Competition for Resources and Space Contributes to the Emergence of Drug Resistance in Cancer

Abstract

Recent experiments reveal targeted therapy of tumours promotes the spread of drug-resistant cancer cells in mixed sensitive-resistant tumours. The hypothesis is that drug-stressed sensitive cells produce diffusible growth factors that stimulate the expansion of drug-resistant cells. A mathematical model employing simple ecological competition and a nonlinear motility law is able to reproduce the magnitude of observed expansion of the resistant populations volume without invoking production of diffusible growth factors. The model shows how the therapy-induced removal of the sensitive population alleviates the competitive pressure on the resistant for resources and space and confirms the in vivo experimental findings, and sheds light onto mechanisms behind the large increase of the drug-resistant cancer cells in the treated tumour.

Appendix

Here is a brief summary of the numerical method used to solve Eqs. 7–11. The main challenge is the treatment of the non-linear flux term. For the 1D problems P1 and P2, the flux term is to rewritten as in [21]

$$\begin{aligned} \frac{\partial }{\partial x}\left( \frac{u}{u+v}\frac{\partial }{\partial x}(u+v)\right) = \frac{u}{u+v}\cdot \frac{\partial ^2}{\partial x^2}(u +v ) +\frac{\partial }{\partial x}(u +v )\cdot \frac{\partial }{\partial x}\left( \frac{u}{u+v} \right) \end{aligned}$$

(15)

and then the equations are integrated in time using the ROWMAP solver [34].

For the 2D problem, a variational formulation scheme is used to discretise the equations in space with Lagrangian P2 finite elements. The non-linear system is integrated in the software environment FreeFem++ [35] by a fully implicit Euler scheme, and the equations are solved iteratively at every time step by Newton’s method. For this purpose, the non-linear diffusion term is approximated as follows for a test function $\psi \in H^1(\varOmega )$. The divergence theorem employing the homogeneous Neumann boundary conditions for $u_j$ gives

$$\begin{aligned} \int _\varOmega \psi \nabla \cdot \left( \frac{u_j(\sum _{i=0}^2\nabla u_i)}{\sum _{i=0}^2 u_i}\right) \,dx = - \int _\varOmega \frac{u_j(\sum _{i=0}^2 \nabla u_i)}{\sum _{i=0}^2 u_i}\nabla \psi \,dx :=-\int _\varOmega F_j\nabla \psi \,dx. \end{aligned}$$

(16)

In order to use Newton’s method for the solution of Eqs. 7–11, $F_j$ in Eq. 16 is expanded in a Taylor series. For a small perturbation $\tilde{u}_i$ in $u_i$, $F_0$, for example, is

$$\begin{aligned} F_0(u_0+\tilde{u}_0, u_1+ & {} \tilde{u}_1,u_2+\tilde{u}_2) \approx \frac{( u_1+ u_2)(\sum _{i=0}^2\nabla u_i)\tilde{u}_0}{(\sum _{i=0}^2 u_i)^2} - \frac{ u_0(\sum _{i=0}^2\nabla u_i)\tilde{u}_1}{( \sum _{i=0}^2 u_i)^2}\\&- \frac{ u_0(\sum _{i=0}^2 \nabla u_i)\tilde{u}_2}{(\sum _{i=0}^2 u_i)^2} + \frac{ u_0(\sum _{i=0}^2\nabla \tilde{u}_i)}{ \sum _{i=0}^2 u_i} + \, \text{ higher } \text{ order } \text{ terms }. \end{aligned}$$

A similar approximation is made for the other two terms $F_1,F_2$.