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Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

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

We propose a model of cancer initiation and progression where tumor growth is modulated by an evolutionary coordination game. Evolutionary games of cancer are widely used to model frequency-dependent cell interactions with the most studied games being the Prisoner’s Dilemma and public goods games. Coordination games, by their more obscure and less evocative nature, are left understudied, despite the fact that, as we argue, they offer great potential in understanding and treating cancer. In this paper we present the conditions under which coordination games between cancer cells evolve, we propose aspects of cancer that can be modeled as results of coordination games, and explore the ways through which coordination games of cancer can be exploited for therapy.

Evolutionary cancer therapy (ECT) delays or forestalls the progression of metastatic cancer by adjusting treatment based on individual patient and disease characteristics. Clinical implementation of ECT can improve patient outcomes but faces technical and cultural challenges. To address those, we propose a systems approach incorporating systems modeling, problem structuring, and stakeholder engagement. This approach identifies and addresses barriers to implementation, ensuring the feasibility of ECT in clinical practice and enabling better metastatic cancer care.

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

Mathematical modeling plays an important role in our understanding and targeting therapy resistance mechanisms in cancer. The polymorphic Gompertzian model, analyzed theoretically and numerically by Viossat and Noble to demonstrate the benefits of adaptive therapy in metastatic cancer, describes a heterogeneous cancer population consisting of therapy-sensitive and therapy-resistant cells. In this study, we demonstrate that the polymorphic Gompertzian model successfully captures trends in both in vitro and in vivo data on non-small cell lung cancer (NSCLC) dynamics under treatment. Additionally, for the in vivo data of tumor dynamics in patients undergoing treatment, we compare the goodness of fit of the polymorphic Gompertzian model to that of the classical oncologic models, which were previously identified as the models that fit this data best. We show that the polymorphic Gompertzian model can successfully capture the U-shape trend in tumor size during cancer relapse, which can not be fitted with the classical oncologic models. In general, the polymorphic Gompertzian model corresponds well to both in vitro and in vivo real-world data, suggesting it as a candidate for improving the efficacy of cancer therapy, for example, through evolutionary/adaptive therapies.

We propose a model of cancer initiation and progression where tumor growth is modulated by an evolutionary coordination game. Evolutionary games of cancer are widely used to model frequency-dependent cell interactions with the most studied games being the Prisoner’s Dilemma and public goods games. Coordination games, by their more obscure and less evocative nature, are left understudied, despite the fact that, as we argue, they offer great potential in understanding and treating cancer. In this paper we present the conditions under which coordination games between cancer cells evolve, we propose aspects of cancer that can be modeled as results of coordination games, and explore the ways through which coordination games of cancer can be exploited for therapy.

Fish populations subject to heavy exploitation are expected to evolve over time smaller average body sizes. We introduce Stackelberg evolutionary game theory to show how fisheries management should be adjusted to mitigate the potential negative effects of such evolutionary changes. We present the game of a fisheries manager versus a fish population, where the former adjusts the harvesting rate and the net size to maximize profit, while the latter responds by evolving the size at maturation to maximize the fitness. We analyze three strategies: i) ecologically enlightened (leading to a Nash equilibrium in game-theoretic terms); ii) evolutionarily enlightened (leading to a Stackelberg equilibrium) and iii) domestication (leading to team optimum) and the corresponding outcomes for both the fisheries manager and the fish. Domestication results in the largest size for the fish and the highest profit for the manager. With the Nash approach the manager tends to adopt a high harvesting rate and a small net size that eventually leads to smaller fish. With the Stackelberg approach the manager selects a bigger net size and scales back the harvesting rate, which lead to a bigger fish size and a higher profit. Overall, our results encourage managers to take the fish evolutionary dynamics into account. Moreover, we advocate for the use of Stackelberg evolutionary game theory as a tool for providing insights into the eco-evolutionary consequences of exploiting evolving resources.

Prostate-specific antigen (PSA) is the most commonly used serum marker for prostate cancer. It plays a role in cancer detection, treatment monitoring, and more recently, in guiding adaptive therapy protocols, where treatment is alternated based on PSA levels. However, the relationship between PSA levels and tumor volume remains poorly understood. Empirical evidence suggests that different cancer cell types produce varying amounts of PSA. Despite this, current mathematical cancer models often assume either that all cell types contribute equally to PSA levels or that only certain subpopulations produce PSA at fixed rates. In this study, we compare Zhang et al.’s classical adaptive therapy protocol with the standard of care, which involves continuous maximum tolerable dose treatment, under different assumptions regarding PSA production. Specifically, we explore the possibility that testosterone-dependent, testosterone-producing, and testosterone-independent cells contribute to PSA production to varying degrees. We use the time to competitive release as a proxy for the time to disease progression. Our findings indicate that adaptive therapy consistently results in a longer time to competitive release compared to the standard of care, regardless of the assumptions about PSA production. However, when testosterone-independent cells are the sole PSA producers, Zhang et al.’s adaptive therapy protocol becomes inapplicable, as PSA levels never fall to half of their initial value, preventing therapy discontinuation. Additionally, we observe that the number and duration of treatment cycles in adaptive therapy are highly sensitive to assumptions about how much each cell type contributes to PSA production. Overall, our results emphasize the need for a deeper understanding of patient-specific PSA dynamics, which could enhance the effectiveness of adaptive therapy in prostate cancer treatment.

A game theory study supported by in vitro experimental data shows that drug treatment of non-small-cell lung cancer cells causes the cells to switch between evolutionary games they play among each other. Moreover, the work calls into question standard assumptions on the fitness costs of drug resistance to cancer cells.