A Stochastic Growth Model for Cancer Cells under Mutation and Metastasis in an Organ

This study has proposed a stochastic model for cancerous growth due to its metastasis in an organ. The birth-and-death and migration processes based on growth and loss rates of pathogenesis of malignant and normal cells are considered. It is assumed that the growth and loss/death rates of both normal and malignant cells follows Poisson processes. The joint probability generating functions in the form of partial differential equation along with statistical measures were derived in terms of system of ordinary differential equations. The Model behaviour was analysed by solving those differential equations and presented graphically. Introduction The carcinogenesis is a complex random multistep process involving initiation, promotion and progression of cells (Tan, 1989, 1987). Transforming normal cells into cancerous cells in any organ is usually initiated with simple mutation and aggravate with abnormal proliferation. Carcinogenesis is highly complex and stochastic, the mutation processes of normal to intermediate stage and intermediate to malignant stage cells are purely random in nature (Serio, 1984). The cancer cell growth has involved more than one stage. While transforming as a malignant cell, it may undergoes birth-and-death process and two mutation processes. It is explained in term of qualitative and epidemiological phenomena through two stage stochastic model (Quinn (1997), Moolvagkar and Venzon (1979), Birkhead (1986), Moolgavkar (1983)) and their objective of carcinogenesis studies are more concerned on finding incidence functions (hazard function). Rao et al., (2014) have studied the growth of cancer cells in the presents of migration and cells growth in the secondary location. The statistical moments are derived explicitly based on homogeneous processes with boundary and initial conditions. The developed model is not solved to a solution so that to obtain the survival function of the cells. In this paper, the above developed model is considered and solved to get the solution to the model to obtain the survival function which in turn to arrive incidence function (Birkhead, 1986). Moreover, the model has the explicit moments obtained from the system of differential equations. Thus, the systems of ordinary differential equations are solved simultaneously using numerical differential equation and presented graphically. Further, same model is extended and regression model approach is attempted to deduce the model to find the behaviour of malignant cells with the assumption of cells growths are deterministic and exponential. The results are presented graphically. Stochastic Model for Growth of Cancerous Cell Let ( ) 11 g t be the growth rate of normal cells in primary tumor; ( ) 21 g t be the growth rate of malignant cells in primary tumor; ( ) 32 g t be the growth rate of immigrant malignant cells in the secondary tumor; ( ) 11 d t be the loss/death rate of normal cells in primary tumor; ( ) 21 d t be the loss/death rate of malignant cells in primary tumor ; ( ) 32 d t be the loss/death rate of malignant cells in secondary tumor; ( ) 11 t t be the transformation rate of normal cells to malignant cells in a primary tumor; ( ) 21 t t be the migration rate of malignant cells in a primary tumor to the secondary tumour; ( ) 32 t t be the emigration rate of malignant cells in the secondary tumour to other part of the body. International Frontier Science Letters Submitted: 2016-01-26 ISSN: 2349-4484, Vol. 8, pp 19-30 Revised: 2016-04-17 doi:10.18052/www.scipress.com/IFSL.8.19 Accepted: 2016-05-23 © 2016 SciPress Ltd., Switzerland Online: 2016-06-29 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ The mechanisms involved in the cell divisions are purely stochastic in nature. Events occurred in non-overlapping interval of time are statistically independent are assumed. Furthermore the following hypotheses are assumed for the model. In a infinitesimal time t ∆ , probability of growth of a normal cell during t ∆ be ( ) ( ) 11 ng t t o t ∆ + ∆ there exist n number of normal cells epoch time t; probability of growth of a malignant cell during t ∆ be ( ) ( ) 21 mg t t o t ∆ + ∆ there exist m number of malignant cells epoch time t; probability of growth of a malignant cell during t ∆ be ( ) ( ) 32 g t t o t ∆ + ∆ there exist m number of malignant cells epoch time t; probability of death of one normal cell during t ∆ be ( ) ( ) 11 nd t t o t ∆ + ∆ there exist n number of normal cells there exist n number of normal cells epoch time t; probability of death of one malignant cell during t ∆ be ( ) ( ) 21 m d t t o t ∆ + ∆ there exist n number of normal cells there exist m number of malignant cells epoch time t; probability of death of immigrant malignant cell during t ∆ be ( ) ( ) 32 d t t o t ∆ + ∆ ; probability of a normal cell transform into malignant cells during t ∆ be ( ) ( ) 11 n t t o t t ∆ + ∆ there exist n number of normal cells; probability of a malignant cell migrate to secondary location during t ∆ be ( ) ( ) 21 m t t o t t ∆ + ∆ there exist m number of malignant cells; probability of a immigrant malignant cell migrate to some other location of body during t ∆ be ( ) ( ) 32 t t o t t ∆ + ∆ ; probability of occurrence of an event other than above said events is ( ) 2 o t ∆ . Let ( ) { } X t , t 0 ≥ be the stochastic counting process of normal cells mechanism and ( ) { } Y t , t 0 ≥ be the stochastic counting process of malignant cells mechanism. Let ( ) ( ) { } X t ,Y t , t 0   ≥   be a joint bivariate stochastic processes of individual stochastic processes of ( ) { } X t , t 0 ≥ and ( ) { } Y t , t 0 ≥ . Such that, ( ) ( ) [ ] { } ( ) n,m P X t ,Y t n,m p t   = =   and marginal processes are ( ) { } ( ) n P X t n p t = = , ( ) { } ( ) m P Y t m p t = = . Let ( ) n,m p t, t t + ∆ be the probability that occurrence of one of the possible cell mechanism in an infinitesimal interval t ∆ , provided there exists ‘n’ number of normal cells and ‘m’ number of malignant cells in the organ epoch time‘t’. Then the differential difference equations of the model (Bharucha-Reid, 1960) are, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) n,m 11 11 11 21 21 21 32 32 32 n,m 11 n 1,m 11 n 1,m 1 21 n,m 1 11 n 1,m 21 21 32 32 n,m 1 32 n,m 1 d p t n g t t d t m g t d t t g t d t t p t dt n 1 g t p t n 1 t p t m 1 g t p t n 1 d t p t m 1 d t t d t t p t g t p t for n,m 1 − + − −


Introduction
The carcinogenesis is a complex random multistep process involving initiation, promotion and progression of cells (Tan, 1989(Tan, , 1987. Transforming normal cells into cancerous cells in any organ is usually initiated with simple mutation and aggravate with abnormal proliferation. Carcinogenesis is highly complex and stochastic, the mutation processes of normal to intermediate stage and intermediate to malignant stage cells are purely random in nature (Serio, 1984). The cancer cell growth has involved more than one stage. While transforming as a malignant cell, it may undergoes birth-and-death process and two mutation processes. It is explained in term of qualitative and epidemiological phenomena through two stage stochastic model (Quinn (1997), Moolvagkar and Venzon (1979), Birkhead (1986), Moolgavkar (1983)) and their objective of carcinogenesis studies are more concerned on finding incidence functions (hazard function). Rao et al., (2014) have studied the growth of cancer cells in the presents of migration and cells growth in the secondary location. The statistical moments are derived explicitly based on homogeneous processes with boundary and initial conditions. The developed model is not solved to a solution so that to obtain the survival function of the cells. In this paper, the above developed model is considered and solved to get the solution to the model to obtain the survival function which in turn to arrive incidence function (Birkhead, 1986). Moreover, the model has the explicit moments obtained from the system of differential equations. Thus, the systems of ordinary differential equations are solved simultaneously using numerical differential equation and presented graphically. Further, same model is extended and regression model approach is attempted to deduce the model to find the behaviour of malignant cells with the assumption of cells growths are deterministic and exponential. The results are presented graphically.

IFSL Volume 8
Further,   converges for x 1, y 1 < < . Multiplying the above differentialdifference equations from (1) to (4) with n m x y and summing over n, m, we get On simplification we obtain 11  11  11  11  11  11   2  21  21  21  21  21  21   1  32  32  32  32  32  32 x, y; t x, y; t x, y;t mx y p t m 1 x y p t y International Frontier Science Letters Vol. 8 21 The above equation (8) is to be solved using the initial condition ( By definition If we expand both side of the eqn. (9) in power of u and v and equate coefficient, We obtain the following system of differential equations, where ( ) i, j m t denotes the statistical moments of order ( ) i, j representing the number of normal cells and number of malignant cells in an organ at time t. The above system of ordinary differential equations is solved numerically using mathematica and results obtained are presented below. The differential equations are in terms of first and second order moments, which are presented for fixed values of parameters such as g 11 =0.00001; d 11 =0.00001; 11 t =0.3; g 21 =0.8; d 21 =1.0x10 -7 ; 21 t = 0.005; g 32 =0.15; d 32 =0.000001; 32 t =1.0x10 -7 over a period of time.

Limiting Behaviour of the Moments
The limiting behaviour of the derived moments are obtained easily by taking limit t → ∞ . In particular the study on the normal cells and malignant cells in the cell's population, obviously our interest is to find the asymptotic performance of the ratio of the average (expected) number of normal cells and average (expected) number of malignant cells in the population (Bharucha-Reid, 1960).
When N 0 =1 and M 0 =0, the ratio is reduced to We observe that in the case of ( ) with initial condition ( )    2  2  21  21  21  21  21  21  11  11  11  11  11  11   dx  dy  y  y  x x y 2  11  11  11  11  2  2  21  21  21  21  21  21  21  21  21  21  21  21   11  11  2  21  21  21  21  21  21 x x dx dy y y y y y y y − β + t + δ β = + − β + δ + t + β + δ + t − β + δ + t + β + δ + t δ + t + − β + δ + t + β + δ + t (30) which is in the form of Riccati first order ordinary differential equation, x = y is the particular solution, and by substitution of It reduces to θ is the integrating constant and for arbitrary a, b, c and real -1<z<1,  The general solution for the model is x, y;0 x y v y y y y 1 y , Elimination of the integration constants, we get This is similar to the model developed by Birkhead (1986), the form obtained in (35) can be used to find the probability that there being no malignant cell at time t after the start of the process is On solving ( ) 1, 0, t φ and derivative of them with respect to t, this enables us to find survival and hazard functions.

Non-homogeneous Processes for Cell Growths with Regression Approach
The mutation parameter t , t t t is also function of time t. If we assume the mutation and migration parameters are constant then it is very simple to derive the statistical moment from the above system of linear differential equations. If  For any organ, it contains active cells (living cells) and inactive cells (benign and necrosis), active cells are participating in the growth and inactive cells are not participating in the growth. Every cells in has the life span and ultimate get died, and then it is replace by the new one. Therefore in the growth process death also to be considered. The normal active cells are having a deterministic growth whereas abnormal active cells do not under a deterministic growth. If it is assumed that the growth of normal cells and malignant cells are deterministic and exponential (Armitage, 1960). Then, the expected number of cells contributed by normal cell to the cell population is

Conclusion
This study can helps to quantification the growth of malignant cells under an abnormal and normal condition, and a proper understanding of growth of different type of cells over a period of time are more helpful in proper monitoring the patient's health and to apply optimum controlling strategies to control growth of cells and maintain the good health. This study has the limitation of obtaining real time data for analysing the model, but an appropriate data can be obtained from the clinical experimental laboratories.