Mitigating non-genetic resistance to checkpoint inhibition based on multiple states of immune exhaustion

Despite the revolutionary impact of immune checkpoint inhibition on cancer therapy, the lack of response in a subset of patients, as well as the emergence of resistance, remain significant challenges. Here we explore the theoretical consequences of the existence of multiple states of immune cell exhaustion on response to checkpoint inhibition therapy. In particular, we consider the emerging understanding that T cells can exist in various states: fully functioning cytotoxic cells, reversibly exhausted cells with minimal cytotoxicity, and terminally exhausted cells. We hypothesize that inflammation augmented by drug activity triggers transitions between these phenotypes, which can lead to non-genetic resistance to checkpoint inhibitors. We introduce a conceptual mathematical model, coupled with a standard 2-compartment pharmacometric (PK) model, that incorporates these mechanisms. Simulations of the model reveal that, within this framework, the emergence of resistance to checkpoint inhibitors can be mitigated through altering the dose and the frequency of administration. Our analysis also reveals that standard PK metrics do not correlate with treatment outcome. However, we do find that levels of inflammation that we assume trigger the transition from the reversibly to terminally exhausted states play a critical role in therapeutic outcome. A simulation of a population that has different values of this transition threshold reveals that while the standard high-dose, low-frequency dosing strategy can be an effective therapeutic design for some, it is likely to fail a significant fraction of the population. Conversely, a metronomic-like strategy that distributes a fixed amount of drug over many doses given close together is predicted to be effective across the entire simulated population, even at a relatively low cumulative drug dose. We also demonstrate that these predictions hold if the transitions between different states of immune cell exhaustion are triggered by prolonged antigen exposure, an alternative mechanism that has been implicated in this process. Our theoretical analyses demonstrate the potential of mitigating resistance to checkpoint inhibitors via dose modulation.

Minimally sufficient experimental design using identifiability analysis

Mathematical models are increasingly being developed and calibrated in tandem with data collection, empowering scientists to intervene in real time based on quantitative model predictions. Well-designed experiments can help augment the predictive power of a mathematical model but the question of when to collect data to maximize its utility for a model is non-trivial. Here we define data as model-informative if it results in a unique parametrization, assessed through the lens of practical identifiability. The framework we propose identifies an optimal experimental design (how much data to collect and when to collect it) that ensures parameter identifiability (permitting confidence in model predictions), while minimizing experimental time and costs. We demonstrate the power of the method by applying it to a modified version of a classic site-of-action pharmacokinetic/pharmacodynamic model that describes distribution of a drug into the tumor microenvironment (TME), where its efficacy is dependent on the level of target occupancy in the TME. In this context, we identify a minimal set of time points when data needs to be collected that robustly ensures practical identifiability of model parameters. The proposed methodology can be applied broadly to any mathematical model, allowing for the identification of a minimally sufficient experimental design that collects the most informative data.

Guiding model-driven combination dose selection using multi-objective synergy optimization

Despite the growing appreciation that the future of cancer treatment lies in combination therapies, finding the right drugs to combine and the optimal way to combine them remains a nontrivial task. Herein, we introduce the Multi-Objective Optimization of Combination Synergy - Dose Selection (MOOCS-DS) method for using drug synergy as a tool for guiding dose selection for a combination of preselected compounds. This method decouples synergy of potency (SoP) and synergy of efficacy (SoE) and identifies Pareto optimal solutions in a multi-objective synergy space. Using a toy combination therapy model, we explore properties of the MOOCS-DS algorithm, including how optimal dose selection can be influenced by the metric used to define SoP and SoE. We also demonstrate the potential of our approach to guide dose and schedule selection using a model fit to preclinical data of the combination of the PD-1 checkpoint inhibitor pembrolizumab and the anti-angiogenic drug bevacizumab on two lung cancer cell lines. The identification of optimally synergistic combination doses has the potential to inform preclinical experimental design and improve the success rates of combination therapies. Jel classificationDose Finding in Oncology.

Mathematical modeling of SARS-CoV-2 viral dynamics and treatment with monoclonal antibodies

The novel coronavirus (SARS-CoV-2) affects primarily the respiratory tract, and if left unchecked can cause a spectrum of pathological manifestations such as pneumonia, acute respiratory distress syndrome, myocardial injury, thromboembolism, and acute kidney injury. Medication strategies have involved minimizing the spread of the virus through antiviral medications (monoclonal antibodies or nucleotide reverse transcriptase inhibitors). Here, we develop a mathematical model that simulates viral dynamics in an untreated individual, and the evaluate the impact that a monoclonal antibody can have on slowing viral replication. Drug pharmacokinetics (PK) was informed by a typical two-compartment PK model with parameters typical of a monoclonal antibody, with a third compartment for the lung included as the drug site of action. The viral dynamics were captured using a simplified model describing uninfected target cells, infected target cells, and viral load in the body. The mechanism of action of the simulated antiviral is based on binding to the virus, thereby preventing it from infecting healthy cells. The model is used to project dosages needed to prevent severe disease under a variety of simulated conditions and subject to realistic constraints. The proposed model can capture a variety of scenarios of longitudinal viral dynamics and assess the impact of antiviral therapy on disease severity and duration. The described approach can be easily adapted to rapidly assess the dosages needed to affect duration and outcome of other viral infections and can serve as part of a fast and efficient scientific and modeling response strategy in the future as needed.

Different costs of therapeutic resistance in cancer: Short- and long-term impact of population heterogeneity

Therapeutic resistance continues to undercut long-term success of many promising cancer treatments. At times, development of therapeutic resistance can come at a fitness cost for the cancer cell population, which could potentially be leveraged to the patient's advantage. A mathematical formulation of such a situation was proposed by Pressley et al. (2020), who discussed two scenarios, namely, when developing therapeutic resistance can come at a cost to proliferative capacity (such as when a drug targets a growth receptor), or to the total tumor carrying capacity (such as when a drug targets neovascularization). Here we expand the analysis of the two models and evaluate both short- and long-term dynamics of a population heterogeneous with respect to resistance. We analyze the four initial distributions with respect to resistance at the time of treatment initiation: uniform, bell-shaped, exponential, and U-shaped. We show that final population composition is invariant to the initial distribution, with a single clone eventually dominating within the population; the value of the resistance parameter of the final clone depends on other system parameters but not on the initial distribution. Transitional behaviors, however, which may have more significant implications for immediate treatment decisions, depend critically on the initial distribution. Furthermore, we show that depending on the mechanism for the cost of resistance (i.e., proliferation vs carrying capacity), increase in natural cell death rate has opposite effects, with higher natural death rate selecting for less resistant cell clones in the long term for proliferation-dependent model, and selecting for more resistant cell clones for carrying capacity-dependent model, a prediction that may have implications for combination therapy with cytotoxic agents. We conclude with a discussion of strengths and limitations of using modeling for understanding treatment trajectory, as well as the promise of model-informed evolutionary steering for improved long-term therapeutic outcomes.

Understanding Metabolic Alterations in Cancer Cachexia through the Lens of Exercise Physiology

Cancer cachexia is one of the leading causes of mortality for late-stage cancer patients. One of its key characteristics is abnormal metabolism and loss of metabolic flexibility, i.e., loss of ability to switch between use of fats and carbohydrates as needed. Here, it is hypothesized that late-stage systemic cancer creates a chronic resource drain on the body that may result in the same metabolic adaptations that occur during intense endurance exercise, activating some of the same mechanisms of nutrient consumption that are supposed to be transient during strenuous physical activity. This hypothesis is evaluated by creating a mathematical model that characterizes the relationships between increased exercise intensity and carbohydrate and fat oxidation. The model is parametrized using published data on these characteristics for a group of professional athletes, moderately active individuals, and individuals with metabolic syndrome. Transitions between different zones of relative nutrient consumption as a function of increased effort are captured through explicitly modeling ventilatory thresholds, particularly VT1 and VT2, where fat is primarily used below VT1, both carbohydrates and fats are used between VT1 and VT2, and where carbohydrates become the primary source of fuel above VT2. A simulation is conducted of projected patterns of nutrient consumption when simulated “effort” remains between VT1 and VT2, or above VT2, and it is proposed that it is the scenario when the simulated effort is maintained primarily above VT2 that most closely resembles metabolic patterns characteristic of cachexia. A discussion of a broader framework for understanding cachectic metabolism using insights from exercise physiology, including potential intervention strategies, concludes this paper.

Mathematical model of a cytokine storm

Cytokine storm is a life-threatening inflammatory response that is characterized by hyperactivation of the immune system, and which can be caused by various therapies, autoimmune conditions, or pathogens, such as respiratory syndrome coronavirus 2 (SARS-CoV-2), which causes coronavirus disease COVID-19. While initial causes of cytokine storms can vary, late-stage clinical manifestations of cytokine storm converge and often overlap, and therefore a better understanding of how normal immune response turns pathological is warranted. Here we propose a theoretical framework, where cytokine storm phenomenology is captured using a conceptual mathematical model, where cytokines can both activate and regulate the immune system. We simulate normal immune response to infection, and through variation of system parameters identify conditions where, within the frameworks of this model, cytokine storm can arise. We demonstrate that cytokine storm is a transitional regime, and identify three main factors that must converge to result in storm-like dynamics, two of which represent individual-specific characteristics, thereby providing a possible explanation for why some people develop CRS, while others may not. We also discuss possible ecological insights into cytokine-immune interactions and provide mathematical analysis for the underlying regimes. We conclude with a discussion of how results of this analysis can be used in future research.

Identifying Safety Thresholds for Immunosuppressive Drugs: Applying Insights from Primary Antibody Deficiencies to Mitigate Adverse Events in Secondary Antibody Deficiencies Using Mathematical Modeling of Preclinical and Early Clinical Data

Immunosuppressive drugs can alleviate debilitating symptoms of autoimmune diseases, but, by the same token, excessive immune suppression can result in an increased risk of infection. Despite the dangers of a compromised immune system, clear definitions of what constitutes excessive suppression remain elusive. Here we review the most common infections associated with primary antibody deficiencies (PADs), such as agammaglobulinemia, common variable immunodeficiency (CVID), and IgA deficiency, as well as infections that are associated with drug-induced or secondary antibody immunodeficiencies (SADs). We identify a number of bacterial, viral, and fungal infections (e.g., Listeria monocytogenes, Staphylococcus sp., Salmonella spp., Escherichia coli, influenza, varicella zoster virus, and herpes simplex virus) associated with both PADs and SADs, and suggest that diagnostic criteria for PADs could be used as a first-line measure to identify potentially unsafe levels of immune suppression in SADs. Specifically, we suggest that, based on PAD diagnostic criteria, IgG levels should remain above 2-3 g/L, IgA levels should not fall below 0.07 g/L, and IgM levels should remain above 0.4 g/L to prevent immunosuppressive drugs from inducing mimicking PAD-like effects. We suggest that these criteria could be used in the early stages of drug development, and that pharmacokinetic and pharmacodynamic modeling could help guide patient selection to potentially improve drug safety. We illustrate the proposed approach using atacicept as an example and conclude with a discussion of the applicability of this approach for other drugs that may induce excessive immune suppression.

Predator-Prey in Tumor-Immune Interactions: A Wrong Model or Just an Incomplete One?

Tumor-immune interactions are often framed as predator-prey. This imperfect analogy describes how immune cells (the predators) hunt and kill immunogenic tumor cells (the prey). It allows for evaluation of tumor cell populations that change over time during immunoediting and it also considers how the immune system changes in response to these alterations. However, two aspects of predator-prey type models are not typically observed in immuno-oncology. The first concerns the conversion of prey killed into predator biomass. In standard predator-prey models, the predator relies on the prey for nutrients, while in the tumor microenvironment the predator and prey compete for resources (e.g. glucose). The second concerns oscillatory dynamics. Standard predator-prey models can show a perpetual cycling in both prey and predator population sizes, while in oncology we see increases in tumor volume and decreases in infiltrating immune cell populations. Here we discuss the applicability of predator-prey models in the context of cancer immunology and evaluate possible causes for discrepancies. Key processes include "safety in numbers", resource availability, time delays, interference competition, and immunoediting. Finally, we propose a way forward to reconcile differences between model predictions and empirical observations. The immune system is not just predator-prey. Like natural food webs, the immune-tumor community of cell types forms an immune-web of different and identifiable interactions.

Estrogen as an Essential Resource and the Coexistence of ER+ and ER– Cancer Cells

Diagnosis of estrogen sensitivity in breast cancer is largely predicated on the ratio of ER+ and ER– cancer cells obtained from biopsies. Estrogen is a growth factor necessary for cell survival and division. It can also be thought of as an essential resource that can act in association with other nutrients, glucose, glutamine, fatty acids, amino acids, etc. All of these nutrients, collectively or individually, may limit the growth of the cancer cells (Liebig’s Law of the Minimum). Here we model estrogen susceptibility in breast cancer as a consumer-resource interaction: ER+ cells require both estrogen and glucose as essential resources, whereas ER– only require the general resource. The model predicts that when estrogen is the limiting factor, other nutrients may go unconsumed and available at higher levels, thus permitting the invasion of ER– cells. Conversely, when ER– cells are less efficient on glucose than ER+ cells, then ER– cells limited by glucose may be susceptible to invasion by ER+ cells, provided that sufficient levels of estrogen are available. ER+ cells will outcompete ER– cells when estrogen is abundant, resulting in low concentrations of interstitial glucose within the tumor. In the absence of estrogen, ER– cells will outcompete ER+ cells, leaving a higher concentration of interstitial glucose. At intermediate delivery rates of estrogen and glucose, ER+ and ER– cells are predicted to coexist. In modeling the dynamics of cells in the same tumor with different resource requirements, we can apply concepts and terms familiar to many ecologists. These include: resource supply points, R∗, ZNGI (zero net growth isoclines), resource depletion, and resource uptake rates. Based on the circumstances favoring ER+ vs. ER– breast cancer, we use the model to explore the consequences of therapeutic regimens that may include hormonal therapies, possible roles of diet in changing cancer cell composition, and potential for evolutionarily informed therapies. More generally, the model invites the viewpoint that cancer’s eco-evolutionary dynamics are a consumer-resource interaction, and that other growth factors such as EGFR or androgens may be best viewed as essential resources within these dynamics.

Bispecific antibodies: A guide to model informed drug discovery and development

Affinity (KD) optimization of monoclonal antibodies is one of the factors that impacts the stoichiometric binding and the corresponding efficacy of a drug. This impacts the dose and the dosing regimen, making the optimum KD a critical component of drug discovery and development. Its importance is further enhanced for bispecific antibodies, where affinity of the drug needs to be optimized with respect to two targets. Mathematical modeling can have critical impact on lead compound optimization. Here we build on previous work of using mathematical models to facilitate lead compound selection, expanding analysis from two membrane bound targets to soluble targets as well. Our analysis reveals the importance of three factors for lead compound optimization: drug affinity to both targets, target turnover rates, and target distribution throughout the body. We describe a method that leverages this information to help make early stage decisions on whether to optimize affinity, and if so, which arm of the bispecific should be optimized. We apply the proposed approach to a variety of scenarios and illustrate the ability to make improved decisions in each case. We integrate results to develop a bispecific antibody KD optimization guide that can be used to improve resource allocation for lead compound selection, accelerating advancement of better compounds. We conclude with a discussion of possible ways to assess the necessary levels of target engagement for affecting disease as part of an integrative approach for model-informed drug discovery and development.

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.

Identifying key questions in the ecology and evolution of cancer

The application of evolutionary and ecological principles to cancer prevention and treatment, as well as recognizing cancer as a selection force in nature, has gained impetus over the last 50 years. Following the initial theoretical approaches that combined knowledge from interdisciplinary fields, it became clear that using the eco-evolutionary framework is of key importance to understand cancer. We are now at a pivotal point where accumulating evidence starts to steer the future directions of the discipline and allows us to underpin the key challenges that remain to be addressed. Here, we aim to assess current advancements in the field and to suggest future directions for research. First, we summarize cancer research areas that, so far, have assimilated ecological and evolutionary principles into their approaches and illustrate their key importance. Then, we assembled 33 experts and identified 84 key questions, organized around nine major themes, to pave the foundations for research to come. We highlight the urgent need for broadening the portfolio of research directions to stimulate novel approaches at the interface of oncology and ecological and evolutionary sciences. We conclude that progressive and efficient cross-disciplinary collaborations that draw on the expertise of the fields of ecology, evolution and cancer are essential in order to efficiently address current and future questions about cancer.

Quantitative Systems Pharmacology Approaches for Immuno-Oncology: Adding Virtual Patients to the Development Paradigm

Drug development in oncology commonly exploits the tools of molecular biology to gain therapeutic benefit through reprograming of cellular responses. In immuno‐oncology (IO) the aim is to direct the patient’s own immune system to fight cancer. After remarkable successes of antibodies targeting PD1/PD‐L1 and CTLA4 receptors in targeted patient populations, the focus of further development has shifted toward combination therapies. However, the current drug‐development approach of exploiting a vast number of possible combination targets and dosing regimens has proven to be challenging and is arguably inefficient. In particular, the unprecedented number of clinical trials testing different combinations may no longer be sustainable by the population of available patients. Further development in IO requires a step change in selection and validation of candidate therapies to decrease development attrition rate and limit the number of clinical trials. Quantitative systems pharmacology (QSP) proposes to tackle this challenge through mechanistic modeling and simulation. Compounds’ pharmacokinetics, target binding, and mechanisms of action as well as existing knowledge on the underlying tumor and immune system biology are described by quantitative, dynamic models aiming to predict clinical results for novel combinations. Here, we review the current QSP approaches, the legacy of mathematical models available to quantitative clinical pharmacologists describing interaction between tumor and immune system, and the recent development of IO QSP platform models. We argue that QSP and virtual patients can be integrated as a new tool in existing IO drug development approaches to increase the efficiency and effectiveness of the search for novel combination therapies.

Immune Suppression in Pregnancy and Cancer: Parallels and Insights

Immune system has evolved to maintain homeostatic balance between effector and regulatory immunity, which is critical to both elicit an adequate protective response to fight pathogens and disease, such as cancer, and to prevent damage to healthy tissues. Transient immune suppression can occur under normal physiological conditions, such as during wound healing to enable repair of normal tissue, or for more extended periods of time during fetal development, where the balance is shifted towards regulatory immunity to prevent fetal rejection. Interestingly, tumors can exhibit patterns of immune suppression very similar to those observed during fetal development. Here some of the key aspects of normal patterns of immune suppression during pregnancy are reviewed, followed by a discussion of parallels that exist with tumor-related immune suppression and consequent potential therapeutic implications.

Delicate Balances in Cancer Chemotherapy: Modeling Immune Recruitment and Emergence of Systemic Drug Resistance

Metronomic chemotherapy can drastically enhance immunogenic tumor cell death. However, the mechanisms responsible are still incompletely understood. Here, we develop a mathematical model to elucidate the underlying complex interactions between tumor growth, immune system activation, and therapy-mediated immunogenic cell death. Our model is conceptually simple, yet it provides a surprisingly excellent fit to empirical data obtained from a GL261 SCID mouse glioma model treated with cyclophosphamide on a metronomic schedule. The model includes terms representing immune recruitment as well as the emergence of drug resistance during prolonged metronomic treatments. Strikingly, a single fixed set of parameters, adjusted neither for individuals nor for drug schedule, recapitulates experimental data across various drug regimens remarkably well, including treatments administered at intervals ranging from 6 to 12 days. Additionally, the model predicts peak immune activation times, rediscovering experimental data that had not been used in parameter fitting or in model construction. Notably, the validated model suggests that immunostimulatory and immunosuppressive intermediates are responsible for the observed phenomena of resistance and immune cell recruitment, and thus for variation of responses with respect to different schedules of drug administration.

Metabolism and Gut Microbiota in Cancer Immunoediting, CD8/Treg Ratios, Immune Cell Homeostasis, and Cancer (Immuno)Therapy: Concise Review

The concept of immunoediting, a process whereby the immune system eliminates immunogenic cancer cell clones, allowing the remaining cells to progress and form a tumor, has evolved with growing appreciation of the importance of cancer ecology on tumor progression. As cancer cells grow and modify their environment, they create spatial and nutrient constraints that may affect not only immune cell function but also differentiation, tipping the balance between cytotoxic and regulatory immunity to facilitate tumor growth. Here, we review how immunometabolism may contribute to cancer escape from the immune system, as well as highlight an emerging role of gut microbiota, its effects on the immune system and on response to immunotherapy. We conclude with a discussion of how these pieces can be integrated to devise better combination therapies and highlight the role of computational approaches as a potential tool to aid in combination therapy design. Stem Cells 2019;37:1273-1280.

Guiding principles for mechanistic modeling of bispecific antibodies

System based pharmacokinetic (PK) models can be used to study and predict the distribution of antibody based drugs into target tissues and assess the pharmacobinding (PB) of the drug to the target and the subsequent pharmacodynamic (PD) changes. In the absence of relevant PD readouts, compounded in cases of novel mechanisms, one can rely on binding between the drug and the target, computed as target occupancy (TO), as a relevant biomarker. This approach assumes that at maximum TO across the dosing interval, the drug-target interaction must demonstrate the intended pharmacology. Such analysis can help set laboratory objectives for protein engineers and chemists and guide them to the appropriate design and binding affinity of the molecule. Analysis of mechanistic models to guide affinity optimization against soluble and membrane-bound targets has been done for monoclonal antibodies (mAbs) (Tiwari et al., The AAPS Journal, 2017). However, comparable understanding of bispecific antibodies (BsAb; drugs with two targets, which are either soluble, membrane-bound, or a combination of the two) is still lacking. We propose to extend the work done by Tiwari et al. (2017) to BsAb. We focus on describing a generic BsAb with two membrane-bound targets, and explore the impact of various parameters on the TO of the BsAb to each target. Performed analysis can guide the optimization of dissociation constant (K_D) of the BsAb, and can also help in identifying druggable targets. Proposed model can be modified and tailored to specific biologics as needed.

From Experiment to Theory: What Can We Learn from Growth Curves?

Finding an appropriate functional form to describe population growth based on key properties of a described system allows making justified predictions about future population development. This information can be of vital importance in all areas of research, ranging from cell growth to global demography. Here, we use this connection between theory and observation to pose the following question: what can we infer about intrinsic properties of a population (i.e., degree of heterogeneity, or dependence on external resources) based on which growth function best fits its growth dynamics? We investigate several nonstandard classes of multi-phase growth curves that capture different stages of population growth; these models include hyperbolic-exponential, exponential-linear, exponential-linear-saturation growth patterns. The constructed models account explicitly for the process of natural selection within inhomogeneous populations. Based on the underlying hypotheses for each of the models, we identify whether the population that it best fits by a particular curve is more likely to be homogeneous or heterogeneous, grow in a density-dependent or frequency-dependent manner, and whether it depends on external resources during any or all stages of its development. We apply these predictions to cancer cell growth and demographic data obtained from the literature. Our theory, if confirmed, can provide an additional biomarker and a predictive tool to complement experimental research.

A Combination of Immune Checkpoint Inhibition with Metronomic Chemotherapy as a Way of Targeting Therapy-Resistant Cancer Cells

Therapeutic resistance remains a major obstacle in treating many cancers, particularly in advanced stages. It is likely that cytotoxic lymphocytes (CTLs) have the potential to eliminate therapy-resistant cancer cells. However, their effectiveness may be limited either by the immunosuppressive tumor microenvironment, or by immune cell death induced by cytotoxic treatments. High-frequency low-dose (also known as metronomic) chemotherapy can help improve the activity of CTLs by providing sufficient stimulation for cytotoxic immune cells without excessive depletion. Additionally, therapy-induced removal of tumor cells that compete for shared nutrients may also facilitate tumor infiltration by CTLs, further improving prognosis. Metronomic chemotherapy can also decrease the number of immunosuppressive cells in the tumor microenvironment, including regulatory T cells (Tregs) and myeloid-derived suppressor cells (MDSCs). Immune checkpoint inhibition can further augment anti-tumor immune responses by maintaining T cells in an activated state. Combining immune checkpoint inhibition with metronomic administration of chemotherapeutic drugs may create a synergistic effect that augments anti-tumor immune responses and clears metabolic competition. This would allow immune-mediated elimination of therapy-resistant cancer cells, an effect that may be unattainable by using either therapeutic modality alone.

Normal Wound Healing and Tumor Angiogenesis as a Game of Competitive Inhibition

Both normal wound healing and tumor angiogenesis are mitigated by the sequential, carefully orchestrated release of growth stimulators and inhibitors. These regulators are released from platelet clots formed at the sites of activated endothelium in a temporally and spatially controlled manner, and the order of their release depends on their affinity to glycosaminoglycans (GAG) such as heparan sulfate (HS) within the extracellular matrix, and platelet open canallicular system. The formation of vessel sprouts, triggered by angiogenesis regulating factors with lowest affinities for heparan sulfate (e.g. VEGF), is followed by vessel-stabilizing PDGF-B or bFGF with medium affinity for HS, and by inhibitors such as PF-4 and TSP-1 with the highest affinities for HS. The invasive wound-like edge of growing tumors has an overabundance of angiogenesis stimulators, and we propose that their abundance out-competes angiogenesis inhibitors, effectively preventing inhibition of angiogenesis and vessel maturation. We evaluate this hypothesis using an experimentally motivated agent-based model, and propose a general theoretical framework for understanding mechanistic similarities and differences between the processes of normal wound healing and pathological angiogenesis from the point of view of competitive inhibition.

Primary and metastatic tumor dormancy as a result of population heterogeneity

Existence of tumor dormancy, or cancer without disease, is supported both by autopsy studies that indicate presence of microscopic tumors in men and women who die of trauma (primary dormancy), and by long periods of latency between excision of primary tumors and disease recurrence (metastatic dormancy). Within dormant tumors, two general mechanisms underlying the dynamics are recognized, namely, the population existing at limited carrying capacity (tumor mass dormancy), and solitary cell dormancy, characterized by long periods of quiescence marked by cell cycle arrest. Here we focus on mechanisms that precede the avascular tumor reaching its carrying capacity, and propose that dynamics consistent with tumor dormancy and subsequent escape from it can be accounted for with simple models that take into account population heterogeneity. We evaluate parametrically heterogeneous Malthusian, logistic and Allee growth models and show that 1) time to escape from tumor dormancy is driven by the initial distribution of cell clones in the population and 2) escape from dormancy is accompanied by a large increase in variance, as well as the expected value of fitness-determining parameters. Based on our results, we propose that parametrically heterogeneous logistic model would be most likely to account for primary tumor dormancy, while distributed Allee model would be most appropriate for metastatic dormancy. We conclude with a discussion of dormancy as a stage within a larger context of cancer as a systemic disease.

Reviewers: This article was reviewed by Heiko Enderling and Marek Kimmel.

Mathematical Modeling of Extinction of Inhomogeneous Populations

Mathematical models of population extinction have a variety of applications in such areas as ecology, paleontology and conservation biology. Here we propose and investigate two types of sub-exponential models of population extinction. Unlike the more traditional exponential models, the life duration of sub-exponential models is finite. In the first model, the population is assumed to be composed of clones that are independent from each other. In the second model, we assume that the size of the population as a whole decreases according to the sub-exponential equation. We then investigate the "unobserved heterogeneity," i.e., the underlying inhomogeneous population model, and calculate the distribution of frequencies of clones for both models. We show that the dynamics of frequencies in the first model is governed by the principle of minimum of Tsallis information loss. In the second model, the notion of "internal population time" is proposed; with respect to the internal time, the dynamics of frequencies is governed by the principle of minimum of Shannon information loss. The results of this analysis show that the principle of minimum of information loss is the underlying law for the evolution of a broad class of models of population extinction. Finally, we propose a possible application of this modeling framework to mechanisms underlying time perception.

Escape from tumor dormancy and time to angiogenic switch as mitigated by tumor-induced stimulation of stroma

A variety of mechanisms have been proposed to explain "cancer without disease", the state of tumor dormancy, characterized by balance in cell proliferation and cell death within a tumor. Here we have investigated a theoretical construct, whereby one of such mechanisms, the time to induction of angiogenesis, or "angiogenic switch", is mitigated by the degree of stromal stimulation by the tumor. We tested this hypothesis and its implications by introducing a mathematical model that captures how angiogenesis regulators, released from the platelet clot, contribute to formation of normal vasculature. We then modified the model to introduce tumor-induced increase in production of angiogenesis regulators and were able to simulate pathological angiogenesis. Through varying parameters governing the degree of tumor-induced stromal stimulation, we were able to qualitatively replicate experimentally observed growth curves for both dormant and actively growing tumors of breast cancer and liposarcoma. In fact, variation of very few parameters was sufficient to replicate any experimentally observed time to angiogenic switch in the available data. Finally, we investigated the effects of tighter binding isoforms of angiogenesis stimulators on neovasculature formation and tumor growth, which may provide an explanation for variations in angiogenesis -dependence in tumors of different tissue origin.

Cancer immunoediting: A process driven by metabolic competition as a predator-prey-shared resource type model

It is a well-established fact that tumors up-regulate glucose consumption to meet increasing demands for rapidly available energy by upregulating a purely glycolytic mode of glucose metabolism. What is often neglected is that activated cytotoxic cells of the immune system, integral players in the carcinogenesis process, also come to rely on glycolysis as a primary mode of glucose metabolism. Moreover, while cancer cells can revert back to aerobic metabolism, rapidly proliferating cytotoxic lymphocytes are incapable of performing their function when adequate resources are lacking. Consequently, it is likely that in the tumor microenvironment there may exist competition for shared resources between cancer cells and the cells of the immune system, which may underlie much of tumor-immune dynamics. Proposed here is a model of tumor-immune-glucose interactions, formulated as a predator-prey-common resource type system. The outcome of these interactions ranges from tumor elimination, to tumor dormancy, to unrestrained tumor growth. It is also predicted that the process of tumor escape can be preceded by periods of oscillatory tumor growth. A detailed bifurcation analysis of three subsystems of the model suggest that oscillatory regimes are a result of competition for shared resource (glucose) between the predator (immune cells) and the prey (cancer cells). Existence of competition for nutrients between cancer and immune cells may provide additional mechanistic insight as to why the efficacy of many immunotherapies may be limited.

Metronomic chemotherapy: an attractive alternative to maximum tolerated dose therapy that can activate anti-tumor immunity and minimize therapeutic resistance

The administration of chemotherapy at reduced doses given at regular, frequent time intervals, termed 'metronomic' chemotherapy, presents an alternative to standard maximal tolerated dose (MTD) chemotherapy. The primary target of metronomic chemotherapy was originally identified as endothelial cells supporting the tumor vasculature, and not the tumor cells themselves, consistent with the emerging concept of cancer as a systemic disease involving both tumor cells and their microenvironment. While anti-angiogenesis is an important mechanism of action of metronomic chemotherapy, other mechanisms, including activation of anti-tumor immunity and a decrease in acquired therapeutic resistance, have also been identified. Here we present evidence supporting a mechanistic explanation for the improved activity of cancer chemotherapy when administered on a metronomic, rather than an MTD schedule and discuss the implications of these findings for further translation into the clinic.

Immune evasion through competitive inhibition: the shielding effect of cancer non-stem cells

It has been recently proposed that the two emerging hallmarks of cancer, namely altered glucose metabolism and immune evasion, may in fact be fundamentally linked. This connection comes from up-regulation of glycolysis by tumor cells, which can lead to active competition for resources in the tumor microenvironment between tumor and immune cells. Here it is further proposed that cancer stem cells (CSCs) can circumvent the anti-tumor immune response by creating a "protective shield" of non-stem cancer cells around them. This shield can protect the CSCs both by creating a physical barrier between them and cytotoxic lymphocytes (CTLs), and by promoting competition for the common resources, such as glucose, between non-stem cancer cells and CTLs. The implications of this hypothesis are investigated using an agent-based model, leading to a prediction that relative CSC to non-CSC ratio will vary depending on the strength of the host immune response. A discussion of possible therapeutic approaches concludes the paper, suggesting that a chemotherapeutic regimen consisting of regular pulsed doses, i.e., metronomic chemotherapy, would yield the best clinical outcome by removing the "protective shield" and thus allowing CTLs to most effectively reach and eliminate CSCs.

Resource consumption, sustainability, and cancer

Preserving a system's viability in the presence of diversity erosion is critical if the goal is to sustainably support biodiversity. Reduction in population heterogeneity, whether inter- or intraspecies, may increase population fragility, either decreasing its ability to adapt effectively to environmental changes or facilitating the survival and success of ordinarily rare phenotypes. The latter may result in over-representation of individuals who may participate in resource utilization patterns that can lead to over-exploitation, exhaustion, and, ultimately, collapse of both the resource and the population that depends on it. Here, we aim to identify regimes that can signal whether a consumer-resource system is capable of supporting viable degrees of heterogeneity. The framework used here is an expansion of a previously introduced consumer-resource type system of a population of individuals classified by their resource consumption. Application of the Reduction Theorem to the system enables us to evaluate the health of the system through tracking both the mean value of the parameter of resource (over)consumption, and the population variance, as both change over time. The article concludes with a discussion that highlights applicability of the proposed system to investigation of systems that are affected by particularly devastating overly adapted populations, namely cancerous cells. Potential intervention approaches for system management are discussed in the context of cancer therapies.

Mixed strategies and natural selection in resource allocation

An appropriate choice of strategy for resource allocation may frequently determine whether a population will be able to survive under the conditions of severe resource limitations. Here we focus on two classes of strategies allocation of resources towards rapid proliferation, or towards slower proliferation but increased physiological and environmental maintenance. We propose a generalized framework, where individuals within a population can use either strategy in different proportion for utilization of a common dynamical resource in order to maximize their fitness. We use the model to address two major questions, namely, whether either strategy is more likely to be selected for as a result of natural selection, and, if one allows for the possibility of resource over-consumption, whether either strategy is preferable for avoiding population collapse due to resource exhaustion. Analytical and numerical results suggest that the ultimate choice of strategy is determined primarily by the initial distribution of individuals in the population, and that while investment in physiological and environmental maintenance is a preferable strategy in a homogeneous population, no generalized prediction can be made about heterogeneous populations.

The emerging "hallmarks" of metabolic reprogramming and immune evasion: distinct or linked?

The role of the immune system in tumor elimination has been shown to be increasingly ambiguous, as many tumors not only escape recognition by the adaptive immune response but also even prime the immune cells to promote tumor growth. This effect is achieved through a number of mechanisms, which include both direct interference with the cells of the adaptive immune response and indirect immunosuppression achieved through modification of the tumor microenvironment. We propose that through upregulation of glycolysis and the consequent lowering of pH in the tumor microenvironment, tumors can take advantage of a pH control system, already exploited by specific immune cell subpopulations, to gain control of the immune system and suppress both cytotoxic and antigen-presenting cells. This is accomplished through the direct competition of tumor cells with actively proliferating glycolytic immune cells for glucose and indirectly through the creation by the tumor of a microenvironment that interferes with maturation and activation of antigen-presenting cells and naïve cytotoxic T cells. Immunosuppressive properties of an acidic microenvironment in the vicinity of the tumor can thus provide additional benefits for upregulation of glycolysis by tumor cells, suggesting that the two emerging "hallmarks of cancer," altered glucose metabolism and immune suppression, are in fact fundamentally linked.

Biological stoichiometry in tumor micro-environments

Tumors can be viewed as evolving ecological systems, in which heterogeneous populations of cancer cells compete with each other and somatic cells for space and nutrients within the ecosystem of the human body. According to the growth rate hypothesis (GRH), increased phosphorus availability in an ecosystem, such as the tumor micro-environment, may promote selection within the tumor for a more proliferative and thus potentially more malignant phenotype. The applicability of the GRH to tumor growth is evaluated using a mathematical model, which suggests that limiting phosphorus availability might promote intercellular competition within a tumor, and thereby delay disease progression. It is also shown that a tumor can respond differently to changes in its micro-environment depending on the initial distribution of clones within the tumor, regardless of its initial size. This suggests that composition of the tumor as a whole needs to be evaluated in order to maximize the efficacy of therapy.

Prisoner's dilemma in cancer metabolism

As tumors outgrow their blood supply and become oxygen deprived, they switch to less energetically efficient but oxygen-independent anaerobic glucose metabolism. However, cancer cells maintain glycolytic phenotype even in the areas of ample oxygen supply (Warburg effect). It has been hypothesized that the competitive advantage that glycolytic cells get over aerobic cells is achieved through secretion of lactic acid, which is a by-product of glycolysis. It creates acidic microenvironment around the tumor that can be toxic to normal somatic cells. This interaction can be seen as a prisoner's dilemma: from the point of view of metabolic payoffs, it is better for cells to cooperate and become better competitors but neither cell has an incentive to unilaterally change its metabolic strategy. In this paper a novel mathematical technique, which allows reducing an otherwise infinitely dimensional system to low dimensionality, is used to demonstrate that changing the environment can take the cells out of this equilibrium and that it is cooperation that can in fact lead to the cell population committing evolutionary suicide.

Myeloid cells in tumour-immune interactions

Despite highly developed specific immune responses, tumour cells often manage to escape recognition by the immune system, continuing to grow uncontrollably. Experimental work suggests that mature myeloid cells may be central to the activation of the specific immune response. Recognition and subsequent control of tumour growth by the cells of the specific immune response depend on the balance between immature (ImC) and mature (MmC) myeloid cells in the body. However, tumour cells produce cytokines that inhibit ImC maturation, altering the balance between ImC and MmC. Hence, the focus of this manuscript is on the study of the potential role of this inhibiting mechanism on tumour growth dynamics. A conceptual predator-prey type model that incorporates the dynamics and interactions of tumour cells, CD8(+) T cells, ImC and MmC is proposed in order to address the role of this mechanism. The prey (tumour) has a defence mechanism (blocking the maturation of ImC) that prevents the predator (immune system) from recognizing it. The model, a four-dimensional nonlinear system of ordinary differential equations, is reduced to a two-dimensional system using time-scale arguments that are tied to the maturation rate of ImC. Analysis shows that the model is capable of supporting biologically reasonable patterns of behaviour depending on the initial conditions. A range of parameters, where healing without external influences can occur, is identified both qualitatively and quantitatively.