## The mathematics of cancer

Most papers, even if they show interesting results, have a limited impact on my work. Papers can yield interesting or relevant methods or results that I will aim to incorporate in my work, or they can present a new perspective on an existing problem that challenges the way I used to think before. In the last 2-3 of years I found papers that sit somewhere between these two cathegories. For instance, I refer to Gerlinger's paper with Swanton as the second best kind of paper you can find: one that may not change the way I see things but presents the worldview that the authors and I share to a wider audience.

Today's paper, the one by Philipp Altrock, Lin Liu and Franziska Michor falls in the cathegory of papers that are just incredibly useful even if none of the material is particularly new to me. The reason is, of course, that this is a review paper. One of the few that Nature Reviews Cancer has published on mathematical modelling in cancer (the other two I know are the one by Sandy Anderson and the one by Helen Byrne). And it is a great summary, covering a good deal of mathematical oncology literature from recent years and highlighting some of the strengths but also weaknesses of that work.

It does not cover all the of it though. It highlights mathematical approaches such as branching and Moran processes, (ordinary and partial) diferential equations, agent-based models as well as one of my favourite tools: evolutionary game theory. Some approaches are neglected. One that comes to mind is Reniak's Inmerse Boundary as well as other more biophysically realistic models. That is understandable given the emphasis of the review on aspects that are likely to drive evolutionary dynamics in cancer like mutations and interactions with the microenvironment.

In conclusion it is a good read that covers most of the relevant literature for those of us concerned on the evolutionary aspects of mathematical oncology. There are some things that a review like this could have covered and that might be material for a follow-up review. One (and I have taken the liberty of using a design by Arturo Araujo to illustrate the point) is that mathematical models of cancer can be used to integrate data and hypotetheses coming from different scales, experimental systems and approaches.

This view of mathematical modelling as a multi-scale glue is what makes it such a great tool to translate biological discoveries into clinical practice. Individual experiments and results alone are unlikely to be substantial enough to change how cancer is treated but through the **glue** of mathematical modelling, a lot of these results can flesh out a bigger idea.

Another one is about how mathematical models of cancer are built, and even more relevant, how do they integrate data. You can either start with some experimental/clinical data and then build a mathematical model onto which to fit such data or... you can start from first principles and see where they take you and then experimentally validate the results. The former uses hypotheses whereas the second generates them.

I guess I should convince somebody to write a complementary review article to follow this one.

Altrock, P., Liu, L., & Michor, F. (2015). The mathematics of cancer: integrating quantitative models Nature Reviews Cancer, 15 (12), 730-745 DOI: 10.1038/nrc4029