A multiscale analysis of nutrient transport and biological tissue growthin vitro

Abstract

In this paper, we consider the derivation of macroscopic equations appropriate to describe the growth of biological tissue, employing a multiple-scale homogenisation method to accommodate explicitly the influence of the underlying microscale structure of the material, and its evolution, on the macroscale dynamics. Such methods have been widely used to study porous and poroelastic materials; however, a distinguishing feature of biological tissue is its ability to remodel continuously in response to local environmental cues. Here, we present the derivation of a model broadly applicable to tissue engineering applications, characterised by cell proliferation and extracellular matrix deposition in porous scaffolds used within tissue culture systems, which we use to study coupling between fluid flow, nutrient transport and microscale tissue growth. Attention is restricted to surface accretion within a rigid porous medium saturated with a Newtonian fluid; coupling between the various dynamics is achieved by specifying the rate of microscale growth to be dependent upon the uptake of a generic diffusible nutrient. The resulting macroscale model comprises a Darcy-type equation governing fluid flow, with flow characteristics dictated by the assumed periodic microstructure and surface growth rate of the porous medium, coupled to an advection-reaction equation specifying the nutrient concentration. Illustrative numerical simulations are presented to indicate the influence of microscale growth on macroscale dynamics, and to highlight the importance of including experimentally-relevant microstructural information in order to correctly determine flow dynamics and nutrient delivery in tissue engineering applications.

A multi-scale analysis of drug transport and response for a multi-phase tumour model

“…The analysis in the current work extends the classical homogenization of flow and transport phenomena along similar lines to the analyses in O'Dea et al (2014) and Penta et al (2014); wherein the growth, flow, and transport dynamics of porous tissues are investigated. Here, however, we additionally consider the dynamics of a multiphase mixture and, as a means of incorporating interstitial growth, we further consider the dynamics of a free interface on which phase transition may occur.…”

“…We remark that the assumption that we may neglect inertial terms is not required a priori for the proceeding analysis. Under the dimensionless scalings set out in section 2.2, we see that inertial terms are in fact relegated to O` 2˘, see O'Dea et al (2014). However, we present the derivation in this manner for the sake of concision, and for consistency with simliar multiphase formulations, Hubbard and Byrne (2013) for instance.…”

“…As such, we proceed by assuming suitable forms for u p0q 1 , u p0q 2 , p 1 , and p 2 and substituting them into the appropriate equations at Op1q and Op q to obtain a pair of coupled, steady Stokes problems that may be solved in order to obtain a description of the flow in Ω, see Davit et al (2013), O'Dea et al (2014), Rubinstein and Torquato (1989), and .…”

“…This transport is most appropriately modelled on the length scale associated with the full extent of the tumour tissue; however, the transport properties of the drug and growth dynamics of the tumour are highly dependent on the evolving microstructural properties of the tumour and its micro-vasculature , Perfahl et al (2011)). In this work we extend the multiscale analyses of O'Dea et al (2014), , and by employing a multiphase model of tumour growth, of the type developed in Breward et al (2002) and , and exploited in the context of vascular tumour growth in Breward et al (2004) and Hubbard and Byrne (2013), in which we explicitly incorporate microscale dynamics into a system of homogenized partial differential equations (PDEs) for tumour growth, response, and transport.…”

“…Examples employing these methods across a range of biological applications include Band and King (2012), Fozard et al (2010), King (2011, 2012), Ptashnyk and Chavarría-Krauser (2010), Roose (2010), andTurner et al (2004). However, of particular interest to the current work is , in which flow and transport equations in a solid tumour and its vasculature are homogenized to obtain a macroscale description of drug transport; and the more recent works of O'Dea et al (2014) and Penta et al (2014) which consider the homogenization of models for growing tissues.…”