Understanding and Predicting the Effect of Cocrystal Components and

DOI: 10.1021/cg9001187

Understanding and Predicting the Effect of Cocrystal Components and pH on Cocrystal Solubility

2009, Vol. 9 3976–3988

Sarah J. Bethune, Neal Huang, Adivaraha Jayasankar, and Naı´ r Rodrı´ guez-Hornedo* Department of Pharmaceutical Sciences, University of Michigan, Ann Arbor, Michigan 48109-1065 Received February 2, 2009; Revised Manuscript Received June 17, 2009

ABSTRACT: Understanding how cocrystal solubility-pH dependence is affected by cocrystal components is important to engineer cocrystals with customized solubility behavior. Equations that describe cocrystal solubility in terms of solubility product, cocrystal component ionization constants, and solution pH are derived for cocrystals with acidic, basic, amphoteric, and zwitterionic components. Studies with carbamazepine-salicylic acid and carbamazepine-4-aminobenzoic acid show that cocrystals of a nonionizable drug achieve pH-dependent solubility when cocrystallized with ionizable coformers. These findings are in good agreement with predicted behavior and provide insight on the ability of coformer to determine the shape of the pH-solubility curve. It is shown that measurement of solution concentrations and pH at the eutectic point, Ctr, is valuable to (a) estimate cocrystal solubility-pH dependence, (b) evaluate the effectiveness of coformer in stabilizing or precipitating cocrystal, and (c) guide cocrystal selection without the time and material consuming determination of full phase solubility diagrams.

Introduction The design of pharmaceutical cocrystals with the objective of meeting aqueous solubility requirements is valuable to guide cocrystal selection and reduce experimental effort. Solubility is a function of crystal lattice and solvation interactions, and the interplay between these factors determines the extent to which solubility is correlated with melting point and enthalpies of fusion.1-3 Pharmaceutical cocrystals include components with a wide range of polarities and ionization properties; thus, in water, a highly polar solvent, solventsolute interactions may have a dominant role in solubility. We recently showed that aqueous solubility of carbamazepine cocrystals range over 3 orders of magnitude, finding that cocrystal solubility is correlated with coformer solubility, and that cocrystal melting point and enthalpies of fusion are not sufficient indicators of aqueous solubility.4 Another study reports good correlation with cocrystal and coformer melting point but weak correlation between cocrystal melting point and aqueous solubility.5 Salt formation has been extensively used to improve drug solubility.6-10 Studies of pharmaceutical salts confirm the breadth of lattice and solvation energies in these solids and thus the varying success of correlations between aqueous solubilities and melting points, counterion hydrophilicities or solubility of precursor acids.1,11,12 In the case of salts and cocrystals whose constituents ionize in solution, pH is an important determinant of solubility. The solubility-pH relationship for salts is well documented, and mathematical models that predict this behavior have been validated.6,7,13-16 Several reports present models that rationalize the solubilitypH behavior of cocrystals by using solubility product and Henderson-Hasselbalch equations, but studies that confirm these predictions are scarce.17-19 A study of gabapentin-3-hydroxybenzoic acid cocrystal found that observed *To whom correspondence should be addressed. Telephone: (734) 7630101. Fax: (734) 615-6162. E-mail: [email protected]. pubs.acs.org/crystal

Published on Web 07/31/2009

solubility-pH relationships followed mathematical models derived for a cocrystal of a zwitterionic drug and acidic coformer.20 Pharmaceutical cocrystals are often designed with acidic, basic, amphoteric, and zwitterionic molecules. These include nonionizable drug with acidic or amphoteric coformers, (carbamazepine cocrystals with benzoic acid, saccharin, salicylic acid, 4-aminobenzoic acid);21-23 basic drugs with acidic coformers (caffeine with maleic acid, glutaric acid, oxalic acid;24 itraconazole with succinic acid, fumaric acid, and malic acid);25 and zwitterionic drugs with acidic coformers (piroxicam with malonic acid and fumaric acid;26 gabapentin with 3-hydroxybenzoic acid),20 among others. Given the wide range of drug and coformer ionization properties, one would expect that cocrystal solubility-pH dependence will greatly vary, even for a family of cocrystals of the same drug. The ability to predict such behavior is important in order to meet targeted aqueous solubilities and customize solubility-pH dependence of cocrystals. With the objective of understanding how ionization properties of cocrystal components modify the solubility-pH dependence of cocrystals, two carbamazepine cocrystals were studied: 1:1 carbamazepine-salicylic acid (CBZ-SLC) and 2:1 carbamazepine-4-aminobenzoic acid monohydrate (CBZ4ABA-HYD). Salicylic acid is a monoprotic acid with pKa of 3.0;27 4-aminobenzoic acid is amphoteric with a range of reported pKa values of 2.2-2.7 for the amine moiety and 4.74.9 for the carboxylic acid moiety.28 Mathematical models for the solubility of cocrystals with different stoichiometries and components of different ionization properties (monoprotic and diprotic acids, amphoteric, and basic) are also derived. These models are extended to predict the pH dependence of cocrystal eutectic points or transition concentrations, Ctr, and compared with experimental measurements. At this point, two solid phases (cocrystal and drug or coformer) coexist in equilibrium with solution and the solution concentration is fixed at a given temperature and pH. Because cocrystal solubility measurements in water or buffered solutions will r 2009 American Chemical Society

Article

Crystal Growth & Design, Vol. 9, No. 9, 2009

be underestimated when a cocrystal is more soluble than one of its components and undergoes a phase transformation, Ctr is shown to be a key measurement from which cocrystal solubility can be estimated.

and eq 5 is rewritten as S cocrystal

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u Ka t ¼ K sp 1þ þ ½H

3977

ð8Þ

Theoretical Section Solubility of a Cocrystal with an Acidic or a Basic Coformer. We have previously shown that cocrystal-solution equilibrium is described by solubility product and that cocrystal solubility decreases with increasing coformer concentration.29,30 If one or both cocrystal components are ionizable, acid or base equilibria will also exist. Consider a 1:1 (drug/coformer) cocrystal RHA where the drug is R and the coformer is HA, a monoprotic acid. The equilibrium reactions for cocrystal dissociation in solution and coformer ionization are given below: RHAsolid hRþHA K sp ¼ ½R½HA

ð1Þ

HAhA - þHþ Ka ¼

½Hþ ½A - ½HA

ð2Þ

where Ksp is the solubility product of the cocrystal, and Ka is the acid ionization constant. Species without subscripts indicate solution phase. The analysis presented here assumes ideal behavior with concentrations replacing activities in the equilibrium constants. This is an approximation with the purpose of establishing general trends, and nonidealities due to complexation, ionic interactions, and solvent-solute interactions will need to be considered for a more rigorous analysis, particularly at high concentrations and ionic strengths. The analytical or total acid concentration, the sum of the ionized and nonionized species, is given by ½AT ¼ ½HAþ½A -

ð3Þ

while total drug, which is nonionizable, is given by ½RT ¼ ½R

ð4Þ

Drug concentration at equilibrium with cocrystal can be expressed in terms of coformer equilibrium concentration, Ksp, Ka, and [Hþ] by substituting [HA] and [A-] from eqs 1 and 2, into eq 3 and rearranging to give ! K sp Ka ½RT ¼ 1þ þ ð5Þ ½AT ½H Total drug concentration at equilibrium, [R]T, is shown as the dependent variable, since it is generally of interest to know how cocrystal solubility varies with coformer concentration and pH. Under these conditions ð6Þ S cocrystal ¼ ½RT When a cocrystal is in equilibrium with solutions of stoichiometry equal to the cocrystal (i.e., there is no excess coformer or drug in solution) then cocrystal solubility is equal to the total concentration of the drug or coformer in solution, ð7Þ S cocrystal ¼ ½RT ¼ ½AT

Under these conditions, solubility is referred to as stoichiometric solubility. Equations 5 and 8 predict that cocrystal solubility will increase with decreasing [Hþ] (increasing pH). Cocrystal solubility is also dependent on cocrystal Ksp and coformer Ka. When pH , coformer pKa, or [Hþ] . Ka, cocrystal solubility approaches its intrinsic solubility (Ksp)1/2. At pH=pKa, or [Hþ]=Ka, the cocrystal solubility is equal to (2Ksp)1/2 or 1.4 times the intrinsic cocrystal solubility. When pH . coformer pKa, or [Hþ] , Ka, cocrystal solubility increases exponentially. The maximum concentration that can be experimentally achieved is, however, limited by the solubilities of drug, coformer, and coformer salts. The reverse pH dependence is predicted for a cocrystal with a nonionizable drug and a basic coformer vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u þ u ½H ð9Þ S cocrystal ¼ tK sp 1þ Ka where Ka is the ionization constant for the conjugate acid of the base. In this case, the solubility increases with decreasing pH (increasing [Hþ]). A valuable implication from this analysis is that cocrystals impart pH-dependent solubility to nonionizable drugs when coformers are ionizable. These predictions are confirmed by studies with carbamazepine cocrystals presented in the Results section. Solubility of Cocrystals with Different Ionization Properties and Stoichiometries. Equations that describe cocrystal solubility dependence on [Hþ], Ksp, and Ka for several types of cocrystals have been derived and are presented in Table 1. The interested reader is directed to the Appendix for the full derivations. These equations describe the cocrystal solubility under stoichiometric conditions, when cocrystal is in equilibrium with solutions of stoichiometry equal to the cocrystal. Table 1 shows that cocrystal solubility is governed by at least two parameters, Ksp and Ka, and one variable, solution pH. In most cases Ka values are known and Ksp can be calculated from experimentally measured cocrystal solubility at one pH. Alternatively, one could target solubility and pH values and calculate the required Ka and Ksp. Most frequently, several cocrystals of a given drug are discovered for which the solubility is not known. From the Ka and Ksp values one could calculate the pH solubility curves and streamline the cocrystal selection process. Theoretical solubility-pH profiles (Figure 1) demonstrate the ability of cocrystals to modify solubility behavior relative to that of the drug. Solubility was calculated for four cocrystals with components that differ in their ionization properties using reported Ka values, experimental cocrystal solubilities, and equations in Table 1. Figure 1a,b shows that cocrystals of a nonionizable drug can exhibit very different solubility-pH behaviors, depending on the coformer ionization properties. A diprotic acid coformer will lead to increases in solubility with pH, as with a monoprotic acid. An amphoteric coformer will result in a U-shaped curve with a solubility minimum in a pH range between the

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Table 1. Cocrystal Solubility Dependence on [Hþ], Ka, and Ksp for Cocrystals with Components That Are Nonionic, Acidic, Basic, Amphoteric, and Zwitterionica cocrystal

solubility equations vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u K K K a1 , H A a1 , H A a2 , H A 2 2 2 þ S ¼ tK sp 1þ ½Hþ ½Hþ 2

RH2A (1:1, nonionic/diprotic acidic)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u K a1, H2 A K a1, H2 A K a2, H2 A 3 K sp t 1þ þ S ¼ 4 ½Hþ ½Hþ 2

R2H2A (2:1, nonionic/diprotic acidic)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u K a1, HAB 3 K sp ½Hþ 1þ þ S ¼t K a2, HAB 4 ½Hþ

R2HAB (2:1, nonionic/amphoteric)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi u u K a;HA ½Hþ t 1þ S ¼ K sp 1þ þ K a, B ½H

BHA (1:1, basic/monoprotic acidic)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !ffi u u 3 K sp K a1, H2 A K a1, H2 A K a2, H2 A ½Hþ t 1þ þ 1þ S ¼ K a, B 4 ½Hþ ½Hþ 2

B2H2A (2:1, basic/diprotic acidic)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi u u K K a, HX a;HA 1þ þ S ¼ tK sp 1þ ½Hþ ½H

HAHX (1:1, monoprotic acidic/monoprotic acidic)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! u u K a, HX K a1, HAB ½Hþ t S ¼ K sp 1þ 1þ þ K a2, HAB ½Hþ ½Hþ

HABHX (1:1, amphoteric/monoprotic acidic)

þ

hABH H2X (1:1, zwitterionic/diprotic acidic) a

ð10Þ

ð11Þ

ð12Þ

ð13Þ

ð14Þ

ð15Þ

ð16Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! u u K a2, - ABHþ K a1, H2 X K a1, H2 X K a2, H2 X ½Hþ ð17Þ þ þ S ¼ tK sp 1 þ 1 þ K a1, - ABHþ ½Hþ ½Hþ ½Hþ 2

Equations apply to a cocrystal in equilibrium with solutions of same stoichiometry as cocrystal (no excess component).

two pKa values. Similar behavior is predicted for a cocrystal of a basic drug and an acidic coformer where the ionizable groups reside in different molecules (Figure 1c). The pH range over which this minimum occurs is dependent on the difference between the two pKa values; the greater the difference, the wider this minimum range will be, as has been shown for zwitterionic solutes.31 This behavior is also predicted for a cocrystal of a zwitterionic drug and an acidic coformer (Figure 1d). Studies with gabapentin-3hydroxybenzoic acid cocrystal showed very good agreement between experimental and predicted pH-dependent behavior.20 Understanding the dependence of cocrystal solubility on both pH and coformer concentration is important to determine phase solubility diagrams and identify thermodynamic stability regions. Table 2 shows equations for cocrystal solubility derived for cocrystals with various ionization properties and stoichiometries. These equations are applicable to cocrystals in equilibrium with solution concentrations of different or equal stoichiometry to the cocrystal. Details of derivations are included in the Appendix. This analysis shows that cocrystal solubility and stability regions can be estimated from a single measurement of cocrystal solubility and pH without the need to experimentally determine a full phase solubility diagram.

Cocrystal Stability and Eutectic Point Dependence on pH. Since cocrystal solubility is dependent on both solution composition and pH, so are the relative thermodynamic stabilities of cocrystal and its components. Figure 2 shows this behavior for a hypothetical nonionizable drug (R) and its 1:1 cocrystal (RHA) with an acidic coformer (HA). The cocrystal solubility is predicted to increase with pH and to decrease as the coformer solution concentration increases, according to eq 5. Reversal of thermodynamic stability can be clearly seen in Figure 2. The stable phase is cocrystal at low pH, and drug at high pH values. Cocrystal and drug solubility curves intersect at the transition concentration (Ctr) or eutectic point. Two important characteristics of this point are that (1) two solid phases (for example, cocrystal and drug) coexist in equilibrium with solution, and (2) solution composition of cocrystal components ([R]tr and [A]tr) is fixed at a given pH and temperature, regardless of the ratio of the two solid phases. Figure 2 shows pH as the only variable determining the eutectic point, since temperature is constant. Ctr has been reported to be a key parameter in establishing the thermodynamic stability regions of cocrystals,4,21,29,30,32,33 including racemic compounds.34 This is the first analysis, to our knowledge, that demonstrates the pH dependence of Ctr. Transition concentrations where other solid phases are in equilibrium with solution, such as cocrystal/coformer or two

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Figure 1. Theoretical solubility-pH profile for (a) 2:1 R2H2A cocrystal calculated using eq 11, (b) 2:1 R2HAB cocrystal calculated using eq 12, (c) 2:1 B2H2A calculated using eq 14, and (d) 1:1 -ABHþH2X cocrystal calculated using eq 17. Drug and coformer pKa values and cocrystal Ksp are included in each graph. Ksp values were either experimentally determined or estimated from published work for the selected cocrystal in each graph (a) carbamazepine-succinic acid,21 (b) carbamazepine-4-aminobenzoic acid hydrate (current work), (c) itraconazole-L-tartaric acid,25 and (d) gabapentin-3-hydroxybenzoic acid.20

cocrystals of different stoichiometries have been presented elsewhere.21,30 These will not be discussed here because the transition concentration between drug and cocrystal is more relevant to phase stability in aqueous solutions where drugs are generally less soluble than coformers. How transition concentrations vary with pH is better appreciated in a two-dimensional plot (Figure 3) of the intersection points in Figure 2. The terms [A]tr and [R]tr represent the coformer and drug concentrations at the transition concentration. The case considered here assumes that drug solubility is independent of pH and coformer concentration; hence, [R]tr is constant. The coformer concentration, [A]tr, can also be expressed in terms of equilibrium constants and [Hþ] by using eq 5 with the appropriate

substitutions as K sp ½Atr ¼ ½Rtr

Ka 1þ þ ½H

! ð26Þ

This equation predicts that coformer transition concentration, [A]tr, increases with pH for the case described in Figures 2 and 3, implying that higher coformer concentrations are necessary to maintain cocrystal stability. Ksp can also be evaluated from measurement of transition concentration at a given pH and in this way estimates the cocrystal phase solubility diagram as a function of pH. The intersection of [A]tr and [R]tr in Figure 3 indicates the

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Bethune et al.

Table 2. Cocrystal Solubility or Equilibrium Drug Concentration Dependence on [Hþ], Coformer Concentration, Ka, and Ksp for Cocrystals with Components that Are Nonionic, Acidic, Basic, Amphoteric, and Zwitterionica cocrystal

solubility equations ½RT ¼

RH2A (1:1, nonionic:diprotic acidic)

K a1, H2 A K a1, H2 A K a2, H2 A þ ½Hþ ½Hþ 2

! ð18Þ

ð19Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u K sp K a1, HAB ½Hþ t þ 1þ ½RT ¼ K a2, HAB ½ABT ½Hþ

R2HAB (2:1, nonionic/amphoteric)

K sp ½BT ¼ ½AT


K a;HA 1þ þ ½H

!

½Hþ 1þ K a, B

ð20Þ

! ð21Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 ! u uK K a1, H2 A K a1, H2 A K a2, H2 A ½Hþ sp þ 1þ 1þ ½BT ¼ t K a, B ½AT ½Hþ ½Hþ 2


½AT ¼

HAHX (1:1, monoprotic acidic/monoprotic acidic)

K sp ½ABT ¼ ½XT

HABHX (1:1, amphoteric/monoprotic acidic)

a

1þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u K sp K a1, H2 A K a1, H2 A K a2, H2 A t þ 1þ ½RT ¼ ½AT ½Hþ ½Hþ 2


hABHþH2X (1:1, zwitterionic/diprotic acidic)

K sp ½AT

K sp ½ABT ¼ ½XT

K sp ½XT

1þ

K a, HX ½Hþ

K a, HX 1þ ½Hþ

!

! 1þ

K a;HA ½Hþ

! ð23Þ

K a1, HAB ½Hþ 1þ þ K a2;HAB ½Hþ

K a1, H2 X K a1, H2 X K a2, H2 X þ 1þ ½Hþ ½Hþ 2

! 1þ

ð22Þ

½Hþ K a1, - ABHþ

!

K a2, - ABHþ þ ½Hþ

ð24Þ ! ð25Þ

Equations apply to a cocrystal in equilibrium with solutions of nonstoichiometric and stoichiometric concentrations.

Figure 2. Theoretical dependence of cocrystal solubility or drug concentration, [R]T, on coformer concentration and pH for a 1:1 RHA cocrystal. Calculated from eq 5 with Ksp =1 10-6 M2 and coformer pKa = 3.0. Solubility of drug, SR, is represented by the yellow plane (SR = 2 10-3 M). Transition concentrations are located at the intersection of the drug solubility with cocrystal solubility.

pH at which the cocrystal stoichiometric solubility is equal to the drug solubility. This is a concept similar to that of pHmax in salts.1,6,35

Figure 3. Drug and coformer transition concentrations, [R]tr and [A]tr, as a function of pH for 1:1 cocrystal RHA, calculated from eq 26 with Ksp=1 10-6 M2, pKa=3.0, and [R]tr=2 10-3 M.

When establishment of true cocrystal-solution equilibrium is not achievable due to phase transformations, the Ctr represents a measurable equilibrium value from which the true cocrystal solubility can be predicted, as

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Table 3. Ctr Dependence on pH for Cocrystals of Different Ionization Properties and Stoichiometriesa cocrystal

coformer transition concentration

RH2A (1:1, nonionic:diprotic acidic)


½Atr ¼

½Atr ¼

K sp ½Rtr

½R2tr

½Atr ¼

HAHX (1:1, monoprotic acidic: monoprotic acidic)

HABHX (1:1, amphoteric/ monoprotic acidic)

hABH H2X (1:1, zwitterionic/ ½Xtr diprotic acidic)

a

K sp

1þ

½R2tr

½Atr ¼


þ

K a1, H2 A K a1, H2 A K a2, H2 A þ ½Hþ ½Hþ 2

K a1, H2 A K a1, H2 A K a2, H2 A þ 1þ ½Hþ ½Hþ 2

K sp

½ABtr ¼

R2HAB (2:1, nonionic: amphoteric)


1þ

drug transition concentration

K sp ½B2o

K a1, HAB ½Hþ þ þ K a2, HAB ½H

K sp ½Bo

1þ

K a;HA ½Hþ

½Xtr ¼

K sp ¼ ½ ABHþ o

K sp ½HAo

K sp ½HABo

1þ

½Rtr ¼ SR

ð28Þ

ð29Þ

½Rtr ¼ SR

ð30Þ

ð31Þ

½Rtr ¼ SR

ð32Þ

ð33Þ

½Btr ¼ ½Bo 1þ

!

!

K a, HX ½Hþ

1þ

ð27Þ

!

K a1, H2 A K a1, H2 A K a2, H2 A þ 1þ ½Hþ ½Hþ 2

½Xtr ¼

!

! ½Btr ¼ ½Bo

ð35Þ

½Hþ K a;B

½Hþ 1þ K a;B

!

K a, HX ½Hþ

ð37Þ

½Atr ¼ ½HAo 1þ

! ð39Þ

K a1, H2 X K a1, H2 X K a2;H2 X þ 1þ ½Hþ ½Hþ 2

½ABtr ¼ ½HABo 1þ

! -

þ

ð41Þ ½ABtr ¼ ½ ABH o 1 þ

! ð34Þ !

K a;HA ½Hþ

ð36Þ !

K a1, HAB ½Hþ þ K a2, HAB ½Hþ ½Hþ K a1, - ABHþ

ð38Þ !

K a2, - ABHþ þ ½Hþ

ð40Þ ! ð42Þ

Equations apply to cocrystal and solid drug in equilibrium with solution. Subscript “o” refers to the solubility of the neutral or nonionic forms.

has been recently demonstrated for cocrystals in aqueous and organic solvents.4 Thus, as the above analysis suggests, cocrystal phase solubility diagram and its pH dependence can be estimated without the large amount of materials and effort required in measuring the full phase diagram. Equations that predict Ctr-pH dependence, where drug and cocrystal are in equilibrium with solution, for cocrystals of varying ionization properties and stoichiometries were derived and are summarized in Table 3. The predictive power of these models is currently being investigated and is evaluated here for two cases: cocrystals of a nonionizable drug with an acidic coformer (carbamazepine-salicylic acid) and an amphoteric coformer (carbamazepine-4-aminobenzoic acid). Materials and Methods Materials. Anhydrous monoclinic carbamazepine (CBZ(III); lot no. 013K1381 USP grade) was purchased from Sigma Chemical Company (St. Louis, MO), stored at 5 °C over anhydrous calcium sulfate and used as received. Salicylic acid (SLC; lot no. 11111KC) and 4-aminobenzoic acid (4ABA; lot no. 05102HD) were purchased from Sigma Chemical Company (St. Louis, MO) and used as received. Water used in this study was filtered through a double deionized purification system (Milli Q Plus Water System from Millipore Co., Bedford, MA). Cocrystal Synthesis. Cocrystals were prepared by the reaction crystallization method at room temperature by adding carbamazepine to nearly saturated solutions of coformer.29,33 CBZSLC was prepared in ethanol while CBZ-4ABA-HYD was prepared in water. Solid phases were isolated and characterized by XRPD.

Measurement of Transition Concentration, Ctr. The transition concentrations, drug and coformer, were measured by HPLC after equilibrating carbamazepine dihydrate and cocrystal in solution at various pH values. For CBZ-4ABA-HYD measurements were carried out at ambient temperature (∼22-23 °C) and at controlled temperature of 25.0 ( 0.1 °C, and for CBZ-SLC at ambient temperature. Solution pH was varied by the addition of small volumes of concentrated HCl or NaOH and the pH at equilibrium was measured. The solid phases at equilibrium were characterized by XRPD. A more detailed description of the method for Ctr measurements has been presented.4 High Performance Liquid Chromatography. The solution concentration of CBZ and coformer was analyzed by Waters HPLC (Milford, MA) equipped with a UV/vis spectrometer detector. Waters’ operation software, Empower, was used to collect and process the data. A C18 Atlantis column (5 μm, 4.6 250 mm; Waters, Milford, MA) at ambient temperature was used to separate the drug and the coformer. The mobile phase was composed of 55% methanol and 45% water with 0.1% trifluoroacetic acid and the flow rate was 1 mL/min using an isocratic method. Injection sample volume was 20 or 50 μL. Absorbance of CBZ, SLC, and 4ABA was monitored at 284, 303, and 284 nm, respectively. X-ray Powder Diffraction. XRPD patterns of solid phases were collected with a benchtop Rigaku Miniflex X-ray diffractometer (Danvers, MA) using Cu KR radiation (λ=1.54 A˚), a tube voltage of 30 kV, and a tube current of 15 mA. Data was collected from 2 to 40° at a continuous scan rate of 2.5° min-1.

Results and Discussion Both cocrystals, CBZ-SLC and CBZ-4ABA-HYD, transformed to carbamazepine dihydrate, CBZ(D) when added to water and at all pH values studied. This indicates that cocrystals reach CBZ solution concentrations above the

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Bethune et al. Table 4. Linear Regression Results and Cocrystal Ksp Values Measured from Ctr Measurements According to eqs 26 and 31 cocrystal

slope ( standard error

R2

Ksp ( standard error

CBZ-SLC (1.75 ( 0.04) 10-3 0.99 (1.13 ( 0.05) 10-6 M2 CBZ-4ABA-HYD (3.70 ( 0.07) 10-3 0.98 (1.21 ( 0.15) 10-9 M3

Figure 4. Experimental and predicted coformer transition concentration, [SLC]tr dependence on pH for CBZ-SLC cocrystal. Curve represents theoretical calculations from eq 26. Measured drug transition concentration [CBZ]tr = 0.00061 ((0.00003) M in this pH range. Figure 6. Experimental and predicted coformer transition concentration, [4ABA]tr, dependence on pH for CBZ-4ABA-HYD. Curves represent theoretical calculations from eq 31. Measured drug transition concentration [CBZ]tr = 0.00057 ((0.00003) M in this pH range.

Figure 5. Plot used to evaluate Ksp for CBZ-SLC cocrystal according to eq 26. Slope is equal to Ksp/[R]tr.

solubility of CBZ(D) and cocrystal-solution equilibrium in aqueous solutions as a function of pH could not be achieved. Cocrystal solubilities were therefore estimated from Ctr measurements following the models described in the Theoretical Section. Drug and coformer transition concentrations where cocrystal and carbamazepine dihydrate are in equilibrium with solution were measured at various pH values. Figure 4 shows the experimental and predicted pH dependence of [SLC]tr for CBZ-SLC, a 1:1 cocrystal of a nonionizable drug and an acidic

coformer. The coformer transition concentration is shown to increase drastically at pH values above 3.0, the pKa of salicylic acid. The measured drug transition concentration, [CBZ]tr, in the pH range studied was 0.00061 ((0.00003) M. This value is slightly higher than the reported carbamazepine dihydrate solubility in water, 0.00053 M.36 Predictions based on eq 26 and parameter values for CBZSLC cocrystal of pKa=3.0 and Ksp=1.1310-6 M2 are in very good agreement with experimental behavior (Figure 4). Ksp was evaluated from the slope (Ksp/[R]tr) of a plot of [A]tr vs (1 þ (Ka/[Hþ])) shown in Figure 5. Linear regression results and Ksp values are presented in Table 4. Figure 6 shows that measured coformer transition concentrations for CBZ-4ABA-HYD follow the predicted behavior for a 2:1 cocrystal with a nonionizable drug and an amphoteric coformer according to eq 31. Coformer transition concentrations are observed to reach a minimum value in a pH range between 3 and 4. The predicted curve was generated with pKa values of 4-aminobenzoic acid pKa1=4.8 and pKa2= 2.6, for the acid and conjugate acid of the base, and cocrystal Ksp =1.2110-9 M3. The measured drug transition concentration, [CBZ]tr, in the pH range studied was 0.00057 ((0.00003) M. Ksp was evaluated from the slope (Ksp/[R]tr2) of a plot of [AB]tr vs (1 þ (Ka1,HAB/[Hþ]) þ ([Hþ]/Ka2,HAB)) shown in Figure 7. The good agreement between the experimental and predicted values for both cocrystals (Figures 4 and 6) demonstrates the predictive power of the models and suggests that Ctr measurements at a single pH provide a good first estimate of cocrystal Ksp when time and sample quantity are limited. Activity coefficient corrections may however be necessary for higher concentrations or ionic strengths.

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3983

Figure 7. Plot used to evaluate Ksp for CBZ-4ABA-HYD cocrystal according to eq 31. Slope is equal to Ksp/([R]tr)2.

As shown in Table 4, Ksp values vary 1000-fold for the two cocrystals studied. From Ksp values under conditions where ionization is negligible, cocrystal solubilities in water were estimated to be 1.0610-3 and 6.810-4 M for CBZ-SLC and CBZ-4ABA-HYD, respectively. These cocrystals are therefore about 2 and 1.3 times more soluble than CBZ(D) at their lowest solubilities. CBZ-4ABA-HYD is a 2:1 cocrystal and for every mole of cocrystal dissolved there are two moles of CBZ. To understand the consequences of the observed pH dependence of Ctr on cocrystal solubility and stability, phase solubility diagrams (PSDs) were generated from measured Ctr’s for both cocrystals according to eqs 5 and 20. Figure 8 demonstrates that cocrystal solubility increases with increasing pH for CBZ cocrystals with SLC or acidic coformers, while solubility decreases as pH approaches the range between pKa values of 4ABA or amphoteric coformers. This implies that as cocrystal solubility increases above drug solubility higher coformer concentrations are needed to maintain cocrystal stability. Thus, pH is shown to be a very important variable determining (1) cocrystal aqueous solubility, (2) the dependence of cocrystal solubility on coformer concentration, and (3) the effectiveness of the coformer in stabilizing or precipitating cocrystal. Turning to the power of the coformer to modulate cocrystal solubility, it is demonstrated that coformer ionization properties and pKa values determine the solubility-pH dependence of cocrystals of a nonionizable drug. Cocrystals are shown to be more soluble than CBZ(D) in solutions without excess coformer even at the pH of lowest cocrystal solubility, that is, pH 1 for CBZ-SLC and pH 4 for CBZ-4ABA-HYD. CBZSLC is estimated to be approximately 2 to 200 times more soluble than CBZ(D) at pH 1 and 7. CBZ-4ABA-HYD is approximately 1.3 times more soluble than CBZ(D) at pH 4 and predicted to be 4 and 7 times more soluble than CBZ(D) at pH 1 and 7. In practice, these solubilities may not be achieved because of transformation to individual components or their salts. However, it is valuable to assess the true solubility advantage of cocrystals, to guide cocrystal selection

Figure 8. Predicted cocrystal solubility (solid lines) for (a) CBZSLC, and (b) CBZ-4ABA-HYD according to eqs 5 and 20 using experimentally measured Ctr values at each pH and pKa,salicylic acid= 3.0; pKa,4-aminobenzoic acid=2.6 and 4.8. Experimentally measured Ctr for drug and coformer are represented by filled circles. The dotted lines represent the stoichiometric solution concentration of cocrystal components.

and formulation/dissolution approaches that harness the cocrystal potential. Conclusions Studies with cocrystals of nonionizable drug and acidic or amphoteric coformers demonstrate that pH-dependent cocrystal solubility can be engineered by the careful selection of cocrystal components. Mathematical models based on cocrystal dissociation and ionization of components predict the measured Ctr dependence on pH and pKa of cocrystal components and explain how cocrystal solubility is influenced by pH, pKa, and coformer concentration. Measurement of Ctr is

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Bethune et al.

valuable to (a) estimate cocrystal solubility, (b) evaluate the effectiveness of coformer in stabilizing or precipitating cocrystal, and (c) guide cocrystal selection without the time and material consuming determination of traditional methods. Acknowledgment. We gratefully acknowledge partial funding from the American Foundation for Pharmaceutical Education, GM07767 National Institute of General Medicine Sciences, Warner Lambert/Park Davis Fellowship and Upjohn Fellowship in Pharmaceutics, College of Pharmacy, University of Michigan. The contents of this publication are solely the responsibility of the authors and do not necessarily represent the official views of funding sources.

For a 1:1 cocrystal, Scocrystal =[A]T =[R]T, under stoichiometric conditions vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u K a1;H2 A K a1;H2 A K a2;H2 A t ðA8Þ S cocrystal ¼ Ksp 1þ þ ½Hþ ½Hþ 2 At the eutectic point eq A7 can be rewritten as K sp ½Atr ¼ ½Rtr

K sp ¼ ½R½H2 A

½Hþ ½HA - ½H2 A

ðA2Þ

HA - hA2 - þHþ ðA3Þ

ðA4Þ

Using equilibrium constants to substitute into eq A4 gives ! K a1;H2 A K a1;H2 A K a2;H2 A K sp ½AT ¼ ðA5Þ 1þ þ ½R ½Hþ ½Hþ 2 Mass balance on [R]T: ½RT ¼ ½R

1þ

K a1;H2 A K a1;H2 A K a2;H2 A þ ½Hþ ½Hþ 2

H2 AhHA - þHþ K a1, H2 A ¼

½Hþ ½HA - ½H2 A

ðA11Þ

HA - hA2 - þHþ K a2;H2 A ¼

½Hþ ½A2 - ½HA -

ðA12Þ

Mass balance on [A]T: ½AT ¼ ½H2 Aþ½HA - þ½A2 -

ðA13Þ

Using equilibrium constants to substitute into eq A13: ! K a1;H2 A K a1;H2 A K a2;H2 A K sp ½AT ¼ 1þ þ ðA14Þ ½Hþ ½R2 ½Hþ 2

½RT ¼ ½R

½Hþ ½A2- ¼ ½HA -

½AT ¼ ½H2 Aþ½HA - þ½A2-

K sp ½RT

ðA10Þ

Mass balance on [R]T:

Mass balance on [A]T:

½AT ¼

K sp ¼ ½R2 ½H2 A

ðA1Þ

H2 AhHA - þHþ

Substituting eq A6 into eq A5:

ðA9Þ

R2 H2 Asolid h2Rsoln þH2 Asoln

This appendix presents the derivation of equations in Tables 1, 2, and 3 that describe how cocrystal solubility, drug and coformer concentrations in equilibrium with cocrystal, and Ctr are influenced by solution pH and cocrystal component pKa. Derivations begin by considering the equilibrium reactions for cocrystal dissociation and ionization for various cocrystal stoichiometries and ionization properties. Ctr’s presented in this appendix refer to the equilibrium between solid drug and cocrystal with solution. Solution complexation and activity coefficient corrections are not considered. Heterogeneous equilibria indicate solid and solution phases as subcripts. All other equilibria and concentrations without subscripts refer to solution phase. Nomenclature is defined in the main text. 1:1 cocrystal with nonionizable drug and diprotic acidic coformer. RH2 Asolid hRsoln þH2 Asoln

K a2, H2 A

!

2:1 cocrystal with nonionizable drug and diprotic acidic coformer.

Appendix

K a1, H2 A ¼

K a1;H2 A K a1;H2 A K a2;H2 A 1þ þ ½Hþ ½Hþ 2

ðA6Þ ! ðA7Þ

Substituting eq A15 into eq A14: ½AT ¼

K sp ½R2T


ðA15Þ ! ðA16Þ

For a 2:1 cocrystal, Scocrystal=[A]T=1/2[R]T, under stoichiometric conditions vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u K a1;H2 A K a1;H2 A K a2;H2 A 3 K sp t ðA17Þ þ 1þ S cocrystal ¼ 4 ½Hþ ½Hþ 2 At the eutectic point eq A16 can be rewritten as ! K a1;H2 A K a1;H2 A K a2;H2 A K sp ½Atr ¼ 1þ þ ½Hþ ½R2tr ½Hþ 2

ðA18Þ

Article


2:1 cocrystal with nonionizable drug and amphoteric coformer. R2 HABsolid h2Rsoln þHABsoln K sp ¼ ½R2 ½HAB

ðA19Þ

HABh- ABþHþ K a1;HAB

½ - AB½Hþ ¼ ½HAB

ðA20Þ

HABHþ hHABþHþ K a2;HAB ¼

½HAB½Hþ ½HABHþ

ðA21Þ

Mass balance on [AB]T: ½ABT ¼ ½HABþ½ - ABþ½HABHþ

ðA22Þ

Using equilibrium constants to substitute into eq A22: ! K sp K a1;HAB ½Hþ ½ABT ¼ ðA23Þ 1þ þ K a2;HAB ½Hþ ½R2

3985

Using equilibrium constants to substitute into eq A31: ! K sp K a;HA ½AT ¼ ðA32Þ 1þ ½B ½Hþ Mass balance on [B]T: ½BT ¼ ½Bþ½BHþ

ðA33Þ

! ½Hþ ½BT ¼ ½B 1þ K a;B ½BT ½B ¼ þ 1þ ½H K a;B

ðA34Þ

ðA35Þ

Substituting eq A35 into eq A32: ! ! K sp ½Hþ K a;HA 1þ ½AT ¼ 1þ K a;B ½BT ½Hþ

ðA36Þ

For a 1:1 cocrystal, Scocrystal = [A]T =[B]T under stoichiometric conditions vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! u u K a;HA ½Hþ t 1þ ðA37Þ S cocrystal ¼ K sp 1þ K a;B ½Hþ

Mass balance on [R]T: ½RT ¼ ½R Substituting eq A24 into eq A23: ½ABT ¼

K sp ½R2T

K a1;HAB ½Hþ 1þ þ þ K a2;HAB ½H

ðA24Þ ! ðA25Þ

For a 2:1 cocrystal, Scocrystal=[A]T=1/2[R]T, under stoichiometric conditions and therefore, eq A25 can be rewritten as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u K a1;HAB ½Hþ 3 K sp t 1þ ðA26Þ þ S cocrystal ¼ K a2;HAB 4 ½Hþ At the eutectic point eq A25 can be rewritten as ! K sp K a1;HAB ½Hþ ½ABtr ¼ 1þ þ K a2;HAB ½Hþ ½R2tr

ðA27Þ

ðA28Þ

½Hþ ½A - ½HA

eq A38 can be rewritten as K sp ½Atr ¼ ½Bo

½Hþ ½B ½BHþ

Mass balance on [A]T: ½AT ¼ ½HAþ½A -

! ðA39Þ

where [B]o is the solubility of the nonionized drug. 2:1 cocrystal with basic drug and diprotic acidic coformer. B2 H2 Asolid h2Bsoln þH2 Asoln ðA40Þ

H2 AhHA - þHþ K a1;H2 A ¼

ðA29Þ

K a2;H2 A ¼

½Hþ ½HA - ½H2 A

ðA41Þ

½Hþ ½A2 - ½HA -

ðA42Þ

BHþ hBþHþ

BHþ hBþHþ K a;B ¼

K a;HA 1þ ½Hþ

HA - hA2 - þHþ

HAhA - þHþ K a;HA ¼

ðA38Þ

By considering the equilibrium between solid drug B with solution Bsolid hBsoln

K sp ¼ ½B2 ½H2 A

1:1 cocrystal with basic drug and monoprotic acidic coformer. BHAsolid hBsoln þHAsoln K sp ¼ ½B½HA

At the eutectic point eq A36 can be rewritten as ! ! K sp ½Hþ K a;HA ½Atr ¼ 1þ þ 1þ K a;B ½Btr ½H

ðA30Þ

K a;B ¼

½Hþ ½B ½BHþ

ðA43Þ

Mass balance on [A]T: ðA31Þ

½AT ¼ ½H2 Aþ½HA - þ½A2 -

ðA44Þ

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Bethune et al.

Using equilibrium constants to substitute into eq A44: ! K a1;H2 A K a1;H2 A K a2;H2 A K sp ðA45Þ ½AT ¼ 2 1þ þ ½Hþ ½B ½Hþ 2

Mass balance on [X]T:

Mass balance on [B]T: ½BT ¼ ½Bþ½BHþ ½BT ¼ ½B 1þ ½B ¼

½Hþ K a;B

ðA46Þ ! ðA47Þ

½BT

þ

ðA48Þ

1þ ½H K a;B

Substituting eq A48 into eq A45: !2 ! K a1;H2 A K a1;H2 A K a2;H2 A K sp ½Hþ 1þ ½AT ¼ 2 1þ þ K a;B ½Hþ ½BT ½Hþ 2 ðA49Þ 1

For a 2:1 cocrystal, Scocrystal=[A]T= /2[B]T, under stoichiometric conditions and therefore, eq A49 can be rewritten as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !2 u u 3 K K a1;H2 A K a1;H2 A K a2;H2 A ½Hþ sp t 1þ 1þ þ S cocrystal ¼ K a;B 4 ½Hþ ½Hþ 2 ðA50Þ At the eutectic point eq A49 can be rewritten as !2 ! K a1;H2 A K a1;H2 A K a2;H2 A K sp ½Hþ ½Atr ¼ 2 1þ 1þ þ K a;B ½Hþ ½Btr ½Hþ 2 ðA51Þ By considering the equilibrium between solid drug B with solution Bsolid hBsoln eq A51 can be rewritten as ½Atr ¼

K sp ½B2o

Using equilibrium constants to substitute into eq A56: ! K sp K a;HA ðA57Þ ½AT ¼ 1þ ½HX ½Hþ


! ðA52Þ

1:1 cocrystal with monoprotic acidic drug and monoprotic acidic coformer. HAHXsolid hHAsoln þHXsoln K sp ¼ ½HA½HX

ðA53Þ

HAhA - þHþ K a;HA ¼

ðA54Þ

HXhX - þHþ K a;HX ¼

K a;HX ½XT ¼ ½HX 1þ ½Hþ ½HX ¼

½H ½X ½HX

Mass balance on [A]T: ½AT ¼ ½HAþ½A -

ðA55Þ

ðA56Þ

1þ

K a;HX ½Hþ

! ðA59Þ

ðA60Þ

ðA61Þ

For a 1:1 cocrystal, Scocrystal = [A]T = [X]T, under stoichiometric conditions vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi u u K K a , HA a;HX ðA62Þ 1þ S cocrystal ¼ tK sp 1þ ½Hþ ½Hþ At the eutectic point eq A61 can be rewritten as ! ! K sp K a;HX K a;HA 1þ ½Atr ¼ 1þ ½Xtr ½Hþ ½Hþ

ðA63Þ

By considering the equilibrium between solid drug HA with solution HAsolid hHAsoln eq A63 can be rewritten as K sp ½Xtr ¼ ½HAo

K a, HX 1þ ½Hþ

! ðA64Þ

where [HA]o is the solubility of the nonionized drug. 1:1 cocrystal with amphoteric drug and monoprotic acidic coformer. HABHXsolid hHABsoln þHXsoln K sp ¼ ½HAB½HX

ðA65Þ

HABh- ABþHþ ½Hþ ½ - AB ½HAB

ðA66Þ

HABHþ hHABþHþ K a2, HAB ¼

-

½XT

ðA58Þ

Substituting eq A60 into eq A57: ! ! K sp K a;HX K a;HA 1þ ½AT ¼ 1þ ½XT ½Hþ ½Hþ

K a1;HAB ¼

½Hþ ½A - ½HA

þ

½XT ¼ ½HXþ½X -

½Hþ ½HAB ½HABHþ

ðA67Þ

HXhX - þHþ K a, HX ¼

½Hþ ½X - ½HX

ðA68Þ

Article


Mass balance on [AB]T: ½ABT ¼ ½HABþ½- ABþ½HABHþ

ðA69Þ

Using equilibrium constants to substitute into eq A69: ! K a1, HAB K sp ½Hþ ½ABT ¼ þ 1þ ðA70Þ K a2, HAB ½HX ½Hþ Mass balance on [X]T: ½XT ¼ ½HXþ½X - K a, HX ½XT ¼ ½HX 1þ ½Hþ ½HX ¼

ðA71Þ ðA72Þ

½XT K 1þ a, HX

ðA73Þ

½Hþ

Substituting eq A73 into eq A70: ! ! K a, HX K a1, HAB K sp ½Hþ 1þ ½ABT ¼ 1þ þ K a2, HAB ½XT ½Hþ ½Hþ ðA74Þ For a 1:1 cocrystal, Scocrystal =[AB]T =[X]T, under stoichiometric conditions vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! u þ u K K ½H a , HX a1 , HAB 1þ þ S cocrystal ¼ tK sp 1þ K a2, HAB ½Hþ ½Hþ ðA75Þ At the eutectic point eq A74 can be rewritten as: ! ! K a, HX K a1, HAB K sp ½Hþ 1þ ½ABtr ¼ 1þ þ K a2, HAB ½Xtr ½Hþ ½Hþ ðA76Þ By considering the equilibrium between solid drug HAB with solution HABsolid hHABsoln eq A76 can be rewritten as K a, HX K sp 1þ ½Xtr ¼ ½HABo ½Hþ

For a 1:1 cocrystal, Scocrystal =[AB]T =[X]T, under stoichiometric conditions S cocrystal ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! u u K a2, - ABHþ K a1, H2 X K a1, H2 X K a2, H2 X ½Hþ tK þ 1 þ þ sp 1 þ K a1, - ABHþ ½Hþ ½Hþ ½Hþ 2

ðA79Þ Equation A78 can be rewritten specifically at the eutectic point, as: ½ABtr ¼

!

! ðA77Þ

where [HAB]o is the solubility of the nonionized form of the drug. 1:1 cocrystal with zwitterionic drug and diprotic acidic coformer. This model has been presented elsewhere.20 The final equation describing the drug concentration dependence on coformer concentration, [Hþ], and Ka values is included below. ! K a1, H2 X K a1, H2 X K a2, H2 X K sp 1þ ½ABT ¼ þ ½XT ½Hþ ½Hþ 2 ! K a2, - ABHþ ½Hþ ðA78Þ 1þ þ K a1, - ABHþ ½Hþ

3987

K sp ½Xtr

1þ

K a1, H2 X K a1, H2 X K a2, H2 X þ ½Hþ ½Hþ 2

! 1þ

½Hþ K a1, - ABHþ

þ

K a2, - ABHþ ½Hþ

!

ðA80Þ -

By considering the equilibrium between solid drug ABHþ with solution þ ABHþ solid h ABHsoln eq A80 can be rewritten as K a1, H2 X K a1, H2 X K a2, H2 X K sp ½Xtr ¼ 1þ þ þ ½ ABH o ½Hþ ½Hþ 2

!

ðA81Þ where [-ABHþ]o is the solubility of the neutral species.

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