Even after a tumor is established, it can early on enter a state of dormancy marked by balanced cell proliferation and cell death. Disturbances to this equilibrium may affect cancer risk, as they may cause the eventual lifetime clinical presentation of a tumor that might otherwise have remained asymptomatic. Previously, we showed that cell death, proliferation, and migration can play a role in shifting this dynamic, making the understanding of their combined influence on tumor development essential. We developed an individual cell-based computer model of the interaction of cancer stem cells and their nonstem progeny to study early tumor dynamics. Simulations of tumor growth show that three basic components of tumor growth-cell proliferation, migration, and death-combine in unexpected ways to control tumor progression and, thus, clinical cancer risk. We show that increased proliferation capacity in nonstem tumor cells and limited cell migration overall lead to space constraints that inhibit proliferation and tumor growth. By contrast, increasing the rate of cell death produces the expected tumor size reduction in the short term, but results ultimately in paradoxical accelerated long-term growth owing to the liberation of cancer stem cells and formation of self-metastases.

Tumours are heterogeneous populations composed of different cells types: stem cells with the capacity for self-renewal and more differentiated cells lacking such ability. The overall growth behaviour of a developing neoplasm is determined largely by the combined kinetic interactions of these cells. By tracking the fate of individual cancer cells using agent-based methods in silico, we apply basic rules for cell proliferation, migration and cell death to show how these kinetic parameters interact to control, and perhaps dictate defining spatial and temporal tumour growth dynamics in tumour development. When the migration rate is small, a single cancer stem cell can only generate a small, self-limited clone because of the finite life span of progeny and spatial constraints. By contrast, a high migration rate can break this equilibrium, seeding new clones at sites outside the expanse of older clones. In this manner, the tumour continually 'self-metastasises'. Counterintuitively, when the proliferation capacity is low and the rate of cell death is high, tumour growth is accelerated because of the freeing up of space for self-metastatic expansion. Changes to proliferation and cell death that increase the rate at which cells migrate benefit tumour growth as a whole. The dominating influence of migration on tumour growth leads to unexpected dependencies of tumour growth on proliferation capacity and cell death. These dependencies stand to inform standard therapeutic approaches, which anticipate a positive response to cell killing and mitotic arrest.

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed "tumor growth paradox"--accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.

Glioblastoma multiforme (GBM) is one of the most aggressive human malignancies with a poor patient prognosis. Ionizing radiation (IR) either alone or adjuvant after surgery is part of standard treatment for GBM but remains primarily non-curative. The mechanisms underlying tumor radioresistance are manifold and, in part, accredited to a special subpopulation of tumorigenic cells. The so-called glioma stem cells (GSCs) are bestowed with the exclusive ability to self-renew and repopulate the tumor, and have been reported to be less sensitive to radiation-induced damage through preferential activation of DNA damage checkpoint responses and increased capacity for DNA damage repair. During each fraction of radiation, non-stem cancer cells (CCs) die and GSCs become enriched and potentially increase in number, which may lead to accelerated repopulation. We propose a cellular Potts model (CPM) that simulates the kinetics of GSCs and CCs in glioblastoma growth and radiation response. We parameterize and validate this model with experimental data of the U87-MG human glioblastoma cell line. Simulations are performed to estimate GSC symmetric and asymmetric division rates and explore potential mechanisms for increased GSC fractions after irradiation. Simulations reveal that, in addition to their higher radioresistance, a shift from asymmetric to symmetric division or a fast cycle of GSCs following fractionated radiation treatment is required to yield results that match experimental observations. We hypothesize a constitutive activation of stem cell division kinetics signaling pathways during fractionated treatment, which contributes to the frequently observed accelerated repopulation after therapeutic irradiation.

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.

BackgroundAlthough altered protocols that challenge conventional radiation fractionation have been tested in prospective clinical trials, we still have limited understanding of how to select the most appropriate fractionation schedule for individual patients. Currently, the prescription of definitive radiotherapy is based on the primary site and stage, without regard to patient-specific tumor or host factors that may influence outcome. We hypothesize that the proportion of radiosensitive proliferating cells is dependent on the saturation of the tumor carrying capacity. This may serve as a prognostic factor for personalized radiotherapy (RT) fractionation.MethodsWe introduce a proliferation saturation index (PSI), which is defined as the ratio of tumor volume to the host-influenced tumor carrying capacity. Carrying capacity is as a conceptual measure of the maximum volume that can be supported by the current tumor environment including oxygen and nutrient availability, immune surveillance and acidity. PSI is estimated from two temporally separated routine pre-radiotherapy computed tomography scans and a deterministic logistic tumor growth model. We introduce the patient-specific pre-treatment PSI into a model of tumor growth and radiotherapy response, and fit the model to retrospective data of four non-small cell lung cancer patients treated exclusively with standard fractionation. We then simulate both a clinical trial hyperfractionation protocol and daily fractionations, with equal biologically effective dose, to compare tumor volume reduction as a function of pretreatment PSI.ResultsWith tumor doubling time and radiosensitivity assumed constant across patients, a patient-specific pretreatment PSI is sufficient to fit individual patient response data (R2 = 0.98). PSI varies greatly between patients (coefficient of variation >128 %) and correlates inversely with radiotherapy response. For this study, our simulations suggest that only patients with intermediate PSI (0.45–0.9) are likely to truly benefit from hyperfractionation. For up to 20 % uncertainties in tumor growth rate, radiosensitivity, and noise in radiological data, the absolute estimation error of pretreatment PSI is <10 % for more than 75 % of patients.ConclusionsRoutine radiological images can be used to calculate individual PSI, which may serve as a prognostic factor for radiation response. This provides a new paradigm and rationale to select personalized RT dose-fractionation.

It remains unclear how localized radiotherapy for cancer metastases can occasionally elicit a systemic antitumor effect, known as the abscopal effect, but historically, it has been speculated to reflect the generation of a host immunotherapeutic response. The ability to purposefully and reliably induce abscopal effects in metastatic tumors could meet many unmet clinical needs. Here, we describe a mathematical model that incorporates physiologic information about T-cell trafficking to estimate the distribution of focal therapy-activated T cells between metastatic lesions. We integrated a dynamic model of tumor-immune interactions with systemic T-cell trafficking patterns to simulate the development of metastases. In virtual case studies, we found that the dissemination of activated T cells among multiple metastatic sites is complex and not intuitively predictable. Furthermore, we show that not all metastatic sites participate in systemic immune surveillance equally, and therefore the success in triggering the abscopal effect depends, at least in part, on which metastatic site is selected for localized therapy. Moreover, simulations revealed that seeding new metastatic sites may accelerate the growth of the primary tumor, because T-cell responses are partially diverted to the developing metastases, but the removal of the primary tumor can also favor the rapid growth of preexisting metastatic lesions. Collectively, our work provides the framework to prospectively identify anatomically defined focal therapy targets that are most likely to trigger an immune-mediated abscopal response and therefore may inform personalized treatment strategies in patients with metastatic disease. Cancer Res; 76(5); 1009-18. Ó2016 AACR.

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