The development of predictive computational models of tumor initiation, growth, and decline is faced with many formidable challenges. Phenomenological models which attempt to capture the complex interactions of multiple tissue and cellular species must cope with moving interfaces of heterogeneous media and the sprouting vascular structures due to angiogenesis and their evolution. They must be able to deliver predictions consistent with events that take place at cellular scales, and they must faithfully depict biological mechanisms and events that are known to be associated with various forms of cancer. In the present work, a ten-species vascular model for the tumor growth is presented which falls within the framework of phase-field (or diffuse-interface) models suggested by continuum mixture theory. This framework provides for the simultaneous treatment of interactions of multiple evolving species, such as tumor cells, necrotic cell cores, nutrients, and other cellular and tissue types that exist and interact in living tissue. We develop a hybrid model that couples the tumor growth with sprouting through angiogenesis. The model is able to represent the branching of new vessels through coupling a discrete model for which the angiogenesis is started upon pre-defined conditions on the nutrient deprivation in the continuum model. Such conditions are represented by hypoxic cells that release tumor growth factors that ultimately trigger vascular growth. We discuss the numerical approximation of the model using mixed finite elements. The results of numerical experiments are also presented and discussed.

This paper presents general approaches for addressing some of the most important issues in predictive computational oncology concerned with developing classes of predictive models of tumor growth. First, the process of developing mathematical models of vascular tumors evolving in the complex, heterogeneous, macroenvironment of living tissue; second, the selection of the most plausible models among these classes, given relevant observational data; third, the statistical calibration and validation of models in these classes, and finally, the prediction of key Quantities of Interest (QOIs) relevant to patient survival and the effect of various therapies. The most challenging aspects of this endeavor is that all of these issues often involve confounding uncertainties: in observational data, in model parameters, in model selection, and in the features targeted in the prediction. Our approach can be referred to as “model agnostic” in that no single model is advocated; rather, a general approach that explores powerful mixture-theory representations of tissue behavior while accounting for a range of relevant biological factors is presented, which leads to many potentially predictive models. Then representative classes are identified which provide a starting point for the implementation of OPAL, the Occam Plausibility Algorithm (OPAL) which enables the modeler to select the most plausible models (for given data) and to determine if the model is a valid tool for predicting tumor growth and morphology (in vivo). All of these approaches account for uncertainties in the model, the observational data, the model parameters, and the target QOI. We demonstrate these processes by comparing a list of models for tumor growth, including reaction–diffusion models, phase-fields models, and models with and without mechanical deformation effects, for glioma growth measured in murine experiments. Examples are provided that exhibit quite acceptable predictions of tumor growth in laboratory animals while demonstrating successful implementations of OPAL.

New directions in medical and biomedical sciences have gradually emerged over recent years that will change the way diseases are diagnosed and treated and are leading to the redirection of medicine toward patientspecific treatments. We refer to these new approaches for studying biomedical systems as predictive medicine, a new version of medical science that involves the use of advanced computer models of biomedical phenomena, high-performance computing, new experimental methods for model data calibration, modern imaging technologies, cutting-edge numerical algorithms for treating large stochastic systems, modern methods for model selection, calibration, validation, verification, and uncertainty quantification, and new approaches for drug design and delivery, all based on predictive models. The methodologies are designed to study events at multiple scales, from genetic data, to sub-cellular signaling mechanisms, to cell interactions, to tissue physics and chemistry, to organs in living human subjects. The present document surveys work on the development and implementation of predictive models of vascular tumor growth, covering aspects of what might be called modeling-and-experimentally based computational oncology. The work described is that of a multi-institutional team, centered at ICES with strong participation by members at M. D. Anderson Cancer Center and University of Texas at San Antonio. This exposition covers topics on signaling models, cell and cell-interaction models, tissue models based on multi-species mixture theories, models of angiogenesis, and beginning work of drug effects. A number of new parallel computer codes for implementing finite-element methods, multi-level Markov Chain Monte Carlo sampling methods, data classification methods, stochastic PDE solvers, statistical inverse algorithms for model calibration and validation, models of events at different spatial and temporal scales is presented. Importantly, new methods for model selection in the presence of uncertainties fundamental to predictive medical science, are described which are based on the notion of Bayesian model plausibilities. Also, as part of this general approach, new codes for determining the sensitivity of model outputs to variations in model parameters are described that provide a basis for assessing the importance of model parameters and controlling and reducing the number of relevant model parameters. Model specific data is to be accessible through careful and model-specific platforms in the Tumor Engineering Laboratory. We describe parallel computer platforms on which large-scale calculations are run as well as specific time-marching algorithms needed to treat stiff systems encountered in some phase-field mixture models. We also cover new non-invasive imaging and data classification methods that provide in vivo data for model validation. The study concludes with a brief discussion of future work and open challenges.

Cancer results from a complex interplay of different biological, chemical, and physical phenomena that span a wide range of time and length scales. Computational modeling may help to unfold the role of multiple evolving factors that exist and interact in the tumor microenvironment. Understanding these complex multiscale interactions is a crucial step towards predicting cancer growth and in developing effective therapies. We integrate different modeling approaches in a multiscale, avascular, hybrid tumor growth model encompassing tissue, cell, and sub-cell scales. At the tissue level, we consider the dispersion of nutrients and growth factors in the tumor microenvironment, which are modeled through reaction-diffusion equations. At the cell level, we use an agent based model (ABM) to describe normal and tumor cell dynamics, with normal cells kept in homeostasis and cancer cells differentiated apoptotic, hypoxic, and necrotic states. Cell movement is driven by the balance of a variety of forces according to Newton's second law, including those related to growth-induced stresses. Phenotypic transitions are defined by specific rule of behaviors that depend on microenvironment stimuli. We integrate in each cell/agent a branch of the epidermal growth factor receptor (EGFR) pathway. This pathway is modeled by a system of coupled nonlinear differential equations involving the mass laws of 20 molecules. The rates of change in the concentration of some key molecules trigger proliferation or migration advantage response. The bridge between cell and tissue scales is built through the reaction and source terms of the partial differential equations. Our hybrid model is built in a modular way, enabling the investigation of the role of different mechanisms at multiple scales on tumor progression. This strategy allows representating both the collective behavior due to cell assembly as well as microscopic intracellular phenomena described by signal transduction pathways. Here, we investigate the impact of some mechanisms associated with sustained proliferation on cancer progression. Specifically, we focus on the intracellular proliferation/migration-advantage-response driven by the EGFR pathway and on proliferation inhibition due to accumulation of growth-induced stresses. Simulations demonstrate that the model can adequately describe some complex mechanisms of tumor dynamics, including growth arrest in avascular tumors. Both the sub-cell model and growth-induced stresses give rise to heterogeneity in the tumor expansion and a rich variety of tumor behaviors.

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