose level decreases drastically. The migration coefficient of tumor cells
corresponds to a noninvasive tumor. We assume that 80% of cell mass is a liquid that turns
into IF when the cell dies. The minimum and initial levels of IF in the tissue are in
physiologically justified range. The parameters of microcirculatory network are estimated so
that its behavior adequately approximates the general features of structure and the dynamics
of functional tumor capillary network [339,383,493], see also Chapter 8. Diffusion
coefficients for bevacizumab and two forms of cisplatin are estimated based on already
defined values for glucose and VEGF and molecule radii of all substances, where the radius
of albumin is used for protein-bound cisplatin.
The parameter of nonspecific binding of cisplatin with tissue elements is varied in
significant range to account for patient-specific effects of the therapy. We also vary the
sensitivity of tumor cells for cisplatin during simulations to study the relative efficacy of
different schemes of drugs administration under various conditions.
10.2.3 Estimate of permeability
Increased permeability is the crucial feature of tumor pathologic microvasculature, which
determines enhanced inflow of metabolites to the tumor, thus leading to its accelerated
growth. It also results in formation of edema. Given the importance of increased
permeability for the results of simulations, we discuss in detail how we obtained
parameters of the microvasculature permeability for all the modeled substances.
The exchange of small lipid-insoluble substances between blood and tissue happens
mainly due to passive diffusion through capillary pores, which can be described by Fick’s
law. For a single pore, which is assumed to be cylindrical and perpendicular to capillary
surface, we thus can write:
Q ¼D
C
C
cap
0
A
tis
;
h
Applications in antitumor therapy 231
https://t.me/medicina_free
where Q is the inflow of substance, D0is its intrapore diffusion coefficient, C
cap
and C
tis
are the concentrations of substance in capillary blood and IF, h is the length of a pore, and
A is the part of pore surface area, available for movement of molecules across it. A
molecule experiences an effect known as the steric exclusion, which means that the
molecule cannot get closer to the pore rim than the length of its hydrodynamic radius.
Hence, it holds
2
A ¼Aða; rÞ¼pðr aÞ
;
where a is the hydrodynamic radius of a substance molecule and r is the radius of a pore.
For the study of a capillary system, it is convenient to introduce the physically measurable
parameter of its permeability P. It is defined as follows:
D0A
Sh
p
;
where A
P ¼
is the sum of pores areas available for the molecules movement across the
p
microvessel wall and S is the surface area of capillaries. This definition leads to the type
of equation, used herein to describe the inflow of all substances in tissue:
Q ¼PSðC
capCtis
Þ:
We regard preexisting microvasculature as consisting of continuous capillaries. The
transfer of blood solutes happens mainly through relatively small pores. At the same time,
capillaries formed as a result of the tumor angiogenesis possess much larger holes called
“fenestrations”, see Fig. 2.8. The diffusion of substances is restricted by pores due to
additional hydrodynamic resistance, which may be quantified using the following
empirically obtained Renkin equation [494]:
5
a
;
r
0
¼D0ðD; a; rÞ¼D$1 2:1
D
a
þ2:09
r
3
a
0:95
r
where D is the free diffusion coefficient of the substance. The original Renkin equation
also includes steric exclusion, which has already been written out in explicit form.
Although real pores are not ideal cylinders, it has been estimated that pores in skeletal and
cardiac muscle restrict diffusion to the same degree as it would have been restricted by
cylindrical pores of radius 4e5 nm [48], so we take 5 nm to be the pores radius of
preexisting capillaries NC. For the modeling purposes, we assume that during
normalization and denormalization of capillaries, i.e., transition from NC to AC and vice
versa, their surface area, width of their walls, and the number of pores remain the same,
whereas radii of all pores change identically. For the known value of permeability P
in
1
232 Chapter 10
https://t.me/medicina_free
case of free diffusion D1, pores radius r1, and molecules radius a1, this approach allows us
to obtain the value of permeability P
P2ðD2; a2; r2Þ
P
1ðD1
under parameters D2, r2, a2:
2
0
ð
D
; a2; r
D
2
2
¼
; a1; r1Þ
0
ðD1; a1; r1ÞAða1; r1Þ
D
1
2
ÞAð
Þ
; r
a
2
2
:
The permeability of continuous capillaries for glucose has been estimated in experiments
and is set to be P
¼ 1:1$105cms [495]. The hydrodynamic radius of glucose is
NC;G
0:36 nm [496], and the one of bevacizumab is estimated to be 4:58 nm [497]. Diffusion
coefficient of bevacizumab is estimated under assumption of inverse proportionality
between diffusion coefficient and molecule radius, using the relevant values for VEGF as
reference ones. This leads to the value of permeability of preexisting capillaries to
bevacizumab P
¼ 1:6$109cms. Hydrodynamic radii of VEFG and free cisplatin are
NC;A
assessed to be 3 and 0:44 nm. For this prediction, one can use the formula r ¼
0:0483$M
molecular mass is approximately 4:5$10
so the value of permeability of normal capillaries for VEGF is P
and for free cisplatin is P
0:386
[498], where M is their molecular mass, expressed is daltons. The
¼ 8$106cms. Hydrodynamic radius of protein-bound
NC;Tf
4
for VEGF [499] and 300 for free cisplatin [500],
NC;V
¼ 6:4$108cms
cisplatin is assumed to be equal to the one of albumin, which is 3:5 nm [49]; permeability
of normal capillaries for cisplatin is P
¼ 2:8$108cms. To obtain the value of
NC;Tb
permeability of angiogenic capillaries to all substances, we need to select the value of
radius of their pores. We choose it to be equal to 7:5 nm. This yields
P
P
¼ 1:2$107cms, P
AC;A
¼ 2:2$105cms, and P
AC;Tf
¼ 7:8$107cms, P
AC;V
¼ 4:5$107cms.
AC;Tb
¼ 2:8$105cms,
AC;G
10.2.4 Initial and boundary conditions, numerical scheme
The system of Eqs. (10.1)e(10.5) is solved in one-dimensional domain of 1:5cm
diameter. Plane geometry is used for computational simplicity, since it does not affect the
qualitative results compared with spherically symmetrical case. The initial and boundary
conditions for all variables are defined to be:
n1ðx; 0Þ¼max0; 0:25h1 ðx=10Þ
ðx; 0Þ¼0
n
2
hðx; 0Þ¼1 n
ðx; 0ÞIFðx; 0Þ
1
IFðx; 0Þ¼0:2
mðx; 0Þ¼0
NCðx; 0Þ¼1
ACðx ; 0Þ¼0
i
2
vn
vx
vn
vx
vh
vx
vIF
vx
vm
vx
vNC
vx
vAC
vx
1
2
ð0; tÞ¼0
ð0; tÞ¼0
ð
0; tÞ¼ 0
ð0; tÞ¼0
ð0; tÞ¼0
ð0; tÞ¼0
ð0; tÞ¼0
n
ðL; tÞ¼0
1
n
ðL; tÞ¼0
2
hðL; tÞ¼1 IF ðL; tÞ
IFðL; tÞ¼0:2
mðL; tÞ¼0
NCðL; tÞ¼1
ACðL; tÞ¼0
Applications in antitumor therapy 233
https://t.me/medicina_free
Gðx; 0Þz5:24
Vðx; 0Þ¼0
Aðx; 0Þ¼0
f
T
ðx; 0Þ¼0 vT
b
ðx; 0Þ¼0 vT
T
vðx; 0Þ¼0vð0; tÞ¼0
vG
ð0; tÞ¼0
vx
vV
ð0; tÞ¼0
vx
vA
ð0; tÞ¼0
vx
f
ð0; tÞ¼0
vx
b
ð0; tÞ¼0
vx
vG
ðL; tÞ¼0
vx
vV
ðL; tÞ¼0
vx
vA
ðL; tÞ¼0
vx
f
vT
vx
b
vT
vx
ðL; tÞ¼0
ðL; tÞ¼0
The initial conditions correspond to the normal tissue with a small colony of tumor cells
located on the left border. The initial distribution of glucose is calculated assuming the steadystate concentration in a normal tissue. The initial distribution of IF is uniform. Its value is
within physiologically justified range. IF inflow into tissue through normal capillaries is
normalized in such a way that its distribution is stationary. Inflow of IF through angiogenic
capillaries is taken to be five times greater. The values of variables of cells, necrosis, and
microvasculature network on the right boundary correspond to the normal tissue.
To speed up the calculations, the equations for VEGF and glucose are solved in the quasistationary approximation. This is justified since these reactions have much higher rates
compared with the rates for other variables. Quasi-stationary approximation leads to a
system of algebraic equations with a tridiagonal matrix. Such matrices are easy to
factorize and invert. For other variables, the splitting with respect to physical processes is
used. Kinetic equations are solved via the fourth-order RungeeKutta method, the
CrankeNicolson scheme is used for the diffusion equations, and convective equations are
solved using flux-corrected transport scheme with explicit antidiffusion stage [501].
10.3 Optimization of protocol for combined antitumor chemo- and
antiangiogenic therapy
10.3.1 Simulation using the basic model parameters
We open this section by Fig. 10.3, which shows the distribution of model variables during
CT by cisplatin combined with AAT by bevacizumab. Two schemes of drug administration
are simulated. In both of them, a dose of cisplatin is injected every 3 weeks, and six
injections are made. In scheme 1, injections of bevacizumab take place simultaneously
with the ones of cisplatin, and its administration continues after the end of CT with the
same periodicity. This scheme corresponds to the standard clinic protocol. In scheme 2,
suggested herein, AAT begins with the last injection of cisplatin and continues as one
injection in every 3 weeks. Fig. 10.3A refers to the day just before the start of therapy, and
thus, it shows the structure of untreated tumor and its microenvironment, which adequately
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