A continuum mechanical framework for modeling tumor growth and treatment in two- and three-phase systems

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.

Tumour growth: Bayesian parameter calibration of a multiphase porous media model based on in vitro observations of Neuroblastoma spheroid growth in a hydrogel microenvironment

“…Note that we model the growing tumour spheroid itself as a viscous fluid, as in the previous tumour models [23]. When averaging the equations, we maintain a strong connection between the microscopic and the macroscopic descriptions by employing the thermodynamically constrained averaging theory (TCAT) [42]. This has the additional advantage that all quantities at the macroscale have a clear relation to their counterparts at the microscale and therefore lend themselves to precise physical interpretation [23].…”

“…Further experimental analyses of how tumour cells interact with their surroundings (ECM and other phases) are clearly necessary. From the computational side however, Miller et al [42] have already proposed a theoretically sound version which considers the interfaces between the phases.…”

Global sensitivity analysis based on Gaussian-process metamodelling for complex biomechanical problems

“…We include three fluid phases: tumour cells, host cells, and interstitial fluid (IF), denoted by superscripts t, h and l, respectively. In addition, the vasculature is modelled as an independent porous network and denoted by superscript v. The governing equations of the model are formulated on the macroscale by employing the Thermodynamically Constrained Averaging Theory (TCAT) [64,65]. Each phase is modelled in an averaged sense based on volume fractions ε α with α denoting an arbitrary phase.…”

Computational Modeling of Microwave Tumor Ablation

“…The macroscale is a phenomenological scale that is far larger than the microscale of cells and voids, but substantially smaller than the system length scale. Macroscale considerations are important because of the possible complications caused by a large number of particles at the molecular scale or the complex microscale structure of biological tissues [ 74 , 75 ]. Good separation of the length scales is pivotal for establishing a macroscale representation identical to the microscale behavior [ 76 , 77 ].…”