The mathematics of cancer: integrating quantitative models

Key Points Mathematical models have become an integral part of cancer biology. They are useful tools for deriving a mechanistic understanding of dynamic processes in cancer. The somatic evolutionary process, which maintains tissues and can initiate cancer, has served as a hallmark of mathematical descriptions of tumours. Mathematical models have helped in the understanding of interactions among homeostatic mechanisms, environmental factors and mutation accumulation that drive tumorigenesis. Using cell-based hierarchical models of tissue structure, theoretical insights have influenced the prediction of the cell of origin of human cancers, which may drive an understanding of metastasis and treatment response. The temporal order of events during tumour development can be recapitulated using mathematical modelling and genomics data sets. Mathematical models have also been used to explore the role of the tumour microenvironment in cancer progression. Such models help to elucidate important microenvironmental barriers to effective cancer treatment and how to overcome them. Metastasis evolution and immunotherapy have attracted increasing interest but still offer a wide range of opportunities for mathematical modelling. In combination with pharmacological considerations, quantitative models have a decisive role in the exploration of novel treatment modalities of cancer. This includes drug scheduling and the effect of combination therapy to avoid the evolution of resistance. The key role of mathematical modelling in the future will not only be to describe what is known, but also to point to gaps in our understanding of which complex interactions drive tumour growth, treatment dynamics and resistance evolution. This Review discusses mathematical modelling approaches in cancer research. These models can complement experimental and clinical studies, but can also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research. Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and