Mathematical Biomarkers

A mathematical approach to predicting the benefit of Adaptive Therapy

Lotka - Volterra Model

This is a simple 2-population Lotka-Volterra tumour model, where $S$ is the number of susceptible cells, and $R$ is the number of resistant cells.

$$ \frac{dS}{dt} = r_{S} S \left(1 - \frac{S+R}{K}\right) \times (1-d_{D}D) - d_{S}S, \\\ \frac{dR}{dt} = r_{R} R \left(1 - \frac{S+R}{K}\right) - d_{R}R $$

Both species follow a modified logistic growth model with growth rates $r_{S}$ and $r_{R}$, where the total population (rather than the species population) is modified by the carrying capacity $K$.

For the susceptible population, this growth rate is also modified by the drug concentration $D$ and the killing rate of the drug $d_{D}$.

Finally, both species have a natural death rate, of $d_{S}$ and $d_{R}$ respectively.

This model is implemented in the LotkaVolterraModel class, which inherits from ODEModel. This parent class sets parameters such as error tolerances for the solver, and then solves the ODE model for each treatment period sequentially.