American Institute of Mathematical Sciences

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November  2011, 16(4): 1119-1136. doi: 10.3934/dcdsb.2011.16.1119

Determination of effective diffusion coefficients of drug delivery devices by a state observer approach

Shalela Mohd Mahali 1, , Song Wang 1, and  Xia Lou 2,

1. 

School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia, Australia
2. 

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

Received  October 2010 Revised  February 2011 Published  August 2011

In this paper we present a state observer approach for the estimation of effective diffusion coefficients of a drug delivery device. In this approach, we construct estimators for the unknown effective diffusion coefficients characterizing the diffusion process of a drug release device using a combination of state observers from the area of adaptive control and the drug diffusion models developed recently by us. We show that the constructed systems are asymptotically stable and the estimators converge to the exact diffusion coefficients. An algorithm is proposed to recursively compute the estimators using a given time series of a release profile of a device. To demonstrate the efficiency and usefulness of this approach, numerical experiments have been performed using experimentally observed drug release profiles of polymeric spherical devices. The numerical results show that the present approach is about 9 times faster than the conventional least squares method when applied to the test problems.
Keywords: State observer, optimization, diffusion coefficient, drug delivery., estimator.
Mathematics Subject Classification: Primary: 49N45; Secondary: 93C4.
Citation: Shalela Mohd Mahali, Song Wang, Xia Lou. Determination of effective diffusion coefficients of drug delivery devices by a state observer approach. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1119-1136. doi: 10.3934/dcdsb.2011.16.1119
References:
[1]

B. Baeumer, L. Chatterjee, P. Hinow, T. Rades, A. Radunskaya and I. Tucker, Predicting the drug release kinetics of matrix tablets, Discrete and Continuous Dynamical Systems - Series B, 12 (2009), 261-277. doi: 10.3934/dcdsb.2009.12.261.

[2]

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.

[3]

D. Chapelle, P. Moireau and P. L. Tallec, Robust filtering for joint state-parameter estimation in distributed mechanical systems, Discrete and Continuous Dynamical Systems, 23 (2009), 65-84.

[4]

D. S. Cohen and T. Erneux, Controlled drug release asymptotics, SIAM Journal on Applied Mathematics, 58 (1998), 1193-1204. doi: 10.1137/S0036139995293269.

[5]

R. Collins, Mathematical modeling of controlled release from implanted drug-impregnated monoliths, Pharmaceutical Science & Technology Today, 1 (1998), 269-276. doi: 10.1016/S1461-5347(98)00063-7.

[6]

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.

[7]

G. J. Crawford, C. R. Hicks, X. Lou, S. Vijayasekaran, D. Tan, T. V. Chirila and I. J. Constable, The Chirila keratoprosthesis: Phase I human clinical trials, Ophthalmology, 109 (2002), 883-889. doi: 10.1016/S0161-6420(02)00958-2.

[8]

T. E. Dabbous, Adaptive control of nonlinear systems using fuzzy systems, J. Ind. Manag. Optim., 6 (2010), 861-880. doi: 10.3934/jimo.2010.6.861.

[9]

M. Dick, M. Gugat and G. Leugering, A strict H1-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction, Numerical Algebra, Control and Optimization, 1 (2011), 225-244.

[10]

S. V. Drakunov and V. J. Law, Parameter estimation using sliding mode observers: application to the Monod kinetic model, Chemical Product and Process Modeling, 2 (2007).

[11]

Q. Gong, I. M. Ross and W. Kang, A pseudospectral observer for nonlinear systems, Discrete and Continuous Dynamical Systems - Series B, 8 (2007), 589-611. doi: 10.3934/dcdsb.2007.8.589.

[12]

J. Gutenwik, B. Nilsson and A. Axelsson, Determination of protein diffusion coefficients in agarose gel with a diffusion cell, Biochemical Engineering Journal, 19 (2004), 1-7. doi: 10.1016/j.bej.2003.09.004.

[13]

C. R. Hicks, G. J. Crawford, X. Lou, T. D. Tan, et al, Cornea replacement using a synthetic hydrogel cornea, AlphaCor: Device, preliminary outcomes and complications, Eye, 17 (2003), 385-392. doi: 10.1038/sj.eye.6700333.

[14]

C. R. Hicks, D. Morrison, X. Lou, G. J. Crawford, A. A. Gadjatsy and I. J. Constable, Orbit implants: Potential new directions, Expert Rev Med Devices, 3 (2006), 805-815. doi: 10.1586/17434440.3.6.805.

[15]

P. A. Ioannou and J. Sun, "Robust Adaptive Control," Prentice-Hall, 1995.

[16]

O. J. Karlsson, J. M. Stubbs, L. E. Karlsson and D. C. Sundberg, Estimating diffusion coefficients for small molecules in polymers and polymer solutions, Polymer, 42 (2001), 4915-4923. doi: 10.1016/S0032-3861(00)00765-5.

[17]

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.

[18]

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.

[19]

K. Nishida, Y. Ando and H. Kawamura, Diffusion coefficients of anticancer drugs and compounds having a similar structure at 30$^\circ$C, J. Colloid & Polymer Science, 261 (1983), 70-73. doi: 10.1007/BF01411520.

[20]

M. Perrier, S. Feyo de Azevedo, E. C. Ferreira and D. Dochain, Tuning of observer-based estimators: Theory and application to the on-line estimation of kinetic parameters, Control Engineering Practice, 8 (2000), 377-388. doi: 10.1016/S0967-0661(99)00164-1.

[21]

J. T. Rafael, S. M. John, I. E. Jonathan, B. Y. Michael, C. Mark and B. Henry, Interstitial chemotherapy of the 9L gliosarcoma: Controlled release polymers for drug delivery in the brain, J. Cancer Research, 53 (1993), 329-333.

[22]

H. Sira-Ramirez, On the sliding mode control of nonlinear systems, Systems & Control letters, 19 (1992), 303-312. doi: 10.1016/0167-6911(92)90069-5.

[23]

J. D. Temmerman, S. Drakunov, H. Ramon, B. Nicolai and J. Anthonis, Design of an estimator for the prediction of drying curves, Control Engineering Practice, 17 (2009), 203-209. doi: 10.1016/j.conengprac.2008.06.002.

[24]

N. Turker and F. Erdogdu, Effects of pH and temperature of extraction medium on effective diffusion coefficient of anthocynanin pigments of black carrot (Daucus carota var. L.), Journal of Food Engineering, 76 (2006), 579-583. doi: 10.1016/j.jfoodeng.2005.06.005.

[25]

K. E. Uhrich, S. M. Cannizaro, R. S. Langer and K. M. Shakesheff, Polymeric systems for controlled drug release, Chem. Rev., 99 (1999), 3181-3198. doi: 10.1021/cr940351u.

[26]

E. A. Veraverbeke, P. Verboven, N. Scheerlinck, M. L. Hoang and B. M. Nicolai, Determination of the diffusion coefficient of tissue, cuticle, cutin and wax of apple, Journal of Food Engineering, 58 (2003), 285-294. doi: 10.1016/S0260-8774(02)00387-4.

[27]

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

[28]

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

[29]

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

[30]

N. Wu , L. Wang, D. C. Tan, M. S. Moochhala and Y. Yang, Mathematical modeling and in vitro study of controlled drug release via a highly swellable and dissoluble polymer matrix: Polyethylene oxide with high molecular weights, Journal of Controlled Release, 102 (2005), 569-581. doi: 10.1016/j.jconrel.2004.11.002.

[31]

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.

[32]

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.

show all references

References:
[1]

B. Baeumer, L. Chatterjee, P. Hinow, T. Rades, A. Radunskaya and I. Tucker, Predicting the drug release kinetics of matrix tablets, Discrete and Continuous Dynamical Systems - Series B, 12 (2009), 261-277. doi: 10.3934/dcdsb.2009.12.261.

[2]

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.

[3]

D. Chapelle, P. Moireau and P. L. Tallec, Robust filtering for joint state-parameter estimation in distributed mechanical systems, Discrete and Continuous Dynamical Systems, 23 (2009), 65-84.

[4]

D. S. Cohen and T. Erneux, Controlled drug release asymptotics, SIAM Journal on Applied Mathematics, 58 (1998), 1193-1204. doi: 10.1137/S0036139995293269.

[5]

R. Collins, Mathematical modeling of controlled release from implanted drug-impregnated monoliths, Pharmaceutical Science & Technology Today, 1 (1998), 269-276. doi: 10.1016/S1461-5347(98)00063-7.

[6]

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.

[7]

G. J. Crawford, C. R. Hicks, X. Lou, S. Vijayasekaran, D. Tan, T. V. Chirila and I. J. Constable, The Chirila keratoprosthesis: Phase I human clinical trials, Ophthalmology, 109 (2002), 883-889. doi: 10.1016/S0161-6420(02)00958-2.

[8]

T. E. Dabbous, Adaptive control of nonlinear systems using fuzzy systems, J. Ind. Manag. Optim., 6 (2010), 861-880. doi: 10.3934/jimo.2010.6.861.

[9]

M. Dick, M. Gugat and G. Leugering, A strict H1-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction, Numerical Algebra, Control and Optimization, 1 (2011), 225-244.

[10]

S. V. Drakunov and V. J. Law, Parameter estimation using sliding mode observers: application to the Monod kinetic model, Chemical Product and Process Modeling, 2 (2007).

[11]

Q. Gong, I. M. Ross and W. Kang, A pseudospectral observer for nonlinear systems, Discrete and Continuous Dynamical Systems - Series B, 8 (2007), 589-611. doi: 10.3934/dcdsb.2007.8.589.

[12]

J. Gutenwik, B. Nilsson and A. Axelsson, Determination of protein diffusion coefficients in agarose gel with a diffusion cell, Biochemical Engineering Journal, 19 (2004), 1-7. doi: 10.1016/j.bej.2003.09.004.

[13]

C. R. Hicks, G. J. Crawford, X. Lou, T. D. Tan, et al, Cornea replacement using a synthetic hydrogel cornea, AlphaCor: Device, preliminary outcomes and complications, Eye, 17 (2003), 385-392. doi: 10.1038/sj.eye.6700333.

[14]

C. R. Hicks, D. Morrison, X. Lou, G. J. Crawford, A. A. Gadjatsy and I. J. Constable, Orbit implants: Potential new directions, Expert Rev Med Devices, 3 (2006), 805-815. doi: 10.1586/17434440.3.6.805.

[15]

P. A. Ioannou and J. Sun, "Robust Adaptive Control," Prentice-Hall, 1995.

[16]

O. J. Karlsson, J. M. Stubbs, L. E. Karlsson and D. C. Sundberg, Estimating diffusion coefficients for small molecules in polymers and polymer solutions, Polymer, 42 (2001), 4915-4923. doi: 10.1016/S0032-3861(00)00765-5.

[17]

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.

[18]

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.

[19]

K. Nishida, Y. Ando and H. Kawamura, Diffusion coefficients of anticancer drugs and compounds having a similar structure at 30$^\circ$C, J. Colloid & Polymer Science, 261 (1983), 70-73. doi: 10.1007/BF01411520.

[20]

M. Perrier, S. Feyo de Azevedo, E. C. Ferreira and D. Dochain, Tuning of observer-based estimators: Theory and application to the on-line estimation of kinetic parameters, Control Engineering Practice, 8 (2000), 377-388. doi: 10.1016/S0967-0661(99)00164-1.

[21]

J. T. Rafael, S. M. John, I. E. Jonathan, B. Y. Michael, C. Mark and B. Henry, Interstitial chemotherapy of the 9L gliosarcoma: Controlled release polymers for drug delivery in the brain, J. Cancer Research, 53 (1993), 329-333.

[22]

H. Sira-Ramirez, On the sliding mode control of nonlinear systems, Systems & Control letters, 19 (1992), 303-312. doi: 10.1016/0167-6911(92)90069-5.

[23]

J. D. Temmerman, S. Drakunov, H. Ramon, B. Nicolai and J. Anthonis, Design of an estimator for the prediction of drying curves, Control Engineering Practice, 17 (2009), 203-209. doi: 10.1016/j.conengprac.2008.06.002.

[24]

N. Turker and F. Erdogdu, Effects of pH and temperature of extraction medium on effective diffusion coefficient of anthocynanin pigments of black carrot (Daucus carota var. L.), Journal of Food Engineering, 76 (2006), 579-583. doi: 10.1016/j.jfoodeng.2005.06.005.

[25]

K. E. Uhrich, S. M. Cannizaro, R. S. Langer and K. M. Shakesheff, Polymeric systems for controlled drug release, Chem. Rev., 99 (1999), 3181-3198. doi: 10.1021/cr940351u.

[26]

E. A. Veraverbeke, P. Verboven, N. Scheerlinck, M. L. Hoang and B. M. Nicolai, Determination of the diffusion coefficient of tissue, cuticle, cutin and wax of apple, Journal of Food Engineering, 58 (2003), 285-294. doi: 10.1016/S0260-8774(02)00387-4.

[27]

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

[28]

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

[29]

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

[30]

N. Wu , L. Wang, D. C. Tan, M. S. Moochhala and Y. Yang, Mathematical modeling and in vitro study of controlled drug release via a highly swellable and dissoluble polymer matrix: Polyethylene oxide with high molecular weights, Journal of Controlled Release, 102 (2005), 569-581. doi: 10.1016/j.jconrel.2004.11.002.

[31]

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.

[32]

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.

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