Optimal control to reach eco-evolutionary stability in metastatic castrate-resistant prostate cancer

Tumor containment: a more general mathematical analysis

Clinical and pre-clinical data suggest that treating some tumors at a mild, patient-specific dose might delay resistance to treatment and increase survival time. A recent mathematical model with sensitive and resistant tumor cells identified conditions under which a treatment aiming at tumor containment rather than eradication is indeed optimal. This model however neglected mutations from sensitive to resistant cells, and assumed that the growth-rate of sensitive cells is non-increasing in the size of the resistant population. The latter is not true in standard models of chemotherapy. This article shows how to dispense with this assumption and allow for mutations from sensitive to resistant cells. This is achieved by a novel mathematical analysis comparing tumor sizes across treatments not as a function of time, but as a function of the resistant population size.

Adaptive Control of Tumor Growth

Cancer treatment optimizations select the most optimum combinations of drugs, sequencing schedules, and appropriate doses that would limit toxicity and yield an improved patient quality of life. However, these optimizations often lack an adequate consideration of cancer's near-infinite potential for evolutionary adaptation to therapeutic interventions. Adapting cancer therapy based on monitored tumor burden and clonal composition is an intuitively sound approach to the treatment of cancer as an inherently complex and adaptive system. The adaptation would be driven by clinical outcome setpoints embodying the aims to thwart therapeutic resistance and maintain a long-term management of the disease or even a cure. However, given the nonlinear, stochastic dynamics of tumor response to therapeutic interventions, adaptive therapeutic strategies may at least need a one-step-ahead prediction of tumor burden to maintain their control over tumor growth dynamics. The article explores the feasibility of adaptive cancer treatment driven by tumor state feedback assuming cell adaptive fitness to be the underlying source of phenotypic plasticity and pathway entropy as a biomarker of tumor growth trajectory. The exploration is undertaken using deterministic and stochastic models of tumor growth dynamics.

Adaptive therapy: a tumor therapy strategy based on Darwinian evolution theory

Cancer progression is a dynamic process of continuous evolution, in which genetic diversity and heterogeneity are generated by clonal and subclonal amplification based on random mutations. Traditional cancer treatment strategies have a great challenge, which often leads to treatment failure due to drug resistance. Integrating evolutionary dynamics into treatment regimens may be an effective way to overcome the problem of drug resistance. In particular, a potential treatment is adaptive therapy, which strategy advocates containment strategies that adjust the treatment cycles according to tumor evolution to control the growth of treatment-resistant cells. In this review, we first summarize the shortcomings of traditional tumor treatment methods in evolution and then introduce the theoretical basis and research status of adaptive therapy. By analyzing the limitations of adaptive therapy and exploring possible solutions, we can broaden people's understanding of adaptive therapy and provide new insights and strategies for tumor treatment.

Evolutionary Dynamics and Intermittent Therapy for Metastatic Cancers

Mathematical models show that the induction period of intermittent cancer therapy drives cancer toward resistance

Towards Optimal Control of Amyloid Fibrillation

Epigallocatechin-3-gallate, as a representative amyloid inhibitors, has shown a promising ability against A[Formula: see text] fibrillation by directly degradating the mature fibrils. Most previous studies have been focusing on its functional mechanisms, meanwhile its optimal dosage has been seldom considered. To solve this critical issue, we refer to the generalized Logistic model for amyloid fibrillation and inhibition and adopt the optimal control theory to balance the effectiveness and cost (or toxicity) of inhibitors. The optimal control trajectory of inhibitors is analytically solved, based on which the influence of model parameters, the difference between the optimal control strategy and several other traditional drug dosing strategies are systematically compared and validated through experiments. It is found that the strategy of multiple-times adding is more suitable for a long-term disease treatment, while single high-dose therapy is preferred for a short-term treatment. We hope our findings can shed light on the rational usage of amyloid inhibitors in clinic.

DDQN-based optimal targeted therapy with reversible inhibitors to combat the Warburg effect

We cover the Warburg effect with a three-component evolutionary model, where each component represents a different metabolic strategy. In this context, a scenario involving cells expressing three different phenotypes is presented. One tumour phenotype exhibits glycolytic metabolism through glucose uptake and lactate secretion. Lactate is used by a second malignant phenotype to proliferate. The third phenotype represents healthy cells, which performs oxidative phosphorylation. The purpose of this model is to gain a better understanding of the metabolic alterations associated with the Warburg effect. It is suitable to reproduce some of the clinical trials obtained in colorectal cancer and other even more aggressive tumours. It shows that lactate is an indicator of poor prognosis, since it favours the setting of polymorphic tumour equilibria that complicates its treatment. This model is also used to train a reinforcement learning algorithm, known as Double Deep Q-networks, in order to provide the first optimal targeted therapy based on experimental tumour growth inhibitors as genistein and AR-C155858. Our in silico solution includes the optimal therapy for all the tumour state space and also ensures the best possible quality of life for the patients, by considering the duration of treatment, the use of low-dose medications and the existence of possible contraindications. Optimal therapies obtained with Double Deep Q-networks are validated with the solutions of the Hamilton-Jacobi-Bellman equation.

Effective dose window for containing tumor burden under tolerable level

A maximum-tolerated dose (MTD) reduces the drug-sensitive cell population, though it may result in the competitive release of drug resistance. Alternative treatment strategies such as adaptive therapy (AT) or dose modulation aim to impose competitive stress on drug-resistant cell populations by maintaining a sufficient number of drug-sensitive cells. However, given the heterogeneous treatment response and tolerable tumor burden level of individual patients, determining an effective dose that can fine-tune competitive stress remains challenging. This study presents a mathematical model-driven approach that determines the plausible existence of an effective dose window (EDW) as a range of doses that conserve sufficient sensitive cells while maintaining the tumor volume below a threshold tolerable tumor volume (TTV). We use a mathematical model that explains intratumor cell competition. Analyzing the model, we derive an EDW determined by TTV and the competitive strength. By applying a fixed endpoint optimal control model, we determine the minimal dose to contain cancer at a TTV. As a proof of concept, we study the existence of EDW for a small cohort of melanoma patients by fitting the model to longitudinal tumor response data. We performed identifiability analysis, and for the patients with uniquely identifiable parameters, we deduced patient-specific EDW and minimal dose. The tumor volume for a patient could be theoretically contained at the TTV either using continuous dose or AT strategy with doses belonging to EDW. Further, we conclude that the lower bound of the EDW approximates the minimum effective dose (MED) for containing tumor volume at the TTV.

Morphodynamical cell state description via live-cell imaging trajectory embedding

Time-lapse imaging is a powerful approach to gain insight into the dynamic responses of cells, but the quantitative analysis of morphological changes over time remains challenging. Here, we exploit the concept of "trajectory embedding" to analyze cellular behavior using morphological feature trajectory histories-that is, multiple time points simultaneously, rather than the more common practice of examining morphological feature time courses in single timepoint (snapshot) morphological features. We apply this approach to analyze live-cell images of MCF10A mammary epithelial cells after treatment with a panel of microenvironmental perturbagens that strongly modulate cell motility, morphology, and cell cycle behavior. Our morphodynamical trajectory embedding analysis constructs a shared cell state landscape revealing ligand-specific regulation of cell state transitions and enables quantitative and descriptive models of single-cell trajectories. Additionally, we show that incorporation of trajectories into single-cell morphological analysis enables (i) systematic characterization of cell state trajectories, (ii) better separation of phenotypes, and (iii) more descriptive models of ligand-induced differences as compared to snapshot-based analysis. This morphodynamical trajectory embedding is broadly applicable to the quantitative analysis of cell responses via live-cell imaging across many biological and biomedical applications.

Evolution-Informed Strategies for Combating Drug Resistance in Cancer

The ever-changing nature of cancer poses the most difficult challenge oncologists face today. Cancer's remarkable adaptability has inspired many to work toward understanding the evolutionary dynamics that underlie this disease in hopes of learning new ways to fight it. Eco-evolutionary dynamics of a tumor are not accounted for in most standard treatment regimens, but exploiting them would help us combat treatment-resistant effectively. Here, we outline several notable efforts to exploit these dynamics and circumvent drug resistance in cancer.

Toward an optimal contraception dosing strategy

Anovulation refers to a menstrual cycle characterized by the absence of ovulation. Exogenous hormones such as synthetic progesterone and estrogen have been used to attain this state to achieve contraception. However, large doses are associated with adverse effects such as increased risk for thrombosis and myocardial infarction. This study utilizes optimal control theory on a modified menstrual cycle model to determine the minimum total exogenous estrogen/progesterone dose, and timing of administration to induce anovulation. The mathematical model correctly predicts the mean daily levels of pituitary hormones LH and FSH, and ovarian hormones E2, P4, and Inh throughout a normal menstrual cycle and reflects the reduction in these hormone levels caused by exogenous estrogen and/or progesterone. Results show that it is possible to reduce the total dose by 92% in estrogen monotherapy, 43% in progesterone monotherapy, and that it is most effective to deliver the estrogen contraceptive in the mid follicular phase. Finally, we show that by combining estrogen and progesterone the dose can be lowered even more. These results may give clinicians insights into optimal formulations and schedule of therapy that can suppress ovulation.

A survey of open questions in adaptive therapy: Bridging mathematics and clinical translation

Adaptive therapy is a dynamic cancer treatment protocol that updates (or 'adapts') treatment decisions in anticipation of evolving tumor dynamics. This broad term encompasses many possible dynamic treatment protocols of patient-specific dose modulation or dose timing. Adaptive therapy maintains high levels of tumor burden to benefit from the competitive suppression of treatment-sensitive subpopulations on treatment-resistant subpopulations. This evolution-based approach to cancer treatment has been integrated into several ongoing or planned clinical trials, including treatment of metastatic castrate resistant prostate cancer, ovarian cancer, and BRAF-mutant melanoma. In the previous few decades, experimental and clinical investigation of adaptive therapy has progressed synergistically with mathematical and computational modeling. In this work, we discuss 11 open questions in cancer adaptive therapy mathematical modeling. The questions are split into three sections: (1) integrating the appropriate components into mathematical models (2) design and validation of dosing protocols, and (3) challenges and opportunities in clinical translation.

Modeling cancer's ecological and evolutionary dynamics

In this didactic paper, we present a theoretical modeling framework, called the G-function, that integrates both the ecology and evolution of cancer to understand oncogenesis. The G-function has been used in evolutionary ecology, but has not been widely applied to problems in cancer. Here, we build the G-function framework from fundamental Darwinian principles and discuss how cancer can be seen through the lens of ecology, evolution, and game theory. We begin with a simple model of cancer growth and add on components of cancer cell competition and drug resistance. To aid in exploration of eco-evolutionary modeling with this approach, we also present a user-friendly software tool. By the end of this paper, we hope that readers will be able to construct basic G function models and grasp the usefulness of the framework to understand the games cancer plays in a biologically mechanistic fashion.

Abiraterone-Docetaxel scheduling for metastatic castration-resistant prostate cancer based on evolutionary dynamics

Patients with metastatic castration-resistant prostate cancer (mCRPC) are divided into three groups based on their response to Abiraterone treatment: best responder, responder, and non-responder. In the latter two groups, successful outcomes may not be achieved due to the development of drug-resistant cells in the tumor environment during treatment. To overcome this challenge, a secondary drug can be used to control the population of drug-resistant cells, potentially leading to a longer period of disease inhibition. This paper proposes using a combination of Docetaxel and Abiraterone in some polytherapy methods to control both the overall cancer cell population and the drug-resistant subpopulation. To investigate the competition and evolution of mCRPC cancer phenotypes, as in previous studies, the Evolutionary Game Theory (EGT) has been used as a mathematical modeling of evolutionary biology concepts.

Evolution-based mathematical models significantly prolong response to abiraterone in metastatic castrate-resistant prostate cancer and identify strategies to further improve outcomes

Background

Abiraterone acetate is an effective treatment for metastatic castrate-resistant prostate cancer (mCRPC), but evolution of resistance inevitably leads to progression. We present a pilot study in which abiraterone dosing is guided by evolution-informed mathematical models to delay onset of resistance.

Methods

In the study cohort, abiraterone was stopped when PSA was <50% of pretreatment value and resumed when PSA returned to baseline. Results are compared to a contemporaneous cohort who had >50% PSA decline after initial abiraterone administration and met trial eligibility requirements but chose standard of care (SOC) dosing.

Results

17 subjects were enrolled in the adaptive therapy group and 16 in the SOC group. All SOC subjects have progressed, but four patients in the study cohort remain stably cycling (range 53-70 months). The study cohort had significantly improved median time to progression (TTP; 33.5 months; p<0.001) and median overall survival (OS; 58.5 months; hazard ratio, 0.41, 95% confidence interval (CI), 0.20-0.83, p<0.001) compared to 14.3 and 31.3 months in the SOC cohort. On average, study subjects received no abiraterone during 46% of time on trial. Longitudinal trial data demonstrated the competition coefficient ratio (αRS/αSR) of sensitive and resistant populations, a critical factor in intratumoral evolution, was two- to threefold higher than pre-trial estimates. Computer simulations of intratumoral evolutionary dynamics in the four long-term survivors found that, due to the larger value for αRS/αSR, cycled therapy significantly decreased the resistant population. Simulations in subjects who progressed predicted further increases in OS could be achieved with prompt abiraterone withdrawal after achieving 50% PSA reduction.

Conclusions

Incorporation of evolution-based mathematical models into abiraterone monotherapy for mCRPC significantly increases TTP and OS. Computer simulations with updated parameters from longitudinal trial data can estimate intratumoral evolutionary dynamics in each subject and identify strategies to improve outcomes.

Funding

Moffitt internal grants and NIH/NCI U54CA143970-05 (Physical Science Oncology Network).

In Silico Investigations of Multi-Drug Adaptive Therapy Protocols

The standard of care for cancer patients aims to eradicate the tumor by killing the maximum number of cancer cells using the maximum tolerated dose (MTD) of a drug. MTD causes significant toxicity and selects for resistant cells, eventually making the tumor refractory to treatment. Adaptive therapy aims to maximize time to progression (TTP), by maintaining sensitive cells to compete with resistant cells. We explored both dose modulation (DM) protocols and fixed dose (FD) interspersed with drug holiday protocols. In contrast to previous single drug protocols, we explored the determinants of success of two-drug adaptive therapy protocols, using an agent-based model. In almost all cases, DM protocols (but not FD protocols) increased TTP relative to MTD. DM protocols worked well when there was more competition, with a higher cost of resistance, greater cell turnover, and when crowded proliferating cells could replace their neighbors. The amount that the drug dose was changed, mattered less. The more sensitive the protocol was to tumor burden changes, the better. In general, protocols that used as little drug as possible, worked best. Preclinical experiments should test these predictions, especially dose modulation protocols, with the goal of generating successful clinical trials for greater cancer control.

Darwinian Approaches for Cancer Treatment: Benefits of Mathematical Modeling

One of the major problems of traditional anti-cancer treatments is that they lead to the emergence of treatment-resistant cells, which results in treatment failure. To avoid or delay this phenomenon, it is relevant to take into account the eco-evolutionary dynamics of tumors. Designing evolution-based treatment strategies may help overcoming the problem of drug resistance. In particular, a promising candidate is adaptive therapy, a containment strategy which adjusts treatment cycles to the evolution of the tumors in order to keep the population of treatment-resistant cells under control. Mathematical modeling is a crucial tool to understand the dynamics of cancer in response to treatments, and to make predictions about the outcomes of these treatments. In this review, we highlight the benefits of in silico modeling to design adaptive therapy strategies, and to assess whether they could effectively improve treatment outcomes. Specifically, we review how two main types of models (i.e., mathematical models based on Lotka-Volterra equations and agent-based models) have been used to model tumor dynamics in response to adaptive therapy. We give examples of the advances they permitted in the field of adaptive therapy and discuss about how these models can be integrated in experimental approaches and clinical trial design.

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.

Evolutionary Dynamics of Treatment-Induced Resistance in Cancer Informs Understanding of Rapid Evolution in Natural Systems

Rapid evolution is ubiquitous in nature. We briefly review some of this quite broadly, particularly in the context of response to anthropogenic disturbances. Nowhere is this more evident, replicated and accessible to study than in cancer. Curiously cancer has been late - relative to fisheries, antibiotic resistance, pest management and evolution in human dominated landscapes - in recognizing the need for evolutionarily informed management strategies. The speed of evolution matters. Here, we employ game-theoretic modeling to compare time to progression with continuous maximum tolerable dose to that of adaptive therapy where treatment is discontinued when the population of cancer cells gets below half of its initial size and re-administered when the cancer cells recover, forming cycles with and without treatment. We show that the success of adaptive therapy relative to continuous maximum tolerable dose therapy is much higher if the population of cancer cells is defined by two cell types (sensitive vs. resistant in a polymorphic population). Additionally, the relative increase in time to progression increases with the speed of evolution. These results hold with and without a cost of resistance in cancer cells. On the other hand, treatment-induced resistance can be modeled as a quantitative trait in a monomorphic population of cancer cells. In that case, when evolution is rapid, there is no advantage to adaptive therapy. Initial responses to therapy are blunted by the cancer cells evolving too quickly. Our study emphasizes how cancer provides a unique system for studying rapid evolutionary changes within tumor ecosystems in response to human interventions; and allows us to contrast and compare this system to other human managed or dominated systems in nature.

The Hallmarks of Cancer as Ecologically Driven Phenotypes

Ecological fitness is the ability of individuals in a population to survive and reproduce. Individuals with increased fitness are better equipped to withstand the selective pressures of their environments. This paradigm pertains to all organismal life as we know it; however, it is also becoming increasingly clear that within multicellular organisms exist highly complex, competitive, and cooperative populations of cells under many of the same ecological and evolutionary constraints as populations of individuals in nature. In this review I discuss the parallels between populations of cancer cells and populations of individuals in the wild, highlighting how individuals in either context are constrained by their environments to converge on a small number of critical phenotypes to ensure survival and future reproductive success. I argue that the hallmarks of cancer can be distilled into key phenotypes necessary for cancer cell fitness: survival and reproduction. I posit that for therapeutic strategies to be maximally beneficial, they should seek to subvert these ecologically driven phenotypic responses.