Dynamics of resistance development to imatinib under increasing selection pressure: a combination of mathematical models and in vitro data

Impact of Resistance on Therapeutic Design: A Moran Model of Cancer Growth

Resistance of cancers to treatments, such as chemotherapy, largely arise due to cell mutations. These mutations allow cells to resist apoptosis and inevitably lead to recurrence and often progression to more aggressive cancer forms. Sustained-low dose therapies are being considered as an alternative over maximum tolerated dose treatments, whereby a smaller drug dosage is given over a longer period of time. However, understanding the impact that the presence of treatment-resistant clones may have on these new treatment modalities is crucial to validating them as a therapeutic avenue. In this study, a Moran process is used to capture stochastic mutations arising in cancer cells, inferring treatment resistance. The model is used to predict the probability of cancer recurrence given varying treatment modalities. The simulations predict that sustained-low dose therapies would be virtually ineffective for a cancer with a non-negligible probability of developing a sub-clone with resistance tendencies. Furthermore, calibrating the model to in vivo measurements for breast cancer treatment with Herceptin, the model suggests that standard treatment regimens are ineffective in this mouse model. Using a simple Moran model, it is possible to explore the likelihood of treatment success given a non-negligible probability of treatment resistant mutations and suggest more robust therapeutic schedules.

Drug-Resistant Cancer Treatment Strategies Based on the Dynamics of Clonal Evolution and PKPD Modeling of Drug Combinations

A method for determining a dosage strategy is proposed to combat drug resistance in tumor progression. The method is based on a dynamic model for the clonal evolution of cancerous cells and considers the Pharmacokinetic/Pharmacodynamic (PKPD) modeling of combination therapy. The proposed mathematical representation models the dynamic and kinetic effects of multiple drugs on the number of cells while considering potential mutations and assuming that no cross-resistance arises. An optimization problem is then proposed to minimize the total number of cancerous cells in a finite treatment period given a limited number of treatments. The dosage schedule, including the amount of each drug to be administered and the timing, is found by solving the optimization problem. This treatment schedule is constrained to achieve a target minimum effectiveness, while also ensuring that the concentration of the drugs, individually and totally, does not exceed a prescribed toxicity threshold. The proposed optimization problem is represented as a Complementary Geometric Programming (CGP) problem. The results show that the solution of the optimization problem for combination therapy is the dosing schedule that leads to tumor eradication at the end of the treatment period. The results also investigate the tumor dynamics for all mutation types when undergoing treatment, showing that single drug therapies can fail to combat the emergence of resistance, while optimized combination therapies can reduce the amount of all mutation types during the course of treatment, thereby combating resistance.

Spatial vs. non-spatial eco-evolutionary dynamics in a tumor growth model

Metastatic prostate cancer is initially treated with androgen deprivation therapy (ADT). However, resistance typically develops in about 1 year - a clinical condition termed metastatic castrate-resistant prostate cancer (mCRPC). We develop and investigate a spatial game (agent based continuous space) of mCRPC that considers three distinct cancer cell types: (1) those dependent on exogenous testosterone (T⁺), (2) those with increased CYP17A expression that produce testosterone and provide it to the environment as a public good (T^P), and (3) those independent of testosterone (T^-). The interactions within and between cancer cell types can be represented by a 3 × 3 matrix. Based on the known biology of this cancer there are 22 potential matrices that give roughly three major outcomes depending upon the absence (good prognosis), near absence or high frequency (poor prognosis) of T^- cells at the evolutionarily stable strategy (ESS). When just two cell types coexist the spatial game faithfully reproduces the ESS of the corresponding matrix game. With three cell types divergences occur, in some cases just two strategies coexist in the spatial game even as a non-spatial matrix game supports all three. Discrepancies between the spatial game and non-spatial ESS happen because different cell types become more or less clumped in the spatial game - leading to non-random assortative interactions between cell types. Three key spatial scales influence the distribution and abundance of cell types in the spatial game: i. Increasing the radius at which cells interact with each other can lead to higher clumping of each type, ii. Increasing the radius at which cells experience limits to population growth can cause densely packed tumor clusters in space, iii. Increasing the dispersal radius of daughter cells promotes increased mixing of cell types. To our knowledge the effects of these spatial scales on eco-evolutionary dynamics have not been explored in cancer models. The fact that cancer interactions are spatially explicit and that our spatial game of mCRPC provides in general different outcomes than the non-spatial game might suggest that non-spatial models are insufficient for capturing key elements of tumorigenesis.

The mathematics of cancer: integrating quantitative models

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

An analysis of strategic treatment interruptions during imatinib treatment of chronic myelogenous leukemia with imatinib-resistant mutations

Chronic myelogenous leukemia (CML) is a cancer of the white blood cells that results from increased and uncontrolled growth of myeloid cells in the bone marrow and the accumulation of these cells in the blood. The most common form of treatment for CML is imatinib, a tyrosine kinase inhibitor. Although imatinib is an effective treatment for CML and most patients treated with imatinib do attain some form of remission, imatinib does not completely eradicate all leukemia cells, and if treatment is stopped, all patients eventually relapse (Cortes, 2005). In Kim (2008), the authors developed a mathematical model for the dynamics of CML under imatinib treatment that incorporates the anti-leukemia immune response, and in Paquin (2011), the authors used this mathematical model to study strategic treatment interruptions as a potential therapeutic strategy for CML patients. Although the authors presented the results of several numerical simulations in Paquin (2011), the studies in that work did not include the possibility of imatinib-resistant mutations or an initial population of imatinib-resistant leukemia cells. As resistance is a significant consideration in any drug treatment, it is important to study the efficacy of the strategic treatment interruption plan in the presence of imatinib resistance. In this work, we modify the delay differential equations model of Kim (2008), Paquin (2011) to include the possibility of imatinib resistance, and we analyze strategic treatment interruptions as a potential therapeutic tool in the case of patients with imatinib-resistance leukemia cells.

Current aspects in resistance against tyrosine kinase inhibitors in chronic myelogenous leukemia

Resistance against tyrosine kinase inhibitors (TKIs) represents a relevant clinical problem in treatment of chronic myelogenous leukemia (CML). On the basis of their activity against the spectrum of BCR-ABL mutations that have shown to be the most prominent mechanism of resistance to imatinib, new TKIs have been classified as second generation (such as nilotinib, dasatinib and bosutinib) or third generation (also cover- ing T315I such as ponatinib) TKIs. However, mutations in BCR-ABL only account for about half of the cases of treatment failure under TKI and other mechanisms either rendering the leukemic cells still dependent of BCR-ABL activity or supporting oncogenic properties of the leukemic cells independent of BCR-ABL signaling have been identified. A detailed understanding of the different underlying resistance mechanisms will be the prerequisite to eventually overcome clinical resistance and for the successful use of tailored combinations of targeted inhibitors in the future.

Effect of interaction of glutathione S-transferases (T1 and M1) on the hematologic and cytogenetic responses in chronic myeloid leukemia patients treated with imatinib

The glutathione S-transferases (GSTs) are phase II xenobiotic metabolizing enzymes known to be involved in the detoxification of carcinogens and anticancer drugs. Individual genetic variation linked to inherited polymorphisms of GSTT1 and GSTM1 leading to a complete loss of enzyme activity could expose subjects to develop cancer or to induce drug resistance. Indeed, despite the impressive results obtained with the imatinib, some patients with chronic myeloid leukemia (CML) fail to achieve the expected results or develop resistance. The present study aimed to examine the impact of GSTT1 and GSTM1 polymorphisms on the response to imatinib in patients with CML. Multiplex polymerase chain reaction was used to detect the genotypes of GSTT1 and GSTM1 in 60 CML patients. We found that side effects were more frequent in patients carrying GSTT1 null when compared to GSTT1 present carriers (31 vs. 16.6 %; χ (2) = 6.2; p = 0.013). The loss of hematologic response was statistically greater in patients carrying the combined genotype GSTT1 present/GSTM1 present (26.3 %) when compared to GSTT1 null/GSTM1 present (12.8 %), GSTT1 present/GSTM1 null (8.3 %) and GSTT1 null/GSTM1 null (0 %), (χ (2) = 18.85; p < 0.001). The complete cytogenetic response was higher in patients harboring the GSTT1 null/GSTM1 null (75 %) compared with GSTT1 null/GSTM1 present (55.6 %), GSTT1 present/GSTM1 null (50 %) and GSTT1 present/GSTM1 present (47.8). On the other hand, the frequency of none cytogenetic responders was more common in patients carrying GSTT1 present/GSTM1 present (34.8 %) when compared to other genotype combinations (χ (2) = 20.99; p = 0.05). Moreover, the GSTT1 present/GSTM1 present appeared to be associated with a final dose of 600 or 800 mg of imatinib, but not significantly. Based on these findings, we find that the interaction between GSTT1 and GSTM1 seems to influence treatment outcome in patients with CML. Therefore, further investigations are required to confirm these results, for better genotype-phenotype correlation.

Functional polymorphism of CYP2B6 G15631T is associated with hematologic and cytogenetic response in chronic myeloid leukemia patients treated with imatinib

In the spite of the impressive results achieved with imatinib in chronic myeloid leukemia (CML) patients, differences in patient's response are observed, which may be explained by interindividual genetic variability. It is known that cytochrome P450 enzymes play a major role in the metabolism of imatinib. The present study aimed to understand the functional impact of CYP2B6 15631G>T polymorphism on the response of imatinib in CML patients and its relation to CML susceptibility. We have genotyped CYP2B6 G15631T in 48 CML patients and 64 controls by PCR-RFLP. CYP2B6 15631G>T was not found to be a risk factor for CML (OR 95 % CI, 1.12, 0.6-2, p > 0.05). Hematologic response loss was higher in patients with 15631GG/TT genotype when compared with 15631GT (36.8 vs. 13.8 %; X (2) = 3.542, p = 0.063). Complete cytogenetic response was higher in 15631GG/GT genotype groups when compared with 15631TT (X (2) = 3.298, p = 0.024). Primary cytogenetic resistance was higher in patients carrying 15631GG/TT genotype when compared with 15631GT carriers (52.6 vs. 17.2 %; X (2) = 6.692, p = 0.010). Furthermore, side effects were more common for patients carrying 15631GG genotypes when compared with GT/TT carriers (36 vs. 13.8 %; X (2) = 8.3, p = 0.004). In light of our results, identification of 15631G>T polymorphism in CML patients might be helpful to predict therapeutic response to imatinib.

A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues

Cancers are rarely caused by single mutations, but often develop as a result of the combined effects of multiple mutations. For most cells, the number of possible cell divisions is limited because of various biological constraints, such as progressive telomere shortening, cell senescence cascades or a hierarchically organized tissue structure. Thus, the risk of accumulating cells carrying multiple mutations is low. Nonetheless, many diseases are based on the accumulation of such multiple mutations. We model a general, hierarchically organized tissue by a multi-compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters. We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy. We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients. This result can contribute to the explanation of very diverse mutation profiles observed by whole genome sequencing of many different cancers.

On the dynamics of neutral mutations in a mathematical model for a homogeneous stem cell population

The theory of the clonal origin of cancer states that a tumour arises from one cell that acquires mutation(s) leading to the malignant phenotype. It is the current belief that many of these mutations give a fitness advantage to the mutant population allowing it to expand, eventually leading to disease. However, mutations that lead to such a clonal expansion need not give a fitness advantage and may in fact be neutral--or almost neutral--with respect to fitness. Such mutant clones can be eliminated or expand stochastically, leading to a malignant phenotype (disease). Mutations in haematopoietic stem cells give rise to diseases such as chronic myeloid leukaemia (CML) and paroxysmal nocturnal haemoglobinuria (PNH). Although neutral drift often leads to clonal extinction, disease is still possible, and in this case, it has important implications both for the incidence of disease and for therapy, as it may be more difficult to eliminate neutral mutations with therapy. We illustrate the consequences of such dynamics, using CML and PNH as examples. These considerations have implications for many other tumours as well.