Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 251

constructs: since this data is difficult, or impossible, to obtain experimentally, it is
an extremely challenging problem to validate the developed theoretical models
against experimental data. However, by validating key features of the model for
which suitable data may be procured, we may employ such models to provide
detailed spatio-temporal information with confidence. Such an approach was
demonstrated in Cummings et al. ( 2009 ) (see Sect. 3.2 ): here the authors validated
the trajectory of a tissue construct within a rotating bioreactor by comparing
theoretical and experimental results, and, having successfully done this, were then
able to use the model to provide detailed spatial information about the nutrient
field surrounding the tissue construct.
• In vitro to in vivo—integration and vascularisation
The models reviewed thus far have focussed on the in vitro generation of tissue
constructs rather than their integration with normal tissue when they are implanted
into patients. Recently, Lutianov and coworkers have formulated a mathematical
model, comprising a system of nonlinear reaction-diffusion equations, to describe
the in vivo regeneration of cartilage by isolated chondrocytes and/or mesenchymal
stem cells that are seeded into a defect in the knee (Lutianov et al. 2011 ). Model
simulations suggest that it takes around eighteen months for chondrocytes to
regenerate a typical defect, of length 10-20 mm and thickness 2-3 mm. The
authors also use their model to demonstrate that mesenchymal stem cells are no
better at regenerating cartilage than chondrocytes. Landman and Cai ( 2007 ) have
also used mathematical modelling to study the integration of tissue constructs into
host tissues. A key aspect of integration is the development of an appropriate
vascular supply to the engineered tissue (Novosel et al. 2011 ). We do not thor-
oughly review theoretical models for the vascularisation of engineered tissues
here, but refer the interested reader to Landman and Cai ( 2007 ) and Lemon et al.
( 2009 ). In contrast to Lutianov et al. who consider avascular tissues, Landman and
Cai develop a model to investigate the feasibility of stimulating the formation of
new blood vessels within the tissue construct to prevent the development of
nutrient-starved regions developing within the tissue construct. Lemon et al.
( 2009 ) described the evolution of different tissue constituents within an artificial
scaffold, including vasculature, via a set of coupled non-linear ordinary differential
equations. Bifurcation analysis was used to determine the extent of scaffold vas-
cularisation as a function of the parameter values. The development of models
which accommodate the formation and establishment of a vascular supply is a
challenging open problem in the field: sophisticated models already exist
describing angiogenesis in, e.g. wound healing and cancer, and it will be
instructive to draw upon these models when considering vascularisation of tissue
engineered constructs (Owen et al. 2009 ; Anderson et al. 2012 ).
In conclusion, despite, or perhaps because of, the many challenges outlined
above, it is clear that tissue engineering is an exciting multidisciplinary field which
is raising many demanding biological questions that can serve as the basis for
fascinating, and equally demanding, mathematical problems for many years to
come. Continued dialogue between mathematical modellers and tissue engineers,