Investigation of solid tumor progression with account of proliferation/migration dichotomy via Darwinian mathematical model

Abstract

A new continuous spatially-distributed model of solid tumor growth and progression is presented. The model explicitly accounts for mutations/epimutations of tumor cells which take place upon their division. The tumor grows in normal tissue and its progression is driven only by competition between populations of malignant cells for limited nutrient supply. Two reasons for the motion of tumor cells in space are taken into consideration, i.e., their intrinsic motility and convective fluxes, which arise due to proliferation of tumor cells. The model is applied to investigation of solid tumor progression under phenotypic alterations that inversely affect cell proliferation rate and cell motility by increasing the value of one of the parameters at the expense of another.It is demonstrated that the crucial feature that gives evolutionary advantage to a cell population is the speed of its intergrowth into surrounding normal tissue. Of note, increase in tumor intergrowth speed in not always associated with increase in motility of tumor cells. Depending on the parameters of functions, that describe phenotypic alterations, tumor cellular composition may evolve towards: (1) maximization of cell proliferation rate, (2) maximization of cell motility, (3) non-extremum values of cell proliferation rate and motility. Scenarios are found, where after initial tendency for maximization of cell proliferation rate, the direction of tumor progression sharply switches to maximization of cell motility, which is accompanied by decrease in total speed of tumor growth.

Full text

Journal of Mathematical Biology
https://doi.org/10.1007/s00285-019-01434-4
Mathematical Biology
Investigation of solid tumor progression with account
of proliferation/migration dichotomy via Darwinian
mathematical model
Maxim Kuznetsov1·Andrey Kolobov1,2
Received: 1 February 2019 / Revised: 21 June 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract
A new continuous spatially-distributed model of solid tumor growth and progression
is presented. The model explicitly accounts for mutations/epimutations of tumor cells
which take place upon their division. The tumor grows in normal tissue and its pro-
gression is driven only by competition between populations of malignant cells for
limited nutrient supply. Two reasons for the motion of tumor cells in space are taken
into consideration, i.e., their intrinsic motility and convective fluxes, which arise due
to proliferation of tumor cells. The model is applied to investigation of solid tumor
progression under phenotypic alterations that inversely affect cell proliferation rate
and cell motility by increasing the value of one of the parameters at the expense of
another.It is demonstrated that the crucial feature that gives evolutionary advantage
to a cell population is the speed of its intergrowth into surrounding normal tissue. Of
note, increase in tumor intergrowth speed in not always associated with increase in
motility of tumor cells. Depending on the parameters of functions, that describe phe-
notypic alterations, tumor cellular composition may evolve towards: (1) maximization
of cell proliferation rate, (2) maximization of cell motility, (3) non-extremum values
of cell proliferation rate and motility. Scenarios are found, where after initial tendency
for maximization of cell proliferation rate, the direction of tumor progression sharply
switches to maximization of cell motility, which is accompanied by decrease in total
speed of tumor growth.
Keywords Tumor progression ·Solid tumor growth ·Spatially distributed modeling ·
Intratumoral heterogeneity ·Tumor cell motility ·Mathematical oncology
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00285-
019-01434- 4) contains supplementary material, which is available to authorized users.
BMaxim Kuznetsov
kuznetsovmb@mail.ru
Extended author information available on the last page of the article
123

M. Kuznetsov, A. Kolobov
Mathematics Subject Classification 35Q92 ·92D15 ·92C17 ·35R09
1 Introduction
Our understanding of nature of cancer evolves with time, a lot of misleading theo-
ries having been produced throughout history (Sudhakar 2009). Although the correct
idea, that cancer arises due to genetic mutations, has been discussed already from the
beginning of the last century (Strong 1958), our knowledge about cancer is still very
far from being complete due to its incredible complexity. Nowadays, driven by the
widespread of oncological diseases, cancer research has become a very large inter-
disciplinary area. One of its relatively novel approaches is mathematical modeling of
tumor progression and tumor treatment, the popularity of which is now growing, not
least due to the rapid development of computer technology. Despite some skepticism,
expressed about mathematical modeling by oncologists, it has already proven to be
able to provide veridical insights into aspects of cancer growth (Boucher et al. 1990;
Gatenby et al. 2006) and to suggest effective optimizations of clinical protocols (Citron
et al. 2003; Chmielecki et al. 2011).
The modern concept of cancer was summarized at the end of the last century in
the famous article “The Hallmarks of Cancer” by Hanahan and Weinberg (2000). It
discusses six common capabilities, acquired by cancer cells. From the mathematical
point of view, four of the hallmarks can be combined into one concept, i.e., that cancer
cells can proliferate unlimitedly under sufficient level of nutrients (these hallmarks
are: limitless replicative potential, self-sufficiency in growth signals, insensitivity to
anti-growth signals and evading apoptosis). This was the crucial aspect already in the
first simplest models of tumor growth, which are governed by ordinary differential
equations and are still often used to fit experimental data (Skehan 1986;Hartetal.
1998). Account for the fifth hallmark—tissue invasion and metastasis—gave rise to
more complicated models, that also consider spatial distribution of cells. This is real-
ized via either continuous approach, i.e., using partial differential equations (Rockne
et al. 2009; Alfonso et al. 2016), or discrete approach, an example of which is cellu-
lar automaton (Jiao and Torquato 2011). The sixth hallmark, sustained angiogenesis,
i.e., formation of new blood vessels, was considered using different methods, includ-
ing complicated multiscale hybrid models (Macklin et al. 2009; Owen et al. 2009;
Kuznetsov et al. 2016).
The paradigm of cancer was expanded less than a decade ago by addition of four
hallmarks (Hanahan and Weinberg 2011), that further broadened the horizons for
mathematical modeling. New models of tumor growth can incorporate such features of
malignant tumors as deregulated metabolism (Kuznetsov and Kolobov 2018), evading
the immune system (Ledzewicz et al. 2012) and tumor-induced inflammation (Ander-
sen et al. 2017). However, probably the most fascinating hallmark for mathematical
modeling is the instability of cancer genome.
Nowadays it is known that each malignant tumor contains a large number of differ-
ent cell populations. This is partly due to very high rate of gene mutations in malignant
cells compared to that in normal ones (Bielas et al. 2006). Also, a comparable role in
intratumoral heterogeneity is played by epigenetic modifications, that result in herita-
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Investigation of solid tumor progression with account...
ble alterations of cells phenotypes without changing their genotypes. The most studied
and probably the most important among such modifications are alterations in DNA
methylation and various modifications of histones (i.e., the proteins, that pack and
order DNA molecules) (Esteller 2008). Via these processes genes can be silenced or
activated. Frequent genetic and epigenetic alterations result in phenotypically hetero-
geneous and constantly changing cellular composition of a tumor. The progression
of disease is therefore determined by evolution of tumor cells under the influence of
selection, one of the driving factors of which is competition for nutritional resources
(Greaves and Maley 2012). The phenomenon of progression of cancer attracts wide
attention since it results in increase of its malignancy with time. Moreover, a great
challenge for cancer treatment is posed by the ability of cancer to acquire resistance to
drugs by propagation of this trait inside a tumor due to drug-induced selective pressure
(Gottesman 2002).
In mathematical modeling different approaches exist for consideration of intratu-
moral phenotypic heterogeneity and tumor progression. In continuous models, the
majority of which are not spatially distributed and thus are governed by ordinary dif-
ferential equations, the simplest approach is inclusion of the probability of transition
of a cancer cell upon its division into another predefined state, denoted by additional
variable, to which altered characteristics correspond. This approach is useful for con-
sideration of competition of small amount of phenotypically different populations,
the features of which differ drastically. The most popular feature, considered by this
method, is acquired drug resistance. Despite its simplicity, this approach allows to
address some of the major problems, frequently met in this kind of investigations,
such as estimating incidence of resistance to a drug under various parameters (Iwasa
et al. 2006) and proposing optimizations for scheduling of chemotherapy (Hadjian-
dreou and Mitsis 2014). In order to account for slow quantitative change of properties,
more complicated methods exist. One of them is to introduce a new variable for every
population which arises due to each mutation/epimutation, which happen upon division
of cells with some probability, and then to obtain new parameters for new variable via
some random alterations of the mother cell’s parameters (Stiehl et al. 2016). Another
method regards only one variable for description of tumor cells, but considers it as a
function of one or more specially introduced phenotypic parameters, which influence
characteristics of cells. Genome/epigenome instability can be expressed via diffu-
sion of cells in the space of phenotypic parameters, which implies an unphysiological
assumption that mutations can occur at any moment of cell cycle rather than only upon
DNA replication (Chisholm et al. 2015; Lorenzi et al. 2016). This can be overcome
by using integro-differential equation with integral kernel defining the probabilities of
various phenotypic shifts upon cell divisions (Lorz et al. 2013). Account for spatial
distribution significantly expands the spectrum of problems, that can be considered.
However, it is rarely met in continuous models, probably due to accompanying signif-
icant increase in computational cost. The existing works of this kind do not consider
mutations/epimutations of cells, multiple populations with various phenotypes being
already present from the beginning of simulations (Lorz et al. 2015; Lorenzi et al.
2018). However, spatial distribution is generally considered in the models, which sim-
ulate dynamics of tumor cells via discrete methods (Anderson et al. 2006; Gerlee and
Anderson 2008; Bouchnita et al. 2017). Such approach is very useful when dealing
123

M. Kuznetsov, A. Kolobov
with small amount of tumor cells, nevertheless, its common disadvantage is, as well,
great computational expense at consideration of later stages of tumor growth.
In this paper we present a new continuous spatially-distributed model of solid
tumor progression, comprised of a system of integro-partial differential equations,
with explicit consideration of mutations/epimutations of tumor cells. It is applied
to investigation of solid tumor progression with account of proliferation/migration
dichotomy, which represents the development of our previous work (Kolobov et al.
2000). Therein, in accordance with experimental studies by Theodorescu et al. (1991),
it has been demonstrated, that tumor can progress towards increase in cell motility
along with decrease in cell proliferation rate. This result is of merely qualitative nature,
and herein we investigate this phenomenon quantitatively. The prominent feature of
the presented model is the account for two reasons of cell motion, i.e., their intrinsic
motility and convective fluxes, which arise due to proliferation of tumor cells in dense
tissue.
2 Darwinian model of solid tumor progression
The model presented here consists of two separate blocks. The first block pro-
vides the mathematical description of changes in tumor cell composition due to
genetic/epigenetic mutations. It is based on the above-discussed use of integro-
differential equation accounting for phenotypic shifts upon divisions of cells, which
is further explained in detail. This block is integrated into the second one, which
represents a spatially-distributed model of growth of solid tumor in normal tissue.
2.1 Consideration of phenotypic instability
Let abe the general phenotypic parameter, which governs a set of characteristics
of malignant cells, that can be expressed numerically, including cell proliferation
rate B(a).Letn(a,t)denote the population density of tumor cells, so that a1
a0n(a,t)da
is the number of cells that possess values of the phenotypic parameter lying between a0
and a1at the moment of time t. We consider ato be scalar for simplicity, since
this is suitable for the purposes of this work. However, the following approach can
be straightforwardly expanded to the case of vector phenotypic parameter. Also, for
convenience, herein a∈[0,1].
We assume that, in a process of division of a cell with certain phenotypic parameter
value A, daughter cells can obtain altered values of it. The probability of obtaining
specific value of ais governed by a predefined function f(a−A), which depends on the
difference between two parameter values. It is reasonable to assume, that this function
has a global maximum at a−A=0, so that daughter cells most frequently receive the
values of the phenotypic parameter, close to A. The following function NewA(a,t)
describes the growth rate of tumor cells population density n(a,t)due only to the
emergence of offsprings of cells possessing phenotypic parameter value A,atthe
moment t:
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Investigation of solid tumor progression with account...
NewA(a,t)=2B(A)n(A,t)·f(a−A).
So far we assume, that cells of each population (i.e., which possess the same value
of phenotypic parameter) divide at constant rate, their proliferation being not limited
by nutrient depletion, overcrowding effect or any other reason. The factor 2 appears
in the formula due to the fact that upon a single cell division two new cells emerge,
phenotypes of which can be altered.
Figure 1illustrates the considered approach in a discrete manner, where Nis the
number of grid points in the permissible range of values for a. Thus, the following
description “converges” to themeaning of presented equations as N→∞. Solid line
depicts the graph NewA(a,t), which represents the distribution of new descendants
of population n(A,t), born over an infinitesimal period of time. The areas under this
curve, painted in different shades of gray, indicate the amounts of new cells of different
populations, that have appeared due to the proliferation of population n(A,t). Dashed
lines represent the distributions of daughter cells of other populations with a= A,
which may be transformed into one another, and into solid line, by shift along X-axis
and either stretch or compression along Y-axis, depending on the ratio of products of
cell proliferation rate and number of cells of two populations. In order to count, how
many new cells, which possess the value of a phenotypic trait A, emerge during this
time period, one needs to sum up the amounts of daughter cells of each population a,
that have undergone phenotypic shifts a−A. They are depicted by overlapping red-
gray areas under the curves in the central grid region. In a strict mathematical form,
this means the following, where all functions are assumed to be integrable:
∀a∈(0,1)New(a,t)
≡1
0
NewA(a,t)dA=1
0
2B(A)n(A,t)·f(a−A)dA.(1)
This concludes the essential part of the supposed approach. In this work, we use
normal distribution for the function, defining the probability of phenotypic shifts,
f(a−A)=1
σ√2πexp −[a−A]2
2σ2,(2)
and we apply the following boundary conditions, which mean that the cells, that are to
leave the permissible range of phenotypic parameter values, take its boundary values:
New(0,t)=
1

0
2B(a)n(a,t)·f(−a)da
+
1

0
2B(a)n(a,t)·⎡
⎣
−a

−∞
f(z)dz⎤
⎦da,
123

M. Kuznetsov, A. Kolobov
2B(A)n(A,t)·f(1/N)
A-3/N A-2/N A-1/N A A+1/N A+2/N A+3/N
N
e
w
A
(a
,
t
)=
2
B
(
A
)
n
(A,
t
)·f
(
a
-
A
)
d
i
s
t
r
i
b
u
t
i
o
n
o
f
n
e
w
c
e
ll
s
a
m
o
u
n
t
o
f
n
e
w
c
e
l
l
s
~ B(a)n(a)
2B(A)n(A,t)·f(0)
values of phenotypic parameter a
Fig. 1 Illustration to the modeling approach for consideration of phenotypic instability of tumor cells. Here
a∈[0,1]is the general phenotypic parameter, its specific values are denoted below the x-axis, where Nis
the number of grid points; NewA(a,t)is the function of growth rate of tumor cells population density n(a,t)
due only to the emergence of new cells, mother cells of which possessed phenotypic parameter value A;
f(a−A)is the function defining the probability of obtaining specific value of aby these new cells; B(a)is
cell proliferation rate. See the text for more explanations
New(1,t)=
1

0
2B(a)n(a,t)·f(1−a)da
+
1

0
2B(a)n(a,t)·⎡
⎣
∞

1−a
f(z)dz⎤
⎦da.(3)
If we had considered a non spatially-distributed case and had ignored cell death,
then the formulation of the problem would be completed by posing initial conditions,
defining the specific function for proliferation rates of different populations B(a)
and considering the following equation for overall change in tumor cells population
density, which accounts for emergence of new daughter cells and for “elimination” of
mother cells upon their division:
∀a∂n(a,t)
∂t=New(a,t)−B(a)n(a,t).
2.2 Tumor growth in tissue
Figure 2shows the block-scheme of interactions between variables in the model of
solid tumor growth in tissue. There are five variables in the model: density of tumor
123

Investigation of solid tumor progression with account...
n(a,r,t) – tumor cells
m(r,t) – necrosis
h(r,t) – normal cells
c(r,t) – capillaries
g(r,t) – glucose
m
cn h
g
Fig. 2 Block-scheme of the model of tumor growth in tissue. White arrows indicate transitions between
variables, green arrows denote stimulating effects that increase the value of a corresponding variable or the
intensity of a corresponding transition, red lines designate inhibitory interactions (color figure online)
cells n(a,r,t), density of normal cells h(r,t), fraction of necrosis m(r,t), concen-
tration of glucose g(r,t)and density of surface area of capillaries c(r,t). Space and
time coordinates rand tare further omitted for notational brevity.
The following set of equations governs the dynamics of tumor cells, normal cells
and necrosis:
Tumor cells: ∂n(a)
∂t=
proliferation with redistribution
 
[New(a)−B(a)n(a)]g
g+g∗
death
 
−Mn(g)n(a)
migration
 
+D(a)Δn(a)
convection
 
−∇(In(a));
Normal cells: ∂h
∂t=
death
 
−Mh(g)h
convection
 
−∇(Ih);
Necrosis: ∂m
∂t=
cell death
 
1

0
Mn(g)n(a)da+Mh(g)h
convection
 
−∇(Im);
where:
Mi(g)=1+tanh([gd−g])
2Mmax
i,i=n,h;
1

0
n(a)da+m+h=1.
(4)
Herein two characteristics of cells are considered to vary between populations,
i.e., their proliferation rate B(a)and intrinsic motility D(a), the choice for which is
explained further in Sect. 3.1.
The term for proliferation of tumor cells takes into account the distribution of new-
born cells over phenotypes, controlled by Eqs. (1)–(3). Rate of tumor cells proliferation
falls under depletion of glucose. The form of the multiplier, which describes this effect,
implies, that proliferation rate of cells is proportional to their uptake of glucose, the
term for which will be discussed further. This assumption is due to the fact that glu-
cose is indispensable substrate for biosynthesis of several types of molecules, without
123

M. Kuznetsov, A. Kolobov
which cell division is impossible (Patra and Hay 2014). Since this multiplier depends
only on glucose level, it can be taken out of parentheses, leaving the function New(a)
unaltered.
Under a significant drop in glucose level, tumor cells, as well as normal ones, die
turning into necrosis. For description of cell death, a sigmoidal function Mi(g)is
used, which implies that cells begin to actively die under a critical level of glucose
in tissue. Tumor cells are known to obtain energy under lack of glucose via multiple
ways: catabolization of glutamine (Phan et al. 2014); utilization of lactate, generated
by other cells during aerobic glycolysis (Sonveaux et al. 2008); and many more,
including individual oxidative pathways for specific cell lines (Moreno-Sánchez et al.
2007). Therefore, the fact that in the model rate of cell death depends on the availability
of only one nutrient is an unavoidable simplification, justified by the fact that levels
of other nutrients would also drastically decrease along with that of glucose.
Tumor cells possess intrinsic motility and are able to migrate throughout the tissue,
which is described via diffusion-like term. We neglect the dependence of motility
of tumor cells on glucose uptake. Invasion of tumor cells into surrounding tissue is
one of the drivers of tumor growth which in existing mathematical models is often
considered as the only one (see for example Swanson et al. 2000; Alfonso et al.
2016). Such models are suitable for consideration of highly invasive tumors, like
glioblastoma. Herein we also consider another reason for tumor growth, i.e., pushing
surrounding tissues away by proliferating tumor cells. This type of growth is crucial to
low-grade compact tumors. During the tumor progression, its growth pattern generally
changes from compact to more invasive, therefore, the growth of real tumor often
represents a combination of these two types. Obviously, it is impossible to model the
growth of a non-invasive tumor via classical reaction–diffusion equations. Moreover,
neglect of convective transport will lead to underestimation of tumor growth speed.
Thus, simultaneous consideration of both types of solid tumor growth provides a
more physiologically correct modeling approach. This approach is realized herein by
inclusion of convective terms, which describe bulk motion of tissue elements.
The velocity field Iis determined by dynamics of tumor cells in tissue. To make its
derivation possible, it is necessary to introduce some additional constitutive assump-
tion. Herein it is the constancy of total density of cells and necrosis, which is normalized
to unity for convenience. This is expressed via the last equation in set of Eq. (4), where
for simplicity it is implied that the volumes of a normal cell and a tumor cell, as well
as of the amount of necrosis, formed in result of their death, are equal, while the volu-
metric fraction of capillary network in tissue in negligible. From the biophysical point
of view this assumption describes dense incompressible tissue, the former character-
istic meaning that total density of cells with necrosis cannot decrease, and the latter
that they cannot increase in any point of space in any moment of time. Equation for
velocity field Ican be obtained by summing up the equations for tumor cells, normal
cells and necrosis. During this procedure, left sides of equations produce the deriva-
tive 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:
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Investigation of solid tumor progression with account...
∇I=
cell proliferation
 
1

0
B(a)n(a)da·g
g+g∗
cell migration
 
+
1

0
D(a)Δn(a)da.
The final expression for Iwill have different forms for different space dimensions
and boundary conditions.
Of note, this approach for consideration of compact tumor growth represents a
special case of a more general method, utilized, e.g., by Byrne and Drasdo (2009).
That method converges to the one presented herein under the assumption that values
of solid pressure, arising due to cell dynamics, are infinitesimal, so they do not hinder
cell proliferation.
The equations for dynamics of glucose and capillaries are as follows:
Glucose: ∂g
∂t=
inflow
 
Pc[gbl −g]
consumption
 
−[Qn
1

0
n(a)da+Qhh]g
g+g∗
diffusion
 
+DgΔg.
Capillaries: ∂c
∂t=
degradation
 
−L[
1

0
n(a)da+m]c
convection
 
−∇(γ Ic);
(5)
Dynamics of glucose is comprised from its inflow from capillaries into the tissue,
consumption by cells and diffusion throughout the tissue. The inflow of glucose is
governed by the process of passive diffusion through the walls of capillaries (Levick
2013). Therefore, the rate of glucose inflow is proportional to the density of capillaries’
surface area and to the difference in glucose concentrations in blood and in tissue.
Direct measurements demonstrate small arterial-venous blood glucose difference not
only in normal tissues, but even across tumors (see for example Eymontt et al. 1965),
so glucose concentration in blood is taken to be constant in the model.
Consumption of glucose by cells is described in traditional way, i.e., via equations
of Michaelis–Menten type. Tumor cells consume glucose much more actively than
normal ones, i.e., Qn>> Qh, since glucose is a crucial energy nutrient for tumor
cells and a major element for their proliferation (Vander Heiden et al. 2009). The rate
of glucose consumption does not vary for different populations of tumor cells, which
is discussed further in Sect. 3.1.
Capillary network degrades inside the tumor—it happens due to mechanical reasons
(Araujo and McElwain 2004), as well as due to various chemical factors (Holash
et al. 1999). Capillary degradation is not considered here in detail, and this process
is described by a rather phenomenological term. Like cells, capillaries move with the
convective fluxes, the speed of their motion being lower than that of cells due to their
connection with each other, i.e., γ<1. Other model parameters are provided and
justified in Sect. 3.2.
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M. Kuznetsov, A. Kolobov
3 Investigation of tumor progression
3.1 Formulation of the task
The list of key properties that foster the prevalence of a cell population in the com-
position of tumor, obviously, includes its high proliferation rate—since it directly
governs the number of cells of this population—and its high intrinsic motility—since
this feature allows cells to leave nutrient-deficient regions inside main tumor mass and
promotes the spread of these cells into the surrounding tissues. As a rule, the degree
of tumor malignancy correlates with these indicators, since under the development of
a tumor, on a large time scale, more and more quickly dividing and rapidly migrating
cells are selected (see for example Louis et al. 2007).
However, for each individual tumor cell, its proliferation rate and motility are
inversely linked, i.e., the more a cell divides, the less it migrates, and vice versa
(Giese et al. 1996). This effect is, in particular, due to the fact that these processes
depend on two parallel metabolic pathways, and each of them relies on glucose: pen-
tose phosphate pathway generates necessary precursors for the synthesis of amino
acids, nucleotides and fatty acids; while glycolysis is the primary pathway for produc-
ing energy, utilized for motion of cancer cells (Shiraishi et al. 2015). It has been shown
by Kathagen-Buhmann et al. (2016) that inhibition of one of the enzymes, related to
either of these two processes, slows down the corresponding process and speeds up
another one (or in some cases at least does not affect the speed of another process
on a detectable level). This suggests that phenotypic alterations, which affect levels
of expression of corresponding enzymes, would lead to analogous effect. Thus, they
would have an ambiguous influence on the evolutionary adaptability of tumor cell
populations by enhancing one of the two crucial properties, that contribute to their
propagation in tumor composition, at the expense of another one.
In light of the above, herein we consider isolated phenotypic alterations of expres-
sion levels of virtual enzymes of either pentose phosphate pathway or glycolityc
pathway. A considered enzyme expression level in each case is denoted by pheno-
typic parameter aand is normalized in such a way, that a=0 refers to one of its
extremum values, corresponding to maximization of usage of glycolysis by a cell, and
thus to minimum cell proliferation rate and maximum motility that can be achieved
by variation of expression level of this enzyme. Vice versa, a=1 refers to maximum
proliferation rate and minimum motility of cells, while with the increase of aB(a)
monotonically increases and D(a)monotonically decreases. Of note, we assume, that
phenotypic alterations only redistribute the flow of glucose between the metabolic
pathways, not affecting the overall glucose consumption rate by cells. Alterations of
an enzyme expression are considered to be governed by normal distribution (see Eq. 2).
As it follows from the results of above mentioned experiments by Kathagen-Buhmann
et al. (2016), identical adjustments of expression levels of various enzymes lead to
quantitatively different changes in cell proliferation rate and motility, so we consider
various possible functions B(a), D(a)of the following general form:
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Investigation of solid tumor progression with account...
B(a)=B0+[B1−B0]·apB,
D(a)=D0+[D1−D0]·apD,
where Bmin ≤B0<B1≤Bmax,
Dmin ≤D1<D0≤Dmax ,
0<pi≤pmax,i=B,D,
(6)
where parameters B0,B1,pB,D0,D1,pDare varied from case to case.
The set of Eqs. (1)–(6) is the complete set of equations for the given study. Its main
question is what the general rules of tumor progression are under such formulation.
3.2 Parameters
The parameters of the model are assessed according to the data of various exper-
iments, if it is possible, and otherwise are estimated in order to reproduce known
characteristics of tumor growth. The basic set of parameters is given in Table 1, where
the following normalization parameters are used to obtain their dimensionless values:
tn=1 h for time, Ln=10−2cm for length, Sn=1 mg/ml for glucose concentration.
Normal capillary surface area density is equal to 100 cm2/cm3, which is its average
value for human muscle (Levick 2013). The value of glucose consumption rate for
human muscle (at rest) is also used in the model. Maximum density of tumor cells
is 3 ×108cells/ml, this value is taken from experimental work on growth of multicel-
lular tumor spheroids by Freyer and Sutherland (1985). Values for proliferation rate
of tumor cells and their glucose consumption rate are also estimated according to data
of this work. Coefficients of tumor cells motility correspond to non-invasive tumors,
being two to three orders of magnitude lower than the values for highly motile glioma
cells, estimated by Swanson et al. (2000). Maximum value of exponents for B(a)
and D(a)is adjusted in order to avoid sharp changes of the value of one parameter
under minor changes of the value of another one during phenotypic shift. Scale param-
eter for phenotypic instability is selected for the direction of tumor progression to be
apparent for almost all considered cases after already 2 to 300 days of tumor growth.
Maximum death rate of normal cells is much higher than that of tumor cells, since
the latter are known to be much less sensitive to metabolic stress (Izuishi et al. 2000).
Rather high value for capillaries degradation rate is dictated by results of the work on
high-resolution imaging of tumor microvasculature by Stamatelos et al. (2014), which
demonstrates that capillaries with adequate blood filling are already very scarce inside
the core of a tumor with the radius as small as 4 mm.
3.3 Numerical solving
For numerical solving the tumor was regarded as consisting of no more than
N+1=101 distinct cell populations n[A], where A=0,1/N,2/N,...,[N−
1]/N,1, within each of which malignant cells are phenotypically identical. Before
the main computations a set of coefficients Fi,i=−N,−N+1,...,−1,0,1,...,
N−1,Nwas introduced, which determine, what fraction of daughter cells of pop-
123

M. Kuznetsov, A. Kolobov
Table 1 Model parameters
Parameter Description Value Dimensionless value Estimations based on
Phenotypic instability of tumor cells
Bmin Minimum proliferation rate 0.01 h−10.01 Freyer and Sutherland (1985)+seetext
Bmax Maximum proliferation rate 0.03 h−10.03 Freyer and Sutherland (1985)+seetext
Dmin Minimum intrinsic motility 2.4×10−6cm2/day 0.001 Swanson et al. (2000)+seetext
Dmax Maximum intrinsic motility 2.4×10−5cm2/day 0.01 Swanson et al. (2000)+seetext
pmax Maximum exponent for B(a)and D(a)2.5 2.5 see text
σScale parameter for phenotypic instability 0.015 0.015 See text
Cell death
Mmax
nMaximum tumor cells death rate 0.002 h−10.002 See text
Mmax
hMaximum normal cells death rate 0.02 h−10.02 See text
Sensitivity to glucose level 5 5 See text
gdCritical level of glucose 0.56 mM 0.1 See text
Capillaries
LDegradation rate 4 ×10−11 ml/cells s 0.012 Stamatelos et al. (2014)+seetext
γNetwork elasticity 0.5 0.5 See Sect. 2.2
123

Investigation of solid tumor progression with account...
Table 1 continued
Parameter Description Value Dimensionless value Estimations based on
Glucose
PPermeability of capillaries 1.1×10−5cm/s 4 Levick (2013)
gbl Blood level 5.56 mM 1 Association (2004)
QnTumor cells consumption rate 1.4×10−16 mol/cell s 27 Freyer and Sutherland (1985)+seetext
QhNormal cells consumption rate 0.46 mg/min 100 ml 0.27 Baker and Mottram (1973)
g∗Michaelis constant for consumption rate 0.04 mM 0.007 Casciari et al. (1992)
DgDiffusion coefficient 2.6×10−6cm 2/s 94 Tuchin et al. (2001)
123

M. Kuznetsov, A. Kolobov
ulation A, that have emerged during a time step, belong to population A+i/N.
These coefficients were calculated as a set of definite integrals of function f(a−A),
which governs the probabilities of phenotypic shifts, over the equal intervals of 1/N
(see Fig. 1), i.e.,:
F0=−er f −0.5
√2σN,
F−N=FN=1
21+er f −1
√2σ,
∀i∈[1,N−1]Fi=1
2er f i+0.5
2√2σN−er f i−0.5
√2σN,
∀i∈[−N+1,−1]Fi=F−i.
The set of equations Eqs. (4)–(5), which governs the tumor growth, was solved in
one-dimensional region with a size of several centimeters. The exact size was selected
for each case in order to be sufficiently small to shorten computational time almost
as much as possible without imposing edge effects. Plane geometry was used for
computational simplicity, since it does not affect the results compared to spherically-
symmetrical case. Initial conditions corresponded to normal tissue with h(x,0)=1,
c(x,0)=1, m(x,0)=0, with a small, 0.1 mm in width, colony of tumor cells of
population with phenotypic parameter a=0.5 situated near the left boundary, where
thus n[0.5](x,0)=0.5, h(x,0)=0.5, c(x,0)=1. Other tumor cells were absent
in the beginning of a simulation: n[i](x,0)=0,i= 0.5. The initial distribution of
glucose g(x,0)was uniform and was calculated as its steady-state concentration in
normal tissue. For all variables zero-flux boundary conditions were set on the left
boundary; for glucose this condition was also set on the right boundary, while values
of other variables on the right boundary were constant, corresponding to normal tissue.
The convective flow speed was set to zero on the left boundary, and free boundary
condition was used for it on the right boundary. These conditions imply that the center
of the tumor is situated on the left boundary, and normal tissue, surrounding the
tumor, moves with constant speed outwards of the computational space through the
right boundary. Our model simulations may thus roughly represent, e.g., growth of
a cancer lump into a cavity, as it may happen in the case of leiomyoma. Boundary
conditions for convective flow speed result in the following equation for it:
I(x,t)=x
0
⎡
⎣
1

0
B(a)n(a,x,t)da·g(x,t)
g(x,t)+g∗⎤
⎦dr
+
1

0
D(a)Δn(a,x,t)da.(7)
Equation for glucose was considered in the quasi-stationary approximation due to
its fast dynamics and was solved using the tridiagonal matrix algorithm. For other
123

Investigation of solid tumor progression with account...
variables, the method of splitting into physical processes was used, i.e., kinetic equa-
tions, diffusion equations and convective equations were solved successively during
each time step. Kinetic equations were solved via second-order Runge–Kutta method,
Crank–Nicolson scheme was used for diffusion equations, and convective equations
were solved using the flux-corrected transport algorithm with explicit anti-diffusion
stage. The last method is introduced in the work by Boris and Book (1973), while
other classical methods are described in many books (see for example Press 2007).
The computational code was implemented in C++ and was run with spatial and tem-
poral discretisations sufficiently fine to not distort the solution.
3.4 Results
3.4.1 Different patterns of tumor progression
For the present study, a hundred sets of values of parameters B0,B1,D0,D1,pB,pD,
which define dependencies of cell proliferation rate and cell motility on phenotypic
parameter, B(a)and D(a), were generated in a random way, so that they satisfy
the conditions of Eq. (6). Then a hundred simulations of 500 days of tumor growth
were performed using these functions. For four of these sets, parametric plots, which
illustrate dependencies between Band D,areshowninFig.3a. For these cases,
Fig. 3b demonstrates the change in time of the average value of phenotypic parameter
throughout all the tumor cells, a, which is calculated as follows:
a(t)=
L

0
1

0
a·n(a,x,t)dadx/
L

0
1

0
n(a,x,t)dadx
As Fig. 3b shows, there is no single direction of tumor progression in terms of
maximizing either proliferation rate or motility of cells. In case 1, from the beginning
of tumor growth, more and more rapidly proliferating populations displace the others
until aapproaches the value, close to unity, overcoming of which is restrained due to
boundary conditions (see Eq. (3)). In case 2, the composition of tumor changes so that
more and more motile populations succeed. Case 3 demonstrates slow change of a,
which resides near its initial value. In this case, distribution of number of tumor cells
over phenotypic parameter n(a,t)=L
0n(a,x,t)dxremainstobesinglemode(see
inlet). Thus, the cells, which possess maximum of either proliferation rate or motility,
turn to be less fit in this case, than the cells with phenotypic value, close to 0.5. In
case 4, the direction of tumor progression, which initially inclines towards increase
in cell proliferation rate, changes after 187 days, eventually leading to domination of
more motile cells. Unlike the previous case, here during the switch in direction of
progression, the distribution n(a)is bimodal. It has maxima at two boundary values
of a, so that the value of aduring this switch is determined by the ratio of numbers of
cells, which possess values of phenotypic parameter, close to the boundary ones.
123

M. Kuznetsov, A. Kolobov
0
0.25
0.5
0.75
1
B,x10
-2
D,x10-3
0 100 200 300 400 500246810
1
1.5
2
2.5
3
(a) (b)
1
2
4
3
days
a
1
2
4
3
a=1
a=0.5
a=0
х
х
х
proliferation rate
increases
motility
increases
a
relative number
of cells
01
a
relative number
of cells
01
Fig. 3 aDependencies between cell proliferation rate Band intrinsic cell motility Dduring phenotypic
shift in four considered cases, where the following functions are used: (1) B(a)=0.014 +0.016 ·a2,
D(a)=0.006 −0.002 ·a1.3;(2)B(a)=0.015 +0.003 ·a2.1,D(a)=0.008 −0.005 ·a0.6;
(3) B(a)=0.015 +0.013 ·a1.5,D(a)=0.009 −0.007 ·a1.8;(4)B(a)=0.012 +0.013 ·a2.3,
D(a)=0.009 −0.006 ·a0.4.bChange of average phenotypic parameter over time during simula-
tion of tumor growth for four cases, shown in (a). Crosses denote the moments, for which distribution of
model variables for case 4 are shown in Fig. 4
3.4.2 Visualization of tumor growth
Figure 4provides snapshots of spatial distributions of model variables for case 4.
In the beginning of the simulation small amount of tumor cells actively proliferate
under abundance of glucose near the left boundary, which represents the center of a
tumor. Since during this period spatial effects play insignificant role, tumor cells, that
proliferate faster, always hold evolutionary advantage in the very beginning of tumor
growth. Increase in number of tumor cells results in significant local degradation of
capillaries, followed by decrease in glucose inflow. Deficiency of glucose leads to
formation of necrosis in tumor center, while proliferating cells concentrate on the
outer region, which possesses sufficient amount of glucose for cell division. Prolif-
erating cells push away the cells, that are located further from the tumor center, that
drives tumor front propagation. Moreover, tumor cells, that migrate towards region
with normal microvasculature density, obtain more glucose, which stimulates their
proliferation and further migration away from tumor center. That represents another
reason for propagation of tumor front. We will refer to the first process as “convective
growth” and to the second process as “intergrowth”.
After the initial period of tumor growth, spatial effects become determinative for
tumor progression. Limited nutrient supply poses selective pressure on tumor cells and
phenotype selection is driven by competition for this supply. Cells that are able to reach
glucose-abundant region, proliferate, while cells in the tumor core die out. As it was
noticed above, this competition may result in different patterns of tumor progression,
and its direction may even change during tumor growth, like in case 4. Figure 4b refers
123

Investigation of solid tumor progression with account...
26 31
30292827
1.0
0.2
0.0
0.4
0.6
0.8
distance to tumor center, mm
day 380
capillaries, c
glucose, g
tumor cells
with necrosis, n+m
1.0
0.2
0.0
0.4
0.6
0.8
3021
distance to tumor center, mm
day 6
1.0
0.2
0.0
0.4
0.6
0.8
10 15
14131211
tumor cells, n
distance to tumor center, mm
day 180
(a) (b)
(c)
Fig. 4 Profiles of tumor cells density n, necrosis fraction m, microcirculatory network surface area density c
and glucose concentration gunder B(a)=0.012 +0.013 ·a2.3,D(a)=0.009 −0.006 ·a0.4(case 4 on
Fig. 3)ona6-th day of tumor growth, b180-th day of tumor growth, c380-th day of tumor growth
to the 180-th day of tumor growth for this case, which is at this moment accompanied
by dominance of more rapidly proliferating tumor cells. Figure 4c demonstrates the
380-th day of simulation, until which more motile cells have already displaced rapidly
proliferating ones in tumor composition, which resulted in reduction in maximum
density of tumor cells, more gentle slope of tumor front and slightly deeper penetration
of tumor cells in adjacent normal tissue.
3.4.3 Reasons for evolutionary advantage of tumor cells
The provided description of tumor growth dynamics allows to suggest that cells, which
are able to migrate away from the main tumor mass and reach nutrient-abundant region,
should be evolutionary advantageous. Therefore, it is the speed of their intergrowth
into normal tissue, that should play crucial role for the evolutionary fitness of a cell
population—since in every point in space convective fluxes shift all the cells with the
same speed, while the speed of intergrowth may vary for different populations. The
intergrowth speed should be an increasing function of both cell proliferation rate Band
motility D. This idea is favored by the fact, that for the well-known Fisher’s equation,
n
t=Bn(1−n)+Dn
xx, which can be regarded as a simple model of tumor growth via
123

M. Kuznetsov, A. Kolobov
only the process of intergrowth, the formula for the wave speed is V≥2√BD (Fisher
1937). Derivation of a similar analytical formula for the set of equations, considered
herein, is a hardly feasible task due to its complexity, i.e., inclusion of convective
terms, interplay between tumor cell populations and consideration of another variables.
However, intergrowth speeds of cell populations can be estimated numerically via the
following approach.
Figure 5a demonstrates the map of tumor front propagation speeds for monoclonal
tumors, the cells of which possess values of Band Din the ranges Bmi n ≤B≤Bmax,
0≤D≤Dmax. The total speed of tumor front can be interpreted as the sum of two
speeds, related separately to convective growth and intergrowth. This is an approxi-
mation, valid only for small enough values of cell motility. Generally, these processes
are not independent, due to the fact that the process of intergrowth redistributes pro-
liferating cells and affects convective fluxes. We define convective speed for a single
cell population as the speed of propagation of a monoclonal tumor with the same
cell proliferation rate and zero cell motility. Further, we estimate intergrowth speed
of a cell population as the total speed of such monoclonal tumor front propagation
minus its convective speed. The map of intergrowth speeds, obtained via this method
for monoclonal tumors is provided in Fig. 5b, where the plots B(D)are repeated for
four considered cases. Note that intergrowth speed on average makes up almost 50%
of total speed of tumor growth within considered range of parameters. Obviously,
in a polyclonal tumor intergrowth speed of a single population may be affected by
simultaneous dynamics of other populations (this effect will be demonstrated further
in Sect. 3.4.4). Nevertheless, intergrowth speed, assessed via the described approach,
serves as a good indicator for direction of tumor progression.
Figure 5c provides the duplication of graphs of average phenotypic parameter by
time for considered cases of tumor progression, along with the heat maps of dependen-
cies of intergrowth speeds of corresponding populations on the value of phenotypic
parameter a. This figure demonstrates, that for cases 1, 2 and 4, in which intergrowth
speed reaches its maximum at one of the boundary values of a, tumor indeed progresses
towards maximization of its overall intergrowth rate, reaching the near-extreme values
of a, overcoming of which is restricted due to boundary conditions. However, in case 3
the value of anoticeably exceeds the value of a, which corresponds to the population,
possessing maximum intergrowth speed.
The qualitative behaviour of aover time for all other simulations in each case
corresponds to one of the four presented cases. The figures, analogous to Fig. 5b, c
for all the cases are provided as Supplementary files. Among 100 cases, in 17 of them
tumor progresses towards maximization of cell proliferation rate, like in case 1; in 51
cases tumor progresses towards maximization of cell motility, among which in 45
cases this direction of evolution is chosen from the early stage of tumor growth, like
in case 2, and in 6 cases the direction of evolution undergoes a distinguishable switch,
like in case 4; and finally, in 32 cases tumor progresses towards non-extremum values
of cell proliferation rate and motility, like in case 3. In all cases of third type the value
of average phenotypic parameter also exceeds the value, which corresponds to the
population with the maximum intergrowth speed.
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Investigation of solid tumor progression with account...
B,x10-2
D,x10-3
1
1.5
2
2.5
3
B,x10-2
D,x10-3
1
1.5
2
2.5
3
(b)
mm/week
0
1
2
4
3
a=1
a=0.5
a=0
0
06810024246810
100 200 300 400 500
0
0.25
0.5
0.75
1
(c) days
a
1
2
4
3
0.321
0.323
0.346
0.380
0.407
Total speed Intergrowth speed
0.404
0.332
0.293
0.258
0.216
0.435
0.450
0.452
0.400
0.181
0.373
0.283
0.269
0.268
0.263
0.455
0
0.25
0.5
0.75
1
a
134
2
Intergrowth speed
proliferation rate
increases
motility
increases
0.2
0.4
0.6
0.8
1.0
1.2
1.4
(a)
Fig. 5 aSpeed of monoclonal tumor growth in dependence of values of its cells’ proliferation rate B
and intrinsic motility D.bEstimated speed of intergrowth of tumor population in surrounding tissue in
dependence of values of Band D. Lines 1–4 designate the considered cases of behaviour of functions B(a)
and D(a)during phenotypic shift, and match the lines, presented in Fig. 3a. cGraph of average phenotypic
parameter over time for four considered cases, which matches the graph presented in Fig. 3b, along with
the heat maps of intergrowth speed of populations for these cases. Bold labels denote maximum values of
intergrowth speed for each case
3.4.4 Under close intergrowth speeds, populations with higher proliferation rate
prevail in tumor composition
The fact that cell populations with higher proliferation rate may prevail in tumor
composition despite lower intergrowth speed, can be explained via consideration of
tumor cell dynamics in case 3. After initial period of tumor growth, a whole spectrum
of cell populations with noticeably different proliferation rates and motilities form a
complex phenotypic structure, illustrated in Fig. 6, where local average phenotypic
value, ˆa, is calculated in the following way:
123

M. Kuznetsov, A. Kolobov
capillaries, c
glucose, g
tumor cells
with necrosis, n
+m
tumor cells, n
0.4
0.3
0.2
0.1
53.0 53.4 53.8 54.2
distance to tumor center, mm
0.418
0.437
0.456
0.475
0.494
0.380
0.399
a
Fig. 6 Distribution of populations along with profiles of model variables at the tumor rim on the 500-th day
of tumor growth under B(a)=0.015 +0.013 ·a1.5,D(a)=0.009 −0.007 ·a1.8(case 3 in Figs. 3,5).
Variation of average phenotypic parameter in space is denoted with different shades of blue (color figure
online)
ˆa(x,t)=
1

0
a·n(a,x,t)da/
1

0
n(a,x,t)da.
The qualitative appearance of this structure persists during tumor growth, accom-
panied by only a very slight increase in aat least until the end of simulation. Such
kind of structure is maintained due to complex interplay between populations of tumor
cells. Distribution of cells with a=0.455, which possess higher nominal intergrowth
speed, has its maximum at the distance of ≈54 mm from the tumor center. There
they actively divide, producing cells of populations with both equal, lower and higher
values of a. Cells with its lower values have higher motility and lower proliferation
rate. However, some of these cells, which due to the process of random migration
have moved several millimeters forward, obtain more glucose and therefore are able
to proliferate faster, that effectively increases their intergrowth speed and stabilizes
their position in front of cells with a=0.455. Nevertheless, the majority of emerging
new cells with a= 0.455, with lower nominal intergrowth speeds, move away from
the outer part of tumor. Since the cells, that possess higher value of a, proliferate faster
under the same levels of glucose, the value of ˆaincreases towards the tumor center.
Thus, the majority of tumor cells correspond to values of a>0.455.
3.4.5 Speed of tumor growth may fall in result of competition between cells
Cases 3 and 4 demonstrate, that tumor progression does not always result in max-
imization of total tumor growth speed. In case 3 cell population with a≈0.7 has
maximum total speed, while tumor progresses towards lower average phenotypic
parameter a≈0.5. In case 4 cell population with a=1 has maximum total speed,
and though in the beginning of tumor progression ainclines towards it, eventually it
switches towards the lowest values of a.
The dynamics of tumor composition for case 4 in shown in Fig. 7, where the average
value of phenotypic parameter ˆais marked for the points, where tumor cell density
123

Investigation of solid tumor progression with account...
g=g*=0.007
(proliferation rate falls two times)
400
0
100
200
300
025
20
0151
530
days
distance to tumor center, mm
0.0
0.2
0.4
0.6
0.8
a
n=0.025
n=0.25
for glucose:
Isolines
for tumor cells:
posion that tumor front would have
without fall in speed of its propagaon
Fig. 7 Distribution of populations during first 400 days of tumor growth under B(a)=0.012 +0.013 ·a2.3,
D(a)=0.009 −0.006 ·a0.4(case 4 on Figs. 3,5). Variation of average phenotypic parameter in space and
time is denoted with different shades of blue. Isolines for indicated values of tumor cell density and glucose
concentration are shown (color figure online)
exceeds 10−6. The graph is arranged so that horizontal slices provide a distribution
of average local phenotypes within the tumor at certain moments of time. As Fig. 5c
indicates, for this case the variation of intergrowth speed of cell populations is rather
moderate in the range from a≈0.4toa=1, which initially drives progression of
tumor towards increase in cell proliferation rate. However, as the average phenotypic
parameter within the main tumor mass increases, a small region at the tumor rim
becomes occupied by cells, which have low values of a. Such cells emerge rarely at
the beginning of tumor growth, however, since their populations have significantly
higher intergrowth speed, they occupy more and more space in glucose-rich tumor
rim, and eventually cut off the nutrient supply for the cells, which have high values
of a, further replacing them within the main tumor mass. Of major interest is the fact
that the speed of tumor front propagation decreases during this process by ≈15%, i.e.,
from 0.62 mm/week at the 180-th day to 0.53 mm/week at the 380-th day. It happens
due to the fact, that remaining tumor cell populations possess lower proliferation rate
(which in this case differs about twice-fold under a=0 and a=1), which results
is significantly lower convective flow speed. Such decline in tumor front propaga-
tion is manifested in 5 out of 6 performed simulations where the direction of tumor
progression undergoes analogous distinguishable switch.
4 Discussion
The results of investigation of solid tumor progression under isolated phenotypic alter-
ations, that inversely affect proliferation rate and motility of cells, were presented.
123

M. Kuznetsov, A. Kolobov
Tumor progression was considered to be driven only by competition between popula-
tions of malignant cells for limited nutrient supply. Two types of tumor growth were
considered simultaneously, i.e., convective growth and intergrowth into surrounding
normal tissue. It was shown, that the crucial feature that renders a cell population evo-
lutionary advantageous is its intergrowth speed. Four qualitatively different patterns
of tumor progression were found, which show that the increase in intergrowth speed
of a tumor, which always accompanies its progression, is not always associated with
increase in motility of its cells. Scenarios were found, in which tumor progression is
accompanied by decrease in total speed of tumor growth.
As we have stated in Sect. 3.4.3, two considered types of growth are not indepen-
dent, since under high motility of cells the process of their intergrowth into surrounding
tissue redistributes the profile of proliferating cells and affects convective fluxes. An
investigation of the interaction of two types of tumor growth is a fascinating task.
However, consideration of growth of highly diffusive tumors using the presented sub-
model of tumor growth under the same basic values of parameters would lead to
unphysiologically explosive tumor growth. Such limitation is associated with a more
general problem of modeling microvasculature network via spatially-distributed vari-
ables, that is unavoidably phenomenological. In reality a small but sufficient local
aggregation of tumor cells will already lead to their competition for nutrients, since
the deeper situated cells will lack them. On the contrary, in a continuous model the
cells are free to grow until rather high numbers of them, because all the variables in
every computational grid are effectively mixed. Moreover, in reality degradation of a
capillary leads to the cessation of downstream blood flow and thus effective shutdown
of the whole branch of capillaries. In case of highly invasive tumors this effect would
be of great importance, however, it is impossible to straightforwardly incorporate it
into the utilized continuous approach.
Explicit consideration of mutations/epimutations of tumor cells upon their divi-
sion results in a system of integro-partial differential equations, numerical solution of
which is rather computationally expensive. In the area of modeling of tumor progres-
sion, there exist at least two more simple approaches, already mentioned in Sect. 1,
which might be used for the considered task, reducing its numerical cost. One of them
is consideration of heterogeneous tumor composition at initial moment and neglect
of mutations/epimutations of cells. However, preliminary simulations utilizing this
approach showed that it is not suitable for the considered task, since it produced qual-
itatively different results depending on the initial phenotypic composition of tumor.
Another method is consideration of diffusion of cells in the space of phenotypic param-
eters, which accounts for cell cycle-independent phenotypic alterations. Preliminary
simulations with this method showed good qualitative correlation with the results
presented herein. Nevertheless, such correlation may fail in cases when significant
alterations in cell proliferation rate accompany tumor growth. An example of such
situation is consideration of antiangiogenic therapy, which reduces the inflow of nutri-
ents to the tumor and thus affects proliferation rate and influences the frequency of
phenotypic shifts. This led to the choice of the utilized approach for the given study,
which may serve as an approbation for its further usage.
It should be noted, that it is an open question, whether some heritable epigenetic
modifications can happen independently of cell division, certain data suggesting this
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Investigation of solid tumor progression with account...
possibility. For example, methylation of DNA is known to be controlled by a family
of enzymes, a part of which is aimed to maintain DNA methylation pattern upon its
replication—thus, epimutations, produced by its errors, certainly happen upon cell
division; however, a part of enzymes perform de novo methylation, and the details
of this process are still not elucidated completely (Robertson 2001). Nevertheless,
certain experimental findings by Velicescu et al. (2002) suggest that cell division is
nevertheless required for de novo methylation, which also speaks in favor of the chosen
approach.
The presented mathematical model is, to the best of our knowledge, the first continu-
ous spatially-distributed model of solid tumor progression, which explicitly considers
mutations/epimutations of tumor cells. It has been used herein for solution of only
one particular problem, and it can serve as a framework for the following studies
focusing on different phenomena of cancer progression. There are a lot of ways to
upgrade the model for such tasks, moreover, the choice of a character of phenotypic
changes, governed by alternating phenotypic parameter, is limited only by the imag-
ination of the researcher. One valuable option is introducing the dependency of the
scale parameter for phenotypic instability σon the phenotypic parameter a, which
accounts for the fact that probability of a mutation upon a single replication of a cell
increases with increase in its malignancy. The submodel of tumor growth in tissue
can be also upgraded, and more complicated versions of it have already been used
in our previous papers (Kolobov and Kuznetsov 2015; Kuznetsov and Kolobov 2017,
2018; Kuznetsov et al. 2018). One of the intriguing options, that we are planning to
implement, is modeling of chemotherapy, that will pose additional selective pressure
on tumor cells.
Acknowledgements The reported study was funded by RFBR according to the research Projects Nos.
16-01-00709, 17-01-00070 and 19-01-00768. Numerical simulations have been prepared with the support
of the “RUDN University Program 5-100”.
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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps
and institutional affiliations.
Affiliations
Maxim Kuznetsov1·Andrey Kolobov1,2
1Division of Theoretical Physics, P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, 53 Leninskii Prospekt, Moscow, Russia 119991
2Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow,
Russia 117198
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