4D Numerical Analysis of Scaffolds: A New Approach - Tissue Engineering: Computer Modeling, Biofabrication and Cell Behavior - page 76

will raise the probability of bond scissions. Also, in the human body, temperature
is kept constant at the homeostatic value of around 37 °C. The influence of the me-
chanical environment in the hydrolysis rate was also reported [ 7 , 30 ]. Loaded fibers
degrade faster than unloaded ones, and the magnitude of degradation depends on
the level of applied stress and the incubation time. Similarly to temperature, stress
also increases the probability of bond scissions. In most applications, the material
is submitted to a stress state. When the stress state remains constant during degra-
dation, the hydrolysis rate must be determined for that particular load case. If any
variation were to occur in the stress state, temperature, or environment, k would no
longer be constant.
In this example, homogeneous degradation with instant diffusion, the hydrolysis
rate, u , is constant, and damage only depends on degradation time. Although, these
considerations are valid in the majority of the cases, in some cases, the hydroly-
sis rate cannot be considered constant. In brief, the hydrolysis rate of the material
( u ) should be determined experimentally in accordance to the degradation environ-
ment of the application. In the characterization section, an example degradation rate
determination will be presented.

5 Further Refinements of Degradation Models

In a complex organism, several substances are responsible for degradation. More
precise models can include each one of these substances. Bioprocess models are of-
ten restricted to the evolution of macroscopic species involved in a reaction scheme
[ 3 ]. Such a reaction scheme describes the main phenomena occurring in the cul-
ture and is typically built of a reduced number of irreversible reactions involving
macroscopic species. For each enzyme and water, its hydrolytic effect is usually
modeled, using a first order differential equation, with different hydrolytic constant
rates and concentrations that must be known. The model used is formally based on
the kinetic mechanism of enzymatic hydrolysis according to the Michaelis-Menten
scheme [ 48 ]:

k 1

−− ←−−
k

Z + S ZS

− 1

(10)

k 2
−→ ZS P + Z

where Z and S represent the enzyme and substrate polymer, respectively and ZS is
the enzyme/substrate complex, P is used to denote the hydrolysis reaction products,
k 1, k

1 and k 2 are rate parameters. k 1 describes the diffusion and adsorption of
the enzyme onto the substrate polymer, k −

−

1 the dissociation of the ZS complex
without degradation (in general, equal to zero) [ 48 ] and k 2 the degradation process.
The degradation is mostly assumed to be the rate-limiting step, because equilibrium
in adsorption is much faster compared to degradation.
In order to perform computer simulation based on these models, the equations
can be discretized using the mixed finite element method for the space and an im-
plicit scheme for the time. Having determined the concentration of the carboxylic