Numerical simulation of drug release from collagen matrices by enzymatic degradation. (English) Zbl 1211.92028
Summary: Biodegradable collagen matrices have become a promising alternative to synthetic polymers as drug delivery systems for sustained release. Previously, a mathematical model describing water penetration, matrix swelling and drug release by diffusion from dense collagen matrices was introduced and tested [cf. F. A. Radu et al., J. Pharm. Sci. 91, 964–972 (2002)]. However, enzymatic matrix degradation influences the drug release as well. Based on experimental studies [cf. {I. Metzmacher}, Enzymatic degradation and drug release behavior of dense collagen implants. Ph.D. thesis, LMU Univ. Munich (2005)] a mathematical model is presented here that describes drug release by collagenolytic matrix degradation. Existence and uniqueness of a solution of the model equations is reviewed. A mixed Raviart-Thomas finite element discretization for solving the coupled system of partial and ordinary differential equations is proposed and analyzed theoretically. The model is verified by a comparison of numerically calculated and experimentally measured data and, in particular, investigated by a parameter sensitivity study. For illustration, some concentration profiles of a two-dimensional simulation are shown.
MSC:
| 92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |
| 37N25 | Dynamical systems in biology |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
| 92C50 | Medical applications (general) |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Software:
UGReferences:
| [1] | Adams R.A.: Sobolev Spaces. Academic Press, New York (1975) · Zbl 0314.46030 |
| [2] | Bastian P., Birken K., Johanssen K., Lang S., Neuß N., Rentz-Reichert H., Wieners C.: UG–A flexible toolbox for solving partial differential equations. Comput. Visualiz. Sci. 1, 27–40 (1997) · Zbl 0970.65129 · doi:10.1007/s007910050003 |
| [3] | Bause M., Knabner P.: Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv. Water Res. 27, 565–581 (2004) · doi:10.1016/j.advwatres.2004.03.005 |
| [4] | Bause M., Merz W.: Higher order regularity and approximation schemes for the monod model of biodegradation. Appl. Numer. Math. 55, 154–172 (2005) · Zbl 1102.92046 · doi:10.1016/j.apnum.2005.02.002 |
| [5] | Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991) · Zbl 0788.73002 |
| [6] | Chen Z.: Expanded mixed finite element methods for quasilinear second-order ellipctic problems II. RAIRO Modél. Anal. Numer. 32, 501–520 (1998) · Zbl 0910.65080 |
| [7] | Cornish-Bowden, A.: Fundamentals of Enzyme Kinetics. Portland Press, London (2004) |
| [8] | Friess W.: Drug Delivery Systems Based on Collagen. Shaker, Aachen (2000) |
| [9] | Fujita H.: Diffusion in polymer-diluent systems. Fortschr. Hochpolym. Forsch. 3, 1–47 (1961) · doi:10.1007/BFb0050514 |
| [10] | Huang A.A.: Kinetic studies on insoluble cellulose-cellulase system. Biotechnol. Bioeng. 17, 1421–1433 (1975) · doi:10.1002/bit.260171003 |
| [11] | Ladyženskaya O.A., Solonnikov V.A., Ural’ceva N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1968) |
| [12] | McLaren A.D.: Enzyme reactions in structurally restricted systems, IV. The digestion of insoluble substrates by hydrolytic enzymes. Enzymologia 26, 237–246 (1963) |
| [13] | Merz W.: Strong solutions for reaction-drift-diffusion problems in semiconductor devices. ZAMM 81, 623–635 (2001) · Zbl 0988.35005 · doi:10.1002/1521-4001(200109)81:9<623::AID-ZAMM623>3.0.CO;2-1 |
| [14] | Merz W.: Global existence result of the monod model. Adv. Math. Sci. Appl. 15, 709–726 (2005) · Zbl 1084.92033 |
| [15] | Metzmacher, I.: Enzymatic Degradation and Drug Release Behavior of Dense Collagen Implants. Ph.D. Thesis, LMU University of Munich, Dr. Hut Verlag, München (2005) |
| [16] | Metzmacher, I., Radu, F.A., Bause, M., Knabner, P., Friess, W.: A model describing the effect of enzymatic degradation on drug release from collagen minirods, Eur. J. Pharm. Biopharm. 67(2), 349–360 (2007). doi: 10.1016/j.ejpb.2007.02.013 · Zbl 1211.92028 |
| [17] | Quarteroni A., Valli A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1997) · Zbl 1151.65339 |
| [18] | Radu F.A., Bause M., Knabner P., Friess W., Lee G.: Modelling of drug release from collagen matrices. J. Pharm. Sci. 91, 964–972 (2002) · doi:10.1002/jps.10098 |
| [19] | Radu, F.A.: Mixed finite element discretization of Richards’ equation: error analysis and application to realistic infiltration problems. Ph.D. Thesis, University of Erlangen-Nürnberg (2004) |
| [20] | Radu F.A., Pop I.S., Knabner P.: Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation. SIAM J. Numer. Anal. 42, 1452–1478 (2004) · Zbl 1159.76352 · doi:10.1137/S0036142902405229 |
| [21] | Rubingh D.N., Bauer M.D.: Catalysis of hydrolysis by proteases at the protein-solution interface. Stud. Polym. Sci. 11, 445–464 (1992) |
| [22] | Thombre A.G., Himmelstein K.J.: A simultaneous transport-reaction model for controlled drug delivery from catalyzed bioerodible polymer matrices. AIChE J. 31, 759–766 (1985) · doi:10.1002/aic.690310509 |
| [23] | Tzafriri A.R.: Mathematical modeling of diffusion-mediated release from bulk degrading matrices. J. Contr. Release 63, 69–79 (2000) · doi:10.1016/S0168-3659(99)00174-1 |
| [24] | Tzafriri A.R., Bercovier M., Parnas H.: Reaction diffusion model of the enzymatic erosion of insoluble fibrillar matrices. Biophs. J. 83, 776–793 (2002) · doi:10.1016/S0006-3495(02)75208-9 |
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