Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 247

constitutive law for the solid component(s) of the tissue. Furthermore, if the tissue
stresses induced by volumetric growth are of interest, as is certainly the case for
the engineering of soft tissues, fluid-based constitutive choices for the cell phase
are inappropriate, since such stresses will dissipate (Araujo and McElwain 2005b ).
Among the first multiphase formulation to accommodate such ideas, Roose
et al. ( 2003 ) presented a poroelastic model for the growth of an avascular tumour
spheroid in which the tissue comprises a porous solid phase, saturated by a fluid
whose flow is governed by Darcy's law. The stress tensor for poroelastic materials
(Biot, 1941 ) was modified by the addition of a term accommodating volumetric
tissue growth. Byrne and Preziosi ( 2003 ) provide a more thorough derivation of
equations appropriate to model tissue as a deformable porous media (comprising a
solid and a liquid phase), as well as considering explicitly environmental influ-
ences on tissue growth.
A comprehensive derivation for a general n-phase mixture is provided by
Araujo and McElwain ( 2005a , b ), who state conservation equations for mass,
linear and angular momentum, and energy; detailed application to a spherically-
symmetric tissue represented as a biphasic mixture of linear-elastic solid (cell)
phase, saturated with an inviscid fluid was given. Such a constitutive assumption
leads to singular behaviour in stress evolution (Jones et al. 2000 ), typically reg-
ularised by the addition of viscosity to the elastic constitutive law (so that the
tissue is modelled as a poroviscoelastic material); Araujo and McElwain show that
anisotropic growth in the direction of least compressive stress is an alternative
method for regularising growth-induced stress. Preziosi and Tosin ( 2009 ) present a
similar multiphase formulation accounting for two different cell types, ECM and
tissue fluid.
Many other authors have given extensive consideration to the derivation of
multiphase models of tissue growth which can accommodate deformation and flow
in a physiologically-accurate way (see e.g. Loret and Simões ( 2005 ); Ricken and
Bluhm ( 2010 ) and references therein); however, the complexity of such models
means that most studies are restricted to biphasic mixtures, or employ simplifying
constitutive laws and/or geometries. An alternative approach is to exploit a CFD
approach (see Sect. 3.2 ) to determine 3D characterisation of the flow and nutrient
transport within cell culture systems. Examples include Kwon et al. ( 2008 ) and
Consolo et al. ( 2011 ), in which the optimisation of conditions for embryonic stem
cells encapsulated within hydrogel beads was considered, when cultured in a
rotating bioreactor (see Sect. 1.1.3 ). Advection and diffusion of oxygen were
coupled to comprehensive multiphase fluid dynamics calculations to investigate
how the rotation speed ensures proper oxygen supply, while maintaining a low-
stress condition. Theoretical predictions of optimal rotation speed were obtained,
ensuring oxygen delivery to the cells while avoiding excessively dense bead
packing and collision with the bioreactor walls. These were employed in in vitro
experiments showing that the bead motion adheres to the in silico analysis and that
such a dynamic culture strategy shows distinct benefits over static culture in terms
of cell number and viability.