DNA AND CELL BIOLOGYVolume 23, Number 10, 2004© Mary Ann Liebert, Inc.Pp. 625–633

Biodynamic Modeling and Simulation of Multistage Cell Mutations

REZA AHANGAR,1 NAWAB ALI,2 KAMRAN IQBAL,3 and KUMUD ALTMAYER4

ABSTRACT

The aim of this study is to present a mathematical computer simulation model for multistage carcinogenesis.The population genetic model is developed based on the reaction diffusion, logistic behavior, and HollingsType II interactions between normal, benign, and premalignant cells. The simple form of the Fisher-Haldane-Wright equation of the genetic model of tumor suppressor gene and oncogenes is used to describe this typeof interaction. Through computer simulation, we observe the behavior, stability, and traveling wave solutionof the premalignant stage mutation as well as its survival under natural selection pressure. As a simple caseof this model, the interaction between normal and tumor cells with one or two stages of mutations is analyzed.

625

INTRODUCTION

THE INITIAL BREAKTHROUGHS in genetic study came with ob-servations made by Darwin and Mendel in the late 19th cen-

tury. The mathematical theory of genetic population was devel-oped by Fisher (1958). The law of genetic equilibrium wasdiscovered independently—and almost simultaneously—by theGerman physician Weinberg and the British mathematicianHardy in the early 20th century. The interactions of mutant cellswith their environment play an important role for survival (Davis,2000). Tumor cells and tumor associate macrophages both re-lease factors which can affect the activity of each other (Sher-man and Portier, 1996). The interaction between neighboringcells by receiving signals was discussed in an article by Leffeland Brash (1996). Mutant cells with a proliferative advantageover normal tissue cells, genetically produce chemicals, whichregulate macrophage proliferation, influx, activation, and com-plex formation (see Witting, 2000; Depazo et al., 2001). Normalcells can be transformed into cancer cells through natural stages,or experimentally, in the lab using chemical agents (Bogen,1989). The effect of migration of mutant cells on tumor growthhas been investigated by Pettet et al. (2001).

Cell proliferation follows the genetic system of haploidmodel organisms rather than diploid models. The interior of cy-toplasmic materials contains biological machinery needed forcell cycle and reproduction. In particular, the chromosome car-

ries a specific nucleotide sequence in a single linear strand ofDNA, the gene, which dictates the synthesis of proteins, whichin turn, determine the characteristics of an organism and cellfunctions (Hoppenstead and Peskin, 2002; Britton, 2003).Genes may also have regulatory binding sites to which proteinsinteract. DNA and protein interaction may bring about geneticchanges that may affect cell population.

We assume that surrounding environmental conditions andthe genetic program are in favor of the mutant cell. The geneticprogram in the mutant cell makes the cell capable of fast ex-ploitation of the space and biomass because of its need to sur-vive and proliferate. The survival of mutant cells with their fastdivision rate is an indication of proper adaptation through ex-ploitation of nutrients and space. This capability for a fast-grow-ing rate in the tissue causes changes in the densities of normaland tumor cells at space-time (x,t). As a result of this conflictand the competition for resources and space, there may be a re-duction in growth rate of normal cells and an increase in tumorcell density rate (Portier and Smith, 2000).

In this paper, we consider the macroscopic evolution of mu-tant cells through the stages leading to the development of thepremalignant tumor. The selective advantages of either the tu-mor suppressor gene or oncogene can be described through theeffect of gene frequency on cell density. We first present theformulation of the problem and discuss our assumptions. Later,we use the mathematical model to study the stability and trav-

1Department of Mathematics, Kansas Wesleyan University, Salina, Kansas.2Graduate Institute of Technology, and 3Department of System Engineering, University of Arkansas at Little Rock, Little Rock, Arkansas.4Department of Mathematics, University of Arkansas at Pine Bluff, Pine Bluff, Arkansas.

eling wave solution of the premalignant stage through compu-tational and numerical methods. Additionally, we considerchanges in DNA and internal forces on the microscopic level,for example, changes in DNA–protein interactions in formu-lating our assumptions.

The organization of this paper is as follows: we first presentthe methodology of a mathematical model of gene–protein in-teraction and cell population growth. We then consider the evo-lution of mutant cells through multiple stages leading to the pre-malignant stage. As special cases of the general model weanalyze one and two stage mutations and study the behavior ofthe resulting system of partial differential equations. We thenuse singular perturbation methods to study the stability and ex-istence of traveling wave solutions. We recognize the need toperform biological experiments to validate the mathematicalmodels; however, the experimental validation part is not un-dertaken at this stage.

MATERIALS AND METHODS

Gene–protein interaction

To describe the gene–protein interactions (Fig. 1), we as-sume that genes are divided into two groups based on the his-tory hereditary trait as well as the effect of governing condi-tions: (1) genes that are prone to mutational change due tovariation in the environment are grouped as G1; (2) genes thatare resistant to the changes of conditions, or are satiated by con-suming the environmental agents E, are grouped as G2.

In a certain time interval �t, the rate of susceptible genes tochanges in their interactions to proteins is represented by a con-stant real number s. There are a certain number u of genes inthe pool that are not susceptible to receive protein. These genesmay have been interacting earlier and now are saturated duringthe time interval �t. The above descriptions can be mathemat-ically formulated as follows: (1) gene pool partitioned in twoalleles with the total numbers G � G1 � G2; (2) total numberof susceptible genes that received protein (plasmids) �s � N � G1 � �t; (3) total number of changes occurred in thegroup G2 � uG2 � �t. Thus, s � N � G1 � �t � u � G2 � �t, u �G2 � s � N � G1 � s � N � (G � G2), and

G2 � .

We similarly assume that the total changes occurring in proteinpopulation is �N � s � N � G1 � �T. Therefore,

� s � N � G1 � s � N � �G � �,s � N � G�u � s � N

�N��t

N � G��us

� � N

and finally

N�(t) � ,

where u is the maximum extraction rate and �us� is the half sat-

uration or resource density at which the extraction rate reachesits maximum.

Gene–protein interactions affecting cell density

To find the effect of gene density on cell population, wepostulate that the rate of changes in cell population is pro-portional to the genes density. According to the gene–proteininteraction (Fig. 1), the gene growth rate is proportional tothe protein concentration. When the protein level is high, thegrowth rate is approximately constant. The reason is that theproteins undergo both chemical and biological process to beabsorbed in DNA.

In the DNA molecule there are regulatory sites where pro-tein molecules bind to regulate the gene expression. When theprotein concentration [E] is low, there is an abundance of bind-ing sites such that the rate of interaction is proportional to theamount of protein. When the protein concentration is high, thebinding sites in DNA are working at maximum capacity, so in-creasing the protein molecules has no additional effect. Becausethe initial velocity (rate) is proportional to [GE], the velocitywill approach a maximum velocity vmax as the protein becomestotally saturated. As a result

� � ⇒ v � vmax (1)

The above relation shows the rate at which the enzymes orproteins are reacting by their interaction with the cells DNA.We thus denote the velocity as v � g[G].

Cell population growth rate affected by gene frequency

To determine how the variation in gene frequency affects thecell population growth rate, assume that the growth rate of celldensity Y is proportional to the cell density function g([G]),where [G] is the gene frequency, therefore

� � � Y � g([G]) (2)

If the gene frequency is a function of time t, relation (2) willassume the following discrete form, that is

� � � Yn � g([Gn]) (3)dY(tn�1)��

dt

dY(t)�

dt

[G]�km � [G]

[G]�km � [G]

[GE]�[Et]

v�vmax

u � N � G�

�us

� � N

AHANGAR ET AL.626

FIG. 1. Gene–protein interactions and changes in a short time interval.

Effect of gene frequency on the cell density in future generations

We would now like to determine how gene frequency af-fects the cell density in future generations, particularly whenone gene has a selective advantage. Let us denote a tumorsuppressor gene and oncogenes by T and S, respectively. De-fine pn and qn to be frequencies of two genes T and S in thenth generation. Assume that one gene has a selective advan-tage � against another gene. This situation is modeled as thefollows:

� . (4)

which represents the relative gene density in the nth genera-tion. When � � 1, there will not be a selection advantage, the relation (4) will produce the Hardy-Weinberg law, that is[Gn�1] � [Gn]. We now introduce

[Gn] �

to represent the relative gene frequency, then

[Gn�1] � (5)

where g([Gn]) � Gn�1 represents a transformation functionfrom generation n to generation (n � 1). We also assume therate of changes in cell density Y(t) is proportional to these ge-netic advantages of T and S. Combining relations (3) and (5),denoting the ratio

� �

which is constant for nth generation, such that

� � (6)

This is the Michaelis-Menton equation relating cell density tothe relative gene frequency. According to this relation, the con-stant real number � represents the maximum extraction rate ofthe resources and the constant � represents half saturation re-source density.

Alternate forms of gene interaction

The Hardy-Weinberg assumptions imply that each geno-type for tumor suppressor gene {T} and oncogene {O} thatis {TT,TO,OO} is equally fit. The expected number of genesan individual contributes to the gene pool of the next gener-ation is independent of genotype. The gene pool can be de-scribed from one generation to the next by the sequence (pn}where pn denotes the proportion of the tumor suppressor genepool of type T immediately before the nth reproduction. Theproportion of the cell generation of gene type T will be pn

2 �pn(1 � pn). Assume genotypes have unequal chances of sur-vivals of � for TT, � for TO, and � for OO. Therefore, thetotal gene pool at the next reproduction time will be propor-tional to wn � �pn

2 � 2�pn(1 � pn) � �(1 � pn)2 and thoseof type T will be proportional to �pn

2 � �pn(1 � pn). Thus,the gene pool proportion of type {T} for the next productionwill be

Y � [Gn]�� � [Gn]

dY�dt

��qn

[Gn]�[Gn] � �

1�qn

pn�qn

pn�qn

1�pn � �qn

pn�1�qn�1

pn�1 � .

This is the famous Fisher-Haldane-Wright (FHW) equation ofmathematical population genetics that can be expressed aspn�1 � pn � g(pn), where g(pn) can be calculated from theabove expression.

Modeling hypotheses

The following hypothesis are used to formulate the mathe-matical model.

H1: multistage mutations. Mutations in DNA cause cancertumors to evolve through stages with the normal stage as ini-tial stage and malignancy the final stage.

H2: diffusion. At every stage of mutation the density ratechanges by diffusion. We also assume that the rate of diffusionat every stage remains constant.

H3: cell’s proliferation process. Assume that both normaland mutant cells have a limited resource environment. Thus,both behave alike logistically within certain parameters.

H4: interactions between cells. In stage i, the cell’s densityis affected by Holling’s type II functional responses with cellsof this stage, previous stage (i � 1), and next stage (i � 1). As-sume that the effect of interactions with any other stages is neg-ligible.

Evolution of the mutant cells

Let Yi represent the density of mutant cells of the ith stageat position (t,x), let Y0 be the density of the initial stage, Yi(i �1,2,3, . . . , n � 1) the density of intermediate stages, and Yn

the density of the final stage; then, the system of partial dif-ferential equations (PDEs) for the density function is given as(see A10 in the Appendix)

� Di�Yi � aiYi �1 � � � �i

� �i�1

(7)

In the above formulation, the constant real numbers Di denotediffusion factors, ai and Ki are logistic parameters, �i and �i

are interaction coefficients, and factors ei and ci (i � 0,1,2, . . . , n � 1) are the satiation factors. In every stage i, the pa-rameters, ai represent the growth rate with unlimited resources.The factor �i is the growth advantage of stage i from the pre-vious stage i � 1, and the parameter �i is the growth advan-tage of the next stage i � 1 from the present stage i. The ini-tial values for the system of PDEs are given as Yi(0) � yi 0 fori � 0,1,2, . . . , n � 1.

The latest stage of mutation

Equation (7) represents the latest stage mutation for k � n.the mutation is in the latest stage of development i � k, then�k�1 � 0 The total number of stages of mutations, which variesfrom place to place, depends on the conditions and type of can-

YiYi�1�1 � ciYi

YiYi�1��1 � eiYi�1

Yi�Ki

�Yi��t

�pn2 � �pn(1 � pn)

���wn

MATHEMATICAL MODELING OF MULTISTAGE CARCINOGENESIS 627

cer. For example, in certain kinds of cancer like melanoma, thetotal number of mutations is n � 5. If the tumor is in the pro-cess of the third stage, then k � 3 is the latest stage. At thistime the undeveloped stage coefficients are considered to bezero, that is �k�1 � �4 � 0. The latest stage cell, in an unfa-vorable environment where surrounding cells are the only avail-able resources, is termed as the premalignant stage. Equation(7) for this stage is defined as

� Dk�Yk � bYk � �k (8)

One- and two-stage mutation models

By Darwin’s Evolution Principle, unfavorable conditionscaused by variation and natural selection may lead the systemto instability. The struggle for existence through this instabil-ity by the individual cell may cause adaptation, and conse-quently, mutation. We are interested in studying one-stage mu-tation model (i.e., when i � 0,1) as well as the two-stage model(when i � 0,1,2), as special cases of the general model (7). As-suming that u represents the density of resources in a closedenvironment consumed by mutant cells v, then the diffusionsystem interaction model for one-stage mutation is given byequations (A5) and (A6) in the Appendix. Similarly, the two-stage mutation model in unfavorable conditions is given byequations (A8) and (A9) in the Appendix.

Premalignant stage

By premalignant mutation, we mean a mutant cell having noresources, other than resources of the surrounding cells or per-vious stage, before metastasizing. Mathematically, the speciesis experiencing the predation mode. This stage may occur nat-urally, after many stages of mutation, or experimentally in thelab by injection of premalignant cells in a normal cell envi-ronment. We use a singular perturbation method for the twostage mutation model in (A9) to study the behavior and stabil-ity of the premalignant stage. The resulting traveling wave so-lution is given by (U(s),V(s),W(s)) where the wave speed is de-pendent on the shape of initial distribution. For example, if weassume that � � 1 and m is very small, then the intersection oftwo traveling waves produces the desired traveling wave solu-tion for nonzero values of m. If we assume � � 0 and m � 1,then the two stage model (A9) simplifies to the single stagemodel (A5) with a similar traveling wave front-type solution.

RESULTS

The aim of this study was to develop a computational modelthat, upon simulation, could predict the changes in cell popu-lation undergoing multistage carcinogenesis. A number of as-sumptions described in the Materials and Methods section wereused to develop mathematical derivations. These assumptionsare based on the established facts and the knowledge ofDNA–protein interactions, and other environmental conditionsthat either favor or prevent mutagenesis during cellular prolif-eration from one stage to the next.

To simulate the numerical solutions to the system of equa-tions for one- or two-stage mutations, we used default parame-

YkYk�1��1 � ekYk�1

Yk�t

ters related to growth, birth, death, mutation, and interactionrates from Kimbell and Conolly (1994). The algorithm can alsobe downloaded from the Web site: www.math.pitt.edu/pub/bardware. The computational simulation of the singularly per-turbed system shows the existence of a traveling wave-type so-lution for one-dimensional tumor growth.

Figure 2 describes the simulation results of the travelingwave solutions U(t, x) of the normal cell density versus time[equation (A12)] when m � 0 and 0 � � 1. This system rep-resents a tumor growth where the premalignant mutation oc-curs within the benign tumor and is initially isolated from hav-ing interaction with normal cells. Figure 3 describes the solutionof (A10) over parameter 0 � m � 1. Figure 4 is a simulationof the normal cell density in the system (A10) for m � 0. Fig-ure 5 is the phase portrait of the solutions V(t, x) and W(t, x)for m � 0 and the diffusion factor D ranging from 0 to 3. Fig-ures 6 through 8 are depictions of the solution to the system ofequations for U(t, x),V(t, x), and W(t, x). These are solutions ofthe normal, benign, and premalignant cell densities, respec-tively, satisfying system (A10) when m � 0, which is equiva-lent to system (A11). Figure 9 demonstrates simulation of pre-malignant growth W(t, x) with respect to normal cell growthover the parameter 0 � � � 1. Figure 10 is simulation of thesolution V(t, x) and W(t, x) satisfying the system (A10) whenm � 0 and D changes from 0 to 3.

DISCUSSION

The present study describes the development of a computa-tional model for multistage carcinogenesis. The model predictsgenetic evolution of benign and premalignant cells from a pop-ulation of normal cell density. This model is based on our hy-potheses, and assumptions that, during cell growth and differ-entiation, both cellular and environmental factors inducemutations in a normal cell population that may lead to genetic

AHANGAR ET AL.628

FIG. 2. Simulation of normal cell density U(t, x) over the pa-rameter epsilon changing between 0 and 1, when the interac-tion factor m � 0.

changes, and therefore, phenotypes. We also incorporated inour systems of equations the assumptions to study the behav-ior of premalignant mutant cells under environmental selectionpressure without considering direct genetic factors for survival.The key factors for evolution of the normal cell to mutant cellare presumed to be a quick variation of the environment, se-lection pressure, and slow adaptation. For example, mathemat-ical and numerical models of tumor angiogenesis were built byByrne and Chaplain (1995) and Goldberg and Rosen (1997).The selective advantage of mutant p53 tumor cells in these mod-els was also studied by Byrne and Gammack (2001).

The present study builds on to earlier results of Ahangar andLin (2003) on multistage carcinogenesis. The results of thisstudy indicate that the cell mutation problem is mathematicallyand computationally tractable. Several cases of one-and two-stage mutation were analyzed and presented in the figures. Al-though biological experiments were not performed to test thevalidity of the assumed parameters, the systems developed werecomputer simulated to obtain predicted solutions under variableconditions.

APPENDIX: MATHEMATICAL FORMULATIONOF ONE- AND TWO-STAGE MODELS OF

CARCINOGENESIS

Let Yi represent the density of mutant cells of the ith stageat position (t, x), such that Y0 is the density of the initial stage,Yi(i � 1,2,3, . . . , n � 1) the density of intermediate stages, andYn the density of the final stage; then, the system of partial dif-ferential equations (PDEs) for the density function is given as

� Di�Yi � aiYi�1 � � � �ig(Yi,Yi�1)

� �i�1G(Yi,Yi�1)(A1)

In the above formulation, the constant real numbers Di denote dif-fusion factors, ai and Ki are logistic parameters, �i and �i are in-

Yi�Ki

Yi�t

teraction coefficients, and g is the hyperbolic interaction function.In every stage i, the parameters, ai represent the growth rate withunlimited resources. The factor �i is the growth advantage of stagei from the previous stage i � 1, and the parameter �i is the growthadvantage of the next stage i � 1 from the present stage i.

The interaction functions

The Holling’s type II interactions of the ith stage mutantcells (density Yi) with the previous stage (density Yi�1), and theinteraction functions with the next stage (density Yi�1) respec-tively, are defined by

g(Yi,Yi�1) � and

g(Yi,Yi�1) � (A2)

where ei and ci (i � 0,1,2, . . . , n � 1) are the satiation factors.The interaction function introduced in this formulation will bereduced to a quadratic form if constant coefficients ei or ci aresignificantly small. In this case, g(Yi,Yi�1) � YiYi�1 org(Yi,Yi�1) � YiYi�1 for i � 0,1,2, . . . , n � 1. With this defi-nition the system of PDE in (A1) becomes

� Di�Yi � aiYi�1 � � � �i

� �i�1 (A3)

The initial and boundary conditions for the system of PDEs aregiven as: Yi(0,x) � Yi0(x),(t, x) � I � � for � � R3 and no flux

on the boundary, i.e., � 0,(i � 1,2, . . . , n),t 0, x � �.

ONE-STAGE MUTATION MODEL

For a single stage mutation, let u represents the density ofresources in a closed environment consumed by mutant cells v,then the diffusion system for one stage mutation is given as:

Yi�n

YiYi�1�1 � ciYi

YiYi�1��1 � eiYi�1

Yi�Ki

Yi�t

YiYi�1�1 � ciYi

YiYi�1��1 � eiYi�1

MATHEMATICAL MODELING OF MULTISTAGE CARCINOGENESIS 629

FIG. 3. Simulation of the densities of normal cells, U(t, x)over the parameter m where 0 � m � 1.

02 4 6 8 10 12 14 16 18 20

0.02

0.015

0.01

0.005

FIG. 4. Simulation of U(t, x) densities of normal cells form � 0 over the parameter epsilon.

AHANGAR ET AL.630

� d1�u � Au�1 � � � B

� d2�v � Cv � D

(A4)

where A, B, C, D and di (i � 1,2) are positive constants. Tosolve this system we introduce new variables as follows: Let

U � Eu,W � , t� � Ct,x� � � �1/2,

� � A/ECK1, � � , � � EK,

and substitute into (A2), to get

Ut � �Uxx � �U (� � U) �

Wt � Wxx � W � �

(A5)

where

� � ,

and for simplicity, we have dropped all symbols of primes. Wenow consider that d1d2, (or � � 0), then the system (A5) ischanged to the nondiffusive ODE system given as

U� � �U(� � U) � and W� � �W � � (A6)

where � 0 and � 1 are fixed real numbers and � �/(� � 1) is a parameter. This system of equations has beenstudied rigorously by many authors, for example, Yanagida

UW�1 � U

UW�1 � U

d1�d2

UW�1 � U

UW�1 � U

D�CE

C�d2

Bv�C

uv�1 � Eu

v�t

uv�1 � Eu

u�k1

u�t

(1994), Owen Sherratt (1997), and particularly, Steve Dun-bar (1986).

TWO-STAGE MUTATION MODEL

Let u, v, and w represent the densities of normal, benign, andmalignant cells, respectively, and assuming two-stage mutationin unfavorable conditions, we get the following system of PDEs

� D0�u � a0u�1 � � � �1

� D1�v � a1v�1 � �� �1 � �2 (A7)

� D2�w � a2w � �2

To rescale system (2.1) we initiate the following steps:

(i) Change the variables: U � e1u,V � e2v,W � �2w/a2, suchthat (u,v,w) � (U/e1,V/e2,a2W/�2), (ut,vt,wt) � (Ut/e1,Vt /e2,a2Wt /�2), and (uxx,vxx,wxx) � (Uxx /e1,Vxx /e2,a2Wxx /�2)

(ii) Change the coordinate system

(t,x) � (,), � a2t, � �a2/D2x,�such that

(Ut,Vt,Wt) � a2(U,V,W), and (Uxx,Vxx,Wxx)

� � �(U,V,W�).

(iii) Redefine the coefficients

� �, � m, 1K0 � �1,e2K1 � �2

� d1, � d2, � a, � ba1�

a2e2K1

a0�a2e1K0

D1�D2

D0�D2

�2�e2a2

�1�e1a2

�1�e2a2

a2�D2

vw�1 � e2v

w�t

vw�1�e2v

uv�1�e1u

v�Ki

v�t

uv�1 � e1u

u�K0

u�t

FIG. 5. Simulation of the densities of V—benign and W—premalignant cells when m � 0 and D changes from 0 to 3.

0.04

0.05

0.3

0.18 0.2

0.7

FIG. 6. Phase portrait of the solution U, V, and W over theparameter epsilon, changes between 0 and 1.

⎧⎪⎨⎪⎩

⎧⎪⎨⎪⎩

⎧⎪⎪⎨⎪⎪⎩

⎧⎪⎨⎪⎩

MATHEMATICAL MODELING OF MULTISTAGE CARCINOGENESIS 631

(iv) Finally, write the transformed equations in the originalcoordinate system (t,x) to get

Ut � d1Uxx � aU(�1�U) � �

Vt � d2Vxx � bV(�2 � V) � m � (A8)

Wt � wxx � w � �

After rescaling, the unit 1 in the factor (1 � V of the denomi-nator represents half satiation. Assuming that the relative unre-stricted growth rate a1 /a2 is not very large or small, whereas a0 /a2

is large and is denoted by 1/�1. The relative migration factor D2

is not very large with respect to the migration rate of benign cellsD1, but is large compared to the diffusion factor D0. Thus,

� �2, � D, � 1/�1, � b.

For the sake of simplicity, and without loosing generality, wereplace �i � � (for i � 1,2) to get

Ut � �Uxx � �1

��U(�1 � U) � �

Vt � DVxx � bV(�2 � V) � m � (A9)

Wt � Wxx � W � �

According to the relation (A9), parameters �,m, and � rep-resent the maximum rate of extraction of environmental re-sources of the cell in every stage.

ANALYSIS USING SINGULAR PERTURBATION METHODS

We use a singular perturbation method for the two-stage mu-tation model in (A9) to study the behavior and stability of the

VW�1 � V

VW�1 � V

UV�1 � U

UV�1 � U

a1�a2

�0�a2

D1�D2

D0�D2

VW�1 � V

VW�1 � V

UV�1 � U

UV�1 � U

premalignant stage. For mathematical simplicity we assume m� � m, and use (A9) to get the following system

�Ut � �2Uxx � U(�1 � U) � �m

Vt � DVxx � bV(�2 � V) � m � (A10)

Wt � Wxx � W � �

where U(t0) � U0,V(t0) � V0,W(t0) � W0 for fixed value of x.The first equation in system (A10) can be expressed in the fol-lowing form

�Ut � �2Uxx � U(�1 � U) � �� (A11)

where �� is a constant real number. This is a singular perturbedequation where U(t,x) is the fast variable while V(t,x) and W(t,x)are slow variables. The parameter � in the first equation mea-sures, on one hand, the relative movement of

� �

and, on the other hand, the effect of the selection pressure onthe reduction of the growth rate �2 with respect to a0 (when�0/�2 � 1/� is large). Since the wave speed is dependent on theshape of initial distribution, Dunbar (1983, 1986) conjecturedthat the stable traveling wave solutions of the system with D �0 are the singular limit solutions of stable traveling wave solu-tions with D � 0. The transversality of the intersection showsthe connections between critical points of three dimensionalphase space. For m � 0,

Ut � �Uxx � U(�1 � U) (A12)1��

D0�D2

UV�1 � U

VW�1 � V

VW�1 � V

UV�1 � U

UV�1 � U

FIG. 7. Simulation of U—normal, V—benign, and W—pre-malignant cell densities over parameter 0 � � � 1 at x � 5.This phase portrait demonstrates w5 verses v5 and versus u5.

FIG. 8. Phase portrait of the solutions U(t, x)—normal, V(t,x)—benign, and W(t, x)—premalignant cell. This simulationproduced over the joint variations over parameters m and D:0 � m � 1,0 � D � 3.

⎧⎪⎪⎨⎪⎪⎩

⎧⎪⎪⎨⎪⎪⎩

⎧⎪⎪⎨⎪⎪⎩

which then decouples from the other two equations in (A10),and represents the Fisher-Kolomogorov model that has a trav-eling wave solution U(s). The remaining equations in (A10) aregiven as:

Vt � DVxx � bV(�2 � V) �

(A13)

Wt � Wxx � W � � .

Steve Dunbar (1996) proved that if D � 0 or 0 D � 1, trav-eling wave solutions also exist for the last two equations of(A10) and are denoted by [V(s), W(s)]. Assume that � � 1 andm is nonzero but very small. The idea of using the transverseintersection of unstable manifold of one equilibrium, with thestable manifold of another equilibrium, will preserve the trav-eling wave solution (central manifold theorem). For small m,the intersection of two traveling wave solutions produces thedesired traveling wave solution for nonzero values of m. This

VW�1 � V

VW�1 � V

argument proves that the traveling wave solution for (A10) ex-ists such that U(s) satisfies (A12) and [V(s), W(s)] satisfies thesystem (A13).

Intersection of two traveling waves

Assume � � 0 and m � 1, then the first equation in (A10)becomes algebraic, U(�1 � U) � 0. Thus, U � 0 or u � �1.To find V and W, substitute these values of U into (A13).Then,

(I) For U � 0,

Vt � DWxx � bV(�2 � V) �

Wt � Wxx � W � �

(A14)

The resulting system (A14), which describes the relationshipbetween V and W for U � 0, is equivalent to the single stagemodel (A5).

(II) For U � �1,

Vt � DVxx � bV(�2 � V) � m �

Wt � Wxx � W � �

(A15)

Rearranging the first equation of (A15) yields

Vt � DVxx � bV(c � V) �

Wt � Wxx � W � �

(A16)

where

c � �2 �

is constant. This is again another form of system (A5), repre-senting the relation between the benign and premalignant cells.According to Dunbar (1986), both of these systems have trav-eling front wave solutions. That is, there are two sets of solu-tions [0,V(s), W(s)] and [�1V(s), W(s)] satisfying (A14) and(A15), respectively.

There is a point x � r (for some real number r) such that thissolution for U jumps from zero to �1, that is

U(t,x) �� for x rfor x � r

(A17)

where at the point x � r, the value of V � V(r) and there is ajump in the value of U from U � 0 to U � �1. After rescalingwith � (x � r)/�, and � t/� we can find the internal layersas a solution of the first equation of (A10) for m � 0,

U � U � U(�1 � U) � m (A18)

This equation has a traveling wave front solution connectingU � 0 to U � �1 with the same speed of the last two equationsof (A10).

UV�1 � U

0��1

m�1��b(1 � �1)

VW�1 � V

VW�1 � V

VW�1 � V

VW�1 � V

�1V�1 � �1

VW�1 � V

VW�1 � V

AHANGAR ET AL.632

35

1

0.05

5

300.15

0.35

FIG. 9. Simulation of premalignant growth W(t, x) with re-spect to normal cell over the parameter 0 � � � 1.

FIG. 10. Simulation of the densities of V-benign and W-pre-malignant cells satisfying system (A9) when m � 0 and Dchanges from 0 to 3.

20

ne

2

0.2

0.35 0.15

0.55

⎧⎪⎨⎪⎩

⎧⎪⎨⎪⎩

⎧⎪⎨⎪⎩

⎧⎪⎨⎪⎩

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Address reprint requests to:Reza Ahangar, Ph.D.

Department of MathematicsKansas Wesleyan University

100 East Claflin AvenueSalina, KS 67401

E-mail: rahangar@kwu.edu or rahangar@alltel.net

MATHEMATICAL MODELING OF MULTISTAGE CARCINOGENESIS 633