Mathematical modelling of drug resistance in anticancer therapies



Polish National Science Centre Grant No. 2016/23/N/ST1/01178 entitled "Mathematical modelling of drug resistance in anticancer therapies":

Streszczenie (PL):

Description (EN):

  • Bodzioch M., Bajger P., Foryś U., Angiogenesis and chemotherapy resistance: optimizing chemotherapy scheduling using mathematical modeling, Journal of Cancer Research and Clinical Oncology, Vol. 147, No. 8 (2021), pp. 2281-2299, doi: 10.1007/s00432-021-03657-9.



    Chemotherapy remains a widely used cancer treatment. Acquired drug resistance may greatly reduce the efficacy of treatment and means to overcome it are a topic of active discussion among researchers. One of the proposed solutions is to shift the therapeutic paradigm from complete eradication of cancer to maintenance, i.e., to treat it as a chronic disease. A concept of metronomic therapy (low chemotherapy doses applied continuously) emerged in early 2000s and was henceforth shown to offer a number of benefits, including targeting endothelial cells and reducing acquired drug resistance. Using mathematical modeling and optimal control techniques, we investigate the hypothesis that lower doses of chemotherapy are beneficial for patients. Our analysis of a mathematical model of tumor growth under angiogenic signaling proposed by Hahnfeldt et al. adapted to heterogeneous tumors treated by combined anti-angiogenic agent and chemotherapy offers insights into the effects of metronomic therapy. Firstly, assuming constant long-term drug delivery, the model suggests that the longest survival time is achieved for intermediate drug doses. Secondly, by formalizing the notion of the therapeutic target being maintenance rather than eradication, we show that in the short term, optimal chemotherapy scheduling consists mainly of a drug applied at a low dose. In conclusion, we suggest that metronomic therapy is an attractive alternative to maximum tolerated dose therapies to be investigated in experimental settings and clinical trials.


  • Bajger P., Bodzioch M., Foryś U., Numerical optimisation of chemotherapy dosage under antiangiogenic treatment in the presence of drug resistance, Mathematical Methods in the Applied Sciences, Vol. 43 (2020), pp. 10671-10689, doi: 10.1002/mma.6958.



    We consider a two-compartment model of chemotherapy resistant tumour growth under angiogenic signalling. Our model is based on the one proposed by Hahnfeldt et al. (1999), but we divide tumour cells into sensitive and resistant subpopulations. We study the influence of antiangiogenic treatment in combination with chemotherapy. The main goal is to investigate how sensitive are the theoretically optimal protocols to changes in parameters quantifying the interactions between tumour cells in the sensitive and resistant compartments, i.e.~the competition coefficients and mutation rates, and whether inclusion of an antiangiogenic treatment affects these results.

    Global existence and positivity of solutions, bifurcations (including bistability and hysteresis) with respect to the chemotherapy dose are studied. We assume that the antiangiogenic agents are supplied indefinitely and at a constant rate. Two optimisation problems are then considered. In the first problem a constant, indefinite chemotherapy dose is optimised to maximise the time needed for the tumour to reach a critical (fatal) volume. It is shown that maximum survival time is generally obtained for intermediate drug dose. Moreover, the competition coefficients have a more visible influence on survival time than the mutation rates. In the second problem, an optimal dosage over a short, 30-day time period, is found. A novel, explicit running penalty for drug resistance is included in the objective functional. It is concluded that, after an initial full dose interval, an administration of intermediate dose is optimal over a broad range of parameters. Moreover, mutation rates play an~important role in deciding which short-term protocol is optimal. These results are independent of whether antiangiogenic treatment is applied or not.


  • Bajger P., Bodzioch M., Foryś U., Singularity of controls in a simple model of acquired chemotherapy resistance, Discrete and Continuous Dynamical Systems Series B, Vol. 24, No. 5 (2019), pp. 2039-2052, doi: 10.3934/dcdsb.2019083.



    This study investigates how optimal control theory may be used to delay the onset of chemotherapy resistance in tumours. An optimal control problem with simple tumour dynamics and an objective functional explicitly penalising drug resistant tumour phenotype is formulated. It is shown that for biologically relevant parameters the system has a single globally attracting positive steady state. The existence of singular arc is then investigated analytically under a very general form of the resistance penalty in the objective functional. It is shown that the singular controls are of order one and that they satisfy Legendre-Clebsch condition in a subset of the domain. A gradient method for solving the proposed optimal control problem is then used to find the control minimising the objective. The optimal control is found to consist of three intervals: full dose, singular and full dose. The singular part of the control is essential in delaying the onset of drug resistance.


  • Bajger P., Bodzioch M., Mathematical model of endothelial cell proliferation and maturation, Mathematica Applicanda, Vol. 46, No. 1 (2018), pp. 3-12, doi: 10.14708/ma.v46i1.6383.



    Blood vessel sprouting (angiogenesis) is one of the hallmarks of cancer. Better quantitative understanding of this process would allow more effective antiangiogenic therapies to be developed. It has been hypothesised that not only the number of endothelial cells, but also the quality of the vasculature play an important role in how chemo- and radiotherapies are delivered to tumour site. Hence in this study a minimally-parametrised mathematical model of endothelial cell proliferation and maturation is developed. Endothelial cells are subdivided into two compartments - mature and immature (or proliferating). The cells are assumed to undergo a self-mediated maturation, while loss of blood vessel quality is mediated by an external growth factor (here VEGF). The model is fitted to experimental data. The model shows how inhibition of VEGF results in better quality vasculature and slower proliferation.


  • Bajger P., Bodzioch M., Log-kill chemotherapy response versus the Norton-Simon hypothesis, Proceedings of the Twenty-Third National Conference on Applications of Mathematics in Biology and Medicine (2017), pp. 9-14.



    This article investigates how the conclusions drawn from mathematical modelling of tumour growth with chemotherapeutic treatment differ for the two choices of chemotherapy response: the Norton-Simon and the log-kill types. Both models are analysed under an assumption of constant, indefinite chemotherapy protocols. Differences between the bifurcation analysis of the two systems are presented. The dependence of the survival time on the competition coefficients between the two types of malignant cells is considered. It is shown that the survival time depends significantly on the ability of the sensitive cells to suppress the resistant population, while the impact of the resistant cells on the sensitive ones is less important. The survival time and optimal dosage is shown to be less dependent on the chemotherapy initiation threshold in the log-kill case than it is under the Norton-Simon hypothesis.