MATHEMATICAL MODELLING AND DYNAMIC SIMULATIONS OF TUMOR CELLS GROWTH WITH INTERVENTION OF VITAMINS

Abstract

The natural sources of the vitamins, which comes from a balanced diet, contribute to the protecting of the body from a progressive cancer. In various epidemiological studies, unhealthy diet patterns have been identified as one of the main risk factors for cancer. This challenges us to dynamically demonstrate the function of the immune system that responds with the appearance of abnormal cells in a tissue by two cases when a person follows a healthy diet and an unhealthy diet. In addition, we also assess the effect of boosting the immune system through the intervention of vitamins on delay tumor cells growth. To achieve these goals, we develop, analyze and simulate mathematical models of the dynamics behavior of the following; immune-healthy diet (IHDM), immune-unhealthy diet (IUNHDM), tumor-immune-unhealthy diet (TIUNHDM), normal-tumor (NTM) and normal-tumor-immune-unhealthy diet (NTIUNHDM). All these models are governed by a system of ordinary differential equations and referred to two or three main populations of normal cells, tumor cells and immune cells. Furthermore, we also investigate the behavior of dynamics models by assuming that a person has begun to change his/her diet to be healthier. The developing models are known as follows; immune-interventionvitamins (IIVM), tumor-immune-vitamins (TIVM), normal-tumor-vitamins (NTVM) and normal-tumor-immune-vitamins (NTIVM). In these models, the intervention effect of vitamins on the dynamic cells is assessed within thirty days dynamically, analytically and numerically. The models are simulated by utilizing the fourth order Runge–Kutta method. From the analysis and simulation models, it can be deduced that the immune system is able to inhibit and eliminate abnormal cells through these three stages: the response stage, the interaction stage, and the recovery stage. Specifically, the failure of the immune system to accomplish the interaction stage occurs when a person follows an unhealthy diet. From analyzing the TIUNHDM and NTM, it was found that the appearance of tumor cells due to the weakening of the immune system, might turn these immune cells into immune cancer cells and has a significant effect on dynamic normal cell cycle. Mathematically, the dynamic behavior of cells is more distinctive by combining all cell populations in NTIUNHDM. The developing models by the intervention of vitamins indicate the following; the immune system can be boosted early through the consumption of a regular intake of vitamins from a natural source of food and drink in which it is simulated by 16% from vitamins per day, prevention of abnormal cells from progressing into tumor when it is simulated by 55% from vitamins per day, and the prevention of tumor cells from upgrading. Furthermore, an occurrence of coexistence between cells in NTVM is obvious when a person consumes regular intake of vitamins which is simulated by 87% from vitamins per day. Regular consumption of vitamins daily, which is simulated by 86% from vitamins, has an important function to support normal cells growth, boost the immune system and delay tumor cells growth. To sum up, even though the response of the immune system is weak, the daily consumption of sufficient vitamins can play an essential role in delaying the development of an early stage of cancer. According to the analysis and the simulation of the research models, it can be deduced that dietary pattern has a significant impact on the functioning of the immune system.