Cancer Adaptive Therapy Models

Increasing numbers of effective therapies are available for metastatic cancers. For example, there are currently 52 drugs approved for use in metastatic prostate cancer. However, of the 37,000 men who are diagnosed with metastatic prostate cancer each year in the US, none are cured.

Although most metastatic cancers now have at least one effective therapy, evolution of resistance almost invariably leads to treatment failure and patient death. Investigating tumor progression during treatment has generally focused on defining the molecular machinery that permits resistance. While often viewed as the result of a “resistance mutation,” many adaptive strategies simply involve increased expression of genes (e.g. xenobiotic metabolism) already in the human genome. Furthermore, clinical treatments to block the resistance mechanism (e.g. the MDR proteins) have generally been unsuccessful reflecting the multiplicity of available strategies. In an evolutionary context, however, the mere presence of a resistant phenotype is not sufficient for treatment failure. Rather, tumor progression requires proliferation of the resistant phenotype at a rate sufficient to produce the billions of cells necessary for clinically evident progressive disease.

Importantly, the necessary molecular, cellular, and population dynamics necessary for cancer progression during treatment are deeply connected because there are often significant phenotypic costs from synthesis, maintenance, and operation of the molecular machinery of resistance that can reduce cellular proliferation and invasion particularly in the substrate-poor environment often present in clinical cancers.

Evolution-based cancer treatment assumes that resistant phenotypes are invariably present prior to initiation of therapy and seeks to exploit evolutionary dynamics to delay or prevent proliferation of these cells. A specific example of this approach is adaptive therapy which is used in clinical settings in which cure is not achievable. Similar to widely accepted practices in pest management, adaptive therapy reduces or periodically withdraws treatment to maintain a stable population of treatment-sensitive cells that can use their fitness advantage (i.e. absence of the cost of resistance) to suppress proliferation of resistant phenotypes. This approach has been investigated in both pre-clinical and clinical conditions. A clinical trial in metastatic castrate-resistant prostate cancer (mCRPC) has been completed and will be discussed. Finally, when cure from a cancer treatment is possible but rare, evolutionary dynamics, based on observations in background extinctions, can theoretically be used to increase the probability of complete eradication of the malignant population.

In this introductory talk, I will review some of the early models related to Adaptive Therapy and discuss the situation now, as well as challenges for the future.

Using a recent clinical trial for prostate cancer, I will discuss adaptive therapy in the context of competition. Methods to enhance competition as well as parallels in treatment of infectious disease will be covered. Emphasis will be placed on highlighting important next steps towards optimizing the benefit and application of adaptive therapy.

Prostate cancer (PCa) is the second most common cancer in American men and the fifth leading cause of death worldwide. Androgen deprivation therapy (ADT) has been the mainstay treatment for advanced prostate cancer (PCa) for the past 70 years. Unfortunately, continuous treatment results in the competitive release of the resistant phenotype, causing the inevitable development of castration-resistant PCa (CRPC). While the addition of chemotherapeutics, such as docetaxel, in metastatic CRPC has been shown to extend survival by between 4.7 and 13.6 months compared to ADT alone, when and how to apply such treatment remains unclear. In fact, administering docetaxel concurrently with ADT prior to the development of CRPC has been shown to extend survival by 8.5 months when compared to ADT alone. We hypothesize that early PSA dynamics can be used to determine the optimal time to administer docetaxel on a per-patient basis. We simulate prostate-specific antigen (PSA) dynamics using a mathematical model of PCa stem and differentiated cell interactions, which is calibrated to longitudinal PSA data from 56 castration naive patients receiving concurrent therapy and 36 castration resistant patients receiving ADT alone followed by concurrent therapy after CRPC development. We use the calibrated model to simulate concurrent therapy administered before and after the development of CRPC and analyze early treatment dynamics to determine the optimal time for concurrent therapy in individual patients. Model simulations based on early response identify when patients receive the most benefit from concurrent docetaxel, demonstrating the feasibility and potential value of clinical trials guided by patient-specific mathematical models of intratumoral evolutionary dynamics.

It can be useful to imagine treating cancer as a game between the two players of the population of cancer cells and the physician. This tumour-doctor game is a special case of the general problem faced by evolving populations. Before we can design and optimize therapies in this setting, it is important to understand (1) the relative power of the two players and (2) the rules of the game that they are playing. As mathematical oncologists, we can draw tools from other disciplines to gain this understanding. Theoretical computer science and information theory teach us that an evolving cancer population is an optimal player and should not be underestimated. Philosophy of science teaches us how to overcome the indirect representations of the modeler and instead directly measure the rules of the game. Finally, experimental biology allows us to design and carry out these measurements as a game assay. This assay helps us see empirically if the games played by cancer have the features (like cost of resistance or competitive release) that we tend to postulate as theorists. I will highlight the contributions of these various fields to mathematical oncology by discussing some of my ongoing work with Ryan McGee, Chia-Hua Lin, Ranjini Bhattacharya, Nathan Farrokhian, and others.

Many naturally occurring cellular populations, from biofilms to solid tumours, are spatially distributed. In these conditions, in contrast to well-mixed liquid environment, cells have not only to compete for nutrients, but also for the precious resource that is free space. Using microbial colonies as model systems to investigate the role of spatial growth in evolution, we found that competition for space results in an excess of high-frequency mutations (jackpots) compared to corresponding well-mixed populations. The characteristic spatial distribution of these mutations leads to most resistant mutants trapped in the bulk of the population, which can be released upon a treatment that kills wild-type cells. Minimal growth is then achieved at intermediate levels of treatment that exploit the natural competition between wild-type and resistant cells in a spatial form of adaptive therapy.

Although metastatic disease is thought to be responsible for about 90% of cancer deaths, there has been relatively little improvement in the understanding and treatment of cancer at this advanced stage. Evolutionary-designed therapies, such as adaptive therapy, can be useful at late stages by exploiting the competition between sensitive and resistant cells. By turning on and off treatment according to the level of a systemic biomarker, the idea is to control tumor burden and avoid resistant takeover. However, the choice to control the tumor over eradication depends on the characteristics of the cancer, and the standard clinical data only vaguely defines some of the general properties of metastatic cancer. Blood biomarkers can be captured frequently but are only a surrogate for a broad measure of total burden, and imaging can capture a different measure of burden yet may miss smaller lesions and may represent a distorted view of those that are captured. Here we use a computational modeling framework to try to understand how adaptive therapy cycling dynamics, tumor burden, and size and number of metastases can elucidate characteristics of metastatic systems in order to develop appropriate treatment strategies for individual patients. We use a system of coupled off-lattice agent-based models to represent individual metastatic lesions. We do not explicitly model the primary lesion, but it is assumed to seed the metastases during the growth phase and to have been subject to local treatment (and removed) prior to the start of the systemic treatment phase of the metastases. Each lesion has its own independent domain in which the cells are subject to space constraints such that proliferation stops when there is no space to divide. However, all lesions are subject to the same systemic treatment, guided by a systemic biomarker, which is represented by the total number of cells in all metastatic lesions. We use data from Moffitt’s metastatic castrate-resistant prostate cancer adaptive therapy trial to evaluate the effects of model parameters. From the first cycle of adaptive therapy (response time when drug is on + regrowth phase when drug is off), we find that the drug response time is generally shorter than the regrowth time for patients. These cycle time distributions are poorly matched in the model if the drug is assumed to act on division but matches well if the drug is on a faster time scale unassociated with division. The model shows that the burden and number of metastases do not clearly associate with cycle times, but metastasis size, age, sensitivity, and cell turnover do. Lastly, the timing and heterogeneity of metastatic seeds can affect how the total burden correlates with the number of visible metastases, as later seeded and less proliferative metastases can go undetected but still contribute to the total burden. The intention of this framework is to provide a more nuanced description of a metastatic system using an initial cycle of adaptive therapy along with clinical data. The results suggest that adaptive therapy cycle dynamics can be informative about the disease state, and therefore the treatment plan, and this could be further improved with more descriptive data on the number and size of metastatic lesions.

Adaptive therapy aims to tackle cancer drug resistance by leveraging intra-tumoral competition between drug-sensitive and resistant cells. So far, most adaptive therapy models have represented competition using a Lotka-Volterra term. However, this is a phenomenological model and only provides limited mechanistic insight. What does the competitive landscape of a resistant cell actually look like, and which factors determine how strongly a cell is impacted by competition? To address these questions we examine a simple, 2-D, on-lattice, agent-based model, in which cells are considered to be either fully drug sensitive or resistant. We compare continuous drug administration and the adaptive schedule pioneered in the first-in-human clinical trial. Previously, we and others have shown that the degree of crowding, the initial resistance fraction, the presence of resistance costs, and the rate of tumour cell turnover are key determinants of the benefit of adaptive therapy. The current model corroborates this conclusion, and we show how these factors alter spatial competition between cells. We draw two main conclusions: i) Turnover is a key modulator of the impact of spatial competition, and ii) not only inter-specific competition but also intra-specific competition of resistant cells with each other is important during adaptive therapy. An implication of this second finding is that the spatial distribution of resistant cells within the tumour is a crucial factor impacting AT. To illustrate this point, we fit the model to prostate-specific antigen data from 67 patients from an intermittent androgen-deprivation therapy trial. This shows how differences in resistance cost and turnover change the tumour's spatial organisation and may explain differences in cycling speed observed. Our work provides insights into how adaptive therapy leverages inter- and intra-specific competition to control resistance, and shows that accurate quantification of competition within a tumour, taking into account both cell-intrinsic factors and the tumour spatial architecture, will be important in assessing the quantitative benefit of adaptive therapy in patients.

The standard of care (SoC) treatment for cancer patients aims to eradicate the tumor by killing the maximum number of cancer cells using the maximum tolerated dose (MTD) of a cancer drug. Continuous therapy at such a high dose selects for resistant cells and the tumor soon becomes refractory to treatment in addition to causing significant side-effects and toxicity to the patient. Adaptive therapy tailors the treatment to an individual patient by monitoring the tumor burden at fixed intervals and adjusting the drug dose accordingly with the aim of maximizing time to progression (TTP). Utilizing a hybrid agent-based model, we simulated adaptive therapy and strive to find some guiding principles to implement adaptive therapy in practice. Dose-adjustment algorithms (cocktail tandem, ping-pong every dose, ping-pong on progression) as well as fixed-dose algorithms (AT-2, intermittent) lead to lower tumor burden and increased TTP relative to MTD. Adaptive therapy works better when cell turnover rates are high. TTP can be increased if a dividing cell can replace a neighboring cell. Adaptive therapy works across a wide spectrum of forward and reverse mutation rates. Taking frequent treatment vacations when no drug is administered if the tumor burden falls below a threshold helps to increase TTP. We found that, in general, lower thresholds of how much the tumor must grow or shrink in order to trigger a dose adjustment, delayed progression the longest, and kept tumor burden at a minimum. The amount that the drug dose is changed mattered less. This suggests that we should change doses when we can detect a change in tumor size, and that optimizing adaptive therapy will likely only be limited by measurement error of our assays.

Evolutionary rescue occurs when a species or population experiences a catastrophic and long-lasting change in environment. In the absence of evolving suitable adaptations to the new conditions, the population goes extinct. Evolutionary rescue becomes a race against time. The speed of evolution determines when the population once again becomes sustainable, and the rate at which the population collapses determines how much time evolution has. Nature is replete with examples of extinctions that might have been forestalled had the species had enough time; flightless and defenseless birds on islands suddenly facing a novel predator; plants suddenly subjected to soil contamination by heavy-metals or salts. Much of the time, most drugs would go on to kill all of the cancer cells; yet evolutionary rescue is the norm rather than the exception for advanced cancers. Cancer cells and their populations possess a number of properties that make them eminently successful at evolutionary rescue in response to therapy: 1) small, pre-existing frequencies of cells that are already sufficiently resistant, 2) extremely large populations that buy time for evolution, 3) unusually rapid evolutionary change as a result of selection for high mutation rates, 4) persister cells or more specifically PACCs (Polyaneuploid cancer cells). In exploring these issues, my goal is to provide some examples and guidance for incorporating the role of evolutionary rescue into models of therapy resistance, and to suggest how such models can then inform evolutionary therapies aiming to contain or striving for cure (extinction therapy).

Evolvability, the capacity for a population to generate heritable variation and respond to natural selection, is a fundamental concept influencing the adaptations and fitness of individual organisms. For many species (e.g., bacteria and yeast), and cancer in particular, evolvability may be a trait that is subjected to natural selection. Genomic instability as a hallmark of cancer suggests that cancer cells often evolve the capacity to evolve more swiftly in response to novel selection gradients. Understanding the evolvability of cancer cells in their continually changing tumor microenvironment may improve our understanding of tumorigenesis and contribute to improved therapeutic strategies. Here, we create a model of competing populations of cancer cells, each with a different evolvability. We then analyze population (=tumor burden) and strategy (=heritable traits) dynamics of the two populations under conditions of cancer initiation and evolutionary rescue (=treatment). We find that more stable environments favor slower evolving species, while unstable environments favor faster evolving ones. We use this knowledge in an adaptive therapy context to find strategies to promote coexistence of the two species by disrupting the environment at intermediate frequencies, allowing for cyclical population dynamics.

In my master thesis I analyzed non-spatially competing treatment-sensitive and treatment-resistant cancer cells using all possible and (qualitatively) different strictly positive competition matrices and four models of growth - Lotka-Volterra, Gompertz, Monod and Lotka-Volterra with a strong Allee effect. The last model of growth does not describe an incurable cancer and the second-to-last model does not describe exponential growth for a vanishingly small population. More importantly, only a competition matrix, and with that a model, that implies a bistable system describes the empirical phenomenon of competitive release and therewith the emergence of large-scale treatment-resistance. Within the just-mentioned descriptive models, or even in the case of any competition between the treatment-resistant and treatment-sensitive cancer cells, I find that minimal treatment, with the constraint that it does stop the reaching of the lethal tumor burden, is the optimal treatment. This maximizes the lifespan and potential for cure, in case that potential is present (it then enhances with time the efficacy of a maximal tolerable dose in the future). Next to these benefits such treatment can have the benefits of being non-cyclical and being less demanding in terms of required information in the form of biomarkers. The only information, or sense, required is to stop reaching the lethal tumor burden. This treatment strategy is summarized as killing as little cancer cells as needed to stay alive in the current moment. To treat with additional intensity can however reduce the cumulative dose, yet this then depends on the model of growth and the parameters. Further analysis and illustration of this is planned for future work. There the specification of the desired trade-off concerning cumulative dose is then relevant and to be studied. Intrinsic stochasticity, uncertainty and spatial effects deserve study as well, and particularly play a crucial role in a balancing call for caution. Analyses have been carried out with carrying capacities above or below the lethal tumor burden, for any combination of cancer cell types and with treatment amounting to the killing of (a proportion of) treatment-sensitive cells. The analysis can also be extended to treatment amounting to reducing the carrying capacity of the treatment-resistant cells, using bifurcation analysis.

The individual cells within a cancer cell population are not all equal. The phenotypic heterogeneity among them can strongly affect disease progression and treatment success. Recent diagnostic advances allow to measure how the characteristics of this heterogeneity change over time. To match these advances, we developed deterministic and stochastic trait-based models that capture important characteristics of the intratumour heterogeneity and allow to evaluate different treatment types that either do or do not interact with this heterogeneity. We focus on growth rate as the decisive characteristic of the intratumour heterogeneity. We find that by modulating the trait distribution of the cancer cell population, a growth rate-dependent treatment delays an eventual relapse compared to a growth rate-independent treatment. As a downside, however, we observe a refuge effect where slower growing subpopulations are less affected by the growth rate-dependent treatment, which may decrease the likelihood of successful therapy. We find that navigating along this trade-off may be achieved by sequentially combining both treatment types if treatment decisions account for the changing trait distribution. Interestingly, even rather large intervals between adaptive treatment changes allow for close-to-optimal treatment results, which may facilitate their practical applicability.

It is not well understood how physiological environmental conditions and nutrient availability influence cancer cell proliferation. Production of oxidized biomass, which requires regeneration of the cofactor NAD+, can limit cancer cell proliferation. However, it is currently unclear which specific metabolic processes are constrained by electron acceptor availability, and how they affect cell proliferation. Here, we use computational and experimental approaches to demonstrate that de novo lipid biosynthesis can impose an increased demand for NAD+ in proliferating cancer cells. While some cancer cells and tumors synthesize a substantial fraction of their lipids de novo, we find that environmental lipids are crucial for proliferation in hypoxia or when the mitochondrial electron transport chain is inhibited. Surprisingly, we also find that even the reductive glutamine carboxylation pathway to produce fatty acids is impaired when cancer cells are limited for NAD+. Furthermore, gene expression analysis of 34 heterogeneous tumor types shows that lipid biosynthesis is strongly and consistently negatively correlated with hypoxia, whereas expression of genes involved in lipid uptake is positively correlated with hypoxia. These results demonstrate that electron acceptor availability and access to environmental lipids can play an important role in determining whether cancer cells engage in de novo lipogenesis to support proliferation.

Current multiple myeloma (MM) treatment guidelines recommend waiting for standard progression criteria to be met before proceeding to the next line of therapy (Rajkumar et al, Blood 2011;117, NCCN Guidelines – MM V2.2019 – Nov 16, 2019 NCCN.org). Assuming relapsed disease is more difficult to treat than disease that is still in remission, a tool that can accurately anticipate relapse 3-6 months before it occurs would be valuable for patient care. Using dosing and serum M-protein time course data from 130 patients from the COMMPASS observational study (NCT01454297-IA9), we developed an individualized adaptive mathematical model of MM induction and relapse in patients taking any combination of bortezomib, lenalidomide, and dexamethasone. We tested the accuracy of this adaptive model to anticipate serum M-protein relapse 3-6 months in advance in a separate ‘validation’ set of 130 COMMPASS patients. For comparison, we also tested the ability of the M-protein ‘velocity’ to predict relapse over the same time window. We derived receiver operating characteristic (ROC) curves to compare the predictors. ROC analysis showed that, at the target true positive rate (TPR) of 80%, the adaptive model-based method has a false positive rate (FPR) of 36%, while the M-protein velocity method has an FPR of 91%. At an FPR of 20%, however, the TPR of the adaptive method was 50% while the TPR for the M-protein velocity was slightly higher at ~61%. While the model-based method outperformed the M-protein ‘velocity,’ neither method achieved the prespecified criteria of 80% sensitivity and specificity in predicting M-protein relapse 3-6 months in advance. We hope to augment the model-based tool with Bayesian priors informed by baseline disease markers as well as incorporating minimal residual disease data as available. Once adequate operating characteristics are demonstrated, our methodology may improve patient level decision making and outcomes in multiple myeloma.

While genetic mutations have long been accepted as a cause of intra-tumour heterogeneity, the role of phenotypic plasticity in drug resistance is becoming increasingly apparent. We study the phenotypic evolution of drug tolerant and drug sensitive sub-populations in non-small cell lung cancer through a non-local partial differential equation structured by age and phenotype. We clarify the role of phenotype plasticity on tumour persistence by recasting the model as a renewal equation and calculating the Malthusian parameter and basic reproduction number, demonstrate the role of phenotype heterogeneity in therapy resistance and use the renewal equation formulation to derive a model informed therapy schedule. We show that this model informed therapeutic schedule results in increased treatment efficacy when compared against the standard of care and drives tumour decay while avoiding the development of drug resistance.

The prostate is an exocrine gland of the male reproductive system dependent on androgens (testosterone and 5α-dihydrotestosterone) for development and maintenance. Since prostate cells and their malignant counterparts require androgen stimulation to grow, prostate cancer (PCa) can be treated by androgen deprivation therapy (ADT). A significant problem in a continuous PCa ADT treatment at maximum doses is the emergence of cell resistance; thus, intermittent adaptive therapy (IAT) is invoked to delay time to progression. Several mathematical models (different for biological resistance mechanisms implemented, response prediction, and optimal dose administered) have been developed in the past years to interpret and forecast IAT treatment. We will present the comparison between 12 intermittent prostate-specific antigens (PSA) dynamical models over the Canadian Prospective Phase II Trial of IAT for locally advanced prostate cancer (Bruchovsky et al., 2006, 2008). Bayesian inference is coupled with model analysis over the space of parameters on and off treatment to individuate each model's strength and weakness in the patients' PSA analysis's primary purpose. Priors are shaped through an interactive process to mimic the in-clinic patient presentation. Finally, we perform a classical Bayesian-model-comparison on the models' global likelihood/evidence to determine the most effective IAT-PSA dynamical model.

Although various mathematical and computational models have been proposed to explain the superiority of particular tumour containment or adaptive therapy strategies, this evolutionary approach to cancer therapy lacks a rigorous theoretical foundation. To address this gap, we have combined extensive mathematical analysis and numerical simulations to establish general conditions under which a containment strategy is expected to control tumor burden more effectively than applying the maximum tolerated dose. We find that when resistant cells are present, an idealized strategy of containing a tumor at a maximum tolerable size maximizes time to treatment failure (that is, the time at which tumor burden becomes intolerable). These results are very general and do not depend on any fitness cost of resistance. We further provide formulas for predicting the clinical benefits attributable to containment strategies in a wide range of scenarios, and we compare outcomes of theoretically optimal treatments with those of more practical protocols. Our results strengthen the rationale for clinical trials of evolutionarily-informed cancer therapy, while also clarifying conditions under which containment might fail to outperform standard of care.

In models that assume a cost of resistance and competition between sensitive and resistant strains, the benefits of strategically reducing treatment depend on maintaining a sufficiently high number of sensitive cells to repress resistant cells. For simple models, all therapies follow the same tradeoff between total cancer cell burden and time to emergence of resistance. We here investigate the generality of this tradeoff by comparing three treatment strategies (metronomic, intermittent and adaptive) in a range of different models, including those that include competition with healthy cells, regulation by the immune system, explicit compartments for resources or growth signals, cell plasticity, or an Allee effect. We establish the conditions under which adaptive therapy provides a feasible way to reduce overtreatment and maintain a more durable response to therapy.

In this talk we shall present numerous discussion points that are of relevance to the concept of evolutionary based adaptive dosing. We shall first briefly discuss the relevance of preclinical studies. We shall then move to the clinic and first discuss selection bias in adaptive therapy studies. Next, we shall discuss the use of poor surrogate biomarkers for making key dosing decisions. Finally, we shall present numerous empirical observations on the relationship between tumour burden markers and clinical end-points such as overall survival. In particular we shall discuss what effect considering tumour burden as a time-dependent covariate has on numerous theoretical studies. The issues raised highlight that the success of evolutionary based adaptive dosing algorithms is not a foregone conclusion and that what is required are well-designed and appropriately analysed randomised control trials. Without such a study we will never know what the benefit, if there is one, of evolutionary based adaptive dosing algorithms.

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. In an ‘extreme’ interpretation of adaptive therapy, 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.

Although the term mathematical oncology was coined by R. Gatenby and P. Maini in 2003, historically the use of mathematical modeling in clinical oncology dates back at least to the 1980's with population pharmacokinetic modeling being applied to clinical trial design and dose individualization. Complemented with pharmacodynamic modeling, these approaches form what is called pharmacometrics. They have a major focus on trying to understand, quantify and predict inter-individual variability of response to pharmacological intervention (either efficacy or toxicity). The aim of my communication is to present a few concrete preclinical or clinical applications of this approach. These have been performed by members of a recently created new unit entitled COMPO (COMPutational pharmacology and clinical Oncology, Inria-Inserm, Center for Cancer Research of Marseille, France) which uniquely gathers clinical oncologists, pharmacists and mathematicians. The three examples that will be presented consist of: 1) clinical dose adaptation of chemotherapy (e.g. cisplatin), or targeted therapy (e.g. sunitinib), 2) the design of one of the first model-driven clinical trials, including Bayesian adaptive treatment individualization and 3) model-driven optimal combination of cytotoxics and anti-angiogenics. To conclude, I will present a few ongoing studies involving the incorporation of higher dimensional data in an approach comining mechanistic modeling and machine learning termed mechanistic learning.

In the first part of this talk, we will discuss how Stackelberg evolutionary game (SEG) theory can help us to frame and design evolutionary therapy for metastatic cancers. Here the physician as a rational player attempts to anticipate and steer eco-evolutionary cancer response by their treatment choice. In the second part of the talk, we will show our new research on fitting the SEG with anonymized data of metastatic non-small cell lung cancer (NSCLC) treated with immune checkpoint inhibition. When well fitted with the patient data, the model could be used for predicting patient prognosis, based on the initial tumor volume estimate. Moreover, it could be used to design evolutionary therapy optimizing pre-specified treatment goals.

Adaptive therapy is an evolution-based treatment approach that aims to maintain tumor volume by employing minimum effective drug doses or timed drug holidays. For successful adaptive therapy outcomes, it is critical to find the optimal timing of treatment switch points. Mathematical models are ideal tools to facilitate adaptive therapy dosing and switch time points. We developed two different mathematical models to examine interactions between drug-sensitive and resistant cells in a tumor. The first model assumes genetically fixed drug-sensitive and resistant populations that compete for limited resources. Resistant cell growth is inhibited by sensitive cells. The second model considers phenotypic switching between drug-sensitive and resistant cells. We calibrated each model to fit melanoma patient biomarker changes over time and predicted patient-specific adaptive therapy schedules. Overall, the models predict that adaptive therapy would have delayed time to progression by 6-25 months compared to continuous therapy with dose rates of 6%-74% relative to continuous therapy. We identified predictive factors driving the clinical time gained by adaptive therapy. The first model predicts 6-20 months gained from continuous therapy when the initial population of sensitive cells is large enough, and when the sensitive cells have a large competitive effect on resistant cells. The second model predicts 20-25 months gained from continuous therapy when the switching rate from resistant to sensitive cells is high and the growth rate of sensitive cells is low. This study highlights that there is a range of potential patient specific benefits of adaptive therapy, depending on the underlying mechanism of resistance, and identifies tumor specific parameters that modulate this benefit.

I will present an evolutionary viewpoint on cancer, seen at the two time scales of (large-time) evolution in the genomes and of (short-time) evolution in the epigenetic landscape of a constituted genome. Drug-induced drug resistance, the biological and medical question I am tackling from a theoretical point of view, related to the plasticity of cancer cells, may be due to biological mechanisms of different natures, mere local regulation, epigenetic modifications (reversible) or genetic mutations (irreversible), according to the extent to which the genome of the cells in the population is affected. In this respect, the modelling framework of adaptive dynamics I will present is more likely to correspond biologically to epigenetic modifications, although eventual induction of emergent resistant cell clones due to mutations under drug pressure is never to be excluded. The built-in targets for theoretical therapeutic control present in the phenotype-structured PDE models I advocate are not supposed to represent well-defined molecular effects of the drugs in use, but rather functional effects, i.e., related to induction of cell death (cytotoxic drugs), or to proliferation in the sense of slowing down the cell division cycle without killing cells (cytostatic drugs). I propose that cell life-threatening drugs (cytotoxics) induce by far more resistance in the highly plastic cancer cell populations than drugs that only limit their growth (cytostatics), and that a rational combination of the two classes of drugs - and possibly others, adding relevant targets to the model - may be optimised to propose therapeutic control strategies to avoid the emergence of drug resistance in tumours. I address this optimal control problem in the context of two populations, healthy and cancer, both endowed with phenotypes evolving with drug pressure, and competing for space and nutrients in a non-local Lotka-Volterra-like way, taking thus into account a constraint of limiting unwanted adverse effects. Finally, I will present some transdisciplinary challenges of cancer modelling that should concern all mathematicians, cell biologists, evolutionary biologists and oncologists, aiming to go beyond the present state of the art in the treatments of cancer.

Drug resistance in cancer chemotherapy is a major cause of treatment failure. The origins of resistance are complex: the resistant phenotype may be present before treatment is initiated (pre-existing), or may be acquired during the course of therapy. Pre-existing resistance is thought to be driven primarily by genetic alterations, but acquired resistance is more complex: does the resistant subpopulation arise spontaneously (genetically or via phenotype-switching), or is it induced by the drug itself? Such questions are difficult to answer, but the consequences may be profound in a clinical setting. In this talk, we present recent work, both mathematical and experimental, on understanding the origins of drug resistance, with special emphasis placed on the more prominent role induced resistance appears to be playing in recent literature. Minimal mathematical models are introduced, and are used to demonstrate that the ability of a drug to promote resistance may have a significant impact on optimal therapy design. Current work towards integrating such models with experimental data is also discussed, and further questions relating to the scheduling of multiple chemotherapies are posed. Future directions and open questions relating to the origins of drug resistance will also be discussed. (This is joint work with James Greene, Jana Gevertz, and Cynthia Sanchez Tapia.)

Despite major strides in the treatment of cancer, the development of drug resistance remains a major hurdle. One strategy which has been proposed to address this is the sequential application of drug therapies where resistance to one drug induces sensitivity to another drug, a concept called collateral sensitivity. Particularly, there is utility in a drug sequence which completes a cycle of such relationships. With such cycles, one could, in theory, generate infinitely long drug sequences which can be used in long term therapy to mitigate the evolution of resistance in a tumor. In this work, we explored the optimal therapeutic strategy using the drugs involved in such a cycle with an arbitrary length, N (>=2). We developed a mathematical model for this research, in which tumor cells are classified as one of N subpopulations represented as { R_i|i =1,2,...,N}. Each subpopulation, R_i , is resistant to Drug i and each subpopulation, R _{ i -1} (or R_N , if i =1), is sensitive to it, so that R_i increases under Drug i as it is resistant to it, and after drug-switching, decreases under Drug i+1 as it is sensitive to that drug(s). Based on the model, we found that there is an initial period of time in which the tumor is `shaped' into a specific makeup of each subpopulation, at which time all the drugs are equally effective ( R* ). After this shaping period, all the drugs are quickly switched with duration relative to their efficacy in order to maintain each subpopulation, consistent with the ideas underlying adaptive therapy. Additionally, we have developed methodologies to administer the optimal regimen under clinical or experimental situations in which no drug parameters and limited information of trackable populations data (all the subpopulations or only total population) are known. The therapy simulation based on these methodologies showed consistency with the theoretical effect of optimal therapy.

Cancer is an evolutionary and ecological process driven by heritable mutations whereby subclones compete on a fitness landscape. Viewing cancer through this eco-evo lens has led to the development of several evolutionary therapy (ET) paradigms. These paradigms have already proven a degree of success in preclinical experiments and clinical trials using an approach known as adaptive therapy (AT). A stable population of treatment-sensitive cells is maintained in order to suppress the growth rate of smaller resistant populations. This adaptive approach means that each patient's treatment is truly personalized on the basis of the tumor's state and response rather than a one-size-fits-all fixed treatment regime. However, there remain several open challenges in designing adaptive therapies. For example, it is not yet clear how to extend these evolutionarily enlightened therapeutic concepts to multiple treatments. Is it evolutionarily optimal to reproduce the single-drug adaptive therapy in a sequential setting or in a concomitant setting? We present a cancer treatment case study (metastatic castrate-resistant prostate cancer) as a point of departure to illustrate three novel concepts to aid the design of multidrug adaptive therapies. First, frequency-dependent "cycles" of tumor evolution can trap tumor evolution in a periodic, controllable loop. Second, the availability and selection of treatments may limit the evolutionary "absorbing region" reachable by the tumor. Third, the velocity of evolution significantly influences the optimal timing of drug sequences. These three conceptual advances provide a path forward for multidrug adaptive therapy.