Bulletin of Mathematical Biology

, Volume 80, Issue 7, pp 1776–1809 | Cite as

Optimal Therapy Scheduling Based on a Pair of Collaterally Sensitive Drugs

  • Nara Yoon
  • Robert Vander Velde
  • Andriy Marusyk
  • Jacob G. Scott
Original Article


Despite major strides in the treatment of cancer, the development of drug resistance remains a major hurdle. One strategy which has been proposed to address this is the sequential application of drug therapies where resistance to one drug induces sensitivity to another drug, a concept called collateral sensitivity. The optimal timing of drug switching in these situations, however, remains unknown. To study this, we developed a dynamical model of sequential therapy on heterogeneous tumors comprised of resistant and sensitive cells. A pair of drugs (DrugA, DrugB) are utilized and are periodically switched during therapy. Assuming resistant cells to one drug are collaterally sensitive to the opposing drug, we classified cancer cells into two groups, \(A_\mathrm{R}\) and \(B_\mathrm{R}\), each of which is a subpopulation of cells resistant to the indicated drug and concurrently sensitive to the other, and we subsequently explored the resulting population dynamics. Specifically, based on a system of ordinary differential equations for \(A_\mathrm{R}\) and \(B_\mathrm{R}\), we determined that the optimal treatment strategy consists of two stages: an initial stage in which a chosen effective drug is utilized until a specific time point, T, and a second stage in which drugs are switched repeatedly, during which each drug is used for a relative duration (i.e., \(f \Delta t\)-long for DrugA and \((1-f) \Delta t\)-long for DrugB with \(0 \le f \le 1\) and \(\Delta t \ge 0\)). We prove that the optimal duration of the initial stage, in which the first drug is administered, T, is shorter than the period in which it remains effective in decreasing the total population, contrary to current clinical intuition. We further analyzed the relationship between population makeup, \(\mathcal {A/B} = A_\mathrm{R}/B_\mathrm{R}\), and the effect of each drug. We determine a critical ratio, which we term \(\mathcal {(A/B)}^{*}\), at which the two drugs are equally effective. As the first stage of the optimal strategy is applied, \(\mathcal {A/B}\) changes monotonically to \(\mathcal {(A/B)}^{*}\) and then, during the second stage, remains at \(\mathcal {(A/B)}^{*}\) thereafter. Beyond our analytic results, we explored an individual-based stochastic model and presented the distribution of extinction times for the classes of solutions found. Taken together, our results suggest opportunities to improve therapy scheduling in clinical oncology.


Cancer Evolution of resistance Dynamical systems Optimal control 


  1. Amirouchene-Angelozzi N, Swanton C, Bardelli A (2017) Tumor evolution as a therapeutic target. Cancer Discov 7(8):805–817CrossRefGoogle Scholar
  2. Atuegwu NC, Arlinghaus LR, Li X, Chakravarthy AB, Abramson VG, Sanders ME, Yankeelov TE (2013) Parameterizing the logistic model of tumor growth by dw-mri and dce-mri data to predict treatment response and changes in breast cancer cellularity during neoadjuvant chemotherapy. Transl Oncol 6(3):256–264CrossRefGoogle Scholar
  3. Berry SJ, Coffey DS, Walsh PC, Ewing LL (1984) The development of human benign prostatic hyperplasia with age. J Urol 132(3):474–479CrossRefGoogle Scholar
  4. Boston EA, Gaffney EA (2011) The influence of toxicity constraints in models of chemotherapeutic protocol escalation. Math Med Biol J IMA 28(4):357–384MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chen J-H, Kuo Y-H, Luh HP (2013) Optimal policies of non-cross-resistant chemotherapy on goldie and coldmans cancer model. Math Biosci 245(2):282–298MathSciNetCrossRefzbMATHGoogle Scholar
  6. Coldman A, Goldie J (1983) A model for the resistance of tumor cells to cancer chemotherapeutic agents. Math Biosci 65(2):291–307CrossRefzbMATHGoogle Scholar
  7. Dhawan A, Nichol D, Kinose F, Abazeed ME, Marusyk A, Haura EB, Scott JG (2017) Collateral sensitivity networks reveal evolutionary instability and novel treatment strategies in ALK mutated non-small cell lung cancer. Sci Rep 7:1232CrossRefGoogle Scholar
  8. Gaffney E (2004) The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling. J Math Biol 48(4):375–422MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gaffney E (2005) The mathematical modelling of adjuvant chemotherapy scheduling: incorporating the effects of protocol rest phases and pharmacokinetics. Bull Math Biol 67(3):563–611MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gerlee P (2013) The model muddle: in search of tumor growth laws. Cancer Res 73(8):2407–2411CrossRefGoogle Scholar
  11. Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22(4):403–434MathSciNetCrossRefGoogle Scholar
  12. Gillies RJ, Verduzco D, Gatenby RA (2012) Evolutionary dynamics unifies carcinogenesis and cancer therapy. Nat Rev Cancer 12(7):487CrossRefGoogle Scholar
  13. Goldie J (1982) Rationale for the use of alternating non-cross-resistant chemotherapy. Cancer Treat Rep 66:439–449Google Scholar
  14. Goldie J, Coldman A (1979) A mathematic model for relating the drug sensitivity of tumors to their spontaneous mutation rate. Cancer Treat Rep 63(11–12):1727–1733Google Scholar
  15. Goldie J, Coldman A (1983) Quantitative model for multiple levels of drug resistance in clinical tumors. Cancer Treat Rep 67(10):923–931Google Scholar
  16. Goldie JH, Coldman AJ (2009) Drug resistance in cancer: mechanisms and models. Cambridge University Press, CambridgezbMATHGoogle Scholar
  17. Holohan C, Van Schaeybroeck S, Longley DB, Johnston PG (2013) Cancer drug resistance: an evolving paradigm. Nat Rev Cancer 13(10):714CrossRefGoogle Scholar
  18. Hutchison DJ (1963) Cross resistance and collateral sensitivity studies in cancer chemotherapy. Adv Cancer Res 7:235–350CrossRefGoogle Scholar
  19. Imamovic L, Sommer MO (2013) Use of collateral sensitivity networks to design drug cycling protocols that avoid resistance development. Sci Transl Med 5(204):204ra132–204ra132CrossRefGoogle Scholar
  20. Jackson TL, Byrne HM (2000) A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. Math Biosci 164(1):17–38MathSciNetCrossRefzbMATHGoogle Scholar
  21. Jonsson VD, Blakely CM, Lin L, Asthana S, Matni N, Olivas V, Pazarentzos E, Gubens MA, Bastian BC, Taylor BS et al (2017) Novel computational method for predicting polytherapy switching strategies to overcome tumor heterogeneity and evolution. Sci Rep 7:44206CrossRefGoogle Scholar
  22. Katouli AA, Komarova NL (2011) The worst drug rule revisited: mathematical modeling of cyclic cancer treatments. Bull Math Biol 73(3):549–584MathSciNetCrossRefzbMATHGoogle Scholar
  23. Kaznatcheev A, Vander Velde R, Scott JG, Basanta D (2017) Cancer treatment scheduling and dynamic heterogeneity in social dilemmas of tumour acidity and vasculature. Br J Cancer 116(6):785CrossRefGoogle Scholar
  24. Komarova NL, Wodarz D (2003) Evolutionary dynamics of mutator phenotypes in cancer. Cancer Res 63(20):6635–6642Google Scholar
  25. Komarova NL, Wodarz D (2005) Drug resistance in cancer: principles of emergence and prevention. Proc Natl Acad Sci USA 102(27):9714–9719CrossRefGoogle Scholar
  26. Legler JM, Feuer EJ, Potosky AL, Merrill RM, Kramer BS (1998) The role of prostate-specific antigen (psa) testing patterns in the recent prostate cancer incidence declinein the united states. Cancer Causes Control 9(5):519–527CrossRefGoogle Scholar
  27. Marusyk A, Almendro V, Polyak K (2012) Intra-tumour heterogeneity: a looking glass for cancer? Nat Rev Cancer 12(5):323CrossRefGoogle Scholar
  28. Monzon FA, Ogino S, Hammond MEH, Halling KC, Bloom KJ, Nikiforova MN (2009) The role of kras mutation testing in the management of patients with metastatic colorectal cancer. Arch Pathol Lab Med 133(10):1600–1606Google Scholar
  29. Nichol D, Jeavons P, Fletcher AG, Bonomo RA, Maini PK, Paul JL, Gatenby RA, Anderson AR, Scott JG (2015) Steering evolution with sequential therapy to prevent the emergence of bacterial antibiotic resistance. PLoS Comput Biol 11(9):e1004493CrossRefGoogle Scholar
  30. Nichol D, Rutter J, Bryant C, Jeavons P, Anderson A, Bonomo R, Scott J (2017) Collateral sensitivity is contingent on the repeatability of evolution, bioRxiv, pp 185892Google Scholar
  31. Palmer AC, Sorger PK (2017) Combination cancer therapy can confer benefit via patient-to-patient variability without drug additivity or synergy. Cell 171(7):1678–1691CrossRefGoogle Scholar
  32. Pisco AO, Brock A, Zhou J, Moor A, Mojtahedi M, Jackson D, Huang S (2013) Non-darwinian dynamics in therapy-induced cancer drug resistance. Nat Commun 4:2467CrossRefGoogle Scholar
  33. Rejniak KA (2005) A single-cell approach in modeling the dynamics of tumor microregions. Math Biosci Eng 2(3):643–655MathSciNetCrossRefzbMATHGoogle Scholar
  34. Scheel C, Weinberg RA (2011) Phenotypic plasticity and epithelial-mesenchymal transitions in cancer and normal stem cells? Int J Cancer 129(10):2310–2314CrossRefGoogle Scholar
  35. Scott J, Marusyk A (2017) Somatic clonal evolution: a selection-centric perspective. Biochimica et Biophysica Acta (BBA) Rev Cancer 1867(2):139–150CrossRefGoogle Scholar
  36. Scott JG, Fletcher AG, Anderson AR, Maini PK (2016) Spatial metrics of tumour vascular organisation predict radiation efficacy in a computational model. PLoS Comput Biol 12(1):e1004712CrossRefGoogle Scholar
  37. Swanson KR, Rockne RC, Claridge J, Chaplain MA, Alvord EC, Anderson AR (2011) Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology. Cancer Res 71(24):7366–7375CrossRefGoogle Scholar
  38. Thomas A, El Rouby S, Reed JC, Krajewski S, Silber R, Potmesil M, Newcomb EW (1996) Drug-induced apoptosis in b-cell chronic lymphocytic leukemia: relationship between p53 gene mutation and bcl-2/bax proteins in drug resistance. Oncogene 12(5):1055–1062Google Scholar
  39. Tomasetti C, Levy D (2010) An elementary approach to modeling drug resistance in cancer. Math Biosci Eng MBE 7(4):905MathSciNetCrossRefzbMATHGoogle Scholar
  40. Werner B, Scott JG, Sottoriva A, Anderson AR, Traulsen A, Altrock PM (2016) The cancer stem cell fraction in hierarchically organized tumors can be estimated using mathematical modeling and patient-specific treatment trajectories. Cancer Res 76(7):1705–1713CrossRefGoogle Scholar
  41. Wilson WH, Teruya-Feldstein J, Fest T, Harris C, Steinberg SM, Jaffe ES, Raffeld M (1997) Relationship of p53, bcl-2, and tumor proliferation to clinical drug resistance in non-hodgkin9s lymphomas. Blood 89(2):601–609Google Scholar
  42. Zhao B, Sedlak JC, Srinivas R, Creixell P, Pritchard JR, Tidor B, Lauffenburger DA, Hemann MT (2016) Exploiting temporal collateral sensitivity in tumor clonal evolution. Cell 165(1):234–246CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  • Nara Yoon
    • 1
  • Robert Vander Velde
    • 2
  • Andriy Marusyk
    • 3
  • Jacob G. Scott
    • 1
  1. 1.Department of Translational Hematology and Oncology ResearchCleveland ClinicClevelandUSA
  2. 2.Department of Molecular Medicine, Morsani College of MedicineUniversity of South FloridaTampaUSA
  3. 3.Department of Cancer Imaging and MetabolismH. Lee Moffitt Cancer Center and Research InstituteTampaUSA

Personalised recommendations