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The role of plasticity and epigenetics in shaping cancer evolution and response to therapy has taken center stage with recent technological advances including single cell sequencing. This roadmap article is focused on state-of-the-art mathematical and experimental approaches to interrogate plasticity in cancer, and addresses the following themes and questions: is there a formal overarching framework that encompasses both non-genetic plasticity and mutation-driven somatic evolution? How do we measure and model the role of the microenvironment in influencing/controlling non-genetic plasticity? How can we experimentally study non-genetic plasticity? Which mathematical techniques are required or best suited? What are the clinical and practical applications and implications of these concepts?

Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology-defined here simply as the use of mathematics in cancer research-complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

Neoplasms change over time through a process of cell-level evolution, driven by genetic and epigenetic alterations. However, the ecology of the microenvironment of a neoplastic cell determines which changes provide adaptive benefits. There is widespread recognition of the importance of these evolutionary and ecological processes in cancer, but to date, no system has been proposed for drawing clinically relevant distinctions between how different tumours are evolving. On the basis of a consensus conference of experts in the fields of cancer evolution and cancer ecology, we propose a framework for classifying tumours that is based on four relevant components. These are the diversity of neoplastic cells (intratumoural heterogeneity) and changes over time in that diversity, which make up an evolutionary index (Evo-index), as well as the hazards to neoplastic cell survival and the resources available to neoplastic cells, which make up an ecological index (Eco-index). We review evidence demonstrating the importance of each of these factors and describe multiple methods that can be used to measure them. Development of this classification system holds promise for enabling clinicians to personalize optimal interventions based on the evolvability of the patient's tumour. The Evo- and Eco-indices provide a common lexicon for communicating about how neoplasms change in response to interventions, with potential implications for clinical trials, personalized medicine and basic cancer research.

Tumor evolution is driven by genetic variation; however, it is the tumor microenvironment (TME) that provides the selective pressure contributing to evolution in cancer. Despite high histopathological heterogeneity within glioblastoma (GBM), the most aggressive brain tumor, the interactions between the genetically distinct GBM cells and the surrounding TME are not fully understood. To address this, we analyzed matched primary and recurrent GBM archival tumor tissues with imaging-based techniques aimed to simultaneously evaluate tumor tissues for the presence of hypoxic, angiogenic, and inflammatory niches, extracellular matrix (ECM) organization, TERT promoter mutational status, and several oncogenic amplifications on the same slide and location. We found that the relationships between genetic and TME diversity are different in primary and matched recurrent tumors. Interestingly, the texture of the ECM, identified by label-free reflectance imaging, was predictive of single-cell genetic traits present in the tissue. Moreover, reflectance of ECM revealed structured organization of the perivascular niche in recurrent GBM, enriched in immunosuppressive macrophages. Single-cell spatial transcriptomics further confirmed the presence of the niche-specific macrophage populations and identified interactions between endothelial cells, perivascular fibroblasts, and immunosuppressive macrophages. Our results underscore the importance of GBM tissue organization in tumor evolution and highlight genetic and spatial dependencies.