Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 232

nutrient-limited growth, (ii) dynamic culture models in which the cells are viewed
as point sources and sinks of metabolites and nutrients respectively, and the cells
occupy no volume; and (iii) dynamic culture models in which the volume fraction
of cells is finite and biochemical and biomechanical interactions with the envi-
ronment are incorporated.

3.1 Modelling Static Tissue Culture

In this section we focus on mathematical models that have been developed to
describe tissue growth in static culture, when the delivery of vital nutrients and
removal of waste products are diffusion-limited. The motivation for many of these
models, particularly those of Galban and Locke ( 1997 , 1999a , b ), stems from
experiments performed by Freed and coworkers in which highly porous polymer
scaffolds of varying thickness were seeded with chondrocytes and immersed in
fluid containing (diffusible) nutrients such as oxygen and glucose (Freed et al.
1993 , 1994 ). The average cell density achieved within the scaffolds was found to
decrease as the thickness of the scaffold increased, leading the authors to conclude
that limitations in nutrient transport, caused by an increase in cell mass, could be
responsible for inhibiting cell growth. As we explain below, the continuum models
of tissue growth in static culture range from phenomenological ones (Galban and
Locke 1997 ) to multiphase models that distinguish between different components
of the developing tissue (Galban and Locke, 1999a , b ). Some models are cast as
moving boundary problems, the free boundary typically delineating regions of the
scaffold that have been colonized by cells and tissue matrix from regions that are
devoid of cells and tissue matrix (Galban and Locke 1997 ; Wilson et al. 2007 ).
A key and unifying feature of these models is the assumption that nutrient
availability is growth-rate limiting.

3.1.1 Quasi-Steady Nutrient Distribution

In an attempt to determine whether Freed et al.'s mechanistic explanation was
consistent with their experimental observations, Galban and Locke developed a
series of mathematical models for the growth of chondrocytes seeded in a porous
polymer matrix (Galban and Locke 1997 , 1999a , b ). In their initial work, two
moving boundary problems were developed to study the influence of nutrient
transport and consumption on cell growth (Galban and Locke 1997 ). They focused
on a single, growth-rate-limiting nutrient (e.g. oxygen or glucose) whose con-
sumption rate was assumed to be highest in regions containing cells. By exploiting
the difference in time-scales for nutrient diffusion within a scaffold (*minutes)
and cell growth (*hours), they were able to make a quasi-steady approximation,
neglecting time-derivatives in the reaction-diffusion equation that defines the
nutrient

distribution within the porous matrix. The position of the moving