Bulletin of Mathematical Biology

, Volume 79, Issue 12, pp 2986–3012 | Cite as

Modeling the Dynamics of Heterogeneity of Solid Tumors in Response to Chemotherapy

  • Heyrim Cho
  • Doron LevyEmail author
Original Article


In this paper, we extend the model of the dynamics of drug resistance in a solid tumor that was introduced by Lorz et al. (Bull Math Biol 77:1–22, 2015). Similarly to the original, radially symmetric model, the quantities we follow depend on a phenotype variable that corresponds to the level of drug resistance. The original model is modified in three ways: (i) We consider a more general growth term that takes into account the sensitivity of resistance level to high drug dosage. (ii) We add a diffusion term in space for the cancer cells and adjust all diffusion terms (for the nutrients and for the drugs) so that the permeability of the resource and drug is limited by the cell concentration. (iii) We add a mutation term with a mutation kernel that corresponds to mutations that occur regularly or rarely. We study the dynamics of the emerging resistance of the cancer cells under continuous infusion and on–off infusion of cytotoxic and cytostatic drugs. While the original Lorz model has an asymptotic profile in which the cancer cells are either fully resistant or fully sensitive, our model allows the emergence of partial resistance levels. We show that increased drug concentrations are correlated with delayed relapse. However, when the cancer relapses, more resistant traits are selected. We further show that an on–off drug infusion also selects for more resistant traits when compared with a continuous drug infusion of identical total drug concentrations. Under certain conditions, our model predicts the emergence of a heterogeneous tumor in which cancer cells of different resistance levels coexist in different areas in space.


Multi-drug resistance Cancer dynamics Diffusion 



The work of DL was supported in part by the National Science Foundation under Grant Number DMS-1713109, by the John Simon Guggenheim Memorial Foundation, by the Simons Foundation, and by the Jayne Koskinas and Ted Giovanis Foundation.


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Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of Mathematics and Center for Scientific Computation and Mathematical Modeling (CSCAMM)University of MarylandCollege ParkUSA

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