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

Abstract

Despite tremendous advances in the field of tissue engineering, a number of obstacles remain that hinder its successful translation to the clinic. One challenge that relates to the use of cells encapsulated in a hydrogel is identifying a hydrogel design that can provide an appropriate environment for cells to successfully synthesize and deposit new matrix molecules while providing a mechanical support that can resist physiological loads at the early stage of implementation. A solution to this problem has been to balance tissue growth and hydrogel degradation. However, identifying this balance is difficult due to the complexity of coupling diffusion, deposition, and degradation mechanisms. Very little is known about the complex behavior of these mechanisms, emphasizing the need for a rigorous mathematical approach that can assist and guide experimental advances. To address this issue, this paper discusses a model for interstitial growth based on mixture theory, that can capture the coupling between cell-mediated hydrogel degradation (i.e., hydrogels containing enzyme-sensitive crosslinks) and the transport of extracellular matrix (ECM) molecules released by encapsulated cells within a hydrogel. Taking cartilage tissue engineering as an example, the model investigates the role of enzymatic degradation on ECM diffusion and its impact on two important outcomes: the extent of ECM transport (and deposition) and the evolution of the hydrogel's mechanical integrity. Numerical results based on finite element analysis show that if properly tuned, enzymatic degradation yields the appearance of a highly localized degradation front propagating away from the cell, which can be immediately followed by a front of growing neotissue. We show that this situation is key to maintaining mechanical properties (e.g., stiffness) while allowing for deposition of new ECM molecules. Overall, our study suggests a hydrogel design that could enable successful tissue engineering (e.g., of cartilage, bone, etc.) where mechanical integrity is important.

Heterogeneity is key to hydrogel-based cartilage tissue regeneration

“…Following this hypothesis and consistent with previous work 16,21 , the stress-strain relation of the ECM is taken to be: $${\mathrm{bold-italic\sigma }}_m=c_s(\lambda tr({\mathrm{bold-italic\epsilon }}_{\mathrm{boldm}})\mathbf{1}+2\mu {\mathrm{bold-italic\epsilon }}_m)$$ where c s is the solid ECM concentration while λ and μ are the Lamè constants expressed in energy per mole. In order to model the porous behaviour of the tissue, a constraint, λ = 0.8 μ is used in this study 25,26 .…”

“…Depending on the local state of the gel, these molecules may diffuse and deposit to create a solid-like matrix where aggrecan and collagen molecules interact to form the neo-tissue that eventually participates in the mechanical stiffness of the construct. To describe these physics, we have constructed a continuum-based multiphasic model 16,21,23 that first considers the coupled transport/mechanics problem in the regions near cells for which the main equations are given below:…”

“…The elastic response of hydrogels is typically captured by Flory’s theory 24 . Due to the small deformations occurring during stress-free growth, the theory can be linearized 21 to yield a stress-stress relation of the form: $${\mathrm{bold-italic\sigma }}_p=2G_{\mathrm{italicgel}}\left(\frac{\nu }{(1-2\nu )}tr({\mathrm{bold-italic\epsilon }}_p)\mathbf{1}+{\mathrm{bold-italic\epsilon }}_p\right)$$ where σ p is the stress tensor, ε p is the strain tensor, G gel is the shear modulus of the swollen gel and 1 is the identity tensor. Note here that since hydrogels are nearly incompressible at short time scales, a Poisson’s ratio close to 0.5 is used in our simulations.…”

“…This operation is performed numerically using the finite element method based on the solution of the cross-link density, ρ x , and the ECM concentration, c s , using constitutive models introduced in the previous section. More details regarding the formulation can be found in 21 . We note that since the composition of the construct evolves in time due to gel degradation and ECM linkage, the average elastic modulus is a function of time (t) and is represented by the function 〈 E 〉( t ).…”

“…The effect of ECM transport, deposition and scaffold degradation on the construct stiffness was also investigated by Sengers et al 19 with a 2D model. Furthermore, the competition between hydrogel degradation and ECM deposition was studied numerically by Dhote et al 20 and Akalp et al 21 . While these models gave important information on the mechanisms of construct evolution, our experimental observations based on poly (ethylene glycol) (PEG) hydrogels suggest that construct growth and mechanical integrity is sensitive to the spatial heterogeneity in hydrogel properties and cell distribution.…”