Modeling, Numerical Simulation and Experimental Investigation of Ion Exchange and Diffusion-Coupled Drug

Transport in Pharmaceutical Polymers

By

Yi Li

A thesis submitted in conformity with the requirements for the degree of Master of Science

Pharmaceutical Science University of Toronto

© Copyright by Yi Li 2017

ii

Modeling, Numerical Simulation and Experimental Investigation

of Ion Exchange and Diffusion-Coupled Drug Transport in

Pharmaceutical Polymers

Yi Li

Master of Science

Pharmaceutical Science

University of Toronto

2017

Abstract

Ionic polymers are used in pharmaceutical products to achieve desired drug release properties.

The ionic interaction between drugs and polymers complicates theoretical prediction of drug

transport adding more difficulties in design of controlled release dosage forms. Therefore, this

thesis investigates drug transport in membranes of Eudragit® RS (RS) and Eudragit® RL (RL)

with different amounts of cationic groups. A mathematical model was developed to describe the

ion-exchange and diffusion-coupled drug loading mechanism. The model was verified by

loading two different anionic drugs, ibuprofen Na and diclofenac Na, into cationic RS/RL

polymer membranes. The dependence of drug loading kinetic and equilibrium were investigated

using numerical simulations and experiments on the different physical chemical parameters.

Thermodynamic and kinetic parameters of the system determined from this work can be used for

the design of experiments to achieve target loading efficacy and efficiency and for prediction of

drug release kinetics of coated dosage forms.

iii

Acknowledgments

I would like to thank my supervisor Dr. Shirley X.Y. Wu for her kind support, patience, and

passing down her knowledge. I would not be able to go through my entire undergraduate and

Master program without her guidance. She always believed in my ability to achieve and gave me

guidance not only in academia, but also life and career planning. She is the most important

person who guided me along the way of acquiring pharmaceutical knowledge. Without her

mentorship, I would not have the knowledge of pharmaceutical science I possess today and have

this thesis written.

I am greatly appreciative to Dr. Paul Grootendorst and Dr. Rob Macgregor for taking their

precious time to attend my committee meetings and provide insightful comments for my

improvement.

My sincere thanks to all my colleagues who contributed a lot on my projects and helped me

overcome the tasks. Thank you Gary Chen and Dr. Jason Li.

I really want to thank my parents, and this thesis is dedicated to Professor Baowu Wei, Zhaitian

Li, Xiuqin Lin, Xiaozhong Jiang, Yu Wei and Zhuo Li. They were dedicated to the ideal that

their later generation should have better life than they do and contribute more to human society

and lead a better future. Their strength, intelligence, and patience in education turned me into

who I am today. I would especially like to thank my grandfather Baowu Wei, who was my first

teacher in calculus, physics and computer science when the technology was not that developed.

His hand-to-hand teaching provided me the precious fundamental knowledge of achieving this

mathematical model. I have taken the opportunities you have provided me and that I have and

will continue to make you proud.

iv

Table of Contents

Acknowledgments.......................................................................................................................... iii

Table of Contents ........................................................................................................................... iv

List of Tables ................................................................................................................................. vi

List of Figures ............................................................................................................................... vii

List of Abbreviations .......................................................................................................................x

Chapter 1 Introduction .....................................................................................................................1

1 Introduction .................................................................................................................................2

1.1 Ion Exchange .......................................................................................................................3

1.1.1 History of Ion Exchange and Its Applications in Pharmaceutics ............................3

1.2 Acrylate Polymers in Pharmaceutics ...................................................................................7

1.2.1 History of Eudragit® ................................................................................................7

1.2.2 Types of Eudragit® Polymers...................................................................................9

1.2.3 Eudragit® RS and Eudragit® RL ............................................................................12

1.3 Mathematical Modeling in Pharmaceutics .........................................................................13

1.4 Goals of This Work ............................................................................................................23

1.5 Synopsis .............................................................................................................................24

Chapter 2 Modeling and Experimental Methods ...........................................................................26

2 Modeling and Experimental Methods .......................................................................................27

2.1 Theoretical Analysis ..........................................................................................................27

2.1.1 Mathematical modeling and derivation of an analytical solution ..........................27

2.2 Nomenclature .....................................................................................................................33

2.3 Methods..............................................................................................................................35

2.3.1 Experimental ..........................................................................................................35

Chapter 3 Results and Discussion ..................................................................................................38

v

3 Results and Discussion ..............................................................................................................39

3.1 Validation of short-time solution of model for long-time numerical evaluations .............39

3.2 Verification of model with experimental data ...................................................................40

3.2.1 Determination of model parameters.......................................................................40

3.2.2 Goodness of fit of model........................................................................................41

3.3 Computational simulation and impacts of different parameters on drug loading

kinetics ...............................................................................................................................43

3.3.1 Influence of loading conditions .............................................................................43

3.3.2 Influence of drug and membrane properties ..........................................................45

3.4 Prediction of loading efficiency and loading level in the membrane ................................46

Chapter 4 Conclusion and Future Perspectives .............................................................................50

4 Conclusions and Future Perspectives ........................................................................................51

4.1 Highlights ...........................................................................................................................51

4.2 Conclusions on the Modelling and Experiments ...............................................................51

4.3 Overall Conclusions and Original Contributions of This Thesis .......................................52

4.4 Limitation of the Work and Future Perspectives ...............................................................52

4.5 Acknowledgements ............................................................................................................53

Appendices .....................................................................................................................................54

vi

List of Tables

Table 1. Commercialized products using Eudragits®. .................................................................... 8

Table 2. The chemical nature, characteristic features, and applications of different types of

Eudragit® ....................................................................................................................................... 11

Table 3. Summary of mechanistic realistic mathematical models under different mechanisms .. 20

Table 4. Summary of important empirical and semi-empirical models for controlled release using

different mathematical approaches ............................................................................................... 21

Table 5. Summary of the parameters of all drug loading conditions. ........................................... 43

Table 6. Summary of the mean square deviation (MSD) and root mean square deviation (RMSD)

values of the model predictions and the experimental data for all drug loading conditions. Each

loading concentrations have three replicates (n = 3). ................................................................... 54

vii

List of Figures

Figure 1. Schematic of ion-exchange mechanism. ......................................................................... 4

Figure 2. Applications of ion-exchange in pharmaceutics. ............................................................. 6

Figure 3. Synthesis of acrylate polymers. ....................................................................................... 9

Figure 4. Chemical structures and different grades of Eudragits®................................................ 10

Figure 5. Chemical structure for Eudragit® RS or Eudragit® RL with a:b:c ratio of 0.1:2:1 and

0.2:2:1 respectively. ...................................................................................................................... 12

Figure 6. Classification system for primarily diffusion controlled drug delivery systems. Only

spherical dosage forms are illustrated, but the classification system is applicable to any type of

geometry. ...................................................................................................................................... 16

Figure 7. Schematic presentation of a swelling controlled drug delivery system containing

dissolved and dispersed drug (stars in circles and solid circles, respectively), exhibiting the

following moving boundaries: (i) swelling front, barrier between the swollen and non-swollen

matrix (ii) diffusion front, separate the swollen matrix containing dissolved and dispersed drug

and the swollen matrix containing dissolved drug only (iii) erosion front, barrier between bulk

and delivery system....................................................................................................................... 17

Figure 8. Degradation mechanisms of biodegradable polymeric nanoparticles: A) bulk erosion,

B) surface erosion. ........................................................................................................................ 18

Figure 9. A schematic diagram of the ion-exchange drug loading mechanism in one half of the

polymer membranes. The blue circles represent the available binding site in the polymer

membrane that can be bounded by either drug (S) or counter ions (Cl-). ..................................... 27

Figure 10. Chemical structures of a) ibuprofen Na and b) diclofenac Na. ................................... 37

Figure 11. a) Fraction of drug remaining in the solution versus time as a function of Langmuir

association constant (K = kads/kdes). b) Comparison of the ion-exchange model with Crank’s

solution of diffusion through a plane sheet at large partition coefficient (Kd) value. ................... 40

viii

Figure 12. The plot of (M0

M∞− 1)

−1

vs. Cb,∞ of ibuprofen Na loading from solutions with four

values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25 mM) into pre-swollen RS100

membranes. ................................................................................................................................... 41

Figure 13. Comparison of the experimental data and model predictions of drug loading into a)

RS100 polymer membranes from ibuprofen Na solution under various of initial loading

concentrations and b) membranes of various RS:RL ratio from 0.15 mM ibuprofen Na solution.

....................................................................................................................................................... 42

Figure 14. Comparison of the experimental data and model predictions of drug loading into

RS95 polymer membranes from a) ibuprofen Na solution and b) diclofenac Na solution. .......... 42

Figure 15. Fraction of drug remaining in the solution versus time as a function of a) volume ratio

of external solution to membrane (λ) and b) initial drug concentration (C0) in the external

solution. ......................................................................................................................................... 44

Figure 16. Fractional drug remaining in solution versus time for membranes with various a)

maximum solute binding capacities (Cmax), b) Langmuir association constants (K= kads / kdes),

and c) drug diffusion coefficients (D) through the membrane. .................................................... 45

Figure 17. Equilibrium drug concentration in the ion-exchange membrane as a function of a) the

volume ratio of solution to membrane (λ) and b) initial loading concentration in the solution (C0).

Cmax = 100 mM, kads = 5×10-3 (mM∙ s)-1, kdes = 1×10-3 s-1, h = 0.02 cm, R = 1 cm. ..................... 48

Figure 18. Drug loading yield as a function of a) the volume ratio of solution to membrane (λ)

and b) initial loading concentration in the solution (C0). Cmax = 100 mM, kads = 5×10-3 (mM∙ s)-1,

kdes = 1×10-3 s-1, h = 0.02 cm, R = 1 cm. ....................................................................................... 48

Figure 19. Drug loading yield (solid lines, left y-axis) and equilibrium drug loading (dashed

lines, right y-axis) as a function of C0 and λ. ................................................................................ 49

Figure 20. Comparison of prediction power between the diffusion model and the ion-exchange-

diffusion coupled model. .............................................................................................................. 55

ix

Figure 21. Determination of model parameters. The plot of (M0

M∞− 1)

−1

vs. Cb,∞ of ibuprofen Na

loading from solutions with four values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25

mM) into pre-swollen RS95 membranes. ..................................................................................... 56

Figure 22. Determination of model parameters. The plot of (M0

M∞− 1)

−1

vs. Cb,∞ of ibuprofen Na

loading from solutions with four values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25

mM) into pre-swollen RS90 membranes. ..................................................................................... 56

Figure 23. Determination of model parameters. The plot of (M0

M∞− 1)

−1

vs. Cb,∞ of diclofenac

Na loading from solutions with four values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25

mM) into pre-swollen RS95 membranes. ..................................................................................... 57

Figure 24. Demonstration of the simulation using MATLAB. Coded by Yi Li. .......................... 57

Figure 25. Demonstration of the simulated data points extraction from MATLAB as matrix ..... 58

Figure 26. Demonstration of raw MATLAB plots simulating effects on loading regarding to a)

different volume ratio and b) different initial loading concentration. Same values of parameters

were used as in Chapter 2. ............................................................................................................ 58

x

List of Abbreviations

GI: gastrointestinal

RS: Eudragit® RS

RL: Eudragit® RL

QAGs: quaternary ammonium groups

PVAP: polyvinyl acetate phthalate

ANN: artificial neural networks

TRO: Toronto Region Operations

DBS: dibutyl sebacate

DDI: double distilled

1

1

Chapter 1 Introduction

2

1 Introduction

An oral controlled release drug delivery system is designed to deliver a drug in a controlled and

predictable manner over a period of time or at a predetermined position in the gastrointestinal

tract. These controlled release dosage forms can reduce the dosing frequency and minimize side

effects on patients. Polymers, hydrophobic, hydrophilic or ionic, each has fundamentally

different hydrophobicity, swelling, erosion and other characteristics, provide flexibility in control

of the main release mechanism. As a key element in oral controlled release dosage forms, ionic

polymers have been applied for drug delivery, taste masking and toxin removal purposes. In an

ion-exchange matrix system, high drug loading and ease of formulation can be achieved. In a

membrane reservoir system, ionic polymer coating can manipulate the drug release rate. Drugs

can also be delivered to targeted GI track position assisted by pH responsive ionic polymers. On

the other hand, well studied ion-exchange theories have been described by several kinetic models

in other fields such as fixed bed columns in chromatography. Although the effects of drug-

polymer interactions on drug release from ionic polymer-based systems have been extensively

studied, very limited work has been reported on the mathematical analysis of the loading and

release kinetics of ionized drug for these important systems. In addition, there is a lack of

modern comprehensive understanding on how ion exchange will affect the kinetics of drug-

polymer interactions on a mechanistic level and by how much on each physical/chemical factor,

such as polymer thickness, surface area, and binding capacity, for better designing

pharmaceutical products. Current formulation developments require time-consuming

experiments to determine the optimal process and formulation parameters for dosage forms to

achieve targeted drug release profiles. Due to the large quantities of excipients and drugs

required, many of these experiments are unfeasible, especially during early stages of drug

development where novel drugs are very scarce and expensive. By developing a mechanistic

model of ion-exchange drug loading for any ionic drugs and polymers, the effects of various

formulation and processing parameters on the overall drug release kinetics can be accurately

extrapolated, significantly reducing the cost and time necessary to create reliable, high-quality

products. A well-validated mechanistic model capable of describing ion-exchange drug loading

and release would be extremely useful in the development of novel formulations. Thus, the

intention of this paper is to present background on classical and modern theoretical concepts of

ion exchange combined with modeling concepts for pharmaceutics. It will proceed by filling the

3

gap in scientific knowledge by providing mechanistic approaches to understand insights of ion-

exchange on polymer-drug interaction, quantitative effects of formulation and processing

parameters on drug loading and release kinetics. Eventually it shall be capable to facilitate the

development of novel formulations by accomplishing significant cost and time reduction to

create reliable, high-quality products.

1.1 Ion Exchange

1.1.1 History of Ion Exchange and Its Applications in Pharmaceutics

Ion-exchangers, by definition, are insoluble solid materials which carry exchangeable cations or

anions. The ions can be exchanged for a stoichiometrically equivalent amount of other ions of

the same sign when the ion-exchanger is in contact with an electrolyte solution [1]. The history

and discovery of ion exchange can be traced back to 1850, credited to two English soil scientists,

Way and Thompson, whose work concerned water-soluble fertilizer salts such as ammonium

sulfate and potassium chloride and found They found their retarded leaching out from soil by the

action of rain water [2, 3]. A few years later, the reversibility of the process was established by

Eichhorn in 1858 [4]. Their discovery was not known until around 50 years later after the

German chemist Gans studied the aluminosilicates and their applications in water softening,

which is still a principle field [5]. Since the late 1920s, natural zeolites and synthetic silicates, for

which Gans introduced by the name permutites, were the only products used for water softening.

Gradually, the permutites were superseded by ion-exchangers prepared from sulphonated coal

(Leibknecht [6] and Smit[7]) . The first complete organic ion-exchanger was synthesized in 1935

by Adam and Holmes, by condensing phenolsulfonic acids with formaldehyde and obtained

resins with ion exchange characteristics [8]. They also proceeded to produce anion exchange

resins through condensation of polyamines with formaldehyde. It was the first time that all

electrolytes were removed from water by a method other than distillation [9]. Following by the

initiation of systematic resin research, by Wolfen in 1935, the field was further extended by

Griessbach in 1939 [10]. Later in the 1940s, more fundamental studies on ion exchange resins

were developed and better resins which had greater stability and larger exchange capacity were

synthesized based on co-polymerization [9, 11-14].

4

Figure 1. Schematic of ion-exchange mechanism. Note: ion-exchange sites also present inside the

cation/anion exchangers.

In pharmaceutics, ion exchange has a rich history for a wide variety of applications. First, it is

important to know that it has been long process for scientists to develop drug delivery systems to

optimize therapeutic effects and to improve patient compliance. There have been tremendous

interests in developing ideal delivery systems that are able to transport drugs to the target site for

absorption or release at a controlled rate over a period of time. Formulation methods that allow

manipulating drug kinetic profiles are based mainly on two principles: physical (diffusion,

erosion, and osmotic pump) and chemical (ion-exchange and drug-polymer

complex/conjugation) [15, 16]. Generally, the chemical systems have advantages including high

drug loading, adjustable release, and protection of unstable drugs. This gives ion-exchange an

opportunity to become significant in formulation technique. Its unique properties such as fast

stoichiometric exchange between solid and liquid phase, adjustable concentration dependent

association/dissociation of ions, and masking mechanisms have been implemented in controlled

release dosage forms. [17-26]. For instance, ion-exchange resins [18, 27-29], ion-exchange fiber

[30-33], and ionic polymer coating [34-36] have been widely applied in oral and transdermal

5

pharmaceutical formulations. Ion exchange applications were first recognized in the early 1940s

when Amberlite IRC-50 (afterwards, Rohm and Haas used the designation IRC to denote ion-

exchange resin chemical grade and the designation IRP to denote the ion exchange resin

Pharmaceutical grade) was introduced and led to successful purification of amino acids,

vitamins, and antibiotics [37]. Later, ion exchange resins were accepted as drug carriers in the

1950s [38]. As an example of innovative formulation improvements using ion exchange resins,

Koff described the use of a castor wax coating to improve the palatability and stability of a

highly acidic cation-exchange resin complexed with amotropine [39]. Around the same time, ion

exchange resins were for the first time used for sustained release and taste masking, due to its

slow uptake and release of alkaloids which was noted by Sauders and Srivastava [18, 40]. Ion

exchange resins were found to attain more continuous and uniform drug release by causing

slower disintegration of the tablets, slower solubilization, and having drugs binding to a solid

carrier from which it is slowly released by the action of the digestive fluids [40]. It was in the

1980s that ion exchange was first introduced for transdermal therapies, which were described in

detail by Jenke [41]. In the past 20 years, more and more innovative pharmaceutical applications

and formulation designs were invented based on ion exchange principles. The field has become

more mature and better developed within the pharmaceutical industry [30, 42-51].

6

Figure 2. Applications of ion-exchange in pharmaceutics.

In summary, based on the history and current understanding, high drug loading and ease of

formulation can be obtained in an ion-exchange matrix system. By simply incubating the

polymer matrix in a drug solution made with deionized water, a large amount of drug can be

easily incorporated into the polymer matrix without requiring a complex loading process and

organic solvent. In addition, the amount of drug loaded can be adjusted by varying initial drug

loading concentration, the incubation time, or the volume ratio of the polymer to the bulk

environment [21, 23]. Ion-exchange polymers also regulate controlled release rate and release

location based on characteristics such as pH sensitivity, effects on disintegration and

solubilization, and unique drug-polymer and polymer-environment ionic interactions. Thus ion-

exchange is a key adjustable component in controlled release dosage forms.

7

1.2 Acrylate Polymers in Pharmaceutics

Polymers and copolymers of (meth)acrylates have had a large impact for decades in the

pharmaceutical industry for various functions such as modifying drug release profiles, protecting

drugs against external influences, improving drug stability, and masking unpleasant tastes [52-

61]. Poly(meth)acrylates are better known by the trade name Eudragit® with the immediate

following letters denoting the different chemical groups (neutral, alkaline, or acid groups) as well

as the functionalities of the polymers or copolymers.

1.2.1 History of Eudragit®

Eudragit® is the brand name for a group of copolymers based on polymethacrylates principally

marketed by the German company Evonik. Eudragit was first introduced in Darmstadt by Rohm

and Hass GmbH in 1953 as a stomach acid resistant coating material for alkaline soluble drugs.

The brand has gradually diversified to include anionic, cationic as well as neutral copolymers

based on methacrylic acid and methacrylic, or acrylic esters or their derivatives in varying

proportions [24]. These polymers allow drugs to be formulated in enteric, protective or

sustained-release formulations to prevent breakdown of the drug until it has reached an area with

an adequate pH in the gastrointestinal (GI) tract. Once the drug reaches its target area, it will

release from the polymer matrix or the coating and be absorbed. Targeted drug release is often

used to prevent dissolution of a drug in an area where the pH is not adequate for absorption, or to

help minimize gastrointestinal tract irritation [24-26].

8

Table 1. Commercialized products using Eudragits®.

Acti

ve

in

gre

die

nt

Tra

de

nam

eM

an

ufa

ctu

red

U

se

d p

oly

me

rsD

isso

luti

on

pH

acam

pro

sate

(I.

N.N

.) c

alc

ium

Cam

pra

l E

CM

erc

kE

udra

git L

30-D

55

>5.5

Am

isulp

ride

Arr

ow

Arr

ow

Generi

cs

Lim

ited

Eudra

git E

100

<5

Beclo

meth

aso

ne

Clip

per

Chie

si P

harm

aceuticals

Eudra

git L

100/5

5>

5.5

dip

ropio

nate

Budeso

nid

eE

nto

cort

Pro

meth

eus

Lab.

Eudra

git L

100-5

5>

5.5

Budenofa

lkD

r. F

alk

Pharm

aE

udra

git L

100 a

nd S

100

6 t

o 7

lithiu

m c

arb

onate

Lis

konum

Sm

ith K

line &

Fre

nch L

abora

tori

es

Lim

ited

Eudra

git E

12.5

om

epra

zole

Om

epra

zole

Gast

ro-R

esi

stant

Capsu

leA

cta

vis

Eudra

git L

30-D

55

>5.5

Mesa

lazi

ne

Cla

vers

al

GS

KE

udra

git L

100

>6

Asa

col

Warn

ner

Chilc

ott

Eudra

git S

100

>7

Asa

col H

DW

arn

ner

Chilc

ott

Eudra

git L

100 a

nd S

100 f

or

oute

r coat

>7

Eudra

git S

100 f

or

inner

coat

Salo

falk

Dr.

Falk

Pharm

aE

udra

git L

100

>6

Mesa

sal

GS

K (

AU

S)

Eudra

git L

100

>6

Calit

ofa

lkD

r. F

alk

Pharm

aE

udra

git L

100

>6

Lia

lda

Cosm

o P

harm

aceuticals

Eudra

git S

100

>7

Mesa

vant

(EU

)C

osm

o P

harm

aceuticals

Eudra

git S

100

>7

Mesr

en M

RT

eva P

harm

aceutical

Eudra

git S

100

>7

Ipocol

Sandoz

Eudra

git S

100

>7

Apri

soS

alix

Pharm

aceuticals

Eudra

git L

100

>6

Sulf

asa

lazi

ne

Colo

-ple

on

Sanofi

-Aventis

Eudra

git L

100-5

5>

5.5

Pepperm

int

Oil

BP

Cole

perm

inM

cN

eil

Pro

ducts

Lim

ited

Eudra

git S

100

>7

Eudra

git L

30-D

55

>5.5

9

Acrylate polymers have wide applications in drug formulations, mostly for controlled release

purposes. Table 1 provides some representative commercialized drugs as examples.

1.2.2 Types of Eudragit® Polymers

Eudragits® are co-polymers synthesized by free-radical polymerization of acrylic acids and

methacrylic acids or their esters such as butyl ester or dimethylaminoethyl ester (Figure 3) [62].

The chemical structures of Eudragit® and its different grades are presented in Figure 4. Different

grades of Eudragit® are commercially available and are supplied in various forms such as dry

powder, granules, aqueous dispersion, or organic solution. The chemical nature, characteristic

features, and applications of different types of Eudragit® have been compiled in Table 2. In

general, Eudragits® can be classified into cationic, anionic, and neutral. The synthesis can be

performed in solvent, bulk, suspension, or emulsion. Variations in chain length can be obtained

via various termination and transfer reactions. The functional properties of methacrylate

copolymers and the final polymer can be adjusted by selecting from a variety of monomers. The

non-functional co-monomers are responsible for steering the polymer properties, and the

functional co-monomers for adjusting the solution profile [62]. As synthetic polymers,

Eudragits® are more reproducible compared to cellulosic derivatives, whose physicochemical

properties vary on the source of raw materials.

Figure 3. Synthesis of acrylate polymers.

10

Eudragit

Grade R1 R2 R3 R4

E CH3 CH2CH2N(CH3)2 CH3 CH3, C4H9

L and S CH3 H CH3 CH3

RL and RS H, CH3 CH3, C2H5 CH3 CH2CH2N(CH3)3+Cl-

NE 30D H, CH3 CH3, C2H5 H, CH3 CH3, C2H5

L 30 D-55

and L 100-55 H, CH3 H H, CH3 CH3, C2H5

Figure 4. Chemical structures and different grades of Eudragits®

11

Table 2. The chemical nature, characteristic features, and applications of different types of

Eudragit®

Categories Eudragit grade Chemical composition Available as Solubility Applications

Cationic

Eudragit E 100 Poly(butyl methacrylate, Granules/98% Soluble in gastric fluid Film coating

(2-dimethyl aminoethyl) to pH 5

methacrylate, methyl

methacrylate) 1:2:1

Eudragit E 12.5 Poly(butyl methacrylate, Organic solution/12.5% Soluble in gastric fluid Film coating

(2-dimethyl aminoethyl) to pH 5

methacrylate, methyl

methacrylate) 1:2:1

Eudragit RL Poly(ethyl acrylate, methyl Granules/97% High permeability Sustained release

100 (Type A) methacrylate, trimethyl

aminoethyl methacrylate

chloride) 1:2:0.2

Eudragit RL PO Poly(ethyl acrylate, methyl Powder/97% High permeability Sustained release

methacrylate, trimethyl

aminoethyl methacrylate

chloride) 1:2:0.2

Eudragit RL 30 D Poly(ethyl acrylate, methyl Aqueous dispersion/30% High permeability Sustained release

methacrylate, trimethyl

aminoethyl methacrylate

chloride) 1:2:0.2

Eudragit RS Poly(ethyl acrylate, methyl Granules/97% Low permeability Sustained release

100 (Type B) methacrylate, trimethyl

aminoethyl methacrylate

chloride) 1:2:0.1

Eudragit RS PO Poly(ethyl acrylate, methyl Powder/97% Low permeability Sustained release

methacrylate, trimethyl

aminoethyl methacrylate

chloride) 1:2:0.1

Eudragit RS 30 D Poly(ethyl acrylate, methyl Aqueous dispersion/30% Low permeability Sustained release

methacrylate, trimethyl

aminoethyl methacrylate

chloride) 1:2:0.1

Anionic

Eudragit L 100 Poly(methacrylic acid, methylPowder/95% Soluble in intestinal Enteric coating

methacrylate) 1:1 fluid from pH ≥ 6

Eudragit L 12.5 Poly(methacrylic acid, methylOrganic solution/12.5% Soluble in intestinal Enteric coating

methacrylate) 1:1 (without plasticizer fluid from pH ≥ 6

Eudragit L 12.5 P Poly(methacrylic acid, methylOrganic solution/12.5% Soluble in intestinal Enteric coating

methacrylate) 1:1 (with1.25% dibutyl fluid from pH ≥ 6

phthalate as plasticizer)

Eudragit L 100-55 Poly(methacrylic acid, ethyl Powder/95% Soluble in intestinal Enteric coating

acrylate) 1:1 fluid from pH ≥ 5.5

Eudragit L 30 D-55 Poly(methacrylic acid, ethyl Aqueous dispersion/30% Soluble in intestinal Enteric coating

(formerly Eudragit L 30 D)acrylate) 1:1 fluid from pH ≥ 5.5

Eudragit S 100 Poly(methacrylic acid, methylPowder/95% Soluble in intestinal Enteric coating

methacrylate) 1:2 fluid from pH ≥ 7

Poly(methacrylic acid, methylOrganic solution/12.5% Soluble in intestinal Enteric coating

methacrylate) 1:2 (without plasticizer) fluid from pH ≥ 7

Poly(methacrylic acid, methylOrganic solution/12.5% Soluble in intestinal Enteric coating

methacrylate) 1:2 (with1.25% dibutyl fluid from pH ≥ 7

phthalate as plasticizer)

Eudragit FS 30 D Methyl acrylate, methyl Aqueous dispersion/30% Soluble above pH 6.8 Enteric coating

methacrylate and methacrylic

acid

Neutral

Eudragit NE 30 D Poly(ethyl acrylate, methyl Aqueous dispersion/30% Swellable, permeable Sustained release

(formerly Eudragit E 30 D)methacrylate) 2:1 with

nonoxynol (1.5%)

Eudragit NE 40 D Poly(ethyl acrylate, methyl Aqueous dispersion/40% Swellable, permeable Sustained release

methacrylate) 2:1 with

nonoxynol (1.5%)

Eudragit NM 30 D Poly(ethyl acrylate, methyl Aqueous dispersion/30% Swellable, permeable Sustained release

methacrylate) 2:1 with PEG

stearyl ether (0.7%)

12

1.2.3 Eudragit® RS and Eudragit® RL

Eudragit® RS (RS) and Eudragit® RL (RL) are commonly used polymers for sustained release

coatings [34-36]. RS and RL are miscible in any ratio and are often incorporated as polymer

blends to modulate membrane permeability [63-65]. Flexibility in formulation design can be

tailored by varying two parameters: polymer ratio and coating quantity. Both polymers are

structurally similar copolymers of methyl methacrylate, ethyl acrylate, and trimethylammonio

ethyl methacrylate chloride (Figure 5). The molar ratios of the monomers are 0.1:2:1 and 0.2:2:1

for RS and RL, respectively.

The polymers are insoluble in physiological conditions, but are able to swell due to the

hydrophilic quaternary ammonium groups (QAGs). The difference in quantity of QAGs between

the two polymers causes varying degrees of interaction between water and drug molecules,

resulting in higher or lower diffusivity for water and drugs within the polymeric networks [34,

66-68]. Due to the higher content of QAGs, RL is more hydrophilic, takes up more water, and

swells to a greater extent than RS. As a result, RL provides much higher permeability compared

to RS. Since QAGs dissociate completely under physiological pH, the swelling and permeability

of the coatings are pH independent [34, 35]. RL and RS are miscible in any ratio and are often

used as polymer blends to modulate membrane permeability [63-65]. In sustained release dosage

forms, the quantity of RS in the blend is much higher as RL features are dominant in these

combinations. Typically, RS/RL ratios are 95:5, 90:10, or 80:20 [34].

Figure 5. Chemical structure for Eudragit® RS or Eudragit® RL with a:b:c ratio of 0.1:2:1 and

0.2:2:1 respectively.

13

For a non-ionized drug, drug release from RS/RL membranes is primarily diffusion controlled

[34]. The polymer membrane controls the rate at which water penetrates into the drug core as

well as the subsequent dissolution and diffusion of the drug [69]. Membrane permeability is

strongly influenced by the ionic strength and the buffer species of the dissolution medium [70-

73]. The QAGs act as ion exchange centers for various anions and the anions exchange with the

chloride counter-ions of the QAGs. The ionic interaction between QAGs and ions in the fluid can

also enhance liquid particle movements and cause influx of water and consequent hydration of

the polymers [70, 71]. Hydration of the membrane allows the polymer chains to relax and form

pores where solutes and drug molecules can diffuse through [70, 71, 74].

1.3 Mathematical Modeling in Pharmaceutics

Scientific modeling is a scientific activity to make a particular feature of the world easier to be

understood, defined, quantified, visualized or simulated by applying existing knowledge. It

requires selecting a relevant situation in the real world and trying to understand, operationalize,

quantify, or even visualize it using different models. A mathematical model is specifically for

describing a system using mathematical concepts. The model can help to explain a system and to

know the effects of different components, which leads to predictions about behavior of the

system [75-77].

Mathematical modeling of drug delivery and prediction of drug release has become increasingly

important in academia and industry. In silico manipulations allow improvements of accuracy and

ease of applications in manufacturing.[78] Computational simulations have already become an

integral part of development in pharmaceutical technology. Knowing the desired drug dosage,

administration, and targeted release profile, mathematical predictions will assist in making good

estimates of the required composition, size, shape, and preparation procedure of the respective

dosage forms.

There are numerous mathematical models and solutions in literature that have been applied to

describe the kinetics of solute transport. In 1855, Adolf Fick derived Fick’s law of diffusion,

which described the relationship between diffusive flux and the concentration under the

assumption of steady state. Fick’s first law determined that flux (solute if it is in solution) goes

14

from high concentration region to low concentration region with a magnitude that is proportional

to the concentration gradient. Fick’s second law predicts how diffusion causes concentration

change with time [79]. This was the first description of particle movements in solution with

concentration gradient and established the fundamental knowledge for all kinds of controlled

release mechanistic modellings in pharmaceuticals research later on. After that, scientists spent a

decade on modeling mass transport in a variety of scenarios. In 1956, Crank summarized

previous work and published equations of diffusion to describe mass transport following

different physical principles and conditions based on Fick’s laws and Carslaw and Jaeger’s heat

conduction in solid. Crank’s book later had a great influence on drug delivery [80]. Takeru

Higuchi, published his model in 1961 for describing drug release from an ointment base (inert

matrix with film geometry) [81]. In 1963, Higuchi published his mechanistic equations of drug

released from a solid dosage form assuming a non-erodible matrix under sink condition [82].

This was the beginning of quantitative analysis of gradual drug release from pharmaceutical

dosage forms, following by Roseman’s further modelling works in controlled release and by Fu’s

cylindrical model of drug release from polymer matrixes in 1970s [83-85]. However, in 1976,

Paul and McSpadden pointed out that Higuchi’s results for planar geometry was off from the

exact results by 11.3% and the discrepancy was removed by them using the exact solution of

semi-infinite system [86]. To improve pharmaceutical modelling, scientists raised the question

that Higuchi’s model was not suitable for describing drug release from tablets with erosion. In

the same year as Paul and Mcspadden, Hopfenberg treated the case of no diffusional contribution

[87]. Baker and Lonsdale presented a brief analysis of the general system including diffusion

[88]. The approximate analytical solution of problems with moving boundaries was given by Lee

in 1980 [89]. From 1985 to 1987, Peppas and Rigter established empirically derived models

applied for specific drug delivery circumstances for both non-swellable and swellable matrixes

under both Fickian and non-Fickian diffusion. Peppas complemented his empirical models in

1989 by adding the coupling of diffusion and relaxation for controlled release [90-95].

Thereafter, new controlled release materials (e.g. hydrogels) and new drug types (e.g. biologics)

were developed and widely applied in pharmaceutical products. As a consequence, new models

that are suitable for more specific scenarios were investigated and developed. Later, detailed

models for diffusion, dissolution, swelling, erosion, precipitation, interparticular interaction and

degradation also offered deeper insights into the underlying drug release mechanisms [60, 61,

96-104]. Specifically, detailed description of dispersed-drug release into a finite medium from

15

sphere with specific boundary layer was reported [102, 105]. More complicated models coupled

multiple factors or having complex release geometry were also generated by Wu and others

[106-108]. Further mathematical models were developed and applied in pharmaceutical

manufacturing. The models are very important for understanding the basic physics of drug

release, allowing for better understanding of drug release mechanisms and assisted formulation

design.

After reviewing the history of pharmaceutical modelling, it is clear that there will not be one

general theory that applies to any type of drug delivery system. There should be different

mathematical models that are applicable to specific scenarios differing in release mechanism,

geometry, drug type, excipient type, and release environment (e.g. in vitro or in vivo). Major

controlled release mechanisms including diffusion, swelling, erosion or degradation, ion

exchange, and osmosis. The following provides a summary of the major currently known release

mechanisms except for ion- exchange which have been introduced at the beginning. Depending

on the application, one or more than one of the mechanisms might be involved [60].

A diffusion controlled release system (Figure 6) considers drug release rate from a device. The

rate was determined by the diffusion of drug molecules through the system. It can be further

classified into membrane–reservoir systems and monolithic (matrix) systems. The first system

contains a drug rich core (reservoir) coated by a membrane. Drug release rate is controlled by the

diffusion of drug molecules from the reservoir through the membrane. Membrane–reservoir

systems usually result in a zero-order release profile as long as the core provides a constant drug

supply (constant activity source). It can also follow first-order release when the core drug

concentration is changed, for example it becomes diluted by imbibed water (non-constant

activity source). In the second system, drugs are uniformly distributed in the matrix. The release

rate relies on the diffusion of drug through the matrix depending on loading level and drug

solubility in the matrix. A monolithic system usually has first-order release profile due to the

increase of diffusional distance and the decrease of drug concentration within the matrix over

period of time, but it is also geometry dependent.

16

Figure 6. Classification system for primarily diffusion controlled drug delivery systems. Only

spherical dosage forms are illustrated, but the classification system is applicable to any type of

geometry.

Swellable controlled release system (Figure 7) has a swellable glassy polymer matrix in

thermodynamically compatible solvent that undergoes transformation from glassy state to the

rubbery state. The matrix forming polymers that remain in the glassy state are rigid, whose drug

diffusion is negligible as compared to that in the rubbery region. Dosage forms made from

swellable hydrophilic polymers will be wetted in an aqueous environment caused by water

penetration. Then, the hydrated polymer chains gradually relax, swell, form pores and become a

gel layer to allow drugs to start to dissolve and diffuse out from the wetted zone. In this system,

as the volume of the device increases, so does the diffusion coefficient in the rubbery zone. The

drug release rate is controlled or altered by the change in polymer morphology by interaction

with the external release medium.

17

Figure 7. Schematic presentation of a swelling controlled drug delivery system containing dissolved

and dispersed drug (stars in circles and solid circles, respectively), exhibiting the

following moving boundaries: (i) swelling front, barrier between the swollen and non-swollen

matrix (ii) diffusion front, separate the swollen matrix containing dissolved and dispersed drug and

the swollen matrix containing dissolved drug only (iii) erosion front, barrier between bulk and

delivery system.

Erosion or degradation systems (Figure 8) are special matrix systems that have erosion or

degradation as the drug release rate limiting step. Different from the solubility dependent

diffusion matrix release systems, erosion systems can control drug release by limiting dissolution

or degradation rate of the matrix forming materials. The system can be further classified into

homogenous and heterogeneous erosion. Homogenous erosion happens when matrix undergoes

bulk degradation resulting in a gradual decrease of the molecular weight of the polymer matrix.

This leads to a higher drug diffusion coefficient in the matrix with time and the matrix will

eventually dissolve or disintegrate to release the remaining drugs. Heterogeneous erosion is

defined as having a rigid and hydrophobic matrix with minimal hydration in the release medium

to release the drug mainly by surface erosion of the matrix.

18

Figure 8. Degradation mechanisms of biodegradable polymeric nanoparticles: A) bulk erosion, B)

surface erosion.

Osmotic- controlled release mechanism is based on the regulation of osmosis through a

semipermeable membrane. Water diffusing across the semipermeable membrane is induced by

an existing chemical potential gradient between the hydrostatic pressure in the tablet core and the

dissolution medium. Since these types of devices have constant volume, the hydrostatic pressure

generated by an influx of water acts like an osmotic pump to force the release of a saturated

solution of the drug through delivery ports.

Depending on the mechanism that dominates the drug release rate, corresponding mathematical

models are derived according to its mechanism, geometry, boundary conditions and etcetera.

Therefore, it is critical to have the right models available for matched conditions to ensure the

validity of models and their power. The variety of known and unknown release mechanisms give

the potential of having different pharmaceutical models. For all these different models, we can

further specify them into two categories: 1) mechanistic realistic theories and 2) empirical and

semi-empirical mathematical models.

A mechanistic realistic mathematical model is based on math and physics equations that describe

real world phenomena; for example, dissolution of drugs or excipients, diffusion mass

transportation, and polymer transition between glassy and rubbery states. For these models, the

equations form basic mathematical theories with physical meaning. In many cases, the more

19

phenomena the system considers, the more complex the math will be, thereby resulting in greater

difficulties of finding solutions for the equations. Either analytical or numerical solutions are

acceptable. If the system is not very complex, analytical analysis can be performed to identify

system-specific parameters in the math equations for describing drug release kinetics. In

analytical solutions, if drug release rate/amount can be separated from all other variables and

parameters on one side of the equation, an explicit solution can be found, which means the

effects of the parameters in particular formulation can be directly seen. On the other hand, if the

drug release rate/amount cannot be separated from other variables and parameter, the solution is

called an implicit solution, which provides less direct effects of the parameters. If the system and

its mathematical model are complex, no analytical solution can be derived and thus one has to

use numerical methods to find the results. However, certain simplifications will be involved and

how to limit errors becomes a concern. For example, first derivatives might be approximated by

finite differences with small time steps and length steps.

20

Summary of the theories:

Model types Subtypes Citations

Theories based on

Fick's law of

diffusion (Figure 6)

Reservoir system: non-constant activity

source (first order release) [109]

Reservoir system: constant activity source

(zero order release) [109]

Modification of reservoir system: non-ideal

(e.g. crack, swelling during release) [110-112]

Monolithic solutions (thin film) [80, 109]

Monolithic solutions (spherical) [80, 113]

Monolithic solutions (cylinder) [78, 114-

116]

Monolithic dispersions (thin film) [81, 117]

Monolithic dispersions (planar and spherical

of homogenous and granular matrix) [82]

Modification of monolithic dispersions and

investigations in parameter effects [118-121]

Miscellaneous [122]

Theories

considering polymer

swelling (Figure 7)

Hydration and glassy to rubbery transitions

[93, 95,

123-125]

swelling plus diffusion simultaneously [126, 127]

Theories

considering polymer

swelling and

polymer and drug

dissolution

model based on polymer disentanglement

and diffusion layer [128-132]

sequential layer model [94, 95,

133, 134]

Theories

considering polymer

erosion/degradation

(Figure 8)

surface (heterogeneous) erosion [89, 135]

bulk (homogeneous) erosion [136]

degradation [137-139]

diffusion coupled with degradation [140-149]

surface erosion coupled with bulk erosion [145, 146]

pH sensitive and crystallization of

degradation products [142]

Table 3. Summary of mechanistic realistic mathematical models under different mechanisms

21

An empirical and semi-empirical mathematical model is based on empirical observations rather

than mathematically describable relationships. These types of theories can be instantly applied to

specific cases when comparing different drug release profiles using a specific parameter. This is

usually used when the mathematical models are not available and its prediction power is low.

However, it is still a great part of pharmaceutical modelling since there were not many

mechanistic models but a great amount of different formulations/applications with unknown and

complicated release mechanisms. The model usually only works under very extreme cases and

specific conditions. Nevertheless, scientists and industries need to pay extra caution and carefully

certify any potential violation of the model assumption whenever using these types of models.

Summary of important empirical and semi-empirical models:

Model types References

Peppas equation [90-92]

Hopfenberg model [87]

Cooney model [150]

Artificial neural network [151-154]

Table 4. Summary of important empirical and semi-empirical models for controlled release using

different mathematical approaches

For ion-exchange, its modelling was not started in the pharmaceutical field. As in other fields,

ion-exchange technology in the 1850s and has been far ahead of its theoretical understanding. It

was not until 1943 when de Vault published his theory of chromatography, that a basis was

provided permitting of a theoretical interpretation of the ion-exchange process [155].

Unfortunately, the process was very poorly understood. To have a complete understanding, it is

essential to understand equilibrium and kinetics. This data also needs to be combined with the

balance of the ion-exchange column expressed in two variables, position and time. Even for a

22

relatively simple case, in reality, the results are already extremely complicated from a

mathematical point of view. Thus, the first model of the ion-exchange chromatography equilibria

study provided by de Vault has failed to fit many of the experimental results. Afterwards,

contributions to ion-exchange mechanism mainly for ion/metal separation using resins or

chromatography have been made. For example, Tompkins and Mayer performed a series of

theoretical analyses of the column separation process [156]; Klinkenberg [157] reviewed heat

transfer analogs and Ketelle [158] introduced the transfer coefficient to overcome the difficulty

of describing the rate of mass transference between phases, rate of adsorption or chemical

reaction in ion-exchange chromatography. More mathematical analysis of ion-exchange has been

studied and analyzed, and physically and mathematically distinguished from general adsorption

using adsorption isotherm in 1958 by Klamer and Krevelen [159-162]. In the past 50 years, more

mathematical models incorporating ion-exchange coupled with other mechanisms (e.g. chemical

reaction [163-165], longitudinal diffusion [166, 167] , mass transportation through pores [168,

169], and etcetera) involved in chromatography applications (e.g. for salt/pH elution [170] and

protein retention [171-173]) were developed.

Mathematical modelling for ion-exchange was not translated and combined into pharmaceutical

applications such as ion-exchange resin and coating for controlled release purposes or sustained

release of transdermal patch until 15 years ago. In pharmaceutical industries, ionic polymers

such as Eudragit®, polyvinyl acetate phthalate (PVAP), and dextran have been widely used for

sustained release and other types of coatings [34-36, 174, 175]. Scientist started trying to use

empirical and semi-empirical models such as the ones developed by machine learning: artificial

neural networks (ANN) to describe kinetics of doxorubicin release from sulfopropyl dextran ion-

exchange microspheres [21, 176]. Effects of salt concentration on drug release of Eudragit® RS

due to ion-exchange was also examined [70]. Later, the first mechanistic model of ion-exchange

for microspheres (resins) were developed allowing for understanding of the dosage form and

drug delivery system [25, 26]. However, in order to design and predict the desired release

kinetics by applying different ionic polymer coating and understanding the relationship between

coating parameters and drug polymer interaction, there is much more to investigate and further

detailed models should be established.

23

1.4 Goals of This Work

Mechanistically, ion-exchange can be described as a two steps reaction with regular diffusion

followed by an adsorption-desorption reaction with stoichiometric exchange of solute with

bounded counter ions [25, 26]. This is different from pure diffusion, where species take place in

the solution outside the particle and within the particle and experience net movement of

molecules from a region of high concentration to a region of low concentration with certain

transport resistance without the adsorption-desorption reaction step [177, 178]. Ion-exchange

behavior has been described by several kinetic models in chromatography such as fixed bed

columns [179-181]. In general, the rate of the reaction at the ion-exchange site is assumed to be

faster than the rate of solute diffusion. Therefore, the reaction is considered as instantaneous.

Also, local equilibrium is assumed to exist at the solid liquid interface. Although the effects of

drug-polymer interactions on drug release from RS/RL polymer-based systems have been

extensively studied, very limited work has been reported on the mathematical analysis of the

loading and release kinetics of ionized drug for these systems.

Current formulation developments require time-consuming experiments to determine the optimal

process and formulation parameters. Many of these experiments require large quantities of

excipients and drugs, which is generally very scarce and expensive at early stages of drug

development, making it unfeasible. By verifying and applying a mechanistic model for ion-

exchange drug loading and release, the effects of various formulation and processing parameters

on the overall drug release kinetics can be quantitatively and accurately predicted, significantly

reducing the cost and time necessary to create reliable, high-quality products. A well-validated

mechanistic model capable of describing ion-exchange drug loading and release would be

extremely useful in the development of novel formulations.

The goal of this thesis is to establish a mathematical model to predict loading kinetics of anionic

drugs into polyacrylate (RS/RL) films and extract thermodynamic and kinetic parameters from

model simulation and experimental data which are useful for prediction of drug release kinetics

in polymer coated dosage forms. I hypothesize the development of a mechanistic model for

describing ion-exchange drug loading will assist formulation design by demonstrating and

predicting the effects of various parameters on drug loading kinetics, and equilibrium can be

predicted using the experiment validated model.

24

Three objectives for this work:

a. To develop a mechanistic model for ion-exchange and diffusion coupled drug loading in slab

geometry

b. To examine the correlations/trends between model–predicted drug loading kinetics with

experimental data

c. To analyze how loading kinetics is affected by different physicochemical properties of the

loading condition

1.5 Synopsis

Chapter 1.1 presents the background information and a comprehensive review of literatures

about ion-exchange in pharmaceutical field relates to the scope of this thesis.

Chapter 1.2 presents the background information and a comprehensive review of literatures

about polyacrylates in pharmaceutical field relate to the scope of this thesis.

Chapter 1.3 presents the background information and a comprehensive review of literatures

about pharmaceutical modelling relate to the scope of this thesis.

Chapter 2 presents:

(1) The mathematical derivation of ion-exchange drug loading mechanism onto polymeric thin

membranes.

(2) Acrylate polymers were used to perform drug loading experiments loaded with ibuprofen Na

and diclofenac Na in vitro.

Chapter 3 presents:

(1) Comparison between computer simulations and experimental data verified high accuracy of

the model, which describes the transportation phenomenon of ion-exchange mechanism better

than other current exist models.

(2) The effects of different physical chemical parameters on the loading kinetic and equilibrium

were investigated using numerical simulations and experiments.

25

Chapter 4 provides conclusions and future perspectives of this thesis.

26

26

Chapter 2 Modeling and Experimental Methods

27

2 Modeling and Experimental Methods

2.1 Theoretical Analysis

2.1.1 Mathematical modeling and derivation of an analytical solution

A kinetic model of drug diffusion with combination of ion exchange binding was developed to

describe the drug loading behavior of thin ionic polymer membrane that is immersed in a well-

stirred drug solution. It is assumed that the membrane is a homogenous matrix. Thus, diffusion

of a solute can be modeled by a single-phase diffusivity whose diffusion coefficient is

independent of drug concentration as well as having negligible transfer resistance and

electrostatic potential effect in the liquid phase. The thin membrane is pre-swollen or pre-

hydrated in DDI water before transferred into the drug solution. Therefore, polymer swelling and

mass transfer due to convection during drug loading are insignificant.

Figure 9 shows the ion-exchange mechanism of drug loading into a thin polymer membrane of

thickness H (m, length) with one side of the membrane’s surface area A (m2, length2) for each of

the two sides of the surface area. The upper and lower halves of the membrane have identical

diffusion and ion-exchange due to the symmetry of the thin membrane along the horizontal axis.

Therefore, only one-half of the membrane was considered, h = H/2 (m, length), where drug ions

exchange with the counterions at polymer-solution interface and diffuse inwards.

Figure 9. A schematic diagram of the ion-exchange drug loading mechanism in one half of the

polymer membranes. The blue circles represent the available binding site in the polymer

membrane that can be bounded by either drug (S) or counter ions (Cl-).

This is a diffusion and ion-exchange controlled drug absorption problem. Owing to a well-stirred

external medium (drug solution), it is assumed that ion exchange takes place at the interface x=h

28

between the membrane and the medium, while with the membrane (i.e., at 0<x<h), drug

diffusion is the rate limiting step, as it is much slower than ion-exchange. This reasonable

assumption was based on (a) the diffusion coefficient of counterions, which are normally small

ions like Na+ and Cl-, is much

larger than that of drug ions, which are normally organic compounds with molecular weights of a

few hundred Dalton; and (b) drug binding onto the polymer of opposite charges is normally

stronger than counterions. The drug concentration in the thin membrane as a function of t (s,

time) and position x (m, length) can be described by Fick’s second law:

∂C

∂t=

D ∂2C

∂x2 (1)

where D is the diffusion coefficient (m2/s, volume of membrane (m3)/distance in membrane

(m)/time (s)), and C is the concentration of the solute in the membrane (mole/ m3=mM, C (x,t)

amount of drug in the membrane (mole)/volume of polymer membrane (m3)). In a well-stirred

finite external solution, the initial and boundary conditions are:

C = 0 at x = 0 and t = 0 (2)

−DA∂C

∂x= V

∂Cb

∂t at x = h (3)

where ∂C

∂x indicates the flux of drug into the membrane of a unit surface area at x=h, Cb(t)

(mole/m3=mM, amount of drug in the external solution (mole)/volume of the external solution

(m3)) is the solute concentration in the external solution, V (m3, volume) is half of the volume of

the external solution, and h (m, length) and A (m2, length2) are the half-thickness and the surface

area of a single side of the thin membrane, respectively. Eq. (2) states the initial conditions and

Eq. (3) describes the mass balance at the interface between the thin membrane and the external

solution. Eq. (3) indicated that the amount of drug absorbed into the membrane equals the

amount of drug reduction in the external solution.

A second-order kinetic binding process is considered for the drug adsorption onto the ionic

groups in the polymer (ligand L) and its desorption from the binding site (SL) by stoichiometric

adsorption-desorption at the interface:

29

S + L kads

⇌kdes

SL (4)

where S is the solute in the external solution, L is the ligand or available binding site in the thin

membrane, and SL is the complex of the solute in the microsphere. kads ((mole/m3)-1/s= mM-1/s,

volume of membrane (m3)/amount of substance (mole)/time (s)), and kdes (m3 of membrane/m3 of

external solution/s, volume of membrane (m3)/volume of external solution (m3)/(time)-1) are the

association and dissociation rate constants, respectively. The respective concentrations of the

above species are Cb for S, (Cmax – C) for L, and C for SL, where Cmax (mole/m3=mM, maximum

amount of bound drug (mole)/volume of polymer membrane (m3)) is the maximum solute

binding capacity of the ion-exchange membrane.

Langmuir kinetics, rather than isotherm, is assumed to present the constitutive relationship

between Cb and C at the interface x=h. The change of Cb with time, a net result of adsorption rate

and desorption rate is expressed as

∂Cb

∂t= kdesC − kadsCb(Cmax − C) at x = h (5)

This is a rate equation induced from ion exchange. The first term of right hand side of the

equation is the rate of desorption and the second one is the rate of adsorption. The difference of

these two rates determines the change of external concentration with time, a variable boundary

condition. The mass balance between the matrix and external medium is described by the

following equations.

The total mass balance at any time is expressed as:

CbV = C0V − A ∫ C ∂xh

0 (6)

where the left hand side is the amount of solute in external volume at time=t, first term of the

right hand side is the total amount of solute in the external medium at time=0, C0 (mole/m3=mM,

initial amount of drug in the external solution (mole)/ volume of the external solution (m3)) is the

initial solute concentration in the external solution. The second term of the right hand side is the

amount of solute absorbed into the membrane at time=t, and A is the surface area of one side of

the membrane.

30

Substituting Eqs. (5) and (6) into (3) gives the boundary condition:

−DA ∂C

∂x= V(kdesC − kadsCb(Cmax − C)) = VkdesC − kads (C0V − A ∫ C ∂x

h

0) (Cmax − C)

at x = h. (7)

In order to convert the variable into dimensionless forms, the following substitutions were

applied:

ζ =x

h, 𝜏 =

Dt

ℎ2 , 𝜃 =𝐶

𝐶𝑚𝑎𝑥 (8)

∂C

∂t=

∂C

∂τ∙

∂τ

∂t ,

∂τ

∂t=

D

h2 (9)

∂C

∂t=

∂

∂τ(θCmax) ∙

D

h2 =DCmax

h2

∂θ

∂τ (10)

∂C

∂x=

∂C

∂ζ∙

∂ζ

∂x=

∂

∂ζ(θCmax) ∙

1

h ,

∂ζ

∂x=

1

h (11)

∂2C

∂x2 =∂

∂x(

∂

∂ζ(θCmax) ∙

1

h) =

Cmax

h

∂

∂x(

∂θ

∂ζ) =

Cmax

h2

∂2θ

∂ζ2 . (12)

Substituting Eqs. (11) and (12) into Eq. (1) reduced Eq. (1) to its dimensionless form:

DCmax

h2

∂θ

∂τ= D

Cmax

h2

∂2θ

∂ζ2 (13)

∂θ

∂τ=

∂2θ

∂ζ2 (14)

Let 𝜆 =𝑉

𝐴ℎ, 𝛼 =

𝑉𝐾𝑑𝑒𝑠ℎ

𝐷𝐴, 𝛽 =

𝑉𝐾𝑎𝑑𝑠ℎ𝐶𝑚𝑎𝑥

𝜆𝐷𝐴, 𝜃0 =

𝐶0𝜆

𝐶𝑚𝑎𝑥 (15)

Eq. (6) becomes:

CbV = C0V − A𝐶𝑚𝑎𝑥ℎ ∫ 𝜃 ∂ζ1

0 (16)

and Eq. (7) becomes:

−𝜕𝜃

𝜕ζ= 𝛼𝜃 − 𝛽( 𝜃0 − ∫ 𝜃𝜕ζ

1

0 )(1 − 𝜃) at ζ = 1 (17)

31

Let:

∂𝜃

∂𝜏=

∂𝜃

∂Y∙

∂Y

∂𝜏 (18)

∂𝜃

∂ζ=

∂𝜃

∂Y∙

∂Y

∂ζ (19)

and substituting Eq. (18) and (19) into Eq. (14) gives:

(∂𝜃

∂Y∙

∂Y

∂𝜏) =

∂

∂ζ(

∂𝜃

∂𝑌∙

∂Y

∂ζ) =

∂2𝜃

∂ζ ∂𝑌∙

∂Y

∂ζ+

∂2𝑌

∂ζ2∙

∂𝜃

∂𝑌 (20)

where Y = 1−ζ

√𝜏 (21)

∂Y

∂𝜏= −

1

2(1 − ζ)𝜏−

3

2 (22)

∂Y

∂ζ= − 𝜏−

1

2 (23)

∂2Y

∂ζ2 = 0 (24)

Substituting Eqs. (21), (22), and (23) into Eq. (20) gives:

(∂𝜃

∂Y∙

∂Y

∂𝜏) =

∂2𝜃

∂ζ ∂𝑌∙

∂Y

∂ζ+

∂2𝑌

∂ζ2 ∙∂𝜃

∂𝑌=

∂2𝜃

∂𝑌∙∂𝑌∙

∂𝑌

∂ζ=

∂2𝜃

∂𝑌2 ∙∂𝑌

∂ζ (25)

∂2𝜃

∂𝑌2 = −(𝑌

2)

∂𝜃

∂Y (26)

Integrating Eq. (26) twice gives:

𝜃 = B1√π erf (Y

2) + B2 (27)

where B1 and B2 are integration constants. B2 can be solved by using the following boundary

condition:

𝜃 = 0 at ζ = 0 and as 𝜏 approaches to 0. (28)

Therefore erf(∞) = 1, and B2 = −B1√π . (29)

32

𝜃 = B1√π [erf (1−ζ

2√𝜏) − 1] (30)

∂𝜃

∂ζ= −

𝐵1𝑒−(ζ−1)2

4𝜏

√𝜏, where −

∂𝜃

∂ζ=

𝐵1

√𝜏 at ζ = 1 (31)

∫ 𝜃𝜕ζ1

0= 𝐵1 (2 (𝑒−

1

4𝜏 − 1) √𝜏 − √𝜋𝑒𝑟𝑓𝑐 (1

2√𝜏)) (32)

Substituting Eqs. (30) - (32) into Eq. (17) gives:

𝐵1

√𝜏= −𝛼𝐵1√𝜋 − 𝛽( 𝜃0 − 𝐵1 (2 (𝑒−

1

4𝜏 − 1) √𝜏 − √𝜋𝑒𝑟𝑓𝑐 (1

2√𝜏)) (1 + B1√π ) (33)

By combining the total mass balance (Eq. (16)) and the concentration profile (Eq.(30)), the

fraction of solute remaining in the solution at any time can be obtained:

Mt

M0= 1 −

1

𝜃0𝐵1 (2 (𝑒−

1

4𝜏 − 1) √𝜏 − √𝜋𝑒𝑟𝑓𝑐 (1

2√𝜏)) (34)

where 𝐵1 is obtained from Eq. (33).

Model parameters Cmax and K can be determined by performing the following derivations. At

equilibrium, when ∂Cb

∂t= 0, Eq. (5) can be converted to the Langmuir isotherm:

C∞ = kadsCb,∞Cmax

kdes+kads Cb,∞=

KCb,∞Cmax

1+KCb,∞ (35)

where C∞ (mole/m3=mM, amount of drug in the polymer membrane at equilibrium (mole)/

volume of polymer membrane (m3)) and Cb,∞ (mole/m3=mM, amount of drug remaining in the

external solution at equilibrium (mole)/ volume of the external solution (m3)) are the equilibrium

solute concentration in the thin polymer membrane and the external solution, respectively, and K

((mole/m3)-1= mM-1, volume of the external solution/amount of substance) is the Langmuir

association constant and defined as

K =kads

kdes (36)

33

Substituting Eq. (35) into Eq. (6) and rearranging the equation at equilibrium gives:

M0

M∞− 1 =

Ah

V

KCmax

1+KCb,∞ (37)

(M0

M∞− 1)

−1

=V

AhKCmax+

VCb,∞

AhCmax (38)

where M∞

M0 is the fraction of solute remaining in the solution at equilibrium. The values of Cmax

and K can be obtained, respectively, from the slope and the ratio of the slope to the intercept of

the plot of (M0

M∞− 1)

−1

vs. Cb,∞.

For an entire membrane, identical diffusions occur simultaneously in both halves and the total

surface area is A’ = 2A, total external volume is 𝑉’ = 2𝑉, and total thickness is 𝐻 = 2ℎ.

2.2 Nomenclature

A surface area of one side of the membrane (a disk shape) (m2, length2)

Bi constant of integration

C solute concentration in the membrane (mM, amount of drug (mole) /volume of polymer

membrane (m3))

C0 initial concentration in the external solution (mM, amount of drug (mole)/volume of the

external solution (m3))

Cb solute concentration in the external solution (mM, amount of drug (mole)/volume of the

external solution (m3))

Cmax maximum solute binding capacity of the ion-exchange membrane (mM, amount of drug

(mole) /volume of polymer membrane (m3))

Cb, ∞ equilibrium solute concentration in the external solution (mM, amount of drug at

equilibrium (mole)/volume of the external solution (m3))

34

C∞ equilibrium solute concentration in the membrane (mM, amount of drug in the membrane at

equilibrium (mole)/volume of polymer membrane (m3))

D solute diffusion coefficient through the polymer (m2/s, volume of membrane (m3)/distance in

membrane (m)/time (s))

Di,w solute diffusion coefficient through the solvent (m2/s, volume of external solution

(m3)/distance in external solutions (m)/time (s))

H thickness of polymer membrane (m, length)

K Langmuir association constant ((mole/m3)-1= mM-1, volume of external solution (m3)/ amount

of substance (mole))

kads association rate constant ((mole/m3)-1/s= mM-1/s, volume of membrane (m3)/ amount of

substance (mole)/time (s),)

kdes dissociation rate constant (m3 of membrane/m3 of external solution/s, volume of membrane

(m3)/volume of external solution (m3)/(time)-1)

Kd partition coefficient

M0 initial amount of solute in the external volume (mole, amount of drug)

Mt amount of solute remaining in the external volume at any time (mole, amount of drug)

M∞ amount of solute remaining in the external solution at equilibrium (mole, amount of drug)

R radius of the disk (m, length)

t time variable (s, time)

x position variable (m, length)

V volume of the external solution (m3, length3)

α dimensionless parameter defined, 𝛼 =𝑉𝐾𝑑𝑒𝑠ℎ

𝐷𝐴

35

β dimensionless parameter defined, 𝛽 =𝑉𝐾𝑎𝑑𝑠ℎ𝐶𝑚𝑎𝑥

𝐷𝐴𝜆

θ dimensionless concentration defined, 𝜃 =𝐶

𝐶𝑚𝑎𝑥

θ0 dimensionless mass ratio defined, 𝜃0 =𝐶0𝜆

𝐶𝑚𝑎𝑥

ζ dimensionless spatial coordinate defined, ζ =x

h

τ dimensionless time defined, 𝜏 =Dt

ℎ2

*Note:

1. All the amount of substance in this study was based on measured mass of substance

(kg)/molecular weight (kg/mole)

2. Due to different experimental conditions, measurements were not always available in SI units.

Before parameter input and calculation, unit conversion needs to be done based on the following

definitions:

1 m 10 dm 100 cm 1000 mm 106 μm

1 m2 100 dm2 104 cm2 106 mm2 1012 μm2

1 m3 1000 dm3 = 1000 L 106 cm3 = 106 mL 109 mm3 = 109 μL 1018 μm3

1 kg 1000 g 106 mg 109 μg

1 M = 1 mole/L =1 mole/dm3

0.001 mole/cm3 1000 mM = 1000 mole/m3

106 μM

1 s 1/60 min 1/3600 h 1/86400 day

2.3 Methods

2.3.1 Experimental

2.3.1.1 Materials

Eudragit® RL 30 D and Eudragit® RS 30 D were generously donated by Evonik Industries

(Darmstadt, Germany). Ibuprofen Na and diclofenac Na and were purchased from Sigma-

Aldrich Canada (Oakville, ON, Canada). Dibutyl sebecate (DBS) was purchased from Morflex

(Greensboro, NC, USA).

36

2.3.1.2 Fabrication of RS/RL Membranes

Free membranes were prepared by mixing Eudragit® RS/RL 30 D with 20% w/w of DBS as

plasticizer based on the dry polymer weight overnight and drying at 37 °C for 24 hours.

Membranes of three different RS: RL ratios were prepared: 100:0 (RS100), 95:5 (RS95), and

90:10 RL (RS90). Samples with diameter of 1.27 cm were cut from the dried membranes for

testing. The membranes were transparent and flexible. Since Eudragit® RS and RL are

structurally similar and miscible in any ratio and the mixing time was long enough, we assumed

the membranes obtained high uniformity.

2.3.1.3 Drug Loading Study

The initial dry weight and thickness of each membrane sample cut from the dried membranes

were measured before immersing in drug solutions. Then, each membrane was pre-swollen in 20

mL of DDI water for 3 days to reach equilibrium swelling under gentle agitation. Assumed there

is no polymer lost during the swelling process.

Drug loading solutions were prepared by dissolving the appropriate amount of either ibuprofen

Na (Figure 10a) or diclofenac Na (Figure 10b) in DDI water. The pre-swollen RS100, RS95, and

RS90 membranes were immersed in 20 mL of ibuprofen Na loading solutions with initial

concentrations of 0.10, 0.15, 0.20, or 0.25 mM. Drug loading study was also performed for RS95

membrane in 20 mL diclofenac Na loading solutions with the same initial concentrations to

verify the effect of different binding kinetics. The membranes were agitated using a horizontal

shaker (75 rpm) at 22 °C. Concentration of drug remaining in the loading solutions was

measured at predetermined time points using UV-vis spectrophotometer until equilibrium was

reached.

37

a) b)

Figure 10. Chemical structures of a) ibuprofen Na and b) diclofenac Na.

2.3.1.4 Statistical Analysis

Each experimental conditions were independently repeated three times (n = 3). Mean square

deviation (MSD), root mean square deviation (RMSD), and coefficient of determination (R2) of

experimental versus model predicted/calculated values were be used to assess model accuracy

and goodness of fit.

38

Chapter 3 Results and Discussion

39

3 Results and Discussion

3.1 Validation of short-time solution of model for long-time numerical evaluations

The solution of the ion-exchange model is comprised of error function and related integrals.

Such solutions are generally suitable for numerical evaluation at short times, or in the early

stages of diffusion. Therefore, computer simulations of drug loading curves at long length of

time and comparison of the ion-exchange model with Crank’s long-time solution of diffusion

through a plane sheet [80] at large partition coefficient (Kd) was made to validate the model’s

predictions of the drug loading curves at extreme lengths of time (Figure 11). Crank’s solution is

defined as follows:

𝑀𝑡

𝑀0= 1 −

𝑉

𝐴𝐻𝐾𝑑

1+𝑉

𝐴𝐻𝐾𝑑

∑

2𝑉

𝐴𝐻𝐾𝑑(1+

𝑉

𝐴𝐻𝐾𝑑)

1+𝑉

𝐴𝐻𝐾𝑑+(

𝑉

𝐴𝐻𝐾𝑑)2𝑞𝑛

2exp (

−𝐷𝑞𝑛2𝑡

(𝑉

2𝐴)

2 )∞𝑛=1 , 𝑡𝑎𝑛𝑞𝑛 = −

𝑉

𝐴𝐻𝐾𝑑𝑞𝑛.

Drug loading curves converged as K (K=Kads/Kdes) increased as shown in the plot of fractional

drug remaining in the solution versus time (Figure 11a). No loading occurred when K is very

small. The parameters used in the computer simulations of drug loading curves at long length of

time were: Cmax = 23 mM, D = 3.15×10-9 cm2/s, C0 = 0.15 mM, h = 0.03 cm, R = 0.75 cm, and V

= 20 mL, while K increased from 0.001 to 105 mM-1. The ion-exchange model closely matched

Crank’s solution for pure diffusion model fitted with a reasonable Kd value (Figure 11b). The

parameters used in the comparison of ion-exchange model and Crank’s solution were the same as

those used previously with the exception of K = 105 mM-1 for ion exchange model and Kd = 40 in

place of K for Crank’s solution. Kd value was optimized by minimizing variance, global

minimum of RMSD using MATLAB TOMLAB global optimization toolbox. When adsorption

rate of the solute is much higher than desorption rate, solute diffusion becomes the rate-limiting

step of loading kinetics. Therefore, at large K values, the loading kinetics are solely diffusion-

dependent and are equal to Crank’s solution with the appropriate Kd value, demonstrating the

predictions made by ion-exchange model are valid even at long lengths of time. Convergence of

K value simulations, and Comparisons between Crank’s model and ion exchange model were

only investigated under industrial interested time windows (manufacturing usually cost less than

40

3 days per batch). At time equals to infinity, which is not the focus of our study, there might be

divergence.

a) b)

Figure 11. a) Fraction of drug remaining in the solution versus time as a function of Langmuir

association constant (K = kads/kdes). b) Comparison of the ion-exchange model with Crank’s solution

of diffusion through a plane sheet at large partition coefficient (Kd) value.

3.2 Verification of model with experimental data

3.2.1 Determination of model parameters

Values of K and Cmax were determined by plotting (M0

M∞− 1)

−1

vs. Cb,∞ (Figure 12). Using Eq.

(37) and (38), Cmax values were determined from the intersection of the line with y axis and K

values from the slope. Diffusivity (D) values of ibuprofen Na in RS90 – 100 membranes and

diclofenac Na through RS95 membranes were obtained through model fitting of a single

replicate of drug loading curve from the corresponding dataset. These fitted values were then

used for verification of the remaining drug loading curves of their respective dataset.

41

Determination of model parameters

Figure 12. The plot of (𝐌𝟎

𝐌∞− 𝟏)

−𝟏𝐯𝐬. 𝐂𝐛,∞ of ibuprofen Na loading from solutions with four values

of initial drug concentrations (0.1, 0.15, 0.2 and 0.25 mM) into pre-swollen RS100 membranes.

For the rest of the experimental groups, model parameters were determined in the same way as

above (See Appendix).

3.2.2 Goodness of fit of model

Predictions of drug loading kinetics made with the ion-exchange model were verified with

experimental data obtained from the drug loading studies (Figure 13 and 14 are representative

curves plotted up to 72 hr, which is a useful time period for pharmaceutical industries since this

particular step is usually less than 3 days in manufacturing cycle). Parameters K, D, and Cmax of

all loading conditions are summarized in Table 5. K and Cmax increased with higher RL content

in the membrane due to the higher amounts of QAGs and therefore, more available binding sites.

As expected, D also increased with higher RL content.

42

a) b)

Figure 13. Comparison of the experimental data and model predictions of drug loading into a)

RS100 polymer membranes from ibuprofen Na solution under various of initial loading

concentrations and b) membranes of various RS:RL ratio from 0.15 mM ibuprofen Na solution.

a) b)

Figure 14. Comparison of the experimental data and model predictions of drug loading into RS95

polymer membranes from a) ibuprofen Na solution and b) diclofenac Na solution.

Comparisons of the model predictions and the experimental data showed very strong agreement

between the model and experiments. The accuracy of the model predictions for each loading

condition was summarized in Table 6 (Appendix). For all loading conditions, MSD ranged from

0.0010 to 0.6545 and RMSD ranged from 0.0026 to 0.0707. The largest deviations between

model predictions and experimental data were from drug loading into RS90 membrane. Due to

higher RL contents, RS90 membranes continuously swelled slightly throughout drug loading and

started disintegrating at later time points, which violates the model assumption on no membrane

swelling and mass lost during the loading process. However, MSD and RMSD values for all sets

of loading conditions are still within reasonable accuracy (< 10% error).

43

Table 5. Summary of the parameters of all drug loading conditions.

Drug Film K (mM-1) D (10-10 cm2/s) Cmax (mM)

Ibuprofen Na

RS100 13 ± 2.6 2.8 32 ± 2.2

RS95 34 ± 3.3 50 40 ± 4.8

RS90 43 ± 2.3 90 62 ± 3.7

Diclofenac Na RS95 16 ± 3.0 4.7 38 ± 1.8

* In Table 5, errors were statistically analyzed based on repeated experiments (n=3) under each

different experimental condition.

Higher initial loading concentration increased the loading efficiency, but decreased loading

efficacy (Figure 13). Higher RL content increased drug loading rate as well as the amount of

drug bound at equilibrium.

The loading rates of diclofenac Na were much slower than ibuprofen Na due to the higher

molecular weight (Figure 14). The extra aromatic ring and bulky side chains of diclofenac Na

caused more steric hindrance, resulting in a 10 folds lower D value in comparison to ibuprofen

Na. For RS95, the K value of ibuprofen Na was approximately twice as much compared to

diclofenac Na.

3.3 Computational simulation and impacts of different parameters on drug loading kinetics

3.3.1 Influence of loading conditions

Computer simulations of drug loading curves under various volume ratios for a fixed membrane

size (λ) and initial loading concentrations (C0) were performed in order to investigate the effects

of loading conditions on drug loading kinetics, specifically, rate of drug loading and fraction of

drug remaining in the solution at equilibrium (M∞/M0) (Figure 15).

44

a) Volume ratio b) Initial loading concentration

Figure 15. Fraction of drug remaining in the solution versus time as a function of a) volume ratio of

external solution to membrane (λ) and b) initial drug concentration (C0) in the external solution.

For fixed membrane radius and thickness, as λ decreased (external volume decrease against

membrane volume with fixed membrane radius and thickness), the loading rate and the M∞/M0

increased. This phenomenon can be attributed to the competition of drug molecules for a fixed

number of binding sites. The available binding sites concentration is determined by maximum

binding sites minus the occupied sites, which will affect the ionic adsorption/desorption kinetics

shown in Eq(4 and 5). The amount of drug in the external solution is increased with higher λ. By

having fixed membrane radius and thickness and λ, having higher C0 (e.g. 1mM), Mt/M0 reaches

plateau earlier around 5 hr while the group with lower C0 (0.05mM) still have significant

decrease on the external drug amount at much later time points, still curving down around 10 hr.

Thus, higher external concentration drives drug loading to completion faster. On the other hand,

the large amount of drugs in the external solution results in a smaller fraction of drug being

loaded into the membrane due to saturation of binding sites. The parameters used in the

computer simulations of drug loading curves under different λ were: Cmax = 100 mM, D = 1×10-8

cm2/s, kads = 5×10-3 (mM∙ s) -1，kdes = 1×10-3 s-1, C0 = 0.25 mM, h = 0.02 cm, and R = 1 cm.

Figure 15a illustrates fractional drug remaining in the solution versus time for different λ.

Drug loading rate and M∞/M0 increased as C0 increased (Figure 15b). Higher C0 brings higher

driving force for drug diffusion into the polymer, which increases the probability of drug

molecules being bound and accelerates drug loading. The parameters used in the computer

simulations of drug loading curves under different C0 were identical as those used under different

λ with the exception of V = 20 mL.

45

3.3.2 Influence of drug and membrane properties

Computer simulations of drug loading curves under various maximum binding capacity (Cmax),

Langmuir association constant (K), and drug diffusion coefficient (D) were performed in order to

investigate their effects on drug loading rate and M∞/M0 (Figure 16).

a) Maximum binding capacity b) Langmuir association constant

c) Drug diffusion coefficient

Figure 16. Fractional drug remaining in solution versus time for membranes with various a)

maximum solute binding capacities (Cmax), b) Langmuir association constants (K= kads / kdes), and c)

drug diffusion coefficients (D) through the membrane.

As Cmax increased, the rate of drug loading increased and Mt/M0 at equilibrium decreased (Figure

16a), indicating that Cmax is an important parameter that determines both the kinetics and

thermodynamics of drug loading process. Cmax provides driving force for diffusion while limiting

the maximum amount of drug that can be loaded by the ion-exchange mechanism. The number

of binding sites determines the amount of drug loaded at equilibrium as well as the loading rate.

Therefore, polymer membranes with higher charge density (Cmax) can achieve faster binding

kinetics and more efficient loading. The parameters used in the computer simulations of drug

46

loading curves under various Cmax were: D = 1×10-8 cm2/s, kads = 5×10-3 (mM∙ s)-1，kdes = 1×10-3

s-1, C0 = 0.25 mM, h = 0.02 cm, R = 1 cm, and V = 20 mL.

Drug loading rate increased and the fraction of drug remaining in the solution at equilibrium

decreased with higher K (Figure 16b). The relative value of kads and kdes represent the affinity of

a drug to a certain polymer material versus solution. Higher drug- membrane affinity offers

faster and higher level of drug loading. The parameters used in the computer simulations of drug

loading curves under various K were identical as those used under various Cmax with the

exception of Cmax = 100 mM in place of kads and kdes.

Changes in D only affected the rate of drug loading (Figure 16c). With a higher D, the drug

loading rate increased and loading reached equilibrium faster. However, equilibrium fraction of

drug remaining in solution stayed the same since it is not determined by the kinetic properties of

the system, but by the thermodynamic properties. The parameters used in the computer

simulations of drug loading curves under different D were identical as those used under various

Cmax with the exception of Cmax = 100 mM in place of D.

3.4 Prediction of loading yield and loading level in the membrane

To obtain desirable drug loading level and yield, the loading conditions need to be manipulated.

λ and C0 have significant impact on the maximum amount of drug that can be extracted from the

solution. For quantitative analysis, the relationship of λ and C0 with the equilibrium drug

concentration in the membrane, C∞, is derived by combining Eq. (6) and (35):

λ𝐶∞

𝐾(𝐶𝑚𝑎𝑥−𝐶∞)+ 𝐶∞ = λ𝐶0 (39)

Further substitution gives:

𝐶𝑚𝑎𝑥𝐾

λ[1+K𝐶0(𝑀∞𝑀0

)]= (

𝑀∞

𝑀0)−1 − 1 (40)

47

The dependence of C∞ and 𝑀∞

𝑀0 on λ and C0 is implicitly described by Eqs. (39) and (40). Their

values can be solved by using numerical methods.

As shown in Figure 17, C∞ quickly increased with higher λ and C0 for a given C0 and λ,

respectively. As λ increased from 20 to 200, the curves became less linear at the beginning and

merged at approximately C0 = 10 mM. A similar trend can also be seen in the plot of C∞ vs. λ

(Figure 17b). The curvature of the plots increases with increasing C0. In addition, as λ increased

from 50 to 200, C∞ scarcely changed with λ at high C0 levels, indicating that drug loading has

reached the maximum equilibrium capacity of the membrane and no more drug can be loaded.

Figure 18 illustrates how λ and C0 affect drug loading yield (1 −𝑀∞

𝑀0). Higher loading yield can

be achieved by lowering λ and C0 for a given drug-polymer system. As λ increased, (1 −𝑀∞

𝑀0)

drastically decreased with increasing C0. This result suggests low volume ratio such as 20 in this

case should be selected for the highest loading yield.

By considering the effects observed in Figure 17 and Figure 18 together, it can be concluded that

the highest drug loading level and the highest loading yield cannot be achieved at the same time.

Therefore, a compromise in either loading levels or loading yield is needed. In cases where the

both highest possible loading level and the loading yield are desired, a plot of (1 −𝑀∞

𝑀0) and C∞

in the same graph is useful. As demonstrated in Figure 19, the intersections of (1 −𝑀∞

𝑀0) and C∞

curves are the highest possible values of both loading level and the loading yield to coexist. The

two curves for λ = 20 provide the highest overall values of loading yield and loading level at C0 =

5 mM.

48

a) b)

Figure 17. Equilibrium drug concentration in the ion-exchange membrane as a function of a) the

volume ratio of solution to membrane (λ) and b) initial loading concentration in the solution (C0).

Cmax = 100 mM, kads = 5×10-3 (mM∙ s)-1, kdes = 1×10-3 s-1, h = 0.02 cm, R = 1 cm.

a) b)

Figure 18. Drug loading yield as a function of a) the volume ratio of solution to membrane (λ) and

b) initial loading concentration in the solution (C0). Cmax = 100 mM, kads = 5×10-3 (mM∙ s)-1, kdes =

1×10-3 s-1, h = 0.02 cm, R = 1 cm.

49

Figure 19. Drug loading yield (solid lines, left y-axis) and equilibrium drug loading (dashed lines,

right y-axis) as a function of C0 and λ.

All the parameters are the same from Figure 17 and 18. Since there is tradeoff between the

highest drug loading level and the highest loading yield, a plot of (1 −𝑀∞

𝑀0) and C∞ in the same

graph demonstrated in Figure 19, the intercepts of (1 −𝑀∞

𝑀0) and C∞ curves are the highest

possible values of both to coexist. It is also important to know that the above simulation only

provides a tool to help us to assess the right loading conditions but not always picking the

intercept. Choices and considerations on some parameters might weighted heavier than others

depending on the cost of the drugs, targeted dose, and reusability of the external drug solution

(e.g. effects from hydrolysis, degradation, instability).

50

Chapter 4 Conclusion and Future Perspectives

51

4 Conclusions and Future Perspectives

4.1 Highlights

a mechanistic mathematical model and its analytical solution for ion-exchange drug

loading into polymer membranes from a finite external volume was provided

the parameters and properties determined from drug loading kinetics model could be

easily applied to predict drug release and assist industries to formulate desired products

cost efficiently

equations obtained can be employed for predicting parameters in drug coating for a given

drug and a desirable drug loading and release kinetics

4.2 Conclusions

A new mechanistic model has been developed to describe drug loading kinetics into ion-

exchange thin membranes from a finite external solution. Model predicted loading curves of

ibuprofen Na and diclofenac Na into pre-swollen RS100, 95, and 90 thin membranes were shown

to strongly agree with experimental data. The results demonstrated that transport of anionic drugs

in RS/RL polymer-based dosage forms follow ion-exchange and diffusion kinetics. Ion-exchange

model developed in this work more accurately described the drug release kinetics than pure

diffusion models that are currently used in formulation development. Numerical analysis

revealed various factors that influence both the kinetics and thermodynamics of drug loading,

including the maximum binding capacity, surface area, membrane thickness,

association/dissociation constants (K), and initial loading concentration. The effects of these

process parameters and material properties on drug loading kinetic were quantified and can be

further extrapolated for application in drug release from coated dosage forms. The simulation

also quantitatively provided us ideas on how different chemical physical parameters involved in

the mass transport affect the equilibrium and kinetics of the drug loading. Suitable polymer

amount or different ratios of polymers in a polymer blend can be predicted to achieve the

optimized drug loading with both satisfied efficacy and efficiency. Although the ion-exchange

model was derived for thin membranes, further modifications can be easily made for other

52

geometries such bead and tablet coating as well as for ionic drug encapsulation kinetic in

nanoparticles. The model can also be employed for designing dosage forms requiring a target

loading content and loading efficiency.

4.3 Original Contributions of This Thesis

This thesis derived an approximate analytical solution for ion-exchange-diffusion coupled drug

loading mechanism for membrane geometry based on fundamental understanding of

physicochemical properties of the polymers and the drugs. This is the first work applying the

ion-exchange-diffusion coupled mathematical model and computer simulation to investigate the

kinetics and equilibration of ionic drug loading onto Eudragit® RL/RS membranes with varying

ratios. The simulated loading kinetics was compared to experimentally measured profiles under

four different initial concentrations for each of two drugs, which is more comprehensive than any

previous study. This work suggested that the model can accurately predict ion-exchange-

diffusion coupled drug-polymer interactions and mass transfer in all the above cases and can be

applied to other ionic drugs and polymers. The information acquired is useful for simplifying

formulation design, increase manufacturing efficiency, and limiting human errors.

4.4 Limitation of the Work and Future Perspectives

Although a mechanistic model has been derived that better described the interaction between

ionic drug and polymer in solid dosage forms, there are some limitations in the studies which

require further investigation. First, this model assumed no polymer swelling or erosion during

the entire process. However, in real life, most of the polymer we used for coating or formulation

design will swell due to hydration or dissolve. Thus, ideally dimensional change with time

should be coupled to this ion-exchange-diffusion coupled model. If swelling or erosion is

included, the complexity of the mathematics would increase, which would cause difficulties in

solving the equations and numerical methods such as finite difference or finite element methods

have to be used (refs Zhou et al). Second, this model only works for polymers that do not

dissolve nor degrade. For example, during the verification, we used water-insoluble Eudragit

RL/RS and we assumed no polymer lost during the entire process. However, many polymers

applied for coating are soluble or degradable. Therefore, dissolution and degradation term should

53

be included. Third, for the value of the parameters, K and Cmax were obtained from loading

experiments based on the mechanistic model. Because this model has mechanistic limitations,

the parameter values might not be the identical to the true values, which means we might have

obtained apparent values for K and Cmax. Different experiments and methods for determining K

and Cmax may be needed.

For future perspective, although the ion-exchange-diffusion model was derived for thin

membranes, further modifications can be made for other geometries such bead and tablet coating

as well as for ionic drug encapsulation kinetic in nanoparticles. Drug release model can also be

developed. Limitations listed above can be addressed. Coupling more mechanisms and establish

a more general and comprehensive model is possible. Analytical solution might not be able to be

obtained, but very accurate numerical simulations can be achieved by using super computers or

novel technologies due to the great improvement of computer science and artificial intelligence.

4.5 Acknowledgements

This work was supported by the Operating grant from Ontario Research Foundation-Research

Excellence (ORF-RE) in partnership with Patheon and Natural Sciences and Engineering

Research Council (NSERC) equipment grants to X.Y. Wu. The Departmental Scholarships to Yi

Li and K. Chen, the NSERC CGS D to J. Li, and Hoffman-La Roche/Rosemarie Hager Graduate

Fellowship to K. Chen are also acknowledged.

54

Appendices

Answer for B1:

𝐵1 =

𝛼√𝜋τ + 1 − √τβ (2 (𝑒−1

4τ − 1) √τ − √𝜋𝑒𝑟𝑓𝑐 (1

2√τ)) ± √(√τβ (2 (𝑒−

14τ − 1) √τ − √𝜋𝑒𝑟𝑓𝑐 (

1

2√τ)) − 𝛼√𝜋τ − 1)

2

+ 4√τ β𝜃0 (√𝜋τ β (2 (𝑒−1

4τ − 1) √τ − √𝜋𝑒𝑟𝑓𝑐 (1

2√τ)))

2√𝜋τ β (2 (𝑒−1

4τ − 1) √τ − √𝜋𝑒𝑟𝑓𝑐 (1

2√τ))

B1 was calculated by MATHEMATICA, only the positive answer was taken and the result was

imported into MATLAB.

Table 6. Summary of the mean square deviation (MSD) and root mean square deviation (RMSD)

values of the model predictions and the experimental data for all drug loading conditions. Each

loading concentrations have three replicates (n = 3).

Ibuprofen Na:

Diclofenac Na:

MSD 0.0856 0.0476 0.0169 0.0042 0.0043 0.0384 0.0051 0.0184 0.0098 0.0092 0.0196 0.0127

RMSD 0.0245 0.0183 0.0109 0.0054 0.0055 0.0164 0.0060 0.0114 0.0083 0.0080 0.0117 0.0094

MSD 0.0347 0.0370 0.0770 0.0010 0.0181 0.1084 0.0032 0.0057 0.0838 0.0072 0.0184 0.0586

RMSD 0.0155 0.0160 0.0230 0.0026 0.0112 0.0273 0.0047 0.0063 0.0240 0.0071 0.0113 0.0201

MSD 0.4245 0.5937 0.2820 0.5214 0.4424 0.3496 0.0297 0.1495 0.2479 0.6545 0.2940 0.3810

RMSD 0.0543 0.0642 0.0442 0.0602 0.0581 0.0517 0.0151 0.0322 0.0415 0.0707 0.0474 0.0539RS90

Co

mp

osi

tio

n

Concentrations (mM)

0.1 0.15 0.2 0.25

RS100

RS95

MSD 0.0100 0.0268 0.0626 0.0291 0.0163 0.0215 0.0147 0.0217 0.0145 0.0629 0.1618 0.0169

RMSD 0.0083 0.0136 0.0208 0.0143 0.0107 0.0123 0.0102 0.0123 0.0100 0.0209 0.0335 0.0108

Concentrations (mM)

0.1 0.15 0.2 0.25

55

Figure 20. Comparison of prediction power between the diffusion model and the ion-exchange-

diffusion coupled model.

In Figure 20, it was obvious that our ion-exchange-diffusion coupling model gave much more

accurate predictions for the drug loading kinetics compared to the diffusion model. The diffusion

model was experiencing a much slower loading kinetic compared to the ion-exchange

mechanism. On the other hand, the experimental data matched well with the kinetic simulated by

ion-exchange-diffusion model.

56

Figure 21. Determination of model parameters. The plot of (𝐌𝟎

𝐌∞− 𝟏)

−𝟏

𝐯𝐬. 𝐂𝐛,∞ of ibuprofen Na

loading from solutions with four values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25 mM)

into pre-swollen RS95 membranes.

Figure 22. Determination of model parameters. The plot of (𝐌𝟎

𝐌∞− 𝟏)

−𝟏

𝐯𝐬. 𝐂𝐛,∞of ibuprofen Na

loading from solutions with four values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25 mM)

into pre-swollen RS90 membranes.

57

Figure 23. Determination of model parameters. The plot of (𝐌𝟎

𝐌∞− 𝟏)

−𝟏

𝐯𝐬. 𝐂𝐛,∞ of diclofenac Na

loading from solutions with four values of initial drug concentrations (0.1, 0.15, 0.2 and 0.25 mM)

into pre-swollen RS95 membranes.

Figure 24. Demonstration of the simulation using MATLAB. Coded by Yi Li.

58

Figure 25. Demonstration of the simulated data points extraction from MATLAB as matrix

a) Volume Ratio b) Initial Loading Concentration

Figure 26. Demonstration of raw MATLAB plots simulating effects on loading regarding to a)

different volume ratio and b) different initial loading concentration. Same values of parameters

were used as in Chapter 2.

All the figures shown in the result section were replotted using extracted data points from

MATLAB seen in Figure 25.

59

59

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