Provides solutions for two- and three-dimensional linear models of controlled-release systems
- Real-world applications are taken from used to help illustrate the methods in Cartesian, cylindrical and spherical coordinate systems
- Covers the modeling of drug-delivery systems and provides mathematical tools to evaluate and build controlled-release devices
- Includes classical and analytical techniques to solve boundary-value problems involving two- and three-dimensional partial differential equations
- Provides detailed examples, case studies and step-by-step analytical solutions to relevant problems using popular computational software
Table of Contents:
Preface ix
Acknowledgements xi
1 Steady-State Analysis of a Two-Dimensional Model for Percutaneous Drug Transport 1
1.1 Separation of Variables in 2-D Cartesian Coordinates1
1.2 Model for Drug Transport across the Skin 3
1.3 Analytical Solution of the Diffusion Model in 2-D Cartesian Systems 4
1.4 Summary 6
1.5 Appendix: Maple, Mathematica, and Maxima Code Listings 6
Problems 10
References 12
2 Constant Drug Release from a Hollow Cylinder of Finite Length in Two Dimensions 13
2.1 Separation of Variables in 2-D Cylindrical Coordinates 13
2.2 Model for Drug Release from a Hollow Cylinder 15
2.3 Analytical Solution of the Transport Model in 2-D Cylindrical Coordinates 15
2.4 Summary 19
2.5 Appendix: Maple Code Listings 19
Problems 20
References 20
3 Analysis of Steady-State Growth Factor Transport Through Double-Layered Scaffolds 23
3.1 Governing Steady-State Transport Equations 23
3.2 Solution Procedure for Transport Through a Two-Layered Scaffold 25
3.3 Concentration Profile of Vascular Endothelial Growth Factor in Two Layers 31
3.4 Summary 32
3.5 Appendix: Maple Code Listings 33
Problems 37
References 38
4 Steady-State Two-Dimensional Diffusion in a Hollow Sphere 39
4.1 Separation of Variables and Legendre Polynomials in 2-D Spherical Coordinates 39
4.2 Model For 2-D Diffusion in a Sphere 43
4.3 Analytical Solution of 2-D Diffusion in Spherical Coordinates 46
4.4 Summary 49
4.5 Appendix: Maple, Mathematica, and Maxima Code Listings 49
Problems 56
References 57
5 Steady-State Three-Dimensional Drug Diffusion through Membranes from Distributed Sources 59
5.1 Separation of Variables in 3-D Cartesian Coordinates 59
5.2 Transport across the Membrane 61
5.3 Analytical Solution of the Diffusion Model in 3-D Cartesian Systems 63
5.4 Summary 68
5.5 Appendix: Maple Code Listings 69
Problems 73
References 73
6 Constant Drug Release from a Hollow Cylinder of Finite Length in Three Dimensions 75
6.1 Separation of Variables in 3-D Cylindrical Coordinates 75
6.2 Model For 3-D Drug Release from a Hollow Cylinder 77
6.3 Analytical Solution of the Transport Model in 3-D Cylindrical Coordinates 78
6.4 Summary 84
6.5 Appendix: Maple Code Listings 85
Problems 87
References 87
7 Sustained Drug Release from a Hollow Sphere in Three Dimensions 89
7.1 Method of Green’s Function in 3-D Spherical Coordinates 89
7.2 Model for Molecular Transport across the Wall of a Hollow Sphere 95
7.3 Analytical Solution of the Transport Model in 3-D Spherical Coordinates 96
7.4 Summary 97
7.5 Appendix: Maple, Mathematica and Maxima Code Listings 98
Problems 105
References 105
8 Analysis of Transient Growth Factor Transport Through Double-Layered Scaffolds 107
8.1 Laplace and Fourier-Bessel-based Methods in 2-D Cylindrical Coordinates 107
8.2 Governing Equations for Transport through Double-Layered Scaffolds 112
8.3 Concentration Profile of Vascular Endothelial Growth Factor in Two Layers 114
8.4 Summary 119
8.5 Appendix: Maple Code Listings 120
Problems 126
References 126
9 Molecular Diffusion through the Stomach Lining and into the Bloodstream 129
9.1 Laplace Transforms, Legendre Functions and Spherical Harmonics129
9.2 Spherical Diffusion in Three Dimensions 132
9.3 Analytical Solution of the Transient Transport Model in 3-D Spherical Coordinates 133
9.4 Summary 138
9.5 Appendix: Maple Code Listings 138
Problems 141
References 143
10 Diffusion-Controlled Ligand Binding to Receptors on Cell Surfaces 145
10.1 Weber’s Integral 145
10.2 Steady-State Diffusion-Limited Ligand Binding 148
10.3 Transient Diffusion-Controlled Ligand Binding in 2-D Cylindrical Coordinates 151
10.4 Summary 156
10.5 Appendix: Maple, Mathematica and Maxima Code Listings 156
Problems 167
References 168
11 Two-Dimensional Analysis of a Cylindrical Matrix Device with a Small Hole For Drug Release 169
11.1 Mathematical Modeling of Drug Transport through the Device 169
11.2 Drug Concentration Profile inside the Matrix 171
11.3 Normalized Cumulative Percentage of Drug Released 177
11.4 Summary 178
11.5 Appendix: Maple Code Listings 178
Problems 182
References 183
12 Three-Dimensional Drug Diffusion through Membranes from Distributed Sources 185
12.1 Governing Equations of the Transport Model 185
12.2 Analytical Solution of the Diffusion Model in 3-D Cartesian Systems 187
12.3 Average Dimensionless Concentration and Flux 194
12.4 Summary 194
12.5 Appendix: Maple and Mathematica Code Listings 195
Problems 207
References 207
13 Effective Time Constant for Two- and Three-Dimensional Controlled-Released Drug-Delivery Models 209
13.1 Effective Time Constant in Controlled-Release Drug-Delivery Systems 209
13.2 Intravitreal Drug Delivery using a 2-D Cylindrical Model 210
13.3 Analysis of a Rectangular Parallelepiped-Shaped Matrix with a Release Area 218
13.4 Summary 225
13.5 Appendix: Maple and Mathematica Code Listings 225
Problems 232
References 232
14 Data Fitting For Two- and Three-Dimensional Controlled- Release Drug-Delivery Models 233
14.1 Data Fitting in Controlled-Release Drug-Delivery Systems 233
14.2 Estimation of Diffusion Coefficient in a Solid Cylinder of Finite Length 234
14.3 Estimation of Diffusion Coefficient in a Rectangular Parallelepiped-Shaped Matrix 240
14.4 Summary 243
14.5 Appendix: Maple and Mathematica Code Listings 244
Problems 256
References 258
15 Optimization of Two- and Three-Dimensional Controlled-Released Drug-Delivery Models 259
15.1 Optimum Design of Controlled-Released Drug-Delivery Systems 259
15.2 Design of a 2-D Cylindrical Dosage Form with a Finite Mass Transfer Coefficient 260
15.3 Design of a Rectangular Parallelepiped-Shaped Matrix with a Finite Mass Transfer Coefficient 265
15.4 Summary 268
15.5 Appendix: Maple and Mathematica Code Listings 268
Problems 282
References 283
Index 285
