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2012, 2(2): 377-393. doi: 10.3934/naco.2012.2.377

Numerical methods for estimating effective diffusion coefficients of three-dimensional drug delivery systems

1. 

School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

2. 

School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009

3. 

Department of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth, WA 6846

4. 

Fakultas Teknologi Industri, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya 60111, Indonesia

Received  October 2011 Revised  January 2012 Published  May 2012

This paper presents a numerical technique in three dimensions for estimating effective diffusion coefficients of drug release devices in rotating and flow-through fluid systems. We first formulate the drug release problems as diffusion equation systems with unknown effective diffusion coefficients. We then develop a numerical technique for estimating the unknown coefficients based on a nonlinear least-squares method and a finite volume discretization scheme for the 3D diffusion equations. Numerical experiments have been performed using experimental data and the numerical results are presented to show that our methods give accurate diffusivity estimations for the test problems.
Citation: Shalela Mohd--Mahali, Song Wang, Xia Lou, Sungging Pintowantoro. Numerical methods for estimating effective diffusion coefficients of three-dimensional drug delivery systems. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 377-393. doi: 10.3934/naco.2012.2.377
References:
[1]

C. Castel, D. Mazens, E. Favre and M. Leonard, Determination of diffusion coefficient from transitory uptake or release kinetics: Incidence of a recirculation loop, Chemical Engineering Science, 63 (2008), 3564-3568. doi: 10.1016/j.ces.2008.03.016.

[2]

O. Corzo and N. Bracho, Determination of water effective diffusion coefficient of sardine sheets during vacuum pulse osmotic dehydration, LWT, 40 (2007), 1452-1458. doi: 10.1016/j.lwt.2006.04.008.

[3]

J. Crank, "The Mathematics of Diffusion, 2nd ed.," Oxford University Press, London, 1975.

[4]

B. Delaunay, Sur la sphére vide, Izv Akad Nauk SSSR, Math and Nat Sci Div, 6 (1934),793-800.

[5]

G. L. Dirichlet, Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, J Reine Angew Math, 40 (1850), 209-227. doi: 10.1515/crll.1850.40.209.

[6]

J. L. Duda, J. S. Vrentas, S. T. Ju and H. T. Liu, Prediction of diffusion coefficients for polymer-solvent systems, AIChE Journal, 28 (1982), 279-285. doi: 10.1002/aic.690280217.

[7]

C. A. Farrugia, Flow-through dissolution testing: A comparison with stirred beaker methods, The chronic ill, 6 (2002), 17-19.

[8]

A. Hukka, The effective diffusion coefficient and mass transfer coefficient of nordic softwoods as calculated from direct drying experiments, Holzforschung, 53 (1999), 534-540.

[9]

K. Lavenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-168.

[10]

X. Lou, S. Wang and S. Y. Tan, Mathematics-aided quantitative analysis of diffusion characteristics of pHEMA sponge hydrogels, Asia-Pac. J. Chem. Eng., 2 (2007), 609-617.

[11]

X. Lou, S. Munro and S. Wang, Drug release characteristics of phase separation pHEMA sponge materials, Biomaterials, 25 (2004), 5071-5080. doi: 10.1016/j.biomaterials.2004.01.058.

[12]

D. W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, SIAM J. Appl. Math., 11 (1963), 431-441. doi: 10.1137/0111030.

[13]

J. J. H. Miller and S. Wang, An exponentially fitted finite element volume method for the numerical solution of 2D unsteady incompressible flow problems, J Comput Phys, 115 (1994) 56-64. doi: 10.1006/jcph.1994.1178.

[14]

S. Mohd Mahali, S. Wang and X. Lou, Determination of effective diffusion coefficients of drug delivery devices by a state observer approach, Discrete and Continuous Dynamical Systems Series B, 16 (2011) 1119-1136.

[15]

S. Mohd Mahali, S. Wang and X. Lou, Numerical methods for estimating effective diffusion coefficients of drug delivery devices in a flow-through system, submitted.

[16]

M. W.-S. Tsang, "Ophthalmic Drug Release from Porous Poly-HEMA Hydrogels," Honours Thesis, University of Western Australia, 2001.

[17]

H. S. Vibeke, L. P. Betty, G. K. Henning and M. Anette, In vivo in vitro correlations for poorly soluble drug, danazol, using the flow-through dissolution method with biorelevant dissolution media, European J Pharmaceutical Sciences, 24 (2005), 305-313.

[18]

S. Wang, S. Mohd Mahali, A. McGuiness and X. Lou, Mathematical models for estimating effective diffusion parameters of spherical drug delivery devices, Theor Chem Acc, 125 (2010), 659-669. doi: 10.1007/s00214-009-0649-2.

[19]

S. Wang and X. Lou, Numerical methods for the estimation of effective diffusion coefficients of 2D controlled drug delivery systems, Optim. Eng.,11 (2010), 611-626. doi: 10.1007/s11081-008-9069-8.

[20]

S. Wang and X. Lou, An optimization approach to the estimation of effective drug diffusivity: from a planar disc into a finite external volume, J Ind Manag Optim, 5 (2009), 127-140.

[21]

D. E. Wurster, V. Buraphacheep and J. M. Patel, The determination of diffusion coefficients in semisolids by Fourier Transform Infrared (Ft-Ir) Spectroscopy, Pharmaceutical Research, 10 (1993), 616-620. doi: 10.1023/A:1018922724566.

[22]

K. Yip, K. Y. Tam and K. F. C. Yiu, An efficient method of calculating diffusion coefficients via eigenfunction expansion, Journal of Chemical Information and Computer Science, 37 (1997), 367-371. doi: 10.1021/ci9604652.

[23]

Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control and Optimization, 1 (2011), 15-34. doi: 10.3934/naco.2011.1.15.

show all references

References:
[1]

C. Castel, D. Mazens, E. Favre and M. Leonard, Determination of diffusion coefficient from transitory uptake or release kinetics: Incidence of a recirculation loop, Chemical Engineering Science, 63 (2008), 3564-3568. doi: 10.1016/j.ces.2008.03.016.

[2]

O. Corzo and N. Bracho, Determination of water effective diffusion coefficient of sardine sheets during vacuum pulse osmotic dehydration, LWT, 40 (2007), 1452-1458. doi: 10.1016/j.lwt.2006.04.008.

[3]

J. Crank, "The Mathematics of Diffusion, 2nd ed.," Oxford University Press, London, 1975.

[4]

B. Delaunay, Sur la sphére vide, Izv Akad Nauk SSSR, Math and Nat Sci Div, 6 (1934),793-800.

[5]

G. L. Dirichlet, Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen, J Reine Angew Math, 40 (1850), 209-227. doi: 10.1515/crll.1850.40.209.

[6]

J. L. Duda, J. S. Vrentas, S. T. Ju and H. T. Liu, Prediction of diffusion coefficients for polymer-solvent systems, AIChE Journal, 28 (1982), 279-285. doi: 10.1002/aic.690280217.

[7]

C. A. Farrugia, Flow-through dissolution testing: A comparison with stirred beaker methods, The chronic ill, 6 (2002), 17-19.

[8]

A. Hukka, The effective diffusion coefficient and mass transfer coefficient of nordic softwoods as calculated from direct drying experiments, Holzforschung, 53 (1999), 534-540.

[9]

K. Lavenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2 (1944), 164-168.

[10]

X. Lou, S. Wang and S. Y. Tan, Mathematics-aided quantitative analysis of diffusion characteristics of pHEMA sponge hydrogels, Asia-Pac. J. Chem. Eng., 2 (2007), 609-617.

[11]

X. Lou, S. Munro and S. Wang, Drug release characteristics of phase separation pHEMA sponge materials, Biomaterials, 25 (2004), 5071-5080. doi: 10.1016/j.biomaterials.2004.01.058.

[12]

D. W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, SIAM J. Appl. Math., 11 (1963), 431-441. doi: 10.1137/0111030.

[13]

J. J. H. Miller and S. Wang, An exponentially fitted finite element volume method for the numerical solution of 2D unsteady incompressible flow problems, J Comput Phys, 115 (1994) 56-64. doi: 10.1006/jcph.1994.1178.

[14]

S. Mohd Mahali, S. Wang and X. Lou, Determination of effective diffusion coefficients of drug delivery devices by a state observer approach, Discrete and Continuous Dynamical Systems Series B, 16 (2011) 1119-1136.

[15]

S. Mohd Mahali, S. Wang and X. Lou, Numerical methods for estimating effective diffusion coefficients of drug delivery devices in a flow-through system, submitted.

[16]

M. W.-S. Tsang, "Ophthalmic Drug Release from Porous Poly-HEMA Hydrogels," Honours Thesis, University of Western Australia, 2001.

[17]

H. S. Vibeke, L. P. Betty, G. K. Henning and M. Anette, In vivo in vitro correlations for poorly soluble drug, danazol, using the flow-through dissolution method with biorelevant dissolution media, European J Pharmaceutical Sciences, 24 (2005), 305-313.

[18]

S. Wang, S. Mohd Mahali, A. McGuiness and X. Lou, Mathematical models for estimating effective diffusion parameters of spherical drug delivery devices, Theor Chem Acc, 125 (2010), 659-669. doi: 10.1007/s00214-009-0649-2.

[19]

S. Wang and X. Lou, Numerical methods for the estimation of effective diffusion coefficients of 2D controlled drug delivery systems, Optim. Eng.,11 (2010), 611-626. doi: 10.1007/s11081-008-9069-8.

[20]

S. Wang and X. Lou, An optimization approach to the estimation of effective drug diffusivity: from a planar disc into a finite external volume, J Ind Manag Optim, 5 (2009), 127-140.

[21]

D. E. Wurster, V. Buraphacheep and J. M. Patel, The determination of diffusion coefficients in semisolids by Fourier Transform Infrared (Ft-Ir) Spectroscopy, Pharmaceutical Research, 10 (1993), 616-620. doi: 10.1023/A:1018922724566.

[22]

K. Yip, K. Y. Tam and K. F. C. Yiu, An efficient method of calculating diffusion coefficients via eigenfunction expansion, Journal of Chemical Information and Computer Science, 37 (1997), 367-371. doi: 10.1021/ci9604652.

[23]

Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra, Control and Optimization, 1 (2011), 15-34. doi: 10.3934/naco.2011.1.15.

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