Part II

Hit and Lead Profiling. Edited by Bernard Faller and Laszlo UrbanCopyright � 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-32331-9

4Solubility and Aggregation

William H. Streng

4.1Importance of Solubility

During the development of pharmaceutically important compounds, the question isoften asked: �what is the solubility of the compound?�While the question appears tobe simple and not too demanding, it can upon further contemplation be very difficultto answer. Not only is it necessary to know from what perspective the question isbeing asked, but it is also necessary to know to what depth of understanding thequestioner expects an answer. If the questioner is someone who has spent muchtime and energy determining the solubility of compounds, then the question morethan likely will be required to include much more detail than if the questioner issomeone who is involved in high throughput screening and is anxious to receive an�it is� or �it is not� soluble response. Having to answer the question can be verydisconcerting.Why is it necessary to ask this question? At what time during the development of

a compound should the question be asked? How correct does the answer need to be?Who is best able to answer the question? There are not necessarily simple ordefinitive answers to these questions.Considerable effort is spent trying to synthesize new compounds. The focus of the

chemist should be on modifying the structures of compounds in ways which willincrease the pharmacological response for a specific activity while minimizing theside effects. Today, chemists need to know about the human genome and the sitesatwhich a specific activity can be achieved. In addition theymust be able tomodify thestructures of the compounds in ways that will permit the compounds to dock at theseactive sites. Almost of equal importance is an understanding of the effect that specificfunctional groups will have on the solubility, stability and ability of the compoundsto distribute between different phases. This last aspect is related to the ability of thecompounds to be transported between, or through, membranes. Addressing thesolubility, if a compound is very insoluble in a series of compounds being considered(even though itmight have amuchgreater absolute pharmacological response), it can

Hit and Lead Profiling. Edited by Bernard Faller and Laszlo UrbanCopyright � 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-32331-9

j71

exhibit a lower response than one which has a smaller absolute pharmacologicalresponse but is more soluble. Conversely, if a compound is extremely soluble andif its route of absorption is transcellular passive diffusion, it might not be easilytransported through membranes to arrive at the active site and therefore very largedoses might be required even though the absolute activity is reasonably good.Therefore, solubility does play an important role in selecting a compound forcontinued development.The question is asked: �how accurately does the solubility need to be determined?�

Here again, the answer is not easy to give. During the early stages of development,any reasonable answer is better than none. It is at this time that testing is frequentlybeing conducted using solvents other than water. It might be acceptable to indicatein a series of compounds that all of the compounds were dissolved in order to obtaina good comparison of their relative activities. Later on during development it isimportant to have a good determination of the solubility. The solubility should bedetermined in a variety of solvents and under different conditions, such as pH andtemperature.There is no specific person who can best answer the question concerning the

solubility. At first a more qualitative answer might be obtained from the discoverychemist who needed to dissolve the compound in the different solvents used in thesynthesis and purification of the compound. Later it should be determined bysomeone who is familiar with the nuances of measuring the solubility and thosefactors which can influence the final result.

4.2Factors Influencing Solubility

If a compound is ideal and is placed in an ideal solvent, the solubility of thecompound can be shown to be simply related to the heat of fusion of the compound.As will be shown, the ideal solubility relationship is given by the van�t Hoff equationand relates the solubility to the heat of fusion of the compound, the melting pointof the compound and the temperature of interest. Because of this simple relation-ship, there is nothing relating to the solvent in which the compound is dissolved.It can therefore be concluded that the solubility should be the same in all solvents.Everyone knows that this just is not the case. Anything which contributes to thesolubility and which is not the heat of fusion of the compound or its melting pointbecome clumped together in the non-ideal part of the solubility. This non-ideal part isformally given by the heat of mixing of the compound with the solvent and hasseveral contributing factors. When someone says that the solubility has heat ofmixing contributions, unless they have done muchmore work to clarify, they are notmaking the job of understanding the solubility behavior any easier, because there canbe many factors which need to be included in this term.What are some of the factors which are included in this term heat of mixing?

When a compound is dissolved a hole must be made in the solvent. This hole mustbe made between the solvent molecules and if the solvent lacks a dipole and is not

72j 4 Solubility and Aggregation

polarizable, little work should be needed to introduce the compound. The closer themolecular volume of the compound is to that of the solvent, the less work that isdone. However, if there is much solvent–solvent interaction and the solvent has adipole or is polarizable,muchmore work is required tomake the hole in the solventto introduce the compound. A solvent which has little solvent–solvent interactionand does not have a dipole is benzene. Conversely, a solvent which has considerablesolvent–solvent interactions and has a dipole is the common solvent water. Withthis in mind, what is the solvent that is most frequently used in the pharmaceuticalindustry? Water.Solvent–solvent interactions have already been mentioned as a possible factor

which can contribute to the heat of mixing. Two other interactions which can bepresent in solution are solute–solvent and solute–solute interactions. In aqueoussolutions these types of interactions are almost always present when consideringthe structures of the molecules of interest in the pharmaceutical industry. Thesemolecules usually possess a dipole or can be polarized and therefore form, at the veryleast, weak long-range interactions with other molecules and/or with water. Thisis not to say that they form a more strongly bonded associated aggregate in solution,but weaker interactions do exist which contribute to the heat of mixing. Because thefunctional parts associated with a pharmaceutically active molecule usually possessa dipole or are polarizable, they orient the water molecules around themselves.By orienting thewatermolecules, they are forming solute–solvent interactions whichaffect the heat of mixing.As is well known, different solvents can have profoundly different abilities to

dissolve or solubilize a specific compound. There are many different classificationschemes used to classify solvents based on their properties. Each onemeets a specificneed. The properties used in a specific scheme normally do not have an abruptchange in their value and in fact change over a wide range. It is therefore somewhatdisconcerting to realize that there is a sense of arbitrary assignment of a value atwhich point the solvent changes from one type to another. Even in the sameclassification scheme there can be overlap of values from one type to another. Oneclassification scheme uses dielectric constant, relative acidity and relative basicityand considers solvents to be: (i) amphiprotic, capable of accepting or donating aproton; (ii) protogenic, acidic; (iii) protophilic, basic; (iv) aprotic, incapable of protontransfer; and (v) non-polar, not possessing a dipole. As a result of using threeproperties where each property can have one of two values, there are a total of eight(23) different classifications for the solvents. Where is this taking us? By knowingsomething about the structure of the molecule and whether it is a base or acid orhas a relatively high or relatively low dielectric constant, some sense of the abilityof a solvent to solubilize the compound can be realized. A very rough rule which isused to help in identifying how well a solvent will solubilize a compound is, �likedissolves like�. While there are many exceptions to this rule, it can be used as a firstestimation.If a molecule is an acid or base it will take on either a positive or negative charge

upon its ionization. The type of solvent used will significantly influence the observedsolubility of the charged species. If the solvent is aprotic or non-polar, the solubility

4.2 Factors Influencing Solubility j73

of the charged species will bemuch less than if the solvent is amphiprotic, protogenicor protophilic. The same effects will be observed if the compound is present as a salt.The enhanced solubilities are due to the solute–solvent charge-dipole type interac-tions. It is important to understand, when trying to decide on an appropriate solventto select, whether the conditions are such that the compound will be present asa charged species.It is safe to say that almost all, if not all, compounds which are of pharmaceutical

importance possessmultiple dipoles. These dipoles interact with any solvent dipoles,resulting in an enhanced solubility in these solvents. These dipoles act in themicroenvironment surrounding the individual components more so than themolecule as a whole (the net dipole for the molecule). The dipoles alter the solventstructure depending on the orientation of the partial charges associated with thedipole. If the charge on the dipole oriented towards the solvent is positive, thenegative end of the solvent dipole orients towards the molecule. If another compo-nent has the negative charge oriented towards the solvent, then the positive end of thesolvent molecule orients towards the molecule. When there are several componentswhich are orienting the solvent molecules differently, there can be a significantde-structuring effect upon the solvent. Similarly, if a solvent has a sufficiently highdipole it can induce or enhance a dipole. This effect not only influences the solubilityof the compound but can also alter its stability.

4.3Methods Used to Determine Solubility

During the development of a compound, differentmethods can be used to determinethe solubility. Because the degree of accuracy of the different methods is notthe same, it is important to recognize how the reported values were determined.Themethods canbe divided into screening tests and conclusive tests but this does notimply that each type does not have its own merit and importance. Either type can bedone using a range of solvents and can include buffers to establish solubilities atspecific pH values. Some of the commercially available instruments adjust thetotal ionic strength to a specific value.When this is done it must be remembered thatthe standard state for these solubilities is not the same as the conventional standardstate of zero total ionic strength. While these differences are usually not criticallyimportant and the values at a non-zero total ionic strength can be corrected to zeroionic strength, these are real differences. Although it is not the purpose of thischapter, the use of these types of instruments to determine the pKa also results in thesame difference in standard states, which can be corrected if needed.Screening tests are those which are used because there is little compound, little

time, or many different compounds to be tested. Many compounds can be synthe-sized using high throughput screening techniques. During this stage biological testsare conducted to ascertain the activity of the compounds. The compounds aresynthesized using 96 well or larger plates. The solubility of the compounds can beestimated using a method in which a known volume of solvent is placed in the

74j 4 Solubility and Aggregation

wells and a light passed through the cell. By the reflection of the light, or lack thereof,an estimation of its solubility can be made. If there is considerable turbidity or lightscattering, the compound is not dissolved. Additional solvent can be added untilthere is no turbidity. Another technique that can be used is to take a sample of thesolution and using microtechniques obtain a UV visible spectrum of the dissolvedmaterial. From the spectra a concentration can be estimated using a Beer�s law plot inwhich a specific functional group is used to quantitate the amount of compounddissolved. Both of thesemethods require small quantities of compound, can be donerapidly, many compounds can be tested within a short period of time and are forthe most part independent of the solvent used, therefore meeting the restraintsfor this stage of development. Some problems associated with these testing proce-dures include: (i) not having a stable polymorph or a crystalline form of thecompound; (ii) having different particle sizes which can influence the rate ofdissolution; (iii) not having a good or representative chromophore which is usedto quantitate the spectral measurement, chemical stability problems; (iv) not havingrelatively pure compounds; or (v) having a residual solvent present in the test solutionwhich is used in the experimental procedure (e.g., DMSO). These problemsnotwithstanding, these methods are valid during this stage of development.More conclusive tests for the solubility usually use traditional procedures in which

the compound is placed in a container along with a solvent and the systemequilibrated for a period of time. As a consequence, these experiments often requiremore amounts of compound and take more time to complete. Depending on theultimate use of the data, these procedures are at times modified to use less amountsof compound and/or less time. For instance, if someone wants to know how muchcompound can be dissolved in a certain period of time, a time restriction can beplaced on the experiment. For example, this is done when the interest is in knowinghow best to prepare a dosage form. If a time restriction is placed on the experiment,it must be realized that independent variables such as particle size and the particularpolymorph used can have significant effects on the final results. Another questionsometimes asked is similar but not exactly the same. When a solution of a specificconcentration is required, then a specific amount of the compound is placed inthe container and a quantity of solvent added which usually results in a solutionconcentration less than saturated. Again, the same concerns dealing with whatparticle size and polymorph is being used are important.When restrictions are not placed on the amount of time or the amount ofmaterial,

different solvents or different quantities of acid or base are usually used as variables.It is not only important to know the solubility of the compounds in aqueous solutionsbut also in other solvents to which the compoundmight be exposed during synthesisand formulation. The solvents usually have a wide range of dielectric constantsand the experimental results provide a solubility profile which can be utilized in theselection of appropriate solvents to use during the development of the compound.Since the compounds almost always selected for development are either weak acidsor weak bases, the solubilities of the compounds will be pH-dependent. The use ofdifferent amounts of acid or base with an excess amount of compound permits thedetermination of a pH-solubility profile.

4.3 Methods Used to Determine Solubility j75

Mention was made of the effects of particle size on the observed solubility. It mayseem strange but different solubilities can be obtained when different particle sizesare used. If experimentally a long time is used to equilibrate the solutions, then theeffect of particle size is not significant; but when short equilibration times areutilized, then the size of the particles can have an effect on the results. The reasonfor this is because the surface energy on a particle is related to the size of the particle.The energy increases with a decrease in the particle size. Therefore enhanceddissolution rates are observed when smaller particle sizes are used. It is observedthat smaller particles dissolve faster than larger particle sizes. If an excess ofcompound is present, the smaller particles dissolve and can supersaturate thesolution. When the supersaturated solution comes out of solution, the size of thelarger particles increases. If the solution is not at equilibrium, the solubility resultsobtained can be higher than the actual solubility due to this supersaturation.Different polymorphs or pseudo-polymorphs will have different solubilities. The

definition of a polymorph is a compound which has exactly the same molecularformula and arrangement of functional groups but the solid state configuration ofthe molecules is different. This includes the solvation of the compound. Oftencompounds with different solvates are compared, for example, a mono-hydrate anda tri-hydrate, and are listed as polymorphs but they are not according to the abovedefinition. When comparing a compound with different solvates of this type they aresometimes referred to as pseudo-polymorphs. Because polymorphic composition isa solid state property and relates to the arrangement of the molecules in the crystal,different polymorphs have different solubilities due to their different crystal ener-gies. Therefore it is important to be sure to knowwhat polymorph is being used in theexperiment and understanding what the effects will be if the solvation is changed.Another factor which can result in different solubilities is the isomeric composi-

tion of a compound. When a compound contains an asymmetric carbon atom, thefour different groups attached to the carbon can be in different arrangements. Thesedifferent arrangements result in the stereochemical behavior of the compound.When there is one asymmetric carbon atom there are two isomeric configurations.These configurations are mirror images of each other. Because the arrangement ofthe groups is different, there can be different intramolecular and intermolecularinteractions. This results in different solute–solute and solute–solvent interactions.The net result of this is that, during the early stages of development, a single isomeris not always available and therefore the solubilities reported can be different thanlaterwhen a relatively pure isomer is available.While this is not something that needsto be of concernwithmany of the compounds being developed, itmust be recognizedwhen a compound does have an asymmetric carbon atom.

4.4Approaches to Solubility

Several different approaches have been made to theoretically determine the solubi-lities of compounds. Before discussing two of these, a brief review of what is meantby ideal solubility is presented.

76j 4 Solubility and Aggregation

The van�t Hoff equation can be used to represent the solubility of a compound.

d ln x2dT

¼ DHs

RT2 ð4:1Þ

Where: X2¼mole fraction solubility; DHS¼ heat of solution of the solute;T¼ temperature; R¼ gas law constant.The heat of solution is related to the heat of fusion and heat ofmixing according to:

DHS ¼ DHf þDHmix ð4:2ÞFor an ideal solution the heat of mixing is zero and therefore Equation (4.1) afterintegrating becomes:

ln x2 ¼ �DHf

RTm�TTmT

� �ð4:3Þ

Where: Tm is the melting temperature of the compound. Also, the heat of fusion isa function of temperature given by:

DHf ¼ DHf ;m�DCPð T m� TÞ ð4:4ÞWhere: DCP¼heat capacity difference between the solid and supercooled liquid;DHf,m¼heat of fusion at the melting point.Substituting Equation (4.4) into Equation (4.1) and integrating results in:

ln x2 ¼ �DHf ;m

RTm�TTmT

� �þ DCP

Rð T m� T

T

��DCP

Rln

T m

Tð4:5Þ

According to Equation (4.5) the ideal solubility of a compound is only dependentupon the heat of fusion, the difference in heat capacity of the solid and supercooledliquid and the melting point of the compound. Since there are no properties ofthe solvent included in the ideal solubility equation, the solubility of a compoundshould be the same in all solvents. This equation overlooks all solute–solvent andsolvent–solvent interactions.One of the approaches to calculating the solubility of compounds was developed by

Hildebrand. In his approach, a �regular� solution involves no entropy change when asmall amount of one of its components is transferred to it froman ideal solution of thesame composition when the total volume remains the same. In other words, a regularsolution can have a non-ideal enthalpy of formation butmust have an ideal entropy offormation. In this theory, a quantity called the Hildebrand parameter is defined as:

d ¼ DUi

Við4:6Þ

Where: d¼Hildebrand parameter; DUi¼molar energy of vaporization of i;Vi¼molar volume of i.With this definition, anexpression for the activity coefficient canbederivedwhich is:

ln g2 ¼V2f21ðd1�d2Þ2

RTð4:7Þ

Where: f1¼ the volume fraction of the solvent f1 ¼ V1x1V1x1 þV2x2

.

4.4 Approaches to Solubility j77

Equation (4.7) is the non-ideal part of the solubility and can be added toEquation (4.3) to obtain an expression for the solubility of a regular solution.

ln x2 ¼ �DHf

RTm�TTmT

� ��V2f21ðd1�d2Þ2

RTð4:8Þ

Because the entropy of formation in Hildebrand theory is ideal, this approachshould be restricted to those systems in which there are no structure effects due tosolute–solvent and solvent–solvent interactions. The implication of this is that thesolute should be non-ionic and not have functional groups which can interact withthe solvent. According to Equation (4.8), the maximum solubility occurs when theHildebrand parameter of the solvent is equal to the Hildebrand parameter ofthe solute. That is, when plotting the solubility versus the Hildebrand parameter,the solubility exhibits a maximum when the solubility parameter of the solvent isequal to the solubility parameter of the solute.Utilizing this same type of approach but incorporating a non-ideal entropy is a

theory called the molecular group surface area approach (MGSA). Instead of usingthe internal energy, DU, the MGSA uses reversible work, W, to represent themolecular pair interactions. An equation for the activity coefficient can be derivedand is given by:

ln g2 ¼sh12A

h2 þsp

12Ap2

kTð4:9Þ

Where: s¼ the surface free energy or surface tension; k¼Boltzmann constant;A¼ the molecular surface area of the solute.The superscripts h and p refer to group contributions due to hydrocarbon and

polar groups, respectively. The polar term is often found to be small while the shapeof the cavity occupied by the solute molecule is irregular and requires a shape factor.Equation (4.9) then becomes:

ln g2 fficsh

12Ah2

kTð4:10Þ

Where: c is the shape factor.Adding Equation (4.10) to Equation (4.3) results in:

ln x2 ¼ �DHf

RTm�TTmT

� �� csh

12Ah2

kTð4:11Þ

Equation (4.11) includes bothnon-ideal entropy andnon-ideal enthalpy and thereforecan be applied to solutions which contain electrolytes and systems which havesolute–solvent and/or solvent–solvent interactions.

4.5Solubility in Non-Aqueous Solvents and Co-Solvents

None of the material presented in the last section was restricted to a specific solventsystem. In fact, the use of the Hildebrand theory should be limited to those systems

78j 4 Solubility and Aggregation

in which there are no significant structure effects related to solute–solvent andsolvent–solvent interactions and is therefore better suited for non-aqueous solvents.Both the Hildebrand and MGSA theory can be applied to co-solvent systems bymodifying the interaction parameters.The Hildebrand parameter for the solvent in Equation (4.8), d1, needs to be

replaced by the value for the mixture determined by multiplying the pure solventvalues by their volume fractions as given below for a two-solvent system.

dmix ¼fidi þ fjdjfi þ fj

ð4:12Þ

Where: i and j represent the two solvents. In addition, the volume fraction of thesolvent needs to be replaced with the sum of the volume fractions of all solvents,therefore Equation (4.8) becomes:

ln x2 ¼ �DHf

RTm�TTmT

� ��V2

Xni

fi

!2

ðdmix�d2Þ2

RTð4:13Þ

Additional terms can be added to this equation in order to correlatewith experimentaldata.When considering the MGSA model, a similar approach can be made. Starting

with Equation (4.11) the mole fraction solubility of a single solvent system can becalculated and labeled x2,1. The mole fraction solubility of a two solvent system canbe given as x2,mix. A linear function of the solubility in the mixed solvent systemcan be calculated according to:

ln X2;mix ¼ ln X2;1 þ k2f2 ð4:14ÞWhere: k2¼ a constant and is obtained by correlating with experimental data.Many systems have been found to correlate with this function but for those which

do not a polynomial expression can be used instead of k2.Even though Hildebrand theory should not apply to solvent systems having

considerable solvent–solvent or solute–solvent interactions, the solubility of com-pounds in co-solvent systems have been found to correlate with the Hildebrandparameter and dielectric constant of the solvent mixture. Often the solubility exhibitsa maximum when plotting the solubility versus either the mixed solvent Hildebrandparameter or the solvent dielectric constant. When comparing different solventsystems of similar solvents, such as a series of alcohols and water, the maximumsolubility occurs at approximately the same dielectric constant orHildebrand parame-ter. This does not mean that the solubilities exhibit the same maximum solubility.

4.6Solubility as a Function of pH

The solubilities of weak acids and bases are dependent upon the pKa value(s) of thecompound, the pH of the solution and the concentration of any counter ions to the

4.6 Solubility as a Function of pH j79

charged species of the weak acid or base. Figure 4.1 is a solubility versus pH profileobtained for a monoprotic weak base, when different amounts of strong acid orstrong base have been added to a constant amount of weak base. In this figure it canbe seen that there are two regions: one at pH values less than the pH value labeledpHmax, designated region 0, and the second at pH values greater than pHmax,designated region 1. It can be shown that for a monoprotic weak base, the solubilityin region 0 is controlled by the solubility product (pKsp) of the charged weak basespecies with the counter ion present in the solution and the pKa of the compound,while in region 1 the solubility is controlled by the neutral species intrinsic solubilityand the pKa. The downturn in the profile at low pH is due to the solubility producteffect of adding more strong acid counter ion to the solution as the pH is decreased.Equations can be derived which can be used to calculate the pKa, pKsp and intrinsic

solubility of the neutral species of a weak base. Because the solubility is controlledby the solubility product in region 0 and the intrinsic solubility of the neutral speciesin region 1, two equations are required to represent the entire profile.

Region 0:

S1;0 ¼� Mþ½ � þ fHþ g

yHþ � KwfHþ gyOH�

� �Mþ½ � þ fHþ g

yHþ � KwfHþ gyOH�

� �2þ 4Ksp

yX� yHBþ

� �1=2

2 fHþ gyBfHþ gyB þKayHBþ

� �ð4:15Þ

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0.10

0.15

0.20

0.25

0.30pH-Solubility Profile of a Weak Base

pH(max)

Sol

ubili

ty (

mol

es/l)

pH

Region 0 Region 1

Figure 4.1 Solubility versus pH for a monoprotic weak base.A constant amount of weak base is considered and the pH isadjusted with a strong acid or strong base.

80j 4 Solubility and Aggregation

Region 1:

S1;1 ¼ 1yB

þ fHþ gKayHBþ

� �fBgS ð4:16Þ

Where: S1,0 and S1,1 represent the solubility of a monoprotic compound in region 0and region 1 respectively; [Mþ ] is the molar concentration of strong base cation; { }represents the activity of the indicated species; Kw, Ksp, Ka are the ionization constantofwater (w), the solubility product of theweakbasewith its counter ion (sp) and aweakbase dissociation constant (a), respectively; yi is the activity coefficient of the indicatedspecies.The pH at which the profile changes from region 0 to region 1 is designated as

pHmax. This pH can be calculated for a monoprotic weak base according toEquation (4.17)

Hþmax

� � ¼� Mþ½ � þ ½Mþ �2 þ 4 1

yHþ þ fBgSKayHBþ

� �Kw

yOH� þ KaKsp

fBgSyX�� �� �1=2

2 1yHþ þ fBgS

KayHBþ

� �ð4:17Þ

Figure 4.1 shows that the solubility is a maximum at pHmax. This occurs because,at this pH, the solution is saturated in both the weak base salt species and the neutralspecies; that is, the solubility is controlled by both the solubility product and theneutral species intrinsic solubility. According to Equation (4.17) an increase in cationconcentration gives an increase in the pH of maximum solubility (a decrease in{Hmax}).Similar to a weak base, the solubility pH profile for a weak acid is shown in

Figure 4.2. This figure is a mirror image of the profile for a weak base.Again, the profile can be divided into two regions. The region where the pH is less

than pHmax is designated region 0 while that region where the pH is greater thanpHmax is designated as region 1. For a weak acid, the intrinsic neutral speciessolubility is controlling the solubility in region 0 and the solubility product ofthe anionic species of the weak acid with its cationic counter ion is controlling thesolubility in region 1. Although the downturn in the profile is not shown in thissimulation, if the calculations had been made at higher pH values, a decrease in thesolubility would have been calculated. This decrease is due to the solubility producteffect which accompanies an increase in the counter ion concentration because ofthe additional strong base present. Equations can be derived which can be used tocalculate the pKa, pKsp and intrinsic solubility of the neutral species of a weak acid.Because the solubility is controlled by the intrinsic solubility of the neutral speciesin region 0 and the solubility product in region 1, two equations are required torepresent the entire profile.

Region 0:

S1;0 ¼ 1yHA

þ Ka

fHþ gyA�

� �fHAgS ð4:18Þ

4.6 Solubility as a Function of pH j81

Region 1:

S1;1 ¼� X�½ � þ fHþ g

yHþ � KwfHþ gyOH�

� �X�½ � þ fHþ g

yHþ � KwfHþ gyOH�

� �2þ 4Ksp

yMþ yA�

� �1=2

2 KayHA

fHþ gyA� þKayHA

� �ð4:19Þ

The pH at which the profile changes from region 0 to region 1 is designated aspHmax. This pH can be calculated for a monoprotic weak base according toEquation (4.20)

Hþmax

� � ¼X�½ � þ ½X��2 þ 4 Ksp

KafHAgSyMþ þ 1yHþ

� �KafHAgS

yA�þ Kw

yOH�

� �� �1=22 Ksp

KafHAgSyMþ þ 1yHþ

� �ð4:20Þ

Figure 4.2 shows that the solubility is a maximum at pHmax. This occurs because,at this pH, the solubility is controlled by both the solubility product and the neutralspecies intrinsic solubility. According to Equation (4.20), an increase in anion concen-tration gives a decrease in the pH of maximum solubility (an increase in {Hmax}).

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0.20

0.25

0.30

pH(max)

pH-Solubility Profile of a Weak Acid

Solu

bility (m

ole

s/l)

pH

Region 0 Region 1

Figure 4.2 Solubility versus pH for a monoprotic weak aid.A constant amount of weak base is considered and the pH isadjusted with a strong acid or strong base.

82j 4 Solubility and Aggregation

4.7Effect of Aggregation Upon Solubility

The self-association of a compound in solution is an effect which should always beconsidered when conducting solubility studies. It is recognized that surfactanttype molecules require this property in order to form micelles. Not as well knownis the fact that aggregation also occurs with the organic compounds of interest to thepharmaceutical industry. There aremany different types of studieswhich can be usedto determine the aggregation number of a compound which does form aggregatesin solution. Some of these are conductivity, calorimetry, osmometry, pH andsolubility. In fact, any type of experiment can be usedwhich ismeasuring a colligativeproperty of the solution; that is, measuring a property which is dependent upon thetotal number of particles in the solution.The solubility profiles obtained for a compound which aggregates have the

different regions as described previously, but themathematical functions describingthe change in solubility with pHaremuchmore complex. The additional equilibriumfor a monoprotic weak base when the charged species is aggregating is:

nHBþ þ pX� > ðHBþ ÞnðX�Þp ð4:21Þ

The equilibrium given by Equation (4.21) has the equilibrium constant:

Kassoc ¼fðHBþ ÞnðX�ÞpgfHBþ gnfX�gp ð4:22Þ

The solubility of the weak base is then given by the equation:

S ¼ ½HBþ �þ ½B� þ n½ðHBþ ÞnðX�Þp� ð4:23Þ

This scheme assumes that aggregation is only occurring with the charged species.When this equilibrium is taken into consideration, the equations which describe thesolubility in the two regions become:

Region 0:

S1;0 ¼ 1yHBþ

þ Ka

fHþ gyB

� �Ksp

fX�g þ nKassoc

yðHBþ ÞnðX�Þp

!fX�gp Ksp

fX�g� �n

ð4:24ÞRegion 1:

S1;1 ¼ 1yB

þ fHþ gKayHBþ

� �fBgs þ n

Kassoc

yðHBþ ÞnðX�Þp

!fX�gp fHþ g

Ka

� �n

fBgns

ð4:25ÞThe contribution of the aggregation to the solubility is given by the last terms inEquations (4.24) and (4.25). Similarly an equation can be derived which gives the pHof maximum solubility, pHmax.

4.7 Effect of Aggregation Upon Solubility j83

ðn�pÞ Kassoc

yðHBþ ÞnðX�Þp

!Kpsp

Ka

fBgs

� �ðp�nÞfHþ

maxgðn�pþ 1Þ !

ð4:26Þ

þ fHþmaxg2

1yHþ

þ fBgsKayHBþ

� �þ Hþ

max

� �Mþ½ �� KaKsp

fBgsyX�þ Kw

yOH�

� �¼ 0

An example of a compound which forms aggregates in the aqueous phase is shownin Figure 4.3. In this figure the data were first treated without assuming aggregateformation and the curve calculated is given by the dash-dot-dash line. Thefigure showsthat there is little correlation between the experimental data and the calculated curveparticularly for the higher concentration data. Fitting the data to Equations (4.24)through (4.26) resulted in the solid line curve and the dotted curve. It can be seen thatthere is very good correlation between the data and the calculated curves.The data were fit to a model which had an aggregate species with the formula

(BHþ )2(X�). Additional studies were conducted using different techniques which

confirmed an aggregate of this composition is present in the solution. In this figurethe dotted line is the calculated curve for which the solution had an additional 0.05MNaCl added. The significant decrease in the solubility when a salt is present havinga common ion with the strong acid or base used to adjust the pH is not restricted to

pH

14121086420

M

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

–– Calculated Experimental 0.05M NaCl Experimental 0.05M NaCl

—•—•••••

•

Calculated profile without association

Figure 4.3 Solubility versus pH for a compound which forms aggregates in solution.

84j 4 Solubility and Aggregation

solutions for which aggregates are present. In fact, this is observed with all solutionsover the pH range where the solubility is controlled by the solubility product betweenthe charged species of the weak acid or base and a counter ion. According to thederivation, the assumption was made that the monomeric species controls thesolubility through its solubility product (Ksp) in Region 0 when aggregation occurs.This follows the convention that the solubility product is written as the lowestcommon ratio of the species involved.However, when there are significant quantitiesof an aggregate in the solution, it might be more correct to express the controllingsolubility product in terms of the aggregate. This would change the interpretationof the calculated parameters but it should not have an effect on the equations usedto determine the solubility.The total solubility of a monoprotic weak acid, when the negative species is

aggregating, is given in the following equation.

S ¼ ½HA� þ ½A��þ p½ðMþ ÞnðA�Þp� ð4:27Þ

The equations for the pHdependency of the solubility in the two regions for aweakacid are given as follows:

Region 0:

S1;0 ¼ 1yHA

þ Ka

fHþ gyA

� �fHAgs þ p

Kassoc

yðMþ ÞnðA�Þp

!fMþ gn Ka

fHþ g� �p

fHAgps

ð4:28ÞRegion 1:

S1;1 ¼ fHþ gKayHA

þ 1yA

� �Ksp

fMþ g þ pKassoc

yðMþ ÞnðA�Þp

!fMþ gn Ksp

fMþ g� �p

ð4:29Þ

An equation can be derived which results in the pH of maximum solubility, pHmax,for a weak acid.

fHþ g2max

Ksp

KafHAgsyMþþ 1

yHþ

� ��fHþ gmax X

�½ �� KafHAgsyA�

þ Kw

yOH�

� �

�ðn�pÞKassocKnspðKafHAgsÞðp�nÞðfHþ gmaxÞðn�pþ 1Þ

yMnAp

¼ 0 ð4:30Þ

The above equilibria were restricted to aggregate formation involving the chargedspecies of the weak acid or base. As with the weak base equations, the assumptionwas also made that the monomeric species is controlling the solubility through itssolubility product. A similar approach can be made when an aggregate is formedwith the neutral species. In addition to these cases, it is possible that both the chargedand neutral species could form aggregates and then the equations would need toinclude both terms. A major assumption in deriving these equations is that thereis only one aggregate species present; that is, all of the aggregates have the same

4.7 Effect of Aggregation Upon Solubility j85

number of weak acid or base molecules. This is usually safe to assume and it is veryuncommon for there to be a significant concentration of more than one aggregatespecies with a specific weak acid or weak base species. Another type of interactionthat can occur is the formation of complexes when a complexing agent has beenadded to the solution. For this case, the same type of approach can be made bysubstituting into the equations the ligand for the common ion species of the strongacid or base. With the necessary equations, it becomes possible to determine theaggregation numbers and equilibrium constants through the use of solubility data.In the case of solubility data, when the experimental data cannot be fit to the knownexpressions assuming no aggregation, there is a high probability that aggregation isoccurring.

4.8Dependence of Dissolution upon Solubility

The rate at which a compound dissolves is dependent upon its surface area,solubility, solution concentration, rate of reaction and transport rate. These quanti-ties are defined as follows: surface area – the surface area of the individual particlesif the compound is not compressed or the surface area of a disk if the compound iscompressed; solubility – the solubility of the polymorphic form in the solid phase;solution concentration – the concentration of the compound in the bulk of thesolution; rate of reaction – the rate at which the solid surface reacts with the solventor dissolution medium; transport rate – the rate at which the compound travelsthrough the diffusion layer. The rate of dissolution, or flux, of a compound can begiven as:

J ¼ kr;tðCs�CbÞ ð4:31ÞWhere: J is the dissolution rate; kr,t is the rate of reaction or transport rate; Cs is thesolubility of the compound; Cb is the concentration in the bulk of the solution.In Equation (4.31) the rate constant is either the reaction rate constant or the

transport rate constant, depending on which rate controls the dissolution process.If the reaction rate controls the dissolution process, then kr,t becomes the rate of thereaction; while if the dissolution process is controlled by the diffusion rate, then kr,tbecomes the diffusion coefficient (diffusivity) divided by the thickness of thediffusion layer. It is interesting to note that both dissolution processes result inthe same formof expression. From this equation the dependence on the solubility canbe seen. The closer the bulk concentration is to the saturation solubility the slowerthe dissolution rate will become. Therefore, if the compound has a low solubility inthe dissolutionmedium, the rate of dissolution will be measurably slower than if thecompound has a high solubility in the same medium.It was mentioned that the solubility is that of the polymorph used to prepare

the solid phase. It is possible to achieve higher dissolution rates by using unstablepolymorphic forms of the compound. For example, if a hydrate is the stablepolymorphic form in the presence of water, an anhydrous form would be more

86j 4 Solubility and Aggregation

soluble and therefore exhibit a faster dissolution rate. If the most stable form of acompound is a hydrate, then the dissolution rate can potentially be increased by usingthe anhydrous polymorphic form. Related to this is the fact that if the compoundis amorphous, the dissolution rate is higher than if a crystalline material is used.Another factor which influences the dissolution rate and is related to the solubility isthe form of the compound when the compound is a weak acid or base. If the neutralform of the compound is used for the solid phase the dissolution rate is slower thanwhen a salt form is used, assuming all other factors remain the same.

4.9Partitioning and the Effect of Aggregation

When two immiscible solvents are placed in contact with each other and a non-ionizable compound is dissolved in one of the solvents, the compound distributesitself between the two solvents. This distribution is referred to as partitioning. Theratio of the concentrations of the compound in each phase is a constant for a specificset of solvents, pH, buffers, buffer concentrations, ionic strength and temperature.This ratio is referred to as a partition coefficient or distribution coefficient and isequal to the ratio of the solubilities in the two solvents.When the compound is aweakacid or base, the distribution of the compound can be shown to be given by thefollowing equation for a monoprotic compound:

kow ¼k0 þ k1

Ka½H�� �

1þ Ka½H�� � ð4:32Þ

Where: kow is the distribution coefficient; k0 and k1 are the intrinsic partitioncoefficient for species 0 and 1 where species 0 is the most protonated.If aggregation occurs in one of the phases, Equation (4.32) needs to be modified.

Let the aggregation be represented by:

nC1 >Cn ð4:33ÞWhere: n is the number ofmonomers in the aggregate; C1 is the concentration of themonomer; Cn is the concentration of the aggregate.For the condition in which there is only one primary monomeric species present

in the aqueous phase and in which the aggregation occurs in the organic phase thefollowing equation can be derived:

kow ¼ nKnkn1ðC1ðwÞÞðn�1Þ ð4:34Þ

Where: Kn is the equilibrium constant for the formation of the aggregate.

Kn ¼ Cn

ðC1Þn

Where: k1 is the intrinsic partition coefficient for the monomeric species; C1(w) isthe concentration of the monomer in the aqueous phase.

4.9 Partitioning and the Effect of Aggregation j87

The following relationship holds when there is only one primary species in theaqueous phase:

log CTðOÞ ¼ log n Knkn1 þ n log CTðwÞ ð4:35Þ

Where: CT(0) is the total concentration in the organic phase; CT(w) is the totalconcentration in the aqueous phase.If the aggregation occurs in the aqueous phase the following equation can be

derived when there is only one primary monomeric species present in the organicphase:

kow ¼ nKn

kn1Cðn�1Þ1ðOÞ

� �� ��1

ð4:36Þ

The following relationship holds when there is only one primary monomericspecies in the organic phase:

log CTðOÞ ¼ log k1� 1nlog n Kn þ 1

nlog CTðwÞ ð4:37Þ

It can be seen in Equations (4.35) and (4.37) that a log–log plot of the totalconcentration in the aqueous phase versus the total concentration in the organicphase should be linear with a slope >1 if aggregation occurs in the organic phase anda slope <1 if aggregation occurs in the aqueous phase. Furthermore, the slope of theline is equal to the number of monomeric molecules forming the aggregate in theorganic phase and is the reciprocal of the number of monomeric molecules formingthe aggregate in the aqueous phase.The assumption made in these derivations is that the aggregate is only present

in one phase and therefore does not partition between the two phases to anyappreciable extent. If the aggregate should partition between the two phases, thenthis approachwould not work and the slope of the line would be close to one. It wouldbe expected that highly charged aggregates could form in the aqueous phase andnot the organic phase and uncharged aggregates would be more apt to form in theorganic phase.An example of this can be seen in Figure 4.4. This is a zwitterionic compoundwith

two pKa values, the first about 4.25 and the second at about 9.0, and therefore overthe pH range of the study there were three possible species which could form. Themeasured log ki values for the three species were 0.4, 0.3 and 0.74 for the cationic,zwitterionic and anionic species, respectively, when not considering aggregation.Experimentally, three pHwere chosen where there was only one monomeric speciespresent: pH 2, pH 7 and pH 11. Four different concentrations of the compoundwere studied: 1.55� 10�4M, 1.01� 10�4M, 4.96� 10�5Mand 1.84� 10�5M. It canbe seen that the slopes of two of the curves are similar while one is different. Theslopes at pH 2 and 7 are 0.86 and 0.97, respectively, while the slope at pH 11 is 0.37.There does not appear to be any aggregation in the organic phase at any pH, butat pH 11 the compound appears to be aggregating in the aqueous phase with anaggregation number of three.

88j 4 Solubility and Aggregation

4.10Solubility in Simulated Biological Fluids

As already mentioned, the solubility of compounds can be influenced by thepresence of other compounds in the solution. This can simply be an ionic strengtheffectwhichresults inchanges to theactivity coefficients;or theadditional compoundscan directly interact with the compound of interest and result in complexes or aggre-gates being formed or salts having lower solubility. The use of simulated biologicalfluids in the determination of solubilities has not been done extensively. The potentialfor some type of interaction increases as the solubility of the compound decreases.While these interactionsresult inan increase in theoverall solubilityof thecompound,it can also have the effect of reducing the transport rate through pharmacologicalmembranes. Some compounds which are used to simulate biological fluids are bileacid salts such as sodium taurocholate. These salts are present in intestinal fluids andare therefore a logical choice to use when trying to simulate intestinal fluids anddetermine the effect biological fluids might have on the overall solubility.Simulated intestinal fluid is associated more with dissolution rate determination

than solubility measurements. Sometimes when a compound has particularly lowsolubility the dissolution is studied in simulated fluids. The intent is to try andproduce an increased dissolution rate. If there is an increased rate it would be due toan interaction between the compound and the dissolution fluid. Sometimes this

-4.2-4.4-4.6-4.8-5.0-5.2-5.4-5.6-5.8-6.0-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

-3.8

pH 2 pH 7 pH 11

log

Co

log Cw

Figure 4.4 Concentration dependence of the partitioning of acompound that forms an aggregate. In this figure, C0 and Cw areCT(0) and CT(w) in Equation (4.35).

4.10 Solubility in Simulated Biological Fluids j89

increased rate is due simply to an increase in the wettability of the compound andtherefore not a specific increase in the solubility; but for poorly soluble compoundsany increase in dissolution rate is often due to specific interactions.Whenever a compound has an increase in solubility when using simulated

biological fluids, some concern should be raised because it can be a forebearer ofchanges to the transport properties. If a compound strongly interacts with somethingin the fluid there is a good chance that the transport rate across the pharmacologicalmembranes decreases. While this decrease might not be bad, it is something thatneeds to be understood and investigated.At other times the compound is found to have significant protein binding. Again

this can lead to an increase in the solubility of the compound. If the binding occursin the blood, it could lead to reduced elimination rate constants. Whenever thepharmacology studies indicate some type of interaction is happening additionalstudies should be conducted to try and elucidate the interactions. If the compoundhas a particularly low solubility it can lend itself to solubility studieswhich canbeusedto determine the interaction equilibriumconstants and effect on the overall solubility.

References

For further reading, the following texts can beused to obtain a more detailed understandingof the material discussed in this chapter.

1 Avdeef, A. (2007) Solubility of sparingly-soluble drugs. In: Special issue.Dressman, J., Reppas, C. (eds) Theimportance of drug solubility. AdvancedDrug Delivery Reviews, 59, 568–590.

2 Avdeef, A., Bendels, S., Tsinman, O.,Tsinman,K.andKansy,M.(2007)Solubility-excipient classification gradient maps.Pharmaceutical Research, 24, 530–545.

3 Avdeef, A., Voloboy, D. and Forman, A.(2006) Dissolution and solubility, inComprehensive Medicinal Chemistry II,Vol. 5, ADME-TOX Approaches(eds B. Testa and H. van de Waterbeemd),Elsevier, Oxford.

4 Avdeef, A. (2001) High-throughputmeasurements of solubility profiles,in Pharmacokinetic Optimization in DrugResearch (eds B. Testa, H. van deWaterbeemd, G. Folkers and R. Guy),Wiley-VCH, Weinheim.

5 Avdeef, A. (2003) Absorption and DrugDevelopment, Solubility, Permeability, and

Charge State, John Wiley & Sons,Hoboken, New Jersey.

6 Connors, K.A. (1987) BindingConstants, The measurement of MolecularComplex Stability, John Wiley & Sons,New York.

7 Grant, D.J.W. and Higuchi, T. (1990)Solubility Behavior of OrganicCompounds, in Techniques of Chemistry,Vol. XXI (ed. W.H. Saunders), John Wiley& Sons, New York.

8 Lewis, G.N., Randall, M., Pitzer, K.S. andBrewer, L. (1961) Thermodynamics,McGraw-Hill Book Company, New York.

9 Popovych, O. and Tomkins, R.P.T. (1981)Nonaqueous Solution Chemistry, JohnWiley& Sons, New York.

10 Shinoda, K. (1978) Translated by P. BecherPrinciples of Solution and Solubility, MarcelDekker, New York.

11 Streng, W.H. (2001) Characterization ofCompounds in Solution, Theory and Practice,Kluwer Academic, Plenum.

12 Yalkowsky, S.H. and Banerjee, S.(1992) Aqueous Solubility, Methods ofEstimation for Organic Compounds,Marcel Dekker.

90j 4 Solubility and Aggregation