Optim Eng (2010) 11: 611–626DOI 10.1007/s11081-008-9069-8

Numerical methods for the estimation of effectivediffusion coefficients of 2D controlled drug deliverysystems

Song Wang · Xia Lou

Received: 29 September 2008 / Accepted: 21 October 2008 / Published online: 6 November 2008© Springer Science+Business Media, LLC 2008

Abstract We present a numerical approach to the estimation of effective diffusioncoefficients and other unknown parameters determining the diffusion process of adrug delivery device, based on experimental data. This approach is to formulate theestimation problem as a nonlinear least-squares problem subject to some constraintsincluding a partial differential equation (PDE) system. Numerical algorithms are de-veloped for both of the solution of the optimization problem and the discretization ofthe PDE system. To show the usefulness of the methods, numerical experiments areperformed using some laboratory observed data. The numerical results show that themethods are accurate and reliable.

Keywords Optimization · Effective diffusion coefficients · Controlled drug release ·Diffusion equation · Finite volume method · Optimum design · Parameter estimation

1 Introduction

Diffusion processes appear in many areas such as engineering, geo-physics, materialscience and bio-medical science. To understand and further control such a process,the corresponding diffusion coefficient needs to be identified which is normally un-known. This is particularly true in the design and development of a controlled drugdelivery system (cf., for example, Abdekhodaie and Cheng 1997; Price et al. 1997;

S. Wang (�)School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway,Crawley, WA 6009, Australiae-mail: swang@maths.uwa.edu.au

X. LouDepartment of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth,WA 6845, Australiae-mail: x.lou@curtin.edu.au

612 S. Wang, X. Lou

Asaokaa and Hirano 2003; Lou et al. 2004, 2007) as the diffusion coefficient is crit-ical to the functionality and performance of the system that needs to provide targetdrugs for a desired period of time and at an effective therapeutic level. Many con-trolled drug delivery systems are made of bio-compatible polymeric materials whicha drug is stored in and then released from by either a diffusion or a chemical degra-dation mechanism, or a combination of both. In a diffusion controlled system, thediffusion coefficient is a measure of the drug transport determined by the physicalproperties including the porosity of the polymer matrix and possible partition of thedrugs. When these properties are fixed, how to accurately estimate the diffusion co-efficient of a polymer matrix becomes crucial for the design of a practical device sothat it has the desired release profile. While a diffusion coefficient can be a functionof space, time and even concentration of substance, in practice, we normally seek fora constant approximation to it which is called the effective diffusion coefficient. Theeffective diffusion coefficient which yields a diffusion process matching the observedone optimally in the least-squares sense at the observation time points represents ef-fectively the properties of the drug molecules and polymer matrix. Discussions on thecorrelation between the effective diffusion coefficient and structure properties such asthe porosity and geometry of a device can be found in, for example, Lou et al. (2004,2007). But this is beyond the scope of this paper because it aims to present our novelnumerical approach to the extraction of effective diffusion coefficients.

There are various existing techniques for the identification of effective diffu-sion coefficients. Most of these methods are based on either empirical or semi-empirical models (cf., for example, Siepmann and Gopferich 2001; Grassi and Grassi2005). Others are based on analytical solutions of diffusion equations obtained forsome special geometries under certain assumptions (cf., for example, Crank 1975;Fu et al. 1976; Abdekhodaie and Cheng 1997; Yip et al. 1997; Lou et al. 2007;Wang and Lou 2007). However, the estimation of diffusion coefficients of devicesby pure numerical methods can not be found in the open literature. For problems ofpractical significance, techniques for effective diffusion estimation based on numer-ical methods for diffusion equations are much more desirable than those based onanalytical solutions the equations. This is because numerical methods such as a finiteelement or finite volume method can handle irregular device geometries.

Apart from the effective diffusion coefficient, two other parameters have beenfound very useful for accurate determination of a diffusion process. These are thewidth of the effective boundary layer and the critical time characterizing the initialburst phenomenon of a diffusion process. This initial phase of a diffusion process,caused normally by excessive drug load near the surface of a device and some freedrugs left on the device surface during the loading process, differs significantly fromthe rest of the process. It can severely pollutes the true effective diffusivity of aprocess if not being separated from the main diffusion process. To our best knowl-edge, estimation of the effective values of these parameters has not been reported.

In this work, we will present a numerical approach in 2D to the estimation of theeffective diffusion coefficient, critical time characterizing the initial burst and widthof a boundary layer for a drug delivering device. In this approach we formulate theparameter extraction problem as an nonlinear optimization problem subject to a setof constraints including a time dependent diffusion equation, we will then present

Numerical methods for diffusion coefficient estimation 613

numerical schemes for solving the optimization problem and the diffusion equation.To show the accuracy and usefulness of the methods, we will perform some numericalexperiments using laboratory experimental data.

The rest of this paper is organized as follows. In the next section, we will presentthe formulation of the problem as a nonlinear least squares problem subject to con-straints including a PDE system. In Sect. 3, we will present some numerical methodsfor solving the optimization problem and for the discretization of the PDE. In Sect. 4we will present numerical results and compare them with those from some existingmethods to demonstrate the accuracy and usefulness of the approach proposed inthis work. Although the approach is presented in two dimensions, the idea can beextended to three dimensions, which will be reported in a forthcoming paper.

2 Formulation of the problem

In this section, we first formulate the drug diffusion from a device to a finite externalvolume as an initial and boundary value problem with several unknown parameters.We will then present an optimization formulation for determining these unknownparameters.

2.1 The diffusion problem

Consider a planar drug delivery device of geometry �d loaded with a drug substance.This device is placed in a planar container of geometry �c filled with liquid (usuallywater). We assume that both �d and �c are open and convex, and have piecewisesmooth boundaries denoted respectively as ∂�d and ∂�c . When stirred, the liquidin part of the region �c \ �d is convection-dominant. Since the velocity of the fluidflow of the liquid can never be known exactly, it is a common practice to assume thatthere is a boundary layer around the device �d so that the fluid is ‘well mixed’ (orconvection-dominant) outside the layer and the diffusion occurs mainly in the layer.

To characterize the boundary layer, we let �l be an open and convex region withpiecewise smooth boundary ∂�l satisfying �d ⊂ �l ⊂ �c. The layer region is givenby �l \ �d . A typical case is illustrated in Fig. 1. Using this notation, the diffusionprocess of the drug from the device into the liquid is determined by the followinginitial and boundary value problem

∂C(x, t)

∂t− D∇2C(x, t) = 0, x ∈ �l \ �d, t ∈ (0, T ], (1)

C(x,0) = U0(x), x ∈ �c, (2)

C(x, t) = U1(t), x ∈ �c \ �l, t ∈ (0, T ], (3)

where C denotes the concentration of the substance, T is a given positive numberspecifying the drug release duration, D is the effective diffusion coefficient, U0 is agiven initial condition of C and U1 is a function of t only, representing the assumptionthat the fluid is ‘well mixed’ in the region outside the boundary layer �l \�d . Clearly,U1(t) is also unknown and the determination of U1(t) is part of the problem (1)–(3).

614 S. Wang, X. Lou

Fig. 1 Geometries of container,device and boundary layer

The width of the boundary layer in the above formulation can be estimated if thevelocity of the fluid is known. However, in practice, it is difficult to estimate thevelocity, therefore the width of the boundary layer also needs to be estimated.

It is worthy to mention that initial burst is commonly observed in a practical re-lease process. An initial burst often appears in a release process due mainly to ex-cessive drug load near the surface of a device and some free drugs left on the devicesurface during the loading process. In this case, the initial release rate is substantiallygreater than that of the rest of the process. On the other hand, the initial release ratecan be substantially smaller than the normal rate if a device is pretreated with theintension to remove the free drug on the device surface. In both cases, the diffusionprocess is divided into two phase in time: the initial burst and a normal diffusion.Substantial error might be resulted if such a process is treated as a single phase. Toovercome this problem, we assume the effective diffusion coefficient is defined by

D ={

D0, 0 < t ≤ tc,D1, tc < t ≤ T ,

where D0 and D1 are two constants and tc is the critical time when the normal diffu-sion process starts. In this case, the diffusion problem (1)–(3) becomes

∂C(x, t)

∂t− Di∇2C(x, t) = 0,

x ∈ �l \ �d, i · tc < t ≤ (1 − i)tc + i · T , i = 0,1 (4)

C(x,0) = U0(x), x ∈ �c, (5)

C(x, t) = U1(t), x ∈ �c \ �l, t ∈ (0, T ], (6)

C(x, t−c ) = C(x, t+c ), x ∈ �c. (7)

The quantities D0,D1 and tc, as well as the width of the boundary layer, are allunknowns and need to be determined.

The initial condition in (2) or (5) is chosen to be

U0(x) ={

K0, x ∈ �d ,0, x ∈ �c \ �d ,

(8)

where K0 is a constant. This initial condition represents a constant concentration inthe device �d and zero concentration outside the device at t = 0.

Numerical methods for diffusion coefficient estimation 615

2.2 The nonlinear least-squares problem

In the diffusion problem (4)–(8) there are unknown parameters D0,D1, tc and thewidth of the boundary layer. To determine these parameters, an experimental or‘ideal’ diffusion profile is needed so that a least-squares method can be used to min-imize the distance between the given and the calculated data. This target profile isnormally given as a time series of the total mass released from the device into theliquid in the time interval [0, t]. Mathematically, it is defined by

M(t) =∫

�c\�d

C(x, t)dx.

To normalize M(t), we let

M∞ = limt→∞M(t),

the total mass released from the device into the liquid in infinite time, and set

R(t) = M(t)

M∞, for t ∈ (0, T ]. (9)

Clearly, 0 ≤ R(t) ≤ 1 for all t ∈ (0, T ]. In what follows, we assume that R is a func-tion of t and the unknown parameters. We comment that, since we are only interestedin the ratio R in the rest of discussion, we may simply assume K0 = 1 in (8) withoutloss of generality.

Given a set of experimental data of the normalized release profile, Re(yk), k =1,2, . . . ,Ne at Ne given sampling time points yk, k = 1,2, . . . ,Ne, we define thefollowing nonlinear least squares problem:

Problem LSQ:

min E(D0,D1, tc, θ) =Ne∑k=1

[Re(yk) − R(yk;D0,D1, tc, θ)

]2wk

subject to (4)–(7),

Dmin ≤ D0 ≤ Dmax,

Dmin ≤ D1 ≤ Dmax,

0 ≤ tc ≤ tmaxc ,

δ ≤ θ ≤ dist(∂�d, ∂�c),

where θ is a parameter measuring the width of the boundary layer, {wk}Ne

1 is a set ofpositive weights, dist(∂�d, ∂�c) denotes the distance between the boundaries of thedevice and container, and Dmin,Dmax, t

maxc and δ are positive constants defining the

lower or upper bounds of the unknowns. A solution to this problem defines a set ofoptimal values of the parameters.

616 S. Wang, X. Lou

3 Numerical methods for solving Problem LSQ

In this section we will present some numerical methods for solving the nonlinearleast-squares problem LSQ. These include gradient based optimization algorithms forthe least squares problem, discretization schemes for the PDE systems, and algorithmfor numerical gradient evaluation.

3.1 The Newton and Lavenberg-Marquardt methods

Newton and Lavenberg-Marquardt methods (cf. Lavenberg 1944; Marquardt 1964)have been successfully used for solving optimum design problems (cf., for ex-ample, Lee et al. 1999). For discussion simplicity, we let u = (u1, u2, u3, u4)

� =(D0,D1, tc, θ)� be the decision variable. Then, the cost in Problem LSQ can be writ-ten as

E(u) =Ne∑k=1

[Re(yk) − R(yk;u)]2wk = (R(u) − Re)�A(R(u) − Re),

where

R(u) = (R(y1;u), . . . ,R(yNe ;u))�,

Re = (Re(y1), . . . ,Re(yNe))�,

A = diag(w1, . . . ,wNe).

Using a Taylor expansion we have

R(u + δu) = R(u) + J (u)δu + O(‖δu‖)δu, (10)

where δu = (δu1, . . . , δu4)�, ‖ · ‖ denotes the Euclidean norm and J is the Jacobian

defined by

J (·) =

⎛⎜⎜⎜⎜⎜⎝

∂R(t1;·)∂u1

∂R(t1;·)∂u2

∂R(t1;·)∂u3

∂R(t1;·)∂u4

∂R(t2;·)∂u1

∂R(t2;·)∂u2

∂R(t2;·)∂u3

∂R(t2;·)∂u4

......

......

∂R(tNe ;·)∂u1

∂R(tNe ;·)∂u2

∂R(tNe ;·)∂u3

∂R(tNe ;·)∂u4

⎞⎟⎟⎟⎟⎟⎠

. (11)

Therefore, omitting the 2nd order term in (10), we have

E(u + δu) ≈ (R(u) + J (u)δu − Re)�A(R(u) + J (u)δu − Re)

= E(u) + δu�J�(u)A(R(u) − Re)

+ (R(u) − Re)�AJ(u)δu + δu�J�(u)AJ (u)δu.

Differentiating E with respect to δu and setting the result to zero, we have

∇E(u + δu) = 2[J�(u)A(R(u) − Re) + J�(u)AJ (u)δu

]= 0,

Numerical methods for diffusion coefficient estimation 617

or (J�(u)AJ (u)

)δu = −J�(u)A(R(u) − Re). (12)

This is Newton’s updating formula. The Lavenberg-Marquardt updating formula is amodification of (12) given by(

J�(u)AJ (u) + λI)

δu = −J�(u)A(R(u) − Re), (13)

where λ ≥ 0 is a positive constant and I denotes the identity matrix. When λ = 0, itreduces to Newton method. When λ > 0, λI is a regularization term when the systemmatrix in (12) is ill-conditioned. Using (13), we define the following algorithm:

Algorithm LM

• Choose a λ > 0 and an initial guess u0 ∈ R4 satisfying the bound constraints in

Problem LSQ.• For k = 0,1, . . . until convergence, evaluate

uk+1 = uk + αδuk,

where δuk solves (13) with u replaced with uk and α ∈ (0,1] is a damping para-meter.

3.2 Evaluation of the Jacobian

Using the definition of R in (9), we have that the ij -th element in the Jacobian matrix(11) is given by

∂R(yi;u)

∂uj

= 1

M∞

∫�c\�d

∂C(x, yi)

∂uj

dx (14)

for i = 1,2, . . . ,Ne and j = 1,2,3,4. Therefore, we need to evaluate the derivative∂C/∂uj in order to determine ∂R/∂uj for any j = 1,2,3,4. For u1 and u2 (or D0and D1) exact gradient formulas can be obtained by differentiating (4)–(6) as follows:

∂Cuj(x, t)

∂t− Di∇2Cuj

(x, t) = δi,(j−1)∇2C(x, t),

x ∈ �l \ �d, i · tc < t ≤ (1 − i)tc + i · T , i = 0,1, (15)

Cuj(x,0) = U

j

0 (x), x ∈ �c, (16)

Cuj(x, t) = 0, x ∈ �c \ �l, t ∈ (0, T ], (17)

Cuj(x, t−c ) = Cuj

(x, t+c ), x ∈ �c (18)

for j = 1,2, where Cuj= ∂C/∂uj , U

j

0 (x) is an unknown function and δi,k is theKroneker delta, i.e., δi,k = 0 if i �= k and 1 if i = k.

The derivatives ∂R/∂u3 and ∂R/∂u4 can only be evaluated numerically by finitedifferencing. Naturally, ∂R/∂u1 and ∂R/∂u2 can also be evaluated numerically. Forbrevity, we put this discussion in Appendix.

618 S. Wang, X. Lou

Fig. 2 (a) Delaunay mesh(solid line) and Dirichlettessellation (dotted line).(b) Notation for edges and nodes

3.3 Discretization

Note that the diffusion coefficient D is normally very small in magnitude (in the rangefrom O(10−7) to O(10−5)). Therefore, the PDE systems are singularly perturbed be-cause of this small parameter. Application of a standard discretization scheme suchas the piecewise linear finite element method often yields solutions with large errors.To overcome this difficulty, we use a finite volume method to discretize the partialdifferential equations in the previous section. The method was originally developedfor solving convection dominated convection-diffusion equations such the semicon-ductor device equations and the incompressible Navier-Stokes equations (cf., for ex-ample, Miller and Wang 1994a). Mathematical analysis of the method can be foundin Miller and Wang (1994b). We now give a brief description of the method appliedto (1).

To construct the finite volume scheme, we first define some meshes on �c . LetQ be a partition of �c consisting of a set of triangles and rectangles. We denote thesets of element vertices and edges of Q by, respectively, X = {xi}N1 and E = {ei}M1 .Without loss of generality we assume that the nodes in X and the edges in E not onor outside ∂�l are numbered {xi}N ′

1 and {ei}M ′1 respectively.

Definition 3.1 The partition Q is a Delaunay mesh if, for every element q ∈ Q, thecircumcircle of the element contains no other vertices in X (cf. Delaunay 1934).

We assume henceforth that Q is a Delaunay mesh.

Definition 3.2 The Dirichlet tessellation D, corresponding to the triangulation Q isdefined by D = {di}N1 , where

di = {x ∈ �c : |x − xi | < |x − xj |, xj ∈ X,j �= i}for all xi ∈ X (cf. Dirichlet 1850).

Given a Delaunay mesh, the corresponding Dirichlet tessellation can be obtainedby connecting all the neighbouring circumcentres of the triangles and rectangles inthe mesh. A typical example is depicted in Fig. 2(a) in which the solid lines formsthe element with a common vertex and the dashed lines forms the Dirichlet tile as-sociated with the vertex. Thus, for each xi ∈ X, the boundary ∂di of the tile di is thepolygon having as its vertices the circumcentres of all triangles and rectangles with

Numerical methods for diffusion coefficient estimation 619

common vertex xi . Each segment of ∂di is perpendicular to one of the edges sharingthe vertex xi (see Fig. 2(a)). The Dirichlet tessellation D is a polygonal mesh dualto the Delaunay mesh Q. For each i = 1,2, . . . ,N ′, integrating (1) over di , using theone-point quadrature rule for the 1st term and applying Green’s formula to the 2ndterm we have

∂Ci(t)

∂t|di | −

∫∂di

D∇C(t) · nds = 0, t ∈ (0, T ], (19)

where Ci(t) denotes the nodal approximation of C(x, t) at xi , |di | denotes the area ofdi and n is the unit vector outward normal to ∂di . We now consider the approximationof the line integral in (19). Let Ii = {j : ei,j ∈ E} denote the index set of neighbouringnodes of xi , where ei,j denotes the edge joining xi and xj , as shown in Fig. 2(b). Since∂di is polygonal and each of its sides is perpendicular to one of the edges joining xi ,we have ∫

∂di

D∇C · nds =∑j∈Ii

∫li,j

D∇C · ei,j ds

≈∑j∈Ii

D(Cj − Ci)

|ei,j | |li,j | (20)

where li,j denotes the segment of ∂di perpendicular to the edge ei,j which is orientedcounterclockwise with respect to xi (see Fig. 2(b)), ei,j denotes the unit vector fromxi to xj , and |li,j | the length of li,j . Replacing the integrand on the LHS of (19) by(20) and dividing the resulting equation by |di | we obtain

∂Ci

∂t+

∑j∈Ii

D|li,j ||ei,j ||di,j | (Ci − Cj ) = 0 (21)

for all i = 1,2, . . . ,N ′, where Ci = Ci(t) and |ei,j | denotes the length of ei,j . Thisfinishes the spatial discretization of the problem.

To discretize the time derivative in (21), we first let tk , k = 0,1, . . . ,K be a se-quence of time points satisfying 0 = t0 < t1 < t2 < · · · < tK = T , where K is a posi-tive integer. For the time discretization, we apply the backward Euler scheme to thesemi-discrete equation (21) to yield the following fully discretized form:

Cki − Ck−1

i

�tk+

∑j∈Ii

Dk|li,j ||ei,j ||di,j | (C

ki − Ck

j ) = 0 (22)

for k = 1,2, . . . ,K and i = 1,2, . . . ,N ′, where �tk = tk − tk−1 and the superscriptk is used to denote nodal approximations at t = tk with C0

i being the given initialcondition in (8) evaluated at xi for all feasible i.

Equation (22) is a linear system for the unknowns Cki for i = 1, . . . ,N ′ and a

fixed k. To solve this linear system, we need to know U1(tk) in order to define theboundary condition for (22) so that (22) can be written in the following matrix form

(�tkB + I )Ck = Ck−1 + �tkkk, (23)

620 S. Wang, X. Lou

where Ck = (Ck1 , . . . ,Ck

N ′)�, B is an N ′ ×N ′ symmetric matrix arising from the sec-ond term of (22), I is the identity matrix and bk a column vector due to the boundarycondition U1(tk). To achieve this, we may use the homogeneous Neumann boundarycondition on ∂�l since ∂U1(t)/∂n = 0. In this case, bk = 0 and we need to increaseN ′ so that it include all the indices of the mesh nodes on ∂�l . However, our numeri-cal experience shows that this approach may cause some computational difficulties astime stepping goes on. In this work, we approximate U1(tk) by the following averageover the region �c \ �l :

Ckl = U1(tk) =

∑Ni=N ′+1 Ck−1

i |di |∑Ni=N ′+1 |di |

, l = N ′ + 1, . . . ,N (24)

with C0l = U0(xl) for all l = 1, . . . ,N . Clearly, the rationale of this approximation is

that at every time step, the concentration outside the region �l is uniform as assumedin the previous formulations.

Since (15) has the same form as that of (1) except for the extra term on its right-hand side, the above discretization is also applicable to (15). Note that the right-handside of (15) is also a Laplace operator, it can be discretized in exactly the same wayas that for the Laplace operator in (1) given above.

Finally, we comment that the system matrix of (23) is a sparse, symmetric, anddiagonally dominant M-matrix. This linear system can be solved either by a directmethod such as the LU decomposition or by an iterative method such as the precon-ditioned conjugate gradient method. Also, the matrix B is time-independent and thusit needs to be generated only once at t = t1. If the matrix is factorized using LU de-composition, it needs to be done only once at t = t1 as well. This will certainly savethe overall computational time.

4 Numerical experiments

We now present some numerical results using the experimental data, (Re(y1), . . . ,

Re(y13)), observed at the 13 time points for four different devices listed in Table 1.(Experimental details including the fabrication and characterization of these devicescan be found in Lou et al. (2004, 2007).) These four test devices are denoted as A1,A2, A3 and A4 respectively. The configuration of each pair of the devices and con-tainers consists of concentric discs with radii r1 and r2 respectively. The parameterθ is chosen from (0,1) so that the boundary layer around the device has the widthr1 + θ(r2 − r1). In all the numerical experiments, we choose wk = 1 in ProblemLSQ for all k = 1,2, . . . ,Ne. Equation (1) is discretized using a triangular mesh withabout 3000 mesh nodes and the step length of the time discretization is chosen to be�tk = tk − tk−1 = 5 seconds for all k = 1,2, . . . with t0 = 0.

To make the problem simpler, we assume that the critical time tc can only take thediscrete values 0, y1, y2, . . . , y6. In this case, the Jacobian matrix J in (11) is at mostof order 3 × 13, depending on the number of unknown parameters. This Jacobianmatrix is evaluated numerically at each time step by divided differences, as described

Numerical methods for diffusion coefficient estimation 621

Tabl

e1

Exp

erim

enta

lval

ues

ofR

for

A1,

A2,

A3

and

A4

Tim

e(h

rs)

0.25

0.5

0.75

11.

52

34

69

2448

72

A1

0.27

090.

4134

0.48

150.

4829

0.48

960.

4948

0.51

340.

5193

0.52

050.

6367

0.82

700.

9479

1.00

00

A2

0.17

050.

3358

0.48

770.

5596

0.58

470.

6307

0.66

380.

6987

0.76

850.

8428

0.92

650.

9868

1.00

00

A3

0.01

750.

0241

0.04

920.

0614

0.21

920.

2530

0.25

840.

2894

0.35

370.

4009

0.69

330.

8300

1.00

00

A4

0.05

940.

1183

0.19

290.

2135

0.39

820.

4490

0.46

420.

6043

0.66

290.

7456

0.91

560.

9566

1.00

00

622 S. Wang, X. Lou

Table 2 Comparison of results from M1 and M2 for the devices A1 and A2

Device LSQ error D or (D0,D1) (cm2/s) tc (hrs)

M1 M2 M1 M2 M2

A1 2.05E–01 1.13E–02 1.09E–05 (3.07E–05, 3.48E–06) 0.75

A2 5.27E–02 1.57E–02 2.56E–05 (3.38E–05, 1.11E–05) 1.5

Fig. 3 Fitted curves of A1 and A2 using M1 and M2

in Appendix. The methods were implemented using Fortran 77 programming lan-guage and all the numerical experiments were performed on a laptop computer underthe Cygwin environment. To make sure the codes work correctly, we have tested thenumerical derivatives with respect to D0 and D1 using the algorithm in Appendixagainst the numerical solutions of (15)–(18) and found that they are in good agree-ment.

We first consider the devices A1 and A2. These two devices were not pre-treatedand thus excessive drug near the device surfaces caused initial bursts in the laboratoryexperiments. Since the liquid was slowly stirred in the experiments, we simply chooseθ = 0.9, i.e., the width of the boundary layer is 90% of the distance from the boundaryof the device to the boundary of the container, and concentrate on the estimationof the diffusion coefficients and the critical time. The computed optimal values ofD0,D1, tc and the least-squares errors for A1 and A2 are listed in Table 2 (denotedas M2). For comparison, we also list the computed results for the case that tc = 0(denoted as M1) in the table. From the table it is clear that our method can handlethe initial burst well. This is also demonstrated by the plots in Fig. 3 from whichit is clear that a single value of diffusion coefficient (M1) does not fit the curves

Numerical methods for diffusion coefficient estimation 623

Table 3 Computed results forA3 and A4 Device LSQ error D (cm2/s) θ

A3 2.84E–02 2.10E–06 0.837

A4 3.83E–02 8.37E–06 0.920

Fig. 4 Fitted curves of A3 and A4

satisfactorily. This is because the initial burst severely affects the computed diffusioncoefficients. However, the method M2 gives very good fits to the experimental data.This observation is also demonstrated by the LSQ errors in Table 2.

We now consider the devices A3 and A4. Since the two devices were treated beforerelease experiments, the initial burst phenomenon is not prominent. Thus, we onlycalculate a single diffusion coefficient, D, and the boundary layer width parameter θ .The computed values are given in Table 3 and the fitted curves are plotted in Fig. 4.From the figure it is seen that the curves fit the corresponding experimental data verywell, which is also demonstrated by the computed least-squares errors.

As mentioned in Sect. 1, there are essentially no reports on the extraction of ef-fective diffusion coefficients and the other parameters by a pure numerical methods.Therefore, we are not able to compare our approach with its close peers. However,there are existing methods for extracting effective diffusion coefficients based on an-alytical solutions. For the geometries of our test problems, two closed forms of an-alytical solution are available. The classic one, given in Crank (1975), Lou et al.(2004) and widely used in practice, is for ‘well-stirred’ liquids which assumes thewidth of a boundary layer near a device is zero. A recent work, outlined in Wang andLou (2007), provides an analytical solution to the diffusion equation which takes intoconsideration of boundary layers and initial burst. We now apply these two methods,

624 S. Wang, X. Lou

Table 4 Results by twoexisting methods CK and WLfor the devices A3 and A4

Device LSQ error D (cm2/s) θ

CK WL CK WL CK WL

A3 4.29E–02 3.13E–02 7.00E–07 3.73E–06 0 0.802

A4 4.00E–02 1.25E–02 2.44E–06 1.18E–05 0 0.936

Table 5 Computed optimalparameters for A4 Device LSQ error (D0,D1) (cm2/s) tc (hrs) θ

A4 1.643E–02 (2.58E–06, 1.03E–05) 0.5 0.932

denoted respectively as CK and WL, to the devices A3 and A4 and list the results inTable 4. Comparing these results with the ones in Table 3 we see that the computedresults by the present method and WL proposed in Wang and Lou (2007) are veryclose to each other, while the analytical method CK in Crank (1975) underestimatesthe effective diffusion coefficients. This is because the analytical solution used in CKis based on the assumption that the width of the boundary layer is zero so that thediffusion process in liquid is faster than that with a boundary layer.

Our last example is again A4. But this time we compute all the four unknownparameters (D0,D1, tc, θ). (Recall tc only takes the discrete values as mentionedbefore.) The computed values are listed in Table 5. Comparing these optimal valueswith those in the last row of Table 3 we see that the results in Table 5 are moreaccurate than those in Table 3. From the table we also see that in this case the initialeffective diffusion coefficient is smaller than the normal one. This is because thedevice was washed prior to the laboratory experiment so that the initial concentrationnear the surface of A4 is lower than that inside the device.

5 Conclusion

In this paper we formulated the parameter extraction problem in optimum drug deliv-ery device design as a nonlinear least squares problem with a PDE and some boundconstraints. Numerical methods for the solution of both the least-squares problem andthe PDE were developed. The approach has been tested using several sets of experi-mental data obtained in our laboratory. The accuracy and usefulness of the methodshave been well demonstrated by the numerical results.

Acknowledgements The authors would like to thank the support from the Australian Research Councilthrough its Discovery Grant No. DP0557148. We would also like to thank the anonymous referees for theirconstructive comments and suggestions.

Appendix: Evaluation of numerical gradients

Though derivatives of C in u1 and u2 (or D0 and D1) can be defined in terms ofthe PDE system (15)–(18) given in Sect. 3.2, they still involve numerical solution

Numerical methods for diffusion coefficient estimation 625

of the PDEs. The evaluation of numerical gradients is desirable since the derivativesof C with respect to tc and θ cannot be analytically defined. We now consider thenumerical evaluation of the gradient ∇uC with respect to u = (D0,D1, tc, θ)�, basedon the discretization schemes in Sect. 3.3 and on the divided differencing.

For each j = 1,2,3,4, let hj > 0 be a chosen perturbation parameter which ismuch smaller than the magnitude of uj but larger than the corresponding mesh pa-rameter of the discretization for the case of j = 3 or 4. Without loss of generality,we assume that the mesh nodes in the spatial partition used in Sect. 3.3 are num-bered in such a way that {xi}Nd

1 is the set of nodes on �d and {xi}N ′′1 is the set of

nodes which are in the perturbation of �l with the perturbation parameter h4. Thenumerical derivatives approximating those in (14) can then be determined using thefollowing algorithm.

Algorithm NG

• Let uj and hj be given for j = 1,2,3,4 and initialize C0 and C0j , j = 1,2,3,4,

with the initial condition U0(x).• For k = 1,2, . . . ,K , perform the following steps:

– Solve (23)–(24) for Ck , using Dk = u1 when tk ≤ u3 and Dk = u2 when tk > u3.– Solve (23)–(24) for Ck

h1, using Dk = u1 + h1 when tk ≤ u3 and Dk = u2 when

tk > u3.– Solve (23)–(24) for Ck

h2, using Dk = u1 when tk ≤ u3 and Dk = u2 + h2 when

tk > u3.– Solve (23)–(24) for Ck

h3, using Dk = u1 when tk ≤ u3 + h3 and Dk = u2 when

tk > u3 + h3.– Solve for Ck

h4the system (23)–(24) with the matrices A and bk being generated

using the mesh on the perturbed region with N ′′ vertices, where Dk = u1 whentk ≤ u3 and Dk = u2 when tk > u3.

– When tk = yi for any i = 1,2, . . . ,Ne, compute

∂R(yi,u)

∂uj

= 1

M∞

N∑l=Nd+1

Ckhj ,l − Ck

l

hj

|dl |, j = 1,2,3,4.

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