Bulletin of Mathematical Biology

, Volume 77, Issue 1, pp 1–22 | Cite as

Modeling the Effects of Space Structure and Combination Therapies on Phenotypic Heterogeneity and Drug Resistance in Solid Tumors

  • Alexander LorzEmail author
  • Tommaso Lorenzi
  • Jean Clairambault
  • Alexandre Escargueil
  • Benoît Perthame
Original Article


Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of selection in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of selection driven by the local cell environment? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a phenotype of resistance to cytotoxic drugs. The model takes explicitly into account the dynamics of resources and anticancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and bang–bang delivery of cytotoxic and cytostatic drugs.


Mathematical oncology Integro-differential equations Space structure Adaptive evolution Cancer Drug resistance Intra-tumor heterogeneity  



The research leading to this paper was (partially) funded by the French “ANR blanche” Project Kibord: ANR-13-BS01-0004. T.L. was supported by the Fondation Sciences Mathématiques de Paris and by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098), and by the FIRB Project—RBID08PP3J.


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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  • Alexander Lorz
    • 1
    • 2
    • 3
    Email author
  • Tommaso Lorenzi
    • 1
    • 2
    • 3
  • Jean Clairambault
    • 1
    • 2
    • 3
  • Alexandre Escargueil
    • 4
    • 5
  • Benoît Perthame
    • 1
    • 2
    • 3
  1. 1.Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.UMR 7598, Laboratoire Jacques-Louis LionsCNRSParisFrance
  3. 3.INRIA-Paris-Rocquencourt, EPC MAMBALe Chesnay CedexFrance
  4. 4.Sorbonne Universités, UPMC Univ Paris 06ParisFrance
  5. 5.UMR_S 938, Laboratory of “Cancer Biology and Therapeutics”INSERMParisFrance

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