PhD_dissertation_Niels_Erik_Olesen

Mechanistic Modelling of Micellization and Aqueous
Cyclodextrin Solutions with Applications in
Biopharmaceutics
Ph.D. dissertation
M.Sc. Eng. Niels Erik Olesen
July 2015
Roskilde University, Denmark
Department of Science, Systems and Models

i
Supervisors
Professor, Ph.D. Peter Westh
Soft Matter and Functional Biomaterials
Department of Science, Systems and Models (NSM)
Roskilde University, Denmark
Divisional director, Ph.D. René Holm
Biologics and Pharmaceutical Science
H. Lundbeck A/S
Copenhagen, Denmark

ii
Preface
The present PhD thesis entitled:
Mechanistic Modelling of Micellization and Aqueous Cyclodextrin Solutions with
Applications in Biopharmaceutics
is submitted to Roskilde University.
The experimental work was conducted at H. Lundbeck A/S, Department of Biologics and
Pharmaceutical Science.
This dissertation is based on the following enclosed appendices:
1) Olesen, N. E., Holm, R., & Westh, P. (2014). Determination of the aggregation number for micelles by
isothermal titration calorimetry. Thermochimica Acta,588, 28-37. Published
2) Olesen, Niels Erik, Peter Westh, and René Holm. "Determination of thermodynamic potentials and
the aggregation number for micelles with the mass-action model by isothermal titration calorimetry: A
case study on bile salts." Journal of colloid and interface science 453 (2015): 79-89. Published
3) Olesen, Niels Erik, Peter Westh, and René Holm. The displacement of drugs from cyclodextrin
complexes by bile salts: modeling of the biopharmaceutical process. Submitted
4) Olesen, Niels Erik, Peter Westh, and René Holm. A biopharmaceutical model to investigate the
impact of cyclodextrin excess on oral drug absorption from aqueous solutions. In preparation.
Additional contributed manuscripts during the Ph.D. (secondary material for which declaration
of co-authorships not have been collected and therefore not is part of the evaluation of this
dissertation):
5) Thermodynamic investigation of the interaction between cyclodextrins and preservatives - application
and verification in a mathematical model to determine the needed preservative surplus in aqueous
cyclodextrin formulations. Holm R., Olesen N. E., Alexandersen S., Mu H. European Journal of
Pharmaceutical Sciences. Submitted
6) Statistical analysis of a method to predict drug-polymer miscibility. Knopp, M. M., Olesen, N. E.,
Rades, T. & Holm, R. Journal of Pharmaceutical Sciences. Submitted
7) Knopp, M. M., Olesen, N. E., Holm, P., Löbmann, K., Holm, R., Langguth, P., & Rades, T. (2015).
Evaluation of Drug–Polymer Solubility Curves Through Formal Statistical Analysis: Comparison of
Preparation Techniques. Journal of pharmaceutical sciences, 104(1), 44-51. Published
8) Knopp, M. M., Olesen, N. E., Holm, P., Langguth, P., Holm, R., & Rades, T. (2015). Influence of
Polymer Molecular Weight on Drug–Polymer Solubility: A Comparison between Experimentally
Determined Solubility in PVP and Prediction Derived from Solubility in Monomer. Journal of
pharmaceutical sciences. Published
9) A comparative study of different differential scanning calorimetric methods for prediction of drug-
polymer solubility. Knopp M. M., Tajber L., Tian Y., Olesen N. E., Jones D. S., Kozyra A., Löbmann K.,
Paluch K., Brennan C. M., Holm R., Healy A. M., Andrews G. and Rades T. Molecular Pharmaceutics.
Submitted
10) An experimental evaluation of powder flow predictions in small-scale equipment based on Jenike’s
hopper design methodology. Søgaard S. V., Olesen N. E, Madsen M. H., Allesø M, Garnaes J,
Rantanen J. Powder Technology. Submitted

iii
Abstract
Cyclodextrins are used as drug solubilizers to enhance oral drug absorption. Clear
pharmaceutical dosing guidelines for the use of cyclodextrins are, however, still requested by the
pharmaceutical industry. In this work, a model to predict the intestinal drug absorption from an
aqueous cyclodextrin solution is proposed. A special focus is on the reported overdosing of
cyclodextrin whereby the absorption is decreased due to a decreased free drug concentration. The
model is based on the underlying biopharmaceutical considerations with emphasis on the
multiple chemical equilibria involving cyclodextrins in the small intestine. Especially the
influence of bile salts is discussed in details as these are thought to be the most important
competitive agent in the equilibrium system.
In the small intestine, bile salts coexist on free and micellar form. A characterization of
micellization of bile salts was therefore initially required as conducted by means of isothermal
titration calorimetry (manuscript 2). Bile salts form a special kind of micelles as their
amphiphilic nature does not originate from a polar head group and an aliphatic chain like
traditional surfactants. They form micelles with aggregation numbers much lower than classical
micelles and their status as true micelles has thus been disputed. Therefore emphasis was put on
describing the chemical thermodynamics of bile salt micellization (manuscript 2) including a
study of the statistical properties of parameter estimation by isothermal titration calorimetry for
micelles (manuscript 1 and 2).
From these studies a chemical equilibrium model was constructed. However, for the drug
compounds studied in this thesis the absorption was only depending on the total bile salt
concentration. The amount of overdosing by cyclodextrins was thus quantified by simple means
by a quantity denoted the intestinal drug solubilizing capacity
SC
totD (manuscript 3). Based on this
knowledge, the equilibrium model was coupled to the kinetics of the drug absorption process.
This dynamical model predicted a decreased absorption rate when cyclodextrins were dosed to
achieve a higher intestinal drug solubilization capacity than needed which correlated with in vivo
studies conducted in rats. By examining the full parameter space of the model it was derived that
overdosing cyclodextrins is unable to affect the fraction absorbed when the so-called
dimensionless dose concentration
*
totD is small (below 0.1) no matter the amount of cyclodextrin
overdosing (manuscript 4). Thereby guidance for when cyclodextrins can be applied without
special precautions is provided.

iv
Resume (Abstract in Danish)
Cyclodextriner anvendes som solubilisatorer til lægemidler for at forbedre den orale
lægemiddelabsorption. Klare farmaceutiske retningslinjer for dosering af cyclodextriner er dog
stadig efterspurgt af lægemiddelindustrien. I denne afhandling foreslås en model til at forudsige
den intestinale lægemiddelabsorption fra en vandig cyclodextrin opløsning. Et særligt fokus er
lagt på den rapporterede overdosering af cyclodextriner, hvorved absorptionen falder som følge
af en nedsat fri lægemiddelkoncentration. Modellen er baseret på bagvedliggende
biofarmaceutiske overvejelser med vægt på de mange kemiske ligevægte, der involverer
cyclodextriner i tyndtarmen. Især galdesaltes indflydelse bliver diskuteret i detaljer, da disse
formodes at være den vigtigste konkurrerende agent for cyclodextrinets solubilisering af
lægemidlet.
I tyndtarmen sameksisterer galdesalte på fri og micellær form. En karakterisering af
micelledannelse af galdesalte blev derfor indledningsvist udført ved hjælp af isotermisk titrerings
kalorimetri (manuscript 2). Galdesalte danner usædvanlige typer af miceller, da galdesaltenes
amfifile natur ikke stammer fra en polær hovedgruppe og en alifatisk kæde som traditionelle
surfaktanter. Micellerne har et meget lavt aggregeringstal og deres status som sande miceller er
således omstridt. En beskrivelse af den kemiske termodynamik for micelledannelse af galdesalte
blev derfor betonet (manuscript 2), herunder en undersøgelse af de statistiske egenskaber for
parameter estimering med isotermisk titrerings kalorimetri for miceller (manuscript 1 and 2).
Fra disse studier blev en kemisk ligevægtsmodel konstrueret. Imidlertid viste det sig, at
absorption kun afhænger af den totale galdesaltskoncentration for lægemidlerne behandlet i
denne afhandling. Mængden af overdosering af cyclodextriner kan således kvantificeres enkelt
ved hjælp af en fysisk størrelse betegnet den intestinale lægemiddel-solubiliserende kapacitet
SC
totD (manuscript 3). Baseret på dette, blev ligevægtsmodellen koblet til kinetikken i
absorptionsprocessen. Denne dynamiske model forudsagde en nedsat absorptionsrate, når
cyclodextriner doseres til at opnå en højere intestinal lægemiddel-solubiliserende kapacitet end
påkrævet og dette korrelerede med in vivo studier udført i rotter. Ved at undersøge modellens
fulde parameterrum blev det vist, at overdosering af cyclodextriner er ude af stand til at påvirke
den absorberede lægemiddel fraktion, når den såkaldte dimensionsløse dosis koncentration
*
totD
er lille, uanset graden af overdosering af cyclodextriner (manuscript 4). Derved kan retningslinjer
foreslås, for hvornår cyclodextriner kan anvendes uden særlige forholdsregler.

Contents
Supervisors....................................................................................................................................... i
Preface............................................................................................................................................. ii
Abstract..........................................................................................................................................iii
Resume (Abstract in Danish)......................................................................................................... iv
Chapter 1: Background and scope .................................................................................................. 1
Chapter 2: Introduction................................................................................................................... 3
Chapter 3: Model building.............................................................................................................. 8
3.1 Model building - introduction............................................................................................... 8
3.2 The pharmaceutical application............................................................................................ 8
3.2.1 Equilibrium part............................................................................................................. 8
3.2.2 Kinetic part - the absorption process ........................................................................... 15
3.2.3 A preview into the coupling of the equilibrium and kinetic part................................. 17
3.3 Micelles............................................................................................................................... 18
3.4 Determination of thermodynamic potentials of micellization ............................................ 21
3.5 Understanding the thermodynamics of chemical reactions involving hydrophobic
substances in aqueous solutions................................................................................................ 23
Chapter 4: Parameter estimation................................................................................................... 32
4.1 Parameter estimation - introduction.................................................................................... 32
4.2 ITC measuring principles.................................................................................................... 33
4.3 Computational aspects ........................................................................................................ 38
4.4 ITC-data supports the model n·S ⇌ Mn for micellization of bile salts............................... 41
4.5 Future outlook..................................................................................................................... 42
4.5.1 Experimental design..................................................................................................... 42
4.5.2 Global fitting versus mixed-effects modelling ............................................................ 48
Chapter 5: Model solution ............................................................................................................ 50
5.1 Model solution - introduction ............................................................................................. 50
5.2 The general mass balance model ........................................................................................ 51
5.3 Impact of cyclodextrin excess on oral drug absorption from aqueous solutions................ 57
5.4 A water pressure analogy.................................................................................................... 62
6: Conclusion ................................................................................................................................ 64
References..................................................................................................................................... 65
Appendices.................................................................................................................................... 74

1
Chapter 1: Background and scope
Drug development is the process of bringing a new chemical entity (NCE) to the market. Due to
the high throughput screening used in modern drug discovery a large number of NCEs makes
their way into the early phases of drug development. Drug molecules are rarely administered
alone, but in a drug formulation together with excipients improving the properties of the drug
product. Biopharmaceutics refers to the process of examining the relation between the
physicochemical properties of the drug molecule, the excipients and the route of administration
on the extent on drug absorption.
To cope with the numerous amounts of drug molecules, an initial screening of their potential to
succeed in the development process (the so-called drugability) is performed. This investigation
of fundamental properties of the drug molecule is one of the task of preformulation [1]. For this
purpose it is desirable to have a formulation, which is flexible, fast to produce and still reflects
the interplay between the drug properties and the final oral route of administration. A solution
based on cyclodextrins is often the choice when dealing with low aqueous soluble compounds in
early preformulation studies of the drugability.
Cyclodextrins are cyclic oligosaccharides with a hydrophilic outer surface and a lipophilic cavity
[2].The solubilisation potential of cyclodextrins can be attributed to their ability to accommodate
nonpolar organic compounds by formation of inclusion complexes as illustrated in Fig. 1.1. The
most common -, -, and -cyclodextrins consist of 6, 7 and 8 glucopyranose units, respectively.
In order to increase the aqueous solubility of the cyclodextrins, a large number of derivatives
such as methylated, hydroxypropylated, and the negatively charged sulfobutylated cyclodextrins,
has been synthesized [3].
Figure 1.1: Illustration of the inclusion of the bile salt glycochenodeoxycholate into the
hydrophobic cavity of γ-cyclodextrin. Reproduced with permission from Holm et al. 2012 [16].

2
This project was conducted at the preformulation department at H. Lundbeck A/S. The
application of cyclodextrins therefore focused on a preformulation setting where cyclodextrins
are applied to increase the apparent solubility of the drug molecule. Cyclodextrins are also used
in marketed drug products but for this type of application there are typically also other
mechanisms of action of cyclodextrins such as improving the dissolution rate, the drug stability
and enhance the intestinal permeability. These mechanisms are, however, outside the scope of
the current work.
Due to the screening process necessitated by the high number of drug molecules, the nature of
preformulation is often to find a problem solution that works in practice rather than finding an
optimal solution for a given drug molecule. The emphasis of mechanistic/mathematical
modelling in this project is somewhat in contradiction with this, as mathematics in general is
concerned with finding an optimal or exact solution to a given problem. Consequently, the
current thesis contains a lot of theoretical results whereas the weight of experimental
observations is smaller and frequently applied as means of validating the theoretical findings.
Preformulation has neither been the primary problem of concern for a large amount of the work
in this thesis. The implications of the current thesis for the preformulation application should
therefore be seen in this light.

3
Chapter 2: Introduction
Oral delivery maintains to be the most frequent route of drug administration. The recent trend of
high throughput assays in drug discovery has, however, resulted in an increasing number of drug
candidates with a poor aqueous solubility [4]. Drug molecules with a low solubility tend to
precipitate and typically have an insufficient and very variable extent of absorption as only drug
molecules on free solvated form are available for uptake over the intestinal membrane. Currently
60-70% of the compounds the pharmaceutical industry develops are classified as having low
solubility [4], [5].
The absorption of a cyclodextrin complexed drug upon oral administration is a relatively
complicated process as outlined in Fig. 2.1. Only the free form of the drug, which in equilibrium
with the complexed species, is available for absorption and hence for providing the
pharmacological effect of the drug. In addition bile salt present in the small intestine has the
ability to displace the complexed drug from the cyclodextrin cavity thus altering the absorption
kinetics [6], [7]. As the bile salt concentration in the intestine is present at supramicellar
concentration the free monomeric bile salt will be in equilibrium with bile salt micelles. To
quantify the displacement effect both the cyclodextrin-bile salt interaction (green circle) and the
micellization of bile salt (red circle) must be taken into account.
A suitable experimental tool to measure the strength of the interactions of the equilibria in Fig.
2.1 is isothermal titration calorimetry (ITC). Previous work by our group has focused on
describing the strength and thermodynamics of the interaction between cyclodextrin and bile
salts by means of ITC [8]–[21] . In this project the attention is turned towards the micelle
formation of bile salts.
The mechanisms depicted in Fig. 2.1 are based on professor Kaneto Uekama’s view on how
competitive agents (such as bile salts) affects the drug uptake [6]. From an aqueous solution, the
problem of cyclodextrin overdosing where a too large degree of complexation decreases the free
Figure 2.1: Schematic presentation of the drug absorption from a cyclodextrin formulation
including the drug displacement by bile salts. The cyclodextrin-bile salt reaction is
encapsulated by the green circle and the micelle formation of bile salts is encapsulated by the
red circle. Modified from Figure 5 in Uekama et al. 1998 [6].

4
drug concentration has been known for the last two decades. The base principles of an optimal
dose criterion were captured in a series of articles by Professor Thorstein Loftsson in the mid
1990’s [22], [23] and have subsequently been elaborated in reviews [6] most lately in 2011 [24].
In general, maximum absorption enhancement is obtained when just enough cyclodextrin is used
to solubilize the entire amount of drug in solution [6]. This total cyclodextrin concentration
denoted SCdrug
totCD 
is shown in Fig. 2.2A. Below SCdrug
totCD 
the drug molecules are in a
suspension which is undesirable from an absorption perspective because some of the drug
molecules will be found as unabsorbable particles. Above SCdrug
totCD 
all drug molecules are in
solution either on free or cyclodextrin complexed form, however, when more cyclodextrin are
added than required to fully solubilize the drug (that is more than SCdrug
totCD 
) a decrease in the
free drug concentration is expected. This decrease is influenced by the interaction between
cyclodextrins and bile salt present in the intestinal environment as illustrated in Fig. 2.2B. These
data shows the fraction of phenacetin remaining in the intestinal lumen when phenacetin is dosed
as a solution at a concentration of 0.01 mM, that is well below its solubility of ~5 mM. The
highest uptake of phenacetin is acheved when this drug is dosed alone ( ), when the drug is co-
adminstered with cyclodextrin the free drug concentration and the flux declines and the fraction
remaining in the lumen in Fig. 2.2B is higher ( ). In addition, the effect of bile salt as a
competitor is shown for fixed concentrations of drug and cyclodextrin and increasing
concentration of the bile salt TC (in ascending order , and ). Thus, an increase in the
concentration of bile salt displaces the the equilibrium system in Fig. 2.1 towards more drug on
free form in accordance with Chaterlier’s principle.
Figure 2.2: A) The relationship between HPβCD concentration and the flux of acetazolamide
from an aqueous formulation. The concentration of acetazolamide was kept constant at 1.0%
(w/v) but the cyclodextrin concentration ranged from 12% to 40% (w/v). An optimum is seen to
occur in the middle of the range for cyclodextrin. Reproduced from Loftsson et al. 2011 [24] with
permission. B) Effect of competitors on the disappearance of phenacetin (0.01 mM) with or
without competitors and cyclodextrin (maltosyl-βCD) from a rat intestinal segment. The legend
shows the molar ratio of phenacin (D) to bile salt (TC) to β-cyclodextrin (βCD). Reproduced
from Ono et al. 2002 [7] with permission.
D:TC:βCD=1:0:500
D:TC:βCD=1:0:0
D:TC:βCD=1:50:500
D:TC:βCD=1:100:500
D:TC:βCD=1:500:500
A B

5
A major challenge in the preformulation work is that cyclodextrins often are dosed in surplus of
SCdrug
totCD 
, either due to uncertainty in determining the level of this dosing or simple
unawareness of the optimal dosing criteria. Classically, only aqueous solutions are applied in
preformulation and this thesis therefore focuses on those. Further, the attention is on the effect of
the free drug concentration on the drug uptake. This mechanism is the simplest and the best
described, but it is important to realize that for the development of the final formulation there are
many other positive effects of cyclodextrin complexation with the drug including improving the
dissolution rate, the drug stability and enhance the permeation rate through the unstirred water
layer as described above [6], [24].
Quantitative biological proceses
In order to inspect the simultaneous influence of the multiple equilibria shown in Fig. 2.1 and the
absorption process a quantitative framework is needed. Biological systems – similar to the one
outlined in Fig. 2.1 – are traditionally studied by a procedure of reductionism; by isolating a part
of a system, fundamental mechanisms can easier be studied experimentally and understood.
However, biology is characterized by numerous mechanisms working in organization and such
complex systems almost always display emergent properties. In this context emergent properties
refer to behavior appearing when mechanisms are put together on a larger scale, which cannot be
realized from the isolated mechanisms. This has motivated the appearance of the “new science of
biology” termed system biology.
This dogma can also be used to shed light on the content of the current project. The main
contribution from the current project is to study phenomena on a larger scale from the interplay
of previously isolated mechanisms for the absorption process and the equilibrium system. To
rationalize such existing scientific knowledge mathematical modelling is a useful framework.
In this thesis mathematical modelling refers to the workflow outlined in Fig. 2.3 [25]. When
prior knowledge of the system under study is available, these physical processes can be
described by mathematical equations (Entry). This translation into a formal system of
mathematical assumptions is the foundation for a mathematical treatment. Next, the scaling of
the model must be chosen, a step known as nondimensionalisation. The dimensionless model
often enables the formulation of a simplified model but if such a simplification is not possible,
the modelling process must be continued with the full model. When a relevant model has been
identified the parameters inherent to the model must be estimated. In the current project this
involves chemical measuring principles. Finally, the now calibrated model can be solved and the
dynamics of the system can be exploited by conducting simulations providing input for new
experiments. The solution to the model can also be used to validate these observations, which
might generate new hypothesis and requires a refinement of the model thus closing the loop. This
process is iterated until a model with desired predictive properties arises.

6
Three main parts can be identified in this workflow as shown in Fig. 2.3. The steps annotated
with 1) are in this thesis referred to as model building. Usually, these steps are attributed to the
discipline of physics where assumptions defining a system are described by mathematical
equations. The steps annotated with 2) are referred to as parameter estimation or model
calibration. Traditionally, this belongs to the scope of the experimental science and is often not
included in the scheme of the modelling process. However, here it has been incorporated since
parameter estimation is a central part of this thesis. Finally, the steps annotated with 3) are
referred to as model solution. The derivation of the solution to a mathematical model typically
belongs to the computational sciences, but to interpret the model solution is typically
interdisciplinary as it requires a broader overview of all the steps in the mathematical modelling
process.
The structure of the current thesis is based on these three phases; 1) model building, 2) parameter
estimation and 3) model solution. However, the classification of the modelling process into these
simple phases is typically too simplistic and a clear distinction is not possible. For instance, an
independent modelling process was dedicated to the model for micellization of bile salt shown in
Fig. 2.1, as the model for this process has been disputed.
The overall purpose of this thesis is to define criteria for optimal use of cyclodextrins as drug
solubilizers in aqueous solutions in a preformulation setting. This involves determining the
Figure 2.3: Illustration of the workflow of mathematical modelling as applied in the current
project. The numbering of the steps refers to: 1) Model building, 2) Parameter
estimation/model calibration and 3) Problem solution.
1) Mathematical
statement of the problem
1) Dimensionless model
1) Simplification of model
3) Solution of simplified
approximate model
(analytical)
2) Estimation of reduced
number of parameters
3) Problem solution
2) Estimation of full
number of parameters
3) Solution of full model
(numerical)
3) Model validation
Entry

7
amount of cyclodextrin needed to yield exactly full solubilisation of the drug in the
gastrointestinal environment (chapter 3 – model building) a description of micellization by bile
salt as measured by ITC (chapter 4 – parameter estimation) and finally an examination of the
expected effect of overdosing on the intestinal drug uptake from considerations of the free drug
concentration and the intestinal permeability (chapter 5 – model solution). In addition, several
perspectives are made in this thesis from material which is not described in the enclosed
manuscripts. This includes a discussion of the chemical origin of the quantities measured by ITC
in chapter 3.5, an outline of an improved method to determine parameters by the ITC in chapter
4.5 and a classification system for the drugs which are relevant for cyclodextrin formulations in
chapter 5.1

8
Chapter 3: Model building
3.1 Model building - introduction
The absorption of a drug can be described to depend on the permeability (P) and the free drug
concentration [D] in accordance with Fick’s 1st
law:  D PJ . In this thesis the permeability is
assumed to be constant, the challenge therefore relies on determining the free drug concentration.
The drug uptake is a time-dynamical phenomena consisting of multiple chemical reactions and
an absorption process. However, as the chemical reactions are expected to evolve on a much
shorter time scale than the absorption process, the chemical reactions are assumed to be in
instantaneous equilibrium. The equations for the chemical equilibrium and the absorption
kinetics can therefore be derived independently and are thus described in two separate
subsections 3.2.1 and 3.2.2.
Besides the model for the pharmaceutical application, a model describing the formation of bile
salt micelles was a major part of this work. Several theories of bile salt micellization have been
put forward in the literature and these models are therefore treated in chapter 3.3.
Finally, the measurement of the chemical thermodynamics in aqueous solutions by ITC is
considered (the enthalpy, the entropy and the Gibbs free energy). The thermodynamic foundation
for the Gibbs free energy of micellization is included in chapter 3.4. This discussion is elaborated
in chapter 3.5 with a treatment of the chemical origin of the enthalpy and entropy of solvation in
lights of solvent reorganization in aqueous solutions.
3.2 The pharmaceutical application
3.2.1 Equilibrium part
The free drug concentration can be modelled form classical chemical theory from conservation
of mass and the law of mass-action. Consider the simple example of a bimolecular reaction
between a cyclodextrin CD and a drug molecule D to form the complex D:CD
D + CD ⇌ D:CD Eq. 3.2.1.1
This equilibrium is characterized by an equilibrium constant defined as KD:CD =
[D:CD]/([CD]·[D]), where [D:CD], [CD] and [D] are the molar concentrations of the complex,
the cyclodextrin and the drug molecule, respectively. Together with the equations for
conservation of mass Dtot = [D] + [D:CD] and CDtot = [CD] + [D:CD], the equilibrium constant
specifies the governing equation of the system
[D:CD]2
– (Dtot + CDtot + 1/ KD:CD)·[D:CD] + Dtot·CDtot = 0 Eq. 3.2.1.2
As the drug molecule has a finite solubility, the amount of drug in surplus of the solubility
concentration Dsol will in equilibrium be on precipitated form. This is also the case when the
drug molecule participate in the equilibrium given by Eq. 3.2.1.1; below a certain total
cyclodextrin concentration  SCdrug
totCD 
the free drug concentration equals its solubility value D =

9
Dsol, whereas above
SCdrug
totCD 
, the free drug concentration is found by solution of Eq. 3.2.1.2.1
The total cyclodextrin concentration where the entire amount of drug is dissolved is found by
solving Eq. 3.2.1.2 together with the condition D = Dsol
 soltot
solCDD
solCDDSCdrug
tot DD
DK
DK
CD 


:
:1
Eq. 3.2.1.3
This principle is illustrated in a recent review by Loftsson and Brewster as shown in Fig. 3.2.1.1.
Figure 3.2.1.1: The concentration of dissolved drug on free and cyclodextrin complexed form
(Stot) and the free drug concentration (Sfree) as a function of total cyclodextrin concentration.
Reproduced from Loftsson et al. 2011 [24] with permission.
When molecules are involved in several equilibria simultaneously, the governing equation will in
similarity with Eq. 3.2.1.2 be a high-order polynomial with roots that are straight-forward to find
by a numerical software. However, as described by the fundamental sentence of algebra there
will be as many roots as the order of the polynomial, but typically only some are physically
realizable. Therefore all solutions must be characterized to identify which ones are of relevance.
For relative simple systems such characterization is feasible in practice, but when these
equilibrium systems increases in complexity, the proof of physical existence and uniqueness of a
root becomes increasingly difficult. In fact, for more complex systems there might be multiple
realizable solutions simultaneously, which give rise to so-called exotic behavior known to occur
in chemical reaction tanks [26].
For the purpose of modelling, it is appealing to have a framework to tell the structure of the
solution in advance. Such a theory was developed in the 1970’s under the name chemical
reaction networks and has become one of the founding principles in system biology.
Chemical reaction networks
The structure of the solution to a chemical reaction network as the one considered in this thesis
(shown in Fig. 2.1) can be described by the Deficiency Zero Theorem. If the deficiency (δ) of a
reaction network is zero, only one physical realizable solution exists. This theorem is therefore
valuable since it ensures existence and uniqueness of the solution. The deficiency of a network is
a nonnegative integer index defined by the formula
1
This treatment is in standard in pharmaceutical science but differ from the definition of a solubility product.

10
sln  Eq. 3.2.1.4
where n is the number of complexes in the network, l is the number of linkage classes and s is
the rank of the network. The definition of these quantities requires some vocabulary as outlined
in the following [26].
Suppose a set of chemical species participate in multiple interconnected chemical equilibria. The
N chemical species can be represented by N orthonormal vectors spanning ℝN
:
e1 = [1,0,0,…,0]
e2 = [0,1,0,…,0] Eq. 3.2.1.5
eN = [0,0,0,…,1]
The number of complexes (n) is defined as the number of reaction complex vectors. These
vectors y1,y2,…,yn are defined as the sum of species vectors for, respectively, the reactants and
the products in a certain reaction.
The rank of the reaction network (s) is defined as the rank of the reaction matrix. This matrix is
found by subtracting the reactant complex vector from the product complex vector for each of
the r reactions. That is, for the reaction yi →yj the corresponding reaction vector is yj-yi. The
reaction matrix is then constructed by listing the reaction vectors under one another to form a
r×N matrix.
Finally, the number of linkage classes (l) can be determined from the standard reaction diagram.
To draw this diagram, complexes are defined in a similar fashion to the complex vectors as the
sum of reactants and the products in a certain reaction. Then each complex is written just once,
and arrows are drawn to indicate a “reacts to” relation in the set of complexes. The linkage class
defines the “separate pieces” of the diagram, i.e. it shows how the various complexes are
“linked” by the reaction arrows.
To illustrate this, the standard reaction diagram for the most basic case of two interacting
equilibria is shown below
D + CD ⇌ D:CD Eq 3.2.1.6
BS + CD ⇌ BS:CD Eq. 3.2.1.7
The number of complexes is n = 4,
 CDBSCDBSCDDDCD :,,:,  Eq. 3.2.1.8
and the number of linkage classes is l = 2
 CDDCDD :, Eq. 3.2.1.9


11
 CDBSCDBS :, Eq. 3.2.1.10
the number of linearly independent reaction vectors (the rank of the network) is s = 2. The
deficiency of this network is therefore δ =4–2–2=0. For any choice of total concentrations and
complexation constants there is one and only one positive solution. The deficiency of any
chemical reaction network can be derived by means of free computational platforms [27], such
as the Chemical Reaction Network Toolbox.2
In this project all chemical equilibrium systems are of deficiency zero and the unique physical
solution can therefore be identified as the largest root of the governing polynomial.
Nondimensionalization
Any physical meaningful quantity must have a dimension to which its magnitude is compared.
The SI-unit system is used as standard for global reference. However, often some characteristic
values of the physical quantities are present in the system under investigation, which provides a
more natural way of scaling the physical quantities. Nondimensionalisation refers to this process
of finding a parameterization intrinsic to the system. These parameters, called dimensionless
groups, controls the behavior of the solution and have several advantages which has been
exploited in this project. For instance, identification of dimensionless groups reduces the number
of independent parameters in the equations and by comparing the magnitudes of the
dimensionless groups, dominant and negligible terms can be identified often leading to
simplifications of the equations. Depending on the knowledge of the system this can even be
done prior to more accurate parameter estimation as indicated by the bifurcation of the flow
diagram in Fig. 2.3.
Nondimensionalization has frequently been applied to systems described by differential
equations. In this project chemical equilibrium systems described by algebraic equations was
subject to nondimensionalization. For such systems a suitable scale is typically found for the
intersection of two species curves or at a transition concentration.
The equilibrium system shown in Fig. 2.1 can be described by the law of mass-action and
conservation of as shown in Fig. 3.2.1.2
Figure 3.2.1.2: The chemical equilibrium system on which the uptake from cyclodextrin
complexes is based. The molecular entities refers to micelle (Mn), bile salt (BS), cyclodextrin
(CD), bile salt-cyclodextrin complex (BS:CD), drug (D), drug-cyclodextrin complex (D:CD) and
n is the aggregation number of the micelle.
2
Available via http://www.crnt.osu.edu/crntwin

12
In manuscript 3 it was shown that the behavior of this system can be characterized by the relative
magnitudes of four dimensionless variables. The first is the dose number, defined as Do=Dtot/Dsol
where Dtot is the total drug concentration and Dsol is the solubility concentration of the drug. The
three others dimensionless variables consists of a product of a complexation constant and a
concentration and are given by solCDD DK : (the so-called complexation efficiency CE),
 soltotCDBS DDK : and KBS:CD·CMC, where KBS:CD is the equilibrium concentration between the
bile salt and the cyclodextrin and CMC is a characteristic concentration of the micellization
process known as the critical micelle concentration (elaborated in chapter 3.3).
Based on rough parameter estimates the number of equations in the model for the equilibrium
system can be reduced before more accurate parameter estimation is initiated. The magnitude of
the quantity  soltotCDBS DDK : can be classified by considering that the dose number
Do=Dtot/Dsol of the drugs, which are relevant to formulate in cyclodextrin formulations, is larger
than 1. In fact many low-soluble drugs will be formulated with 1Do . The mean binding
constant between bile salt and HPβCD at 37 °C is approximately -1
: mM20CDBSK for the bile
composition in both humans and rats [11], [28]. The magnitude of the second quantity
CMCK CDBS: is quantified from the CMC for the mixed micelle, which is estimated to 1 mM at
physiological conditions [29]–[31]. The magnitude of the complexation efficiency
solCDD DK :CE  can be estimated, as it is known that the majority of complexes formed by
cyclodextrins have a binding constant below 10 mM-1
[32]. It can therefore be assumed that the
complexation efficiency is much smaller in comparison to the two other dimensionless variables,
i.e.  soltotCDBSsolCDD DDKDK  :: and CMCKDK CDBSsolCDD ::  .
As a result of this, the amount of cyclodextrin to fully solubilize the drug in the presence of bile
is given by:
  totsoltot
solCDD
solCDDSCdrug
tot BSDD
DK
DK
CD 


:
:1
Eq. 3.2.1.11
The derivation of Eq. 3.2.1.11 is elaborated in manuscript 3. In essence the assumptions
CMCKDK CDBSsolCDD ::  and  soltotCDBSsolCDD DDKDK  :: implies that before totCD reaches
the amount of cyclodextrin to fully solubilize the drug (denoted SCdrug
totCD 
) all bile salt will be
complexed with cyclodextrins and there will therefore be depletion of free bile salts and
micelles.
To quantify when overdosing with cyclodextrins occur, the behavior of the system can be
simulated as a function of CDtot. In this case an exact choice of parameters is required, however,
as long as the choice of parameters is done in accordance with the criteria described above, the
dynamics of the system will not depend on the exact values. For illustration, danazol was chosen

13
as a model compound. Danazol (Mw=337.46 g/mol) has an aqueous solubility of 0.61 µg/mL
(1.8×10-3
mM) at 37 °C and a complexation constant with HPβCD on 61.9 mM-1
at 37 °C [33].
In Fig. 3.2.1.3 the species concentrations in the equilibrium system are shown as a function of
CDtot. To visualize the dynamical range of all species in the system, the value of the free drug
concentration is shown on the left vertical axis (multiplied by 10-3
) and the other species are
shown on the right axis. From Fig. 3.2.1.3 it is seen that a transition in free drug concentration
occurs at mM9.44SCdrug
totCD in agreement with the approximate solution resulting from the
nondimensionalization.
Figure 3.2.1.3: Simulated species concentration for equilibrium system shown in Fig. 2.1 as a
function of total cyclodextrin concentration. The curves represent the free concentration of drug
[D], precipitated drug Dprec, drug-cyclodextrin complex [D:CD], cyclodextrin [CD], bile salt
[BS], bile salt-cyclodextrin complex [BS:CD] and bile salt on micellar form n·[Mn]. A transition
concentration for [D] is clearly seen at the concentration predicted by SCdrug
totCD 
at the right-most
dashed line, whereas the left-most dashed line shows the total cyclodextrin concentration
required to fully solubilize the drug in absence of the bile salts Eq. 3.2.1.3. Parameter values are
KBS:CD=30 mM-1
, KD:CD=61.9 mM-1
, BStot=15 mM, , n=10, CMC=1, Dtot=3 mM, Dsol=0.0018
mM. The free drug concentration is shown on the left vertical axis (multiplied by 10-3
) and the
other species are shown on the right axis.
In the following section a model to derive the free drug concentration during intestinal transit is
derived. In this context it is desirable to regard the total drug concentration as the independent
variable as this will decline in the course of the drug absorption process.

14
Equilibrium model for the free drug concentration
By rearranging Eq. 3.2.1.11 we can derive the intestinal drug solubilisation capacity SC
totD , which
provides the maximal total drug concentration that can be solubilized by the amount of
cyclodextrin in the solution in the presence of bile salts:
  soltottot
solCDD
solCDDSC
tot DBSCD
DK
DK
D 


:
:
1
Eq. 3.2.1.12
When the drug is absorbed across the intestinal membrane it results in a decrease in the total
drug concentration in the intestinal lumen. Conversely, it can be assumed that the total
cyclodextrin concentration is constant, based upon their low extent of intestinal absorption [34].
If the dosed cyclodextrin solution is formulated such that the drug initially is fully solubilized,
the free drug concentration will be below the solubility value when some of the drug is absorbed.
Therefore, to develop an insight into this system it is desirable to describe the free drug
concentration as a function of total drug concentration remaining in the intestinal lumen.
In manuscript 3 it was shown that when the four dimensionless quantities are of a certain
magnitude, namely CMCKDK CDBSsolCDD ::  and  soltotCDBSsolCDD DDKDK  :: and
1:  solCDD DK and 1 soltot DDDo , then the free drug concentration can be shown to be a
linear function of the total drug concentration:
  







SC
tottotsol
SC
tottottotSC
tot
sol
tot
DDD
DDD
D
D
D
,
,
D Eq. 3.2.1.13
The derivation of Eq. 3.2.1.13 is elaborated in manuscript 3, in summary the criteria mentioned
above implies that the concentrations of most species are negligible and hence
  tottot BSCD CD and   totDCD:D as seen in Fig. 3.2.1.3, which together with the
expression for the drug-cyclodextrin complexation constant       CD: :  CDDKCDDD
demonstrates the existence of proportionality in Eq. 3.2.1.13.
To illustrate these findings danazol is considered again. The complexation efficiency of danazol
is CE = 61.9 mM-1
 1.8×10-3
mM ≈ 0.11 and it is therefore reasonable to apply Eq. 3.2.1.13 for
this drug molecule. In Fig. 3.2.1.4 the free drug concentration as a function of the amount
absorbed is seen. Fig. 3.2.1.4a shows the case where the total cyclodextrin concentration is dosed
at SCdrug
totCD 
with a danazol dose of Dtot(0) = 3 mM. Initially, the total drug concentration is
exactly at the “drug solubilizing capacity” and the entire amount of drug molecules will therefore
be in solution. When the drug is absorbed, corresponding to a decrease in the total drug
concentration, the free drug concentration declines as seen in Fig. 3.2.1.4a. If cyclodextrins are

15
dosed to an amount where the total cyclodextrin concentration is in surplus of SCdrug
totCD 
all drug
will still be in solution, but the free drug concentration will at any value be lower than by dosing
at SCdrug
totCD 
as seen in Fig. 3.2.1.4b. This overdosing of cyclodextrins will therefore result in
lower fractions of drug absorbed.
Figure 3.2.1.4: Illustration of the effect of drug absorption on the free drug concentration.
Concentration of free drug, drug cyclodextrin complex and drug on precipitated form with initial
values of (a) SCdrug
tottot CDCD 
 i.e. CDtot = 44.9 mM, (b) SCdrug
tottot CDCD 
 i.e. CDtot = 53.9 mM.
To visualize the dynamical range of all species in the system, the value of the free drug
concentration is shown on the left vertical axis (multiplied by 10-3
) and the other species are
shown on the right axis.
In summary, the nondimensionalisation has been applied in this section to show which of the
parameters in the equilibrium system the drug absorption truly depends on. This emergent
behavior results in a very valuable simplification which barely could be predicted by considering
the isolated mechanisms alone.
3.2.2 Kinetic part - the absorption process
To describe the intestinal uptake of a drug, the algebraic equations derived for the chemical
equilibrium system above must be coupled to the kinetics of the absorption process described by
differential equations.
In the context of biopharmaceutics it is desired to be able to predict the drug concentration in the
systemic blood circulation. However, from the drug permeates the intestinal wall until it reaches
the systemic circulation it must pass the liver. During the liver transit many drugs are subject to
an extensive metabolism greatly reducing the amount of drug reaching the blood. In this work
the aim is only to predict the drug fraction, which passes the intestinal wall, denoted the fraction
absorbed (Fa).
a b

16
The absorption process is in this work described by a simple one-compartment model based on
the following understanding of the oral absorption process. When a drug is dosed orally in a
solid form the tablet will first have to disintegrate into smaller particles. For standard immediate
release tablets this disintegration is assumed to be fast. The disintegrated particles must
afterwards dissolve into free solvated drug molecules in the aqueous gastrointestinal
environment. A majority of new drug candidates are poorly water-soluble and for these
compounds no more drug molecules than the solubility limit can be solvated in equilibrium. The
final step in the absorption process is that the free drug molecules pass the intestinal wall.
In Fig. 3.2.2.1 the so-called bucket representation is shown depicting the one-compartment
model of the absorption process. Three compartments connected in series are present: a solid
compartment where the drug molecules enter, a dissolved compartment which is confined on the
top by the solubility and an absorbed compartment. The flux between the solid and the dissolved
compartment is governed by the dissolution rate (kdiss) while the flux between the dissolved and
absorbed compartment is governed by the absorption rate (kabs).
Figure 3.2.2.1: Rate-limiting steps in oral absorption of a drug represented by the bucket model.
Upper compartment represent solid drug, middle compartment represent dissolved drug and
bottom compartment represent absorbed drug. a) Dissolution rate-limited absorption, b)
permeability limited absorption and c) solubility-permeability limited absorption.
The absorption process can be divided into three classes (a, b and c in Fig. 3.2.2.1) depending on
the origin of rate-limitation. Several authors give their own statement of these classes often with
some ambiguity, but the one outlined below largely follows the definition by Sugano in a recent
biopharmaceutics textbook [35]. In addition the potential impact of cyclodextrin on the drug
uptake is mentioned for each class based on the assumption that cyclodextrin only modifies the
apparent solubility of the drug [36].
Dissolution rate-limited absorption (DRL) - Fig. 3.2.2.1a: In this case the dissolution rate of the
drug is much slower than the permeation rate. Once the drug molecules are dissolved they
instantly permeate the intestinal membrane and get absorbed into the body. In this case the rate
Absorption
rate
Dissolution
rate
Absorption
rate
Dissolution
rate
Absorption
rate
Dissolution
rate
a b c

17
of drug absorption is determined by the dissolution rate. Impact of cyclodextrin: A potential
benefit of cyclodextrin on the absorption can be obtained by fully dissolving the drug.
Permeability limited absorption (PL) - Fig. 3.2.2.1b: In this case the drug particles dissolve
immediately and completely. Mathematically this is a special case as the solid compartment is
empty and there will be no effect of increasing the dissolution rate constant. The amount
absorbed will in this case depend on the total drug concentration and the absorption rate constant.
Impact of cyclodextrin: Addition of cyclodextrin is expected to affect the drug absorption
negatively.
Solubility-permeability limited absorption (SPL) - Fig. 3.2.2.1c: In this case, the dissolution rate
is faster than the absorption rate and free drug molecules have accumulated in the dissolved
compartment at the solubility. If this concentration of [D] = Dsol is maintained throughout the
transit time the maximal absorbable dose is obtained given by MAD = kabs·Dsol·Ttransit. Impact of
cyclodextrin: Addition of cyclodextrin is expected to affect the drug absorption negatively.
In this project the effect of cyclodextrin on dissolution rate-limited drugs is analyzed, as this is
the situation where a potential benefit must be expected from the use of cyclodextrin in the
formulation.
3.2.3 A preview into the coupling of the equilibrium and kinetic part
A large part of mechanistic modelling is about developing an intuitive understanding of the
system. To this end it might be beneficial at the current stage to provide some qualified guesses
of what the effect of cyclodextrin on the drug absorption is. Such intuition is provided in Fig.
3.2.3.1.
Fig. 3.2.3.1a shows the free drug concentration of a poorly soluble drug compound as a function
of the total cyclodextrin concentration. Four samples are prepared indicated by the four circles.
When the total drug concentration in the gut decreases during the absorption of the drug the free
drug concentration declines as shown in Fig. 3.2.3.1b. The 5 dashed lines in Fig. 3.2.3.1b are iso-
contour for the absorption rate constant. The amount absorbed can be read off on the 1st
axis
from the intersection between the dashed lines and the curves for the free drug concentration. As
the absorption rate increases proportionally to the free drug concentration the iso-contour lines
are not straight but bended. It must therefore be expected that the absorption is a nonlinear
relationship depending at least on the absorption rate, the degree of overdosing and the solubility
of the drug. The drug absorption process will be elaborated in Chapter 5 – model solution.

18
Figure 3.2.3.1: (a) The initial free drug concentration as a function of the total cyclodextrin
concentration, the black line is the mathematical exact solution whereas the blue line is the
approximation for CE << 1 as elaborated in manuscript 4. The four circles illustrate the initial
free drug concentration for different doses of cyclodextrin. (b) Free drug concentration as a
function of total drug concentration given by Eq. 3.2.1.13 from the four samples, illustrating the
concentration during the time course of the absorption process. The black dashed lines are
contour lines for the amount absorbed for five hypothetical absorption rate constants.
3.3 Micelles
A micelle is a self-assembled aggregate of molecules in an aqueous solution. The molecules
constituting a micelle are amphiphilic thus having one part which likes water and one part
disliking water. At low concentrations the amphiphilic molecules are molecularly dispersed and
solvated by the water molecules but above a certain concentration, known as the critical micelle
concentration (CMC) the amphiphiles self-associate into small aggregates with a finite
aggregation number (n) and are thus typically consisting of 50-100 molecules as shown in Fig.
3.3.1.
Figure 3.3.1: The concentration of free monomers  S (grey line) and the concentration of
monomers in the micelles n·[Mn] (black line) as a function of total surfactant concentration Stot
for n=100. The species concentration at CMC is marked with circles.
a b

19
Historically, there have been different approaches to understand the behavior of micelles as
recently reviewed by Romsted [37]. One key principle governing the behavior of a micelle is
Tanford’s principle of opposing forces, which specifies that hydrophobic forces at the interface
between the hydrophobic part of the amphiphilic molecule and water induce the molecules to
associate, whereas repulsive forces3
at the surface of the micelles oppose the association.
Tanford’s principle dictates that the associating forces derive from the decrease in exposed
hydrophobic area to the surrounding water [38]. Proportionality between the volume of a micelle
(Vmicelle) and the aggregation number (n) therefore exists4
:
nVmicelle  Eq. 3.3.1
On the other hand, the dissociating forces are associated with the surface of the micelles. From
the so-called square-cube law it is given that the surface area of the micelle (Amicelle) is
proportional to (Vmicelle)2/3
, and hence:
3/23/2
interface nVA micelle  Eq. 3.3.2
As a consequence of this, each monomer takes up a certain optimal surface area of the micelle
and the opposing forces causes the energy to vary parabolically around this area, when the
micelles are spherical [35–37]. Therefore, the mean size of spherical micelles is relatively
insensitive to the total concentration and the micelles are consequently almost monodisperse (i.e.
the micelles consists of n monomers). The sudden change in the behavior of the system at CMC
is therefore a direct consequence of Tanford’s principle [39].
Two interpretations of micelles have historically been dominant. One conception is that the
abrupt change in behavior at CMC is due to a phase transition Above CMC two macroscopic
phases are thus in equilibrium with each other:
monomermicelle n   Eq. 3.3.3
where μmicelle is the chemical potential of the micelle, μmonomer is the chemical potential of the
monomer and n is the aggregation number.
The second model is known as the mass-action model. For non-ionic surfactants the mass-action
model interprets micellization as a two-state chemical equilibrium between micelles Mn and its
constituent, the bile salt monomers, S [40]:
n·S⇌Mn, Eq. 3.3.4
The equilibrium is characterized by a micellization constant defined by:
 
 n
non
K
S
Mn
 Eq. 3.3.5
3
Repulsive forces can for instance originate from electrostatic interactions on the headgroup of an amphiphile
molecule
4
Assuming that the volume each monomer takes up as a function of aggregation number is constant

20
where [Mn] and [S] are the concentrations of micelles and free bile salt monomers, respectively.
Combining Eq. 3.3.5 with the equation for mass conservation, the system is fully specified by:
Stot = [S] + n·Knon
·[S]n
Eq. 3.3.6
It must be emphasized that the two-state reaction n·S⇌Mn in the mass-action model only is an
approximation. In reality, a certain size distribution of micelles with aggregation number other
than n exists. However, in contradiction with the phase model, the mass-action model predicts
that the transition at CMC will be smooth for low aggregation numbers and the intersection
between the species curves (cf. Fig. 3.3.1) will no longer be at twice the value of the CMC. This
is depicted in Fig. 3.3.2 where the concentration of monomers on free [S] and micellar form
n·[Mn] are shown as a function of total surfactant concentration Stot. The illustration has been
simplified by scaling the 1st
axis in such a way that the upper limit is set to the total
concentration at the equivalence concentration where the species curves intersect [S] = n·[Mn],
that is n non
tot nKS 
 1equi
2 . It can be shown that this transformation results in identical species
curves for a specific value of n independent of the value of K. All concentrations are made
dimensionless by normalizing them with the total concentration at the equivalence concentration
equi
totS . The circles mark the fractions of surfactants and micelles at the value given by equi
totSCMC .
Figure 3.3.2: The dimensionless concentration of monomers on free form  equi
totSS (grey line)
and on micellar form   equi
totn SMn (black line ) as a function of dimensionless total surfactant
concentration equi
tottot SS for 10050,25,10,5,3,n . The CMC for the two species are marked with
circles.
The mass-action model is somewhat counter-intuitive as in lights of classical molecular theory it
is very unlikely that more than 3 molecules collide simultaneously to form a new complex.
Classically, a chemical self-association process is described as a solute interacting with itself to
form dimers, trimers, and so on in a stepwise association manner, as depicted by the reaction
scheme shown below.

21
2·S ⇌S2
S + S2 ⇌ S3

S + Sn-1 ⇌ Sn
Different types of cooperativity5
can be modelled by arranging the relative magnitudes of the
stepwise association constants, which can ultimately lead to the all-or-none response n·S⇌Mn.
Currently, there is a major part of the literature applying this model of bile salt micellization
[40]–[46]. An explanation for this might be that hydrogen bonds are believed to stabilize bile salt
micelles, however, recent results from molecular simulations suggest that hydrogen bonds do not
drive the association [47]. In my opinion, there is no reason to believe that micellization of bile
salt should follow fundamental other principles than dictated by the classical micelle theory. As
least for the studies conducted in this work the accuracy of ITC do not seem to allow an
estimation of the size distribution of the micelles and a more detailed model therefore appears
premature (cf. chapter 4 – parameter estimation). A classical review of bile salt micelles
conducted by Donald M. Small can be found in [29].
The consequences of the two different interpretations from the phase model and the mass-action
model, respectively, are discussed in manuscript 2 and summarized below in chapter 3.4
3.4 Determination of thermodynamic potentials of micellization
The pseudo phase-model approximation to the mass-action model
In this section the standard free energy change of demicellization for the mass-action model
(MAM) and the pseudo phase-model (PPM) are discussed. The standard free energy change of
demicellization per monomer is defined as:
n
n
G monomermicelle
demic

  
 Eq. 3.4.1
with SI-unit in kJ per mol of monomer.
Phillips [48] defined the CMC as the inflection point of a solution property with respect to the
total concentration. For ITC this criterion is defined as d3
[S]/dStot
3
=0 [49] and enables a method
to determine the binding constant from CMC and n as derived in the supporting material of
manuscript 2:
   
n
n
CMCS
n
nn
nn
n
B
K
tot




















1
2
2
2
22
2
CMC
12
21

Eq. 3.4.2
5
Cooperativity denotes that the affinity for a molecule to become part of the aggregate is facilitated by the
molecules already present in the aggregate

22
where   CMCStot
B


is the free counter-ion concentration at CMC. The Gibbs free energy of
demicellization is therefore given by
 
     

























12
2
ln
22
2
ln
1
lnln
1
ln
22
2
nn
n
n
RT
n
nn
RT
n
n
BRTCMCRT
n
n
K
n
RT
MAMG
CMCS
demic
tot


Eq. 3.4.3
The phase model was originally derived on its own basis, but it has later been shown that it can
be interpreted as an approximation to the mass-action model for large n. When n is large the two
last terms become negligibly small and can be omitted. In addition it was shown in manuscript 2
that the free counter-ion concentration at CMC is approximately equal to CMC i.e.
  CMCB
CMCStot



. The approximation applied by the phase model is therefore given by:
    CMCRTPPMGdemic ln1  
Eq. 3.4.4
For micelles behaving in accordance with the mass-action model, the phase model is therefore
only an approximation, which will become asymptotically equal to the mass-action model as the
micelle becomes a true macroscopic phase i.e. for n .
In Fig. 3.4.1 the mass-action model   KnRTMAMGdemic ln 
and the phase model
  CMCRTPPMGdemic ln 
are compared at T=20 °C for β = 0. Fig. 3.4.1a shows the relation
for the Gibbs energy change of demicellization as a function of n = 2–100 and CMC=1–10 mM
for the mass-action model (thick lines) and the phase model (thin lines) while Fig. 3.4.1b shows
the error in percentage of using the mass-action model compared to the phase model
   
 MAMG
MAMGPPMG
demic
demicdemic



 as a function of n = 2–100 and CMC=1–10 mM. Fig. 3.4.1c and
3.4.1d are a magnifications of Fig. 3.4.1a and 3.4.1b, respectively, for a narrower interval of
aggregation numbers n = 3–10 relevant for bile salt.

23
Figure 3.4.1: a) Level curves for the Gibbs energy change per monomer for n = 2–100 in kJ/mol
at T=20 °C calculated based on the mass-action model KnRTGMAM ln 
, where
 
n
n
nn
nn
n
K













1
2
2
2
22
2
M1
CMC
12
2
(thick lines) and on the phase model 






M1
ln
CMC
RTGPPM

(thin lines). b) percentage error by the phase model    
 demicG
demicGdemicG
MAM
MAMPPM



 , c) same
as Fig. 3.4.1a but with n = 3–10, d) same as Fig. 3.4.1b but with n = 3–10.
Normally, values from the phase model are reported on the mole fraction scale, but here they are
stated on the molarity scale in accordance with the mass-action model [38], [50]. The phase
model predicts that K is independent of n, which graphically corresponds to the horizontal lines
in Fig. 3.4.1a and 3.4.1c. However, according to the mass-action model, K depends strongly on n
for low aggregation numbers and the two models become inconsistent. From the right panel in
Fig. 3.4.1 it is seen that the smallest errors are found for high n and low CMC in the lower right
corners of the plots, whereas for small values of n it is necessary to know the specific value of
the aggregation number, to estimate ΔG° from CMC.
3.5 Understanding the thermodynamics of chemical reactions involving
hydrophobic substances in aqueous solutions
Water as a solvent is from a physical perspective the central theme in this PhD project. Most of
the scientific challenges encountered in this work have a common origin in the special solvation
properties of water. Examining these properties can thus favour an understanding of both the low
solubility nature of nonpolar drug molecules, the inclusion of nonpolar substances into the
cyclodextrin cavity and the formation of micelles.
To understand a chemical equilibrium, thermodynamic potentials are useful quantities. The
thermodynamic parameters which are directly derived from an ITC experiments are the enthalpy
ΔH°, entropy ΔS° and Gibbs free energy ΔG°

24
ΔH° = ΔG° + T·ΔS° Eq. 3.5.1
In manuscript 1, 2 and 3 the thermodynamic potentials have been measured for the formation of
inclusion complexes between bile salt and cyclodextrins and for the formation of micelles. These
reactions involve molecules which are partly hydrophobic6
. In aqueous solutions, reactions often
have large variations in ΔH° and ΔS° when we change some molecular parameter such as the
temperature or the hydrophobic areas of the involved species. This has been observed for both
micellization of bile salts [52] and complexation between cyclodextrins and bile salts [17]. On
contrary, the variation in ΔG° is much smaller thus resulting in an approximate compensation
between the entropy and enthalpy. There is therefore a decoupling between the enthalpic
contributions to the reaction and the underlying “driving force” – a phenomenon known as
entropy-enthalpy compensation. Therefore, a sound understanding of the origin of these
thermodynamic potentials is required to facilitate a correct interpretation.
In the following a brief introduction to the theory underlying the interpretation of
thermodynamic potentials in water is discussed. Initially the focus is on the simplest chemical
process that is the solvation of a nonpolar molecule in water. For this process the classical
textbook interpretation of the hydrophobic effect is summarized and the theory derived by
Professor Arieh Ben-Naim explaining the entropy-enthalpy compensation is motivated. These
theories are subsequently applied to shed light on the more complicated processes of inclusion
complex formation and micellization.
Solvation of hydrophobic molecule
Classical interpretation of the hydrophobic hydration – the hydrophobic effect:
In the liquid phase, water molecules form a dynamic structure of intermolecular hydrogen bonds.
The solvation of a hydrophobic (nonpolar) molecule by water is associated with the formation of
a structured cage of water molecules surrounding the nonpolar compound. The hydrogen bonds
between water molecules in the solvent cage are reoriented tangentially to the surface of the
solute molecule. Due to the newly formed bonds this process is enthalpic favorable ΔH < 0, but
on the other hand entropic unfavorable ΔS < 0 since the water molecules in the solvent cage have
restricted mobility and thus increases the “order” of the system. However, when many solute
molecules cluster together, the total exposed surface area decreases and more water molecules
are free to move. The net effect of formation of large clusters of hydrophobic molecules is then a
decrease in the organization of the solvent and therefore a net increase in entropy of the system.
This increase in entropy of the solvent is large enough to assist a spontaneous association of the
hydrophobic molecules resulting in formation of small particles from the solute molecules and
ultimately precipitation. This is the reason why non-polar molecules have a poor aqueous
solubility [39]. Such processes, which are driven by a greater disorder of the solvent are denoted
hydrophobic interactions [51].
The explanation given above of “structure making” and “structure breaking” in the water is the
classical interpretation of the hydrophobic effect, which can be dated back to Frank and Evans’
proposal of iceberg formation in 1945 [53]:
6
A hydrophobic molecule is defined by having a positive Gibbs energy of transfer from a non-polar to polar solvent
[51].

25
“When a rare gas atom or nonpolar molecule dissolves in water at room temperature, it modifies
the water structure in the direction of greater “crystallinity” – the water so to speak builds a
microscopic iceberg around it.”
However, according to the interpretation by Ben-Naim this picture is too simple and largely
misleading. In fact the bare association between structure and entropy is often wrong. It is the
strength in binding energies between the water molecules that explains the solvation entropy and
not the change of structure itself [54]. To understand these arguments, a consideration of Ben-
Naim’s central claim is required, namely that a simple principle is able to explain many of the
unique properties of water as a solvent. In fact, this principle is of such importance that Ben-
Naim simply refers to this mechanism as the principle of liquid water stating that:
“the unique properties of water can be explained by the packing of water in such a way that low
local density is correlated with strong binding energy and high local density is correlated with
weak binding energy”
In the section below the derivation of Ben-Naim’s theorem for entropy-enthalpy compensation is
summarized for the solvation of a nonpolar molecule. The results are similar to the idea
suggested by Lumry and Rajender 1970 [55], who proposed that a reaction A → B taking place
in an aqueous was coupled to a second reaction of the water molecules in which n water
molecules in state W1 underwent a transition to state W2, i.e. n(H2O)W1 → n(H2O)W2. By this
reasoning the overall enthalpy and entropy changes is
ΔH = ΔHA→B + n·ΔHW1→W2 Eq. 3.5.2
ΔS = ΔSA→B + n·ΔSW1→W2 Eq. 3.5.3
Lumry and Rajender suggested that in the normal experimental temperature range of 250-320 K,
the Gibbs free energy for the coupled reaction in the water is approximately zero ΔGW1→W2≈0,
but that the enthalpy n·ΔHW1→W2 and entropy n·ΔSW1→W2 terms are of substantial role in Eq. 3.5.2
and 3.5.3 [55], [56]. However, by including Ben-Naim’s principle of liquid water, it becomes
unnecessary to assume that ΔGW1→W2≈0, entropy-enthalpy compensation in fact emerges from
packing of water into the two classes.
The entropy-enthalpy compensation theorem by Ben-Naim
According to Ben-Naim, the principle of liquid water is necessary and sufficient for any
successful model for water (pair potential) in order to demonstrate water-like behavior.
Historically there have been two types of models to describe water; continuum models and
mixture models. For many years the mixture-model approach was based on various choices of
models for each of the components comprising the mixture, hence a “mixture of models”.
However, more recently it has been shown that the principle of liquid water can emerge from a
single form of the pair potential; the distinction between continuum and mixture modes is
therefore obsolete according to Ben-Naim.
The entropy-enthalpy compensation theorem summarized below, was derived by Ben-Naim in
1965 from an exact two-structure, mixture-model of water [57], however, according to Ben-

26
Naim it can be derived without relying on the classical mixture-model of water [54]. The two-
structure, mixture-model of water assumes a distribution of water molecules into a mixture of
two water components, which can be modelled as a simple equilibrium as depicted in Fig. 3.5.1.
Figure 3.5.1: Illustration of the principle of liquid water. Low local density is correlated with
strong binding energy and high local density is correlated with weak binding energy. Cover
image copied from Ben-Naim 2010 with permission [54].
The chemical equilibrium for an exact two-structure, mixture-model of water is given by
L ⇌ H Eq. 3.5.4
where L refers to water molecules with low density and high interaction energy and H refers to
water molecules with high density and low interaction energy.
These two entities are assumed to be in equilibrium. Their chemical potential therefore satisfy
the condition
HL   Eq. 3.5.5
From conservation of mass the total number of water molecules is given by
Nw = NL + NH Eq. 3.5.6
where N is number of molecules and the subscript refers to the total (w) or type of water
molecules (L) and (H).
Introducing a hydrophobic specie in water displace the equilibrium L ⇌ H. Upon addition of Ns
solute molecules to a constant amount of solvent molecules Nw, the chemical potential of the
solute can be expanded in a total differential
  r
S
f
S
NS
L
HL
NNsNs
S
wwLw
N
N
N
G
N
G
 
























,
Eq. 3.5.7
Here the term
wL NNS
f
S
N
G
,








 refers to the contribution to the chemical potential when the
reaction HL  is “frozen-in” (f) and  
wNS
L
HL
r
S
N
N








  refers to the contribution to the
chemical potential from “relaxation” (r) allowing the reaction L ⇌ H to find a new equilibrium.

27
By substituting the condition for a chemical equilibrium (Eq. 3.5.5) into Eq. 3.5.7, we
have
f
SS   Eq. 3.5.8
The chemical potential of the solute will thus not be affected by the relaxation term. From a
similar argument about the entropy and enthalpy it can be shown that the relaxation terms of
these two quantities always compensate each other in equilibrium
  r
S
f
S
NS
L
HL
NNsNs
HH
N
N
HH
N
H
N
H
wwLw
























,
Eq. 3.5.9
  r
S
f
S
NS
L
HL
NNsNs
SS
N
N
SS
N
S
N
S
wwLw
























,
Eq. 3.5.10
From the condition for a chemical equilibrium HL   (Eq. 3.5.5), we have
LLLHHH STHSTH   Eq. 3.5.11
By a rearrangement    
ww NS
L
LH
NS
L
LH
N
N
SST
N
N
HH 















 it is seen that the relaxation part
of the entropy and enthalpy compensate each other in equilibrium
r
S
r
S STH  Eq. 3.5.12
Ben-Naim refers to this as the exact entropy-enthalpy compensation law. Note that it is the
equilibrium condition between the two types of water molecules HL   that results in the
compensation between relaxation parts of the entropy and enthalpy. It is thus not an assumption
we make at this stage in the derivation as in Lumry and Rajender’s approach for Eq. 3.5.2 and
3.5.3, where we must assume that ΔGW1→W2≈0, but a property that emerges from a deeper level
of organization of the system which we have reason to believe is there according to Ben-Naim’s
principle of liquid water.
It can be shown that the result derived above for the partial molar quantities (∂H/∂NS)Nw and
(∂S/∂NS)Nw also is true for the thermodynamic parameters ΔH and ΔS. When relatively large
structural changes in the solvent takes place for a chemical reaction in water (which may have
other enthalpic and entropic contributions than the one discussed above), that is, whenever the
“relaxation” terms are much larger than the “frozen-in” terms ΔHr
>> Hf
and T·ΔSr
>> T·Sf
then
the reaction will be subject to an approximate entropy-enthalpy compensation
ff
STGH  
Eq. 3.5.13
HL  

28
For small non-polar solutes, it is expected that the water component with low local density is
stabilized by the addition of the solutes s by enhancing the formation of hydrogen bonds between
water molecules.7
Figure 3.5.2: The surroundings of H and L molecules. The L has more room in its immediate
surroundings to accommodate a solute s. Reproduced from Ben-Naim 2010.
The shift towards more of the L component results in a decrease in the enthalpy. Due to the
equilibrium between the L and H component this effect is however compensated by the entropy
when the equilibrium is allowed to relax. The total observed entropy does therefore not explain
much of the Gibbs free energy as assumed in the “ice-berg model”, where the change in Gibbs
free energy is caused by an increase in entropy from a more structured water network
surrounding the solute.
The entropy-enthalpy compensation has nonetheless not been widely acknowledged as witness
from the review of Chodera and Mobley [58]. Although according to Ben-Naim [54], the
entropy-enthalpy compensation theorem has been “reproved” several times by several authors
using different nomenclature and different notations, maybe in the most famous form by
Grunwald and Steel [59]. Several authors state that entropy-enthalpy compensation is largely
unexplained and some that “In principle, no explicit relationship between the enthalpy change
and the entropy change can logically be derived from fundamental thermodynamics” [60]. It is
not uncommon to find statements in textbooks in chemistry referring to the hydrophobic
interaction, as a major driving force in folding of macromolecules, the binding of substrate to
enzymes and most other molecular interactions in biology but these arguments are disputed by
Ben-Naim.
Interpretation of the thermodynamic potentials in light of Ben-Naim’s theory
In the following the theory for the entropy-enthalpy compensation by Ben-Naim will very briefly
be applied to shed light on a thermodynamic interpretation of the interaction between
cyclodextrins and bile salts and micellization of bile salts.
Cyclodextrins
Cyclodextrins interacts with bile salts by formation of inclusion complexes, whereby the bile salt
molecule enters the hydrophilic cavity of the cyclodextrin. A recent review by Biedermann et al.
[61] explains the driving force for the host-guest complexation of cyclodextrins by the ejection
7
This is not to be confused with the situation for polar molecules such as ions, where the strong electric field near
their surface, force the water molecules to orient their dipole moments towards the charge of the ion.

29
of high-energy water in the cyclodextrin cavity by the guest molecule. As shown in Fig. 3.5.3,
water molecules inside the cavity are confined and participate in fewer hydrogen bonds than the
average of ~3.6 for water molecules in the bulk [61]. By inclusion of a guest molecule into the
cavity the water molecules are released to the bulk to form additional hydrogen bonds yielding
an enthalpic advantage.
Figure 3.5.3: Snapshot from MD simulation for a β-CD with 5 cavity water molecules.
Hydrogen bonds are represented by a dashed line. Reproduced from Biedermann et al 2014 [61]
with permission.
However in the work by Biedermann et al., the events taking place in the bulk water when the
guest leaves for the benefit of the cavity was largely overlooked as this work focused on
describing the driving force for the inclusion complex formation and not the entropy-enthalpy
compensation. Measurements of the complexation of cyclodextrin with small hydrophobic
molecules such as bile salts by ITC are usually accompanied by entropy-enthalpy compensation.
In light of Ben-Naim’s theory this seems reasonable as the inclusion of a guest into the
cyclodextrin cavity has similarities with a desolvation process and a perturbation of the
equilibrium L ⇌ H in the bulk water can be expected to follow when the guest is removed. This
approach was taken by Liu and Guo [56] who extended the derivation above for a single solute
to a system including a cyclodextrin and a guest molecule. However, in this approach it should
be emphasized that the high-energy water trapped in the cyclodextrin cavity should not be
included in the equilibrium condition μL=μH (as this state ceases to exist when the guest expels
the water molecules and therefore not is an equilibrium state). Therefore the high-energy water is
an essential driving force for complexation. Further discussion of the entropy-enthalpy
compensation for cyclodextrins can be found in [32], [56], [60], [62].
Micelles
According to classical interpretation of thermodynamics the hydrophobic interactions are
important in the formation of micelles, as the entropy change is positive even though the
molecules are clustering together (and thus are more “ordered”) [51]. However, the textbook by
Israelachvili states that the origin of the driving force for the hydrophobic interaction still is
unknown [39]. Ben-Naim does as such not treat micelles, but mentions that the hydrophobic
effects clearly is important in processes involving long chain hydrophobic molecules such as
formation of micelles (i.e. when the frozen-in part of Eq. 3.5.7 becomes large).
ITC measurements of micellization, such as those conducted in manuscript 2, typically show
entropy-enthalpy compensation as a function of temperature. A model of micellization should
therefore be able to show this property. One model which incorporated this was recently

30
proposed by Fisicaro et al. which included an argument about cavity reduction and a resulting
expansion of the volume of the solvent. In fact, the classical hydrophobic effect has also been
subject to a compensating argument, that is: the reduced mobility of the water molecules in the
solvation shell is entropic disadvantageous, but the water-water hydrogen bonds in the solvation
shell are strengthened and favored by enthalpy. At higher temperatures, the water molecules
become more mobile leading to a compensating effect of the entropy and enthalpy.
Summary and perspectives on hydrophobic processes in aqueous solutions
The scope of the current thesis has only allowed a very brief discussion of the different
interpretations of the hydrophobic effects, but in the light of recent contributions by Ben-Naim
and others the entropy production might be of different origin than in the classical textbook
interpretation of the hydrophobic effect. These points are important for a sound interpretation of
the measured enthalpy response in ITC.
One example of how the knowledge of thermodynamics in aqueous solutions can be employed
was given in manuscript 2 as described in the following. Frequently it is assumed, that the value
of the isobaric heat capacity  pp THC  can be described as a linear function of the
hydrophobic surface area that gets exposed to water during this process [63][52]. As the
aggregation number for bile salts is considerably smaller than the aggregation number for more
regular micelles, bile salts constitutes an opportunity to observe the variation in ΔCp with the
dehydrated hydrophobic surface area. In fact, it can be derived that the heat capacity depends on
the aggregation as follows:
31
nbaCp  Eq. 3.5.14
When the aggregation number is as small as for bile salt micelles, we may be able to observe this
variation as shown in Fig. 3.5.5 and elaborated in manuscript 2.
Figure 3.5.5: The estimated ΔCp as a function of the average aggregation number for each type
of bile salt. Filled symbols, experimental values in water, open symbols, experimental values in
150 mM NaCl.

31
The discussion in the current section 3.5 about the origin of the thermodynamic potentials is the
theoretical foundation for all in vitro measurements by ITC performed in this project and should
therefore be considered according to the workflow of the modelling process in Fig. 2.3. It should
also be noted that Ben-Naim’s model is based on solvent reorganization, but several other
mechanisms have been proposed [64]. Considering the diversity of chemical reactions it is in my
opinion likely that a single mechanism is not able to explain all of them.

32
Chapter 4: Parameter estimation
4.1 Parameter estimation - introduction
A statistical estimator is a mathematical rule for calculating an estimate of a given quantity based
on observed data. In an overdetermined equation system where there are more equations than
unknown parameters, the least-square estimator is the method of choice for parameter estimation.
The least-square estimator is found by minimizing the residuals sum of squares:
Eq. 4.1.1
where yi is the ith observation, f(xi,θ) is the model function (the subject of chapter 3 – model
building) and θ is a vector of parameters.
Inference from the least-square estimator is usually derived in terms of a point estimate (the most
likely estimate) and interval estimates (expressing the uncertainty about the estimate). Two
important properties related to statistical inference is bias, i.e. does the point estimate in general
deviate from the true value and variance, i.e. how certain is the point estimate. The least-square
estimator will under certain conditions obtain an unbiased estimate and is optimal in the sense
that it has the lowest variance among the group of unbiased estimators.
In this project the aggregation number n and thermodynamic potentials for micellization of bile
salts have been estimated by means of isothermal titration calorimetry (ITC). As the model
function is nonlinear, as described in chapter 3 – model building, this requires
numerical methods. Such pre-implemented software to determine the aggregation number in
combination with the thermodynamic potentials from ITC data has to the best of my knowledge
not been available in the past8
. Currently it is therefore beyond standard practice in most
publications to determine the aggregation number [65], [66]. Recently, two computational
platforms have specialized in analysis of ITC experiments called AFFINImeter9
and ITCsy10
.
ITCsy does not include a model of micellization whereas such a model currently is under
development at AFFINImeter at the time of writing. Nonetheless, these platforms do collect
much of the state-of-the-art knowledge about parameter estimation by ITC and therefore
represent a good reference point for a scientific discussion addressing this issue.
The overall aim of the parameter estimation conducted in this project is to quantify the
equilibrium system shown in Fig. 2.1. However, as the results in chapter 3 – model building
showed, the full equilibrium system in Fig. 2.1 is not expected to be necessary to describe the
drug uptake. Rather, the free drug concentration can be quantified from the drug solubilisation
8
Two manufactures of isothermal titration calorimeters exists: MicroCal Inc. (Malvern Instruments Ltd, UK) and
TA Instruments, Inc, USA. In this project I have used the VP-ITC 200 from MicroCal, however the data analysis
software for the VP-ITC is no longer supported by MicroCal.
9
AFFINImeter is a commercial software developed by a spin-off company of the University of Santiago de
Compostela, Spain and provided as a web application at https://www.affinimeter.com/
10
ITCsy is a sister-program to SEDPHAT developed by: Dynamics of Macromolecular Assembly Section,
Laboratory of Cellular Imaging and Macromolecular Biophysics, National Institute of Biomedical Imaging and
Bioengineering, National Institutes of Health, Bethesda, MD 20892, USA. It can be downloaded free of charge from
https://sedfitsedphat.nibib.nih.gov/software/default.aspx
    2
; 
i
ii xfyRSS 
 ;ixf

33
capacity     soltottotsolCDDsolCDD
SC
tot DBSCDDKDKD  :: 1 , which only depends on the total bile salt
concentration as shown by the nondimensionalization in chapter 3 – model building. More
experimental data showing that bile salt micelles can be described by the mechanism n·S ⇌ Mn
might still strengthen the modelling process. But largely, the current chapter contains material
which is independent of the overall pharmaceutical aim and is therefore majorly self-contained.
The basic measuring principles for isothermal titration calorimetry are introduced in chapter 4.2.
This includes a demonstration of the dynamics of the response for both non-ionic and ionic
micelles with a special emphasis on low aggregation numbers as this is the case for bile salt
micelles. Chapter 4.3 discusses the scaling of the model function to speed up the computational
time and ensure convergence. These aspects are applied on bile salt micelles in chapter 4.4
verifying the model n·S ⇌ Mn for the micellization process. Finally, chapter 4.5 makes some
future outlooks for parameter estimation with ITC. This involves procedures which currently not
are implemented. Chapter 4.5.1 discusses optimization of the experimental design for measuring
micelles with ITC and chapter 4.5.2 discusses parameter estimation where a single set of
parameters is estimated simultaneously on multiple ITC titrations which in the past have been
denoted global fitting.
4.2 ITC measuring principles
Isothermal titration calorimetry is a method to measure the heat generated or absorbed by a
reaction in a liquid solution at constant temperature. In an ITC-experiment, the titrant is injected
sequentially into a cell filled with an aqueous buffer. Each injection produces a signal peak,
which returns to the baseline after a couple of minutes. The heat for an injection (dQ) is obtained
by first estimating the baseline and then integrating the signal-baseline difference over the
duration of the peak. The full sequence typically consists of 20 to 30 peaks, in which titrant is
injected into the cell containing the remains of all preceding injections [67]. Micellization can be
investigated by ITC in a so-called dilution experiment, where a solution of concentrated micelles
above CMC is titrated into a buffer. Initially, the micelles dissociate upon injection, but as
subsequent injections increase the total concentration in the reaction cell, dissociation ceases.
To extract information from an ITC experiment a functional relationship – known as a binding
isotherm – is applied. Mathematically this is expressed as a system of two coupled equations.
The first is a constitutive equation relating the experimentally controlled variable (i.e. the total
surfactant concentration in the reaction cell, Stot) to the measured heat response dQ from the ITC.
The second equation is a model of the chemical reaction incorporating conservation of mass and
the reaction mechanism in terms of the micellization constant.
The constitutive equation
The constitutive equation can be derived by considering that the heat detected by the ITC is
proportional to the change in number of molecules in the micelles, i.e.
. Usually the heat is normalized by the increase in total surfactant
concentration, and we therefore write
 ncelldemic MdnVHdQ 

34
Eq. 4.2.1
where dQ is the heat, ΔHmic is the enthalpy of micellization, Vcell is the volume of the reaction
cell, Stot is the total surfactant concentration, and [Mn] is the concentration of micelles each with
an aggregation number of n monomers.
If the concentration of micelles in the syringe is much larger than CMC, Eq. 4.2.1 can be
approximated by
Eq. 4.2.2
This is the model normally assumed in the literature for a dilution experiment [49], [68], but as
shown in manuscript 2 it is only an approixmation and in reality the enthalpy depends on the
syringe concentration. This concentration dependence will, however, be neglected here. Notice
that Eq. 4.2.2 is stated in infinitesimal form for mathematical convenience whereas a real ITC
experiment measures a discrete heat quantity. In addition, the concentrations in the heat-detecting
reaction cell will be affected by volume displacement for the perfusion-type ITC applied in this
project, as material will be expelled to the overflow volume in the course of an experiment. To
address these two points, the ITC-model must be discrete11
and the considered concentrations should
only include the part of the surfactants, which are in the heat detecting reaction cell. Details of these
topics can be found in the literature [67], [69], [70]. For illustrative purposes the infinitesimal forms
are considered below.
The equation for the chemical reaction
To associate the ITC-signal described by Eq. 4.2.2 with the underlying chemical parameters we
need an equation for the chemical reaction. Different models for micellization have been
discussed in chapter 3.3 – model building. In this section the mass-action model n·S ⇌ Mn is
elaborated. The case of non-ionic micelles has previously been discussed; another frequently
occurring case is ionic micelles. The dynamics of the ITC response for both types of micelles has
been the subject of manuscript 2 and will be summarized below. The link between the
constitutive equation described above and the equation for the chemical reaction is given by the
derivative in concentration d[S]/dStot in Eq. 4.2.2. As ΔHdemic is assumed to be constant in Eq.
4.2.2 the observed ITC-signal will simply be a normalized version of the derivative
d[S]/dStot=1/(Vcell·ΔHdemic)·dQ/dStot, and this quantity will be referred to as the normalized
enthalpogram. For simplicity the analysis of the ITC-signals dependence on the chemical
reaction will therefore be based on the normalized enthalpogram in the following section.
For non-ionic micelles the equation for the chemical reaction reads Stot=S+n·K·[S]n
, by implicit
differentiation of this equation it is seen that the normalized enthalpogram is given by
Eq. 4.2.3
11
As outlined in manuscript 2, parameter estimation based on the infinitesimal equations is erroneous.
 
tot
n
miccell
tot dS
Md
nHV
dS
dQ

 
tot
demiccell
tot dS
Sd
HV
dS
dQ

 
  12
1
1


 n
tot SKndS
Sd

35
A suitable nondimensionalization is found where the concentration of free monomers [S] equals
the concentration of monomers as micelles n·[Mn]. This concentration is given by
. Introducing the dimensionless concentrations   n
nKSS 
 1
and
n
tottot nKSS 
 1
and inserting into Eq. 4.2.3 yields    1
11 
 n
tot SndSSd .
Dynamics of the ITC measurements for non-ionic and ionic micelles
Non-ionic micelles
In Fig. 4.2.1a the concentration of monomers on free [S] and micellar form n[Mn] are shown as a
function of total surfactant concentration Stot. Fig. 4.2.1b shows the first derivative of the curves
shown in Fig. 4.2.1a, which are normalized enthalpograms as explained above.
Figure 4.2.1: (a) The dimensionless concentration of monomers on free form   equi
totSS (grey
line) and on micellar form   equi
totn SMn (black line) as a function of dimensionless total
surfactant concentration equi
tottot SS for . The CMC for the two species are
marked with circles. (b) Normalized enthalpogram d[S]/dStot=1/(Vcell·ΔHdemic)·dQ/dStot as a
function of dimensionless total surfactant concentration equi
tottot SS for . The
value of the normalized enthalpogram at CMC is marked with circles.
Of particular importance for bile salts with low aggregation numbers is the illustration in Fig.
4.2.1 that lower values of n lead to a broader transition region. An ITC-titration typically ends
close to the equivalence concentration and at this point it can be shown that the slope of the
monomer concentration is given by d[S]/dStot=1/(n+1), n>2. Thus, when the aggregation number
is low as for bile salts there will be a significant increase in monomer concentration as a function
of Stot even far above the CMC. It is also seen that the nondimensionalisation only affects the
horizontal scale and not the vertical scale of Fig. 4.2.1a as only the right-hand side of Eq. 4.2.3
was affected by the dimensionless variables.
Ionic micelles
For ionic micelles the monomers are typically fully ionized, but the high electric field strength at
the surface of the micelles will cause adsorption of some proportion of the free counter-ions.
nequi
tot KnS 
 1
10050,25,10,5,3,n
10050,25,10,5,3,n

36
Therefore, the mass-action model interprets the micellization as a chemical equilibrium process
between the charged micelles Mn
-(1-β)·n
and its constituents, the bile salt monomers S-
and bound
counter-ions B+
:
n·S-
+β·n·B+
⇌Mn
-(1-β)·n
Eq. 4.2.4
where n is the average aggregation number and β is the average degree of counter-ion binding to
the micelle [71].
The equilibrium is characterized by an ionic micellization constant defined by:
Eq. 4.2.5
where [Mn
-(1-β)·n
], [S-
], [B+
] are the concentrations of micelles, free bile salt monomers and free
counter-ions, respectively. Combining Eq. 4.2.5 with the two equations for conservation of mass
for the surfactants and for the counter-ions, the system is fully specified by:
Eq. 4.2.6
To illustrate this system, simulations can be conducted for various values of β with an initial
counter-ion concentration of zero as shown in Fig. 4.2.2. To ease visualization and comparison
the plots are again shown as a function of the total normalized surfactant concentration
 -equi
tottot SS , where the normalization is chosen as the total surfactant concentration, at which the
free concentration of surfactants [S-
] and micelle [Mn
-(1-β)·n
] are in equivalence, that is
, where is the counter-ion concentration at the equivalence
concentration. It can be shown that this transformation results in identical species curves for a
specific value of n and β regardless the value of K-
.
The left column in Fig. 4.2.2 shows the concentration of the individual species and the right
column shows the normalized enthalpograms equal to the first derivative of the free surfactant
concentration d[S]/dStot. The influence of β, depends on n and the system is therefore sketched
for two values of n, namely n = 5 (Fig. 4.2.2a and 4.2.2b) and n = 30 (Fig. 4.2.2c and 4.2.2d).
The main difference is found in the post-micellar region.
 
    nn
n
K 





BS
M )1(
n
     n
tot BSnKSS


 
 n
n
equitot BnKS  
 1-equi
2

 
equiB

37
Figure 4.2.2: Influence on degree of counter-ion binding on micellization simulated by the
mass-action model for a system with zero initial concentration of counter-ions. (a) Species
concentrations for n=5 and , (b) Normalized enthalpogram for n=5 and,
, (c) Species concentrations for n=30 and , (d) Normalized
enthalpogram for n=30 and . The species concentrations are shown as the
fractions of free monomers [S-
] (grey lines), counter-ions [B+
] (grey dashed lines) and monomers
in the micelles n· [Mn
-(1-β)·n
] (black lines). All figures are plotted as a function of total surfactant
concentration normalized by the equivalence concentration .
In Fig. 4.2.2c it is seen that the concentration of free surfactants [S-
] actually decreases for all
simulated values of β except when β = 0 and β = 1, that is in all cases where the micelles are
charged. This phenomenon is in accordance with experimental data and has previously been
pointed out by Moroi [72] and may result in negative values in the normalized enthalpogram
above CMC as shown in Fig. 4.2.2d. Notice, that for the micelle has no net charge and the
enthalpograms becomes identical to a system with and twice the aggregation number as can
easily be derived from Eq. 4.2.6.
 1,43,21,41,0
 1,43,21,41,0  1,43,21,41,0
 1,43,21,41,0
)(equi
tottot SS
1
0

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