Stochastic Modeling of Inter & Intra Stage Dependent Cancer Growth under Chemotherapy through Trivariate Poisson Processes

This study has proposed a Stochastic Model for cancer growth under chemotherapy with the assumptions of the growth, transition and loss parameters of different stages are inter and intra dependent. A trivariate Poisson process approach has been adopted for modeling the three stage cancer growth by considering the stages of cells in the tumor namely normal cell, mutant cell and malignant cell in the presence and absence of chemotherapy during time ‘t’. Stochastic differential equations were obtained and the three dimensional joint probability functions along with related statistical measures were derived. Model behavior was analysed through numerical data.


Introduction
The spread of cancer is more vigorous when a normal cell will transformed in to a mutant and further to a malignant cell. The growth, spread and loss processes of different types of cells are purely random and influenced by many known and unknown factors. The cell division and control mechanism of cancerous cells are out of the genetic instructions of any living body. Continuous proliferation of cells within each stage, transformation of cells from one stage to another stage leads to enormous growth within the limited membrane structures of a tissue may cause the formation of tumours. Development of secondary and invasion of cancerous cells through metastasis is again subject to many random issues and uncertainties. Usually the violation of genetic instruction of the cell will be initiated with conversion of a normal cell in to a typical and erratic behaved cell. Further, this erratic behaved/ mutant cell will have a faster growth of division as well as continuous and unending proliferation leads to formation of malignant cell. Hence, the cell division behaviour is varying from one stage to other stage.
Quantification of cancer growth through the mathematical models was initiated by Mayneord (1932). Tumor resistance to chemotherapy by Coldmann et al. (1983), effects of drug resistance in the presence of chemotherapy by Birkhead et al. (1984), chemotherapy of experimental tumors by Coldman et al. (1986), cancer progression and response of chemotherapy by Sandeep sanga et al. Stochastic modeling on pathophysiology of cancer and its Biology is more appropriate to study the cancer growth and loss processes as it is influenced by many uncertain factors. This study is on stochastic modeling of cancer cell growth in three stages namely normal cell, mutant cell and malignant cell by using trivariate Poisson process in chemotherapy. The process of presence and absence of drug with indexed variables is introduced by assuming the growth of tumor during drug administration and drug vacations are complementary. This model is useful to study the growth of cancer cell as overall phenomena of chemotherapy.

Stochastic Model for Cancer Growth during Chemotherapy
A Stochastic model for three stage cancer cell growth in chemotherapy is developed with the following assumptions. Let the events occurred in non-overlapping intervals of time are statistically independent. Let Δt be an infinitesimal interval of time. Let there be 'n' normal cells, 'm' mutant cells, 'k' malignant cells initially at time't'. Let assuming 0 and 1 when the patient is in drug presence and absence respectively.

Let
be the rate of growths; be the rate of death of cells; be the rate of transformation of cells from k to k+1 stage. where, 'I' be the stage of cells, i=1,2,3 for normal, mutant, malignant cells respectively; 'j' be the state of drug, j=0,1 (absence, presence); 'k' be the transformation of cells k to k+1 stage, k=1,2. All the parameters follow poisson processes and, let postulates of the model are.
Let be individual stochastic processes of normal cell, mutant cell and malignant cell. Such that, be the tri-variate process such that processes will be Let us define,

BSMaSS Volume 10
The probability of occurrence of other than the above events during an infinitesimal interval of time be the joint probability of existence of 'n' normal cells, 'm' mutant cells and 'k' malignant cells in a tumor during chemotherapy per unit time 't'. Then differentialdifference equations of the model are:

Differential Equations and Statistical Measures
Let denote the moments of order of normal cells, mutant cells, malignant cells at time 't'.