Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Design of personalized cancer treatments by use of optimal control problems: The case of chronic myeloid leukemia

1 Mathematics Research Institute of the University of Valladolid (IMUVa) and Faculty of Economics, University of Valladolid, Spain
2 The Irma H. Russo, MD Breast Cancer Research Laboratory, Fox Chase Cancer Center, Philadelphia, USA

Special Issues: Understanding biology with mathematics and simulations

The advances in the mathematical explanation of the dynamics underlying treated cancer has opened the door to the mathematical design of optimal therapies. In parallel, the improvements and cost reductions in experimentation and data analysis techniques have made the formulation of personalized therapies possible. However, the design of cancer therapies making use of optimal control theory has not fully considered this possibility in detail. In this paper we contribute to the existing literature by analyzing the diverse alternatives that optimal therapy models offer to design personalized treatments. Taking as the starting point the Chronic Myeloid Leukemia (CML) optimal therapy model in [25], we design personalized optimal therapy models for patients with: CML; CML with intrinsic and/or induced resistance to the administered drug; CML and suffering high drug toxicity and/or allergy to the administered drug; and CML with presence of adverse factors. Along the paper we show that the clinical and medical applicability -the ultimate objective of this biomathematical research- of our proposed personalized models relies on the joint and proper use of the implemented calibration, simulation, and mathematical approaches and techniques. All the theoretical results generated by our personalized optimal therapy models are corroborated by clinical evidence.
  Article Metrics

Keywords system of difference equations; optimal control problem; chronic myeloid leukemia; imatinib therapy; calibration; personalized treatment

Citation: Pedro José Gutiérrez-Diez, Jose Russo. Design of personalized cancer treatments by use of optimal control problems: The case of chronic myeloid leukemia. Mathematical Biosciences and Engineering, 2020, 17(5): 4773-4800. doi: 10.3934/mbe.2020261


  • 1. A. J. Lotka, Contribution to the Theory of Periodic Reaction, J. Phys. Chem., 14 (1910), 271-274.
  • 2. A. J. Lotka, Analytical Note on Certain Rhythmic Relations in Organic Systems, Proc. Natl. Acad. Sci. U.S.A., 6 (1920), 410-415.
  • 3. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, 1925.
  • 4. V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. R. Accad. Naz. Lincei, 2 (1926), 31-113.
  • 5. V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 118, (1926), 558-560.
  • 6. V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Paris: Gauthier-Villars, 1931.
  • 7. L. A. Loeb, K. R. Loeb, J. P. Anderson, Multiple mutations and cancer, Proc. Natl. Acad. Sci. U.S.A., 100 (2003), 776-781.
  • 8. G. Magombedze, W. Garira, E. Mwenje, C. P. Bhunu, Optimal control for HIV-1 multi-drug therapy, Int. J. Comp. Math., 88 (2011), 314-340.
  • 9. N. Tarfulea, A mathematical model for HIV treatment with time-varying antiretroviral therapy, Int. J. Comp. Math., 88, (2011), 3217-3235.
  • 10. Y. Koizumi, S. Iwami, Mathematical modeling of multi-drugs therapy: A challenge for determining the optimal combinations of antiviral drugs, Theor. Biol. Med. Mod., 11 (2014), 41.
  • 11. H. T. Banks, Modeling and Control in the Biomedical Sciences, Springer, 1975.
  • 12. J. D. Murray, Mathematical Biology I: An Introduction, Springer, 2002.
  • 13. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003.
  • 14. C. W. Clark, Mathematical Bioeconomics. The Optimal Management of Renewable Resources, John Wiley & Sons, Inc., 1990.
  • 15. U. Ledzewicz, H. Schättler, A. Friedman, E. Kashdan, Mathematical methods and models in Biomedicine, Springer, 2013.
  • 16. M. Eisen, Mathematical Methods and Models in the Biological Sciences, Prentice Hall, 1988.
  • 17. P. J. Gutiérrez Diez, I. H. Russo, J. Russo, The Evolution of the Use of Mathematics in Cancer Research, Springer, 2012.
  • 18. A. Świerniak, U. Ledzewicz, H. Schättler, Optimal Control for a Class of Compartmental Models in Cancer Chemotherapy, Int. J. Appl. Math. Comp. Sci., 13 (2003), 357-368.
  • 19. J. M. Murray, Optimal control for a cancer chemotherapy problem with general growth and cost functions, Math. Biosci. 98 (1990), 273-287.
  • 20. J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. Biosci., 100 (1990), 49-67.
  • 21. S. Nanda, H. Moore, S. Lenhart, Optimal control of treatment in a mathematical model of chronic myelogenous leukemia, Math. Biosci., 210 (2007), 143-156.
  • 22. B. E. Aïnseba, C. Benosman, Optimal control for resistance and suboptimal response in CML, Math. Biosci., 227 (2010), 81-93.
  • 23. K. Schepers, E. M. Pietras, D. Reynaud, J. Flach, M. Binnewies, T. Garg, et al., Myeloproliferative neoplasia remodels the endosteal bone marrow niche into a self-reinforcing leukemic niche, Cell Stem Cell, 13 (2013), 285-299.    
  • 24. S. Kapoor, V. P. Rallabandi, C. Sakode, R. Padhi, P.K. Roy, A patient-specific therapeutic approach for tumour cell population extinction and drug toxicity reduction using control systems-based dose-profile design, Theor. Biol. Med. Mod., 10 (2013), 68.
  • 25. P. J. Gutiérrez-Diez, M. Á.,López-Marcos, J. Martínez-Rodríuez, J. Russo, The effects of time valuation in cancer optimal therapies: A study of chronic myeloid leukemia, Theor. Biol. Med. Mod., 16 (2019), 10.
  • 26. M. Baccarani, F. Castagnetti, G. Gugliotta, G. Rosti, A review of the European LeukemiaNet recommendations for the management of CML, Ann. Hematol., 94 (2015), S141-S147.
  • 27. M. W. Deininger, J. G. Hodgson, N. P. Shah, J. E. Cortes, D. W. Kim, F. E. Nicolini, et al., Compound mutations in BCR-ABL1 are not major drivers of primary or secondary resistance to ponatinib in CP-CML patients, Blood, 127 (2016), 703-712.
  • 28. F. Michor, T. P. Hughes, Y. Iwasa, S. Branford, N. P. Shah, C. L. Sawyers, et al., Dynamics of chronic myeloid leukaemia, Nature, 435 (2005), 1267-1270.
  • 29. D. Murray Lyon, Does the reaction to adrenalin obey Weber's Law?, J. Pharmacol. Exp. Ther., 21 (1923), 229-235.
  • 30. M. Inoue, K. Kaneko, Weber's law for biological responses in autocatalytic networks of chemical reactions, Phys. Rev. Lett., 107 (2011), 048301.
  • 31. H. W. Sinn, Weber's law and the biological evolution of risk preferences: The selective dominance of the logarithmic utility function, Gen. Pap. Risk Ins. Theor., 28 (2002), 87-100.
  • 32. R. Capdeville, S. Silverman, Imatinib: A targeted clinical drug development, Semin. Hematol., 40 (2003), 15-20.
  • 33. M. Bonifacio, F. Stagno, L. Scaffidi, M. Krampera, F. Di Raimondo, Management of Chronic Myeloid Leukemia in Advanced Phase, Front. Oncol., 9 (2019), 1132.
  • 34. S. Soverini, A. Hochhaus, F. E. Nicolini, F. Gruber, T. Lange, G. Saglio, et al., BCR-ABL kinase domain mutation analysis in chronic myeloid leukemia patients treated with tyrosine kinase inhibitors: Recommendations from an expert panel on behalf of European LeukemiaNet. Blood, 118 (2011), 1208-1215.
  • 35. I. Galinsky, S. Buchanan, Guide to Interpreting Disease Responses in Chronic Myeloid Leukemia, J. Adv. Prac. Oncol., 3, (2012), 225-236.
  • 36. S. Prabhu, D. Saadat, M. Zhang, L. Halbur, J. P. Fruehauf, S. T. Ong, A novel mechanism for BCR-ABL action: BCR-ABL-mediated induction of the eIF4F translation initiation complex and mRNA translation, Oncogene, 26 (2007), 1188-1200.
  • 37. L. Hu, L. Pu, D. Yang, C. Zhang, H. Wang, Y. Ding, et al., How to detect the rare BCR-ABL (e14a3) transcript: A case report and literature review, Oncol. Lett., 14 (2017), 5619-5623.
  • 38. S. I. Ismail, R. G. Naffa, A. F. Yousef, M. T. Ghanim, Incidence of BCR-ABL fusion transcripts in healthy individuals, Mol. Med. Rep. 9 (2014), 1271-1276.
  • 39. T. Hughes, G. Saglio, A. Quintás-Cardama, M. J. Mauro, D. W. Kim, J. H. Lipton, et al., BCR-ABL1 mutation development during first-line treatment with dasatinib or imatinib for chronic myeloid leukemia in chronic phase, Leukemia, 29 (2015), 1832-1838.    
  • 40. J. Kaeda, D. O'Shea, R. M. Szydlo, E. Olavarria, F. Dazzi, D. Marin, et al., Serial measurement of BCR-ABL transcripts in the peripheral blood after allogeneic stem cell transplantation for chronic myeloid leukemia: An attempt to define patients who may not require further therapy, Blood, 107 (2006), 4171-4176.
  • 41. M. Houshmand, G. Simonetti, P. Circosta, V. Gaidano, A. Cignetti, G. Martinelli, et al., Chronic myeloid leukemia stem cells, Leukemia, 33 (2019), 1543-1556.
  • 42. F. Loscocco, G. Visani, S. Galimberti, A. Curti, A. Isidori, BCR-ABL Independent Mechanisms of Resistance in Chronic Myeloid Leukemia, Front. Oncol., 9 (2019), 939.
  • 43. H. Kitamura, Y. Tabe, T. Ai, K. Tsuchiya, M. Yuri, S. Misawa, et al., A new highly sensitive real-time quantitative-PCR method for detection of BCR-ABL1 to monitor minimal residual disease in chronic myeloid leukemia after discontinuation of imatinib, PLoS One, 14 (2019), e0207170.
  • 44. R. Arora, R. D. Press, Measurement of BCR-ABL1 transcripts on the International Scale in the United States: Current status and best practices, Leuk. Lymphoma, 58 (2017), 8-16.
  • 45. Physycians' Desk Reference, PDR Network (ed) 64 edition, LLC at Montvale, 2010.
  • 46. A. Quintás-Cardama, J. Cortes, H. Kantarjian, Biology of Chronic and Acute Myeloid Leukemia, in The Molecular Basis of Cancer, Elsevier, (2008), 371-383.
  • 47. L. Löf, L. Arngården, U. Olsson-Strömberg, B. Siart, M. Jansson, J. S. Dahlin, et al., Flow Cytometric Measurement of Blood Cells with BCR-ABL1 Fusion Protein in Chronic Myeloid Leukemia, Sci. Rep. 7 (2017), 623.
  • 48. D. Raspadori, P. Pacelli, A. Sicuranza, E. Abruzzese, A. Iurlo, D. Cattaneo, et al., Flow Cytometry Assessment of CD26+ Leukemic Stem Cells in Peripheral Blood: A Simple and Rapid New Diagnostic Tool for Chronic Myeloid Leukemia, Cytometry Part B, 96 (2019), 294-299.
  • 49. F. E. Craig, K. A. Foon, Flow cytometric immunophenotyping for hematologic neoplasms, Blood, 111 (2008), 3941-3967.
  • 50. D. Campana, Minimal residual disease in acute lymphoblastic leukemia, Semin. Hematol., 46 (2009), 100-106.
  • 51. D. Campana, Role of minimal residual disease monitoring in adult and pediatric acute lymphoblastic leukemia, Haematol./Oncol. Clin. N.A., 23 (2009), 1083-1098.
  • 52. N. Shah,Medical management of CML, Hematology, 1 (2007), 371-375.
  • 53. P. K. Bhamidipati, H. Kantarjian, J. Cortes, A. M. Cornelison, E. Jabbour, Management of imatinib-resistant patients with chronic myeloid leukemia, Ther. Adv. Hematol., 4 (2013), 103-117.
  • 54. M. S. Marcolino, E. Boersma, N. C. D. Clementino, A. V. Macedo, A. D. Marx-Neto, M. H. C. R. Silva, et al., Imatinib treatment duration is related to decreased estimated glomerular filtration rate in chronic myeloid leukemia patients, Ann. Oncol., 22 (2011), 2073-2079.
  • 55. M. Baccarani, J. Cortes, F. Pane, D. Niederwieser, G. Saglio, J. Apperley, et al., Chronic myeloid leukemia: An update of concepts and management recommendations of European LeukemiaNet, J. Clin. Oncol., 27 (2009), 6041-6051.
  • 56. R. A. Everett, Y. Zhao, K. B. Flores, Y. Kuang, Data and implication based comparison of two chronic myeloid leukemia models, Math. Biosci. Eng., 10 (2013), 1501-1518.
  • 57. C. Fava, G. Saglio, Can we and should we improve on frontline imatinib therapy for chronic myeloid leukemia?, Semin. Hematol., 47 (2010), 319-326.
  • 58. P. Valent, Imatinib-resistant chronic myeloid leukemia (CML): Current concepts on pathogenesis and new emerging pharmacologic approaches, Biol. Targets Ther., 1 (2007), 433-448.
  • 59. R. Bhatia, Chronic Myeloid Leukemia, in Hematology, seventh edition, Elsevier, (2018), 1055-1070.
  • 60. T. R. Randolph, Myeloproliferative neoplasms, in Rodak's Hematology Clinical Applications and Principles, St. Louis, Missouri: Saunders, (2015), 561-590.
  • 61. N. Bouizem, M. Helal, B. Aïnseba, A. Lakmeche, Leukemia mathematical model, ITM Web Conf., 4 (2015), 01006.
  • 62. A. Hochhaus, S. Kreil, A. S. Corbin, P. La Rosée, M. C. Müller, T. Lahaye, et al., Molecular and chromosomal mechanisms of resistance to imatinib (STI571) therapy, Leukemia, 16 (2002), 2190-2196.
  • 63. V. Karavasilis, A. Reid, R. Sinha, J. S. De Bono, Cancer drug resistance, in Cancer Drug Design and Discovery, (S. Neidle ed.), (2008), Elsevier.
  • 64. M. Baccarani, G. Saglio, J. Goldman, A. Hochhaus, B. Simonsson, F. Appelbaum, et al., Evolving concepts in the management of chronic myeloid leukemia: recommendations from an expert panel on behalf of the European Leukemia Net, Blood, 108 (2006), 1809-1820.
  • 65. R.P. Nelson Jr, K. Cornetta, K. E. Ward, S. Ramanuja, C. Fausel, L.D. Cripe, Desensitization to imatinib in patients with leukemia, Ann. Allergy, Asthma, Immunol., 97 (2006), 216-222.
  • 66. M. Albayrak, H. Celebi, A. Albayrak, E. S. Can, V. Aslan, B. Onec, et al., Serious skin reaction associated with imatinib in a patient with chronic myeloid leukemia, Eurasian J. Med., 43 (2011), 192-195.
  • 67. V. Chou, S. McClelland, D. Resnick, M. Lee-Wong, Successful Desensitization of an Adult with Type I Hypersensitivity to Imatinib, Internet J. Asthma, Allergy Immunol., 4 (2004), 2.
  • 68. E. Faber, M. Divoká, I. Skoumalová, M. Novák, I. Marešová, K. Mičová, et al., A lower dosage of imatinib is sufficient to maintain undetectable disease in patients with chronic myeloid leukemia with long-term low-grade toxicity of the treatment, Leuk. Lymphoma, 57 (2016), 370-375.
  • 69. Y. Zhu, S. X. Qian, Clinical efficacy and safety of imatinib in the management of Ph(+) chronic myeloid or acute lymphoblastic leukemia in Chinese patients, OncoTargets Ther., 7 (2014), 395-404.
  • 70. European Leukemia Net, 2020. Available from: https://www.leukemia-net.org/content/leukemias/cml/euro_and_sokal_score/index_eng.html.
  • 71. L. C. Kuntegowdanahalli, G. B. Kanakasetty, A. H. Thanky, L. Dasappa, L. A. Jacob, S. B. Mallekavu, et al., Prognostic and predictive implications of Sokal, Euro and EUTOS scores in chronic myeloid leukaemia in the imatinib era-experience from a tertiary oncology centre in Southern India, Ecancermedicalscience, 10 (2016), 679.
  • 72. S. Chhikara, S. Sazawal, K. Singh, R. Chaubey, H. Pati, S. Tyagi, et al., Comparative analysis of the Sokal, Euro and European Treatment and Outcome Study score in prognostication of Indian chronic myeloid leukemia-chronic phase patients on imatinib, S.A. J. Cancer, 7, (2018), 258-262.
  • 73. J. Hasford, M. Baccarani, V. Hoffmann, J. Guilhot, S. Saussele, G. Rosti, et al., Predicting complete cytogenetic response and subsequent progression-free survival in 2060 patients with CML on imatinib treatment: the EUTOS score, Blood, 118 (2011), 686-692.
  • 74. M. Baccarani, F. Castagnetti, G. Gugliotta, G. Rosti, A review of the European Leukemia Net recommendations for the management of CML, Ann. Hematol., 94 (2015), S141-S147.
  • 75. M. Schemionek, T. Spieker, L. Kerstiens, C. Elling, M. Essers, A. Trumpp, et al., Leukemic spleen cells are more potent than bone marrow-derived cells in a transgenic mouse model of CML, Leukemia, 26, (2012), 1030-1037.
  • 76. L.O. Ngolet, I. Kocko, A. Elira-Dokekias, Pregnancy and Accelerated Phase of Myeloid Chronic Leukemia Treated with Imatinib: A Case Report from a Developing Country, Case Rep. Hematol., (2016), 6104948.
  • 77. E. Fimmel, Y. S. Semenov, A. S. Bratus, On optimal and suboptimal treatment strategies for a mathematical model of leukemia, Math. Biosci. Eng., 10 (2013), 151-165.
  • 78. M. Bocchia, A. Sicuranza, E. Abruzzese, A. Iurlo, S. Sirianni, A. Gozzini, et al., Residual Peripheral Blood CD26+ Leukemic Stem Cells in Chronic Myeloid Leukemia Patients During TKI Therapy and During Treatment-Free Remission, Front. Oncol., 8 (2018), 194
  • 79. G. Caocci, B. Martino, M. Greco, E. Abruzzese, M. M. Trawinska, S. Lai, et al., Killer immunoglobulin-like receptors can predict TKI treatment-free remission in chronic myeloid leukemia patients, Exp. Hematol., 43 (2015), 1015-1018.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved