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

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.
Citation: Shalela Mohd Mahali, Song Wang, Xia Lou. Determination of effective diffusion coefficients of drug delivery devices by a state observer approach. Discrete & 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. doi: 10.3934/dcdsb.2009.12.261. Google Scholar

[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. doi: 10.1016/j.ces.2008.03.016. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[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. doi: 10.1016/S0161-6420(02)00958-2. Google Scholar

[8]

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

[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, 1 (2011), 225. Google Scholar

[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). Google Scholar

[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. doi: 10.3934/dcdsb.2007.8.589. Google Scholar

[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. doi: 10.1016/j.bej.2003.09.004. Google Scholar

[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. doi: 10.1038/sj.eye.6700333. Google Scholar

[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. doi: 10.1586/17434440.3.6.805. Google Scholar

[15]

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

[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. doi: 10.1016/S0032-3861(00)00765-5. Google Scholar

[17]

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

[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. Google Scholar

[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. doi: 10.1007/BF01411520. Google Scholar

[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. doi: 10.1016/S0967-0661(99)00164-1. Google Scholar

[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. Google Scholar

[22]

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

[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. doi: 10.1016/j.conengprac.2008.06.002. Google Scholar

[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. doi: 10.1016/j.jfoodeng.2005.06.005. Google Scholar

[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. doi: 10.1021/cr940351u. Google Scholar

[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. doi: 10.1016/S0260-8774(02)00387-4. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00214-009-0649-2. Google Scholar

[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. doi: 10.1007/s11081-008-9069-8. Google Scholar

[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. doi: 10.1016/j.jconrel.2004.11.002. Google Scholar

[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. doi: 10.1023/A:1018922724566. Google Scholar

[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. doi: 10.1021/ci9604652. Google Scholar

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. doi: 10.3934/dcdsb.2009.12.261. Google Scholar

[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. doi: 10.1016/j.ces.2008.03.016. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[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. doi: 10.1016/S0161-6420(02)00958-2. Google Scholar

[8]

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

[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, 1 (2011), 225. Google Scholar

[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). Google Scholar

[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. doi: 10.3934/dcdsb.2007.8.589. Google Scholar

[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. doi: 10.1016/j.bej.2003.09.004. Google Scholar

[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. doi: 10.1038/sj.eye.6700333. Google Scholar

[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. doi: 10.1586/17434440.3.6.805. Google Scholar

[15]

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

[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. doi: 10.1016/S0032-3861(00)00765-5. Google Scholar

[17]

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

[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. Google Scholar

[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. doi: 10.1007/BF01411520. Google Scholar

[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. doi: 10.1016/S0967-0661(99)00164-1. Google Scholar

[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. Google Scholar

[22]

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

[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. doi: 10.1016/j.conengprac.2008.06.002. Google Scholar

[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. doi: 10.1016/j.jfoodeng.2005.06.005. Google Scholar

[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. doi: 10.1021/cr940351u. Google Scholar

[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. doi: 10.1016/S0260-8774(02)00387-4. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00214-009-0649-2. Google Scholar

[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. doi: 10.1007/s11081-008-9069-8. Google Scholar

[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. doi: 10.1016/j.jconrel.2004.11.002. Google Scholar

[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. doi: 10.1023/A:1018922724566. Google Scholar

[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. doi: 10.1021/ci9604652. Google Scholar

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