Mathematical models of targeted cancer therapy

Abstract

Improved understanding of the molecular underpinnings of cancer initiation and progression has led to the development of targeted cancer therapies. The importance of these new methods of cancer treatment necessitates further research into the dynamic interactions between cancer cells and therapeutic agents, as well as a means of analysing their relationship quantitatively. The present review outlines the application of mathematical modelling to the dynamics of targeted cancer therapy, focusing particular attention on chronic myeloid leukaemia and its treatment with imatinib (Glivec).

Figure 1

Chronic myeloid leukaemia patient data. The figure shows the leukaemic cell burden in a patient without resistance (A), in a patient whose therapy is stopped (B) and in a patient developing resistance (C). Imatinib therapy commences at day 0, and the percentage of leukaemic cells in peripheral blood is measured by quantitative PCR of the BCR-ABL oncogene normalised by the values of BCR. (A) Upon initiation of imatinib treatment, the leukaemic cell burden declines bi-phasically. The first slope reflects the depletion of differentiated cancer cells and the second slope reflects the depletion of leukaemic progenitors. (B) If therapy is stopped, the leukaemic cell load returns to levels at or beyond pretreatment baseline because imatinib is not capable of decreasing the abundance of leukaemic stem cells. (C) Resistance evolves in many patients after a variable period of successful therapy. The patient shown developed the resistance mutation M351T (methionine-threonine substitution at position 351) before day 100 of therapy. Figure adapted from Michor et al (2005).

Figure 2

Model dynamics. The figure shows the dynamics of treatment without resistance mutations (A), when therapy is stopped (B), and with resistance mutations (C). The rows show the simulation output for stem cell dynamics (SC), progenitors (PC), differentiated (DC) and terminally differentiated cells (TC), and the ratio of BCR-ABL over BCR in percent (green curve). Wild-type cells are shown in black, leukaemic cells in blue and resistant leukaemic cells in red. (A) Imatinib therapy is started at day 0 and leads to a bi-phasic decline of the percentage of leukaemic cells. Leukaemic stem cells continue to expand at a slow rate. (B) Discontinuation of treatment (broken line) leads to a rapid relapse of leukaemic cells to levels beyond pretreatment baseline because leukaemic stem cells were not depleted during imatinib therapy. (C) Evolution of resistance leads to an increase in the percentage of cancer cells. Parameter values are d₀=0.003, d₁=0.008, d₂=0.05, d₃=1, a_x=0.8, b_x=5, c_x=100, r_y=0.008, a_y=2a_x, b_y=2b_x, c_y=c_x, r_z=0.023. During therapy, we have a_y′=a_y/100, b_y′=b_y/750, c_y′=c_y, a_z=a_y′=a_y, b_z=b_y′=b_y and c_z=c_y′=c_y. In (c), we have z₀(0)=10 and u=4.10⁻⁸. Figure adapted from Michor et al (2005).