General diffuse-interface theories and an approach to predictive tumor growth modeling

J. Tinsley Oden, Andrea Hawkins, Serge Prudhomme

Research output: Contribution to journalArticlepeer-review

165 Scopus citations


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 CahnHilliard 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.

Original languageEnglish (US)
Pages (from-to)477-517
Number of pages41
JournalMathematical Models and Methods in Applied Sciences
Issue number3
StatePublished - Mar 2010
Externally publishedYes

Bibliographical note

Funding Information:
The support of this work by the National Science Foundation under grant CNS-0540033 is gratefully acknowledged. We also acknowledge support of our work on statistical calibration and validation of computational models through DOE grant DE-FC52-08NA28615. Finally, we would also like to express our thanks to Professor Paolo Podio-Guidugli, who read an earlier draft of this work and whose helpful suggestions led to improvements in the presentation.


  • Bayesian statistics
  • Calibration and validation
  • Diffuse-interface
  • Mixture theory
  • Predictive models
  • Tumor growth
  • Uncertainty quantification

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics


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