In the last post we discussed how the concept of Adaptive Therapies (**AT**s) depends critically on **competition** for resources between chemotherapy resistant cells and chemotherapy sensitive cells. This competition is what allows resistance to be kept in check when treatment is not applied. The balance between this competition and overall population growth is what drives the success of a good AT.

Here is an example of what happens when we apply chemotherapy in a conventional way, that is, the treatment is applied for as long as the overall tumor population responds to it. Take a look at the featured image in this post to have an idea of how the colors correspond to the number of pumps (that correlate with resistance and slow proliferation). In this simulation the mutation rate was set to 10% to allow for faster dynamics:

A longer version can be seen here but I only show the action every 50 time steps:

Here is how growth is affected shortly after the application of the treatment:

The plots show how as the chemotherapy is applied, the population initially decreases but, as the average number of pumps increase (which decrease proliferation but increase resistance to chemo) the population becomes increasingly resistant. *But is this resistance something that comes from a pre-existing population or does it evolve de novo?* In the case of this model the answer is complicated. The following plot shows the evolution of the tumor size and its heterogeneity over time (in a different but equivalent simulation in a smaller grid so the number of cells is smaller) over time using a Shannon index:

I am not clear that the Shannon index is the best metric but it is widely accepted and will keep using it unless a reader comes with a better suggestion. The plot shows that heterogeneity increases initially but remains low for a time as selection goes for cells that resemble the phenotype the simulation was seeded with (with 1 pump). At timestep=1000 the treatment is applied and, even though population size is reduced, heterogeneity increases once again.

This begs the question: **how sensitive are these results to the heterogeneity of the tumor** at the time of application of chemotherapy? Heterogeneity explains why chemotherapy does not kill all the tumor cells, at least initially. In this model there are three sources of heterogeneity in the tumor population: **1**. the mutation rate; **2**. the strength of selection in the absence of chemotherapy (weak selection would explain the presence of resistant cells in the absence of treatment); and **3**. the strength of selection during treatment (weak selection would give time to sensitive cells to produce progeny that might be more resistant). Let’s see now a simulation where we make it more difficult for the tumor to evolve resistance: starting only with sensitive cells, 1% mutation rate, higher cost for cells with pumps (stronger selection in the absence of treatment) and high chemo efficacy (stronger selection against sensitive cells in the presence of treatment).

Yay, this time the tumor did not have a chance! How does the Shannon index of this simulation compare to the previous simulation? It is much smaller! It is about 0.71 (compared to close to 2.7 in the previous simulation).

So we can, at least in theory, treat our simulated tumors in a way that resistance will not emerge. But that requires us to focus on aspects of cancer evolution for which we have little or no control. More likely than not, efficacy of the treatment, delivery and even microenvironmental protection (see the work by Meads, Gatenby and Dalton at NRC for a comprehensive review) will increase the chances for resistance to emerge. So what happens when we allow for a section of the microenvironment (centered in the middle of the computational domain) to confer protection to the tumour cells, regardless of their sensitivity to chemotherapy?

It seems that in most occasions we can assume that hitting the computer simulated tumour as hard as we can with chemo might not be a great solution. What else can we do? I suspect you know what I will be suggesting in my next post but here is a hint:

[…] a very simple math model to explore the role of heterogeneity in the efficacy of ATs [here, here and […]

Fun post, David. A quick question about the legend in the header; is it really the case that all the colours of black, yellow, orange, pink, and red correspond to 1 to 10 pumps? If so, why are they marked by different colours? I would also find it very helpful if you included the specific functions that translated from number-of-pump and presence/absence of therapy to growth rate and death rate.

The other part I want to focus on is when you write:

And then use the bump in the Shannon index as a justification for this being complicated. I am not sure if that is a reasonable justification. The Shannon index captures both the number of different mutants and how close the distribution is to uniform. When you turn on chemotherapy, you end up hitting the most common cells the hardest (since the most common cells are the ones with the fewest pumps) and so their proportion decreases rapidly, and since probabilities must sum to 1 the other ones increase towards the uniform. So the bump is coming from distribution becoming more uniform not anything to do with new mutations.

If you wanted to make this apparent in the graph, you could plot the log (using the same base as you use for calculations of Shannon index) of the number of distinct mutants on the same plot. The gap between log-mutants and Shannon index will give you a measure of how far the population is from uniform distribution and the log-mutants will give you the number of mutants. After you turn on therapy, you will see a bump in the Shannon index, but nothing super exciting will happen to the log-mutants.

I would suggest the direct way as a way to track the effect of

de novomutations. Introduce an extra binary tag to your agents that is set to zero if the mutant arouse before therapy was turned on and to one if the mutant arouse after therapy was turned on; then track the populations of zero and one mutants.As a final suggestion. I think that for models and measures that are this complicated, it is useful to compare them to controls. A control for heterogeneity that I would consider is the neutral model: mutation rate and mechanism is the same, but the number of pumps has no effect on fitness with or without therapy — the pumps become just an arbitrary number that can increase due to mutation. You could even combine the control and the non-control in the same model by introducing a second numerical trait (maybe “useless pumps”?) that has no effect on fitness (with and without therapy) and mutates independently of (but according to the same rate and mechanism) the fitness-affecting pumps. You can then compare the fitness-affecting-pump heterogeneity to the fitness-unaffecting-pump heterogeneity (according to both the Shannon index and the log-of-number-of-distinct-types index).

Nothing is a quick question when it comes from you Artem! But in any case no, the colors represent the entire spectrum of possibilities, that is 1-100. More answers to come later but I am sure to use some of your suggestions.