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Abstract Multiple myeloma (MM) is an osteolytic malignancy that is incurable due to the emergence of treatment resistant disease. Defining how, when and where myeloma cell intrinsic and extrinsic bone microenvironmental mechanisms cause relapse is challenging with current biological approaches. Here, we report a biology-driven spatiotemporal hybrid agent-based model of the MM-bone microenvironment. Results indicate MM intrinsic mechanisms drive the evolution of treatment resistant disease but that the protective effects of bone microenvironment mediated drug resistance (EMDR) significantly enhances the probability and heterogeneity of resistant clones arising under treatment. Further, the model predicts that targeting of EMDR deepens therapy response by eliminating sensitive clones proximal to stroma and bone, a finding supported by in vivo studies. Altogether, our model allows for the study of MM clonal evolution over time in the bone microenvironment and will be beneficial for optimizing treatment efficacy so as to significantly delay disease relapse.

Abstract Spatial agent‐based models are frequently used to investigate the evolution of solid tumours subject to localized cell–cell interactions and microenvironmental heterogeneity. As spatial genomic, transcriptomic and proteomic technologies gain traction, spatial computational models are predicted to become ever more necessary for making sense of complex clinical and experimental data sets, for predicting clinical outcomes, and for optimizing treatment strategies. Here we present a non‐technical step by step guide to developing such a model from first principles. Stressing the importance of tailoring the model structure to that of the biological system, we describe methods of increasing complexity, from the basic Eden growth model up to off‐lattice simulations with diffusible factors. We examine choices that unavoidably arise in model design, such as implementation, parameterization, visualization and reproducibility. Each topic is illustrated with examples drawn from recent research studies and state of the art modelling platforms. We emphasize the benefits of simpler models that aim to match the complexity of the phenomena of interest, rather than that of the entire biological system. Our guide is aimed at both aspiring modellers and other biologists and oncologists who wish to understand the assumptions and limitations of the models on which major cancer studies now so often depend.

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

Abstract Bone-forming osteoblasts and -resorbing osteoclasts control bone injury repair, and myeloid-derived cells such as monocytes and macrophages are known to influence their behavior. However, precisely how these multiple cell types coordinate and regulate each other over time within the bone marrow to restore bone is difficult to dissect using biological approaches. Conversely, mathematical modeling lends itself well to this challenge. Therefore, we generated an ordinary differential equation (ODE) model powered by experimental data (osteoblast, osteoclast, bone volume, pro- and anti-inflammatory myeloid cells) obtained from intra-tibially injured mice. Initial ODE results using only osteoblast/osteoclast populations demonstrated that bone homeostasis could not be recovered after injury, but this issue was resolved upon integration of pro- and anti-inflammatory myeloid population dynamics. Surprisingly, the ODE revealed temporal disconnects between the peak of total bone mineralization/resorption, and osteoblast/osteoclast numbers. Specifically, the model indicated that osteoclast activity must vary greatly (> 17-fold) to return the bone volume to baseline after injury and suggest that osteoblast/osteoclast number alone is insufficient to predict bone the trajectory of bone repair. Importantly, the values of osteoclast activity fall within those published previously. These data underscore the value of mathematical modeling approaches to understand and reveal new insights into complex biological processes.

Myeloid-derived monocyte and macrophages are key cells in the bone that contribute to remodeling and injury repair. However, their temporal polarization status and control of bone-resorbing osteoclasts and bone-forming osteoblasts responses is largely unknown. In this study, we focused on two aspects of monocyte/macrophage dynamics and polarization states over time: 1) the injury-triggered pro- and anti-inflammatory monocytes/macrophages temporal profiles, 2) the contributions of pro- versus anti-inflammatory monocytes/macrophages in coordinating healing response. Bone healing is a complex multicellular dynamic process. While traditional in vitro and in vivo experimentation may capture the behavior of select populations with high resolution, they cannot simultaneously track the behavior of multiple populations. To address this, we have used an integrated coupled ordinary differential equations (ODEs)-based framework describing multiple cellular species to in vivo bone injury data in order to identify and test various hypotheses regarding bone cell populations dynamics. Our approach allowed us to infer several biological insights including, but not limited to,: 1) anti-inflammatory macrophages are key for early osteoclast inhibition and pro-inflammatory macrophage suppression, 2) pro-inflammatory macrophages are involved in osteoclast bone resorptive activity, whereas osteoblasts promote osteoclast differentiation, 3) Pro-inflammatory monocytes/macrophages rise during two expansion waves, which can be explained by the anti-inflammatory macrophages-mediated inhibition phase between the two waves. In addition, we further tested the robustness of the mathematical model by comparing simulation results to an independent experimental dataset. Taken together, this novel comprehensive mathematical framework allowed us to identify biological mechanisms that best recapitulate bone injury data and that explain the coupled cellular population dynamics involved in the process. Furthermore, our hypothesis testing methodology could be used in other contexts to decipher mechanisms in complex multicellular processes.

Over 40 years ago, seminal papers by Armstrong and McGehee and by Levins showed that temporal fluctuations in resource availability could permit coexistence of two species on a single resource. Such coexistence results from non-linearities or non-additivities in the way resource supply translates into fitness. These reflect trade-offs where one species benefits more than the other during good periods and suffers more (or does less well) than the other during less good periods, be the periods stochastic, unstable population dynamics, or seasonal. Since, coexistence based on fluctuating conditions has been explored under the guises of “grazers” and “diggers,” variance partitioning, relative non-linearity, “opportunists” and “gleaners,” and as the storage effect. Here we focus on two phenotypes, “cream skimmers” and “crumb pickers,” the former having the advantage in richer times and the latter in less rich times. In nature, richer and poorer times, with regular or stochastic appearances, are the norm and occur on many time scales. Fluctuations among richer and poorer times also appear to be the norm in cancer ecosystems. Within tumors, nutrient availability, oxygen, and pH can fluctuate stochastically or periodically, with swings occurring over seconds to minutes to hours. Despite interest in tumor heterogeneity and how it promotes the coexistence of different cancer cell types, the effects of fluctuating resource availability have not been explored for cancer. Here, in the context of pulsed resources, we (1) develop models of foraging consumers who experience pulsed resources to examine four types of trade-offs that can promote coexistence of phenotypes that do relatively better in richer versus in poorer times, (2) establish that conditions in tumors are conducive for this mechanism, (3) propose and empirically explore biomarkers indicative of the two phenotypes (HIF-1, GLUT-1, CA IX, CA XII), and (4) and compare cream skimmer and crumb picker biology and ecology in nature and cancer to provide cross-disciplinary insights into this interesting, and, we argue, likely very common, mechanism of coexistence.

Dormancy is an inactive period of an organism’s life cycle that permits it to survive through phases of unfavorable conditions in highly variable environments. Dormancy is not binary. There is a continuum of dormancy phenotypes that represent some degree of reduced metabolic activity (hypometabolism), reduced feeding, and reduced reproduction or proliferation. Similarly, normal cells and cancer cells exhibit a range of states from quiescence to long-term dormancy that permit survival in adverse environmental conditions. In contrast to organismal dormancy, which entails a reduction in metabolism, dormancy in cells (both normal and cancer) is primarily characterized by lack of cell division. “Cancer dormancy” also describes a state characterized by growth stagnation, which could arise from cells that are not necessarily hypometabolic or non-proliferative. This inconsistent terminology leads to confusion and imprecision that impedes progress in interdisciplinary research between ecologists and cancer biologists. In this paper, we draw parallels and contrasts between dormancy in cancer and other ecosystems in nature, and discuss the potential for studies in cancer to provide novel insights into the evolutionary ecology of dormancy.

Summary: Cancers are the result of eco-evolutionary processes fueled by heritable phenotypic diversification and driven by environmentally dependent selection. Space represents a key growth-limiting ecological resource, the ability to explore this resource is likely under strong selection. Using agent-based modeling, we explored the interplay between phenotypic strategies centered on gaining access to new space through cell-extrinsic degradation of extracellular matrix barriers and the exploitation of this resource through maximizing cell proliferation. While cell proliferation is a cell-intrinsic property, newly accessed space represents a public good, which can benefit both producers and non-producers. We found that this interplay results in ecological succession, enabling emergence of large, heterogeneous, and highly proliferative populations. Even though in our simulations both remodeling and proliferation strategies were under strong positive selection, their interplay led to sub-clonal architecture that could be interpreted as evidence for neutral evolution, warranting cautious interpretation of inferences from sequencing of cancer genomes.

Advanced cancers, such as prostate and breast cancers, commonly metastasize to bone. In the bone matrix, dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone remodeling. In the presence of a cancerous microenvironment, the topology of this network changes. In those situations, osteocytes often appear to be either overdifferentiated (i.e., there are more dendrites than healthy bone) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, histological sections from metastatic breast cancer xenografted mice show that number of osteocytes per unit area is different between healthy bone and cancerous bone. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We considered variations of our model to study specific physiological hypotheses related to osteoblast differentiation; for example predicting how changing biological parameters, such as rates of bone secretion, rates of cancer formation, and rates of osteoblast differentiation can allow for qualitatively different network topologies. We then used our model to explore how commonly applied therapies such as bisphosphonates (e.g., zoledronic acid) impact osteocyte network formation.

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

The Hybrid Automata Library (HAL) is a Java Library developed for use in mathematical oncology modeling. It is made of simple, efficient, generic components that can be used to model complex spatial systems. HAL's components can broadly be classified into: on- and off-lattice agent containers, finite difference diffusion fields, a GUI building system, and additional tools and utilities for computation and data collection. These components are designed to operate independently and are standardized to make them easy to interface with one another. As a demonstration of how modeling can be simplified using our approach, we have included a complete example of a hybrid model (a spatial model with interacting agent-based and PDE components). HAL is a useful asset for researchers who wish to build performant 1D, 2D and 3D hybrid models in Java, while not starting entirely from scratch. It is available on GitHub at https://github.com/MathOnco/HAL under the MIT License. HAL requires the Java JDK version 1.8 or later to compile and run the source code.

Natural selection in cancer often results in the emergence of increasingly malignant tumor cells that display many if not all of the hallmarks of cancer. One of the most important traits acquired during cancer progression is angiogenesis. Tumor cells capable of secreting pro-angiogenic factors can be seen as cooperators where the improved oxygenation, nutrient delivery and waste disposal resulting from angiogenesis could be seen as a public good. Under this view, the relatively costly secretion of molecular signals required to orchestrate angiogenesis would be undertaken exclusively by cooperating tumor cells but the benefits of angiogenesis would be felt by neighboring tumor cells regardless of their contribution to the process. In this work we detail a mathematical model to better understand how clones capable of secreting pro-angiogenic factors can emerge in a tumor made of non-cooperative tumor cells. Given the importance of the spatial configuration of the tumor in determining the efficacy of the secretion of pro-angiogenic factors as well as the benefits of angiogenesis we have developed a spatial game theoretic approach where interactions and public good diffusion are described by graphs. The results show that structure of the population affects the evolutionary dynamics of the pro-angiogenic clone. Specifically, when the benefit of angiogenesis is represented by sigmoid function with regards to the number of pro-angiogenic clones then the probability of the coexistence of pro-angiogenic and angiogenesis-neutral clones increases. Our results demonstrate that pro-angiogenic clone equilibrates into clusters that appear from surrounding vascular tissues towards the center of tumor. These clusters appear notably less dense after anti-angiogenic therapy.

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

Small changes to the parameters of a system can lead to abrupt qualitative changes of its behavior, a phenomenon known as bifurcation. Such instabilities are typically considered problematic, however, we show that their power can be leveraged to design novel types of mechanisms. Hysteresis mechanisms use transient changes of system parameters to induce a permanent improvement to its performance via optimal equilibrium selection. Optimal control mechanisms induce convergence to states whose performance is better than even the best equilibrium. We apply these mechanisms in two different settings that illustrate the versatility of bifurcation mechanism design. In the first one we explore how introducing flat taxation could improve social welfare, despite decreasing agent “rationality,” by destabilizing inefficient equilibria. From there we move on to consider a well known game of tumor metabolism and use our approach to derive potential new cancer treatment strategies.

Since the discovery of tumour initiating cells (TICs) in solid tumours, studies focussing on their role in cancer initiation and progression have abounded. The biological interrogation of these cells continues to yield volumes of information on their pro-tumourigenic behaviour, but actionable generalised conclusions have been scarce. Further, new information suggesting a dependence of tumour composition and growth on the microenvironment has yet to be studied theoretically. To address this point, we created a hybrid, discrete/continuous computational cellular automaton model of a generalised stem-cell driven tissue with a simple microenvironment. Using the model we explored the phenotypic traits inherent to the tumour initiating cells and the effect of the microenvironment on tissue growth. We identify the regions in phenotype parameter space where TICs are able to cause a disruption in homeostasis, leading to tissue overgrowth and tumour maintenance. As our parameters and model are non-specific, they could apply to any tissue TIC and do not assume specific genetic mutations. Targeting these phenotypic traits could represent a generalizable therapeutic strategy across cancer types. Further, we find that the microenvironmental variable does not strongly affect the outcomes, suggesting a need for direct feedback from the microenvironment onto stem-cell behaviour in future modelling endeavours.

During embryogenesis, multicellular animals are shaped via cell proliferation, cell rearrangement, and apoptosis. At the end of development, tissue architecture is then maintained through balanced rates of cell proliferation and loss. Here, we take an in silico approach to look for generic systems features of morphogenesis in multicellular animals that arise as a consequence of the evolution of development. Using artificial evolution, we evolved cellular automata-based digital organisms that have distinct embryonic and homeostatic phases of development. Although these evolved organisms use a variety of strategies to maintain their form over time, organisms of different types were all found to rapidly recover from environmental damage in the form of wounds. This regenerative response was most robust in an organism with a stratified tissue-like architecture. An evolutionary analysis revealed that evolution itself contributed to the ability of this organism to maintain its form in the face of genetic and environmental perturbation, confirming the results of previous studies. In addition, the exceptional robustness of this organism to surface injury was found to result from an upward flux of cells, driven in part by cell divisions with a stable niche at the tissue base. Given the general nature of the model, our results lead us to suggest that many of the robust systems properties observed in real organisms, including scar-free wound-healing in well-protected embryos and the layered tissue architecture of regenerating epithelial tissues, may be by-products of the evolution of morphogenesis, rather than the direct result of selection.