Abstract
The growth and treatment of tumors is an important problem to society that involves the manifestation of cellular phenomena at length scales on the order of centimeters. Continuum mechanical approaches are being increasingly used to model tumors at the largest length scales of concern. The issue of how to best connect such descriptions to smaller-scale descriptions remains open. We formulate a framework to derive macroscale models of tumor behavior using the thermodynamically constrained averaging theory (TCAT), which provides a firm connection with the microscale and constraints on permissible forms of closure relations. We build on developments in the porous medium mechanics literature to formulate fundamental entropy inequality expressions for a general class of three-phase, compositional models at the macroscale. We use the general framework derived to formulate two classes of models, a two-phase model and a three-phase model. The general TCAT framework derived forms the basis for a wide range of potential models of varying sophistication, which can be derived, approximated, and applied to understand not only tumor growth but also the effectiveness of various treatment modalities.
Highlights
Tumor growth and treatment is an area of science of significant interest to society
The thermodynamically constrained averaging theory (TCAT) approach provides a means to formulate such models, which are founded upon conservation principles, thermodynamics, and a set of mathematical theorems
Several conclusions follow from this work: 1. Macroscale continuum mechanical approaches provide a means to describe tumor formation, growth, and various treatment modalities at a length scale relevant to human systems of concern
Summary
Tumor growth and treatment is an area of science of significant interest to society. Ideally, scientists wish to understand fundamental aspects of tumor growth in sufficient detail to enable accurate mathematical models of the behavior at the length scale of interest in humans, which is on the order of centimeters. One might endeavor, for example, to understand the processes and reactions that lead to the damage and repair of DNA, gene variations important for specific types of cancer, the role of environmental factors, and interactions among contributing factors. Within this context, important fundamental understanding, such as the hallmarks of cancer [24,25], emerges. Tumor growth is often described based upon purely statistical representations of empirical fits to observations [7,8,27,34,58,62] While such fits to data may be good, mechanistic understanding is lacking from such approaches. Empirical fits are not based on system physics and provide an insufficient basis fundamentally to describe factors affecting tumor growth and to make meaningful, mechanistically based descriptions of how fundamental changes in a system will affect tumor growth
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