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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.

Background:Tumours are made up of a mixed population of different types of cells that include normal structures as well as ones associated with the malignancy, and there are multiple interactions between the malignant cells and the local microenvironment. These intercellular interactions, modulated by the microenvironment, effect tumour progression and represent a largely under-appreciated therapeutic target. We use observations of primary tumour biology from prostate cancer to extrapolate a mathematical model. Specifically, it has been observed that in prostate cancer three disparate cellular outcomes predominate: (i) the tumour remains well differentiated and clinically indolent – in this case the local stromal cells may act to restrain the growth of the cancer; (ii) early in its genesis the tumour acquires a highly malignant phenotype, growing rapidly and displacing the original stromal population (often referred to as small cell prostate cancer) – these less common aggressive tumours are relatively independent of the local microenvironment and (iii) the tumour co-opts the local stroma – taking on a classic stromagenic phenotype where interactions with the local microenvironment are critical to the cancer growth.Methods:We present an evolutionary game theoretical construct that models the influence of tumour–stroma interactions in driving these outcomes. We consider three characteristic and distinct cellular populations: stromal cells, tumour cells that are self-reliant in terms of microenvironmental factors and tumour cells that depend on the environment for resources, but can also co-opt stroma.Results:Using evolutionary game theory we explore a number of different scenarios that elucidate the impact of tumour–stromal interactions on the dynamics of prostate cancer growth and progression, and how different treatments in the metastatic setting can affect different types of tumours.Conclusion:The tumour microenvironment has a crucial role in selecting the traits of the tumour cells that will determine prostate cancer progression. Equally important treatments like hormone therapy affect the selection of these cancer phenotypes making it very important to understand how they impact prostate cancer's somatic evolution.

Drug resistance remains an elusive problem in cancer therapy, particularly for novel targeted therapies. Much work is focused upon the development of an arsenal of targeted therapies, towards oncogenic driver genes such as ALK-EML4, to overcome the inevitable resistance that develops over time. Currently, after failure of first line ALK TKI therapy, another ALK TKI is administered, though collateral sensitivity is not considered. To address this, we evolved resistance in an ALK rearranged non-small cell lung cancer line (H3122) to a panel of 4 ALK TKIs, and performed a collateral sensitivity analysis. All ALK inhibitor resistant cell lines displayed significant cross-resistance to all other ALK inhibitors. We then evaluated ALK-inhibitor sensitivities after drug holidays of varying length (1–21 days), and observed dynamic patterns of resistance. This unpredictability led us to an expanded search for treatment options, where we tested 6 further anti-cancer agents for collateral sensitivity among resistant cells, uncovering possibilities for further treatment, including cross-sensitivity to standard cytotoxic therapies, as well as Hsp90 inhibitors. Taken together, these results imply that resistance to targeted therapy in non-small cell lung cancer is highly dynamic, and also one where there are many opportunities to re-establish sensitivities where there was once resistance. Drug resistance in cancer inevitably emerges during treatment; particularly with novel targeted therapies, designed to inhibit specific molecules. A clinically-relevant example of this phenomenon occurs in ALK-positive non-small cell lung cancer, where targeted therapies are used to inhibit the ALK-EML4 fusion protein. A potential solution to this may lie in finding drug sensitivities in the resistant population, termed collateral sensitivities, and then using these as second-line agents. This study shows how the evolution of resistance in ALK-positive lung cancer is a dynamic process through time, one in which patterns of drug resistance and collateral sensitivity change substantially, and therefore one where temporal regimens, such as drug cycling and drug holidays may have great benefit.

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Many tumors are hierarchically organized and driven by a sub-population of tumor initiating cells (TICs), or cancer stem cells. TICs are uniquely capable of recapitulating the tumor and are implied to be highly resistant to radio- and chemotherapy. Macroscopic patterns of tumor expansion before treatment and tumor regression during treatment are tied to the dynamics of TICs. Until now, quantitative information about the fraction of TICs from macroscopic tumor burden trajectories could not be inferred. In this study, we generated a quantitative method based on a mathematical model that describes hierarchically organized tumor dynamics and patient-derived tumor burden information. The method identifies two characteristic equilibrium TIC regimes during expansion and regression. We show that tumor expansion and regression curves can be leveraged to infer estimates of the TIC fraction in individual patients at detection and after continued therapy. Furthermore, our method is parameter-free; it solely requires knowledge of a patient’s tumor burden over multiple time points to reveal microscopic properties of the malignancy. We demonstrate proof of concept in the case of chronic myeloid leukemia (CML), wherein our model recapitulated the clinical history of the disease in two independent patient cohorts. Based on patient-specific treatment responses in CML, we predict that after one year of targeted treatment, the fraction of TICs increases 100-fold and continues to increase up to 1000-fold after five years of treatment. Our novel framework may significantly influence the implementation of personalized treatment strategies and has the potential for rapid translation into the clinic.

The strong correlation between IDH1 mutation status and the MRI-based invasion profile suggests that use of our tumor growth model may lead to noninvasive clinical detection of IDH1 mutation status and thus lead to better treatment planning, particularly prior to surgical resection, for contrast-enhancing gliomas.