Biomedical Engineering Reference
In-Depth Information
polymerization. In order to account for water uptake and drug release, two additional
constituents (water and drug) constitute the mixture. Our approach is based on basic
and widely established physical laws: (i) constituents diffuse individually accord-
ingly to Fick's law of diffusion, (ii) hydrolysis is accounted as chemical reactions
that result in the production/destruction of polymeric chains and water consumption,
and (iii) balances of mass of constituents yield coupled partial differential equations
that govern the reaction-diffusion system. Constitutive relationships characterizing
the diffusivity of each constituent and the scission reaction rates must be specified;
once known, the problem is closed and can be solved given initial and boundary con-
ditions. Previously, we have proposed several broad assumptions in the derivation of
phenomenologically reasoned constitutive relationships; nonetheless, their sufficient
generality resulted in a general class of mathematical models that describe the poly-
mer degradation and erosion. Our approach truly unified the behaviours of surface
and bulk erosion and the nondimensionalization of the governing equations allowed
the identification of the Thiele modulus, the ratio between characteristic timescales
of diffusion and reaction, which is a key parameter in conferring the continuous shift
between bulk or surface erosion behaviour to the solution of the governing equations.
Our model would have immediate direct impact in the design of biodegradable
implants if these phenomenologically derived general constitutive behaviours were
better characterized in regard to particular polymeric systems of interest. One pos-
sible strategy would be to perform an unprecedented series of experiments with the
goal of characterizing the diffusivity of each constituent (water, drug, monomers,
oligomers, etc) in a changing media (as the network degrades and erodes). To over-
come this unfeasible plan, we have developed a coupling between the macroscale
of the biodegradable polymer bulk, which is governed by the reaction-diffusion sys-
tem, and the microscale of chemical reactions and molecular diffusion, which with
the aid of atomistic simulations characterizes locally the diffusion of constituents in
the polymer bulk accordingly to its changing microstructure. Atomistic or molecular
dynamics simulations cannot yet provide further insight into the rates of reactions
taking place, but on the other hand, have been able to characterize the diffusion
coefficient of molecules in a polymeric network with quite success (cf. [18] and ref-
erences therein). With these new tools, we are able to provide unique information
into the macroscale model in the specification of the local diffusivities of mixture
constituents. More precisely, the macroscale model provides local molecular con-
figurations (i.e. set of polymer chains and water ) as input for the microscale model,
which then outputs the local diffusivities of the constituents at the macroscale level.
Unfortunately, molecular dynamics simulations are extremely expensive computa-
tionally, and hence this dynamic coupling strategy would not result in a practical
solution. To this end, we developed lumping strategies to parametrize the range of
local molecular configurations and statically couple both scales.
