Closed-form Solutions for Drug Transport through Controlled-Release Devices in Two and Three Dimensions
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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
1
STEADY-STATE ANALYSIS OF A TWO-DIMENSIONAL MODEL FOR PERCUTANEOUS DRUG TRANSPORT
1.1 SEPARATION OF VARIABLES IN 2-D CARTESIAN COORDINATES
The Laplace's equation in two-dimensional Cartesian coordinates takes the form
(1.1)which is solved to give
(1.2)where and are arbitrary functions of and , respectively, and . This solution can be obtained in Maple using the command pdsolve. However, Eq. (1.2) is rarely used, in practice. Instead, the method of separation of variables is adopted. The goal of this technique is to reduce the original problem into a system of ordinary differential equations in one variable (Rice & Do, 1995).
A solution of Eq. (1.1) is . These two functions satisfy the equations
(1.3)and
(1.4)where represents an arbitrary constant. After solving Eqs. (1.3) and (1.4) for and , we obtain
(1.5)The solution (1.5) is expressed as an additive separation of variables (Cherniavsky, 2010). Another method for solving Eq. (1.1) is the use of a multiplicative separation of variables such that . In this case, and satisfy the following ordinary differential equations:
(1.6)and
(1.7)After solving Eqs. (1.6) and (1.7), the solution is
(1.8)Given that is an arbitrary constant, it is possible to apply the principle of superposition (Farlow, 1993) to get
(1.9)The discrete form of Eq. (1.9) is
(1.10)The types of boundary conditions determine the choice of Eq. (1.9) or (1.10). In cases where Eqs. (1.5) and (1.9) are both solutions, their sum is also a solution:
(1.11)or
(1.12)after using the discretized form of Eq. (1.9).
1.2 MODEL FOR DRUG TRANSPORT ACROSS THE SKIN
The steady-state drug transport across the skin is described by Laplace's equation (1.1). The drug is contained in a patch of length (Fig. 1.1). During treatment, the drug concentration in the reservoir remains unchanged (Simon & Ospina, 2013). Two segments perpendicular to the skin surface, and , are chosen in this application. There is no exchange of material with the environment except at the skin/capillary boundary. A first-order elimination kinetics is observed at the interface. After using the dimensionless variables and constants,
(1.13)Figure 1.1 Diagram of the drug absorption model.
the boundary conditions are
(1.14) (1.15) (1.16) (1.17)where
(1.18)and "Heaviside(y - a)" is the step function defined as
(1.19)The coefficients and are the drug diffusivity and clearance at the skin/capillary boundary; and are the concentrations in the reservoir and in the skin, respectively. Also,
(1.20)Note that Eq. (1.14) is equivalent to the following three conditions:
(1.21) (1.22)and
(1.23)1.3 ANALYTICAL SOLUTION OF THE DIFFUSION MODEL IN 2-D CARTESIAN SYSTEMS
We look for a solution to Eq. (1.1) of the form (i.e., multiplicative separation of variables). Eq (1.8) is used in this case. Condition (1.15) leads to
(1.24)Replacing Eq. (1.24) in Eq. (1.8) and applying condition (1.16) yield
(1.25)leading to
(1.26)Equation (1.8) becomes
(1.27)Given that there is a solution for every value of , we write
(1.28)Applying the principle of superposition for linear equations, we have
(1.29)The use of condition (1.17) in Eq. (1.29) results in
(1.30)Therefore, the concentration is
(1.31)Finally, after applying Eq. (1.14) to Eq. (1.31), we have
(1.32)The solution to the problem, defined by Eqs. (1.1), (1.14), (1.15), (1.16), (1.17), is given by Eqs. (1.31) and (1.32). Using Eq. (1.32), it is possible to develop successive approximations. For example, a zero-order solution can be obtained by setting with . In this case, Eq. (1.31) reduces to
(1.33)The coefficient is calculated from Eq. (1.32) resulting in the following "zero-order" approximation of the concentration (Fig. 1.2):
(1.34)Figure 1.2 Normalized drug concentration in the skin: , , , and .
1.4 SUMMARY
The method of separation of variables was applied to solve Laplace's equation in two dimensions. In this technique, the partial differential equation is reduced to ordinary differential equations in one variable. The principle of linear superposition was implemented to add the solutions of the subproblems and generate the solution of the initial PDE model. This procedure helps derive the spatial distribution of drug across the skin.
1.5 APPENDIX: MAPLE, MATHEMATICA, AND MAXIMA CODE LISTINGS
1.5.1 Maple Code: steadytwo.mws
________________________________________________________________ > restart:with(VectorCalculus):with(PDETools); > eq:=Laplacian(c(x,y),cartesian[x,y])=0; > eq1:=alpha(y)*c(0,y)+beta(y)*Eval(diff(c(x,y),x),x=0)=delta(y): > eq2:=Eval(diff(c(x,y),y),y=-L[d])=0: > eq3:=Eval(diff(c(x,y),y),y=L[u])=0: > eq4:=Eval(diff(c(x,y),x),x=1)+w*c(1,y)=0: > eq5:=pdsolve(eq,HINT=f(x)*g(y)): > eq6:=factor(build(eq5)): > eq7:=eval(diff(rhs(eq6),y),y=-L[d])=0: > > eq8:=isolate(eq7,_C4): > eq9:=factor(subs(eq8,eq6)): > eq10:=subs(_C3=1,eq9): > eq11:=factor(combine((eq10),sin)): > eq12:=eval(diff(rhs(eq11),y),y=L[u])=0: > eq13:=sin(_c[1]^(1/2)*(L[u]+L[d]))=0: > > eq14:=_c[1]^(1/2)*(L[u]+L[d])=n*Pi: > eq15:=isolate(eq14,_c[1]): > eq16:=simplify(subs(eq15,eq11),power,symbolic): > eq17:=subs(_C1=A[n],_C2=B[n],c=c[n],eq16): > eq18:=c(x,y)=Sum(rhs(eq17),n=0..infinity); > > eq22:=subs(c=c[n],eq4): > > eq23:=subs(x=1,eq17): > eq24:=factor(eval(subs(Eval=eval,subs(eq23,subs(eq17,eq22))))): > eq25:=simplify(simplify(factor(isolate(eq24,B[n])),power,symbolic),exp): > eq26:=collect(subs(eq25,eq17),A[n]): > eq26A:=subs(n=0,simplify(series(rhs(eq26),n=0,4))): > eq27:=c(x,y)=eq26A+Sum(rhs(eq26),n=1..infinity): > > eq19:=eq1: > > eq19A:=eval(subs(x=0,eq27)): > eq20:=factor(eval(subs(Eval=eval,subs(eq19A,subs(eq27,eq19))))): > ============= Zero-order approximation====================== > eq28:=factor(eval(subs(Sum=sum,subs(A[n]=0,eq20)))): > eq29:=simplify(subs(y=z,Int(lhs(eq28),y=-L[d]..L[u])=Int(rhs(eq28),y=-L[d]..L[u]))): > eq30:=isolate(eq29,A[0]): > eq31:=eval(subs(Sum=sum,subs(A[n]=0,eq27))): > eq32:=subs(eq30,eq31): > ñas:=alpha(y)=Heaviside(y)-Heaviside(y-L[c]): > ñasA:=beta(y)=alpha(y)-1: > ñasB:=delta(y)=alpha(y): > eq32A:=subs(y=z,ñas): > eq32B:=subs(y=z,ñasA): > eq32C:=subs(y=z,ñasB): > eq32D:=eval(subs(Int=int,subs(eq32A,subs(eq32C,eq32B,eq32)))) assuming L[c]>0 and L[d]>0 and L[u]>0 and L[u]>L[c]; > ______________________________________________________________________
1.5.2 wxMaxima Code: steadytwo.wxm
_____________________________________________________________________ (%i1) eq: diff(C(x,y),x,2)+diff(C(x,y),y,2)=0; (%i2)...
