The vast majority of cancer next-generation sequencing data consist of bulk samples composed of mixtures of cancer and normal cells. To study tumor evolution, subclonal reconstruction approaches based on machine learning are used to separate subpopulation of cancer cells and reconstruct their ancestral relationships. However, current approaches are entirely data-driven and agnostic to evolutionary theory. We demonstrate that systematic errors occur in subclonal reconstruction if tumor evolution is not accounted for, and that those errors increase when multiple samples are taken from the same tumor. To address this issue, we present a novel approach for model-based subclonal reconstruction that combines data-driven machine learning with evolutionary theory. Using public, synthetic and newly generated data, we show the method is more robust and accurate than current techniques in both single-sample and multi-region sequencing data. With careful data curation and interpretation, we show how the method allows minimizing the confounding factors that affect non-evolutionary methods, leading to a more accurate recovery of the evolutionary history of human tumors..

The increasing rate of antibiotic resistance and slowing discovery of novel antibiotic treatments presents a growing threat to public health. Here, we consider a simple model of evolution in asexually reproducing populations which considers adaptation as a biased random walk on a fitness landscape. This model associates the global properties of the fitness landscape with the algebraic properties of a Markov chain transition matrix and allows us to derive general results on the non-commutativity and irreversibility of natural selection as well as antibiotic cycling strategies. Using this formalism, we analyze 15 empirical fitness landscapes of E. coli under selection by different β-lactam antibiotics and demonstrate that the emergence of resistance to a given antibiotic can be either hindered or promoted by different sequences of drug application. Specifically, we demonstrate that the majority, approximately 70%, of sequential drug treatments with 2–4 drugs promote resistance to the final antibiotic. Further, we derive optimal drug application sequences with which we can probabilistically ‘steer’ the population through genotype space to avoid the emergence of resistance. This suggests a new strategy in the war against antibiotic–resistant organisms: drug sequencing to shepherd evolution through genotype space to states from which resistance cannot emerge and by which to maximize the chance of successful therapy.

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.