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While a large and growing literature exists on mathematical and computational models of tumor growth, to date tumor growth models are largely qualitative in nature, and fall far short of being able to provide predictive results important in life-and-death decisions. This is largely due to the enormous complexity of evolving biological and chemical processes in living tissue and the complex interactions of many cellular and vascular constituents in living organisms. Several new technologies have emerged, however, which could lead to significant progress in this important area: (i) the development of so-called phase-field, or diffuse-interface models, which can be developed using continuum mixture theory, and which provide a general framework for modeling the action of multiple interacting constituents. These are based on generalizations of the Cahn–Hilliard models for spinodal decomposition, and have been used recently in certain tumor growth theories; (ii) the emergence of predictive computational methods based on the use of statistical methods for calibration, model validation, and uncertainty quantification; (iii) advances in imaging, experimental cell biology, and other medical observational methodologies; and (iv) the advent of petascale computing that makes possible the resolution of features at scales and at speeds that were unattainable only a short time in the past. Here we develop a general phenomenological thermomechanical theory of mixtures that employs phase-field or diffuse interface models of surface energies and reactions and which provides a framework for generalizing existing theories of the types that are in use in tumor growth modeling. In principle, the framework provides for the effects of M solid constituents, which may undergo large deformations, and for the effect of N - M fluid constituents, which could include highly nonlinear, non-Newtonian fluids. We then describe several special cases which have the potential of providing acceptable models of tumor growth. We then describe the beginning steps of the development of Bayesian methods for statistical calibration, model validation, and uncertainty quantification, which, with further work, could produce a truly predictive tool for studying tumor growth. In particular, we outline the processes of statistical calibration and validation that can be employed to determine if tumor growth models, drawn from the broad class of models developed here, are valid for prediction of key quantities of interest critical to making decisions on various medical protocols. We also describe how uncertainties in such key quantities of interest can be quantified in ways that can be used to establish confidence in predicted outcomes.

In this work, we propose to extend the Arlequin framework to couple particle and continuum models. Three different coupling strategies are investigated based on the L 2 norm, H 1 seminorm, and H 1 norm. The mathematical properties of the method are studied for a one-dimensional model of harmonic springs, with varying coefficients, coupled with a linear elastic bar, whose modulus is determined by simple homogenization. It is shown that the method is wellposed for the H 1 seminorm and H 1 norm coupling terms, for both the continuous and discrete formulations. In the case of L 2 coupling, it cannot be shown that the Babuška-Brezzi condition holds for the continuous formulation. Numerical examples are presented for the model problem that illustrate the approximation properties of the different coupling terms and the effect of mesh size.

The idea that one can possibly develop computational models that predict the emergence, growth, or decline of tumors in living tissue is enormously intriguing as such predictions could revolutionize medicine and bring a new paradigm into the treatment and prevention of a class of the deadliest maladies affecting humankind. But at the heart of this subject is the notion of predictability itself, the ambiguity involved in selecting and implementing effective models, and the acquisition of relevant data, all factors that contribute to the difficulty of predicting such complex events as tumor growth with quantifiable uncertainty. In this work, we attempt to lay out a framework, based on Bayesian probability, for systematically addressing the questions of Validation, the process of investigating the accuracy with which a mathematical model is able to reproduce particular physical events, and Uncertainty quantification, developing measures of the degree of confidence with which a computer model predicts particular quantities of interest. For illustrative purposes, we exercise the process using virtual data for models of tumor growth based on diffuse-interface theories of mixtures utilizing virtual data.

The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly ill-posed Navier-Stokes equation into a well-posed set of PDE's. Spectral eddyviscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasi-linear, that they fail to be regularizations of the Navier-Stokes equations. We then propose a new spectral hyper-viscosity model that regularizes the Navier-Stokes equations while being spectrally accurate. We finally review scale-similarity models and two-scale subgrid viscosity models. A new energetically coherent scale-similarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that two-scale methods are mathematically justified in the sense that, when applied to linear non-coercive PDE's, they actually yield convergence in the graph norm. Mathematics Subject Classification (2000). 65N30, 65M, 76D05, 76F.