Biomedical Engineering Reference
In-Depth Information
Erosion is a much more complex phenomenon to model, not only because of
the interplay between different physical mechanisms as well as due to the dramatic
changes that occur in the polymer as it erodes. The choice of effective modelling
tools is, however, not straightforward, and two main approaches can be currently
identified: models based on differential equations that consider the erodible mate-
rial as a continuum and stochastic models that describe degradation and erosion as a
probabilistic event (cf. [40] for comprehensive review). In the scope of the determin-
istic approach, Heller and Baker [17] pioneered with a simple model for degrada-
tion from bulk eroding polymers consisting of steady state water diffusion coupled
with a reaction equation describing the kinetics of the degradation mechanism. Lee
[25] proposed a simplified model for surface erosion and drug release from poly-
mer films based on the movements of two fronts, a diffusion front and an erosion
front. Thombre, Joshi, and Himmelstein [20, 46, 47] proposed a comprehensive the-
ory for drug release, water penetration, and erosion and corroborated the theoretical
findings with experimental results. Similar methods based on diffusion equations
that account degradation and erosion in more complex systems have been devel-
oped since [5, 26, 36]. On the other hand, stochastic models complemented with
Monte Carlo simulations to simulate surface or bulk eroding polymers have been
developed (cf. Zygourakis [50] and Gopferich and co-workers [12, 13, 14]). Erosion
is described as being a probabilistic event and the polymer bulk as a grid of pixels.
By removing eroded pixels from the grid, the stochastic evolution of a polymeric
matrix was obtained and experimentally measurable parameters, such as porosity
and weight loss, were calculated. Erosion fronts and a distinction between erosion
modes were inferred from the results and their fit to experimental data allowed the
determination of erosion rate constents. Although such models have shown good
performance because of their versatility to account a multitude of phenomena occur-
ring due to degradation and erosion (e.g. the formation of voids inside the polymer
bulk as well as in the treatment of moving erosion fronts), their associated compu-
tational cost is generally much larger than with the solution of partial differential
equations. Nonetheless, the common difficulties associated with erosion modelling
are still present: (i) the necessity of choosing a priori the mode of erosion to model
for, (ii) the difficulty arising from modelling preferential degradation of the amor-
phous phase, and (iii), the most difficult aspect, the incorporation of changes in the
microstructure caused by erosion , which are usually specified within phenomeno-
logical reasoning. While the first two aspects have been tackled to some extent, the
latter is still an open problem that will need a huge amount of insightful theoretical
modelling and careful experimental characterization.
The authors have introduced a general class of mixture models to study water up-
take, degradation, erosion, and drug release from degradable polydisperse polymeric
matrices [43]. The model is comprehensive starting from individual polymer scission
reaction all the way up to the macroscale diffusion, allows for the systematic char-
acterization of the mass loss during the erosion process, and unifies both bulk and
surface extremes of the erosion mode spectrum. The mixture is characterized by a
finite number of constituents describing the polydisperse polymeric system, i.e. each
representing collection of chains whose size belongs to a finite interval of degree of
