The Impact of Cell Density and Mutations in a Model of Multidrug Resistance in Solid Tumors

Abstract

In this paper we develop a mathematical framework for describing multidrug resistance in cancer. To reflect the complexity of the underlying interplay between cancer cells and the therapeutic agent, we assume that the resistance level is a continuous parameter. Our model is written as a system of integro-differential equations that are parametrized by the resistance level. This model incorporates the cell-density and mutation dependence. Analysis and simulations of the model demonstrate how the dynamics evolves to a selection of one or more traits corresponding to different levels of resistance. The emerging limit distribution with nonzero variance is the desirable modeling outcome as it represents tumor heterogeneity.

A mathematical approach to differentiate spontaneous and induced evolution to drug resistance during cancer treatment

“…Mathematical modeling studies in particular have been used to explore both broad and detailed aspects of cancer drug resistance, as reviewed in [43,7,25]. The fundamental question of how the presence of drug resistant cells influences tumor dynamics and treatment outcomes has been thoroughly explored in mathematical models under a wide variety of assumptions [14,37,39,47,58,40,23,24,70,16,45,60,63,4,22,33,44,52,65,3,32,80,13,26,29,46,49,51,57,59,77,18,68,1,9,10,20]. Cancer models have also been utilized to assess how various underlying mechanisms contribute to the resistant phenotype [80,13,59,18,20], and to calculate the probability that drug resistance emerges within a specified time frame, be it before or during cancer treatment …”

“…As examples, pre-existing BCR-ABL kinase domain mutations confer resistance to the tyrosine kinase inhibitor imatinib in chronic myeloid leukemia patients [66,36], and pre-existing MEK1 mutations confer resistance to BRAF inhibitors in melanoma patients [8]. Many mathematical models have considered how the presence of preexisting resistant cells impact cancer progression and treatment [37,39,58,40,23,24,70,16,60,63,4,22,44,3,32,80,29,49,57,59,77,68,9,10].On the other hand, acquired drug resistance broadly describes the case in which drug resistance develops during the course of therapy from a population of cells that were initially drug sensitive [34]. The term "acquired resistance" is really an umbrella term for two distinct phenomena, which complicates the study of acquired resistance.…”

“…On the one hand there is resistance that is randomly acquired during the course of treatment, be it due to random genetic mutations or stochastic non-genetic phenotype switching [65]. This spontaneous form of acquired resistance has been considered in many mathematical models [14,39,47,58,40,23,24,45,4,22,33,44,52,32,80,26,46,51,57,10]. On the other hand, drug resistance can be induced (caused) by the drug itself [62,61,69,65,64].…”

Deterministic Mathematical Modelling for Cancer Chronotherapeutics: Cell Population Dynamics and Treatment Optimization

“…Quite recently, continuous models structured according to a phenotypic trait and based on integrodifferential equations, are common in ecology and are based on Darwinian selection principles, applied to the problem of emergence-and its overcoming by combinations of cytotoxic and cytostatic drugs-of drug resistant subpopulations in cancer cell populations, have been proposed [57,71,77]. They offer the advantage of allowing the possibility to represent slow evolution, according to the expression of a phenotype rather than by jumps due to pointwise mutations, of a cancer cell population, a feature which makes them amenable to represent epigenetic modifications on which drug-induced drug resistance is likely to rely when it is reversible.…”

Transfer of Drug Resistance Characteristics Between Cancer Cell Subpopulations: A Study Using Simple Mathematical Models

“…Many models have studied some of the key phenomena involved from the theoretical point of view (See, e.g., the review by Lorz et al (2013) or Lavi et al (2013), Greene et al (2014) and references therein).…”