Some simple mathematical models of tumor growth

Abstract

The principal motivation of this final project is the interest in applied mathematics. Of all the courses I have studied the ones that have arouse my curiosity the most are Mathematical models and Dynamic systems and Differential equations. In particular, what I liked was the chance to extract conclusions from other sciences as physics or economics by mathematical study. For that reason, I have based my project on these fields. We also have to say these courses are much related to the mathematical modelling, so with this project I will be able to strength and increase my knowledge in all of these subjects. When the topic was proposed to me, it seemed interesting the possibility of introduce myself in the cancer’s study, since it is so present in the current society. Besides, it attracted attention to me the relation it could have with mathematics. Lots of times, what happens is that in different fields such as physics or chemistry, a topic is investigated but not in a deep mathematical way. In other words, it might be said that the relations between the different branches should be more joined for a better analysis of both. Therefore, this project is based on the understanding of several growth models of tumors based on differential equations and it is attempted to give a better mathematical explanation of the proposed models in the chapter 1 of the paper: Some mathematical models of tumor growth. Since I had only an unclear idea of what the cancer was at a biological level, it has been required to study in depth some references the paper was offering to understand the biological part. The project consist of three parts. The first one gives a brief introduction on what the cancer is, points out the types of tumors and which ones are studied and gathers some information of the impact of the cancer in our society. Furthermore, it gives an explanation of the relation of cancer and mathematics, and the important vocabulary required for a better understanding of the whole study. The second part explains the exhibited models biologically on the paper using its references and others that I thought they were interesting to extend the information. Beginning with the simplest models of one variable and continuing with three models of two variables. Adding references where necessary, for the interest of the reader to study the topic in depth. The aim of the third and final part is to understand mathematically the models exposed previously, show alternative proofs and extend results of, proof details that are left in, use a program to exemplify the results and obtain conclusions from a biological point of view.