Modelling cancer with ordinary differential equations

Question

How can I model cancer spread and cancer treatment using first order differential equations?

I will be investigating this topic in my high school math paper, but am unsure of where to begin.

Answer 1

There are many applications of differential equations (e.g. ode, pde, sde, sdpde, etc.) in the study of cancers. You could search the profile of Professor Yang Kuang who is one of the leading figures in mathematical oncology.

Here is one of the easier models (ode) to understand and apply https://www.mdpi.com/2076-3417/9/1/36. If you want model that directly deals with cancer growth, this is often most accurately modeled with pde, for instance melanoma, ... (on Professor Kuang's research on his website).

You can find essentially existing mathematical models for practically any cancer type. They can model the cancer in a physiological level, a macroscopic level or multi-scale level (which connects the molecular level to the macroscopic realization of the cancer). There are also phenomenological model (e.g. model that is formulated from what is observed rather than from known physical/biological laws), mechanistic model and hybrid model. The subject is very rich and is at an intersection of many different fields.

Both heavy mathematical analysis and computational examination can be carried out to study these models. But it is often best study with real (or even synthetic) data, so clinicians would trust your model more. But for a school project, some simulations and understanding of the model would likely to suffice.

For a book reference, Professor Brian Ingalls writes a very readable text called "Mathematical Modeling in Systems Biology". The book (and his website) contains an abundant of example and code for you to learn from.