Closed-form Solutions for Drug Transport through Controlled-Release Devices in Two and Three Dimensions

Laurent Simon, PhD, is Associate Professor of Chemical Engineering and served as the Associate Director of the Pharmaceutical Engineering Program at New Jersey Institute of Technology. Dr. Simon is the author of Laboratory Online, a series of educational and interactive modules that help engineers build a strong understanding of drug delivery technologies and their underlying engineering principles. During his time at NJIT, Dr. Simon has received the Excellence in Teaching Award, Master Teacher Designation, Newark College of Engineering Saul K. Fenster Innovation in Engineering Education Award and a Distinguished Teaching Award from the American Society of Engineering Education (ASEE). Juan Ospina is currently an Assistant Professor at EAFIT University in the Logic and Computation Group, Physics Engineering Program. He has published numerous articles on the topic of mathematical physics.

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