Cancer treatment as a game: integrating evolutionary game theory into the optimal control of chemotherapy

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.

Biomarkers or biotargets? Using competition to lure cancer cells into evolutionary traps

Background and Objectives

Cancer biomarkers provide information on the characteristics and extent of cancer progression and help inform clinical decision-making. However, they can also play functional roles in oncogenesis, from enabling metastases and inducing angiogenesis to promoting resistance to chemotherapy. The resulting evolution could bias estimates of cancer progression and lead to suboptimal treatment decisions.

Methodology

We create an evolutionary game theoretic model of cell-cell competition among cancer cells with different levels of biomarker production. We design and simulate therapies on top of this pre-existing game and examine population and biomarker dynamics.

Results

Using total biomarker as a proxy for population size generally underestimates chemotherapy efficacy and overestimates targeted therapy efficacy. If biomarker production promotes resistance and a targeted therapy against the biomarker exists, this dynamic can be used to set an evolutionary trap. After chemotherapy selects for a high biomarker-producing cancer cell population, targeted therapy could be highly effective for cancer extinction. Rather than using the most effective therapy given the cancer's current biomarker level and population size, it is more effective to 'overshoot' and utilize an evolutionary trap when the aim is extinction. Increasing cell-cell competition, as influenced by biomarker levels, can help prime and set these traps.

Conclusion and Implications

Evolution of functional biomarkers amplify the limitations of using total biomarker levels as a measure of tumor size when designing therapeutic protocols. Evolutionarily enlightened therapeutic strategies may be highly effective, assuming a targeted therapy against the biomarker is available.

Evolutionary analysis of replicator dynamics about anti-cancer combination therapy

The emergence and growth of drug-resistant cancer cell subpopulations during anti-cancer treatment is a major challenge for cancer therapies. Combination therapies are usually applied for overcoming drug resistance. In the present paper, we explored the evolution outcome of tumor cell populations under different combination schedules of chemotherapy and p53 vaccine, by construction of replicator dynamical model for sensitive cells, chemotherapy-resistant cells and p53 vaccine-resistant cells. The local asymptotic stability analysis of the evolutionary stable points revealed that cancer population could evolve to the population with single subpopulation, or coexistence of sensitive cells and p53 vaccine-resistant cells, or coexistence of chemotherapy-resistant cells and p53 vaccine-resistant cells under different monotherapy or combination schedules. The design of adaptive therapy schedules that maintain the subpopulations under control is also demonstrated by sequential and periodic application of combination treatment strategies based on the evolutionary velocity and evolutionary absorbing regions. Applying a new replicator dynamical model, we further explored the supportive effects of sensitive cancer cells on targeted therapy-resistant cells revealed in mice experiments. It was shown that the supportive effects of sensitive cells could drive the evolution of cell population from sensitive cells to coexistence of sensitive cells and one type of targeted therapy-resistant cells.

Optimal chemotherapy counteracts cancer adaptive resistance in a cell-based, spatially-extended, evolutionary model

Most aggressive cancers are incurable due to their fast evolution of drug resistance. We model cancer growth and adaptive response in a simplified cell-based (CB) setting, assuming a genetic resistance to two chemotherapeutic drugs. We show that optimal administration protocols can steer cells resistance and turned it into a weakness for the disease. Our work extends the population-based (PB) model proposed by Orlando et al. (Physical Biology, 2012), in which a homogeneous population of cancer cells evolves according to a fitness landscape. The landscape models three types of trade-offs, differing on whether the cells are more, less, or equal effective when generalizing resistance to two drugs as opposed to specializing to a single one. The CB framework allows us to include genetic heterogeneity, spatial competition, and drugs diffusion, as well as realistic administration protocols. By calibrating our model on Orlando et al.'s assumptions, we show that dynamical protocols that alternate the two drugs minimize the cancer size at the end of (or at mid-points during) treatment. These results significantly differ from those obtained with the homogeneous model---suggesting static protocols under the pro-generalizing and neutral allocation trade-offs---highlighting the important role of spatial and genetic heterogeneities. Our work is the first attempt to search for optimal treatments in a CB setting, a step forward toward realistic clinical applications.

Identifies Immune Feature Genes for Prediction of Chemotherapy Benefit in Cancer

Chemotherapy is still the most fundamental treatment for advanced cancers so far. Previous studies have indicated that immune cell infiltration (ICI) index could serve as a biomarker to predict chemotherapy benefit in breast cancer and colorectal cancer. However, due to different responses of tumor infiltrating immune cells (TIICs) to chemotherapy, the prediction efficiency of ICI index is not fully confirmed by now. In our study, we first extended this conclusion in 7 cancers that high ICI index could certainly indicate chemotherapy benefit (P<0.05). But we also found the fraction of different TIICs and the interaction of TIICs were varies greatly from cancer to cancer. Therefore, we executed correlation and causal network analysis to identify chemotherapy associated immune feature genes, and fortunately identified six co-owned immune feature genes (CD48, GPR65, C3AR1, CD2, CD3E and ARHGAP9) in 10 cancers (BLCA, BRCA, COAD, LUAD, LUSC, OV, PAAD, SKCM, STAD and UCEC). Base on this, we developed a chemotherapy benefit prediction model within six co-owned immune feature genes through random forest classifying (AUC =0.83) in cancers mentioned above, and validated its efficiency in external datasets. In short, our work offers a novel model with a shrinking panel which has the potential to guide optimal chemotherapy in cancer.

Combination treatment optimization using a pan-cancer pathway model

The design of efficient combination therapies is a difficult key challenge in the treatment of complex diseases such as cancers. The large heterogeneity of cancers and the large number of available drugs renders exhaustive in vivo or even in vitro investigation of possible treatments impractical. In recent years, sophisticated mechanistic, ordinary differential equation-based pathways models that can predict treatment responses at a molecular level have been developed. However, surprisingly little effort has been put into leveraging these models to find novel therapies. In this paper we use for the first time, to our knowledge, a large-scale state-of-the-art pan-cancer signaling pathway model to identify candidates for novel combination therapies to treat individual cancer cell lines from various tissues (e.g., minimizing proliferation while keeping dosage low to avoid adverse side effects) and populations of heterogeneous cancer cell lines (e.g., minimizing the maximum or average proliferation across the cell lines while keeping dosage low). We also show how our method can be used to optimize the drug combinations used in sequential treatment plans—that is, optimized sequences of potentially different drug combinations—providing additional benefits. In order to solve the treatment optimization problems, we combine the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) algorithm with a significantly more scalable sampling scheme for truncated Gaussian distributions, based on a Hamiltonian Monte-Carlo method. These optimization techniques are independent of the signaling pathway model, and can thus be adapted to find treatment candidates for other complex diseases than cancers as well, as long as a suitable predictive model is available.

Combination therapies are a promising approach to counter complex diseases such as cancers. Two key difficulties in the design of effective cancer combination therapies are the large number of available drugs and the heterogeneity of cancers which render exhaustive laboratory studies impractical. In recent years, sophisticated signaling pathway models that can predict responses to combination treatments at a molecular level have been developed. This motivates the question of how one can leverage mechanistic models to identify candidates for novel combination treatments. In this paper we propose a combination treatment optimization framework which employs a large-scale pan-cancer pathway model. We formulate treatment optimization problems for single cell lines and heterogeneous populations of cancer cells. We further investigate sequential treatment plans and combine an existing evolutionary algorithm with an efficient Hamiltonian Monte-Carlo based sampling scheme. During extensive simulation studies our approach identified combination therapies which are predicted to be more effective than conventional treatments. We hope that one day in silico experiments will be used to identify a small set of promising treatment candidates which can then form a starting point for laboratory studies, allowing for an efficient use of limited resources and accelerated discovery of effective therapies.

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.

How Should Cancer Models Be Constructed?

Choosing and optimizing treatment strategies for cancer requires capturing its complex dynamics sufficiently well for understanding but without being overwhelmed. Mathematical models are essential to achieve this understanding, and we discuss the challenge of choosing the right level of complexity to address the full range of tumor complexity from growth, the generation of tumor heterogeneity, and interactions within tumors and with treatments and the tumor microenvironment. We discuss the differences between conceptual and descriptive models, and compare the use of predator-prey models, evolutionary game theory, and dynamic precision medicine approaches in the face of uncertainty about mechanisms and parameter values. Although there is of course no one-size-fits-all approach, we conclude that broad and flexible thinking about cancer, based on combined modeling approaches, will play a key role in finding creative and improved treatments.

Cancer as a form of life: Musings of the cancer and evolution symposium

Advanced cancer is one of the major problems in oncology as currently, despite the recent technological and scientific advancements, the mortality of metastatic disease remains very high at 70-90%. The field of oncology is in urgent need of novel ideas in order to improve quality of life and prognostic of cancer patients. The Cancer and Evolution Symposium organized online October 14-16, 2020 brought together a group of specialists from different fields that presented innovative strategies for better understanding, preventing, diagnosing, and treating cancer. Today still, the main reasons behind the high incidence and mortality of advanced cancer are, on one hand, the paucity of funding and effort directed to cancer prevention and early detection, and, on the other hand, the lack of understanding of the cancer process itself. I argue that besides being a disease, cancer is also a form of life, and, this frame of reference may provide a fresh look on this complex process. Here, I provide a different angle to several contemporary cancer theories discussing them from the perspective of "cancer-forms of life" (i.e. bionts) point of view. The perspectives and the several "bionts" introduced here, by no means exclusive or comprehensive, are just a shorthand that will hopefully encourage the readers, to further explore the contemporary oncology theoretical landscape.

Treatment-induced evolutionary dynamics in nonmetastatic locally advanced rectal adenocarcinoma

Multi-modal treatment of non-metastatic locally advanced rectal adenocarcinoma (LARC) includes chemotherapy, radiation, and life-altering surgery. Although highly effective for local cancer control, metastatic failure remains significant and drives rectal cancer-related mortality. A consistent observation of this tri-modality treatment paradigm is that histologic response of the primary tumor to neoadjuvant treatment(s), which varies across patients, predicts overall oncologic outcome. In this chapter, we will examine this treatment response heterogeneity in the context of evolutionary dynamics. We hypothesize that improved understanding of eco-evolutionary pressures rendering small cancer cell populations vulnerable to extinction may influence treatment strategies and improve patient outcomes. Applying effective treatment(s) to cancer populations causes a "race to extinction." We explore principles of eco-evolutionary extinction in the context of these small cancer cell populations, evaluating how treatment(s) aim to eradicate the cancer populations to ultimately result in cure. In this chapter, we provide an evolutionary rationale for limiting continuous treatment(s) with the same agent or combination of agents to avoid selection of resistant cancer subpopulation phenotypes, allowing "evolutionary rescue." We draw upon evidence from nature demonstrating species extinction rarely occurring as a single event phenomenon, but rather a series of events in the slide to extinction. We posit that eradicating small cancer populations, similar to small populations in natural extinctions, will usually require a sequence of different external perturbations that produce negative, synergistic dynamics termed the "extinction vortex." By exploiting these unique extinction vulnerabilities of small cancer populations, the optimal therapeutic sequences may be informed by evolution-informed strategies for patients with LARC.

What Is the Storage Effect, Why Should It Occur in Cancers, and How Can It Inform Cancer Therapy?

Intratumor heterogeneity is a feature of cancer that is associated with progression, treatment resistance, and recurrence. However, the mechanisms that allow diverse cancer cell lineages to coexist remain poorly understood. The storage effect is a coexistence mechanism that has been proposed to explain the diversity of a variety of ecological communities, including coral reef fish, plankton, and desert annual plants. Three ingredients are required for there to be a storage effect: (1) temporal variability in the environment, (2) buffered population growth, and (3) species-specific environmental responses. In this article, we argue that these conditions are observed in cancers and that it is likely that the storage effect contributes to intratumor diversity. Data that show the temporal variation within the tumor microenvironment are needed to quantify how cancer cells respond to fluctuations in the tumor microenvironment and what impact this has on interactions among cancer cell types. The presence of a storage effect within a patient’s tumors could have a substantial impact on how we understand and treat cancer.

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

In the absence of curative therapies, treatment of metastatic castrate-resistant prostate cancer (mCRPC) using currently available drugs can be improved by integrating evolutionary principles that govern proliferation of resistant subpopulations into current treatment protocols. Here we develop what is coined as an 'evolutionary stable therapy', within the context of the mathematical model that has been used to inform the first adaptive therapy clinical trial of mCRPC. The objective of this therapy is to maintain a stable polymorphic tumor heterogeneity of sensitive and resistant cells to therapy in order to prolong treatment efficacy and progression free survival. Optimal control analysis shows that an increasing dose titration protocol, a very common clinical dosing process, can achieve tumor stabilization for a wide range of potential initial tumor compositions and volumes. Furthermore, larger tumor volumes may counter intuitively be more likely to be stabilized if sensitive cells dominate the tumor composition at time of initial treatment, suggesting a delay of initial treatment could prove beneficial. While it remains uncertain if metastatic disease in humans has the properties that allow it to be truly stabilized, the benefits of a dose titration protocol warrant additional pre-clinical and clinical investigations.

The Evolution and Ecology of Resistance in Cancer Therapy

Despite the continuous deployment of new treatment strategies and agents over many decades, most disseminated cancers remain fatal. Cancer cells, through their access to the vast information of the human genome, have a remarkable capacity to deploy adaptive strategies for even the most effective treatments. We note there are two critical steps in the clinical manifestation of treatment resistance. The first, which is widely investigated, requires molecular machinery necessary to eliminate the cytotoxic effect of the treatment. However, the emergence of a resistant phenotype is not in itself clinically significant. That is, resistant cells affect patient outcomes only when they succeed in the second step of resistance by proliferating into a sufficiently large population to allow tumor progression and treatment failure. Importantly, proliferation of the resistant phenotype is by no means certain and, in fact, depends on complex Darwinian dynamics governed by the costs and benefits of the resistance mechanisms in the context of the local environment and competing populations. Attempts to target the molecular machinery of resistance have had little clinical success largely because of the diversity within the human genome-therapeutic interruption of one mechanism simply results in its replacement by an alternative. Here we explore evolutionarily informed strategies (adaptive, double-bind, and extinction therapies) for overcoming treatment resistance that seek to understand and exploit the critical evolutionary dynamics that govern proliferation of the resistant phenotypes. In general, this approach has demonstrated that, while emergence of resistance mechanisms in cancer cells to every current therapy is inevitable, proliferation of the resistant phenotypes is not and can be delayed and even prevented with sufficient understanding of the underlying eco-evolutionary dynamics.

Factors Influencing Private Hospitals' Participation in the Innovation of Biomedical Engineering Industry: A Perspective of Evolutionary Game Theory

The innovation of the biomedical engineering (BME) industry is inseparable from its cooperation with medical institutions. China has considerable medical institutions. Although private hospitals account for more than half of Chinese medical institutions, they rarely participate in biomedical engineering industry innovation. This paper analyzed the collaborative relationship among biomedical engineering enterprises, universities, research institutes, public hospitals and private hospitals through evolutionary game theory and discussed the influence of different factors on the collaborative innovation among them. A tripartite evolutionary game model is established which regards private hospitals as a stakeholder. The results show that (1) the good credit of private hospitals has a positive effect on their participation in collaborative innovation; (2) it is helpful for BME collaborative innovation to enhance the collaborative innovation ability of partners; (3) the novelty of innovation projects has an impact on BME collaborative innovation. The specific impacts depend on the revenue, cost and risk allocation ratio of innovation partners; (4) the higher the practicability of innovation projects, the more conducive to collaborative innovation.

Can Environmental Manipulation Help Suppress Cancer? Non-Linear Competition Among Tumor Cells in Periodically Changing Conditions

It has been shown that the tumor population growth dynamics in a periodically varying environment can drastically differ from the one in a fixed environment. Thus, the environment of a tumor can potentially be manipulated to suppress cancer progression. Diverse evolutionary processes play vital roles in cancer progression and accordingly, understanding the interplay between these processes is essential in optimizing the treatment strategy. Somatic evolution and genetic instability result in intra-tumor cell heterogeneity. Various models have been developed to analyze the interactions between different types of tumor cells. Here, models of density-dependent interaction between different types of tumor cells under fast periodical environmental changes are examined. It is illustrated that tumor population densities, which vary on a slow time scale, are affected by fast environmental variations. Finally, the intriguing density-dependent interactions in metastatic castration-resistant prostate cancer (mCRPC) in which the different types of tumor cells are defined with respect to the production of and dependence on testosterone are discussed.

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

Purpose

Drug resistance is a major impediment to the success of cancer treatment. Resistance is typically thought to arise from random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself via either genetic changes or epigenetic alterations. This relatively novel notion of resistance complicates the already challenging task of designing effective treatment protocols.

Materials and Methods

To better understand resistance, we have developed a mathematical modeling framework that incorporates both spontaneous and drug-induced resistance.

Results

Our model demonstrates that the ability of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. We have also proven that the induction parameter in our model is theoretically identifiable and propose an in vitro protocol that could be used to determine a treatment’s propensity to induce resistance.

Optimizing adaptive cancer therapy: dynamic programming and evolutionary game theory

Recent clinical trials have shown that adaptive drug therapies can be more efficient than a standard cancer treatment based on a continuous use of maximum tolerated doses (MTD). The adaptive therapy paradigm is not based on a preset schedule; instead, the doses are administered based on the current state of tumour. But the adaptive treatment policies examined so far have been largely ad hoc. We propose a method for systematically optimizing adaptive policies based on an evolutionary game theory model of cancer dynamics. Given a set of treatment objectives, we use the framework of dynamic programming to find the optimal treatment strategies. In particular, we optimize the total drug usage and time to recovery by solving a Hamilton-Jacobi-Bellman equation. We compare MTD-based treatment strategy with optimal adaptive treatment policies and show that the latter can significantly decrease the total amount of drugs prescribed while also increasing the fraction of initial tumour states from which the recovery is possible. We conclude that the use of optimal control theory to improve adaptive policies is a promising concept in cancer treatment and should be integrated into clinical trial design.

Mathematical modeling and computational prediction of cancer drug resistance

Diverse forms of resistance to anticancer drugs can lead to the failure of chemotherapy. Drug resistance is one of the most intractable issues for successfully treating cancer in current clinical practice. Effective clinical approaches that could counter drug resistance by restoring the sensitivity of tumors to the targeted agents are urgently needed. As numerous experimental results on resistance mechanisms have been obtained and a mass of high-throughput data has been accumulated, mathematical modeling and computational predictions using systematic and quantitative approaches have become increasingly important, as they can potentially provide deeper insights into resistance mechanisms, generate novel hypotheses or suggest promising treatment strategies for future testing. In this review, we first briefly summarize the current progress of experimentally revealed resistance mechanisms of targeted therapy, including genetic mechanisms, epigenetic mechanisms, posttranslational mechanisms, cellular mechanisms, microenvironmental mechanisms and pharmacokinetic mechanisms. Subsequently, we list several currently available databases and Web-based tools related to drug sensitivity and resistance. Then, we focus primarily on introducing some state-of-the-art computational methods used in drug resistance studies, including mechanism-based mathematical modeling approaches (e.g. molecular dynamics simulation, kinetic model of molecular networks, ordinary differential equation model of cellular dynamics, stochastic model, partial differential equation model, agent-based model, pharmacokinetic–pharmacodynamic model, etc.) and data-driven prediction methods (e.g. omics data-based conventional screening approach for node biomarkers, static network approach for edge biomarkers and module biomarkers, dynamic network approach for dynamic network biomarkers and dynamic module network biomarkers, etc.). Finally, we discuss several further questions and future directions for the use of computational methods for studying drug resistance, including inferring drug-induced signaling networks, multiscale modeling, drug combinations and precision medicine.

Tumor Clearance Analysis on a Cancer Chemo-Immunotherapy Mathematical Model

Mathematical models may allow us to improve our knowledge on tumor evolution and to better comprehend the dynamics between cancer, the immune system and the application of treatments such as chemotherapy and immunotherapy in both short and long term. In this paper, we solve the tumor clearance problem for a six-dimensional mathematical model that describes tumor evolution under immune response and chemo-immunotherapy treatments. First, by means of the localization of compact invariant sets method, we determine lower and upper bounds for all cells populations considered by the model and we use these results to establish sufficient conditions for the existence of a bounded positively invariant domain in the nonnegative orthant by applying LaSalle's invariance principle. Then, by exploiting a candidate Lyapunov function we determine sufficient conditions on the chemotherapy treatment to ensure tumor clearance. Further, we investigate the local stability of the tumor-free equilibrium point and compute conditions for asymptotic stability and tumor persistence. All conditions are given by inequalities in terms of the system parameters, and we perform numerical simulations with different values on the chemotherapy treatment to illustrate our results. Finally, we discuss the biological implications of our work.

Natural Selection Between Two Games with Applications to Game Theoretical Models of Cancer

Evolutionary game theory has been used extensively to study single games as applied to cancer, including in the context of metabolism, development of resistance, and even games between tumor and treatment. However, the situation when several games are being played against each other at the same time has not yet been investigated. Here, we describe a mathematical framework for analyzing natural selection not just between strategies, but between games. We provide theoretical analysis of situations of natural selection between the games of Prisoner's dilemma and Hawk-Dove, and demonstrate that while the dynamics of cooperators and defectors within their respective games is as expected, the distribution of games changes over time due to natural selection. We also investigate the question of mutual invasibility of games with respect to different strategies and different initial population composition. We conclude with a discussion of how the proposed approach can be applied to other games in cancer, such as motility versus stability strategies that underlie the process of metastatic invasion.

Including Blood Vasculature into a Game-Theoretic Model of Cancer Dynamics

For cancer, we develop a 2-D agent-based continuous-space game-theoretical model that considers cancer cells’ proximity to a blood vessel. Based on castrate resistant metastatic prostate cancer (mCRPC), the model considers the density and frequency (eco-evolutionary) dynamics of three cancer cell types: those that require exogenous testosterone ( T + ), those producing testosterone ( T P ), and those independent of testosterone ( T - ). We model proximity to a blood vessel by imagining four zones around the vessel. Zone 0 is the blood vessel. As rings, zones 1–3 are successively farther from the blood vessel and have successively lower carrying capacities. Zone 4 represents the space too far from the blood vessel and too poor in nutrients for cancer cell proliferation. Within the other three zones that are closer to the blood vessel, the cells’ proliferation probabilities are determined by zone-specific payoff matrices. We analyzed how zone width, dispersal, interactions across zone boundaries, and blood vessel dynamics influence the eco-evolutionary dynamics of cell types within zones and across the entire cancer cell population. At equilibrium, zone 3’s composition deviates from its evolutionary stable strategy (ESS) towards that of zone 2. Zone 2 sees deviations from its ESS because of dispersal from zones 1 and 3; however, its composition begins to resemble zone 1’s more so than zone 3’s. Frequency-dependent interactions between cells across zone boundaries have little effect on zone 2’s and zone 3’s composition but have decisive effects on zone 1. The composition of zone 1 diverges dramatically from both its own ESS, but also that of zone 2. That is because T + cells (highest frequency in zone 1) benefit from interacting with T P cells (highest frequency in zone 2). Zone 1 T + cells interacting with cells in zone 2 experience a higher likelihood of encountering a T P cell than when restricted to their own zone. As expected, increasing the width of zones decreases these impacts of cross-boundary dispersal and interactions. Increasing zone widths increases the persistence likelihood of the cancer subpopulation in the face of blood vessel dynamics, where the vessel may die or become occluded resulting in the “birth” of another blood vessel elsewhere in the space. With small zone widths, the cancer cell subpopulations cannot persist. With large zone widths, blood vessel dynamics create cancer cell subpopulations that resemble the ESS of zone 3 as the larger area of zone 3 and its contribution to cells within the necrotic zone 4 mean that zones 3 and 4 provide the likeliest colonizers for the new blood vessel. In conclusion, our model provides an alternative modeling approach for considering density-dependent, frequency-dependent, and dispersal dynamics into cancer models with spatial gradients around blood vessels. Additionally, our model can consider the occurrence of circulating tumor cells (cells that disperse into the blood vessel from zone 1) and the presence of live cancer cells within the necrotic regions of a tumor.

Fractionated follow-up chemotherapy delays the onset of resistance in bone metastatic prostate cancer

Prostate cancer to bone metastases are almost always lethal. This results from the ability of metastatic prostate cancer cells to co-opt bone remodeling leading to what is known as the vicious cycle. Understanding how tumor cells can disrupt bone homeostasis through their interactions with the stroma and how metastatic tumors respond to treatment is key to the development of new treatments for what remains an incurable disease. Here we describe an evolutionary game theoretical model of both the homeostatic bone remodeling and its co-option by prostate cancer metastases. This model extends past the evolutionary aspects typically considered in game theoretical models by also including ecological factors such as the physical microenvironment of the bone. Our model recapitulates the current paradigm of the "vicious cycle" driving tumor growth and sheds light on the interactions of heterogeneous tumor cells with the bone microenvironment and treatment response. Our results show that resistant populations naturally become dominant in the metastases under conventional cytotoxic treatment and that novel schedules could be used to better control the tumor and the associated bone disease compared to the current standard of care. Specifically, we introduce fractionated follow up therapy - chemotherapy where dosage is administered initially in one solid block followed by alternating smaller doeses and holidays - and argue that it is better than either a continuous application or a periodic one. Furthermore, we also show that different regimens of chemotherapy can lead to different amounts of pathological bone that are known to correlate with poor quality of life for bone metastatic prostate cancer patients.

Modeling treatment-dependent glioma growth including a dormant tumor cell subpopulation

Background

Tumors comprise a variety of specialized cell phenotypes adapted to different ecological niches that massively influence the tumor growth and its response to treatment.

Methods

In the background of glioblastoma multiforme, a highly malignant brain tumor, we consider a rapid proliferating phenotype that appears susceptible to treatment, and a dormant phenotype which lacks this pronounced proliferative ability and is not affected by standard therapeutic strategies. To gain insight in the dynamically changing proportions of different tumor cell phenotypes under different treatment conditions, we develop a mathematical model and underline our assumptions with experimental data.

Results

We show that both cell phenotypes contribute to the distinct composition of the tumor, especially in cycling low and high dose treatment, and therefore may influence the tumor growth in a phenotype specific way.

Conclusion

Our model of the dynamic proportions of dormant and rapidly growing glioblastoma cells in different therapy settings suggests that phenotypically different cells should be considered to plan dose and duration of treatment schedules.

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.

Suppressive drug combinations and their potential to combat antibiotic resistance

Antibiotic effectiveness often changes when two or more such drugs are administered simultaneously and unearthing antibiotic combinations with enhanced efficacy (synergy) has been a longstanding clinical goal. However, antibiotic resistance, which undermines individual drugs, threatens such combined treatments. Remarkably, it has emerged that antibiotic combinations whose combined effect is lower than that of at least one of the individual drugs can slow or even reverse the evolution of resistance. We synthesize and review studies of such so-called 'suppressive interactions' in the literature. We examine why these interactions have been largely disregarded in the past, the strategies used to identify them, their mechanistic basis, demonstrations of their potential to reverse the evolution of resistance and arguments for and against using them in clinical treatment. We suggest future directions for research on these interactions, aiming to expand the basic body of knowledge on suppression and to determine the applicability of suppressive interactions in the clinic.

Homeostasis Back and Forth: An Ecoevolutionary Perspective of Cancer

The role of genetic mutations in cancer is indisputable: They are a key source of tumor heterogeneity and drive its evolution to malignancy. But, the success of these new mutant cells relies on their ability to disrupt the homeostasis that characterizes healthy tissues. Mutated clones unable to break free from intrinsic and extrinsic homeostatic controls will fail to establish a tumor. Here, we will discuss, through the lens of mathematical and computational modeling, why an evolutionary view of cancer needs to be complemented by an ecological perspective to understand why cancer cells invade and subsequently transform their environment during progression. Importantly, this ecological perspective needs to account for tissue homeostasis in the organs that tumors invade, because they perturb the normal regulatory dynamics of these tissues, often coopting them for its own gain. Furthermore, given our current lack of success in treating advanced metastatic cancers through tumor-centric therapeutic strategies, we propose that treatments that aim to restore homeostasis could become a promising venue of clinical research. This ecoevolutionary view of cancer requires mechanistic mathematical models to both integrate clinical with biological data from different scales but also to detangle the dynamic feedback between the tumor and its environment. Importantly, for these models to be useful, they need to embrace a higher degree of complexity than many mathematical modelers are traditionally comfortable with.

Cancer treatment scheduling and dynamic heterogeneity in social dilemmas of tumour acidity and vasculature

Background:

Tumours are diverse ecosystems with persistent heterogeneity in various cancer hallmarks like self-sufficiency of growth factor production for angiogenesis and reprogramming of energy metabolism for aerobic glycolysis. This heterogeneity has consequences for diagnosis, treatment and disease progression.

Methods:

We introduce the double goods game to study the dynamics of these traits using evolutionary game theory. We model glycolytic acid production as a public good for all tumour cells and oxygen from vascularisation via vascular endothelial growth factor production as a club good benefiting non-glycolytic tumour cells. This results in three viable phenotypic strategies: glycolytic, angiogenic and aerobic non-angiogenic.

Results:

We classify the dynamics into three qualitatively distinct regimes: (1) fully glycolytic; (2) fully angiogenic; or (3) polyclonal in all three cell types. The third regime allows for dynamic heterogeneity even with linear goods, something that was not possible in prior public good models that considered glycolysis or growth factor production in isolation.

Conclusions:

The cyclic dynamics of the polyclonal regime stress the importance of timing for anti-glycolysis treatments like lonidamine. The existence of qualitatively different dynamic regimes highlights the order effects of treatments. In particular, we consider the potential of vascular normalisation as a neoadjuvant therapy before follow-up with interventions like buffer therapy.

Changing mutational and adaptive landscapes and the genesis of cancer

By the time the process of oncogenesis has produced an advanced cancer, tumor cells have undergone extensive evolution. The cellular phenotypes resulting from this evolution have been well studied, and include accelerated growth rates, apoptosis resistance, immortality, invasiveness, and immune evasion. Yet with all of our current knowledge of tumor biology, the details of early oncogenesis have been difficult to observe and understand. Where different oncogenic mutations may work together to enhance the survival of a tumor cell, in isolation they are often pro-apoptotic, pro-differentiative or pro-senescent, and therefore often, somewhat paradoxically, disadvantageous to a cell. It is also becoming clear that somatic mutations, including those in known oncogenic drivers, are common in tissues starting at a young age. These observations raise the question: how do we largely avoid cancer for most of our lives? Here we propose that evolutionary forces can help explain this paradox. As humans and other organisms age or experience external insults such as radiation or smoking, the structure and function of tissues progressively degrade, resulting in altered stem cell niche microenvironments. As tissue integrity declines, it becomes less capable of supporting and maintaining resident stem cells. These stem cells then find themselves in a microenvironment to which they are poorly adapted, providing a competitive advantage to those cells that can restore their functionality and fitness through mutations or epigenetic changes. The resulting oncogenic clonal expansions then increase the odds of further cancer progression. Understanding how the causes of cancer, such as aging or smoking, affect tissue microenvironments to control the impact of mutations on somatic cell fitness can help reconcile the discrepancy between marked mutation accumulation starting early in life and the somatic evolution that leads to cancer at advanced ages or following carcinogenic insults. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer?, edited by Dr. Robert A. Gatenby.

Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases

While chemoresistance in primary tumors is well-studied, much less is known about the influence of systemic chemotherapy on the development of drug resistance at metastatic sites. In this work, we use a hybrid spatial model of tumor response to a DNA damaging drug to study how the development of chemoresistance in micrometastases depends on the drug dosing schedule. We separately consider cell populations that harbor pre-existing resistance to the drug, and those that acquire resistance during the course of treatment. For each of these independent scenarios, we consider one hypothetical cell line that is responsive to metronomic chemotherapy, and another that with high probability cannot be eradicated by a metronomic protocol. Motivated by experimental work on ovarian cancer xenografts, we consider all possible combinations of a one week treatment protocol, repeated for three weeks, and constrained by the total weekly drug dose. Simulations reveal a small number of fractionated-dose protocols that are at least as effective as metronomic therapy in eradicating micrometastases with acquired resistance (weak or strong), while also being at least as effective on those that harbor weakly pre-existing resistant cells. Given the responsiveness of very different theoretical cell lines to these few fractionated-dose protocols, these may represent more effective ways to schedule chemotherapy with the goal of limiting metastatic tumor progression.

Exploiting evolutionary principles to prolong tumor control in preclinical models of breast cancer

Conventional cancer treatment strategies assume that maximum patient benefit is achieved through maximum killing of tumor cells. However, by eliminating the therapy-sensitive population, this strategy accelerates emergence of resistant clones that proliferate unopposed by competitors-an evolutionary phenomenon termed "competitive release." We present an evolution-guided treatment strategy designed to maintain a stable population of chemosensitive cells that limit proliferation of resistant clones by exploiting the fitness cost of the resistant phenotype. We treated MDA-MB-231/luc triple-negative and MCF7 estrogen receptor-positive (ER(+)) breast cancers growing orthotopically in a mouse mammary fat pad with paclitaxel, using algorithms linked to tumor response monitored by magnetic resonance imaging. We found that initial control required more intensive therapy with regular application of drug to deflect the exponential tumor growth curve onto a plateau. Dose-skipping algorithms during this phase were less successful than variable dosing algorithms. However, once initial tumor control was achieved, it was maintained with progressively smaller drug doses. In 60 to 80% of animals, continued decline in tumor size permitted intervals as long as several weeks in which no treatment was necessary. Magnetic resonance images and histological analysis of tumors controlled by adaptive therapy demonstrated increased vascular density and less necrosis, suggesting that vascular normalization resulting from enforced stabilization of tumor volume may contribute to ongoing tumor control with lower drug doses. Our study demonstrates that an evolution-based therapeutic strategy using an available chemotherapeutic drug and conventional clinical imaging can prolong the progression-free survival in different preclinical models of breast cancer.

Dynamics of cancer progression and suppression: A novel evolutionary game theory based approach

In this paper, a novel mathematical approach is proposed for the dynamics of progression and suppression of cancer. We define mutant cell density, ρ(μ) (μ × ρ), as a primary factor in cancer dynamics, and use logistic growth model and replicator equation for defining the dynamics of total cell density (ρ) and mutant fraction (μ), respectively. Furthermore, in the proposed model, we introduce an analytical expression for a control parameter D (drug), to suppress the proliferation of mutants with extra fitness level σ. Lastly, we present a comparison of the proposed model with some existing models of tumour growth.

Integrating Evolutionary Game Theory into Mechanistic Genotype-Phenotype Mapping

Natural selection has shaped the evolution of organisms toward optimizing their structural and functional design. However, how this universal principle can enhance genotype-phenotype mapping of quantitative traits has remained unexplored. Here we show that the integration of this principle and functional mapping through evolutionary game theory gains new insight into the genetic architecture of complex traits. By viewing phenotype formation as an evolutionary system, we formulate mathematical equations to model the ecological mechanisms that drive the interaction and coordination of its constituent components toward population dynamics and stability. Functional mapping provides a procedure for estimating the genetic parameters that specify the dynamic relationship of competition and cooperation and predicting how genes mediate the evolution of this relationship during trait formation.

Edge effects in game-theoretic dynamics of spatially structured tumours

Cancer dynamics are an evolutionary game between cellular phenotypes. A typical assumption in this modelling paradigm is that the probability of a given phenotypic strategy interacting with another depends exclusively on the abundance of those strategies without regard for local neighbourhood structure. We address this limitation by using the Ohtsuki-Nowak transform to introduce spatial structure to the go versus grow game. We show that spatial structure can promote the invasive (go) strategy. By considering the change in neighbourhood size at a static boundary--such as a blood vessel, organ capsule or basement membrane--we show an edge effect that allows a tumour without invasive phenotypes in the bulk to have a polyclonal boundary with invasive cells. We present an example of this promotion of invasive (epithelial-mesenchymal transition-positive) cells in a metastatic colony of prostate adenocarcinoma in bone marrow. Our results caution that pathologic analyses that do not distinguish between cells in the bulk and cells at a static edge of a tumour can underestimate the number of invasive cells. Although we concentrate on applications in mathematical oncology, we expect our approach to extend to other evolutionary game models where interaction neighbourhoods change at fixed system boundaries.

Can oncology recapitulate paleontology? Lessons from species extinctions

Although we can treat cancers with cytotoxic chemotherapies, target them with molecules that inhibit oncogenic drivers, and induce substantial cell death with radiation, local and metastatic tumours recur, resulting in extensive morbidity and mortality. Indeed, driving a tumour to extinction is difficult. Geographically dispersed species of organisms are perhaps equally resistant to extinction, but >99.9% of species that have ever existed on this planet have become extinct. By contrast, we are nowhere near that level of success in cancer therapy. The phenomena are broadly analogous--in both cases, a genetically diverse population mutates and evolves through natural selection. The goal of cancer therapy is to cause cancer cell population extinction, or at least to limit any further increase in population size, to prevent the tumour burden from overwhelming the patient. However, despite available treatments, complete responses are rare, and partial responses are limited in duration. Many patients eventually relapse with tumours that evolve from cells that survive therapy. Similarly, species are remarkably resilient to environmental change. Paleontology can show us the conditions that lead to extinction and the characteristics of species that make them resistant to extinction. These lessons could be translated to improve cancer therapy and prognosis.

Chemotherapy in heterogeneous cultures of cancer cells with interconversion

Recently, the interconversion between differentiated and stem-like cancer cells has been observed. Here, we model the in vitro growth of heterogeneous cell cultures in the presence of interconversion from differentiated cancer cells to cancer stem cells (CSCs), showing that, by targeting only CSC with cytotoxic agents, it is not always possible to eradicate cancer. We have determined the kinetic conditions under which cytotoxic agents in in vitro heterogeneous cultures of cancer cells eradicate cancer. In particular, we have shown that the chemotherapeutic elimination of in vitro cultures of heterogeneous cancer cells is effective only if it targets all cancer cell types, and if the induced death rates for the different subpopulations of cancer cell types are large enough. The quantitative results of the model are compared and validated with experimental data.

Nonstandard personalized medicine strategies for cancer may lead to improved patient outcomes

Cancer is an evolutionary process that is driven by mutation and selection. Tumors are genetically unstable, and research has shown that this is the most efficient way for cancers to evolve. Genetic instability leads to genetic heterogeneity and dynamic change within a single individual's tumor, in turn leading to therapeutic resistance. Cancer treatment has also evolved from an empirical science of killing dividing cells to the current era of 'personalized medicine', exquisitely targeting the molecular features of individual cancers. However, current personalized medicine regards a single individual's cancer as largely uniform and static. Moreover, from a strategic perspective, current personalized medicine thinks primarily of the immediate therapy selection. Ongoing research suggests that new, nonstandard personalized treatment strategies that plan further ahead and consider intratumoral heterogeneity and the evolving nature of cancer (due to genetic instability) may lead to the next level of therapeutic benefit beyond current personalized medicine.

Evolutionary game theory for physical and biological scientists. II. Population dynamics equations can be associated with interpretations

The use of mathematical equations to analyse population dynamics measurements is being increasingly applied to elucidate complex dynamic processes in biological systems, including cancer. Purely 'empirical' equations may provide sufficient accuracy to support predictions and therapy design. Nevertheless, interpretation of fitting equations in terms of physical and biological propositions can provide additional insights that can be used both to refine models that prove inconsistent with data and to understand the scope of applicability of models that validate. The purpose of this tutorial is to assist readers in mathematically associating interpretations with equations and to provide guidance in choosing interpretations and experimental systems to investigate based on currently available biological knowledge, techniques in mathematical and computational analysis and methods for in vitro and in vivo experiments.

Turning ecology and evolution against cancer

The fight against cancer has drawn researchers from a wide variety of disciplines, ranging from molecular biology to physics, but the perspective of an ecological theorist has been mostly overlooked. By thinking about the cells that make up a tumour as an endangered species, cancer vulnerabilities become more apparent. Studies in conservation biology and microbial experiments indicate that extinction is a complex phenomenon, which is often driven by the interaction of ecological and evolutionary processes. Recent advances in cancer research have shown that tumours, like species striving for survival, harbour intricate population dynamics, which suggests the possibility to exploit the ecology of tumours for treatment.