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\AtBeginDocument{{\noindent\small
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conference 10, 2003, pp. 33--53.\newline
ISSN: 1072-6691. http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\thanks{Published February 28, 2003.}
\vspace{9mm}}
\begin{document}
\setcounter{page}{33}
\title[\hfilneg EJDE/Conf/10\hfil Multistage evolutionary model]
{Multistage evolutionary model for carcinogenesis mutations}
\author[Reza Ahangar \& Xiao-Biao Lin\hfil EJDE/Conf/10\hfilneg]
{Reza Ahangar \& Xiao-Biao Lin}
\address{Reza Ahangar\hfill\break
Math Department, Kansas Wesleyan University, Salina, KS 67401, USA}
\email{rahangar@kwu.edu or rahangar@alltel.net}
\address{Xiao-Biao Lin\hfill\break
Math Department, North Carolina State University,
Raleigh, NC 27695-8205, USA}
\email{xblin@math.ncsu.edu}
\date{}
\thanks{R. Ahangar would like to thank University of Central
Arkansas, Conway, AR, for giving me \hfill\break\indent
a visiting position, where most of
this work was performed.}
\thanks{X.-B. Lin was partially supported by grant DMS-9973105
from National Science Foundation}
\subjclass[2000]{92C30, 62P10, 92D10, 92D25}
\keywords{Cancer, mutation, interaction, travelling waves,
singular perturbation.}
\begin{abstract}
We developed a mathematical model for carcinogenesis mutations
based on the reaction diffusion, logistic behavior,
and interactions between normal, benign, and premalignant mutant
cells. We adopted a deterministic view of the multistage evolution
of the mutant cell to a tumor with a fast growth rate, and its
progress to a malignant stage. In a simple case of this model,
the interaction between normal and tumor cells with one or two
stages of mutations was analyzed. The stability of the dynamical
system and the travelling wave solutions for different stages of
evolution of mutant cells were investigated. We observed the effect
of variation and natural selection in shaping the carcinogenesis
malignant mutation.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\numberwithin{equation}{section}
\section{Formulations of Carcinogenesis Mutations}
\subsection*{Introduction}
In 1971, Alfred Knudson proposed a theory that united the two forms of
retinoblastoma under genetic mutations. He explained that the two
mutations
would occur, one after another either during embryonic development or
shortly
after birth, in one of the cells of the retina. This elementary stage of
mutation could be inherited. Consequently, a rare somatic mutation is
required to trigger the explosive outgrowth of a tumor. The pair of genes
inside \textit{the thirteenth human chromosome} are deactivated
during this mutation.
Cell Kinetic Multistage (CKM) cancer risk studied by Bogen (1989) is based
on the assumption that cell proliferation follows the exponential growth
and
geometric model. In this approach, biological evidence indicates that
precancerous cells may typically proliferate geometrically. A computer
simulation of cell growth governed by stochastic processes, developed by
Conolly and Kimbell, assumes that the normal cell is growing exponentially
(Kimbell 1993). A stochastic process model for one, two, and three-stage
transformation toward malignancy was developed for embryonic and adult
mice using the Gompertzian pattern (Mao and Wheldon 1994) and another by
Portier (2000) for two-stage.
In this paper, we use the evolution principles of variation,
natural selection, and reproduction to construct our model.
In particular, we shall study how cell interaction affects and causes
changes in birth, death, and mutation rates. When selection pressure
from the surrounding environment is harsh, cells go through a
competition stage for nutrients, space and other resources causing them
to alter genetic programming in order to survive under new environmental
conditions.
Under environmental pressure, the new DNA program allows cells to
exploit biomasses, needed for their growth, from other resources in the
body. This internal DNA program change for the cell's survival is
called \textit{adaptation}. The birth of a new cell with a reprogramming
of
genetic makeup for adaptation is called a \textit{mutation}.
In the following steps, we will demonstrate factors and conditions which
will affect both normal and tumor cell growth rates either directly or
indirectly.
\noindent1. \textbf{Oxygen and nutrients through blood vessels:} Folkman
and
colleagues demonstrated that solid tumors establish their own blood
supplies
by encouraging the growth of new blood vessels into the tumor tissue to
grow beyond the small size of approximately one cubic millimeter (Folkman,
1971, Gimbron et al 1972, and Golrderg 1997). In the absence of a blood
supply, the diffusion of oxygen and nutrients across numerous cell layers
limits the tumor size.
\noindent2. \textbf{Body's Immune System: }Owen and Sherratt studied the
interactions between macrophage, tumor cells, and biochemical regulators
(Owen 1997). They showed that tumors contain a high proportion of
macrophages, a type of white blood cell which can have a variety of
effects
on the tumor, leading to a delicate balance between growth promotion and
inhibition and the ability of macrophage to kill mutant cells. Macrophages
also are able to lyse tumor cells over normal cells.
\noindent3. \textbf{Programmed Death or Cell's Birth and death process}:
An
article in the ``New York Academy of Science'' also reviewed the idea of
``programmed death'' in which the cell uses signals from neighboring cells
to either commit suicide or to stay alive (Raff and Wish 1996). ``A cell
on
the verge of becoming cancerous is surrounded by normal cells that undergo
\textit{apoptosis} when damaged. These dying cells leave some space into
which the mutated cell can grow by inducing more healthy cells to kill
themselves'' (Leffel and Brash 1996).
Normal cells are programmed to divide under certain conditions and do not
live forever because they are programmed to die. The birth and death of
cells is one of the most fascinating features of DNA programming.
The gene's program can sometimes increase the growth of normal cells
(oncogene) and sometimes limit cell growth (tumor suppressor gene).
\textit{ Proto-Oncogenes} operate like accelerated pedals in cell
proliferation, and
tumor suppressor genes work like brakes (Weinberg 1998). Researchers
believe
that the tumor - suppressor gene or P53 protein normally stops a
DNA-damaged
cell from reproducing until it has had time to make repairs. Cells that
become irreparably damaged rely on their death program for the greater
good
of the organism. If repairs are made, then P53 allows the cell cycle to
continue. But if the damages are too serious to be patched, P53 activates
other genes that cause the cell to self-destruct. For example, increasing
levels of P53 protein in the cell causes an increase in the process of
mutant cell suicide and ultimately a reduction of cancer cells. This
cellular proofreading serves to erase genetic mistakes. Undetected genetic
error leads the mutation process. The article presented by (Byrne 2001)
demonstrates a microscopic model about tumor control by protein P53.
\noindent4. \textbf{Evolution:} Evolution is another important factor in
cell proliferation and mutation. It is based on the \textit{principles of
variation and natural selection.} Variation involves changing the
conditions
of the environment and the cell's internal forces. These forces are
imposed
on cells to alter DNA programs to adopt new conditions. The mutant cell
may
survive by extracting nutrients from the surrounding environment. Thus the
interactions of mutant cells with their environment play an important role
for survival (Davis 2000).
\noindent5. \textbf{Interaction through Signals, Biochemical Regulators
and Enzymes: }
Tumor cells and tumor-associated macrophages both release factors that
can affect each other. An introduction to interactions between a
cell and its neighboring cells by receiving signals was discussed in the
July 1996 issue of ``Scientific American''.
Mutant cells with a proliferative advantage over normal tissue cells
produce
a generic chemical which regulates macrophage proliferation, influx,
activation, and complex formation (Michelson and Leith 1997, Dopazo et al
2001, and Witting 2001).
\noindent6. \textbf{Interaction between a cell and its environment based
on the principle of struggle for existence:} We will study a macroscopic
model of normal
and tumor cells in this paper, but the microscopic factors like signal
functions which affect the cell division, birth, death, and mutations
will not be considered in our model.
In this step, it is assumed that surrounding environmental conditions and
the genetic program are in favor of the mutant cell. The genetic program
in
the mutant cell makes the cell capable of fast exploitation of the space
and
biomass because of its need to survive and proliferate. The survival of
mutant cells with their fast division rate is an
indication of proper adaptation, exploitation, and availability of
nutrients and space.
The capability for a
fast-growing rate in the tissue causes changes in the densities of normal
and tumor cells at space-time $(t,x)$. Through this conflict and
competition
in using resources and space, there may be a reduction in normal cell
growth
rate and an increase in tumor cell density rate.
In the following formulation, we will consider the \textit{macroscopic
evolution of mutant cells} through the stages to the development of a
premalignant tumor. Consideration of all changes in DNA and internal
forces
on the microscopic level that affect cell birth, death, and mutations
through the signal functions are beyond the scope of this paper. The
effect
of migration of mutant cells on tumor growth is investigated by (Pettet et
al 2001).
\subsection*{Modeling and Formulation}
Assume that cell density depends on the following hypotheses
\noindent\textbf{H1: Multistage Mutations:} Mutations in DNA
cause cancer tumors to evolve through stages with the normal stage as the
initial stage and malignancy as the final stage.
\noindent\textbf{H2: Diffusion:} At every stage of mutation, the rate of
density changes by diffusion and the rate of
diffusion at every stage remains constant.
\noindent\textbf{H3: Cell Proliferation process:} Both
normal and mutant cells have a limited resource environment. Thus, both
behave alike logistically within certain parameters.
\noindent\textbf{H4: Interactions Between Stages of Mutant Cells: }In all
stages, cell density will be affected by quadratic interactions with
cells of the current, previous, and next stage. Assume that the effect of
interactions within other stages are negligible.
Let us consider Y$_{i}$ as a density of mutant cells of the i-th stage at
position $(t,x)$ where $i=0,1,2,3,\dots,n$. We accept all hypotheses, H1
through H4, to develop a multistage model for mutant cell densities
Y$_{i}$
at stage $i=0,1,2,\dots,n$, where Y$_{0}$ will be the initial stage, $%
Y_{i}(i=1,2,3,\dots,n-1)$ the density of intermediate stages, and Y$_{n}$
represents the density of the final stage. The system of
equations for the density function is
\begin{equation}
\frac{\partial Y_{i}}{\partial t}=D_{i}\Delta
Y_{i}+a_{i}Y_{i}(1-\frac{Y_{i}%
}{K_{i}})+\eta_{i}Y_{i}Y_{i-1}-\mu_{i+1}Y_{i}Y_{i+1}, \tag{1.1}
\end{equation}
for $i=1,2,\dots,n-1$.
In this formulation the constant real numbers $D_{i}$ are diffusion
factors,
$a_{i}$ and $K_{i}$ are logistic parameters, and $\eta_{i}$ and $\mu_{i}$
are interaction coefficients for the i-th stage. In every stage i, the
parameters $a_{i}$ represent the growth rate with unlimited resources. The
factor $\eta_{i}$ is the \textit{growth advantage of stage} i from the
previous stage $i-1$, and the parameter $\mu_{i}$ is the \textit{growth
advantage of the next stage} $i+1$ from the present stage $i$.
\subsection*{Latest Stage}
Note that system (1.1) does not include the normal stage,
$i=0$, and the final stage, $i=n$. The positive integer $n$
is the number of mutations leading
to the latest stage and is dictated by many factors, including nature, the
body,
and the type of cancer.
The current stage which has not reached malignancy is called the
latest stage. Suppose that the mutation is in the latest stage of
development $0\leq $ $i0,x\in\partial\Omega.
$$
The boundary conditions are to be interpreted as
\textit{no flux}
conditions. This means that the mutation is not in the metastasized stage
and there is no migration of cells across $\partial\Omega$. For all of
these
stages, $\Omega$ is considered the only habitat for cancer cells.
\subsection*{Formulation of Two-Stage Mutation}
For simplicity we consider a two-stage mutation model: benign and the
latest
stage of premalignancy. When all conditions are in favor of the mutant
cell, the latest stage may lead to malignant mutation. However, we are
interested in studying the latest stage under selection pressure.
To demonstrate the mathematical form of the selection pressure, we adopt
equation (1.4) for the latest stage. Thus the two-stage model with
premalignancy as latest stage will be in the following form
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=D_{0}\Delta u+a_{0}u(1-\frac{u}{K_{0}})-\mu
_{1}uv, \\
\frac{\partial v}{\partial t}=D_{1}\Delta v+a_{1}v(1-\frac{v}{K_{1}})+\eta
_{1}uv-\mu_{2}vw, \\
\frac{\partial w}{\partial t}=D_{2}\Delta w-a_{2}w+\eta_{2}vw. %
\end{gathered} \tag{1.7}
\end{equation}
The initial conditions are $u(0)=u_{0},v(0)=v_{0},$and $%
w(0)=w_{0}$ with no flux on the boundary.
\section{One-Stage Mutation Interacting System}
Epidemiologists succeeded in converting normal cells to cancer cells by
introducing single oncogenes into them. A single oncogene was enough to
create a malignant cancer cell in one hit (Weinberg 1996). We accept the
one-stage model for the purpose of its simplicity of evolution in
developing the cancer cell. This type of transformation of normal cell to
a full blown cancer cell is possible experimentally in the laboratory by
using chemical agents (Weinberg 1998). However, cancer formation is a
complex process involving a long sequence of steps, rather than a simple
one-hit event that converts a fully normal cell into a highly malignant
one
in a single step. When we eliminate all intermediate stages, the
mathematical form of (1.7) include the initial, the intermediate, and
the latest
stage of evolution toward cancer cells.
We will present a single hit mutation for the one-stage model. Michelson
and
Leith (1997) studied this type of interaction between tumor cells with no
diffusion factors of angiogenesis in tumor growth control. By certain
assumptions which are dependent on the nature of the tumor on one side and
the genetic program with environmental conditions on the other side, this
model will be reduced to a particular case of tumor growth for which
enormous work has been done during the past few decades in mathematical
biology. For example, in the one-stage model when there is no diffusion
factor, systems (1.3) and (1.4) will be reduced to Lotka-Volterra
equations
or to the logistic case where it will be the Pearl-Verhulst model.
\subsection*{One-Stage Carcinogenesis Mutation in an Unfavorable
Environment}
Our mathematical form should be able to explain the principles of
evolution:
variation, natural selection, and survival of the fittest. In the
following
model, we assume that mutant cells go through the selection pressure in an
unfavorable environment. The more aggressive mutant cells are able to
exploit the environment and the resources of cells of previous stages and
have a better chance to survive. Since we are eliminating the intermediate
stages, the environment and normal cells provide resources for the latest
stage.
Assume that the initial stage cells are the only resource for the growth
of
the mutant cells. Then the quadratic interaction behaves as a predator-prey
model. In this case, the body's immune system is very strong and is
harshly
active against the growth of the tumor cell, that is, the density of the
mutant cell in the absence of nutrients is negatively proportional to its
size.
When the environment is not in favor of the mutant cell, then according to
one of the principles of evolution, the mutant cell will face selection
pressure. If it cannot adapt to a new condition, it will not survive. The
mutant cells that can use the resources of normal cells have a better
chance
of survival. With these conditions imposed on the mutant cells, the model
will be
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=D_{1}\Delta u+au(1-\frac{u}{K_{1}})-r_{1}uv,\\
\frac{\partial v}{\partial t}=D_{2}\Delta v-ev+r_{2}uv,
\end{gathered} \tag{2.1}
\end{equation}
where $e,r_{1},$ and $r_{2}$ are positive constant real numbers. Dunbar
(1983) studied this model. See also Smitalova and Sujan (1991). By
rescaling (2.1) in one dimension, that is,
\begin{gather*}
U=\frac{u}{K_{1}},\quad V=\frac{r_{1}}{a},t'=at,\quad
x'=x(D_{2}/a)^{-1/2}=(\frac{a}{D_{2}})^{1/2}x \\
D=\frac{D_{1}}{D_{2}},\quad\gamma=\frac{r_{2}K_{1}}{a},\quad\delta
=\frac{ e}{r_{2}K_{1}},
\end{gather*}
we obtain
\begin{equation}
\begin{gathered}
U_{t}=DU_{xx}+U(1-U-V), \\
V_{t}=V_{xx}+\gamma(U-\delta)V.
\end{gathered} \tag{2.2}
\end{equation}
The necessary condition for the survival of mutant cells in this stage is
$ 0<\delta=\frac{e}{r_{2}K_{1}}<1$. This means that the tumor growth factor
$e $ cannot be very large, but its interaction capability against normal
cells $r_{2}$ and the carrying capacity of normal cells will help the tumor
to survive. The travelling wave exists in (2.2) when $D=\dfrac{D_{1}}{D_{2}}$
is very small and negligible. Thus,
\[
U(x,t)=w(x+ct),\quad V(x,t)=z(x+ct)
\]
where the wave with a single variable $s=x+ct$ is moving in a stationary
system with the speed $c$. Substituting in (2.2) yields $\quad$%
\begin{equation}
\begin{gathered}
cw'=Dw''+w(1-w-z), \\
cz'=z''+\gamma z(w-\delta).
\end{gathered} \tag{2.3}
\end{equation}
The second order nonlinear system can be substituted by a first order
system of ODE with respect to the variable $s$
\begin{equation}
\begin{gathered}
w'=(1/c)w(1-w-z), \\
z'=y, \\
y'=cy-\gamma z(w-\delta).
\end{gathered} \tag{2.4}
\end{equation}
For system (2.4) there are two pairs of critical points whose
connections will be the solution to the system. The non-negative solutions
satisfying the condition
\begin{equation}
\begin{gathered}
w(-\infty)=1,\quad w(\infty)=\delta, \\
z(-\infty)=0,z(\infty)=1-\delta
\end{gathered} \tag{2.5}
\end{equation}
are called type I solutions. Non-negative solutions of (2.5) satisfying
\begin{equation}
\begin{gathered}
w(-\infty)=0,\quad w(\infty)=\delta, \\
z(-\infty)=0,z(\infty)=1-\delta
\end{gathered}\tag{2.6}
\end{equation}
are called type II solutions. The following conclusion for $D=0$ can be
shown (Dunbar 1983).
\begin{theorem} \label{thm2.1}
The travelling wave front solutions $(w(s),z(s))$
of system (2.2) satisfying condition (2.6) (type II) exist if
$0x_{0}, \end{cases} \tag{5.6}
\end{equation}
where at the point $x=x_{0}$, the value of $\widetilde{V}=V(x_{0})$, and
there is a jump for the value of $U$ from $U=0$ to
$\widetilde{U}=1- \widetilde{V}$.
To study the solution of the first equation, after rescaling
\[
\xi=(x-x_{0})/\varepsilon,\tau=t/\varepsilon,
\]
the internal layer will satisfy the ``\textit{slow variable}'' system
\begin{equation}
U_{\tau}=U_{\xi\xi}+U(1-U-m\widetilde{V}). \tag{5.7}
\end{equation}
This equation has a travelling wave front solution connecting $U=0$ to
$\widetilde{U}=1-V(x_{0})$ with the same speed of the last two equations.
\section{Concluding Remarks}
The following is a brief review of our results.
\noindent\textbf{1- Variation and Natural Selection:} The tumor formed by
a one-stage mutation may approach a stable equilibrium state. By one of the
evolution principles of ``variation'', the environment, and
consequently,
the DNA algorithm will change. This equilibrium state persists until new
conditions force the mutant cells to develop an adaptation procedure to
survive. Some conditions are not favorable for mutant cells, but those
that
can adapt to the new conditions will survive. These changes are inevitable
and may cause a sequence of mutations.
If the cell is in the latest stage of its mutation, and has not reached
the final malignant stage under unfavorable conditions such as lack of
oxygen and nutrients, it is in the \textit{premalignant stage}.
\noindent\textbf{2- Mathematical Representations for Stages:} A
deterministic and macroscopic model representing the evolution of
carcinogenesis mutation using parabolic PDEs was developed.
\noindent\textbf{3- Stability and instability:} In order to study the
chance of malignant mutation, we have studied the instability of
premalignant mutations. This will lead us to understand the
chance
of another mutation toward the malignant stage. Particularly, our interest
has been to study the evolution of premalignant cells when unfavorable
conditions are imposed on the mutant \textit{cells by selection
pressures}.
\noindent\textbf{4- One-stage with Diffusion factor in Unfavorable
Conditions:} If the environment is not in favor of mutant cells, the
following system can represent the normal and tumor densities and their
interactions$\quad$%
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=D_{1}\Delta u+au(1-\frac{u}{K_{1}})-r_{1}uv,\\
\frac{\partial v}{\partial t}=D_{2}\Delta v-ev+r_{2}uv.%
\end{gathered} \tag{6.1}
\end{equation}
When $D=\frac{D_{1}}{D_{2}}$ is very small, we have the following statement.
\begin{theorem} \label{thm6.1}
The travelling wave front solutions (w(s),z(s)) of system (2.2)
satisfying condition (2.6) (type II) exists if
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\end{document}