Mathematical Modeling of Desorption-Diffusion Controlled Drug Release from Polymer Matrices Rami Tzafriri Institute of Computer Science and The Department of Neurobiology, The Hebrew University, Jerusalem, Israel This research was supported by a grant from the Isreali MOS The model of Singh et al. (1994) The binding of charged polypeptides onto a hydrogel matrix is modeled as Langmuir adsorption and the release of free drug is assumed to be governed by simple diffusion from a 1D slab. (1) M k C M max M k M a d t (2) C D 2C P M t t x 2 (3) (4) C(0) C0 , M (0) C 0 x 0, 2L 0 x 2L KC0 1 KC0 , K ka kd Qualitative Analysis This problem involves 2 typical time scales: 2 L D ffid and des 1 kd Accordingly, two limiting cases are of interest: Instantaneous desorption Fast diffusion diff L2k d 1 des D L2 kd 1 D Instantaneous desorption and linear binding In this limit M KC KC 1 KC and C D 2C eff t x 2 with Deff D 1 KM max P This can be solved analytically and has been used to model the release of polylysine and gentamicine from collagen matrices Analytical solution of the linear binding case For small L (films, microspheres) we are more likely to 2 L kd encounter the case of fast diffusion: 1 D for which no analytical solution is available in the literature and the experimental results can only be analyzed by solving the equations numerically, which is rather cumbersome. However, it turns out that Eqs. (1)-(4) can be solved analytically in the case linear binding: KC 1 kaC kd In this case: (1`) M k M d t Integrating this yields: M M (0) exp k t K d M max C0 exp k d t and (2`) C D 2C k P M max C exp k t a 0 2 d t x The system (2`), (3)-(4) can be solved analytically for slabs, spheres, long cylinders and rectangular parallelepipeds, and predicts a bi-phasic release profile. Example: lysozyme release from a gelatin film Kuijpers et al. (1998) measured the release of lysozyme from cross-linked gelatin films into a PBS solution under perfect sink conditions. They considered the case: L 0.03cm, P 250 mg/ml, C0 0.5 mg/ml and estimated: Dwater 1.110 6 cm2 / s , K M max 0.020 mg/ml which implies that 83.3% of the initial drug load is adsorbed onto the gelatin matrix. Experimental estimate: Fitted values: D 0.39 10 6 cm2 / s , K M max 0.016 mg/ml kd 0.0633 h-1 2 L kd 0.163 1 D adsorbed drug (mg/mg) 0.25 0.2 0.15 0.1 0.05 0 -0.05 0 -5 5 10 15 free drug (mg/ml) Figure 2: experiment vs. fit relative lysozyme release K M max 0.020 mg/ml Figure 1: Adsorption isotherm 150 100 50 0 0 50 100 150 time (h) According to this fit 19.6% of the initial drug load is released in a short diffusive burst at a rate of 3.85/h, whereas 80.4% is adsorbed and released at a rate of 0.06/h.