## Drug holidays and losing resistance with replicator dynamics

September 2, 2016 6 Comments

A couple of weeks ago, before we all left Tampa, Pranav Warman, David Basanta and I frantically worked on refinements of our model of prostate cancer in the bone. One of the things that David and Pranav hoped to see from the model was conditions under which adaptive therapy (or just treatment interrupted with non-treatment holidays) performs better than solid blocks of treatment. As we struggled to find parameters that might achieve this result, my frustration drove me to embrace the advice of George Pólya: “If you can’t solve a problem, then there is an easier problem you can solve: find it.”

In this case, I opted to remove all mentions of the bone and cancer. Instead, I asked a simpler but more abstract question: what qualitative features must a minimal model of the evolution of resistance have in order for drug holidays to be superior to a single treatment block? In this post, I want to set up this question precisely, show why drug holidays are difficult in evolutionary models, and propose a feature that makes drug holidays viable. If you find this topic exciting then you should consider registering for the 6th annual Integrated Mathematical Oncology workshop at the Moffitt Cancer Center.^{[1]} This year’s theme is drug resistance.

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## Spatializing the Go-vs-Grow game with the Ohtsuki-Nowak transform

June 30, 2017 by Artem Kaznatcheev Leave a comment

Recently, I’ve been thinking a lot about small projects to get students started with evolutionary game theory. One idea that came to mind is to look at games that have been analyzed in the inviscid regime then ‘spatialize’ them and reanalyze them. This is usually not difficult to do and provides some motivation to solving for and making sense of the dynamic regimes of a game. And it is not always pointless, for example, our edge effects paper (Kaznatcheev et al, 2015) is mostly just a spatialization of Basanta et al.’s (2008a) Go-vs-Grow game together with some discussion.

Technically, TheEGG together with that paper have everything that one would need to learn this spatializing technique. However, I realized that my earlier posts on spatializing with the Ohtsuki-Nowak transform might a bit too abstract and the paper a bit too terse for a student who just started with EGT. As such, in this post, I want to go more slowly through a concrete example of spatializing an evolutionary game. Hopefully, it will be useful to students. If you are a beginner to EGT that is reading this post, and something doesn’t make sense then please ask for clarification in the comments.

I’ll use the Go-vs-Grow game as the example. I will focus on the mathematics, and if you want to read about the biological or oncological significance then I encourage you to read Kaznatcheev et al. (2015) in full.

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Filed under Analytic, Commentary, Models, Reviews, Technical Tagged with mathematical oncology, replicator dynamics, spatial structure