Optimal control for resistance and suboptimal response in CML

The Contribution of Evolutionary Game Theory to Understanding and Treating Cancer

Evolutionary game theory mathematically conceptualizes and analyzes biological interactions where one's fitness not only depends on one's own traits, but also on the traits of others. Typically, the individuals are not overtly rational and do not select, but rather inherit their traits. Cancer can be framed as such an evolutionary game, as it is composed of cells of heterogeneous types undergoing frequency-dependent selection. In this article, we first summarize existing works where evolutionary game theory has been employed in modeling cancer and improving its treatment. Some of these game-theoretic models suggest how one could anticipate and steer cancer's eco-evolutionary dynamics into states more desirable for the patient via evolutionary therapies. Such therapies offer great promise for increasing patient survival and decreasing drug toxicity, as demonstrated by some recent studies and clinical trials. We discuss clinical relevance of the existing game-theoretic models of cancer and its treatment, and opportunities for future applications. Moreover, we discuss the developments in cancer biology that are needed to better utilize the full potential of game-theoretic models. Ultimately, we demonstrate that viewing tumors with evolutionary game theory has medically useful implications that can inform and create a lockstep between empirical findings and mathematical modeling. We suggest that cancer progression is an evolutionary competition between different cell types and therefore needs to be viewed as an evolutionary game.

The effects of time valuation in cancer optimal therapies: a study of chronic myeloid leukemia

BACKGROUND

The mathematical design of optimal therapies to fight cancer is an important research field in today's Biomathematics and Biomedicine given its relevance to formulate patient-specific treatments. Until now, however, cancer optimal therapies have considered that malignancy exclusively depends on the drug concentration and the number of cancer cells, ignoring that the faster the cancer grows the worse the cancer is, and that early drug doses are more prejudicial. Here, we analyze how optimal therapies are affected when the time evolution of treated cancer is envisaged as an additional element determining malignancy, analyzing in detail the implications for imatinib-treated Chronic Myeloid Leukemia.

METHODS

Taking as reference a mathematical model describing Chronic Myeloid Leukemia dynamics, we design an optimal therapy problem by modifying the usual malignancy objective function, unaware of any temporal dimension of cancer malignance. In particular, we introduce a time valuation factor capturing the increase of malignancy associated to the quick development of the disease and the persistent negative effects of initial drug doses. After assigning values to the parameters involved, we solve and simulate the model with and without the new time valuation factor, comparing the results for the drug doses and the evolution of the disease.

RESULTS

Our computational simulations unequivocally show that the consideration of a time valuation factor capturing the higher malignancy associated with early growth of cancer and drug administration allows more efficient therapies to be designed. More specifically, when this time valuation factor is incorporated into the objective function, the optimal drug doses are lower, and do not involve medically relevant increases in the number of cancer cells or in the disease duration.

CONCLUSIONS

In the light of our simulations and as biomedical evidence strongly suggests, the existence of a time valuation factor affecting malignancy in treated cancer cannot be ignored when designing cancer optimal therapies. Indeed, the consideration of a time valuation factor modulating malignancy results in significant gains of efficiency in the optimal therapy with relevant implications from the biomedical perspective, specially when designing patient-specific treatments.

Evaluating optimal therapy robustness by virtual expansion of a sample population, with a case study in cancer immunotherapy

Cancer is a highly heterogeneous disease, exhibiting spatial and temporal variations that pose challenges for designing robust therapies. Here, we propose the VEPART (Virtual Expansion of Populations for Analyzing Robustness of Therapies) technique as a platform that integrates experimental data, mathematical modeling, and statistical analyses for identifying robust optimal treatment protocols. VEPART begins with time course experimental data for a sample population, and a mathematical model fit to aggregate data from that sample population. Using nonparametric statistics, the sample population is amplified and used to create a large number of virtual populations. At the final step of VEPART, robustness is assessed by identifying and analyzing the optimal therapy (perhaps restricted to a set of clinically realizable protocols) across each virtual population. As proof of concept, we have applied the VEPART method to study the robustness of treatment response in a mouse model of melanoma subject to treatment with immunostimulatory oncolytic viruses and dendritic cell vaccines. Our analysis (i) showed that every scheduling variant of the experimentally used treatment protocol is fragile (nonrobust) and (ii) discovered an alternative region of dosing space (lower oncolytic virus dose, higher dendritic cell dose) for which a robust optimal protocol exists.

Optimized Treatment Schedules for Chronic Myeloid Leukemia

Over the past decade, several targeted therapies (e.g. imatinib, dasatinib, nilotinib) have been developed to treat Chronic Myeloid Leukemia (CML). Despite an initial response to therapy, drug resistance remains a problem for some CML patients. Recent studies have shown that resistance mutations that preexist treatment can be detected in a substantial number of patients, and that this may be associated with eventual treatment failure. One proposed method to extend treatment efficacy is to use a combination of multiple targeted therapies. However, the design of such combination therapies (timing, sequence, etc.) remains an open challenge. In this work we mathematically model the dynamics of CML response to combination therapy and analyze the impact of combination treatment schedules on treatment efficacy in patients with preexisting resistance. We then propose an optimization problem to find the best schedule of multiple therapies based on the evolution of CML according to our ordinary differential equation model. This resulting optimization problem is nontrivial due to the presence of ordinary different equation constraints and integer variables. Our model also incorporates drug toxicity constraints by tracking the dynamics of patient neutrophil counts in response to therapy. We determine optimal combination strategies that maximize time until treatment failure on hypothetical patients, using parameters estimated from clinical data in the literature.

Targeted therapy using imatinib, nilotinib or dasatinib has become standard treatment for chronicle myeloid leukemia. A minority of patients, however, fail to respond to treatment or relapse due to drug resistance. One primary driving factor of drug resistance are point mutations within the driving oncogene. Laboratory studies have shown that different leukemic mutants respond differently to different drugs, so a promising way to improve treatment efficacy is to combine multiple targeted therapies. We build a mathematical model to predict the dynamics of different leukemic mutants with imatinib, nilotinib and dasatinib, and employ optimization techniques to find the best treatment schedule of combining the three drugs sequentially. Our study shows that the optimally designed combination therapy is more effective at controlling the leukemic cell burden than any monotherapy under a wide range of scenarios. The structure of the optimal schedule depends heavily on the mutant types present, growth kinetics of leukemic cells and drug toxicity parameters. Our methodology is an important step towards the design of personalized optimal therapeutic schedules for chronicle myeloid leukemia.

Evolutionary Dynamics of Chronic Myeloid Leukemia Progression: the Progression-Inhibitory Effect of Imatinib

The t(9;22) translocation that causes chronic myeloid leukemia (CML) drives both transformation and the progression process that eventually results in the disease changing to acute leukemia. Constitutively activated Bcr-Abl signaling in CML creates high levels of reactive oxygen species (ROS) that produce 8-oxo-guanine in DNA; this is mutagenic and causes chronic phase (CP) progression to blast phase (BP). We modeled three types of mutations involved in this progression: mutations that result in myeloid progenitor cells proliferating independently of external growth factors; mutations causing failure of myeloid progenitor cells to differentiate; and mutations that enable these cells to survive independently of attachment to marrow stroma. We further modeled tyrosine kinase inhibitors (TKI) as restoring myeloid cell apoptosis and preventing ROS-driven mutagenesis, and mutations that cause TKI resistance. We suggest that the unusually low rate of resistance to TKI arises because these drugs deplete ROS, which in turn decrease mutation rates.

Minimizing metastatic risk in radiotherapy fractionation schedules

Metastasis is the process by which cells from a primary tumor disperse and form new tumors at distant anatomical locations. The treatment and prevention of metastatic cancer remains an extremely challenging problem. This work introduces a novel biologically motivated objective function to the radiation optimization community that takes into account metastatic risk instead of the status of the primary tumor. In this work, we consider the problem of developing fractionated irradiation schedules that minimize production of metastatic cancer cells while keeping normal tissue damage below an acceptable level. A dynamic programming framework is utilized to determine the optimal fractionation scheme. We evaluated our approach on a breast cancer case using the heart and the lung as organs-at-risk (OAR). For small tumor [Formula: see text] values, hypo-fractionated schedules were optimal, which is consistent with standard models. However, for relatively larger [Formula: see text] values, we found the type of schedule depended on various parameters such as the time when metastatic risk was evaluated, the [Formula: see text] values of the OARs, and the normal tissue sparing factors. Interestingly, in contrast to standard models, hypo-fractionated and semi-hypo-fractionated schedules (large initial doses with doses tapering off with time) were suggested even with large tumor α/β values. Numerical results indicate the potential for significant reduction in metastatic risk.

Etiology and treatment of hematological neoplasms: stochastic mathematical models

Leukemias are driven by stemlike cancer cells (SLCC), whose initiation, growth, response to treatment, and posttreatment behavior are often "stochastic", i.e., differ substantially even among very similar patients for reasons not observable with present techniques. We review the probabilistic mathematical methods used to analyze stochastics and give two specific examples. The first example concerns a treatment protocol, e.g., for acute myeloid leukemia (AML), where intermittent cytotoxic drug dosing (e.g., once each weekday) is used with intent to cure. We argue mathematically that, if independent SLCC are growing stochastically during prolonged treatment, then, other things being equal, front-loading doses are more effective for tumor eradication than back loading. We also argue that the interacting SLCC dynamics during treatment is often best modeled by considering SLCC in microenvironmental niches, with SLCC-SLCC interactions occurring only among SLCC within the same niche, and we present a stochastic dynamics formalism, involving "Poissonization," applicable in such situations. Interactions at a distance due to partial control of total cell numbers are also considered. The second half of this chapter concerns chromosomal aberrations, lesions known to cause some leukemias. A specific example is the induction of a Philadelphia chromosome by ionizing radiation, subsequent development of chronic myeloid leukemia (CML), CML treatment, and treatment outcome. This time evolution involves a coordinated sequence of > 10 steps, each stochastic in its own way, at the subatomic, molecular, macromolecular, cellular, tissue, and population scales, with corresponding time scales ranging from picoseconds to decades. We discuss models of these steps and progress in integrating models across scales.

dNTP Supply Gene Expression Patterns after P53 Loss

Loss of the transcription factor p53 implies mRNA losses of target genes such as the p53R2 subunit of human ribonucleotide reductase (RNR). We hypothesized that other genes in the dNTP supply system would compensate for such p53R2 losses and looked for this in our own data and in data of the Gene Expression Omnibus (GEO). We found that the de novo dNTP supply system compensates for p53R2 losses with increases in RNR subunit R1, R2, or both. We also found compensatory increases in cytosolic deoxycytidine kinase (dCK) and thymidine kinase 1 (TK1) and in mitochondrial deoxyguanosine kinase (dGK), all of the salvage dNTP supply system; in contrast, the remaining mitochondrial salvage enzyme thymidine kinase 2 (TK2) decreased with p53 loss. Thus, TK2 may be more dedicated to meeting mitochondrial dNTP demands than dGK which may be more obligated to assist cytosolic dNTP supply in meeting nuclear DNA dNTP demands.

Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity

We derive and analyse a deterministic model for the transmission of malaria disease with mass action form of infection. Firstly, we calculate the basic reproduction number, R(0), and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of malaria, R(0)<1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory we derive the conditions under which it is optimal to eradicate the disease and examine the impact of a possible combined vaccination and treatment strategy on the disease transmission. When eradication is impossible, we derive the necessary conditions for optimal control of the disease using Pontryagin's Maximum Principle. The results obtained from the numerical simulations of the model show that a possible vaccination combined with effective treatment regime would reduce the spread of the disease appreciably.

A multicellular basis for the origination of blast crisis in chronic myeloid leukemia

Chronic myeloid leukemia (CML) is characterized by a specific chromosome translocation, and its pathobiology is considered comparatively well understood. Thus, quantitative analysis of CML and its progression to blast crisis may help elucidate general mechanisms of carcinogenesis and cancer progression. Hitherto, it has been widely postulated that CML blast crisis originates mainly via cell-autonomous mechanisms such as secondary mutations or genomic instability. However, recent results suggest that carcinogenic transformation may be an inherently multicellular event, in departure from the classic unicellular paradigm. We investigate this possibility in the case of blast crisis origination in CML. A quantitative, mechanistic cell population dynamics model was employed. This model used recent data on imatinib-treated CML; it also used earlier clinical data, not previously incorporated into current mathematical CML/imatinib models. With the pre-imatinib data, which include results on many more blast crises, we obtained evidence that the driving mechanism for blast crisis origination is a cooperation between specific cell types. Assuming leukemic-normal interactions resulted in a statistically significant improvement over assuming either cell-autonomous mechanisms or interactions between leukemic cells. This conclusion was robust with regard to changes in the model's adjustable parameters. Application of the results to patients treated with imatinib suggests that imatinib may act not only on malignant blast precursors, but also, to a limited degree, on the malignant blasts themselves.