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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.

A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy–Forchheimer–Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes extensions of goal-oriented methods for a posteriori error estimation and control of numerical approximation to a class of highly-nonlinear problems in computational solid mechanics. An updated Lagrangian formulation of the dynamical, large-deformation response of structures composed of strain-rate-sensitive elastomers and elastoplastic materials is developed. To apply the theory of goal-oriented error estimation, a backward-in-time dual formulation of these problems is derived, and residual error estimators for meaningful quantities of interest are established. The target problem class is that of axisymmetric deformations of layered elastomer-reinforced shells-of-revolution subjected to shock loading. Extensive numerical results on solutions of representative problems are given. It is shown that extensions of the theory of goal-oriented error estimation can be developed and applied effectively to a class of highly-nonlinear, multi-physics problems in solid and structural mechanics. [Copyright &y& Elsevier]

The ultimate goal of cancer treatment utilizing thermotherapy is to eradicate tumors and minimize damage to surrounding host tissues. To achieve this goal, it is important to develop an accurate cell damage model to characterize the population of cell death under various thermal conditions. The traditional Arrhenius model is often used to characterize the damaged cell population under the assumption that the rate of cell damage is proportional to exp(-EaRT), where Ea is the activation energy, R is the universal gas constant, and T is the absolute temperature. However, this model is unable to capture transition phenomena over the entire hyperthermia and ablation temperature range, particularly during the initial stage of heating. Inspired by classical statistical thermodynamic principles, we propose a general two-state model to characterize the entire cell population with two distinct and measurable subpopulations of cells, in which each cell is in one of the two microstates, viable (live) and damaged (dead), respectively. The resulting cell viability can be expressed as C(tau,T)=exp(-Phi(tau,T)kT)(1+exp(-Phi(tau,T)kT)), where k is a constant. The in vitro cell viability experiments revealed that the function Phi(tau,T) can be defined as a function that is linear in exposure time tau when the temperature T is fixed, and linear as well in terms of the reciprocal of temperature T when the variable tau is held as constant. To determine parameters in the function Phi(tau,T), we use in vitro cell viability data from the experiments conducted with human prostate cancerous (PC3) and normal (RWPE-1) cells exposed to thermotherapeutic protocols to correlate with the proposed cell damage model. Very good agreement between experimental data and the derived damage model is obtained. In addition, the new two-state model has the advantage that is less sensitive and more robust due to its well behaved model parameters.