A New ODE-Based Model for Tumor Cells and Immune System Competition

Abstract

Changes in diet are heavily associated with high mortality rates in several types of cancer. In this paper, a new mathematical model of tumor cells growth is established to dynamically demonstrate the effects of abnormal cell progression on the cells affected by the tumor in terms of the immune system’s functionality and normal cells’ dynamic growth. This model is called the normal-tumor-immune-unhealthy diet model (NTIUNHDM) and governed by a system of ordinary differential equations. In the NTIUNHDM, there are three main populations normal cells, tumor cell and immune cells. The model is discussed analytically and numerically by utilizing a fourth-order Runge–Kutta method. The dynamic behavior of the NTIUNHDM is discussed by analyzing the stability of the system at various equilibrium points and the Mathematica software is used to simulate the model. From analysis and simulation of the NTIUNHDM, it can be deduced that instability of the response stage, due to a weak immune system, is classified as one of the main reasons for the coexistence of abnormal cells and normal cells. Additionally, it is obvious that the NTIUNHDM has only one stable case when abnormal cells begin progressing into early stages of tumor cells such that the immune cells are generated once. Thus, early boosting of the immune system might contribute to reducing the risk of cancer.

Data Driven Mathematical Model of Colon Cancer Progression

“…Although there are many mathematical models for cancer [34][35][36][37][38][39][40][41][42][43], one of the outstanding challenges in mathematical modeling of cancers is the existence of many unknown parameters and the limited number of data sets. For this reason, the approach of many of these mathematical models is to assume some values for parameters, use estimated parameters from other diseases, or vary the parameters and initial conditions within biologically-feasible values in order to investigate their effects on the results.…”

“…While there are many papers that use mathematical models for colon cancer progression [34][35][36][37][38][39][40][41][42][43], only a few have attempted to include immune interaction in their model. Models such as [40][41][42] define a system of ordinary differential equations (ODEs) that describe the interactions between cancerous cells and various sub-populations of immune cells (including NK cells, CD8+ T cells, lymphocytes, natural death cells and interleukins) and explore how these interactions can influence tumor growth over time. While time course data for the growth of untreated tumors are not currently easily available to verify models such as [40], other models such as [41] include simulations of treatment plans that can be compared on the population-level to results from previous clinical trials.…”

“…Models such as [40][41][42] define a system of ordinary differential equations (ODEs) that describe the interactions between cancerous cells and various sub-populations of immune cells (including NK cells, CD8+ T cells, lymphocytes, natural death cells and interleukins) and explore how these interactions can influence tumor growth over time. While time course data for the growth of untreated tumors are not currently easily available to verify models such as [40], other models such as [41] include simulations of treatment plans that can be compared on the population-level to results from previous clinical trials. To generate population-level simulation results while acknowledging the different responses to treatment that can arise from differing patient immune profiles, this study selects a range of parameter values to simulate 64 unique "virtual patients" for which to solve the system of ODEs describing potential treatments.…”

Data-Driven Mathematical Model of Osteosarcoma

“…Although some studies include bone modeling, osteoblast cells, or osteosarcoma treatments [ 50 , 51 , 52 , 53 , 54 ], to the best of our knowledge, there is currently no mathematical model explaining the progression of osteosarcoma tumors. The relationship between immune cells and tumor cells have been used as an alternative approach in the mathematical modeling of different cancers types in some studies [ 55 , 56 , 57 , 58 ]. Objective of this study is to build a data-driven model for the progression of osteosarcoma tumors that considers immune cell interactions with tumor cells.…”

Data Driven Mathematical Model of FOLFIRI Treatment for Colon Cancer

“…The effect of radiotherapy and chemotherapy on tumor growth has been modelled using partial differential equations (PDEs), ordinary differential equations (ODEs) and linear quadratic models in breast and brain tumors [ 41 , 42 , 43 ]. Immune cell interactions with tumor cells are used as an alternative approach for the mathematical modeling of the cancer treatments in a few studies that pharmacokinetic ODEs are defined to predict the optimal dosing regimen and the combined effect of chemotherapy and immunotherapy [ 44 , 45 , 46 , 47 ]. Many of these models such as [ 45 ] can not be verified because of lack of time course data for the growth of treated and/or untreated tumors.…”