SELECTION AND ASSESSMENT OF PHENOMENOLOGICAL MODELS OF TUMOR GROWTH

We address general approaches to the rational selection and validation of mathematical and computational models of tumor growth using methods of Bayesian inference. The model classes are derived from a general diffuse-interface, continuum mixture theory and focus on mass conservation of mixtures with up to four species. Synthetic data are generated using higher-order base models. We discuss general approaches to model calibration, validation, plausibility, and selection based on Bayesian-based methods, information theory, and maximum information entropy. We also address computational issues and provide numerical experiments based on Markov chain Monte Carlo algorithms and high performance computing implementations.

[1]  J. Hopfield,et al.  From molecular to modular cell biology , 1999, Nature.

[2]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[3]  Christian P. Robert,et al.  The Bayesian choice : from decision-theoretic foundations to computational implementation , 2007 .

[4]  Kumbakonam R. Rajagopal,et al.  ON A HIERARCHY OF APPROXIMATE MODELS FOR FLOWS OF INCOMPRESSIBLE FLUIDS THROUGH POROUS SOLIDS , 2007 .

[5]  Juan Soler,et al.  ON THE ASYMPTOTIC THEORY FROM MICROSCOPIC TO MACROSCOPIC GROWING TISSUE MODELS: AN OVERVIEW WITH PERSPECTIVES , 2012 .

[6]  H. Frieboes,et al.  Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method. , 2008, Journal of theoretical biology.

[7]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[8]  Nicola Bellomo,et al.  On the foundations of cancer modelling: Selected topics, speculations, and perspectives , 2008 .

[9]  Federica Cavallo,et al.  2011: the immune hallmarks of cancer , 2011, Cancer Immunology, Immunotherapy.

[10]  D. Madigan,et al.  Bayesian Model Averaging for Linear Regression Models , 1997 .

[11]  Andrea Hawkins-Daarud,et al.  Toward a predictive model of tumor growth , 2011 .

[12]  R. M. Bowen.  Part I – Theory of Mixtures , 1976 .

[13]  J. Tinsley Oden,et al.  Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth , 2012, Journal of Mathematical Biology.

[14]  J. Beck,et al.  Model Selection using Response Measurements: Bayesian Probabilistic Approach , 2004 .

[15]  R. Baierlein.  Probability Theory: The Logic of Science , 2004 .

[16]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[17]  D. L. Sean McElwain,et al.  A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues I: A General Formulation , 2005, SIAM J. Appl. Math..

[18]  J. Tinsley Oden,et al.  GENERAL DIFFUSE-INTERFACE THEORIES AND AN APPROACH TO PREDICTIVE TUMOR GROWTH MODELING , 2010 .

[19]  Alexander R A Anderson,et al.  Quantifying the Role of Angiogenesis in Malignant Progression of Gliomas: in Silico Modeling Integrates Imaging and Histology Nih Public Access Author Manuscript Introduction , 2011 .

[20]  Adrian E. Raftery,et al.  Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors , 1999 .

[21]  Karl W. Schulz,et al.  The Parallel C++ Statistical Library 'QUESO': Quantification of Uncertainty for Estimation, Simulation and Optimization , 2011, Euro-Par Workshops.

[22]  Keming Yu,et al.  Bayesian Mode Regression , 2012, 1208.0579.

[23]  Xiangrong Li,et al.  Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching , 2009, Journal of mathematical biology.

[24]  Albert Lai,et al.  Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model. , 2009, Cancer research.

[25]  Ivo Babuška,et al.  A systematic approach to model validation based on Bayesian updates and prediction related rejection criteria , 2008 .

[26]  James L. Beck,et al.  Bayesian Updating and Model Class Selection for Hysteretic Structural Models Using Stochastic Simulation , 2008 .

[27]  D. Hanahan,et al.  The Hallmarks of Cancer , 2000, Cell.

[28]  Vittorio Cristini,et al.  Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach , 2010 .

[29]  James O. Berger,et al.  Objective Bayesian Methods for Model Selection: Introduction and Comparison , 2001 .

[30]  H. Jeffreys.  Some Tests of Significance, Treated by the Theory of Probability , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[31]  Omar Ghattas,et al.  From SIAM News , Volume 43 , Number 10 , December 2010 Computer Predictions with Quantified Uncertainty , Part II , 2010 .

[32]  Luigi Preziosi,et al.  Cancer Modelling and Simulation , 2003 .

[33]  Vittorio Cristini,et al.  Physical oncology: a bench-to-bedside quantitative and predictive approach. , 2011, Cancer research.

[34]  Sai Hung Cheung,et al.  PARALLEL ADAPTIVE MULTILEVEL SAMPLING ALGORITHMS FOR THE BAYESIAN ANALYSIS OF MATHEMATICAL MODELS , 2012 .

[35]  Kristoffer G. van der Zee,et al.  Numerical simulation of a thermodynamically consistent four‐species tumor growth model , 2012, International journal for numerical methods in biomedical engineering.

[36]  D. Hanahan,et al.  Hallmarks of Cancer: The Next Generation , 2011, Cell.

[37]  H. Frieboes,et al.  Computer simulation of glioma growth and morphology , 2007, NeuroImage.