Abstract
Four methods are proposed for modeling diffusion in heterogeneous media where diffusion and partition coefficients take on differing values in each subregion. The exercise was conducted to validate finite element modeling (FEM) procedures in anticipation of modeling drug diffusion with regional partitioning into ocular tissue, though the approach can be useful for other organs, or for modeling diffusion in laminate devices. Partitioning creates a discontinuous value in the dependent variable (concentration) at an intertissue boundary that is not easily handled by available general-purpose FEM codes, which allow for only one value at each node. The discontinuity is handled using a transformation on the dependent variable based upon the region-specific partition coefficient. Methods were evaluated by their ability to reproduce a known exact result, for the problem of the infinite composite medium (Crank, J. The Mathematics of Diffusion, 2nd ed. New York: Oxford University Press, 1975, pp. 38–39.). The most physically intuitive method is based upon the concept of chemical potential, which is continuous across an interphase boundary (method III). This method makes the equation of the dependent variable highly nonlinear. This can be linearized easily by a change of variables (method IV). Results are also given for a one-dimensional problem simulating bolus injection into the vitreous, predicting time disposition of drug in vitreous and retina. © 2000 Biomedical Engineering Society.
PAC00: 8710+e, 0270Dh, 4266Ew
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Araie, M., and D. M. Maurice. The loss of fluorescein, fluo-rescein glucuronide and fluorescein isothiocyanate dextran from the vitreous by the anterior and retinal pathways. Exp. Eye Res. 52:27-39, 1991.
Axelsson, O., and I. Gustafsson. An efficient finite element method for nonlinear diffusion problems. Bull. Greek Math. Soc. 32:45-61, 1991.
Boderke, P., H. E. Bodde, M. Ponec, M. Wolf, and H. P. Merkle. Mechanistic and quantitative prediction of aminopep-tidase activity in stripped human skin based on the HaCaT cell sheet model. J. Invest. Dermatology 3:180-184, 1998.
Boderke, P., K. Schittkowski, M. Wolf, and H. P. Merkle. Modeling of diffusion and concurrent metabolism in cutane-ous tissue, J. Theor. Biol. 204:393-407, 2000.
Castellan, G. W. Physical Chemistry, 2nd ed. Reading, MA: Addison-Wesley, 1971, pp. 321-323.
Crank, J. The Mathematics of Diffusion, 2nd ed. New York: Oxford University Press, 1975, pp. 38-39.
Dutra do Camo, E. G.. Finite element spaces with disconti-nuity capturing. I: Transport problems with boundary layers. Comput. Methods Appl. Mech. Eng. 105:299-314, 1993.
Fatt, I. Flow and Diffusion in the Vitreous Body of the Eye. Bull. Math. Biol. 37:85-90, 1975.
Friedrich, S., Y.-L. Cheng, and B. Saville. Finite element modeling of drug distribution in the vitreous humor of the rabbit eye. Ann. Biomed. Eng. 25:303-314, 1997.
Gienger, G., A. Knoch, and H. P. Merkle. Modeling and numerical computation of drug transport in laminates: model case evaluation of transdermal delivery. J. Pharm. Sci. 75:9-15, 1986.
Hughes, T. J. R., M. Mallet, and A. Mizukami. A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput. Methods Appl. Mech. Eng. 54:341-355, 1986.
Hughes, T. J. R., and M. Mallet. A new finite element for-mulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng. 58:329-336, 1986.
Leaf, G. K., M. Minkoff, G. D. Byrne, D. Sorensen, T. Bleakney, and J. Salzman. DISPL: A Software Package for One and Two Spatially Dimensioned Kinetics-Diffusion Problems. Argonne National Laboratory ANL-77-12 Rev. 1, November 1978.
Moshfegh, A. Z., S. M. Mirbagheri, and M. Shirinparvar. 2D computer simulation of diffusion in solids: Cu/Si and C/Fe systems. Mater. Manuf. Processes 12:95-105, 1997.
Nikolakopoulou, G. A., D. Edelson, and N. L. Schryer. Modeling chemically reacting flow systems-II. An adaptive spatial mesh technique for problems with discontinuities and steep fronts. Comput. Chem. (Oxford) 6:93-99, 1982.
Ohtori, A., and K. Tojo. In vivo/in vitro correlation of intra-vitreal delivery of drugs with the help of computer simula-tion. Biol. Pharm. Bull. 17:283-290, 1994.
Schittkowski, K. Parameter estimation in one-dimensional time-dependent partial differential equations. Optimization Meth. Software 7:165-210, 1997.
Schittkowski, K. Parameter Estimation in a Mathematical Model for Substrate Diffusion in a Metabolically Active Cu-taneous Tissue. Prog. Opt. 11:183-204, 2000.
Schittkowski, K. PDEFIT: A FORTRAN code for parameterestimation in partial differential equations. Optimization Meth. Software 10:539-582, 1999.
Tezduyar, T. E., and Y. J. Park. Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations. Comput. Methods Appl. Mech. Eng. 59:307-325, 1986.
Tojo, K., K. Nakagawa, Y. Morita, and A. Ohtori. A pharmacokinetic model of intravitreal delivery of gancyclovir. Eur. J. Pharm. Biopharm. 47:99-104, 1999.
Usmani, A. S. An h-adaptive SUPG-FEM solution of the pure advection equation. Appl. Numer. Math. 26:193-202, 1998.
Usmani, A. S. Solution of steady and transient advection problems using an h-adaptive finite element method. Int. J. Comput. Fluid Dyn. 11:249-259, 1999.
Zhou, Y., and X. Y. Wu. Finite element analysis of diffu-sional drug release from complex matrix systems. I. Complex geometries and composite structures. J. Controlled Release 49:277-288, 1997.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Missel, P.J. Finite Element Modeling of Diffusion and Partitioning in Biological Systems: The Infinite Composite Medium Problem. Annals of Biomedical Engineering 28, 1307–1317 (2000). https://doi.org/10.1114/1.1329886
Issue Date:
DOI: https://doi.org/10.1114/1.1329886