The present study deals with the mathematical modeling of crosslinking kinetics of polymer–phenol conjugates mediated by the Horseradish Peroxidase (HRP)-hydrogen peroxide (H2O2) initiation system. More specifically, a dynamic Monte Carlo (MC) kinetic model is developed to quantify the effects of crosslinking conditions (i.e., polymer concentration, degree of phenol substitution and HRP and H2O2 concentrations) on the gelation onset time; evolution of molecular weight distribution and number and weight average molecular weights of the crosslinkable polymer chains and gel fraction. It is shown that the MC kinetic model can faithfully describe the crosslinking kinetics of a finite sample of crosslinkable polymer chains with time, providing detailed molecular information for the crosslinkable system before and after the gelation point. The MC model is validated using experimental measurements on the crosslinking of a tyramine modified Hyaluronic Acid (HA-Tyr) polymer solution reported in the literature. Based on the rubber elasticity theory and the MC results, the dynamic evolution of hydrogel viscoelastic and molecular properties (i.e., number average molecular weight between crosslinks, Mc, and hydrogel mesh size, ξ) are calculated.
In this work, a computational systems approach, comprising models at different time and length scales, is developed to elucidate the fundamental chemical, physical, transport, and biological processes in relation to the design of a novel nose-to-brain drug delivery system for the treatment of neurodegenerative diseases. In particular, a kinetic model is established to describe the formation of a hydrogel patch, via the enzymatic cross-linking of tyramine-modified hyaluronic acid, on the olfactory cleft. A fluid transport model is employed to describe the flow of a cross-linkable viscoelastic polymer solution through an applicator in terms of tube and reactive fluid characteristics (e.g., tube geometry, volumetric flow rate, and viscosity). The spreading of the deposited HA droplet on a mucus substrate is modeled via the development of a dynamic droplet deformation model. Subsequently, a dynamic drug release model is formulated to quantify the release rate of an active pharmaceutical ingredient (e.g., long-acting insulin analogues) from a distributed population of drug-loaded polymeric carriers embedded into a hydrogel matrix in terms of molecular and morphological properties of the hydrogel−drug carriers system. Finally, the drug flux from the hydrogel−mucosa interface to the olfactory bulb via the epithelium and lamina propria olfactory sublayers is modeled with a series of dynamic mass transport models. The various mathematical models are integrated together following a multiscale modeling approach to aid the identification of key design system parameters and material properties that can lead to the optimization of the complex drug carriers−hydrogel, droplet deposition, film formation, and drug delivery system to achieve a desired therapeutic effect over a two-week delivery time.
In this study, two different modelling approaches, namely, a deterministic and a stochastic one, are developed to model the enzymatic cross‐linking of polymer–phenol conjugates. A comprehensive kinetic mechanism is postulated to describe the elementary reactions in the cross‐linking of polymer–phenol chains in the presence of the horseradish peroxidase (HRP)–H2O2 initiation system. In the first approach, a moments‐based model is derived to account for the conservation of all molecular species and leading moments of the number chain length distribution (NCLD) in the reactive system. In the second approach, a stochastic Monte Carlo kinetic model is formulated to follow the time evolution of a sample of cross‐linkable polymer chains and calculate the weight chain length distribution (WCLD). From the numerical solutions of both models, the dynamic evolution of the concentrations of all the reactive species, the gelation onset time, the sol and gel mass fractions as well as the number and weight average molecular weights of the cross‐linkable polymer chains are calculated. The two derived models are validated using experimental kinetic measurements on the enzymatic cross‐linking of tyramine‐modified hyaluronic acid and carboxymethyl‐chitin. It is shown that both models can accurately predict the gelation onset time of the two cross‐linkable systems over a wide range of variations in HRP and H2O2 concentrations. Finally, the MC model predictions on the weight average number of polymer chains in the cross‐linked molecules are compared to Flory's analytical solution on the tetrafunctional cross‐linking of polymer chains of uniform length.
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