Numerical Methods for the Estimation of Effective Diffusion … · flow-through fluid systems....

Numerical Methods for the Estimation ofEffective Diffusion Coefficients and otherParameters of Controlled Drug Delivery

Systems

This thesis is

presented to the

School of Mathematics & Statistics

for the degree of

Doctor of Philosophy

of

The University of Western Australia

By

Shalela Mohd Mahali

January 2012

I miss you...

Haryani Muhd Mahali

-Al-Fatihah-

Abstract

In this thesis we develop various numerical tools for estimating unknown parameters

that characterise the diffusion property of a polymeric drug device in controlled drug

delivery. Two types of fluid systems are considered in this work: the rotating fluid

system and the flow-through fluid system. Based on the consideration of effects

from the initial burst and boundary layer phenomena, three mathematical models

are developed for the parameter estimation problem. They are the basic model

(BM), initial burst model (IB) and boundary layer model (BL). The latter two

models can also be combined to form the initial burst and boundary layer model

(IB+BL). In these models, up to four unknown parameters need to be determined.

These are the diffusion coefficient in the initial burst phase, diffusion coefficient after

the initial burst, width of the boundary layer and the time of the initial burst.

We first develop analytical solutions for the diffusion process of a drug from a

spherical device to a finite external volume. In these solutions, we assume that the

container of the system is spherical and concentric with the spherical device. The

formula for the ratio of the mass released in a given time interval and the total

mass released in infinite time is also derived for both BM and IB models. We then

propose an optimisation approach to the estimation of the parameters based on a

nonlinear least-squares method and the developed analytical solutions.

A new observer approach method is developed for the parameter estimation prob-

lems. In this approach, we construct estimators for the unknown effective diffusion

coefficients characterising the diffusion process of a drug release device using a com-

bination of state observers from the area of adaptive control and the developed drug

v

diffusion models. We show that the constructed systems are asymptotically stable

and the estimators converge to the exact diffusion coefficients. An algorithm is pro-

posed to recursively compute the estimators using a given time series of a release

profile of a device. The numerical results show that this approach is much faster

than the conventional least squares method when applied to the test problems.

We then present a full numerical approach to the estimation of effective diffu-

sion coefficients of drug diffusion from a device into a container in a flow-through

fluid system. Compared to the rotating fluid system considered earlier, this system

has a source and a sink condition due to a fluid flowing through the system at a

constant rate. In this approach we first formulate the drug delivery problem as an

initial boundary value problem containing the diffusion equation. We then propose

a continuous nonlinear least-squares problem containing the system as a constraint

to estimate the unknown parameters. The nonlinear optimisation problem is dis-

cretised by applying a finite volume scheme in space and an implicit time stepping

scheme to the equation system, yielding a finite-dimensional nonlinear least-squares

problem.

Finally, we extend the full numerical technique to three dimensions for esti-

mating effective diffusion coefficients of drug release devices in both rotating and

flow-through fluid systems. The 3-dimensional full numerical technique is crucial

for solving the parameter estimation problems in their real 3D geometries.

Extensive numerical experiments have been performed using experimental data

from various polymeric devices for all the methods developed in this research to

demonstrate their performance. All the numerical results show that our methods

are efficient and accurate and thus they provide useful tools for solving real-world

problems in optimum design of drug delivery devices.

vi

Acknowledgements

Iwould like to first thank my supervisor, Professor Song Wang, for his guidance

during my graduate studies. His ongoing advices have been invaluable for me

not only as a postgraduate student, but also as a future researcher. I also wish to

thank my external supervisor from Curtin University, Associate Professor Xia Lou,

for her help on providing me with the experimental data and chemistry background

knowledge for drug delivery. I am very grateful for the opportunity to be involved

in this research which has been initially explored by both of them. My gratitude

extends to School of Mathematics & Statistics, The University of Western Australia

for providing me with facilities to perform my research. I am also grateful for

the scholarship by Ministry of Higher Education Malaysia. Without their financial

support, pursuing my PhD here is almost impossible. To my fellow graduate students

at this school and from my home country, thank you for the moral support that

has helped me to keep my positive spirit throughout this journey. I also would

like to thank all my family members, especially my parents Mohd Mahali Sarifeh

and Siti Rohani Zainudin for their continuous support and encouragement. Their

unconditional love has been a source of my strength despite our exhaustive grief

on the loss of our beloved family members during my studies. Finally, a foremost

appreciation to my other half, Mohd Nazeri Nanafi, for having patiently waited and

supported me for the last four years.

vii

List of Publications

Publications that arise from this study:

1. 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: Theory, Computation, and Modeling (Theo-

retica Chimica Acta), 125 (2009), 659–669.

2. S. Mohd Mahali, S. Wang, and X. Lou, Determination of effective diffusion

coefficients of drug delivery devices by a state observer approach,Discrete and

Continuous Dynamical Systems - Series B, 16 (2011), 1119 - 1136.

3. S. Mohd Mahali, S. Wang, X. Lou and S. Pintowantoro, Numerical methods

for estimating effective diffusion coefficients of 3-dimensional drug delivery

systems, Numerical Algebra, Control & Optimization, 2 (2012), 377-393.

4. S. Mohd Mahali, S. Wang and X. Lou, Numerical methods for estimating

effective diffusion coefficients of drug delivery devices in a flow-through system,

submitted.

ix

Contents

Abstract v

Acknowledgements vii

List of Publications ix

List of Abbreviations and Symbols xx

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Aims of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Brief Description of the Research . . . . . . . . . . . . . . . . . . . . 8

1.3.1 The Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 The Mathematical Models . . . . . . . . . . . . . . . . . . . . 10

1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mathematical Models for Estimating Diffusion Parameters of Spher-

ical Drug Delivery Devices 13

2.1 The Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 The Basic Model and its Analytical Solution . . . . . . . . . . 14

2.1.2 Total Mass Released in [0, t] from the Device . . . . . . . . . . 20

2.1.3 The Initial Burst . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . 24

2.2.1 Testing the Mathematical Models . . . . . . . . . . . . . . . . 24

xi

2.2.2 The Effect of Initial Burst . . . . . . . . . . . . . . . . . . . . 25

2.2.3 Determination of the Diffusion Coefficients . . . . . . . . . . . 27

2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 A State Observer Approach for Estimating Diffusion Parameters

of Drug Delivery Devices 31

3.1 Construction of the Estimators . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Estimator EI for the Basic Model . . . . . . . . . . . . . . . . 32

3.1.2 Estimator EII for the Basic Model . . . . . . . . . . . . . . . 36

3.1.3 Extension of the Estimators to the Initial Burst Model . . . . 37

3.2 Convergence of the Methods . . . . . . . . . . . . . . . . . . . . . . . 41


3.3.1 Estimator EI . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2 Estimator EII and Comparisons of EI and EII . . . . . . . . . 46

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Numerical Methods for Estimating Diffusion Parameters of 2-dimensional

Flow-Through Drug Delivery System 53

4.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Time Discretisation . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.2 Determination of boundary conditions for (54) . . . . . . . . . 61

4.3 Estimation of the Unknown Parameters . . . . . . . . . . . . . . . . . 63


4.4.1 Unweighted Least-squares Method . . . . . . . . . . . . . . . . 66

4.4.2 Weighted Least-squares Method . . . . . . . . . . . . . . . . . 66

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Numerical Methods for Estimating Diffusion Parameters of 3-dimensional

Drug Delivery Systems 73

5.1 The Drug Delivery Systems . . . . . . . . . . . . . . . . . . . . . . . 75

xii

5.1.1 Rotating Fluid System . . . . . . . . . . . . . . . . . . . . . . 75

5.1.2 Flow-through System . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 The Width of the Diffusion Layer . . . . . . . . . . . . . . . . . . . . 76

5.3 The Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.1 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.2 Estimation of the unknown parameters . . . . . . . . . . . . . 83


5.4.1 Case I: Spherical Devices in Cylindrical Containers in the Ro-

tating Fluid System . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2 Case II: Cylindrical Devices in Cylindrical Containers in the

Rotating Fluid System . . . . . . . . . . . . . . . . . . . . . . 89

5.4.3 Case III: Cylindrical Devices in Cylindrical Containers in the

Flow-through Fluid System . . . . . . . . . . . . . . . . . . . 90

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Conclusions and Future Works 93

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Appendices 111

A Experimental Data for Spherical Polymeric Devices 111

B Experimental Data for Flow-through Experiment 113

C Experimental Data for Cylindrical Polymeric Devices 137

D Analytical evaluation for integrals in (34) and (36) 139

xiii

List of Tables

1 Results from Models BM and IB for all the tested devices . . . . . . . 25

2 Results from Estimator EI. . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Results from Estimator EII . . . . . . . . . . . . . . . . . . . . . . . 48

4 Error in the discrete L2-norm squared for Estimators EI and EII . . . 49

5 CPU time (in seconds) used by EI, EII and WLSQ . . . . . . . . . . 51

6 Results by the unweighted least-squares method. . . . . . . . . . . . . 66

7 Results by the weighted least-squares method, WLSQe . . . . . . . . 69

8 Results from the weighted least-squares method, WLSQm . . . . . . 69

9 Computed diffusion coefficients D and least-squares error E for Data

S2080-05 using various meshes. . . . . . . . . . . . . . . . . . . . . . 86

10 Computed updates at selected iterations for Data S2080-05 with m =

n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

11 Optimal results from ANAL and the FULL3D for BM in Case I. . . . 88

12 Optimal results from ANAL and FULL3D for Model IB in Case I. . . 89

13 Optimal results from FULL3D for BL in Case I. . . . . . . . . . . . . 89

14 Optimal results from ANAL, FULL2D and FULL3D for Model BM

in Case II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

15 Optimal results from FULL3D for BM, IB and BL using Data A4 in

Case II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

16 Optimal results from FULL2D and the FULL3D for Model BM in

Case III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

17 Experimental data of Mt/M∞ (or Re) for spherical polymeric devices 111

xv

18 Experimental data of concentration value in a flow–through system

for disc/cylindrical polymeric devices . . . . . . . . . . . . . . . . . . 113

19 Experimental data of Mt/M∞ (or Re) for spherical polymeric devices 137

xvi

List of Figures

1 A spherical device with radius rd placed in a container with radius rc 15

2 Fitted curves by Models BM and IB for S2080-10 and S4060-10 . . . 26

3 Fitted curves by IB for devices loaded with 0.5 wt% (a) and 1.0 wt%

(b) drug solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 The optimal effective diffusion coefficients of the devices for drug

loading 0.5 wt% and 1.0 wt% . . . . . . . . . . . . . . . . . . . . . . 27

5 Predicted drug release profiles for 0.5wt% drug solution using Esti-

mator EI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Predicted drug release profiles for 1.0wt% drug solution using Esti-

mator EI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Predicted ratio of the mass release using BM and IB models and

Estimator EII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8 The optimal effective diffusion coefficients of the devices for drug load-

ing 0.5wt% and 1.0wt% drug solution using Least-squares method,

Estimator EI and Estimator EII . . . . . . . . . . . . . . . . . . . . . 50

9 Geometries of the container, the device and the diffusion layer . . . . 55

10 An example of the Delaunay mesh (solid line) and the corresponding

Voronoi polygons (dotted line) with notation for edges and nodes . . 59

11 Curve fitting for data set A using uWLSQ for BM and IB. . . . . . . 67

12 Predicted drug release profiles for data between 0 to 2 hours and 2 to

25 hours for A and B using uWLSQ method. . . . . . . . . . . . . . . 67

xvii

13 Predicted drug release profiles for data between 0 to 2 hours and 2 to

50 hours for C and D using uWLSQ method. . . . . . . . . . . . . . . 68

14 The fitted curves by uWLSQ, WLSQm and WLSQe for data sets B

and D using BM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

15 The fitted curves by uWLSQ, WLSQm and WLSQe for data sets B

and D using IB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

16 The geometries of cylindrical (a) and spherical (b) devices and con-

tainers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

17 The flow-through system used in the experiment. . . . . . . . . . . . 77

18 The illustration of the cylindrical (a) and spherical (b) boundary layer. 78

19 3D Delaunay mesh and Dirichlet Tessellation (a), A Voronoi polygon

(b), Edge from xi to xj and the corresponding facet li,j (c). . . . . . 80

20 Computed release profiles after the selected numbers of iterations with

m = n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

21 Computed diffusion coefficients for Data S2080-05 starting from 3

different initial guesses on the mesh with m = 2, n = 2. . . . . . . . . 88

xviii

List of Abbreviations and Symbols

BM Basic Model

IB Initial Burst Model

BL Boundary Layer Model

CPU Central Processing Unit

WLSQ Weighted least-squares

uWLSQ Unweighted least-squares

PHEMA Poly(2-hydroxyethyl methacrylate)

ANAL Analytical method

FULL2D Full numerical method for the 2D diffusion problem

FULL3D Full numerical method for the 3D diffusion problem

D Effective diffusion coefficient

D0 Effective diffusion coefficient during an initial burst

D1 Effective diffusion coefficient after an initial burst

tc Critical time that separates the initial burst phase and

the normal phase

Ωd Region for a drug device

Ωl Region for the union of the drug device and the diffusion

layer

Ωc Region for a full of liquid container

rd Radius for a drug device

rl Radius for the union of the drug device and the diffusion

layer

rc Radius for a full of liquid container

xx

hd Height of a drug device

hl Height of the diffusion layer

hc Height of a full of liquid container

θ Ratio of the diffusion layer width over the whole external

liquid

M0 Initial mass of drug loading

Mt Total mass of drug released from the device in the time

interval [0, t]

M∞ or M∞ Total mass of drug released from the device in infinite

time

R Ratio of the mass released in a time interval over the

mass released in infinite time

Me or Re Experimental data of the release profiles

Vd Volume of a drug device

Vl Volume of the union of the drug device and the diffusion

layer

Vc Volume of a full of liquid container

C Concentration

xi ith node

x x-coordinate of node x

y y-coordinate of node x

z z-coordinate of node x

ei,j Edge connecting node xi to node xj

ei,j Unit vector from node xi to node xj

di Dirichlet tessellation

li,j A line segment or a facet of boundary of di connecting

the circumcenters of two elements sharing ei,j

xxi

Chapter 1

Introduction

1.1 Literature Review

Traditionally, drugs are consumed in the form of a solid or liquid, orally or as an

injection. The conventional ways of drug consumption not only require the patient

to frequently take the drug to maintain the drug concentration at the effective level,

but also at the same time put the patient at a high risk of drug poisoning and

other side effects. As an example, the concentration of a drug rises abruptly right

after the drug is taken in a pill form, in which the rises can be briefly above the

toxic level, then drop below the effective level [20]. The pharmaceutical field has

become increasingly innovative in its efforts to dispense patients from this burden.

An advance over simple fast-acting chemical compounds that are dispensed orally

or as injectables, are formulations that control the rate and period of drug delivery

(i.e., time-release medications) and target specific areas of the body for treatment,

which have become increasingly common and complex [93]. This development surely

can reduce the dose intake efforts on the part of the patients and also provide more

effective treatment.

The use of polymer materials as drug carriers is one of the most used technologies

in controlled drug delivery [19, 45, 44, 60, 59, 78, 90]. The ease of polymer process-

ing and the flexibility of manipulation to have the right characteristics are among

1

1.1. LITERATURE REVIEW

the reasons why polymeric drug devices have become popular. The designed drug

devices are not only suitable to be orally consumed [70], but also can be implanted

directed in the targeted area [7, 98] for optimal treatment. The polymeric drug deliv-

ery field, with its recent developments, was critically reviewed in [58]. Based on the

Deborah number or the time of degradation compared to the human lifetime, the

polymeric drug devices are categorised into biodegradable and non-biodegradable

polymers [66].

The drug delivery system used in this study is based on a porous matrix made

of poly(2- hydroxyethyl methacrylate) (PHEMA) hydrogels. PHEMA is well known

for its biomedical applications as contact lenses, intraocular lenses and cardiovascu-

lar implants [76, 79]. Materials based on PHEMA absorb large amounts of water

without dissolving, and in their swollen state they behave like typical gels. There-

fore, the term hydrogels is commonly employed for them. In most applications

PHEMA hydrogels refer to the crosslinked polymers produced by bulk polymerisa-

tion which are transparent and contain a homogeneous polymer matrix containing

pores measured in nanometres. Although polymers of this type allow the diffusion

of various solutes, their transport properties are limited by effective mean pore size,

or mesh diameters, within the polymer. They are more suited for such applications

as contact lenses, in which a combination of optical clarity and limited diffusive

characteristics is required [72].

Chapter 4 in [66] presents mathematical models that characterise the kinetics

of drug release from polymeric devices. Similar to [2], drug release is classified

into three mechanisms: a diffusion controlled system, a swelling controlled system,

and an erosion controlled system. Basically, the diffusion and swelling controlled

systems are for a non-biodegradable polymer matrix, while the diffusion and erosion

controlled systems are for a biodegradable polymer matrix. A swelling controlled

system is to give more controllability for the drug delivery with a low diffusivity

drug in polymer. The matrix swelling will increase drug mobility, which is also

called the enhanced diffusion situation. The erosion mechanism of biodegradable

2

INTRODUCTION

polymer matrix which depends on the erosion number as suggested in [9], will also be

dominated by diffusion if the erosion rate is slow [2]. Hence, among the mechanisms

stated in the literature for controlled drug delivery, we may conclude that diffusion

is the major mechanism.

Considering that diffusion is the most important mechanism to control drug

release [85], diffusion controlled drug devices have become the most widely used

drug delivery system [75]. In a diffusion controlled device, the delivery of drugs is

largely dependent on the diffusion property of the drug in the developed device. In

general, this controlled system can be divided into a reservoir system and a matrix

system. In a reservoir system, a drug reservoir is surrounded by a polymer matrix

shell. In a matrix system, the drug is incorporated in the polymer matrix [2].

In a diffusion-controlled device, the delivery of drugs is largely dependent on the

diffusion property of the drug in a constructed device, which is often characterised

by the diffusion coefficient of the drug in the material. A diffusion coefficient is a

very important parameter in the design of a controlled drug delivery system. It is

critical to the functionality and performance of a system that needs to provide target

drug for a desired period of time and at an efficient therapeutic level. In theory, the

coefficient can be a function of the space, time, concentration, or even temperature

in the system [29]. However, in practice we usually seek a constant approximation

to the coefficient, which is called the effective diffusion coefficient. This effective

diffusion coefficient is a measure of the drug transport determined by the physical

properties including the porosity of the polymer matrix, possible partition of the

drugs and interactions, if any, between the drug and the polymer matrix. For a given

device, drug release profiles from the device into a finite volume during a period of

time can be determined through laboratory experiments. The effective diffusion

coefficient of the drug is then estimated by setting up a mathematical model for the

diffusion process of the device and comparing the numerical value from the model

and the experimentally observed drug release data [95]. The extraction of drug

diffusion parameters especially the effective diffusion coefficient is important for the

3


optimal design of a polymeric device to deliver drugs in a controlled fashion [59].

Estimation of diffusion coefficients or unknown parameters in general, has, in

fact, attracted much attention from both academia and practitioners (e.g. [53, 6,

28, 39, 82, 13, 49]) due to the important roles in numerous areas such as food science

and technology [17, 89, 91], chemical engineering and chemistry [11, 40, 52, 101],

and pharmaceutical science and engineering [69, 100], just to name a few. In par-

ticular, in controlled drug delivery, accurate determination of diffusion parameters

is important for controlling and optimising a drug delivery process so as to engi-

neer effective devices and/or formulae that show better drug efficacy and lower side

effect with improved drug responses. In the last two decades, some efforts have

been made by pharmacists, biomedical scientists, engineers and mathematicians in

order to achieve these results [14, 16, 96, 95, 94, 99, 5]. For a given device, drug

release profiles from the device into a finite volume during a period of time can be

determined through laboratory experiments. The estimation of the effective diffu-

sion coefficient of the drug during the process involves two tasks. One task is to set

up a mathematical model for the diffusion process of the device and the other is to

numerically estimate the effective diffusion coefficient based on the model and some

given information such as experimentally observed drug release data.

Based on the information available from the literature and background of the

experiment, mathematical models can be developed for the drug delivery systems.

By developing these models, the process of drug release can be better understood.

This facilitates, in turn, the improvement of the system. Mathematical models are

also able to save time and reduce costs in designing drug devices by reducing the

required laboratory experimentation [84]. The fact that mathematical modeling

of drug delivery is a field of significant academic and economic importance is true

not only in the biopharmaceutical disciplines [84], but also in the increasingly active

tissue engineering research field where the development of three dimensional scaffolds

meeting the requirements of cell migration, tissue growth, and the transportation of

nutritious chemicals such as growth factors is still a challenge [62]. An ideal delivery

4

INTRODUCTION

requires a device to supply and release therapeutic agents to a desired location with

a precise therapeutic dose for a prolonged period of time [38].

In general, a diffusion process is governed by a diffusion equation with appro-

priate initial and boundary conditions. In the context of drug release, analyti-

cal solutions are available for some special cases with regular device geometries

[42, 1, 59, 96, 95]. Though these methods provide useful tools for estimating effec-

tive diffusion coefficients, they can hardly be extended to problems with irregular

geometries. Empirical and semi-empirical models of drug delivery have also been

proposed (see, for example, [84, 60]). However, these methods lack mathematical

rigorous. The Higuchi model is among the earliest models to quantitatively repre-

sent drug delivery [46, 47] and is widely used due to its simplicity [32, 8, 35]. In a

recent paper, Siepmann et al. [83] revisit the Higuchi equation and discussed some of

the misuse of the equation. This equation was originally derived for a 1-dimensional

problem and is applicable to the case where the initial concentration of the drug

loaded in the device is substantially higher than the solubility of the drug, which

is also called the dispersed matrix system. The other popular empirical solution in

drug delivery problem is the power law equation [54] where the fractional release of

the drug is exponentially related to the release time. The values of the exponential

rate, n, for regular geometries have been determined. However only n = 0.5 and

n = 1 are physically realistic, whereas other values of n indicate anomalous trans-

port kinetics [81, 80]. Over the years, many other mathematical formulations for

drug delivery systems have been developed empirically, semi-empirically, and also

analytically. Despite the numerous suggested mathematical models, most of them

only consider a well-stirred system. A phenomenon called initial burst that is known

to occur in the delivery process is also usually neglected as in [33].

The initial burst phenomenon happens due to excessive drug substance near the

device’s surface. If the device is not treated, there is a chance this phenomenon

can cause drug toxicity in the patient. Hence, pharmacists have designed drug

devices that can avoid this phenomenon [48]. During the design of a drug device,

5


this phenomenon should be considered while collecting the data from the laboratory.

Ignoring this phenomenon when extracting the effective diffusion coefficient from the

experimental data would cause a huge error. The initial burst phenomenon causes

an initial release rate higher than the rest of the drug release process, therefore the

data misrepresent the normal drug release for the device. Following the suggested

method in [94], the formulations in this research also divide the effective diffusion

coefficient into two constants, representing the two phases of the release. The critical

time that separates these two phases also becomes one of the unknowns that need

to be determined. Such a model not only handles the case when the initial release is

faster, but also when the initial release is slower than the rest of the process, which

can happen due to the over-treatment of a device.

In most of the solutions available in the literature, the systems considered were

well-stirred. This consideration leads to a uniform concentration assumption through-

out the liquid. However, well-stirred systems can never appear in practice because

the magnitude of the flow velocity on the device’s boundary should be zero due

to the so-called no-slip boundary condition . In order to acknowledge the no-slip

boundary condition in a well-stirred or partially well-stirred system, we assume there

is a layer immediately adjacent to the device’s surface, forming a boundary dividing

the liquid into two subregions as in [94]. The first subregion is defined around the

device where the concentration is not uniform. This convection dominant layer is

called diffusion layer or boundary layer in this thesis. The region from the outer

boundary of the boundary layer to the boundary of the system is convection dom-

inant (or well-stirred) in which the drug concentration is uniform. The width of

the diffusion/boundary layer is one of the unknown parameters considered in this

research which will be approximated using a numerical method.

Although the modelling of drug delivery systems is rapidly improving and head-

ing to better development of the drug delivery in the tissue [55, 73], estimating the

optimal parameters in the models has not attracted much attention in the phar-

maceutical field. On the other hand, many optimisation methods for estimating

6

INTRODUCTION

the best unknown parameters are available in the literature. Mathematical tools,

particularly numerical partial differential equation and optimisation techniques have

been used successfully and extensively in optimum designs of many engineering de-

vices such as semiconductor devices (cf., for example, [3, 43, 97]). Chapter 2 in [30]

discusses the basic steps to estimate the values of coefficients in models by fitting

the models to the available data. Even for a non-constant unknown parameter, the

approach for its identification is briefly discussed in [68]. As an example, by writing

the problem into a nonlinear least-squares problem, various well established meth-

ods can be used such as the Gauss–Newton method [51], the Levenberg–Marquardt

method [56, 63], and many others for parameter estimation. Despite the success of

these techniques in many areas, reports on the systematic use of advanced mathe-

matical tools in the design of controlled drug delivery devices are scarce in the open

literature, except for some simple models with known analytical solutions of the

diffusion equation (cf., for example, [16, 27, 61]).

Although the above mentioned methods are well-established and give good accu-

racies, solving the problem iteratively needs a huge Central Processing Unit (CPU)

time consumption. The computational cost will further increase if the problem

is discretized, in multiple dimensional and involves several number of parameters

.Therefore, in the current fast-paced world, this can be troublesome. In some situ-

ation, fast and low-cost solutions are more important than accurate solutions [74].

Although the need for fast results usually occur in factories, the military, or robotics-

based problems, there is no doubt that the same priority applies in the competitive

field of pharmaceuticals. Therefore, without disregarding the well-established meth-

ods, new methods that can reduce computational costs need to be developed. The

Broyden method has been used to reduce the computational cost in computing

the Jacobian for the gradient-based optimisation method [71]. A discussion about

achieving faster results in numerical solution has been included in [4]. Although

current advances in computational machines such as the development of supercom-

puters [24] can help in achieving faster results, such machines are expansive and

7

1.2. AIMS OF THE RESEARCH

not usually available in every institution. This implementation of the methods on

supercomputers also requires a special coding technique.

In this PhD research, we developed drug delivery models that consider the ini-

tial burst phenomenon as well as boundary layer effects. The models were solved

analytically and also numerically in this research. The estimation of the unknown

parameters was achieved first using a non-linear least squares method and then using

a new observer approach method to save CPU time. We summarise the objectives

of this research in the next section.

1.2 Aims of the Research

The aims of this research are

i To derive analytical solutions for spherical drug delivery devices taking into

consideration of the initial burst effect.

ii To develop mathematical models for drug delivery in a flow-through system.

iii To estimate the optimal unknown parameters in the developed models using

the least-squares method.

iv To design a new method based on the observer approach to reduce the CPU

time used in the least-squares method in estimating the optimal unknown

parameters in the developed models.

v To extend the 2-dimensional numerical tools in estimating the effective diffu-

sion coefficients in [96] to 3-dimensions.

1.3 Brief Description of the Research

The diffusion coefficient plays the most important role in characterising a polymeric

drug device. Therefore, in the development of the device, the determination of the

8

INTRODUCTION

coefficient is crucial. In this research, drug delivery models are developed mathe-

matically. The models are then solved analytically for spherical devices considering

the initial burst phenomenon and numerically for more general devices considering

both initial burst and the boundary layer effects. A few other unknown parameters

occur during the mathematical modelling and need to be estimated together with

the diffusion coefficients based on the laboratory data for the tested devices.

1.3.1 The Experiments

Polymeric Sphere Devices in a Rotating Fluid System Most of the mathe-

matical results derived in this PhD research are dedicated to extracting the effective

diffusion coefficients of spherical polymeric drug devices designed in the Chemical

Laboratory in Curtin University. The preparation of the the devices is explained in

our published work [95]. They were then placed in the centre of a container which

has an air-tight sealant filled with enough deionised water and placed upon an or-

bital shaker (Chiltern Scientific) at a speed of 45 rpm. At preset points of time,

an amount of the drug solution was removed from a marked location and further

diluted for quantitative analysis of released drug concentration, Mt, using a UV–Vis

spectrometer.

Polymeric Cylinder/Disc Devices in a Flow-Through System Originally,

this experiment is to simulate drug delivery in eyes [88]. With a different exper-

imental setup compared to the rotating fluid system, some adjustment needed to

be done in the mathematical models. A sink condition as well as the mass loss in

the system needed to be considered. The preparation of the devices and the experi-

mental apparatus was explained in [88]. Compared to the rotating fluid system, the

external liquid in this flow-through system is simultaneously being pumped in and

pumped out resulting in a mass loss in the region. Button-shaped drug devices were

used in this experiment.

9

1.3. BRIEF DESCRIPTION OF THE RESEARCH

1.3.2 The Mathematical Models

Basically three mathematical models for drug delivery processes are developed in this

research, starting with a diffusion equation with one unknown diffusion coefficient,

D. Most of the mathematical solutions to drug delivery systems available in the

literature only consider this basic model. In this research, the basic model is further

modified to handle additional phenomena which occur in the process such as the

effects of the initial burst and the boundary layer. The models can be summarised

as follows.

i Basic Model (BM): This model consists only one unknown parameter which

is the (constant) effective diffusion coefficient, D.

ii Initial Burst Model (IB): In this model, the diffusion process is divided into

two phases with two different effective diffusion coefficients:

D =

D0, 0 < t ≤ tc,

D1, t > tc,

where tc > 0 is the critical time. Besides the two unknown diffusion coefficients

D0 and D1, tc is also an unknown parameter in this model. If tc = 0 (or ∞),

this model reduces to the basic model. This model is used to handle the so-

called Initial Burst Phenomenon in laboratory experiments in which excessive

drug left on the surface of a device during the drug load process causes the

diffusion to be much faster in a short initial period of time than in the rest of

the process.

iii Boundary Layer Model (BL): Denoting Ωd as a region for a drug device located

in a full of fluid container, Ωc, we divide the liquid region into two subregions.

The one around the device, called the boundary layer, is a diffusion dominated

region and the subregion outside the boundary layer is convection-dominant so

that the drug concentration is uniform in it. If we use Ωl to denote the union

10

INTRODUCTION

of the device and the boundary layer, then the two subregions are respectively

Ωl \ Ωd and Ωc \ Ωl. In this case, we may introduce one unknown parameter

to be defined later in this thesis.

The developed models are solved analytically for Models BM and IB or numer-

ically for all models to find the drug release profiles. The unknown parameters are

extracted by fitting the laboratory data and the solutions of the models. The de-

signed methods not only aim at accuracy of the estimation (nonlinear least-squares),

but also a reduced computational time (observer approach), indicated by CPU time.

1.4 Outline of the Thesis

This thesis consists of six chapters including Chapter 1: Introduction and Chapter 6:

Conclusions. In Chapter 2, we develop two mathematical models for drug delivery

from a spherical polymeric drug devices into an external volume. The analytical

solutions for each model are then derived. The new solutions include the effect of

the initial burst phenomenon which is expected to happen in the experiments. The

unknown parameters are then estimated using the built-in least-squares solver in

Matlab. Some numerical tests are shown in the final section of the chapter to show

the performance of each model. To reduce the CPU time used in the estimation

stage, we propose in Chapter 3 a new observer approach method. The numerical

tests are included at the end of the chapter to show the CPU time improvement

by using the method. Moving to a more complex liquid system, Chapter 4 presents

numerical solutions for drug delivery in a flow-through system. We show some

numerical results using weighted and unweighted least-squares methods to show

the performance of the numerical solutions. Finally in Chapter 5, we extend the

numerical solution to 3-dimensions. The 3D numerical solutions are then applied to

all the solutions discussed in the previous chapters. Some numerical tests are done

to show the usefulness of the solution.

11

1.4. OUTLINE OF THE THESIS

12

Chapter 2

Mathematical Models for

Estimating Diffusion Parameters

of Spherical Drug Delivery Devices

This chapter introduces the mathematical models of drug delivery for a spher-

ical device. Since this study is based on a porous matrix made of PHEMA

hydrogels, where the drug delivery from such material is diffusion driven, we may

represent the diffusion process by a diffusion equation with appropriate initial and

boundary conditions. However, solving such a diffusion problem analytically is very

difficult. Analytical and approximate solutions to several simple models can be

found in [10, 18, 34, 37, 84]. Some widely used models such as those in [18] are

based on the assumption that the liquid in the diffusion region is ‘well-stirred’, i.e.,

the concentration of the substance in the liquid is uniform which is not always

true. Therefore, in this chapter, we consider a drug delivery system that diffusion

dominant in the whole region (i.e unstirred liquid).

Once a diffusion model has been established, one needs to determine the effective

diffusion coefficient using the model. A classical trial-and-error process is neither

optimal nor automatic. In the previous work [96] , a model has been proposed

for the estimation of effective diffusion coefficients and other critical parameters of

13

2.1. THE MATHEMATICAL METHODS

PHEMA devices of a 2D disc geometry. The model was used in conjunction with

a nonlinear least-squares method. Unlike existing ones such as those in [18], this

model can handle the initial burst effect. In the present chapter, we extend the

techniques in [96] to devices of a spherical geometry (Figure 1). We first propose a

basic mathematical model governing the diffusion process of a drug from a spherical

device into a finite volume. The model is then extended to a model that considers

the initial burst effect. The analytical solutions for both models are obtained to

provide explicit expressions for evaluating the concentration value at a given time

on the region. From the solutions, formulae to represent the mass diffused from the

device into the external volume in a given period of time can be evaluated. The

unknown parameters in the formulae which are the effective diffusion coefficients of

the drug from the devices and the critical time of the initial burst are determined

by an optimisation technique.

This chapter is organised as follows. In Section 2.1 analytical solutions are

derived and formulae for the ratio of the mass released in a given time interval and

the total mass released in infinite time are obtained for both the basic and the initial

burst models. The approach has been tested using experimental data for spherical

devices made of porous PHEMA hydrogels. The effectiveness and accuracy of the

method are well demonstrated by the numerical results in Section 2.2. The model

is then used to determine the diffusion parameters including the effective diffusion

coefficients a series of devices that vary in both the porous structure and the drug

loading levels.

2.1 The Mathematical Methods

2.1.1 The Basic Model and its Analytical Solution

A spherical device with radius rd preloaded with an amount of drug, M0 is consid-

ered. We assume that (1) the device is placed in a sphere container of radius rc

filled with water so that the device and the container are concentric, as depicted

14

MATHEMATICAL MODELS FOR ESTIMATING DIFFUSION PARAMETERS

OF SPHERICAL DRUG DELIVERY DEVICES

Figure 1: A spherical device with radius rd placed in a container with radius rc

in Figure 1; and (2) the release process is diffusion-dominant and radial because of

symmetry, i.e., the concentration of drug in liquid is uniform for a fixed r. Based on

the considerations, the diffusion process of this problem is governed by the following

diffusion equation in spherical coordinates:

∂C(r, t)

∂t= D

(∂2C(r, t)

∂r2+

2

r

∂C(r, t)

∂r

)

, 0 < r < rc, t > 0, (1)

∂C(rc, t)

∂r= 0, t > 0, (2)

C(r, 0) = H(r) (3)

where D is a constant and C(r, t) is the unknown concentration. We assume that

at t = 0, the concentration is uniform in the device and zero in liquid, i.e,

H(r) =

M0/Vd, 0 < r < rd,

0, rd < r < rc

(4)

where Vd = 4πr3d/3 is the volume of the device. To solve this problem, we use the

technique of separation of variables as outlined below.

15


Let C(r, t) = u(t)v(r). (eqn:1) then becomes

u′v = D

(

uv′′ +2

ruv′)

= Du

(

v′′ +2

rv′)

From this we haveu′

Du=v′′ + 2

rv′

v= −λ,

where λ > 0 is a constant to be determined. The above expression contains two

equations:

u′ + λDu = 0, (5)

v′′ +2

r+ λv = 0 (6)

(5) has the fundamental solution u = e−λDt and (6) is a Bessel’s equation of the

form

y′′ + (d− 1)y′/x+ (λ− µ/x2)y = 0

with d = 3 and µ = 0. The fundamental solution to this equation is

v(r) = j0

(

r√λ)

where j0(z) =sin zz

is the 0th order spherical Bessel function. Therefore the solution

of (1) is of the form

Cλ(r, t) = j0(r√λ)e−λDt, (7)

where λ is a parameter called the eigenvalue of the problem. To determine λ, we

apply the boundary condition as in (2) to (7) to get

∂Cλ(rc, t)

∂r= j′0(rc

√λ)√λe−λDt = 0.

This implies j′0(rc√λ) = 0. Let αn > 0 be such that

j′0(αn) =αn cosαn − sinαn

α2n

= 0 for n = 1, 2, ... (8)

16



Then, we have rc√λ = αn or

λn =α2n

r2cfor n = 1, 2, ...

Substituting the above λn into (7) gives

Cλn(r, t) = j0(

rαn

rc)e−Dα2

nt/r2c

which is a solution to (1) for each n = 1, 2, ....

When λ = 0, (6) has the general solution

v =B0

r+ A0 (9)

with additive constant A0 and B0. (note α0 = 0 is also a root of (8) and thus

λ = 0 is the eigenvalue corresponding to this root.) Thus, (9) represents the steady

state solution to (1)–(3). Applying the boundary condition (2)–(9) gives B0 = 0.

Therefore, combining the fundamental solutions to Equation (5) and (6) and using

the superposition principle, we have the following series solution to (1) :

C(r, t) =

∞∑

n=0

Anj0

(αnr

rc

)

e−Dα2nt/r

2c , (10)

where An’s are coefficients to be determined. (Recall that αn = 0 and j0(0) = 1.)

Note that the steady-state solution of the problem when t → ∞ is C(r,∞) =

M0/Vc, where Vc = 4πr3c/3 is the volume of the container. Therefore, we have, from

(10),

A0 =M0/Vc. (11)

Before determining the coefficients An, n = 1, 2, ..., we first derive simpler mathe-

matical expressions for∫ rc0j0

(rαm

rc

)

· j0(

rαn

rc

)

r2dr. For this purpose, we consider

the following possible cases:

17


Case I : m 6= n, αm, αn > 0

∫ rc

0

j0

(rαm

rc

)

· j0(rαn

rc

)

r2dr =

∫ rc

0

rc sin(

rαm

rc

)

rαm·rc sin

(rαn

rc

)

rαnr2dr

=r2c

αmαn

∫ rc

0

sin

(rαm

rc

)

sin

(rαn

rc

)

dr

=r2c

αmαn

(

−rcαm cos(αm) sin(αn)− αn sin(αm) cos(αn)

(αm − αn)(αm + αn)

)

=−r3c

αnα3m − αmα3

n

[αm cos(αm) sin(αn)− αn sin(αm) cos(αn)

− sin(αm) sin(αn) + sin(αm) sin(αn)]

=−r3c

αnα3m − αmα3

n

[sin(αn)(αm cos(αm)− sin(αm))

− sin(αm)(αn cos(αn)− sin(αn))]

= 0,

since αm and αn are roots of α cos(α)− sin(α) = 0.

Case II : m = 0, αn > 0

∫ rc

0

j0(0) · j0(rαn

rc

)

r2dr =

∫ rc

0

rc sin(

rαn

rc

)

rαn

r2dr

=rcαn

∫ rc

0

r sin

(rαn

rc

)

dr

=rcαn

∫ αn

0

rcαnu sin(u)

rcαn

du

(

u =rαn

rc

)

=r3cα3n

∫ αn

0

u sin(u)du

=r3cα3n

(sin(u)− u cos(u))|αn

0

=r3cα3n

(sin(αn)− αn cos(αn))

= 0.

18



Case III : m = n, αn > 0

∫ rc

0

j20

(rαn

rc

)

r2dr =

∫ rc

0

r2c sin2(

rαn

rc

)

r2α2n

r2dr

=r2cα2n

∫ rc

0

sin2

(rαn

rc

)

dr

=r3cα3n

∫ αn

0

sin2(u)du

(

u =rαn

rc

)

=r3cα3n

(

−cos(u) sin(u)

2+u

2

)∣∣∣∣

αn

0

= − r3c2α3

n

(cos(αn) sin(αn) + αn)

= − r3c2α3

n

(αn − αn cos2(αn) + αn) (sin(αn) = αn cos(αn))

=r3c2α2

n

(1− cos2(αn)) =r3c2α2

n

sin2(αn)

=r3c2α2

n

α2n cos

2(αn) =r3c2cos2(αn).

As a conclusion of the above derived solutions, we may write

∫ rc

0

j0

(rαm

rc

)

· j0(rαn

rc

)

r2dr =

0, m 6= n, αm, αn ≥ 0,

r3c2cos2 αn, m = n, αn > 0.

(12)

We now use the initial condition (3) and the above results to determine An, for

n = 1, 2, .... Applying the initial condition (3) to (10) and using (11) we have

C(r, 0) =M0

Vc+

∞∑

n=1

Anj0

(rαn

rc

)

= H(r)

where H is the function defined in (4). Multiplying both sides of the above equation

by r2j0(rαm/rc) for any m = 1, 2, ..., integrating the resulting equation from 0 to rc

19


and using (4), where H(r) is zero from rd to rc, and (12), we have

Am.r3c2cos2 αm =

3M0

4πr3d

∫ rd

0

r2j0

(

rαm

rc

)

dr

=3M0

4πr3d

∫ σαm

0

r3cα3m

u sinudu (u = rαm/rc, σ = rd/rc)

=3M0

4πr3d

r3cα3m

[sin u− u cosu]σαm

0

=3M0

4πr3d

r3cα3m

[sin(σαm)− (σαm) cos(σαm)]σαm

0

=3M0

4πr3d

r3cα3m

(σαm)2j1(σαm)

=3M0

4πσαmj1(σαm),

where j1(z) = (sin z)/z2 − (cos z)/z is the first order spherical Bessel function. We

thus have

Am =3M0

2πσαmr3c cos2 αm

j1(σαm), m = 1, 2, · · · . (13)

Substituting (11) and (13) into (10) we get

C(r, t) =M0

Vc+

3M0

2πσr3c

∞∑

n=1

j1(σαn)

αn cos2 αn

j0

(αnr

rc

)

e−Dα2nt/r

2c . (14)

This is an analytical solution to (1)-(3) in the region defined by 0 < r < rc and

0 < t <∞.

2.1.2 Total Mass Released in [0, t] from the Device

We now derive the total mass released from the device into the container in the time

interval [0, t], denoted as Mt. For clarity, we let

Kn(t) =j1(σαn)

αn cos2 αne−Dα2

nt/r2c .

20



Multiplying both sides of (14) by r2 sinφdθdφdr and integrating the resulting equa-

tion over the region: 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π and rd ≤ r ≤ rc , we have

Mt = 4π

∫ rc

rd

C(r, t)r2dr

=M0(Vc − Vd)

Vc+

6M0

σr3c

∞∑

n=1

Kn(t)

∫ rc

rd

j0

(αnr

rc

)

r2dr

=M0(Vc − Vd)

Vc+

6M0

σr3c

∞∑

n=1

Kn(t)

×∫ αn

σαn

r3cα3n

u sinudu

(

u =rαm

rc, σ =

rdrc

)

=M0(Vc − Vd)

Vc+

6M0

σr3c

∞∑

n=1

Kn(t)r3cα3n

[sin u− u cosu]αn

σαn

=M0(Vc − Vd)

Vc− 6M0σ

∞∑

n=1

Kn(t)

αnj1(σαn)

=M0(Vc − Vd)

Vc− 6M0σ

∞∑

n=1

j21(σαn)

α2n cos

2 αn

e−Dα2nt/r

2c .

(15)

In the above we used (8). When t→ ∞, we have

Mt → M∞ =M0

Vc(Vc − Vd)

which is the total mass released from the device into the external volume in the time

interval [0,∞]. Dividing both sides of (15) by M∞ gives

Mt

M∞= 1− 6σ

1− σ3

∞∑

n=1

j21(σαn)

α2n cos

2 αne−Dα2

nt/r2c

= 1− 6σ

1− σ3

∞∑

n=1

j21(σαn)

sin2 αn

e−Dα2nt/r

2c .

(16)

This is a formula for the ratio of the mass released from the device into the liquid

during the time interval [0, t] and the total mass release from the device in infinite

time. We comment that the deduction of (16) is based on the assumptions that

the device and the container are concentric as depicted in Figure 1 and that the

21


diffusion in the liquid is homogeneous. These assumptions are normally satisfied

in ideal laboratory conditions. When the assumptions are not satisfied, the diffu-

sion problems (1)-(3) can only be solved by a full numerical method which will be

discussed in Chapter 5.

2.1.3 The Initial Burst

As briefly mentioned in Chapter 1, a burst often appears in the initial phase of a

release process. This is because, during the drug load process, some free drugs are

left on the device surface. In this case, the initial release rate is substantially greater

than that during the rest of the process. On the other hand, the initial release rate

may also be much smaller than the normal rate if a device is pre-washed to remove

the free drugs on the device surface. In both cases, it is desirable to identify the

initial burst and its effect on the diffusion process. To characterise the initial burst,

we assume that the effective diffusion coefficient is a piecewise constant in time, i.e.,

D =

D0, 0 < t ≤ tc,

D1, t > tc,

(17)

where D0 and D1 are constants and tc is the threshold time. All of these parameters

are yet to be determined. From the previous section, we see that when 0 < t ≤ tc ,

the concentration C(r, t) is given by (14) with D = D0. Using the same argument

as that for (10) we have

C(r, t) =∞∑

0

Anj0

(αnr

rc

)

e−D1α2nt/r

2c , t > tc, (18)

where An’s are coefficients to be determined. Using the same argument employed

for determining A0 in (10) we have A0 = M0/Vc. The continuity condition at tc

22



that C(r, t−c ) = C(r, t+c ) for all admissible r gives

∞∑

n=0

Anj0

(αnr

rc

)

e−D1α2ntc/r

2c = A0 +

3M0

2πσr3c

×∞∑

n=0

j1(σαn)

αn cos2 αn

j0

(αnr

rc

)

e−D0α2ntc/r

2c .

Matching the coefficients on both sides of the above equality, we have

A0 = A0 =M0

Vcand An =

3M0

2πσr3c

j1(σαn)

αn cos2 αne−(D0−D1)α2

ntc/r2c (19)

for n ≥ 1. Combining this with (18) we have the expression for C(r, t) when t > tc.

It is clear that when 0 < t ≤ tc,Mt

M∞

is given by (16) with D = D0. Using the

same argument as that for (16), it is easy to show from (18) and (19) that

Mt

M∞= 1− 6σ

(1− σ3)

∞∑

n=1

j21(σαn)

sin2 αn

e−α2n(D1(t−tc)c+D0t)/r2c .

for t > tc.

In conclusion, by representing R(t, D0, D1) =Mt

M∞

, the mass ratio for this case

is given by

R(t, D0, D1) =

1− 6σ

1− σ3

∞∑

n=1

j21(σαn)

sin2(αn)e−Dα2

nt/r2c , 0 < t ≤ tc,

1− 6σ

1− σ3

∞∑

n=1

j21(σαn)

sin2(αn)e−α2

n(D1(t−tc)+D0tc)/r2c , t > tc

(20)

for any given tc. Hence we consider the following two different models, which are

the basic model (BM) when tc = 0 and the initial burst model (IB) when tc > 0.

In what follows, we use Re(t) := R(t, D0, D1) to denote an exact (or experimentally

measured) mass ratio for notational clarity.

23

2.2. NUMERICAL RESULTS AND DISCUSSIONS

2.2 Numerical Results and Discussions

2.2.1 Testing the Mathematical Models

In this section, we will test the models established in the previous section using

some experimental data. Three porous PHEMA spherical devices, denoted respec-

tively as S2080, S3070 and S4060 will be tested which were cast following special

specifications. The formulations were selected to produce devices that chemically

identical but structurally different. The final two digits in the devices’ name repre-

sent the percentage of water which is mixed with the remainder percentage of HEMA

monomer in producing the polymeric devices. As an example, S2080 consists 20%

of HEMA and 80% of water. The devices were loaded with drugs with two types

of concentration which are 0.5 and 1.0 wt%. Therefore, in total, six data will be

considered in the numerical tests; S2080-05, S3070-05, S4060-05,S2080-10, S3070-10

and S4060-10 where the suffixes -05 and -10 are to denote the two concentration lev-

els respectively. The details on the manufacturing of the devices and experiments on

the drug release profiles can be found in [95] with detail on the physical properties of

the polymer hydrogels discussed in [60, 59]. Each experimental release profile time

series contains 12 data points, (ti, Re(ti)) for i = 1, 2, ..., 12. Appendix A lists the

time series of all the devices.

The series of solutions obtained contain up to three unknown parameters D0, D1

and tc. To determine these parameters, a nonlinear least-squares algorithm is used.

The algorithm is to minimise the fitting error

E(tc, D0, D1) =K∑

k=1

(Re(tk)− RN(tk, tc, D0, D1))2wk, (21)

where wk is a positive constant and Re(tk) is the experimentally measured value of

the ratio Mt

M∞

(or Mt

M∞

) at tk for k = 1, 2, ..., K. The quantity RN in (21) is the

sum of the first N terms of an exact solution of the ratio. For instance, RN for the

solution in Section 2.1.3 is given by

24



RN (t, tc, D0, D1) =

1− 6σ1−σ3

∑Nn=1

j21(σαn)

sin2 αne−D0α2

nt/r2

2 , 0 < t ≤ tc,

1− 6σ1−σ3

∑Nn=1

j21(σαn)

sin2 αne−α2

n(D1(t−tc)+D0tc)/r2c , t > tc.

For simplicity, we assume that tc only takes values from the discrete set t1, t2, ..., tK .

For all the tests below, we choose N = 62. The first 62 roots of (8) are calculated

numerically using Matlab.

To avoid possible local minima, the least-squares problem is solved using the

following initial starting points

D0 = D1 = 10−5/2i for i = 1, 2, ..., 10

The weights in (21) are chosen to be

wk = K(tk − tk−1)/tK for k = 1, 2, ..., K with t0 = 0.

2.2.2 The Effect of Initial Burst

Table 1: Results from Models BM and IB for all the tested devices

Data Diffusivity (10−6 cm2 s−1) tc (hour) Least-squares error (10−2)BM IB IB BM IB

S2080-05 2.78 (3.25, 2.63) 4.5 0.84 0.76S3070-05 1.87 (2.84, 1.64) 4.5 1.92 1.33S4060-05 1.55 (0.90, 1.69) 4.5 5.51 4.95S2080-10 15.61 (41.27, 3.65 ) 1.0 2.81 0.29S3070-10 2.47 (3.33, 2.20) 4.5 0.77 0.47S4060-10 2.04 (8.21, 1.88) 0.5 1.32 0.76

In order to see the effect of initial burst, the ratio of the mass released Mt/M∞

of all devices, collected at 12 time points from 0.5 hour to 72.8 hour (Appendix

A ) were fitted with both Models BM and IB. The first six time points, (t1 =

0.5 hour, t2 = 1.0 hour, ..., t6 = 4.5 hour) have been tested as the candidates for the

optimal critical time, tc. The fitted curves using the models are displayed in Figure

25


0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

M(t

)/M

(inf)

S2080−10 ExperimentS2080−10 BMS2080−10 IBS4060−10 ExperimentS4060−10 BM S4060−10 IB

Figure 2: Fitted curves by Models BM and IB for S2080-10 and S4060-10

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

M(t

)/M

(inf)

S2080−05 ExperimentS2080−05 IBS3070−05 ExperimentS3070−05 IBS4060−05 ExperimentS4060−05 IB

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

M(t

)/M

(inf)

S2080−10 ExperimentS2080−10 IBS3070−10 ExperimentS3070−10 IBS4060−10 ExperimentS4060−10 IB

(a) (b)

Figure 3: Fitted curves by IB for devices loaded with 0.5 wt% (a) and 1.0 wt% (b)drug solutions

26



2 for experimental data of S2080-10 and S4060-10. The fittings by IB are more

satisfactory than those by Model BM. The more adequate approximation by IB is

also demonstrated by the smaller values of the lease squares errors in comparison

with BM in Table 1. This improvement is most obvious in Data S2080-10 where

the least squares error of IB is ten-fold smaller than BM. In addition, by comparing

D0 and D1 of all the data from IB in Table 1, we could conclude that the initial

burst effects are more apparent in devices with higher concentration drug loading.

We plot the fittings of all experimental data by IB in Figure 3 a and b to show the

accuracy of our estimations.

2.2.3 Determination of the Diffusion Coefficients

Applying the mathematical models to the experimental data of all investigated de-

vices has indicated that IB yields better fitting and approximation results when an

initial burst of drugs occurs. Fittings of all experimental data by this model are

shown in Fig. 3 (a) and (b).

S2080 S3070 S40601.5

2

2.5

3

3.5

4x 10−6

Devices

Diff

usio

n C

oeffi

cien

ts (

cm2 /s

)

0.5 wt% drug solution1.0 wt% drug solution

Figure 4: The optimal effective diffusion coefficients of the devices for drug loading0.5 wt% and 1.0 wt%

27

2.3. CONCLUSIONS

The computed parameters listed in Table 1 demonstrate that the corrected ef-

fective diffusion coefficient, D1, of S2080 is greater than that of S3070, and greater

still than that of S4060 at both drug loading levels. The descending trend of D1

in devices loaded with 1.0 wt% drug solutions is more significant than that in de-

vices loaded with 0.5 wt% drug solutions. Considering that the initial burst effect

is seemed only obvious in 1.0 wt% drug loaded devices, we plot D from BM for

devices with drug loading 0.5 wt% and D1 from IB for devices with drug loading

1.0% in Figure 4 to further illustrate the pattern of the diffusion coefficient towards

the devices’ formulation. These observations are coincide with the fact that S2080

has a more porous structure than the other two devices and are also agreement with

previous study on disc geometry [96]. We have also noticed that the initial burst

effect in S2080 is more significant than the other two devices as shown in Table 1,

indicating that the drugs are more prone to burst from S2080 due to its softer and

more porous nature.

2.3 Conclusions

In this chapter, we have developed a full mathematical model for extracting effec-

tive parameters such as diffusion coefficients and critical time of initial burst that

determine the release process of a drug from a spherical device into a finite vol-

ume. Explicit expressions for the analytical solutions of these models have been

obtained which contain the parameters as unknown decision variables. A nonlinear

least-squares method is then used for finding the optimal solutions to these param-

eters, yielding an optimal fit to a set of experimental data. Numerical experiments

have been performed using laboratory observed data of three drug release devices

made of porous hydrogel polymers with two different drug loading levels to show

the accuracy and usefulness of the models. The results demonstrated that the full

mathematical model can effectively identify the drug initial burst effect, if any, and

therefore can more accurately determine the diffusion parameters that govern a true

28



diffusion process, whilst the basic model is effective only for the uncontaminated ex-

perimental data. The computed diffusion parameters are explicable in terms of the

drug loading concentrations and the porous structure of the devices and are gener-

ally consistent with the results obtained from previous studies on the disc geometry.

Full numerical methods such as those in [96, 103] are presented in Chapter 4 and 5

for estimating effective diffusion parameters of drugs from hydrogel devices of more

general geometries. In the following chapter, a new faster method will be introduced

as an alternative to the nonlinear least squares method used in the current chapter.

29

2.3. CONCLUSIONS

30

Chapter 3

A State Observer Approach for


of Drug Delivery Devices

Given an experimental release profile, an empirical or analytical diffusion model

can be used in conjunction with a nonlinear optimisation algorithm to de-

termine unknown diffusion coefficient. A least-squares method is usually used to

minimise the error between the experimental and the computed drug release profiles

at the given observation time points [60, 59, 96, 95]. This approach involves solv-

ing highly nonlinear least-squares problems with non-trivial constraints and thus

it is normally computationally expensive, particularly for problems with multiple

unknown parameters.

In adaptive control, state observer estimators are often used for ‘on-line’ iden-

tification of unknown parameters in a system in which an estimator is ‘trained’ at

each instant of time by a given input-response time series so that the error in the

estimator converges to zero [21]. Using this idea, Drakunov et al. [28] defined an

observer for estimating the state of a dynamic system when only some outputs of the

system are accessible for measurements. Recently, Temmerman et al. [87] proposed

an estimator to identify the effective diffusion coefficient of a food diffusion process

31

3.1. CONSTRUCTION OF THE ESTIMATORS

and thus to predict the drying curves. Observer-based estimation strategies have

also been proposed for various other problems [12, 36, 77].

In this chapter we present a state observer approach for the estimation of effec-

tive diffusion coefficients of a drug delivery device. In this approach, we construct

estimators for the unknown effective diffusion coefficients characterising 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 in Chapter 2(i.e [95]).

We comment that though we use the diffusion models in Chapter 2 for the rest of

our discussions, the ideas to be presented can also be used for any other empirical

or analytical diffusion models under certain conditions such as the resulting systems

are observable.

The rest of this chapter is organised as follows. In Section 3.1, we derive estima-

tors for the unknown parameters in drug delivery models from the previous chapter

and propose algorithms to numerically evaluate the estimators using a given time

series of a release profile of a device. In Section 3.2, the stability and convergence

of constructed dynamical systems will be discussed. The numerical results using a

set of laboratory observed drug release profile data will be presented in Section 3.3

to show the efficiency and effectiveness of the methods. The improvement of this

new method in term of the CPU time saved in comparison with the least squares

method is also shown in the numerical results.

3.1 Construction of the Estimators

3.1.1 Estimator EI for the Basic Model

In the basic model, there is only one unknown parameter – the effective diffusion

coefficient D. We now construct an estimator for D. An estimator is proposed in

[87], based on the difference between the exact and the estimated total masses at

time t. In this work, we construct an estimator based on the difference between the

integrals of the estimated total masses at time t, as given below.

32

A STATE OBSERVER APPROACH FOR ESTIMATING DIFFUSION

PARAMETERS OF DRUG DELIVERY DEVICES

Given a constant t0 > 0, let D(t) denotes an estimator for D at time t ≥ t0 and

consider the difference between the integrals of the estimated and the exact mass

ratios given by

E(t) =

∫ t

0

e(τ, D(t))dτ (22)

for t ≥ t0, where

e(τ, D(t)) = Re(τ)−R(τ, D(t)) (23)

for any non-negative τ and t. Differentiating E(t) with respect to t gives

E(t) = e(t, D(t))−(∫ t

0

∂R(τ, D(t))

∂Ddτ

)

˙D(t)

= e(t, D(t))−H(t, D(t))˙D(t), (24)

where

H(t, D(s)) =

∫ t

0

∂R(τ, D(s))

∂Ddτ (25)

for any t, s ≥ 0. Clearly, (24) defines a first-order system for E(t). Following [87],

we choose D(t) satisfying

˙D(t) = H−1(t, D(t))× L(E(t))× sgn(E(t)), (26)

where sgn(z) denotes the sign of z and L(E(t)) is a weighting function satisfying

L(z) ≥ 0, ∀z ≥ 0, and L(0) = 0.

Clearly, the choice of L(z) is non-unique. In this work, we chose

L(E(t)) = β|E(t)| (27)

for some constant β > 0. In this case, (26) becomes

˙D(t) = βH−1(t, D(t))|E(t)|sgn(E(t)) = βH−1(t, D(t))E(t). (28)

33


A solution to the system (24) and (28) defines the difference E(t) between the

integrals of the estimated and exact mass ratios and the estimator D(t) of D. We

expect that E(t) → 0 and D(t) → D as t → ∞ as will be discussed later in this

chapter. In the rest of this section, we will present an algorithm for numerically

computing D(t). We start this discussion by evaluating the integrals involved in

(22) and (25).

From (20) we have that, for any t, s ≥ 0,

I(t, D(s)) :=

∫ t

0

R(τ, D(s))dτ

=

∫ t

0

(

1− 6σ

1− σ3

∞∑

n=1

j21(σαn)

sin2(αn)e−D(s)α2

nτ/r2c

)

dτ

=

[

τ +6σr2c

D(s) (1− σ3)

∞∑

n=1

j21(σαn)

α2n sin

2(αn)e−D(s)α2

nτ/r2c

]t

τ=0

= t− 6σr2c

D(s) (1− σ3)

∞∑

n=1

j21(σαn)

α2n sin

2(αn)

(

1− e−D(s)α2nt/r

2c

)

. (29)

Replacing D in (20) for R with D(s) and differentiating the resulting equation

with respect to D gives

∂R(t, D(s))

∂D=

6tσ

r2c (1− σ3)

∞∑

n=1

α2nj

21(σαn)


nt/r2c

for any t, s ≥ 0. Using integration by parts we have from the above

H(t, D(s)) =

∫ t

0

(

6τσ

r2c (1− σ3)

∞∑

n=1

α2nj

21(σαn)


nτ/r2c

)

dτ

=6στ

−D(s) (1− σ3)

∞∑

n=1

j21(σαn)


nt/r2c

∣∣∣

t

τ=0

− 6σr2c

D2(s) (1− σ3)

∞∑

n=1

j21(σαn)

α2n sin

2(αn)e−D(s)α2

nτ/r2c

∣∣∣

t

τ=0

34



=t

D(s)(R(t, D(s))− 1)

+6σr2c

D2(s) (1− σ3)

∞∑

n=1

j21(σαn)

α2n sin

2(αn)

(

1− e−D(s)α2nτ/r

2c

)

=1

D(s)

(

tR(t, D(s))− I(t, D(s)))

, (30)

where I(t, D(s)) is the integral given in (29).

We are now ready to present our algorithm for solving the nonlinear system

(24) and (28) for E(t) and D(t). Note that Re(t) is unknown, but in practice,

approximate values of Re(t) at distinct time points can be obtained by labora-

tory experiments. For a given experimental drug release profile time series (ti, Rie),

i = 1, 2, ..., N , we propose the following algorithm to calculate the successive ap-

proximations Di, i = 1, 2, ..., N + 1 .

Algorithm 3.1.1. 1. Choose positive constants β, D and D satisfying 0 < D <

D, and choose an initial guess D0 ∈ [D,D]. Perform the following steps for

i = 0, 1, ..., N .

(a) Evaluate

Ei =1

2

i∑

k=1

(Rk−1e +Rk

e)(tk − tk−1)− I(ti, Di), (31)

where I(t, D) is defined in (29).

(b) Use (30) to calculate H(ti, Di).

(c) Evaluate˙Di = βEi/H(ti, D

i).

(d) Compute Di+1 using the following forward Euler scheme and the given

lower and upper bounds:

Di+1 = maxmin ˙

Di(ti+1 − ti) + Di, D, D.

2. Set D = DN+1.

35


In Algorithm 3.1.1, the integral∫ ti0Re(τ)dτ is approximated by the trapezoidal

quadrature rule. Hence, we have (31) to approximate (22). To avoid local minima,

we may repeat Algorithm 3.1.1 using a series of initial guesses D0. The upper

and lower bound D and D are used in the algorithm to avoid overshooting of the

numerical approximation Di+1 in the ith iteration. For our case, we choose (D,D) =

(10−8, 10−4), because the effective diffusion coefficients of practical significance are

within this range.

3.1.2 Estimator EII for the Basic Model

A more conventional error indicator (or estimator) is the following cost function

E(t) =

∫ t

0

|e(τ, D(t))|pdτ (32)

for t ≥ t0 > 0, where p ≥ 1 is a constant and e(τ, D(t)) is defined in (23). Clearly,

this cost function is the pth power of the Lp norm of e(τ, D(t)). Although p ≥ 1 is

arbitrary, in the rest of this work, we are only interested in the case that p ∈ 1, 2.

Differentiating E with respect to t gives

E(t) = |e(t, D(t))|p − p

(∫ t

0

sgn(e(τ, D(t)))|e(τ, D(t))|p−1∂R(τ, D(t))

∂Ddτ

)

˙D(t)

=: |e(t, D(t))|p −H(t, D(t))˙D(t), (33)

where

H(t, D(t)) = p

(∫ t

0

sgn(e(τ, D(t)))|e(τ, D(t))|p−1∂R(τ, D(t))

∂Ddτ

)

and sgn(z) denote the sign of z. Following the idea in the previous subsection, we

choose D(t) and L(E(t)) as given in (26) and (27) respectively so that D(t) satisfies

(28) with E(t) defined by (32). Then, (33) and (28) form a nonlinear system for

E(t) and D(t).

36



To numerically compute D(t), we may also use Algorithm 3.1.1. However, in

this case, we are no longer able to work out analytically the integrals in (32) and

(33). Instead, a numerical quadrature rule such as the trapezoidal rule used in (31)

needs to be used for evaluating the integrals numerically.

3.1.3 Extension of the Estimators to the Initial Burst Model

We now consider the extension of the estimators to the IB model. For simplicity,

we assume that tc takes only a finite number of discrete time points (usually, the

first few observation time points in a laboratory experiment) as suggested in [95]

and thus we can find an optimal tc by exhaustive search. Let us first consider the

extension of Estimator EI.

Estimator EI

Let D0(t) and D1(t) be approximations to D0 and D1 at time t respectively. For

any given tc, we define

E(t) =

∣∣∣∣

∫ t∗

0

e(τ, D0(t), D1(t))dτ

︸︷︷︸

E1

∣∣∣∣+

∣∣∣∣

∫ t∗

tc


︸︷︷︸

E2

∣∣∣∣=: |E1(t)|+ |E2(t)|

(34)

for t ≥ t0 > 0, where t∗ = mintc, t, t∗ = maxtc, t and

e(τ, D0(t), D1(t)) = Re(τ)− R(τ, D0(t), D1(t)). (35)

37


Differentiating E with respect to t gives

E(t) = sgn(E1(t))e(t, D0(t), D1(t))1− sgn(t− tc)

2

+sgn(E2(t))e(t, D0(t), D1(t))1 + sgn(t− tc)

2

−[

sgn(E1(t))

∫ t∗

0

∂R(τ, D0(t), D1(t))

∂D0

dτ

+sgn(E2(t))

∫ t∗

tc

∂R(τ, D0(t), D1(t))

∂D0

dτ

]

˙D0(t)

−sgn(E2(t))

(∫ t∗

tc

∂R(τ, D0(t), D1(t))

∂D1

dτ

)

˙D1(t)

=: e(t, D0(t), D1(t))−H0(t, D0(t), D1(t))˙D0(t)

−H1(t, D0(t), D1(t))˙D1(t) (36)

almost everywhere for t ≥ t0 , where

e(t, D0(t), D1(t)) = sgn(E1(t))e(t, D0(t), D1(t))1− sgn(t− tc)

2

+sgn(E2(t))e(t, D0(t), D1(t))1 + sgn(t− tc)

2, (37)

H0(t, D0(t), D1(t)) = sgn(E1(t))

∫ t∗

0

∂R(τ, D0(t), D1(t))

∂D0

dτ

+sgn(E2(t))

∫ t∗

tc

∂R(τ, D0(t), D1(t))

∂D0

dτ,

H1(t, D0(t), D1(t)) = sgn(E2(t))

(∫ t∗

tc

∂R(τ, D0(t), D1(t))

∂D1

dτ

)

.

Similar to the choice (28), we choose

˙D0(t) = βH−1

0 (t, D0(t), D1(t))E(t), (38)

˙D1(t) = βH−1

1 (t, D0(t), D1(t))E(t), (39)

where β is a positive constant.

Clearly, (36), (38) and (39) form a nonlinear ODE system for E(t), D0(t) and

38



D1(t), though the right-hand side of (36) is not continuous. As in Subsection 3.1.1,

some integrals in (34) and (36) can be evaluated exactly. We leave this discussion

to the Appendix D.

Estimator EII

In this case, we define

E(t) =

∫ t

0

∣∣∣e(τ, D0(t), D1(t))

∣∣∣

p

dτ (40)

for a constant p ∈ 1, 2, where e(τ, D0(t), D1(t)) is defined in (35). Differentiating

E(t) with respect to t gives

E(t) = |e(t, D0(t), D1(t))|p

− p

(∫ t

0

sgn(e(τ, D0(t)), D1(t))|e(τ, D0(t), D1(t))|p−1∂R(τ, D0(t), D1(t))

∂D0

dτ

)

︸︷︷︸

H0

× ˙D0(t)

− p

(∫ t

0

sgn(e(τ, D0(t)), D1(t))|e(τ, D0(t), D1(t))|p−1∂R(τ, D0(t), D1(t))

∂D1

dτ

)

︸︷︷︸

H1

× ˙D1(t)

=: |e(t, D0(t), D1(t))|p −H0(t, D(t))˙D0(t)−H1(t, D(t))

˙D1(t) (41)

for t ≥ t0 > 0. We now choose˙D0(t) and

˙D1(t) in the same way as given in (38)

and (39) with E(t) given in (40). Unlike the case of Estimator EI, in this case, the

integrals in (40) and (41) can only be evaluated numerically by a quadrature rule.

For any given initial guess (D00, D

01), we may compute (Di

0, Di1) for i = 1, 2, ..., N+

1 with a slightly modified version of Algorithm 3.1.1 for both EI and EII as shown

follows.

Algorithm 3.1.2. 1. Choose positive constants β, D and D satisfying 0 < D <

D, and choose an initial guess (D00, D

01) ∈ [D,D]. Perform the following steps

39


for i = 0, 1, ..., N .

(a) Evaluate

Ei =1

2

i∑

k=1

(Rk−1e +Rk

e)(tk − tk−1)− I(ti, Di0, D

i1),

where I(t, D0, D1) is defined in Appendix D for Estimator EI. Whereas

full numerical quadrature rule is used in calculating Ei for Estimator EII

as follows

Ei =1

2

i∑

k=1

(Rk−1e +Rk

e)(tk − tk−1)−

1

2

i∑

k=1

(R(tk−1, Di0, D

i1) +R(tk, D

i0, D

i1))(tk − tk−1).

(b) Calculate H0(ti, Di0, D

i1) andH1(ti, D

i0, D

i1) based on the derived formulae

in Appendix D for Estimator EI. For Estimator EII, H0(ti, Di0, D

i1) and

H1(ti, Di0, D

i1) are numerically evaluated by applying numerical quadra-

ture rule to integration terms in (41).

(c) Evaluate˙Di

0 = βEi/H0(ti, Di0, D

i1). Only when ti > tc, evaluate

˙Di

1 =

βEi/H1(ti, Di0, D

i1) .

(d) Compute Di+10 using the following forward Euler scheme and the given

lower and upper bounds:

Di+10 = max

min ˙

Di0(ti+1 − ti) + Di

0, D, D.

(e) Similarly, compute Di+11 using the following formula

Di+11 =

Di+10 , for 0 ≤ ti ≤ tc

maxmin ˙

Di1(ti+1 − ti) + Di

1, D, D

for ti > tc.

40



2. Set (D0, D1) = (DN+10 , DN+1

1 ).

3.2 Convergence of the Methods

We now consider the convergence of the method. In [87], the authors discuss briefly

the convergence of their estimator, which is based on the difference Re(t)−R(t, D(t))

if applied to our case. However, they have not provided a rigorous mathematical

analysis. When their estimator is used for our case, we have difficulties to prove the

convergence for the estimated diffusion coefficient, though the system is asymptoti-

cally stable. One reason for this may be because there are multiple solutions for the

diffusion coefficient. Intuitively, since Re(t) and R(t, D(t)) represent respectively the

experimentally observed and computed mass ratios, Re(t) → 1 and R(t, D(t)) → 1

as t → ∞ for any choice of D > 0. Therefore, it is possible that there are multiple

solutions.

In this work, we shall present a rigorous convergence analysis for our estimators

using the Lyapunov function method which is often used in stability and convergence

analysis [25]. For brevity, we only consider the system formed by (36)–(39). The

discussions for other systems are either identical or very similar to the discussion

for (36)–(39).

We first show that E(t) = 0 as t → ∞ by proving that the feedback control of

the form (26) is a sliding mode control and the system (36)–(39) is in sliding mode

[28, 86]. Substituting the observers in (38) and (39) into (36) we have the following

feedback control system:

E(t) = E(t)− 2βE(t) (42)

for t ≥ t0 > 0, where E(t) = e(t, D0(t), D1(t)). For the solution of this system, we

have the following theorem.

Theorem 3.2.1. When β is sufficiently large, the solution to (42) with any initial

condition E(t0) = E0 6= 0 satisfies limt→∞E(t) = 0.

41

3.2. CONVERGENCE OF THE METHODS

Proof. Consider the following energy function:

V (E(t)) =1

2E2(t),

where E is defined in (34). Differentiating V with respect to t and using (42) we

have

V = E(t)E(t) = E(t)(E(t)− 2βE(t)

)(43)

for sufficiently large t. From (37) and (35) we see that E(t) is a linear combination

of Re(t)− R(t, D0(t), D1(t)). However, using (20) we have

Re(t)−R(t, D0(t), D1(t)) → 0 as t→ ∞.

Therefore, E(t) is bounded for any t ≥ t0.

Now, when E(t) > 0, E(t)− 2βE(t) < 0 if β is sufficiently large. On the other

hand, if E(t) < 0, E(t)− 2βE(t) > 0 when β is sufficiently large. Combining these

we have from (43) that

V (E(t)) < 0 if E 6= 0.

This implies that (42) satisfies the conditions of sliding mode. Thus, V is strictly

monotonically decreasing when t increases and E 6= 0. Using a standard Lyapunov

theorem we have that E(t) → 0 as t→ ∞ (cf., for example, [50]).

We now proceed to show that the estimated parameters D0 and D1 converge to

the exact diffusion coefficients D0 and D1 respectively when t→ ∞. For Estimator

EII, it is intuitively true that that E(t) → 0 implies (D0(t), D1(t)) → (D0, D1) when

t → ∞. This is because in Estimator EII, E(t) defined in (32) or (40) is the pth

power of the Lp-norm of Re(t) − R(t, D0(t), D1(t)). However, this property is not

so obvious for Estimator EI from the definition of E in (22) or (34). Therefore,

the following Theorem 3.2.2 establishes the convergence of (D0(t), D1(t)) for this

estimator, which also covers the convergence of D in BM.

42



Theorem 3.2.2. Let (E(t), D0(t), D1(t)) be the solution to (36)–(39) and D0 and

D1 be the exact diffusion coefficients determining the release mass ratio Re(t). Then,

we have

limt→∞

(D0(t)−D0) = 0 and limt→∞

(D1(t)−D1) = 0.

Proof. For any t > tc, using (20) we have from (34)

E(t) =

∣∣∣∣

∫ tc

0


∣∣∣∣+

∣∣∣∣

∫ t

tc


∣∣∣∣

=

∣∣∣∣∣

∞∑

n=1

Kn

∫ tc

0

(

e−D0α2nτ/r

2c − e−D0(t)α2

nτ/r2c

)

dτ

∣∣∣∣∣

+

∣∣∣∣∣

∞∑

n=1

Kn

∫ t

tc

(

e−α2n(D1(τ−tc)+D0tc)/r2c − e−α2

n(D1(t)(τ−tc)+D0(t)tc)/r2c)

dτ

∣∣∣∣∣

=

∣∣∣∣∣

∞∑

n=1

Kn

(

− r2cα2n

)(

e−D0α2nτ/r

2c − e−D0(t)α2

nτ/r2c

)∣∣∣∣∣

tc

τ=0

+

∣∣∣∣∣

∞∑

n=1

Kn

(

− r2cα2n

)(

e−α2n(D1(τ−tc)+D0tc)/r2c − e−α2

n(D1(t)(τ−tc)+D0(t)tc)/r2c)∣∣∣∣∣

t

τ=tc

=

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

(

1

D0

(1− e−D0α2ntc/r

2c )− 1

D0(t)(1− e−D0(t)α2

ntc/r2c )

)∣∣∣∣∣

+

∣∣∣∣

∞∑

n=1

Knr2cα2n

(1

D1

(

e−α2nD0tc/r2c − e−α2

n[D1(t−tc)+D0tc]/r2c

)

− 1

D1(t)

(

e−α2nD0(t)tc/r2c − e−α2

n[D1(t)(t−tc)+D0(t)tc]/r2c

))∣∣∣∣

=:

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

Pn(t)

∣∣∣∣∣+

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

Qn(t)

∣∣∣∣∣, (44)

where

Kn =6σj21(σαn)

(1− σ3) sin2 αn

and the definitions of Pn(t) and Qn(t) are self-explanatory. Letting t → ∞ and

using Theorem 3.2.1 we have from (44)

limt→∞

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

Pn(t)

∣∣∣∣∣= 0 = lim

t→∞

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

Qn(t)

∣∣∣∣∣. (45)

43

3.2. CONVERGENCE OF THE METHODS

Using the Mean Value Theorem we have from the definition of Pn

Pn(t) =

(

1

D0

(1− e−D0α2ntc/r

2c )− 1

D0(t)(1− e−D0(t)α2

ntc/r2c )

)

=D0(t)−D0

D0D0(t)(1− e−D0α2

ntc/r2c )− 1

D0(t)

(

e−D0α2ntc/r

2c − e−D0(t)α2

ntc/r2c

)

=D0(t)−D0

D0D0(t)(1− e−D0α2

ntc/r2c )− D0 − D0(t)

D0(t)

α2ntcr2c

exp

(

−α2ntcr2c

ξ(t)

)

= (D0(t)−D0)

1− exp

(

−D0α2ntc

r2c

)

D0D0(t)+α2ntc exp

(

−α2ntcr2cξ(t)

)

r2cD0(t)

︸︷︷︸

fn(t)

=: (D0(t)−D0)fn(t),

where ξ(t) is a point between D0 and D(t). Therefore, from (45) we have

limt→∞

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

Pn(t)

∣∣∣∣∣= lim

t→∞|D0(t)−D0|

∣∣∣∣∣

∞∑

n=1

Knr2cα2n

fn(t)

∣∣∣∣∣= 0,

implying limt→∞ |D0(t)−D0| = 0.

Using a similar argument as the above one we can show that limt→∞ |D1(t) −D1| = 0.

The convergence results in Theorems 3.2.1 and 3.2.2 are established for the con-

tinuous system. On the other hand, in computations, we need to discretise the

system using a trapezoidal quadrature rule and Euler’s difference scheme. Hence,

there are two levels of approximation involved. However, since both the trapezoidal

quadrature rule and Euler’s difference scheme are convergent, the solution to the

discretized system is still converges to that of the continuous one which further

converges to the exact parameters as proved in Theorem 3.2.2.

44




In this section, we present some numerical results to demonstrate the usefulness

and efficiency of the estimators using a set of experimental data for three polymeric

devices loaded with 0.5wt% and 1wt% drug solutions. In what follows we use S2080,

S3070 and S4060 to denote the three devices respectively and use suffixes -05 and

-10 to denote the two concentration levels respectively.

We avoid possible local minima by repeating Algorithm 3.1.1 with ten initial

guesses

D0 = 10−5/2i i = 1, 2, ..., 10

for BM, and similarly repeating Algorithm 3.1.2 with the following initial guesses

D00 = D0

1 = 10−5/2i, i = 1, 2, ..., 10

for IB. Note that, in most of the cases, the initial guess for D1 is redefined with a

current value of D0 as suggested in Algorithm 3.1.2. For tc, we assume that it take

discrete values from the set of the first six experimental observation time points in

hours, i.e.,

tc ∈ 0.5, 1.0, 1.5, 2.0, 3.0, 4.5.

The choice of β in Algorithm 3.1.1 is rather empirical. In our numerical experi-

ments, we chose β ∈ [10−6, 5× 10−6]. More specifically, the numerical results given

below were obtained using β = 5 × 10−6 for BM and β = 2.5 × 10−6 for IB. For

brevity, we only consider Estimators EI and EII with p = 1.

3.3.1 Estimator EI

We first consider the numerical performance of EI. Define the discrete relative error

for EI as

relEI =12

∑12k=1

(Rk−1

e +Rke

)(tk − tk−1)− I(t12, D0, D1)

12

∑12k=1 (R

k−1e +Rk

e) (tk − tk−1)

45


with t0 = 0 and R0e = 0, where I(t, D0, D1) is defined in the Appendix D(or in

(29) in the case D0 = D1). The computed effective diffusion coefficients, relative

errors and tc for the data sets using Models BM and IB are listed in Table 2. From

the table it is seen that the relative errors are in the range from 8.6E-4 to 1.3E-2,

indicating that EI gives very good fittings to the data. From the table we also see

that the IB model gives better results than the BM in term of the relative errors

in almost all of the data. The results in Table 2 also show that it is possible that

D0 is smaller than D1, i.e., the initial release rate is lower than the normal rate.

The devices were pre-washed to remove excessive drug on their surfaces and when a

device is over-washed, the initial diffusion process is slower than its normal process.

Table 2: Results from Estimator EI.

Data Diffusivity (×10−6 cm2 s−1) tc (hours) relEI(×10−3)BM IB IB BM IB

S2080-05 2.75 (0.61, 3.48) 2.0 9.03 7.48S3070-05 2.02 (1.21, 2.15) 4.5 10.63 1.99S4060-05 1.50 (0.01, 1.72) 2.0 1.51 1.06S2080-10 7.46 (14.70, 6.84) 0.5 1.00 1.16S3070-10 2.46 (1.90, 2.52) 3.0 6.49 1.43S4060-10 2.18 (0.89, 2.32) 3.0 12.41 0.87

To further demonstrate the usefulness of the method, we plot the fitted release

profiles in Figures 5 and 6 using the optimal values from BM. The experimental

times series are also depicted in the figure for reference. From the figure we see that

all the six curves fit the corresponding experimental data very well, showing that EI

results in accurate approximations to the effective diffusion coefficients.

3.3.2 Estimator EII and Comparisons of EI and EII

We now demonstrate the numerical performance of EII. For brevity, we only test

EII with p = 1, and define the following discrete relative error:

relEII =12

∑12k=1

(|Rk−1

e −R(tk−1, D0, D1)|+ |Rke − R(tk, D0, D1)|

)(tk − tk−1)

12

∑12k=1 (R

k−1e +Rk

e) (tk − tk−1).

46



0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

M(t

)/M

(in

f)

S2080−05 ExperimentS2080−05 BMS3070−05 ExperimentS3070−05 BMS4060−05 ExperimentS4060−05 BM

Figure 5: Predicted drug release profiles for 0.5wt% drug solution using EstimatorEI

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

M(t

)/M

(in

f)

S2080−10 ExperimentS2080−10 BMS3070−10 ExperimentS3070−10 BMS4060−10 ExperimentS4060−10 BM

Figure 6: Predicted drug release profiles for 1.0wt% drug solution using EstimatorEI

47


Table 3 contains the computed results from EII. Most of the computed diffusion

coefficients are similar to those from EI for BM. However the IB model gives com-

puted values slightly different from those using EI. Similar to the case of EI, the

results using IB model are mostly better than those using BM model. To illustrate

the computed results further, we plot the fitted release curves for devices S2080-10

and S4060-10 in Figure 7 using both Models BM and IB. From the figure we see

that the computed results from EII fit the experimental data very well using both

BM and IB.

Table 3: Results from Estimator EII

Data Diffusivity (×10−6 cm2 s−1) tc (hours) relEII(×10−2)BM IB IB BM IB

S2080-05 2.49 (4.48, 2.65) 4.5 4.43 2.62S3070-05 1.65 (19.01, 1.45) 0.5 5.46 3.95S4060-05 1.49 (0.01, 1.47) 1.0 7.68 7.36S2080-10 15.16 (57.58, 4.01) 1.0 3.87 2.05S3070-10 2.47 (2.50, 2.20) 4.5 3.07 2.84S4060-10 1.89 (3.77, 1.74) 3.0 3.65 2.26

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

M(t

)/M

(inf)

S2080−10 ExperimentS2080−10 BMS2080−10 IBS4060−10 ExperimentS4060−10 BMS4060−10 IB

Figure 7: Predicted ratio of the mass release using BM and IB models and EstimatorEII

48



From the computed relative errors in Tables 2 and 3 it would be thought that

EI gives much better results than EII. However, this is not the case as relEII is not

comparable with relEI. To compare the two methods EI and EII, we use following

discrete L2-norm squared:

WLSQ Error =

12∑

k=1

(Rk

e − R(tk, D0, D1))2 (tk − tk−1)

t12.

To compare the performances of EI and EII, we list the computed errors in the

above discrete norm from the methods in Table 4. From the table it is seen that the

computed errors from both EI and EI are comparable, demonstrating that the two

types of estimators are of similar accuracy.

Table 4: Error in the discrete L2-norm squared for Estimators EI and EII

Data WLSQ Error(×10−3)BM IB

EI EII EI EIIS2080-05 0.70 1.01 2.11 1.05S3070-05 1.77 2.05 2.35 2.99S4060-05 4.64 4.68 5.22 5.10S2080-10 4.05 2.35 2.93 0.84S3070-10 0.64 0.64 0.96 0.68S4060-10 1.23 1.26 2.69 0.75

Figure 8 shows the resulted optimal diffusion coefficients using WLSQ, Estimator

EI and Estimator EII. The dotted lines represent results from BM for data with

0.5 wt% drug concentration. The optimal diffusion coefficients from all the three

methods especially EI and WLSQ are very closed with each other. The solid lines

which represent the optimal D1 from IB for data with 1.0wt% drug concentration,

show the similarity for all the results except for device S2080. However the huge

D1 from EI is not fairly compared with the results from the other two methods

because EI estimate shorter initial burst period (tc = 0.5 hour) instead of 1.0 hour

in the other methods. Recall that in designing the estimators, we assume that tc is a

discrete variable. Therefore, there is a possibility the best tc is in between (0.5, 1.0)

49


S2080 S3070 S40601

2

3

4

5

6

7x 10−6

Devices

Diff

usio

n C

oeffi

cien

ts (

cm2 /s

)

WLSQ (0.5wt%)EI (0.5wt%)EII (0.5wt%)WLSQ (1.0 wt%)EI (1.0 wt%)EII (1.0 wt%)

Figure 8: The optimal effective diffusion coefficients of the devices for drug loading0.5wt% and 1.0wt% drug solution using Least-squares method, Estimator EI andEstimator EII

hour for this case. However,although it is possible to construct an estimator for tc

as a continuous unknown variable,this consideration may lead to an additional error

because of the application of interpolation technique. This interpolation technique is

needed to find the exact value, Re(t) at a non-observation time point in evaluating

estimator for the continuous tc. Therefore we conclude that considering tc as a

discrete variable is the best and most practical due to it’s simplicity and ability

to avoid additional error due to the interpolation. Hence the presented results are

optimal with respect to each estimator.

Overall, Figure 8 shows the descending trend of the effective diffusion coefficients

when the percentage of HEMA in the devices formulation is increasing. Similar to

observation from the previous chapter, the descending trend is more significant in

devices loaded with 1.0 wt% drug solutions than the devices loaded with 0.5 wt%

drug solutions.

To further demonstrate the the numerical performances of estimators EI and

EII, we list the CPU time in seconds required for computing the effective diffusion

50



Table 5: CPU time (in seconds) used by EI, EII and WLSQ

Data BM IBEI EII WLSQ EI EII WLSQ

S2080-05 0.20 0.79 1.07 2.06 7.65 8.66S3070-05 0.16 0.83 0.95 2.04 7.55 8.10S4060-05 0.16 0.80 0.83 1.98 7.53 8.57S2080-10 0.20 0.83 1.39 2.06 7.69 9.68S3070-10 0.17 0.83 0.96 2.12 7.59 8.56S4060-10 0.19 0.81 0.89 2.01 7.55 7.98Average 0.18 0.82 1.01 2.04 7.59 8.59

coefficients in Table 5. From this table we see that the computational costs required

by EI are much less than those required by EII in all the situations. This is mainly

because in EI, most of the integrals can be evaluated analytically, while in EII,

integrals are evaluated numerically using the trapezoidal quadrature rule. Therefore,

the latter requires more CPU time than the former. For comparison, we also list the

CPU costs in seconds required by the weighted least squares method (WLSQ) used

in [95] in Table 5. It is clear that both EI and EII require less CPU time than the

WLSQ method. In fact, when compared to WLSQ, EI saves more than 80% CPU

time in BM and about 75% in IB model, whereas the CPU time saving for EII is

almost 20% in BM and 12% in IB model.

3.4 Conclusions

We have constructed two estimators, EI and EII, for the estimation of effective

diffusion coefficients in the drug diffusion models developed in [95]. Two types of

models have been considered in this chapter which are the basic model BM with

one unknown parameter and the IB model with three unknown parameters. We

have demonstrated the stability of and convergence of the methods. Numerical

experiments based on a set of laboratory experimental data have been carried out

to show the efficiency and effectiveness of the methods. The numerical results show

that our estimators give accurate estimations for the unknown diffusion coefficients

51

3.4. CONCLUSIONS

and our new estimators require up to 6 times less computational time compared to

the conventional least squares method.

52

Chapter 4

Numerical Methods for


of 2-dimensional Flow-Through

Drug Delivery System

Full numerical methods have been proposed for estimating unknown parameters

in full diffusion and convection-diffusion equations (cf. [101, 94, 57, 15]).

Though the method in [94] can handle irregular geometries of drug delivery devices

and fluid circulation, it is designed for drug delivery systems without source and

sink conditions. On the other hand, it has been pointed out that a system with a

continuous fluid flowing through it is much closer to in vivo conditions than one with

a circulatory fluid [31, 92]. In this chapter, we will develop mathematical models

for such a drug release system used in [88] and numerical methods for solving the

models.

We present a numerical approach to the estimation of effective diffusion coef-

ficients of drug diffusion from a device into a container with a source and sink

condition due to a fluid flowing through the system at a constant rate. In this

approach we first formulate an initial boundary value problem to represent drug

53

4.1. FORMULATION OF THE PROBLEM

diffusion in a such system. An approximation technique is proposed for the eval-

uation of the concentration function containing a number of unknown parameters.

Having solved this system, we devise the estimation of the unknown parameters as

a finite-dimensional nonlinear least-squares problem.

This chapter is organised as follows. In the next section, we will first describe

the drug release system motivated by the experimental setting for ophthalmic drug

delivery simulation [88]. We then develop a system of equations representing the

drug delivery in a flow-through system for both with and without initial burst ef-

fects. In Section 4.2, we will propose some discretization schemes for the continuous

diffusion problem to form a problem in finite dimensions. Based on this finite di-

mensional problem, we propose, in Section 4.3, a nonlinear least-squares problem

for the unknown parameters. In Section 4.4, we will present some numerical results

using experimental data in [88] to demonstrate the usefulness and accuracy of our

approach.

4.1 Formulation of the Problem

In the experimental setting used in [88], a drug device was placed in a container filled

with liquid. This setting is called a diffusion cell. An inlet and an outlet connection

attached to this diffusion cell to allow fluid flowing through the system. Fresh fluid

from a reservoir was pumped into the diffusion cell at a constant rate q, and the

mixed fluid in the diffusion cell was pumped out at the same rate q. The inlet and

outlet were positioned far apart and the liquid in the container is well-stirred using

a magnetic stirrer.

Mathematically, let us consider a drug release device of geometry Ωd loaded with

a drug. This device is placed in a container of geometry Ωc filled with liquid (water)

as depicted in Figure 9. This is a 2D analogue of the aforementioned experimental

setting. As mentioned before, the liquid is stirred, and hence we assumed that it is

convection-dominant in part of the region Ωc \ Ωd. In this case, there exists a sub-

region region of Ωc \ Ωd, denoted as Ωl \ Ωd, around the device, in which the mass

54

NUMERICAL METHODS FOR ESTIMATING DIFFUSION PARAMETERS OF

2-DIMENSIONAL FLOW-THROUGH DRUG DELIVERY SYSTEM

transfer is dominated by the diffusion process. We refer this region as a diffusion

layer. The mass transfer is convection-dominant outside this layer, and thus the drug

concentration in region Ωc \Ωl is almost uniform. Colour indicator was used in [88]

to show that the system is well-stirred. However, as we stated in the earlier chapter,

well-stirred is impossible in practice due to the no-slip condition. Therefore, instead

of consider the whole system is well-stirred, we consider the existence of the diffusion

layer, Ωl \ Ωd although it is very thin in this case. In addition, we assume that there

is an exchange of fresh fluid and fluid in the container due to a flow through the

diffusion cell at a constant rate (a source-sink condition). This exchange causes a

mass loss, but there is no net fluid loss from the system. For simplicity, we assume

the mass loss due to the sink condition only occurs in the uniform concentration

region, Ωc \ Ωl.

Figure 9: Geometries of the container, the device and the diffusion layer

Let C(x, t) denote the concentration of the drug at any point x ∈ Ωc and time

t. Then, mathematically, C should satisfies the the following convection-diffusion

equation:∂C (x, t)

∂t−D∇2C (x, t) + a · ∇C (x, t) = f(t) (46)

55

4.1. FORMULATION OF THE PROBLEM

where a is the velocity vector of the fluid in Ωc, D is the diffusion coefficient, and

f(t) represents the sink term.

Note that a is not known exactly. However, in the diffusion layer, Ωl \ Ωd, the

magnitude of a is very small relative to the diffusion coefficient, because we have

assumed that the mass transfer process is diffusion-dominant in Ωl \ Ωd. Therefore,

C in this layer can be determined by the following diffusion equation

∂C (x, t)

∂t−D∇2C (x, t) = 0 x ∈ Ωl \ Ωd. (47)

On the other hand, we have |a| ≫ D in the region outside the diffusion layer.

Therefore, dividing both sides of (46) by |a| we have

1

|a|∂C (x, t)

∂t− D

|a|∇2C (x, t) +∇C (x, t) =

f(t)

|a| .

From this we have

∇C (x, t) ≈ 1

|a|

[

f(t)− ∂C (x, t)

∂t

]

.

In the region Ωc \ Ωl we assume the liquid is well-mixed so that C is uniform, or

∇C (x, t) = 0. Hence, from the above equation we have Ωc \ Ωl is

C(x, t) = C(t), anddC(t)

dt= f(t) (48)

for x ∈ Ωc\Ωl and t > 0, where C(t) is the unknown concentration to be determined.

The source-sink term f(t) is a combination of concentration increasing rate due

to the drug diffusion and the concentration decreasing rate due to the pumping. Let

P (t) denote the concentration increasing rate and we can write f as

f(t) = P (t)− qC(t)

Vc − Vl(49)

where q is the constant pumping rate and Vc and Vl are respectively the areas of Ωc

and Ωl. The last term in (49) represents the mass loss rate per unit area at time t.

56



Combining (47), (48) and (49), and considering the initial burst effect, we formu-

late the flow-through system as the following initial and boundary value problem:

∂C (x, t)

∂t−Di∇2C (x, t) = 0, x ∈ Ωl,

i · tc < t ≤ (1− i)tc + i · T, i = 0, 1 (50a)

C (x, 0) =

M0

Vd, x ∈ Ωd,

0, x ∈ Ωc \ Ωd,

(50b)

C (x, t) = C(t), x ∈ Ωc \ Ωl, (50c)

dC (t)

dt= P (t)− qC(t)

Vc − Vl, x ∈ Ωc \ Ωl. (50d)

In this work we assume that q(t) ≥ 0 is either constant or piecewise constant. In

what follows, we refer the mathematical model represented by (50) to as the basic

model (BM) when tc = 0 with D = D0 is the single diffusion coefficient and as the

initial burst model (IB) otherwise with D0 and D1 as defined previously in (17).

The BM has one unknown diffusion coefficient D and IB contains three unknown

parameters, D0, D1 and the critical time tc.

Given a desired or experimental total mass as function of time in the liquid,

we define an infinite-dimensional least-squares problem to determine D in BM or

(D0, D1, tc) in IB. However, since (50) need to be discretized in practical computa-

tions, we will omit this discussion and introduce a least-squares problem in finite

dimensions for determining the unknown parameters later in this chapter.

4.2 Discretisation

Before discussing how to determine the unknown parameters in (50), we first dis-

cretise the PDE system (50) using a finite volume scheme in space and a finite

difference scheme in time. Since the diffusion coefficient D is normally very small in

57

4.2. DISCRETISATION

magnitude,the PDE systems are singularly perturbed. Hence, the use of finite ele-

ment method often yields solutions with large errors [94]. This is the reason for the

finite volume method to be used for this diffusion problem. A detailed description

of the finite volume method can be found in [64]. Below we give a brief account of

this method applied to the system (50).

Let the solution domain Ωc be partitioned into a mesh S consisting triangles

and rectangles with N vertices and M edges. The set of of the vertices and edges

of S are denoted by X = xiN1 and E = eiM1 respectively. We assume that S

is a Delaunay mesh, that is, the mesh satisfies that the interior of the circumcircle

of each element in S contains no vertices of X [22]. Without loss of generality, we

assume that the the mesh nodes in X are numbered in such a way that xiNd

1 is

the set of nodes in Ωd, xiNl

1 the set of nodes in Ωl, xiNc

i=Nl+1 are in Ωc \ Ωl and

xiNi=Nc+1 are on the boundary ∂Ωc of Ωc. Note that the union of all the elements

of this partition form a computational domain, denoted as Ωhc , approximating Ωc

due to the approximation of the curved boundary by edges.

Dual to this Delaunay mesh S, we define a second mesh by connecting the

circumcenters of the two elements in S sharing an edge in E. The dual mesh is

consists of polygons called Voronoi polygons (or Dirichlet tessellations) [22]. In

general, for each mesh node xi, we may define the Dirichlet tessellation as follows

di = x ∈ Ωhc : |x− xi| < |x− xj |,xj ∈ X, , j 6= i.

Figure 10 is an example of the local structure of a Delaunay mesh and Voronoi

polygons associated with a node xi in which ei,j denotes the edge connecting xi and

its neighbour xj , di denotes the Voronoi polygon associated with xi, and li,j denotes

the segment of ∂di connecting the circumcenters of the two elements sharing ei,j.

For each i = 1, 2, ..., Nl, integrating (50a) over di gives

∫

di

[∂C (x, t)

∂t−D∇2C (x, t)

]

dΩ = 0.

58



Figure 10: An example of the Delaunay mesh (solid line) and the correspondingVoronoi polygons (dotted line) with notation for edges and nodes

Applying the one-point quadrature rule to the first term and integrating the second

term by parts in the above expression, we have

∂Ci

∂t|di| −

∫

∂di

D∇C · nds = 0 (51)

where Ci(t) denotes an approximation of C(xi, t), |di| is the area of di and n is the

unit vector normal to ∂di. Let Ii denote the index set of all the neighbouring nodes

of xi. Then, for each i, ∂di consists of a finite number of segments li,j, j ∈ Ii. The

line integral in (51) can be approximated by the following finite differences

∫

∂di

D∇C · nds =∑

j∈Ii

∫

li,j

D∇C · ei,jds

≈∑

j∈Ii

D(Cj − Ci)

|ei,j||li,j|

where |li,j| and |ei,j| denote the lengths of li,j and ei,j respectively and ei,j denotes

the unit vector from xi to xj. Replacing the line integral in (51) by the above

approximation and dividing the resulting equation by |di| , we have

∂Ci

∂t+∑

j∈Ii

D|li,j||ei,j||di|

(Ci − Cj) = 0 (52)

59

4.2. DISCRETISATION

for i = 1, 2, ..., Nl. This is the semi-discretised form of the (50a).

4.2.1 Time Discretisation

We now consider the time discretisation of (67). Let tk, k = 0, 1, ..., K be a set of

points in [0, T ] satisfying 0 = t0 < t1 < · · · < tK = T . Applying the backward Euler

scheme on this mesh to (67) gives

Cki − Ck−1

i

tk+∑

j∈Ii

D|li,j||ei,j||di|

(Cki − Ck

j ) = 0

for k = 1, 2, ..., K and i = 1, 2, ..., Nl, where tk = tk − tk−1 and Cki is an unknown

approximation to Ci(tk) with C0i defined by the given initial condition. Rearranging

the above equation, we have

Cki +tk

∑

j∈Ii

D|li,j||ei,j||di|

(Cki − Ck

j ) = Ck−1i (53)

for k = 1, 2, ..., K and i = 1, 2, ..., Nl.

Equation (53) is a linear system for Cki for i = 1, 2, ..., Nl and a fixed k. To solve

this system, it is necessary to define boundary conditions. From (50c) we see that

Cki = C(tk) on or outside ∂Ωl (or for i > Nl). Therefore, (53) can be written in the

following matrix form:

(tkB + I)Ck = Ck−1 +tkbk (54)

for k = 1, 2, ..., K, where Ck =

(Ck

1 , Ck2 , ..., C

kNl

)T, B is an Nl × Nl matrix cor-

responding to second term in (53), I is the Nl × Nl identity matrix and bk is an

Nl × 1 matrix representing the boundary condition in terms of C in (50c) and (50d)

which will be discussed in detail below. For the above system, we have the following

theorem.

Theorem 4.2.1. The system matrix of (54) is an M-matrix.

60



The proof of this theorem can be found in [65]. Because of the above theorem,

(54) is uniquely solvable for any k and the discretisation satisfies a discrete maximum

principle.

4.2.2 Determination of boundary conditions for (54)

When we apply the forward Euler’s finite difference scheme to (50d) on the mesh

tk, k = 0, 1, ..., K defined in the previous subsection, we have

Ck − Ck−1

tk= P k−1 − qCk−1

Vc − Vl

for k = 1, 2, ..., K. Recall Vc − Vl is the area of Ωc \ Ωl. Using the partition dual to

S define before, we define the following approximation

Vc − Vl ≈N∑

i=Nl+1

|di|.

Combining the above two approximations, we define the following scheme for (50d):

Ck = Ck−1 +tkP k−1 − qtkCk−1

∑Ni=Nl+1 |di|

(55)

for k = 1, 2, ..., K − 1 with the initial condition, C0 = 0.

Now it remains to define an approximation to P k−1. Note P (t) is the rate of

drug concentration increase due to the flux across the the boundary ∂Ωl of the

diffusion layer. This rate is unknown exactly. However, the term ∆tkPk−1 in the

RHS of (55) represents the concentration increase in Ωc \ Ωl from tk−1 to tk, and

so Ck−1 +tkP k−1 is the concentration in Ωc \ Ωl at tk without counting the mass

loss in the period (tk−1, tk) due to the sink condition. In [94], such a problem (drug

diffusion without sink condition) has been solved in 2-dimensions. In the paper,

the uniform concentration in region Ωc \ Ωl was approximated by averaging the

concentration from the previous time step over the region. Based on this idea, we

may define the concentration by solving system (54) for i = 1, 2, ..., Nc (i.e., in the

61

4.2. DISCRETISATION

region Ωc), instead of i = 1, 2, ..., Nl only. This leads to the following linear system

(

tkB+ I)

Ck= C

k−1+tkb

k(56)

for k = 1, 2, ..., K, where Ck=(Ck

1 , Ck2 , ..., C

kNc

)T, B is an Nc × Nc matrix

corresponding to second term in (53), I is the Nc ×Nc identity matrix and bkis an

Nc × 1 matrix representing the boundary condition in terms of ˆCk on the boundary

∂Ωc.

Using the solution to the above system, we define the approximation as follows

Ck−1 +tkP k−1 = ˆCk =

∑Ni=Nl+1C

k−1i |di|

∑Ni=Nl+1 |di|

.

Substituting this representation into (55), we finally have

Ck =

∑Ni=Nl+1C

k−1i |di| − qtkCk−1

∑Ni=Nl+1 |di|

. (57)

The above process is summarised in the following algorithm:

Algorithm 4.2.1. For k = 1, 2, ..., K,

1. Evaluate Ck using (57) with C0 defined by the given initial condition.

2. solve the following matrix system

(

tkB + I)

Ck= C

k−1+tkb

k

for Ck=((Ck)T , Ck

Nl+1, ..., CkNc

)Twith the boundary condition Ck

i = Ck for

i = Nc, Nc + 1, ..., N , where B is an Nc × Nc matrix corresponding to the

second term in (53) for i = 1, 2, ..., Nc, Ck−1

=((Ck−1)T , Ck, ..., Ck

)T, I is the

Nc×Nc identity matrix and bkis an Nc×1 matrix representing the boundary

condition in terms of Ck on the boundary ∂Ωc.

62



With this discretisation, we are ready to pose an optimisation problem for de-

termining the unknown parameters in (50) in the following section.

4.3 Estimation of the Unknown Parameters

Let u denote the vector of unknown parameters in the two models, i.e., u = (D)

for BM and u = (D0, D1, tc)T for IB. We let Np denote the number of components

in u. To determine u, a desired or experimental release profile needs to be given.

In what follows, we let

(M1e ,M

2e , ...,M

Ke

e )T (58)

be a given time series of total mass in the liquid region Ωc \ Ωd obtained by manip-

ulating the concentration value at the observation time points tek, k = 1, 2, ..., Ke by

a laboratory experiment, where Ke is an integer.

Without loss of generality, we assume that the experimental observation time

points tekKe

k=1 is a subsection of the set of mesh points tkK1 defined in the previous

section. At time tek, using the meshes defined in the previous section we approximate

the mass in the liquid region as follows

Mk (u) :=N∑

Nd+1

Cki (u)|di| ≈

∫

Ωc\Ωd

C (x, tek) dx (59)

where Ck(u) is the solution from Algorithm 1 for a fixed u. To estimate u we

consider the following weighted least-squares problem:

minumin≤u≤umax

E (u) =Ke∑

k=1

[Mk

e −Mk(u)]2wk (60)

subject to

Eq.(59) with Ck(u) defined by Algorithm 1,

where wk’s are weights and umin and umax are given lower and upper bounds on u.

63

4.3. ESTIMATION OF THE UNKNOWN PARAMETERS

This problem will be referred to as WLSQ. We may rewrite the cost function in

(60) in the following matrix form

E(u) = (M(u)−Me)TW (M(u)−Me) (61)

where

Me =(M1

e ,M2e , ...,M

Ke

e

)T,

M(u) =(M1(u),M2(u), ...,MKe(u)

)T,

W = diag (w1, w2, ..., wKe) .

To determine an update δu for u, we use the following Taylor expansion:

M(u+ δu) = M(u) + J(u)δu+O(‖δu‖)δu, (62)

J(u) := ∂M(u)∂u

is the Ke ×Np Jacobian matrix defined by

Jk,j =∂M k(u)

∂ujk = 1, 2, ..., Ke, j = 1, 2, ..., Np,

where || · || denotes the Euclidean norm and O(‖δu‖) denotes a term which has a

magnitude comparable to ‖δu‖. Omitting the term O(‖δu‖)δu and substituting

the truncated (62) into (61) gives

E(u+ δu) = (M(u)+ J(u)δu−Me)TW (M(u)+ J(u)δu−Me) .

It is trivial to show that the gradient of E with respect to δu is

∇E(u+ δu) = 2J(u)TW (M(u)+ J(u)δu−Me) .

Setting ∇E(u+ δu) = 0 we have

J(u)TWJ(u)δu = J(u)TW (M(u)−Me) .

64



Based on this, we define the following Levenberg-Marquardt correction formula [56,

63]:(J(u)TWJ(u) + λI

)δu = J(u)TW (M(u)−Me) (63)

where λ ≥ 0 is a constant and I the Np ×Np identity matrix.

For any s = 0, 1, ..., let δu denote the current approximation to the solution of

the minimisation problem given above. Then, the next iterate is defined by

us+1 = u

s + αδus

where 0 < α ≤ 1 is a damping parameter and δus solves (63) with u = us.

To conclude this section, we comment in computation, the Jacobian matrix J(u)

can be evaluated using divided differences as discussed in [94].


We have tested our methods using four experimental data sets, denoted respectively

as A, B, C and D, given in Appendix B which is taken from previous work by

[88]. The maximum time for series A and B are 24 hours, while those for C and

D are about 50 hours with the time interval of 3 minutes for all of them. Each of

these series provide an estimation of the mass in (58). In what follows, we will use

the square of the following relative weighted l2-norm of M − M e to measure the

performance of our methods:

||M −M e||2,r =√∑Ke

k=1 (Mke −Mk)2w(k)

∑Ke

k=1(Mke )

2wk

, (64)

where wk’s are the weights in (60). When wk = 1/Ke for k = 1, 2, ..., Ke, (64)

becomes an unweighted relative error of M −M e in the usual Euclidean norm and

the WLSQ in the previous section becomes an unweighted least-squares problem

(uWLSQ). To simplify our computation, we assume that the critical time tc only

65


takes a set of discrete time points, which are

tc = 0.2i hour, i = 1, 2, ..., 10.

Below we will discuss the performances of WLSQ and uWLSQ separately.

4.4.1 Unweighted Least-squares Method

Table 6: Results by the unweighted least-squares method.

Device Diffusivity (×10−6 cm2 s−1) tc (hours) ||M −M e||22,r(×10−2)BM IB IB BM IB

A 3.53 (4.12, 2.89) 2.00 4.04 2.35B 1.61 (1.80, 0.84) 1.20 14.80 9.67C 1.68 (1.73, 1.31) 2.00 2.88 2.23D 1.02 (1.33, 0.67) 0.60 11.20 4.18

Table 6 contains the computed unknown parameters using uWLSQ for the four

experimental data sets. From the table we see that IB gives better fitting than BM

in terms of the relative errors. To visualise the fitting quality, we plot the fitted

curves for A using the two models, along with the experimental data, in Figure

11, showing that the fitting is qualitatively very good. Fitting qualities for other

data sets are very similar to that for A. Though the graph looks pretty good, the

computed relative errors in Table 6 show that the fitting quality may not be that

great. This can be seen from the plots in Figure 12 and Figure 13 in which we re-plot

the curves and data in different time scales. From these figures we see that large

fitting errors occurs in first half an hour when the variations of the data are large.

This is understandable since uWLSQ has uniform weights and large errors only

occur in a small subinterval. To remedy this, weighted least-squares are desirable.

4.4.2 Weighted Least-squares Method

There are several ways to define the weights for least-squares problems. For our case,

we wish to put more weights at data points which have large magnitudes compared

66



0 5 10 15 20 250

0.25

0.5

0.75

1

1.25

1.5

1.75

2x 10−4

Time t in hours

Mt

Experimental Data AuWLSQ BMuWLSQ IB

Figure 11: Curve fitting for data set A using uWLSQ for BM and IB.

0 0.5 1 1.5 20

0.20.40.60.8

11.21.41.61.8

2x 10−4

Time t in hours

Mt


5 10 15 200

0.4

0.8

1.2

1.6

2

2.4x 10−5

Time t in hours

Mt


(a) (b)

0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5x 10−4

Time t in hours

Mt

Experimental Data BuWLSQ BMuWLSQ IB

5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.41.5x 10

−5

Time t in hours

Mt

Experimental Data BuWLSQ BMuWLSQ IB

(c) (d)

Figure 12: Predicted drug release profiles for data between 0 to 2 hours and 2 to 25hours for A and B using uWLSQ method.

67


0 0.5 1 1.5 20

1

2

3

4

5

6

7

8x 10−4

Time t in hours

Mt

Experimental Data CuWLSQ BMuWLSQ IB

10 20 30 400

0.20.40.60.8

11.21.41.61.8

2x 10−4

Time t in hours

Mt

Experimental Data CuWLSQ BMuWLSQ IB

(a) (b)

0 0.5 1 1.5 20

1

2

3

4

5

6

7x 10−4

Time t in hours

Mt

Experimental Data DuWLSQ BMuWLSQ IB

10 20 30 400

0.10.20.30.40.50.60.70.80.9

11.11.2x 10

−4

Time t in hours

Mt

Experimental Data DuWLSQ BMuWLSQ IB

(c) (d)

Figure 13: Predicted drug release profiles for data between 0 to 2 hours and 2 to 50hours for C and D using uWLSQ method.

to the rest. Therefore, we propose the following two weighting sets:

• Weights proportional to the total of mass, Mke :

wk =Mk

e∑Ke

j=1Mje

for k = 1, 2, ..., Ke.

• Weights proportional to the error from the first optimisation iteration of the

unweighted least-squares method

wk =

1Ke, for Iteration 1

(Mke −Mk)

2

∑Kej=1(M

je−Mj)

2 , for Iteration No. ≥ 2

68



for k = 1, 2, ..., Ke and Mk is the resulting mass after the first iteration of

uWLSQ.

The weighted least-squares methods corresponding to these two systems are denoted

as WLSQm and WLSQe. Both of these two weighting sets satisfy that∑Ke

k=1wk = 1.

Table 7: Results by the weighted least-squares method, WLSQe


A 4.40 (4.50, 3.50) 0.60 0.52 0.05B 3.34 (3.48, 1.87) 0.60 2.72 1.43C 1.90 (1.96, 0.97) 1.00 0.72 0.15D 1.68 (1.19, 0.64) 0.40 2.70 0.57

Table 8: Results from the weighted least-squares method, WLSQm


A 4.12 (4.33, 3.19) 0.20 1.69 0.69B 2.14 (2.42, 1.51) 0.20 12.30 9.38C 1.71 (1.61, 1.86) 0.20 2.19 1.79D 1.31 (1.44E, 0.83) 0.40 6.48 3.10

Table 7 and 8 show the results from WLSQm and WLSQe respectively. The

fitting errors are smaller in Table 7 and 8 than those in Table 6, though they may

not absolutely comparable because different weight sets are were used for computing

them. To further see the improvements, we plot fitted curves by uWLSQ, WLSQm

and WLSQe for Devices B and D using BM in Figure 14. From the figure we

see the data in the two hours are better fitted by WLSQm and WLSQe than by

uWLSQ. Figure 15 contains the computed curves by the three methods for Devices

B and D using IB. From this figure, we see that WLSQm produced the best fitting

performance among others.

69


0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−4

Time t in hours

Mt

Experimental Data BuWLSQ BMWLSQe BMWLSQm BM

5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.41.5x 10

−5

Time t in hours

Mt

Experimental Data BuWLSQ BMWLSQe BMWLSQm BM

(a) (b)

0 0.5 1 1.5 20

1

2

3

4

5

6

7x 10−4

Time t in hours

Mt

Experimental Data DuWLSQ BMWLSQe BMWLSQm BM

10 20 30 400

0.10.20.30.40.50.60.70.80.9

11.11.2x 10

−4

Time t in hours

Mt

Experimental Data DuWLSQ BMWLSQe BMWLSQm BM

(c) (d)

Figure 14: The fitted curves by uWLSQ, WLSQm and WLSQe for data sets B andD using BM.

70



0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−4

Time t in hours

Mt

Experimental Data BuWLSQ IBWLSQe IBWLSQm IB

5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.41.5x 10

−5

Time t in hours

Mt

Experimental Data BuWLSQ IBWLSQe IBWLSQm IB

(a) (b)

0 0.5 1 1.5 20

1

2

3

4

5

6

7x 10−4

Time t in hours

Mt

Experimental Data DuWLSQ IBWLSQe IBWLSQm IB

10 20 30 400

0.10.20.30.40.50.60.70.80.9

11.11.2x 10

−4

Time t in hours

Mt

Experimental Data DuWLSQ IBWLSQe IBWLSQm IB

(c) (d)

Figure 15: The fitted curves by uWLSQ, WLSQm and WLSQe for data sets B andD using IB.

71

4.5. CONCLUSIONS

4.5 Conclusions

We have developed two mathematical models for drug delivery in a flow-through sys-

tem and some schemes for the discretisation of these models. Using the discretised

systems, we define a weighted least-squares problem for the determination of the un-

known effective diffusion coefficients and other parameters. Numerical experiments

using laboratory experimental drug release profiles have been carried out and the

numerical results showed that our approach gives accurate and useful estimations

on the unknown parameters.

72

Chapter 5

Numerical Methods for


of 3-dimensional Drug Delivery

Systems

Methods for estimating effective diffusion coefficients can be found in the open

literature which involve optimisation techniques and solution of diffusion

equations. Analytical solutions to diffusion problems are available for some problems

with simple geometries [18]. However, diffusion problems of practical significance

can not usually be solved analytically. Although we have solved analytically the drug

diffusion problem for spherical devices in Chapter 2, the geometry of the container

has been assumed to be spherical and concentric with the device which are not the

real situation in the experiment. Empirical diffusion models have also been proposed

to estimate effective diffusion coefficients, but they do not have solid mathematical

backgrounds. Recently, a full numerical method for estimating effective diffusion

coefficients has been proposed in 2-dimensions [94]. Since real drug release problems

are always 3-dimensional, full numerical methods in 3-dimensions are desirable for

solving realistic problems.

73

In Chapter 2 and 3 we have considered the external liquid as unstirred. The ex-

periment however used a stirrer which resulted the liquid to be partly well-stirred.

Therefore, the region in the external liquid should be divided into a convection dom-

inant region (region with uniform concentration) and a diffusion dominant region

(region around the device which also called a boundary layer) as we did in Chapter

4. However in Chapter 4, since the liquid is known to be well-stirred, we simply

assumed the diffusion layer was very thin. In this chapter, we treat the width of

the diffusion layer as an unknown. As we mentioned in the introduction chapter, we

refer the model that considers this unknown as the boundary layer model (BL). In

this chapter, we apply this model to the drug delivery system discussed in Chapter

2 and 3 which are referred as the rotating fluid system. Whereas, we stick to the

idea that the width of the diffusion layer in the flow-through system discussed in

Chapter 4 is known and very small.

This chapter presents a full numerical technique in 3-dimensions for estimating

effective diffusion coefficients of drug release devices in rotating and flow-through

fluid systems. In Section 5.1, We first distinguish two types of the fluid systems in

drug delivery. The drug release problems have been formulated as a diffusion equa-

tion systems with unknown parameters. Before proceeding to the numerical method,

we first introduce the width of the boundary layer as an unknown in Section 5.2. We

then develop a numerical technique for estimating the unknown parameters based

on a nonlinear least-squares method and finite volume and difference discretisation

schemes for the 3D diffusion equation in Section 5.3. Finally, in Section 5.4 Numeri-

cal experiments have been performed using various laboratory generated time series

of release profiles. The numerical results are presented to show that our method gives

accurate diffusivity estimations for the test problems. For discussion simplicity, we

use cylindrical and the spherical geometries to demonstrate the method. However,

the proposed method works also for problems with other 3-dimensional geometries.

74


3-DIMENSIONAL DRUG DELIVERY SYSTEMS

5.1 The Drug Delivery Systems

Throughout the thesis, we have covered drug delivery in two types of fluid systems.

In the rest of our discussion, we refer the fluid system in Chapter 2 and 3 as a

rotating fluid system and the fluid system in Chapter 4 as a flow-through system.

Details on the systems are given as follow.

5.1.1 Rotating Fluid System

A rotating fluid system is a closed fluid system in which the medium is stirred

with a magnetic stirrer and not replaced by fresh medium. It is sometimes called a

circulatory fluid system. Research on drug release rates in such a system has been

done in [59, 60]. The development of the mathematical models and the estimation

of the unknown parameters for this system can be found in [95, 96, 94, 67].

Figure 16 show the device-container settings of cylindrical and spherical devices

used in the drug release experiments. The diffusion problem in Cartesian coordinates

for a general geometry setting in 3D can be written as

∂C(x, t)

∂t−Di∇2C(x, t) = 0, x ∈ Ωl,

i · tc < t ≤ (1− i)tc + i · T, i = 0, 1 (65a)

C(x, 0) =

M0/Vd, x ∈ Ωd

0, x ∈ Ωc \ Ωd

(65b)

C(x, t) = C(t), x ∈ Ωc \ Ωl t > 0, (65c)

where C(x, t) is the drug concentration, Vd denotes the volume of the device Ωd,

C(t) denote the unknown uniform concentration in the well-mixed region and T is a

large positive number. This model is similar to the one used in [94] for 2D problems.

75

5.2. THE WIDTH OF THE DIFFUSION LAYER

(a) (b)

Figure 16: The geometries of cylindrical (a) and spherical (b) devices and containers.

5.1.2 Flow-through System

A flow-though system is one in which the mixed medium is pumped out of the

system at a constant rate and fresh medium is pumped into it at the same rate.

Therefore, there is a loss of drug mass due to the exchange of mixed and fresh

media. Compared to the rotating fluid system, the flow-through system is better

related to in vivo [31, 92]. Figure 17 depicts a typical flow-through system with a

cylindrical device which was used in [88]. The fluid in this system is well-stirred as

indicated by the colour indicator in the experiment and the mass loss is assumed to

occur only in the uniform concentration region which is outside the diffusion layer.

A 2-dimensional drug diffusion problem of this system has been solved numerically

in the previous chapter. The drug diffusion in the system is determined by the initial

and boundary value problem as derived in (50).

5.2 The Width of the Diffusion Layer

We introduce the boundary layer model (BL) in this section. Consider that a device

of geometry Ωd loaded with a drug is placed in a container of geometry Ωc filled

with a liquid such as water. The drug in the device will diffuse into the liquid. For

76



Figure 17: The flow-through system used in the experiment.

a cylindrical container, Ωc is defined as

Ωc = (r, φ, h) : 0 ≤ r < rc, 0 ≤ φ < 2π, 0 < h < hc.

Whereas we may define Ωd as

Ωd = (r, φ, h) : 0 ≤ r < rd, 0 ≤ φ < 2π, 0 < h < hd

for a cylindrical device and

Ωd = (r, φ, ψ) : 0 ≤ r < rd, 0 ≤ φ < 2π, 0 ≤ ψ ≤ π

for a spherical devices.

In BL, the liquid region is divided into two subregions. The one around the

device, called the diffusion layer, is a diffusion dominated region and the subregion

outside this layer is convection-dominant so that the drug concentration is uniform

in it. If we use Ωl to denote the union of the device and the boundary layer, then the

two subregions are respectively Ωl \ Ωd and Ωc \ Ωl. In this case, we may introduce

one unknown parameter, θ, where 0 < θ ≤ 1 is a parameter that represents the ratio

of the diffusion layer width over the whole external liquid.

77

5.2. THE WIDTH OF THE DIFFUSION LAYER

(a) (b)

Figure 18: The illustration of the cylindrical (a) and spherical (b) boundary layer.

For the device-container setting depicted in Figure 18(a), we introduce θ, hl and

rl satisfying rd < rl < rc and

θ =rl − rdrc − rd

=hl − hdhc − hd

.

Using the cylindrical coordinate system centred at the centre of the bottom surface

of the container (and the device), the domain Ωl in this case is then defined as

Ωl = (r, φ, h) : 0 ≤ r < rd + θ(rc − rd), 0 ≤ φ < 2π, 0 < h < hd + θ(hc − hd).

For the device-container setting depicted in Figure 18(b), we introduce parame-

ters θ and rl satisfying rd < rl < rc and

θ =rl − rdrc − rd

.

Using this θ and the spherical coordinate system centred at the centre of the spherical

device, we define the region Ωl as

Ωl = (r, φ, ψ) : 0 ≤ r < rd + θ(rc − rd), 0 ≤ φ < 2π, 0 ≤ ψ ≤ π ∩ Ωc.

78



5.3 The Numerical Method

In this section we present a discretisation method for the systems given in the

previous section. This discretisation is based on a finite volume scheme in space

and an implicit finite difference scheme in time which was used in Chapter 4 for the

flow-through system and in [94] for the rotating fluid system, both in 2-dimensions.

A similar approach applied in this chapter, with a few modifications to extend the

method to 3-dimensions.

5.3.1 Discretisation

In this subsection we use the finite volume method proposed in [65] for the spatial

discretisation of the diffusion equations in (65) and (50). In what follows, we will use

the cylindrical region depicted in Figures 16 and 17 to demonstrate the finite volume

method. We start this discussion with the partitioning of the solution domain.

Since the solution domain Ωc is cylindrical, we partition it into polygonal prisms

as follows. Without loss of generality, we assume that the cylinder is parametrised

by x2+y2 ≤ r2c and 0 ≤ z ≤ hc. Let (xm, ym), m = 1, 2, ..., Nxy be a set of Nxy nodes

in x2 + y2 ≤ r2c for a positive integer Nxy. Using this set of nodes, we construct a

2D Delaunay triangulation (or mesh) consisting of triangles and rectangles ([23]).

Then, the interval (0, hc) in the z-direction is divided into subintervals with Nz mesh

nodes 0 = z1 < z2 < · · · < zNz= hc. Using the triangulation in the xy-direction

and the partition in the z-direction we define a partition for the cylinder so that it

contains either triangular or rectangular prisms. This mesh contains N = Nxy ×Nz

mesh nodes (xm, ym, zn) for m = 1, 2, ..., Nxy and n = 1, 2, ..., Nz, and is referred to

as a Delaunay mesh for the solution domain Ωc. We re-order these nodes using a

single subscript in such a way that xiNd

1 is the set of nodes in Ωd, xiNl

i=1 are in

the region Ωl, xiNc

i=Nl+1 are in Ωc \ Ωl and xiNi=Nc+1 are on the boundary ∂Ωc of

Ωc. Note that the union of all the elements of this partition form a computational

domain, denoted as Ωhc , approximating Ωc due to the approximation of the curved

boundary by facets.

79

5.3. THE NUMERICAL METHOD

We now define another mesh dual to the Delaunay one defined above. For each

mesh node xi := (xm, ym, zn), we define the Dirichlet tessellation ([26]) (or Voronoi

polyhedron) as follows

di = (x, y, z) ∈ Ωhc : (x− xm)

2 + (y − ym)2 < (x− xp)

2 + (y − yp)2,

zn − (zn+1 − zn)/2 < z < zn + (zn+1 − zn)/2; ∀p = 1, 2, ..., Nxy, p 6= m.

Clearly, di consists of points of Ωhc closer to xi than to other mesh nodes. A typical

local structure of the Delaunay mesh and Dirichlet tessellations/Voronoi polyhedra

is depicted in Figure 19 (a) where solid lines represent the Delaunay mesh and the

dotted lines represent the corresponding Dirichlet tessellation.

(a)

(b) (c).

Figure 19: 3D Delaunay mesh and Dirichlet Tessellation (a), A Voronoi polygon(b), Edge from xi to xj and the corresponding facet li,j (c).

Figure 19 (b) is an example of the Voronoi polyhedron, di, associated with a node

xi with 7 neighbouring nodes xjk , k = 1, 2, ..., 7. Each facet of the boundary ∂di

of a tile di is perpendicular to an edge connecting xi and its neighbour xj . Let ei,j

80



and li,j denote respectively the edge from xi to xj and the facet of di perpendicular

to ei,j, as illustrated in Figure 19 (c).

We follow the similar technique explained in Section 4.2 to discretise the diffusion

equations. For each i = 1, 2, ..., Nl (i.e i : xi ∈ Ωl), we integrate the diffusion

equation in (65a) or (50a) over the Voronoi polygon di

∫

di

[∂C(t) (x, t)

∂t−D∇2C(t) (x, t)

]

dΩ = 0.

Then, by aplying the one-point quadrature rule and integration by parts to the

above equation, we have

∂Ci(t)

∂t|di| −

∫

∂di

D∇C(t) · ndS = 0 (66)

where Ci(t) denotes an approximation of C(xi, t), |di| is the volume of di and n

denotes the unit vector outward-normal to ∂di. Let Ii denotes the index set of all

the neighbouring nodes of xi. For each i, ∂di consists of a finite number of facets

li,j for j ∈ Ii, as depicted in Figure 19(b). Let ei,j denote the unit vector from xi

to xj. From the construction of the two meshes we have that ei,j coincides with the

unit vector outward normal to the facet li,j. Therefore, the surface integral in (66)

can be approximated by the following finite differences

∫

∂di

D∇C(t) · ndS =∑

j∈Ii

∫

li,j

D∇C(t) · ei,jdS

≈∑

j∈Ii

D(Cj(t)− Ci(t))

|ei,j||li,j|,

where |li,j| denotes the area of the facet li,j and |ei,j| denotes the length of edge ei,j.

This approximation is then substituted into the surface integral in (66). We have

the following semi-discretised form of the diffusion equation after dividing both sides

81


of the resulting equation by |di| ,

∂Ci(t)

∂t+∑

j∈Ii

D|li,j||ei,j||di|

(Ci(t)− Cj(t)) = 0 (67)

for i = 1, 2, ..., Nl.

From the semi-discretised form, we proceed to the time discretisation as discussed

in Subsection 4.2.1 using the Backward Euler method. Upon defining boundary

conditions which are different for the flow-through system and the rotating fluid

system, the discretised diffusion system can be written in the matrix form as in

(54). We continue with discussing the Nl × 1 matrix representing the boundary

condition in terms of C, bk, for both fluid systems.

In the formulations of (65c) and (50c), we assumed that the concentration in the

region outside the boundary layers (i.e. in the well-mixed region) is uniform, i.e.,

C(x, t) = C(t). However, C(t) is unknown and needs to be determined numerically.

To achieve this, we refer the definitions of the diffusion layer for spherical and

cylindrical devices as explained in Section 5.2.

The drug concentration outside the boundary layer for the rotating fluid system

(65) is simply defined as the average concentration over the region given by

C(t) =1

|Ωc \ Ωl|

∫

Ωc\Ωl

C(x, t)dΩ, (68)

where |Ωc \ Ωl| denotes the volume of Ωc \ Ωl. The concentration C(t) defines the

boundary condition for (54). Note that C(x, t) is unknown and thus (68) and (54)

are coupled systems for Ck and Ck := C(tk).

Recall the assumption that the mesh nodes are ordered in such a way that

xiNi=Nl+1 is the set of nodes outside Ωl. Thus, to determine Ck and Ck numerically,

we decouple (68) and (54) using the following algorithm:

Algorithm 5.3.1. For k = 1, 2, ..., K,

82



1. evaluate

Ck =

∑Ni=Nl+1C

k−1i |di|

∑Ni=Nl+1 |di|

(69)

with C0 = 0 defined by the initial condition, and

2. solve(

tkB + I)

Ck= C

k−1+tkb

k(70)

for Ck=((Ck)T , Ck

Nl+1, ..., CkNc

)Twith the boundary condition Ck

i = Ck for

i = Nc, Nc + 1, ..., N , where B is an Nc × Nc matrix corresponding to the

second term in (53) for i = 1, 2, ..., Nc, Ck−1

=((Ck−1)T , Ck, ..., Ck

)T, I is the

Nc×Nc identity matrix and bkis an Nc×1 matrix representing the boundary

condition in terms of Ck on the boundary ∂Ωc.

The approximation of C(t) in the region Ωc \Ωl for the flow-through system can

be achieved using a decoupling algorithm similar to Algorithm 5.3.1 with a different

approximation for C, which is Algorithm 4.2.1.

5.3.2 Estimation of the unknown parameters

In this section, we formulate the unknown parameter estimation problem as a least-

squares problem that best approximates the observed experimental data. Once

again, we use u to denote the vector of unknown parameters. Four different choices

of u are u = (D) for BM, u = (D0, D1, tc)T for IB, u = (D0, D1, θ)

T for BL and

u = (D0, D1, tc, θ)T for the combination of IB and BL.

In the previous chapter, we have introduced (58) as the given time series of the

total mass in the liquid region Ωc \ Ωd recorded during a laboratory flow-through

experiment at the pre-defined experimental time points tek, k = 1, 2, ..., Ke with Ke a

positive integer with tekKe

k=1 is a subset of the time mesh points. The value is then

approximated at tek using (59). However, in the rotating fluid system, it is easy to

83


see that the total mass released from the device in infinite time is

M∞ =M0

(Nc∑

i=Nd+1

|di|)

/

(Nc∑

i=1

|di|)

≈ M0

|Ωc|× |Ωc \ Ωd|.

Therefore, the total mass in (59) can be normalised as follows

Rk(u) :=Mk (u)

M∞=

∑Nc

i=Nd+1 Cki (u)|di|

M0(∑Nc

i=Nd+1 |di|)

/(∑Nc

i=1 |di|) ≈

∫

Ωc\ΩdC (x, tek) dx

∫

Ωc\Ωd

M0

|Ωc|dx

.

Similarly, the normalised experimental data are given by

Rke :=

Mke

M∞, k = 1, 2, ..., Ke. (71)

In computation, we may approximate M∞ in (71) by MKee when the last laboratory

observation time point teKeis sufficiently large. Clearly, both Re → 1 and R → 1 as

t approaches infinity.

To determine the unknown u, we post the following weighted least-squares prob-

lem:

minumin≤u≤umax

E (u) =

∑Ke

k=1

[Rk

e − Rk(u)]2wk for the rotating fluid system

∑Ke

k=1

[Mk

e −Mk(u)]2wk for the flow-through system

where Ckused in Rk (respectively in Mk) is determined by (69) and (70) (re-

spectively (57) and (70)), wk’s are weights and umin and umax are given lower and

upper bounds on u. Recent advances in numerical methods for solving nonlinear

least-squares were discussed in [102]. To solve this least-squares problem, we use

the Levenberg-Marquardt method [56, 63]. This method was presented in Section

4.3 for the flow-through system and can be applied to the rotating fluid system by

simply replacing the total mass value with the normalised form of it.

84




In this section we use our method developed in the previous section to estimate

effective diffusion coefficients and other unknown parameters using some laboratory

generated drug diffusion time series . We divide the numerical experiments into three

cases – sphere devices in a rotating fluid system, cylindrical devices in a rotating

fluid system and cylindrical devices in a flow-through fluid system. Numerical results

will be compared with the results from other methods developed by us. For ease

of discussion, we use FULL3D, FULL2D and ANAL to denote, respectively, the

full 3D numerical method developed in this paper, the full 2D numerical method

developed in [94] and the analytical solutions developed in [96] and [95] for simplified

cylindrical and spherical device-container geometries.

In our numerical experiments on FULL3D, we use meshes consisting of triangular

and rectangular prisms with mesh sizes equal to

∆x = ∆y = rc/(4× 2m), ∆z = hc/(4× 2m), ∆t = 40/2n(seconds), (72)

where m and n are non-negative integer, rc and hc denote respectively the radius

and height in centimetres of the cylindrical container. All computations for FULL3D

were carried out in Fortran 77 double precision under Linux environment.

5.4.1 Case I: Spherical Devices in Cylindrical Containers in

the Rotating Fluid System

For this section, we use the same experimental data as in [95] which is Appendix

A. This data set contains experimentally observed time series for 6 different de-

vices denoted respectively by S2080-05, S3070-05, S4060-05, S2080-10, S3070-10

and S4060-10, of which the suffixes ‘-05’ and ‘-10’ represents two different drug load

levels – 5 wt% and 10 wt%. For details of these devices, we refer to [95]. The analyt-

ical solutions developed in Chapter 2 ([95]) and Chapter 3 ([67]) were based on the

assumption that both the device and container are concentric spheres, though the

85


Table 9: Computed diffusion coefficientsD and least-squares error E for Data S2080-05 using various meshes.

m = n 0 1 2 3D (×10−6 cm2 s−1) 6.05 3.90 3.58 3.53

E(×10−2) 4.36 1.56 1.56 1.49

container used in experiments was cylindrical. Using the full numerical method in

3 dimensions (FULL3D) suggested in this chapter, we are able to solve the problem

in its real geometries – a spherical device in a cylindrical container.

To make sure that our method and computer code work properly, we have per-

formed a convergence study for BM (only one unknown parameter D) using the time

series S2080-05. We use ‖δD‖ ≤ 10−8 as the stopping criterion for the Levenberg-

Marquardt method and the initial guess for D is chosen to be D0 = 10−6. The

optimal results for the meshes with m = n = 0, 1, 2, 3 in (72) are shown in Table

9. As can be seen, the computed least-squares errors decreases as the mesh sizes

decrease. To further study the performance of the numerical method, we consider

the convergence of the method in terms of the nonlinear iterations for the mesh

with m = n = 4 in (72). Table 10 shows the convergence history of the update δD

and the effective diffusion coefficient D and Figure 20 displays the convergence of

the pointwise approximation of the release profile for a few iterations. Clearly, both

show that as the number of iterations increases, both δD and the least-squares error

decrease.

Table 10: Computed updates at selected iterations for Data S2080-05 with m = n =3.

Iter 1 Iter 3 Iter 5 Iter 11 OptimalδD (×10−7) - 12.29 7.19 0.14 0.07

D (×10−6 cm2 s−1) 1.00 2.22 3.09 3.53 3.53E (×10−2) 49.25 7.74 2.00 1.49 1.49

In order to check the numerical stability of the developed scheme with respect

to initial guesses, we solve the least-squares problem for data set S2080-05 using

the mesh with m = 2, n = 2 in (72) starting from 3 different initial guesses D0 =

10−6, 5×10−6 and 10−5. The computed optimal effective diffusion coefficients against

86



0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

Time t in hours

Mt/M

∞

Experimental Data S2080−05

Iteration 1

Iteration 2

Iteration 3

Iteration 5

Iteration 11

Figure 20: Computed release profiles after the selected numbers of iterations withm = n = 3.

the number of iterations are plotted in Figure 21, showing that, starting from the 3

initial guesses, the computed diffusion coefficients all converge to roughly the same

point.

We are now ready to present the results from FULL3D and compare them with

those from other methods. All the results computed by FULL3D have been obtained

using the mesh with m = 2 and n = 2 in (72). We first consider the application of

FULL3D to BM. Table 11 contains the computed diffusion coefficients and the least-

squares errors for the 6 experimental times series using FULL3D and the analytical

solutions derived in [95] (ANAL). From the table we see that computed diffusion

coefficients and the least-squares errors from both methods are very similar to each

other, indicating that both methods give consistent results.

Table 12 contains the results using the FULL3D and ANAL for IB in which

the diffusion coefficient is of the form (17) containing three unknown parameters.

In our work, we assume that tc only takes values from the first few experimental

observation time points. Again, the results from both methods are similar. From

the table we see that the least-squares errors are smaller than those listed in Table

11. Also, the initial burst phenomenon is not obvious for devices with suffix ‘-05’,

87


0 1 2 3 4 5 6 7 8 9 10 11 120

0.5

1

1.2x 10−5

Iteration

D (

cm2 s

−1)

D0=1×10−6

D0=5×10−6

D0=1×10−5

Figure 21: Computed diffusion coefficients for Data S2080-05 starting from 3 differ-ent initial guesses on the mesh with m = 2, n = 2.

though there are several noticeable differences in tc for these devices. The initial

burst is very prominent for devices with the suffix ‘-10’ as what we discovered in the

earlier chapters. This is because these devices have an initial drug loading twice as

much as that of the devices with the suffix ‘-05’.

Finally, we list the computed results from the two methods for BL in Table 13

in which θ is the parameter defined in Section 3 characterising the width of the

boundary layer region. Though there are slight differences in the computed θ from

the two methods, the computed D and least-squares errors are very close to each

other.

Table 11: Optimal results from ANAL and the FULL3D for BM in Case I.

D(×10−6 cm2 s−1) Error (×10−2)Data ANAL FULL3D ANAL FULL3DS2080-05 3.08 3.58 1.48 1.56S3070-05 2.21 2.60 4.67 4.05S4060-05 1.58 1.84 3.21 4.17S2080-10 22.02 24.72 10.74 9.19S3070-10 2.91 3.36 1.47 1.18S4060-10 2.73 3.21 3.54 2.99

88



Table 12: Optimal results from ANAL and FULL3D for Model IB in Case I.

(D0, D1)(×10−6 cm2 s−1) tc(s) Error (×10−2)Data ANAL FULL3D ANAL FULL3D ANAL FULL3DS2080-05 (4.00, 2.93) (3.31,3.97 ) 1800 10800 1.38 1.44S3070-05 (2.58, 1.74) (1.57, 3.06) 16200 3600 3.96 3.36S4060-05 (2.22, 1.48) (1.66, 2.02) 1800 16200 2.92 3.97S2080-10 (60.43, 5.78) (68.11, 7.13) 1800 1800 0.90 0.85S3070-10 (5.78, 2.44) (5.87, 2.92) 1800 1800 0.51 0.43S4060-10 (7.88, 1.99) (7.93, 2.43) 1800 1800 0.61 0.56

Table 13: Optimal results from FULL3D for BL in Case I.

D(10−6 cm2 s−1) Ratio of the width layer, θ Error (×10−2)S2080-05 3.48 0.85 1.58S3070-05 2.56 0.84 4.06S4060-05 1.61 0.73 4.76S2080-10 26.22 0.86 9.20S3070-10 3.41 0.88 1.18S4060-10 3.28 0.88 3.00

5.4.2 Case II: Cylindrical Devices in Cylindrical Containers

in the Rotating Fluid System

In [96] and [94] we developed respectively an analytical method (ANAL) and a

full numerical method (FULL2D) for the 2D diffusion estimation problem in which

the device and container are concentric discs. The case considered in [96, 94] is a

simplified version of the one depicted in Figure 16(a). In this subsection, we solve

the full-scale problem in 3D using FULL3D and compare the results with those from

ANAL and FULL2D. For brevity, we will solve the problem using only two sets of

experimental data labelled with A3 and A4. The time series A3 and A4 can be

found in [94].

Computed diffusion coefficients and the least-squares errors for BM are listed

in Table 14 using the three methods. From the table we see that all the methods

have comparable least-squares errors. Unlike the situation for the spherical devices,

the estimated diffusion coefficients from FULL3D are smaller than the those from

the other two methods. We believe that the results from FULL3D is more accurate

89


because in the 3D formulation (65), as well as in practice, the diffusion occurs in the

outward normal directions of both the side and top surfaces of the cylinder, while

in the 2D models it occurs only in the outward normal direction of the side surface

direction. This shows that the 2D methods over-estimate the effective diffusion

coefficients and thus the present 3D method is important for accurately estimating

the effective diffusion coefficients for cylindrical devices.

Table 14: Optimal results from ANAL, FULL2D and FULL3D for Model BM inCase II.

D (×10−6 cm2 s−1) Error (×10−2)Data ANAL FULL2D FULL3D ANAL FULL2D FULL3DA3 2.49 2.41 1.13 4.11 2.95 3.00A4 9.01 10.09 4.47 1.30 1.32 1.36

To further show the usefulness of FULL3D, we solve the diffusion estimation

problem in all the three models by FULL3D using the time series A4 and record the

results in Table 15. Comparing these results with those in [94, Table 5] we see that

the 2-dimensional numerical method FULL2D over-estimates the effective diffusion

coefficients.

Table 15: Optimal results from FULL3D for BM, IB and BL using Data A4 in CaseII.

(D0, D1)(×10−6 cm2 s−1)Ratio of thewidth layer,θ

tc hour Error (×10−2)

BM 4.47 - - 1.36IB (3.40, 4.71) - 0.75 1.29BL 4.48 0.8949 - 1.36

5.4.3 Case III: Cylindrical Devices in Cylindrical Contain-

ers in the Flow-through Fluid System

We now demonstrate the accuracy and usefulness of FULL3D for the flow-through

fluid system (50). The experimental time series for two different devices, denoted

90



as A and B respectively, are taken from [88] which is in Appendix B. The problems

have also been solved previously using FULL2D and the results from both methods

are listed in Table 16. Similar to Case II, the computed diffusion coefficients from

FULL3D are smaller than those from FULL2D because of the reason mentioned in

Case II above.

Table 16: Optimal results from FULL2D and the FULL3D for Model BM in CaseIII.

D (×10−6 cm2 s−1) Error (×10−2)DATA FULL2D FULL3D FULL2D FULL3D

A 4.28 1.44 5.09 8.54B 2.02 0.757 2.62 3.97

5.5 Conclusions

We have developed a 3-dimensional numerical scheme to estimate the unknown

parameters in drug delivery problems. Two fluid systems have been considered

which are the rotating fluid system and the flow-through system. We have tested

the numerical method using sphere– and cylinder–shaped devices for the rotating

fluid system, and only cylinder–shaped devices for the flow-through system. Re-

sults from the new 3-dimensional numerical scheme was then compared with the

previous available analytical and 2–dimensional numerical solutions. Comparison

between the analytical solution and the 3–dimensional numerical solution for the

sphere device shows that both methods agree with each other. This not only proves

that the analytical method is correct, but also means that the 3-dimensional nu-

merical method is working well. In this case, only the geometry of the container

is considered differently, the geometry of the device is however identical in both

solutions. This is the reason why both methods give the similar solution. On the

contrary, the cylinder devices and containers were considered as discs in the pre-

vious solutions. Therefore, in both fluid systems, the resulted diffusion coefficients

from the 3–dimensional numerical methods are relatively smaller compared to the

91

5.5. CONCLUSIONS

results from the previous disc’s analytical solution and the 2-dimensional numerical

method. This is due to the different consideration in defining the device’s geometry.

The 3-dimensional numerical method has the ability to solve drug delivery for ir-

regular devices’ geometry. In this way, more drug delivery problem could be solved

and extended to more complicated problems. In addition, the accuracy could be

improved especially for the estimation of the boundary layer by refining the mesh.

The only constraint for this implementation is the expensive computational cost

which the improvement will be another challenge. Alternatively, the experiment set

up might be adjusted to suit the 2–dimensional method or the analytical solution.

92

Chapter 6

Conclusions and Future Works

6.1 Conclusions

We have developed analytical and numerical methods for the estimation of effective

diffusion coefficients and other parameters determining the diffusion processes from

polymeric devices into external volumes. Numerical experiments have been carried

out using laboratory generated drug release times series to test the performances of

these methods and the numerical results presented demonstrate that the methods

are accurate and practically useful.

We started our research by deriving an analytical solution to the drug delivery

from a spherical device into a finite external volume. The diffusion problem was first

written as a diffusion equation in spherical coordinates with appropriate boundary

and initial conditions. The analytical solution of such diffusion problem is hard

to derive. In this work, we have assumed that both the device and container are

concentric spheres. Although analytical solutions for some diffusion problems can

be found in [18], these solutions were based on the assumption that the liquid in the

container is well-stirred which is practically impossible due to the no-slip condition

in fluid mechanics. On the other hand, in our solution, we considered the fluid of

the diffusion system as unstirred. The technique of separation of variables was then

used to solve the diffusion equation, resulting in a series solution to the diffusion

93

6.1. CONCLUSIONS

equation. An expression for the total mass released from a device into an external

volume for a time interval [0, t] was obtained by integrating the concentration over

the region outside the device. Using this expression, we have derived the formula for

the ratio of the mass released by dividing the expression by the total mass released

in the time interval [0,∞]. This ratio defines the release profile of a drug delivery

system.

Beside the basic model that has only one unknown diffusion coefficient, we have

also analytically solved the initial burst model containing two diffusion coefficients

and an unknown critical time point. In the initial burst model, each diffusion coef-

ficient represents a phase of the diffusion process. The total diffusion process was

divided into two phases in order to handle initial burst effect that happens in prac-

tice. Usually, the initial release rate is expected to be higher than the rest of the

process because of reasons such as the presence of excessive drug on a device’s sur-

face. However, slower initial release rate could also happen due to an over-treated

procedure prior to a release experiment. The time that separates the two phases

was considered as another unknown parameter that needs to be determined together

with the diffusion coefficients. This model usually gives better results than those

from the basic model.

To determine the unknown parameters numerically, a nonlinear least-squares

technique is used in conjunction with the developed analytical models. Our numeri-

cal experiments have shown that this approach together with our analytical solutions

gives very good fittings for the release profiles and thus provides accurate estima-

tions for the unknown parameters. However, the nonlinear least-squares method is

usually CPU-time consuming, especially for problems with more than one unknown

parameter. To remedy this, we took an initiative to develop a new approach to

reducing the CPU time consumption in estimating the unknown parameters using

the developed models. This approach is based on the idea of ’on-line training’ in

adaptive control.

State observer estimators are used in adaptive control for ‘on-line’ identification

94

CONCLUSIONS AND FUTURE WORKS

of unknown parameters in a system. In this work, we developed two types of estima-

tors and used them in conjunction with the analytical solutions we derived earlier for

estimating the unknown parameters. Using the Lyapunov theorem, we were able to

prove the stability of our estimators. In addition, we have established a convergence

theorem showing that as the time increases, our estimated diffusion coefficients con-

verge to the exact ones. Our numerical experiments have shown that our observer

approach method not only gives accurate estimations for the unknown diffusion co-

efficients, but also requires much less CPU time than the nonlinear least-squares

method.

We then a numerical method for estimating the unknown parameter in a more

complex drug delivery system, called a flow-through system. In the system, called a

rotating system, we considered earlier in this thesis, the fluid is not replaced by any

fresh fluid and thus there is no drug mass loss in the process. However, in a flow-

through system, the fluid is continuously replaced by fresh fluid, providing a sink

condition in the system. This is done by pumping in fresh fluid into and pumping

the mixed fluid out of the system at the same rate. While the volume of liquid in

the system remains unchanged, there is mass loss in the system that needs to be

considered. This system is better related to in vivo compared to the rotating fluid

system, thus important to be considered in effort to solve more real problem. In

fact, our mathematical model for drug delivery in this system was developed based

on an experimental setting in [88] which was prepared to simulate conditions in eyes.

In the experiment, a colour indicator was used to show the uniform concentration

of the external fluid. However, although the system was visibly well-stirred, we

assume that there is a thin layer (i.e. diffusion layer) around the device where the

concentration is not uniform due to the no-slip condition. Therefore we divided

the region into two sub-regions: a diffusion dominant region (i.e. diffusion layer)

around the device and a convection dominant region outside the diffusion layer. In

this work, we developed a full numerical method for the problem in 2-dimensions

for simplicity. Numerical determination of the boundary conditions was proposed

95

6.2. FUTURE WORK

taking into consideration of the mass loss and stirring in the system. The numerical

scheme was based on a finite volume method for the spatial discretisation and the

backward Euler method for time discretisation. The system matrix of the dscretized

system from the method is an M-matrix. The unknown parameters in the system

were then estimated using a weighted least-squares method with three different

weights. Numerical experiments have been performed to demonstrate the accuracy

and usefulness of this method.

Finally, we extended the full 2D numerical scheme to 3-dimensions. Recall that

we introduced a diffusion layer and convection dominant layer in the numerical solu-

tions and apply the full 3D numerical method to the unknown parameter estimation

problem with a rotating fluid system. In this development, we assume the width

of the diffusion layer is also an unknown parameter be determined. Therefore, the

full numerical technique can be used for solving not only the basic and initial burst

models considered earlier, but also the boundary layer model. This full numerical

technique in 3-dimensions is able to solve the problems in their real 3D geometries.

Numerical results were presented to show the accuracy of the full 3D method.

To summarize, we have developed various analytical and numerical methods

for in both 2D and 3D for estimating unknown effective diffusion coefficients and

other parameters determining the diffusion process of a drug delivery device. These

numerical methods provide accurate and efficient numerical tools for designing drug

delivery devices with desired release rates.

6.2 Future Work

The current study has been focused on the drug delivery in simple laboratory exper-

iments using regular shaped devices. Although we have considered the flow-through

system which is better related to in-vivo, we suggest that more complicated systems

can be considered in the future. Additional phenomena that appear in human body

can also be taken into consideration one by one depending on how the drug is in-

tended to be consumed. The shape of the device also can be taken as a decision

96

CONCLUSIONS AND FUTURE WORKS

variable(s) in order to achieve an optimal delivery. In our work, we only introduced

the observer approach method in the analytical solutions. It can also be used in con-

junction with the full numerical methods. The full 3D numerical method developed

is CPU-time consuming and thus advanced numerical techniques such as adaptive

mesh refinement and multigrid methods can be used to reduce its computational

costs.

97

6.2. FUTURE WORK

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110

Appendix A

Experimental Data for Spherical

Polymeric Devices

Table 17: Experimental data of Mt/M∞ (or Re) for spherical polymeric devices

Time (hour) S2080-05 S3070-05 S4060-05 S2080-10 S3070-10 S4060-100.50 0.193 0.129 0.158 0.577 0.232 0.2641.00 0.245 0.121 0.212 0.671 0.274 0.2701.50 0.272 0.191 0.230 0.675 0.309 0.3272.00 0.309 0.274 0.260 0.686 0.308 0.3743.00 0.302 0.469 0.286 0.697 0.388 0.3884.50 0.483 0.517 0.332 0.732 0.467 0.4756.87 0.606 0.582 0.365 0.866 0.589 0.55924.87 0.808 0.773 0.654 0.934 0.834 0.76232.70 0.899 0.805 0.781 0.963 0.865 0.83350.95 0.973 0.906 0.971 0.989 0.919 0.93055.78 0.997 0.972 0.973 0.980 0.947 0.93172.80 1.000 1.000 1.000 1.000 1.000 1.000

111

Appendix B

Experimental Data for

Flow-through Experiment

Table 18: Experimental data of concentration value in a flow–through system fordisc/cylindrical polymeric devices

Time Data A Data B Data C Data D(hour) (g/ml) (g/ml) (g/ml) (g/ml)0.05 4.289E-5 1.606E-5 1.469E-4 1.332E-40.10 3.325E-5 3.581E-5 1.717E-4 1.602E-40.15 2.344E-5 2.236E-5 1.634E-4 1.372E-40.20 1.941E-5 1.763E-5 1.545E-4 1.143E-40.25 1.719E-5 1.518E-5 1.467E-4 9.828E-50.30 1.568E-5 1.357E-5 1.377E-4 8.689E-50.35 1.458E-5 1.242E-5 1.271E-4 7.701E-50.40 1.366E-5 1.157E-5 1.192E-4 7.040E-50.45 1.292E-5 1.089E-5 1.157E-4 6.612E-50.50 1.216E-5 1.033E-5 1.091E-4 6.216E-50.55 1.149E-5 9.703E-6 1.037E-4 5.819E-50.60 1.090E-5 9.232E-6 9.596E-5 5.520E-50.65 1.039E-5 8.757E-6 9.011E-5 5.251E-50.70 9.931E-6 8.386E-6 8.608E-5 5.017E-50.75 9.511E-6 8.037E-6 8.234E-5 4.789E-50.80 9.138E-6 7.679E-6 7.865E-5 4.580E-50.85 8.867E-6 7.437E-6 7.559E-5 4.311E-50.90 8.623E-6 7.075E-6 7.323E-5 4.127E-50.95 8.386E-6 6.836E-6 7.078E-5 3.970E-5

113

1.00 8.168E-6 6.604E-6 6.805E-5 3.826E-51.05 7.967E-6 6.427E-6 6.712E-5 3.604E-51.10 7.809E-6 6.139E-6 6.560E-5 3.507E-51.15 7.644E-6 5.869E-6 6.387E-5 3.427E-51.20 7.488E-6 5.662E-6 6.226E-5 3.362E-51.25 7.332E-6 5.480E-6 6.049E-5 3.290E-51.30 7.190E-6 5.159E-6 6.011E-5 3.237E-51.35 7.037E-6 4.642E-6 5.894E-5 3.169E-51.40 6.893E-6 4.523E-6 5.728E-5 3.076E-51.45 6.776E-6 4.554E-6 5.585E-5 3.002E-51.50 6.653E-6 4.454E-6 5.441E-5 2.891E-51.55 6.514E-6 4.318E-6 5.377E-5 2.821E-51.60 6.394E-6 4.267E-6 5.279E-5 2.779E-51.65 6.293E-6 4.177E-6 5.174E-5 2.698E-51.70 6.192E-6 4.048E-6 5.031E-5 2.609E-51.75 6.080E-6 3.971E-6 4.942E-5 2.539E-51.80 5.988E-6 3.933E-6 4.844E-5 2.479E-51.85 5.889E-6 3.796E-6 4.737E-5 2.433E-51.90 5.791E-6 3.751E-6 4.625E-5 2.387E-51.95 5.696E-6 3.654E-6 4.530E-5 2.343E-52.00 5.608E-6 3.552E-6 4.411E-5 2.294E-52.05 5.518E-6 3.459E-6 4.341E-5 2.248E-52.10 5.433E-6 3.392E-6 4.262E-5 2.212E-52.15 5.349E-6 3.340E-6 4.195E-5 2.191E-52.20 5.273E-6 3.281E-6 4.105E-5 2.172E-52.25 5.185E-6 3.183E-6 4.015E-5 2.156E-52.30 5.107E-6 3.133E-6 3.951E-5 2.133E-52.35 5.034E-6 3.045E-6 3.889E-5 2.113E-52.40 4.963E-6 2.990E-6 3.779E-5 2.097E-52.45 4.902E-6 2.959E-6 3.752E-5 2.077E-52.50 4.843E-6 2.920E-6 3.697E-5 2.065E-52.55 4.783E-6 2.888E-6 3.617E-5 2.054E-52.60 4.713E-6 2.860E-6 3.538E-5 2.041E-52.65 4.650E-6 2.829E-6 3.463E-5 2.024E-52.70 4.588E-6 2.764E-6 3.402E-5 2.004E-52.75 4.528E-6 2.731E-6 3.349E-5 1.990E-52.80 4.473E-6 2.707E-6 3.314E-5 1.975E-52.85 4.421E-6 2.668E-6 3.264E-5 1.952E-52.90 4.372E-6 2.627E-6 3.210E-5 1.936E-52.95 4.316E-6 2.602E-6 3.170E-5 1.919E-53.00 4.270E-6 2.572E-6 3.152E-5 1.907E-53.05 4.231E-6 2.510E-6 3.183E-5 1.891E-53.10 4.183E-6 2.422E-6 3.145E-5 1.865E-53.15 4.143E-6 2.401E-6 3.083E-5 1.862E-5

114

EXPERIMENTAL DATA FOR FLOW-THROUGH EXPERIMENT

3.20 4.088E-6 2.377E-6 3.049E-5 1.841E-53.25 4.045E-6 2.352E-6 2.981E-5 1.820E-53.30 4.007E-6 2.328E-6 2.956E-5 1.805E-53.35 3.965E-6 2.315E-6 2.918E-5 1.789E-53.40 3.914E-6 2.280E-6 2.880E-5 1.771E-53.45 3.875E-6 2.280E-6 2.854E-5 1.758E-53.50 3.831E-6 2.247E-6 2.810E-5 1.737E-53.55 3.785E-6 2.184E-6 2.760E-5 1.721E-53.60 3.746E-6 2.161E-6 2.731E-5 1.707E-53.65 3.710E-6 2.145E-6 2.685E-5 1.693E-53.70 3.677E-6 2.120E-6 2.648E-5 1.679E-53.75 3.630E-6 2.118E-6 2.603E-5 1.665E-53.80 3.585E-6 2.116E-6 2.566E-5 1.650E-53.85 3.551E-6 2.087E-6 2.521E-5 1.630E-53.90 3.521E-6 2.074E-6 2.479E-5 1.613E-53.95 3.480E-6 2.055E-6 2.438E-5 1.606E-54.00 3.447E-6 2.027E-6 2.393E-5 1.595E-54.05 3.421E-6 2.002E-6 2.359E-5 1.577E-54.10 3.388E-6 1.974E-6 2.338E-5 1.565E-54.15 3.359E-6 1.912E-6 2.289E-5 1.555E-54.20 3.331E-6 1.916E-6 2.260E-5 1.532E-54.25 3.304E-6 1.912E-6 2.223E-5 1.528E-54.30 3.272E-6 1.906E-6 2.189E-5 1.515E-54.35 3.247E-6 1.893E-6 2.154E-5 1.505E-54.40 3.222E-6 1.880E-6 2.131E-5 1.489E-54.45 3.194E-6 1.864E-6 2.101E-5 1.476E-54.50 3.161E-6 1.850E-6 2.089E-5 1.462E-54.55 3.150E-6 1.827E-6 2.067E-5 1.450E-54.60 3.123E-6 1.808E-6 2.029E-5 1.436E-54.65 3.093E-6 1.792E-6 2.013E-5 1.418E-54.70 3.065E-6 1.768E-6 1.981E-5 1.404E-54.75 3.044E-6 1.757E-6 1.938E-5 1.386E-54.80 3.021E-6 1.747E-6 1.899E-5 1.371E-54.85 2.996E-6 1.727E-6 1.876E-5 1.364E-54.90 2.968E-6 1.718E-6 1.835E-5 1.360E-54.95 2.945E-6 1.709E-6 1.795E-5 1.351E-55.00 2.923E-6 1.688E-6 1.761E-5 1.347E-55.05 2.900E-6 1.676E-6 1.743E-5 1.362E-55.10 2.875E-6 1.668E-6 1.723E-5 1.340E-55.15 2.852E-6 1.650E-6 1.701E-5 1.292E-55.20 2.827E-6 1.636E-6 1.684E-5 1.279E-55.25 2.805E-6 1.626E-6 1.666E-5 1.276E-55.30 2.780E-6 1.612E-6 1.646E-5 1.354E-5

115


116



117


118


11.80 1.302E-6 7.311E-7 7.065E-6 6.501E-611.85 1.296E-6 7.291E-7 7.035E-6 6.488E-611.90 1.292E-6 7.313E-7 6.997E-6 6.483E-611.95 1.286E-6 7.252E-7 6.969E-6 6.465E-612.00 1.280E-6 7.197E-7 6.957E-6 6.472E-612.05 1.273E-6 7.168E-7 7.004E-6 6.464E-612.10 1.269E-6 7.047E-7 6.985E-6 6.444E-612.15 1.263E-6 7.019E-7 7.003E-6 6.378E-612.20 1.255E-6 6.990E-7 7.015E-6 6.526E-612.25 1.247E-6 6.922E-7 7.008E-6 6.559E-612.30 1.245E-6 6.889E-7 6.987E-6 6.531E-612.35 1.232E-6 6.828E-7 6.992E-6 6.492E-612.40 1.222E-6 6.742E-7 6.983E-6 6.494E-612.45 1.219E-6 6.753E-7 6.951E-6 6.477E-612.50 1.208E-6 6.702E-7 6.921E-6 6.337E-612.55 1.197E-6 6.666E-7 6.900E-6 6.294E-612.60 1.189E-6 6.635E-7 6.935E-6 6.258E-612.65 1.185E-6 6.612E-7 6.919E-6 6.242E-612.70 1.179E-6 6.586E-7 6.903E-6 6.657E-612.75 1.170E-6 6.543E-7 6.879E-6 6.425E-612.80 1.167E-6 6.480E-7 6.869E-6 6.182E-612.85 1.154E-6 6.431E-7 6.856E-6 6.017E-612.90 1.148E-6 6.407E-7 6.831E-6 5.975E-612.95 1.138E-6 6.359E-7 6.811E-6 5.995E-613.00 1.135E-6 6.360E-7 6.817E-6 5.954E-613.05 1.128E-6 6.335E-7 6.782E-6 5.927E-613.10 1.123E-6 6.302E-7 6.744E-6 5.910E-613.15 1.118E-6 6.272E-7 6.726E-6 5.944E-613.20 1.121E-6 6.212E-7 6.705E-6 5.978E-613.25 1.118E-6 6.155E-7 6.667E-6 5.931E-613.30 1.113E-6 6.160E-7 6.631E-6 5.941E-613.35 1.109E-6 6.163E-7 6.578E-6 5.879E-613.40 1.102E-6 6.141E-7 6.432E-6 5.911E-613.45 1.095E-6 6.137E-7 6.220E-6 5.882E-613.50 1.093E-6 6.102E-7 6.139E-6 5.890E-613.55 1.086E-6 6.023E-7 6.166E-6 5.940E-613.60 1.085E-6 6.015E-7 6.201E-6 5.867E-613.65 1.077E-6 5.982E-7 6.185E-6 5.880E-613.70 1.073E-6 5.959E-7 6.097E-6 6.006E-613.75 1.066E-6 5.938E-7 5.999E-6 5.997E-613.80 1.059E-6 5.938E-7 5.999E-6 5.968E-613.85 1.052E-6 5.896E-7 6.056E-6 5.926E-6

119


120



121


122



123

22.50 4.301E-7 2.307E-7 3.281E-6 3.695E-622.55 4.277E-7 2.317E-7 3.272E-6 3.687E-622.60 4.297E-7 2.307E-7 3.264E-6 3.686E-622.65 4.243E-7 2.288E-7 3.261E-6 3.679E-622.70 4.215E-7 2.306E-7 3.265E-6 3.929E-622.75 4.201E-7 2.444E-7 3.287E-6 4.135E-622.80 4.195E-7 2.481E-7 3.304E-6 3.765E-622.85 4.210E-7 2.445E-7 3.295E-6 3.620E-622.90 4.179E-7 2.429E-7 3.284E-6 3.531E-622.95 4.128E-7 2.442E-7 3.268E-6 3.485E-623.00 4.119E-7 2.412E-7 3.257E-6 3.470E-623.05 4.085E-7 2.388E-7 3.252E-6 3.426E-623.10 4.072E-7 2.374E-7 3.241E-6 3.427E-623.15 4.058E-7 2.351E-7 2.872E-6 3.414E-623.20 4.024E-7 2.347E-7 2.654E-6 3.407E-623.25 4.024E-7 2.334E-7 2.660E-6 3.423E-623.30 4.019E-7 2.317E-7 2.651E-6 3.411E-623.35 3.980E-7 2.331E-7 2.629E-6 3.425E-623.40 3.974E-7 2.301E-7 2.618E-6 3.430E-623.45 3.980E-7 2.282E-7 2.608E-6 3.435E-623.50 3.983E-7 2.264E-7 2.607E-6 3.431E-623.55 3.954E-7 2.267E-7 2.604E-6 3.415E-623.60 3.989E-7 2.232E-7 2.607E-6 3.507E-623.65 4.003E-7 2.233E-7 2.648E-6 3.438E-623.70 3.958E-7 2.243E-7 2.660E-6 3.437E-623.75 3.864E-7 2.243E-7 2.651E-6 3.426E-623.80 3.828E-7 2.226E-7 2.638E-6 3.448E-623.85 3.800E-7 2.209E-7 2.625E-6 3.479E-623.90 3.795E-7 2.216E-7 2.608E-6 3.499E-623.95 3.706E-7 2.223E-7 2.618E-6 3.494E-624.00 3.704E-7 2.610E-6 3.505E-624.21 – – 2.605E-6 3.497E-624.26 – – 2.907E-6 3.324E-624.31 – – 2.921E-6 3.263E-624.36 – – 2.938E-6 3.262E-624.41 – – 2.958E-6 3.252E-624.46 – – 2.965E-6 3.230E-624.51 – – 2.951E-6 3.225E-624.56 – – 2.924E-6 3.194E-624.61 – – 2.910E-6 3.188E-624.66 – – 2.902E-6 3.188E-624.71 – – 2.903E-6 3.160E-624.76 – – 2.898E-6 3.420E-6

124


24.81 – – 2.895E-6 3.186E-624.86 – – 2.882E-6 3.025E-624.91 – – 2.859E-6 2.956E-624.96 – – 2.845E-6 2.924E-625.01 – – 2.853E-6 2.902E-625.06 – – 2.849E-6 2.890E-625.11 – – 2.858E-6 2.884E-625.16 – – 2.860E-6 2.874E-625.21 – – 2.830E-6 2.861E-625.26 – – 2.820E-6 2.844E-625.31 – – 2.842E-6 2.835E-625.36 – – 2.838E-6 2.837E-625.41 – – 2.838E-6 2.838E-625.46 – – 2.816E-6 2.855E-625.51 – – 2.803E-6 2.874E-625.56 – – 2.794E-6 2.891E-625.61 – – 2.798E-6 2.906E-625.66 – – 2.830E-6 2.914E-625.71 – – 2.846E-6 2.914E-625.76 – – 2.852E-6 2.909E-625.81 – – 2.855E-6 2.893E-625.86 – – 2.842E-6 2.751E-625.91 – – 2.825E-6 2.767E-625.96 – – 2.818E-6 2.806E-626.01 – – 2.838E-6 2.836E-626.06 – – 2.838E-6 2.834E-626.11 – – 2.810E-6 2.835E-626.16 – – 2.807E-6 2.834E-626.21 – – 2.835E-6 2.832E-626.26 – – 2.808E-6 2.818E-626.31 – – 2.819E-6 2.821E-626.36 – – 2.846E-6 2.822E-626.41 – – 2.855E-6 2.843E-626.46 – – 2.832E-6 2.854E-626.51 – – 2.838E-6 2.862E-626.56 – – 2.834E-6 2.861E-626.61 – – 2.827E-6 2.861E-626.66 – – 2.817E-6 2.855E-626.71 – – 2.807E-6 2.842E-626.76 – – 2.808E-6 2.860E-626.81 – – 2.813E-6 2.850E-626.86 – – 2.829E-6 2.826E-626.91 – – 2.828E-6 2.816E-6

125


126



127


128



129


130



131


132



133

44.11 – – 1.757E-6 1.369E-644.16 – – 1.760E-6 1.368E-644.21 – – 1.755E-6 1.355E-644.26 – – 1.759E-6 1.353E-644.31 – – 1.771E-6 1.356E-644.36 – – 1.767E-6 1.349E-644.41 – – 1.761E-6 1.348E-644.46 – – 1.756E-6 1.338E-644.51 – – 1.757E-6 1.340E-644.56 – – 1.755E-6 1.341E-644.61 – – 1.758E-6 1.334E-644.66 – – 1.753E-6 1.328E-644.71 – – 1.744E-6 1.329E-644.76 – – 1.744E-6 1.326E-644.81 – – 1.742E-6 1.324E-644.86 – – 1.751E-6 1.321E-644.91 – – 1.745E-6 1.323E-644.96 – – 1.755E-6 1.322E-645.01 – – 1.776E-6 1.318E-645.06 – – 1.785E-6 1.317E-645.11 – – 1.791E-6 1.310E-645.16 – – 1.780E-6 1.309E-645.21 – – 1.789E-6 1.304E-645.26 – – 1.788E-6 1.303E-645.31 – – 1.774E-6 1.301E-645.36 – – 1.761E-6 1.293E-645.41 – – 1.773E-6 1.293E-645.46 – – 1.766E-6 1.293E-645.51 – – 1.758E-6 1.288E-645.56 – – 1.749E-6 1.286E-645.61 – – 1.735E-6 1.281E-645.66 – – 1.733E-6 1.278E-645.71 – – 1.736E-6 1.275E-645.76 – – 1.735E-6 1.275E-645.81 – – 1.731E-6 1.271E-645.86 – – 1.739E-6 1.269E-645.91 – – 1.743E-6 1.263E-645.96 – – 1.751E-6 1.259E-646.01 – – 1.758E-6 1.260E-646.06 – – 1.764E-6 1.254E-646.11 – – 1.762E-6 1.250E-6

134


46.16 – – 1.741E-6 1.253E-646.21 – – 1.740E-6 1.284E-646.26 – – 1.748E-6 1.272E-646.31 – – 1.746E-6 1.272E-646.36 – – 1.743E-6 1.264E-646.41 – – 1.737E-6 1.254E-646.46 – – 1.749E-6 1.248E-646.51 – – 1.750E-6 1.261E-646.56 – – 1.745E-6 1.279E-646.61 – – 1.738E-6 1.272E-646.66 – – 1.736E-6 1.266E-646.71 – – 1.727E-6 1.264E-646.76 – – 1.724E-6 1.261E-646.81 – – 1.721E-6 1.257E-646.86 – – 1.724E-6 1.254E-646.91 – – 1.724E-6 1.251E-646.96 – – 1.721E-6 1.250E-647.01 – – 1.726E-6 1.244E-647.06 – – 1.721E-6 1.245E-647.11 – – 1.724E-6 1.240E-647.16 – – 1.726E-6 1.237E-647.21 – – 1.728E-6 1.234E-647.26 – – 1.720E-6 1.230E-647.31 – – 1.710E-6 1.229E-647.36 – – 1.723E-6 1.220E-647.41 – – 1.743E-6 1.198E-647.46 – – 1.742E-6 1.193E-647.51 – – 1.750E-6 1.189E-647.56 – – 1.743E-6 1.186E-647.61 – – 1.742E-6 1.190E-647.66 – – 1.729E-6 1.190E-647.71 – – 1.724E-6 1.185E-647.76 – – 1.723E-6 1.184E-647.81 – – 1.727E-6 1.186E-647.86 – – 1.738E-6 1.186E-647.91 – – 1.749E-6 1.183E-647.96 – – 1.769E-6 1.188E-648.01 – – 1.774E-6 1.182E-648.06 – – 1.766E-6 1.180E-648.11 – – 1.764E-6 1.176E-648.16 – – 1.763E-6 1.191E-648.21 – – 1.755E-6 1.179E-648.26 – – 1.756E-6 –48.31 – – 1.768E-6 –

135

Appendix C

Experimental Data for Cylindrical

Polymeric Devices

Table 19: Experimental data of Mt/M∞ (or Re) for spherical polymeric devices

Time (hour) A3 A40.25 0.0175 0.17220.50 0.0241 0.17930.75 0.0492 0.18621.00 0.0614 0.27601.50 0.2192 0.37522.00 0.2530 0.40603.00 0.2584 0.48884.00 0.2894 0.61346.00 0.3537 0.61619.00 0.4009 0.701524.00 0.6933 0.933048.00 0.8300 0.984572.00 1.0000 1.0000

137

Appendix D

Analytical evaluation for integrals

in (34) and (36)

We have from Equation (20)

I(t, D0(s), D1(s)) :=

∫ t

0

R(τ, D0(s), D1(s))dτ

=

∫ t

0

(

1− 6σ1−σ3

∞∑

n=1

j21(σαn)

sin2(αn)e−D0(s)α2

nτ/r2

2

)

dτ, 0 ≤ t ≤ tc,

∫ tc0

(

1− 6σ1−σ3

∞∑

n=1

j21(σαn)

sin2(αn)e−D0(s)α2

nτ/r2

2

)

dτ+

+∫ t

tc

(

1− 6σ

1− σ3

∞∑

n=1

j21(σαn)

sin2(αn)e−α2

n(D1(s)(τ−tc)+D0(s)tc)/r22

)

dτ, t > tc,

=

I(t, D0(s)), 0 ≤ t ≤ tc

I(tc, D0(s)) + τ |tτ=tc

+6σr2

2

(1−σ3)

∞∑

n=1

j21(σαn)

α2n sin

2(αn)

e−α2n(D1(s)(τ−tc)+D0(s)tc)/r22

D1(s)|tτ=tc t > tc,

139

=

I(t, D0(s)), 0 ≤ t ≤ tc,

I(tc, D0(s)) + (t− tc)+

− 6σr22

(1−σ3)

∞∑

n=1

j21(σαn)

α2n sin

2(αn)

e−D0(s)α2ntc/r

2

2

(

1− e−α2n(D1(s)(t−tc))/r22

)

D1(s)t > tc,

where I(t, D(s)) is defined in (29).

Differentiating (20) with respect to D0 and D1, respectively, and integrating the

resulting expressions, we have the following analytical expressions forH0(t, D0(s), D1(s))

and H1(t, D0(s), D1(s)):

H0(t, D0(s), D1(s))

=

H(t, D0(s)), 0 ≤ t ≤ tc,

H(tc, D0(s))−∫ tc0R(τ, D0(s))dτ+

+ tcD1(t)

(

R(t, D0(s), D1(s))−R(tc, D0(s)))

, t > tc

and

H1(t, D0(s), D1(s))

=

0, 0 ≤ t ≤ tc

1D1(t)

(

(t− tc)R(t, D0(s), D1(s))−∫ t

tcR(τ, D0(s), D1(s))dτ

)

, t > tc.

140

Numerical Methods for the Estimation of Effective Diffusion … · flow-through fluid systems....

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