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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

Abstract

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.

Keywords

Multi-drug resistance Cancer dynamics Diffusion 

Notes

Acknowledgements

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.

References

  1. Amir ED, Davis KL, Tadmor MD, Simonds EF, Jacob Levlne H, Bendall SC, Shenfeld DK, Krishnaswamy S, Nolan GP, Pe’er D (2013) viSNE enables visualization of high dimensional single-cell data and reveals phenotypic heterogeneity of leukemia. Nat Biotechnol 31(6):545–552CrossRefGoogle Scholar
  2. Anderson AR, Chaplain M (1998) Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull Math Biol 60(5):857–899CrossRefzbMATHGoogle Scholar
  3. Anderson AR, Weaver AM, Cummings PT, Quaranta V (2006) Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127(5):905–915CrossRefGoogle Scholar
  4. Ariffin AB, Forde PF, Jahangeer S, Soden DM, Hinchion J (2014) Releasing pressure in tumors: What do we know so far and where do we go from here? Cancer Res 74:2655–62CrossRefGoogle Scholar
  5. Birkhead BG, Rakin EM, Gallivan S, Dones L, Rubens RD (1987) A mathematical model of the development of drug resistance to cancer chemotherapy. Eur J Cancer Clin Oncol 23:1421–1427CrossRefGoogle Scholar
  6. Brodie EDIII (1992) Correlational selection for color pattern and antipredator behavior in the garter snake Thamnophis ordinoides. Evolution 46:1284–1298CrossRefGoogle Scholar
  7. Corbett T, Griswold D, Roberts B, Peckham J, Schabel F (1975) Tumor induction relationships in development of transplantable cancers of the colon in mice for chemotherapy assays, with a note on carcinogen structure. Cancer Res 35:2434–2439Google Scholar
  8. de Bruin EC, Taylor TB, Swanton C (2013) Intra-tumor heterogeneity: lessons from microbial evolution and clinical implications. Genome Med 5:1–11CrossRefGoogle Scholar
  9. de Pillis LG, Radunskaya AE, Wiseman CL (2005) A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res 65:7950–7958CrossRefGoogle Scholar
  10. de Pillis LG, Savage H, Radunskaya AE (2014) Mathematical model of colorectal cancer with monoclonal antibody treatments. Br J Med Med Res 4:3101–3131CrossRefGoogle Scholar
  11. Fodal V, Pierobon M, Liotta L, Petricoin E (2011) Mechanisms of cell adaptation: When and how do cancer cells develop chemoresistance? Cancer J 17(2):89–95CrossRefGoogle Scholar
  12. Foo J, Michor F (2014) Evolution of acquired resistance to anti-cancer therapy. J Theor Biol 355:10–20CrossRefGoogle Scholar
  13. Garvey CM, Spiller E, Lindsay D, Ct Chiang, Choi NC, Agus DB, Mallick P, Foo J, Mumenthaler SM (2016) A high-content image-based method for quantitatively studying context-dependent cell population dynamics. Sci Rep 6(29752):1–12Google Scholar
  14. Gerlinger M, Rowan AJ, Horswell S, Larkin J, Endesfelder D, Gronroos E, Martinez P (2012) Intratumor heterogeneity and branched evolution revealed by multiregion sequencing. N Engl J Med 366:883–892CrossRefGoogle Scholar
  15. Gillet JP, Gottesman MM (2010) Mechanisms of multidrug resistance in cancer. Methods Mol Biol 596:47–76CrossRefGoogle Scholar
  16. Goldie JH, Coldman AJ (1979) A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat Rep 63:1727–1733Google Scholar
  17. Goldie JH, Coldman AJ (1983a) A model for resistance of tumor cells to cancer chemotherapeutic agents. Math Biosci 65:291–307CrossRefzbMATHGoogle Scholar
  18. Goldie JH, Coldman AJ (1983b) Quantative model for multiple levels of drug resistance in clinical tumors. Cancer Treat Rep 67:923–931Google Scholar
  19. Gottesman MM (2002) Mechanisms of cancer drug resistance. Annu Rev Med 53:615–627CrossRefGoogle Scholar
  20. Gottesman MM, Fojo T, Bates SE (2002) Multidrug resistance in cancer: role of ATP-dependent transporters. Nat Rev Cancer 2(1):48–58CrossRefGoogle Scholar
  21. Greene J, Lavi O, Gottesman MM, Levy D (2014) The impact of cell density and mutations in a model of multidrug resistance in solid tumors. Bull Math Biol 74:627–653MathSciNetCrossRefzbMATHGoogle Scholar
  22. Grothey A (2006) Defining the role of panitumumab in colorectal cancer. Community Oncol 3:6–10Google Scholar
  23. Hanahan D, Weinberg RA (2011) Hallmarks of cancer: the next generation. Cell 144(5):646–674CrossRefGoogle Scholar
  24. Iwasa Y, Nowak MA, Michor F (2006) Evolution of resistance during clonal expansion. Genetics 172:2557–2566CrossRefGoogle Scholar
  25. Jabin PE, Schram RS (2016) Selection-mutation dynamics with spatial dependence pp 1–21Google Scholar
  26. Kimmel M, Swierniak A, Polanski A (1998) Infinite-dimensional model of evolution of drug resistance of cancer cells. J Math Syst Estim Control 8:1–16MathSciNetzbMATHGoogle Scholar
  27. Komarova N (2006) Stochastic modeling of drug resistance in cancer. Theor Popul Biol 239(3):351–366MathSciNetCrossRefGoogle Scholar
  28. Lavi O, Gottesman MM, Levy D (2012) The dynamics of drug resistance: a mathematical perspective. Drug Resist Update 15(1–2):90–97CrossRefGoogle Scholar
  29. Lavi O, Greene J, Levy D, Gottesman MM (2013) The role of cell density and intratumoral heterogeneity in multidrug resistance. Cancer Res 73:7168–7175CrossRefGoogle Scholar
  30. Lavi O, Greene J, Levy D, Gottesman MM (2014) Simplifying the complexity of resistance heterogeneity in metastatic cancer. Trend Mol Med 20:129–136CrossRefGoogle Scholar
  31. Lorz A, Lorenzi T, Hochberg ME, Clairambault J, Perthame B (2013) Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies. ESAIM Math Model Numer Anal 47:377–399MathSciNetCrossRefzbMATHGoogle Scholar
  32. Lorz A, Lorenzi T, Clairambault J, Escargueil A, Perthame B (2015) Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull Math Biol 77:1–22MathSciNetCrossRefzbMATHGoogle Scholar
  33. Medema JP (2013) Cancer stem cells: the challenges ahead. Nat Cell Biol 15(4):338–344CrossRefGoogle Scholar
  34. Michor F, Nowak MA, Iwasa Y (2006) Evolution of resistance to cancer therapy. Curr Pharm Des 12:261–271CrossRefGoogle Scholar
  35. Minchinton AI, Tannock IF (2006) Drug penetration in solid tumours. Nat Rev Cancer 6:583–592CrossRefGoogle Scholar
  36. Mirrahimi S, Perthame B (2015) Asymptotic analysis of a selection model with space. J Math Pures Appl 104:1108–1118MathSciNetCrossRefzbMATHGoogle Scholar
  37. Mumenthaler SM, Foo J, Choi NC, Heise N, Leder K, Agus DB, Pao W, Michor F, Mallick P (2015) The impact of microenvironmental heterogeneity on the evolution of drug resistance in cancer cells. Cancer Inform 14:19–31Google Scholar
  38. Panagiotopoulou V, Richardson G, Jensen OE, Rauch C (2010) On a biophysical and mathematical model of Pgp-mediated multidrug resistance: understanding the space-time dimension of MDR. Eur Biophys J 39:201–211CrossRefGoogle Scholar
  39. Panetta JC (1998) A mathematical model of drug resistance: heterogeneous tumors. Math Biosci 147:41–61CrossRefzbMATHGoogle Scholar
  40. Rainey PB, Travisano M (1998) Adaptive radiation in a heterogeneous environment. Nature 2:69–72CrossRefGoogle Scholar
  41. Roose T, Chapman SJ, Maini PK (2007) Mathematical models of avascular tumor growth. SIAM Rev 49(2):179–208MathSciNetCrossRefzbMATHGoogle Scholar
  42. Schättler H, Ledzewicz U (2015) Optimal control for mathematical models of cancer therapies, 1st edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  43. Simon R, Norton L (2006) The norton-simon hypothesis: designing more effective and less toxic chemotherapeutic regimens. Nat Clin Pract Oncol 3:406–407CrossRefGoogle Scholar
  44. Swierniak A, Kimmel M, Smieja J (2009) Mathematical modeling as a tool for planning anticancer therapy. Eur J Pharmacol 625(1–3):108–121CrossRefGoogle Scholar
  45. Teicher BA (2006) Cancer drug resistance. Humana Press, TotowaCrossRefGoogle Scholar
  46. Tomasetti C, Levy D (2010) An elementary approach to modeling drug resistance in cancer. Math Biosci Eng 7:905–918MathSciNetCrossRefzbMATHGoogle Scholar
  47. Trédan O, Galmarini CM, Patel K, Tannock IF (2007) Drug resistance and the solid tumor microenvironment. J Natl Cancer Inst 99:1441–1454CrossRefGoogle Scholar
  48. Vaupel P, Kallinowski F, Okunieff P (1989) Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review. Cancer Res 49:6449–6465Google Scholar
  49. Wosikowski K, Silverman JA, Bishop P, Mendelsohn J, Bates SE (2000) Reduced growth rate accompanied by aberrant epidermal growth factor signaling in drug resistant human breast cancer cells. Biochimica et Biophysica Acta 1497(2):215–226CrossRefGoogle Scholar
  50. Wu A, Loutherback K, Lambert G, Estévez-Salmerón L, Tlsty TD, Austin RH, Sturm JC (2013) Cell motility and drug gradients in the emergence of resistance to chemotherapy. Proc Natl Acad Sci 110(40):16,103–16,108CrossRefGoogle Scholar
  51. Yu P, Mustata M, Peng L, Turek JJ, Melloch MR, French PM, Nolte DD (2004) Holographic optical coherence imaging of rat osteogenic sarcoma tumor spheroids. Appl Opt 43:4862–4873CrossRefGoogle Scholar

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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