Emergence of Anti-Cancer Drug Resistance: Exploring the Importance of the Microenvironmental Niche via a Spatial Model

Abstract

Practically, all chemotherapeutic agents lead to drug resistance. Clinically, it is a challenge to determine whether resistance arises prior to, or as a result of, cancer therapy. Further, a number of different intracellular and microenvironmental factors have been correlated with the emergence of drug resistance. With the goal of better understanding drug resistance and its connection with the tumor microenvironment, we have developed a hybrid discrete-continuous mathematical model. In this model, cancer cells described through a particle-spring approach respond to dynamically changing oxygen and DNA damaging drug concentrations described through partial differential equations. We thoroughly explored the behavior of our self-calibrated model under the following common conditions: a fixed layout of the vasculature, an identical initial configuration of cancer cells, the same mechanism of drug action, and one mechanism of cellular response to the drug. We considered one set of simulations in which drug resistance existed prior to the start of treatment, and another set in which drug resistance is acquired in response to treatment. This allows us to compare how both kinds of resistance influence the spatial and temporal dynamics of the developing tumor, and its clonal diversity. We show that both pre-existing and acquired resistance can give rise to three biologically distinct parameter regimes: successful tumor eradication, reduced effectiveness of drug during the course of treatment (resistance), and complete treatment failure. When a drug resistant tumor population forms from cells that acquire resistance, we find that the spatial component of our model (the microenvironment) has a significant impact on the transient and long-term tumor behavior. On the other hand, when a resistant tumor population forms from pre-existing resistant cells, the microenvironment only has a minimal transient impact on treatment response. Finally, we present evidence that the microenvironmental niches of low drug/sufficient oxygen and low drug/low oxygen play an important role in tumor cell survival and tumor expansion. This may play role in designing new therapeutic agents or new drug combination schedules.

Appendix

Before testing various mechanisms of tumor resistance, our model has been calibrated to 1) achieve a stable gradient of oxygen when no cancer cells are present, as would be the case in healthy tissue; 2) generate a tumor cluster that completely fills the available space when no drug is applied as would take place in non-treated tumors; this will result in another stable gradient of oxygen with hypoxic areas located far from the vasculature; and 3) completely eliminate the tumor during the treatment when the cells do not acquire resistance. These three self-calibration steps are discussed in this section.

First, the influence of oxygen and drug are normalized so that S _ξ  = S _γ  = 1. Then we determine the oxygen diffusion coefficient $\mathcal{D}_{\xi }$ and oxygen boundary conditions that lead to a (numerically) stable gradient of oxygen when no cancer cells are present; that is with no cellular uptake. Several boundary conditions were considered, however, the best results in terms of irregular gradient stabilization and the extent of hypoxic areas were achieved for the sink-like conditions with $\varpi =$ 0.45 (see Section 2.1). The resulting stable oxygen gradient is shown in Fig. 9a and the relative 2-norm error between two oxygen concentrations generated in two consecutive time steps is shown in Fig. 9c. The obtained oxygen gradient served as an initial condition for the reaction-diffusion equation for oxygen.

Next, a tumor cell oxygen uptake rate p _ξ and a hypoxia threshold Thr _hypo were selected to allow for tumor growth with a small subpopulation of hypoxic cells, and for generation of a (numerically) stable gradient of oxygen when the tumor reaches its stable configuration. This population of tumor cells, including the hypoxic cell fraction and tumor clonal composition, serves as a control case (with no treatment) shown in Fig. 1b. For the stable tumor population the (numerically) stable oxygen gradient is shown in Fig. 9b, and the relative 2-norm error of the oxygen changes over 25, 000 iterations is shown in Fig. 9d.

Fig. 9

Oxygen distribution and its stabilization error curves. (a,b) Numerically stable gradients of oxygen in a domain with no cancer cells (a), and in a domain where non-treated cancer cells uptake oxygen (b). The greyscale contours indicate oxygen distribution levels. (c,d) Relative 2-norm error of oxygen changes $\|\eta (\mathbf{x},t +\varDelta t) -\eta (\mathbf{x},t)\|_{2}$ calculated for 25000 iterations showing its numerical stability for cases (a) and (b), respectively.

Finally, the drug diffusion coefficient $\mathcal{D}_{\gamma }$, drug uptake rate p _γ , and death threshold Thr _death were determined so that the cluster of tumor cells with no resistance is eradicated. This ensures that for the chosen drug parameters, the drug is effective when there are no resistant tumor cells. This case is discussed in Section 3.1. All parameters determined by the procedure described here are listed in Table 2.