Tuning Reaction and Diffusion Mediated Degradation of Enzyme‐Sensitive Hydrogels

Abstract

Enzyme-sensitive hydrogels are promising for cell encapsulation and tissue engineering, but result in complex spatiotemporal degradation behavior that is characteristic of reaction-diffusion mechanisms. An experimental and theoretical approach is presented to identify dimensionless quantities that serve as a design tool for engineering enzyme-sensitive hydrogels with controlled degradation patterns by tuning the initial hydrogel properties and enzyme kinetics.

Tuning tissue growth with scaffold degradation in enzyme-sensitive hydrogels: a mathematical model

“…In this study, this front is characterized in two ways: its speed is defined as that of the boundary ( ρ * = β ) that separates the completely degraded and non-degraded gel regions while its width is defined as the distance between the boundary ( ρ = ρ c ) and the point at which ρ * = 0.99. Similar observations are reported both numerically and experimentally in (33). …”

“…5d,e) clearly show nonlinear relationships between the hydrogel design and the degradation behavior, but generally suggest that increasing ${\tilde{r}}_e$ and ${\kappa }_e^{\ast }$ results in a sharper and faster degradation front. We note here that these results cannot be generalized as different trends may be observed for different values of the network connectivity β (33). A more general understanding of this system may be found by studying the role of β on the width and speed of the degradation front, yielding three-dimensional maps in the ($r_e^{\ast }$, ${\kappa }_e^{\ast }$, β ) space.…”

“…The change of cross-link density ρ , defined as the number of cross-links per volume of hydrogel in the dry state (i.e. when no solvent is present) can therefore be written: $$\frac{D\rho }{Dt}=-k^{\prime }{c}_e\frac{\rho }{\rho +k^{″}}$$ where D/Dt represents the material time derivative, c e is the enzyme concentration while k ′ = k cat ( J 0 – 1) is the degradation rate constant, k ″ = K m ( J 0 – 1) is the cross-link density at which the reaction rate is at half-maximum, while k cat and K m are Michaelis-Menten kinetic constants (32,33), and J 0 is the initial swelling ratio after cell encapsulation, but before degradation occurs. Cross-link density will eventually reach the critical (or reverse-gelation) point ρ c at which the polymer network loses its overall connectivity.…”

“…Two observations can be made based on equations (2.2) and (2.3). First, enzymes, due to their small hydrodynamic radius (~60-85 Å (33)) may enter the hydrogel space but their diffusion can be strongly hindered for tightly cross-linked gels. ECM molecules (in particular collagen and aggrecans) are however characterized by relatively large sizes, on the order of 200 Å and larger, and are therefore unable to be transported within the gel.…”

“…The flux boundary condition at the cell boundary is similarly normalized as $q_{\alpha }^{0^{\ast }}=q_{\alpha }^0{L}^2∕\left(D_m^{\infty }{c}_m^0\right)$ and the properties of the ECM are written in relation to the polymer stiffness and RT as: $${\mu }^{\ast }=\frac{C_m^0\mu }{RT},{\lambda }^{\ast }=\frac{\lambda }{\mu }$$ where Poisson's ratio is v = 0.5(1 + λ *). Note here that the non-dimensional critical cross-link density ${\rho }_c^{\ast }$ is also known as the network connectivity parameter β (33,57). This parameter is important to the degradation process as decrease in its value would imply an increase in the number of crosslink to be cleaved before reverse gelation.…”