Mathematical Modelling of Regeneration of a Tissue-Engineered Trachea - Computational Modeling in Tissue Engineering - page 420

To the extent that the mathematical model presented in this chapter does
encapsulate the main mechanisms guiding tracheal regeneration, a major difficulty
is how to determine the values of the model parameters as they pertain to real
tissue. Justification for the choice of some of the parameter values listed in ( 44 )
was given but others were chosen on an ad hoc basis to demonstrate typical
characteristics of the model solutions. Further work needs to be done, using expert
knowledge and experimental measurements, to obtain more reliable values for the
parameters. If the amount of experimental data available is limited then this will
place restrictions on parameter identifiability for such complicated models [ 64 ]. In
any case, it may be impractical and unethical to obtain parameter values from
direct measurements of real tracheal tissue. An alternative is to use in vitro models
[ 52 , 59 ]. Thus in the example of the need to determine the mobility of MSCs
within the cartilage, quantified by the parameter D msc in the model, new in vitro
experiments could be made of MSCs invading cartilage, the results of which could
have much more general significance to the understanding of cartilage regenera-
tion. Just as biology stimulates research in mathematics, the need to quantify key
elements in the mathematical model properly can motivate new and insightful
biological experiments.
A challenging aspect of developing mathematical models for use in tissue
engineering is the need to account for the mechanical stresses that occur within
growing tissues. This problem has stimulated previous work that our group has
carried out on modelling of tissue growth into porous biomaterials, that takes into
account the forces generated between cells and their surroundings [ 46 ], and the
forces acting on membranes of tissue that grows from host tissue into scaffolds
[ 44 ]. An interesting aspect of mathematical modelling of tracheal regeneration is
the need to link inflammation and changes in ECM to stenosis, this corresponding
to an increase in the size of the domain of the model. This was done in the present
case by formulating the model equations as a moving boundary problem, with the
dilation of tissue elements being related to the rate of ECM secretion and degra-
dation [see Sect. 3.2 and Eq. ( 24 )]. This approach may have applicability to other
biological modelling problems, since the mechanisms that regulate the extent of
tissue boundaries in an organism are fundamental to the understanding of tissue
development and disease.
The work carried out by our group represents only a small step forward in
developing of mathematical modelling of tissue-engineered tracheal regeneration.
The model currently has limitations to its suitability for guiding the development
of therapies, and further work must be done to overcome these limitations. The
present model is better suited for generating hypotheses for how regeneration
works, identifying key processes that require further elucidation, and finding ways
forward towards developing more complex and realistic models. This chapter has
highlighted an important use for mathematical models: they can allow knowledge
of different biological mechanisms to be integrated within a single theoretical
framework, thereby allowing an understanding of how these mechanisms interact
within real tissue. Modelling results can yield interesting and unexpected phe-
nomena which can suggest possible explanations for experimental outcomes and