Estimation of the Solubility of Metastable Polymorphs: A Critical

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Review

Estimation of the solubility of metastable polymorphs. A critical review. Lucrece Nicoud, Filippo Licordari, and Allan S. Myerson Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.8b01200 • Publication Date (Web): 07 Sep 2018 Downloaded from http://pubs.acs.org on September 9, 2018

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Crystal Growth & Design

Estimation of the solubility of metastable polymorphs. A critical review. Lucrèce Nicoud, Filippo Licordari, and Allan S. Myerson∗ Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA-02139 E-mail: [email protected] Phone: +1-617-452-3790 Abstract Solubility measurements of metastable polymorphs are often complicated by solventmediated transformation toward more stable forms. In this review, we first summarize potential experimental methods to estimate the solubility of metastable polymorphs. Then, we discuss a methodology based on a thermodynamic model, which allows solubility predictions from (i) solubility data of the stable form, and (ii) solid-state properties of the stable and metastable forms.

Introduction Polymorphism is the ability of a species to exist in different crystalline structures, due to differences in packing arrangement and/or in molecular conformation. Different polymorphs may strongly differ in terms of several properties, such as solubility, color, shape, hygroscopy, filterability, or compactability. 1–5 Here, we focus on the solubility, which is an essential property for the design of crystallization processes. Solubility is also key to the performances of pharmaceutical drugs because a poor aqueous solubility is associated with a low bioavailability. 6,7 The differences in solubility between polymorphs result from differences in lattice 1

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energy, and more precisely from differences in the balance between the attractive forces that hold the solid together and the disruptive forces that bring solute molecules in solution. The polymorph with the lowest solubility is necessarily the most stable at the considered temperature. Indeed, a polymorph with a higher solubility would eventually undergo solventmediated transformation, i.e., dissolve and recrystallize as another polymorph characterized by a lower solubility, as the system strives to reach equilibrium. It is generally advisable to formulate the most stable polymorph to avoid any transformation issues. However, in the case of poorly soluble drugs, it may be desirable to produce a metastable form with a view to improving bioavailability. 8 The aforementioned solvent-mediated transformation may prevent accurate solubility measurements of metastable polymorphs and had led to the emergence of the expression “kinetic solubility” in the literature. 9 However, solubility is by no means a kinetic property and its true value is certainly not affected by the time course of a polymorphic transformation. Measuring the solubility of polymorphs requires rigor and precision to circumvent the issues associated with solvent-mediated transformation. In this review, we first summarize potential experimental methods to estimate the solubility of metastable polymorphs. Then, we discuss a methodology to predict their solubility based on a thermodynamic model. This model is based on (i) solubility data of the stable form and (ii) solid-state thermodynamic properties of the stable and metastable forms. Finally, we briefly mention in silico approaches for solubility prediction.

Experimental methods Presentation of the methods Solubility is defined as the maximum amount of a substance that can be dissolved in a liquid under given operating conditions. Solubility values differ enormously depending on the substance of interest, the composition of the liquid and temperature. The experimental 2


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determination of solubility for stable systems is relatively straightforward and already well documented in the literature. 10–12 Two main approaches may be identified, namely isothermal and nonisothermal. Isothermal methods are the most precise and primarily consist of two steps: (i) saturating the solution with the solute, and (ii) analyzing the concentration of the dissolved solute. For point (i), an excess of solid is placed in contact with the selected solvent at a given temperature in tightly sealed containers and the slurry is stirred until thermodynamic equilibrium is reached. The time necessary to reach equilibrium depends on several factors including the crystal structure, the solvent, the temperature, the crystal size and the agitation speed. As a rough indication, 24 h is usually sufficient when dissolving a fine powder under vigorous stirring. For point (ii), the solute concentration can be determined by a variety of analytical techniques. In the case of a pure solute characterized by a sufficiently high solubility, the solute concentration can for instance be analyzed by gravimetry after removing the solid by filtration. In the presence of impurities, it is preferable to use high performance liquid chromatography (HPLC). The concentration in the liquid can also be estimated without prior filtration with attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy. In nonisothermal approaches, a known mass of solute is equilibrated with a know mass of solvent at a given temperature, and the temperature is then progressively increased while stirring the slurry. 13 The saturation temperature is defined as the temperature at which the last crystals dissolve, which can for instance be estimated from turbidity measurements, thus eliminating the need to analyze the composition of the liquid phase. Too high heating ramps would result in overestimated values of the saturation temperature (and thus underestimated values of the solubility), and it is thus essential to employ a sufficiently slow temperature ramp to be as close as possible to equilibrium. It is recommended to perform measurements at several temperature ramps and extrapolate the results to an infinitely slow heating ramp. Most often, such experiments are performed as a series of heating-cooling cycles. Measuring the solubility of metastable polymorphs is more challenging due to the poten-

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tial solvent-mediated transformation toward a more stable form. The latter occurs due to the difference in solubility between the two forms. When the solute concentration is equal to the solubility of the metastable form, the supersaturation of the stable form is still positive, so that the stable form will eventually start crystallizing. This leads to a decrease in the solute concentration below the solubility of the metastable form, which in turn triggers the dissolution of the latter. If the dissolution of the metastable form is faster than the crystallization of the stable form, the solute concentration remains practically equal to the solubility of the metastable form during the transformation. On the other hand, if the dissolution is slow, the solute concentration drops below the solubility of the metastable form. Once all the crystals of the metastable form have been transformed, the stable form continues crystallizing until reaching equilibrium. The solvent-mediated transformation of a number of systems has been studied theoretically and experimentally, most often employing a combination of microscopy, ATR-FTIR and Raman spectroscopy. 14–19 It was found that the crystallization of the stable form often starts on the surface of crystals of the metastable form, a phenomenon referred to as cross-nucleation. 17,20–22 The rate at which solvent-mediated transformation occurs has been shown to strongly depend on the operating conditions, such as temperature, solvent, and agitation. 23,24 A change in these conditions can lead to a change in the rate-limiting step, i.e., either dissolution of the metastable form or crystallization of the stable form. For some systems, no solvent-mediated transformation was observed even after several days 25–27 and in such cases, the solubility measurements of metastable polymorphs does not present any additional difficulty as compared to those of stable polymorphs. However, when the rate of polymorphic transformation is fast as compared to the rate of dissolution of the metastable crystals, the common experimental methods mentioned previously cannot be employed as such. This led to incomplete solubility data sets (e.g., missing polymorphic forms, limited range of temperatures or solvents) 26,28–31 and potentially inaccurate results, which motivated the writing of this review. Three possible methods have been identified to measure the solubility of metastable polymorphs and a description of each method is given

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in the following: M1: approach the solubility from an undersaturated solution, i.e., by dissolving crystals of the metastable form under stirring. 32–40 It is highly recommended to monitor the concentration in the liquid phase not only after a given amount of time, as it is often done to evaluate the solubility of stable polymorphs, but during the entire course of the dissolution processs, for instance with ATR-FTIR. In any case, it is absolutely necessary to characterize the polymorphic form of the solid, for instance with powder X-ray diffraction (PXRD) or Raman spectroscopy. If the recrystallization of the stable form is sufficiently slow, the concentration in the liquid phase monotonically increases before flattening out, and the solubility can be estimated as the concentration value at the plateau. 32–34 Otherwise, the concentration reaches a maximum and then decreases toward the solubility of a more stable form. It is often assumed that the concentration maximum corresponds to the solubility of the metastable form. 35–37 However, this may lead to an underestimated value of the true solubility if the recrystallization of a more stable form starts before complete dissolution is achieved. A more rigorous approach consists of estimating the solubility of the metastable form by fitting the beginning of the experiment with an appropriate dissolution model. 39 Considering a first order process, the time evolution of the solute concentration can be described by dcj /dt = kd (c∗j − cj ), where cj is the concentration of the dissolved polymorph j, c∗j its solubility and kd the dissolution rate constant. In any case, it is advised to use crystals of the metastable form at the highest possible purity since traces of the stable form may induce the fast crystallization of the stable form due to secondary nucleation. 41 M2: approach the solubility from a supersaturated solution, i.e., by crystallizing the metastable form. 32 Once more, it is necessary to monitor the concentration in the liquid phase and to characterize the polymorphic form. This method requires identifying operating conditions where a significant amount of the metastable form can be crystallized (e.g., by varying the solute concentration, stirring speed, amount of seeds). If the solvent-mediated transformation is limited by the crystallization of the stable form (and not by the dissolution

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of the metastable form), the concentration in the liquid phase reaches a plateau at the solubility of the metastable form before it drops down toward the solubility of a more stable form. M3: prepare several solutions at increasing concentrations above the solubility of the stable form, add a few crystals of the metastable polymorph in each solution and track those with microscopy. 42 The solubility of the metastable form is estimated as the lowest concentration where dissolution of the crystals does not occur (ignoring the potential apparition of a more stable form). Such method is sometimes referred to as a “bracketing” method. These three methods have their own strength and limitations. Method M1 is the most commonly used and is readily applicable in the case of a slow polymorphic transformation, i.e., when the solute concentration reaches a plateau during the dissolution experiment before dropping toward the solubility of a more stable form. However, when the transformation is fast, i.e., when the solute concentration reaches a clear maximum during the dissolution experiment, this method must be complemented with further investigations before concluding on the solubility value. It is recommended to use either Method M3 and/or a dissolution model to validate the results. Method M2 requires identifying operating conditions where a significant amount of the metastable form can be crystallized and where the solvent-mediated transformation is limited by the recrystallization of the stable form. In these conditions, the solute concentration reaches a plateau before dropping to the solubility of a more stable form, and the concentration value at plateau provides a good estimate of the solubility of the metastable form. Method M3 can always be applied but requires preparing samples of various concentrations and its precision is limited by the accuracy of the prepared samples.

Illustration of the methods In this section, we illustrate the three methods described previously considering paracetamol in ethanol as a model system. Experiments were performed for the purpose of this review in order to discuss the three methods with the same system. In ethanol, paracetamol forms 6


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two polymorphs: form I (the most stable form under usual temperature conditions, which is commercially available) and form II (which is metastable). Form II has a better compression behavior than form I, 43 but tends to rapidly convert to form I. The presence of various additives, and in particular metacetamol, was shown to allow the crystallization and stabilization of form II. 44 Here, form II was prepared by cooling a solution containing 300 mg paracetamol and 30 mg metacetamol per g ethanol from 50 ◦ C to 0 ◦ C. In the following, all the concentrations are reported in mg of solute per g of solvent.

Concentration [mg/g]

(a) M1: dissolution (20 oC) 300 250 200 150 100

0

0.5

1

1.5

2

2.5

3

Time [h] Concentration [mg/g]

(b) M2: crystallization (0 oC) 300 250 200 150 100

0

2

4

Time [h]

6

8

130 mg/g

(c) M3: bracketing (0 oC)

140 mg/g

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 min

1 min

2 min

0 min

20 min

40 min

Figure 1: Illustration of three methods to estimate the solubility of polymorphs. Experiments were performed with paracetamol in ethanol. Figure 1(a) shows the results of the dissolution of crystals of form II in an undersaturated 7



solution (method M1) at 20 ◦ C. The concentration in the liquid phase was monitored with ATR-FTIR (ReactIR, Metler Toledo) and the solid form was determined by offline Raman spectroscopy (Raman WorkStation, Kaiser Optical systems). It was found that at around 45 min, the concentration in the liquid phase reaches a maximum at approximately 202 mg/g, then it decreases due to the crystallization of form I. The value of 202 mg/g is slightly below the value determined from method M3 (≈210 mg/g), thus suggesting that the dissolution of form II is almost complete before form I starts appearing. The temperature of 20 ◦ C was selected here to illustrate a case where solvent-mediated transformation is fast, and thus stress the importance of monitoring both the concentration in the liquid phase and the polymorph content. At lower temperatures, the polymorphic transformation was found to be kinetically hindered. Figure 1(b) shows the results of a seeded crystallization experiment of form II (method M2) at 0 ◦ C. Once more, the concentration in the liquid phase was monitored with ATRFTIR and the solid form was determined by Raman spectroscopy. It was found that during the first hour, the concentration in the liquid phase drops due to the crystallization of form II almost exclusively. Then a first plateau is reached at 143 mg/g, which corresponds to the solubility of form II and is in good agreement with the value determined with method M3 (≈140 mg/g). During this first plateau, form II transforms into form I via dissolution and recrystallization. Once all the crystals of form II have been transformed, the concentration in the liquid drops toward the solubility of form I. It is worth mentioning that this method could be employed only at low temperatures (below 10 ◦ C), where it was possible to crystallize form II. At higher temperatures, seeding the crystallizer with crystals of form II was not sufficient to induce the crystallization of form II. The crystallization of form I was so fast compared to that of form II that the concentration in the liquid phase would directly drop to the solubility of form I without reaching an intermediate plateau. Finally, Figure 1(c) illustrates method M3. Solutions at 130 mg/g and 140 mg/g were prepared, which are both above the solubility of form I at 0 ◦ C (125.5 mg/g). Few crys-

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tals of form II were added to each solution and tracked with high resolution microscopy (Blaze900 probe, BlazeMetrics) at 0 ◦ C. At 130 mg/g, it is seen that the crystals dissolve within few minutes, indicating that the selected concentration lies below the solubility of form II. At 140 mg/g instead, no dissolution is observed during the time frame of the experiment. To allow a faster screening of solution concentrations (by increments of 5 mg/g) and temperatures, similar experiments were performed with an inverted microscope platform (Axio Observer Z.1m, Zeiss). This device has a lower resolution but a higher throughput than the Blaze probe. For each temperature, the solubility of form II was estimated as the lowest concentration where dissolution of the crystals did not occur within approximately 16 h. The results are shown with red circles in Figure 2(a). As a comparison, the solubility values of form I measured by gravimetry are represented with blue circles. These results are consistent with other literature data. 45 Form I is the most stable over the whole temperature range, and paracetamol is thus said to be a monotropic system. On the other hand, when the solubility curves cross, the system is referred to as enantiotropic. The panels (b) and (c) of Figure 2 show the measured solubility of two polymorphic forms of chloramphenicol palmitate in water and sulfathiazole in 1-propanol, respectively, as a function of temperature (symbols). The solubility data of chloramphenicol palmitate were measured by Muramatsu et al 39 using method M1 coupled with a dissolution model. Due to the very low solubility of choramphenicol palmitate in water, a radiotracer method was employed to accurately measure concentrations. On the other hand, the solubility data of sulfathiazole were acquired by Munroe et al, 29 also using method M1. The concentration in the liquid phase was not monitored but the polymorphic form was verified with PXRD after 48 h of equilibration. Due to fast solvent-mediated transformation, the solubility of only two forms out of five could be assessed with this technique. Here, we label the polymorphs of sulfthiazole according to the nomenclature of Munroe, which follows that of the Cambridge Structural Database. A comparison between the different existing nomenclatures has been

9



(a) Paracetamol Solubility [mg/g]

600

Form II 400 200

Form I

0 -20

0

20

40

60

(b) Chloramphenicol palmitate Solubility [mg/g]

10-4

Form α 1

Form β

0.5

0

0

10

20

30

40

50

60

(c) Sulfathiazole 10

Solubility [mg/g]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8 6 4

Form III

2 0

Form IV 0

10

20

30

40

Temperature [oC]

50

60

Figure 2: Solubility of two polymorphs of (a) paracetamol in ethanol, (b) chloramphenicol palmitate in water and (c) sulfathiazole in propanol. Symbols represent experimental data, while lines represent simulations from the thermodynamic model. Experimental solubilities of paracetamol are original, whereas those from chloramphenicol palmitate and sulfathiazole are taken from Muramatsu 39 and Munroe, 29 respectively. Blue lines represent fittings to the blue symbols, whereas red lines are predictions keeping the binary interaction parameters previously fitted. The shaded red areas correspond to the confidence interval obtained by varying the fusion enthalpy by ±5%. established in the literature. 46 In the next Section, we discuss the possibility to describe the set of experimental data presented in Figure 2 with a thermodynamic model.

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Predictions from a thermodynamic model Presentation of the method In a saturated solution, the activity of a polymorph j dissolved in the liquid phase (denoted aLj ) is equal to the activity of that polymorph in the pure solid state (denoted aSj ): aLj = aSj

(1)

A suitable standard state to express aSj as a function of thermodynamic parameters is that of a pure supercooled liquid at the temperature of the solution and at some specified pressure, for instance the atmospheric pressure. Note that with this choice of standard state, the activity of the solid is not equal to 1, as is common practice in chemistry. It can be demonstrated that: 47,48 Z T ∆Hjf dln aSj 1 = + ∆Cp,j dT dT RT 2 RT 2 TjT r

(2)

where R is the ideal gas constant, T is the temperature, TjT r is the triple point temperature of j, ∆Hjf its enthalpy of fusion and ∆Cp,j the difference between the molar heat capacities of the hypothetical supercooled melt and the solid. For the sake of simplicity, it is often assumed that (i) ∆Cp,j is independent of temperature, and (ii) TjT r is close to the fusion temperature of j at atmospheric pressure. On the other hand, the activity of the solute in the liquid phase can be written as:

aLj = γj (x∗j , T )x∗j

(3)

where x∗j is the solubility expressed in mole fraction of the solute in solution, and γj is the activity coefficient that depends both on T and x∗j . By combining Equations 1, 2, and 3 and the applying the two aforementioned approxima-

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tions, it follows that the solubility of any polymorph j at a temperature T can be expressed as: 47,48 x∗j (T ) =

x∗,id j (T ) γj (x∗j , T )

(4)

where x∗,id = aSj is commonly referred to as the ideal solubility and is given by: j

x∗,id j (T ) = exp

∆Hjf R

1 − f T Tj 1

!

∆Cp,j − R

Tjf Tjf ln − +1 T T

!! (5)

Importantly, it is observed in Equation 4 that the solubility of a polymorph can be broken down into two parts: one that depends exclusively on the properties of the solid phase (through Tjf , ∆Hjf , ∆Cp,j ), and the other one that depends exclusively on the properties of the liquid phase (through γj ). The two terms in Equation 5 are not of equal importance, with the term in ∆Hjf being most generally dominant with respect to the term in ∆Cp,j . The comparison between the solubility measured at a given temperature with the corresponding ideal solubility estimated with Equation 5 provides an estimate of the activity coefficient at saturation. There are several models to describe the variations of the activity coefficient with composition and temperature in a binary mixture, such as the Wilson, 49 NRTL, 50 or UNIQUAC 51 models to name just a few. As an example, we only report here the one by Wilson: ln (γj ) = −ln(1 − Asj xs ) − xs

xj Asj xs Ajs − 1 − Asj xs 1 − Ajs xj

(6)

where the solvent mole fraction is given by xs = 1−xj and the binary interaction parameters by:  Vs gsj − gjj    Asj = 1 − V exp − RT j  V g − gss   Ajs = 1 − j exp − js Vs RT

(7)

The quantities gjs = gsj , gss , and gjj represent the interaction energy between solute–solvent, solvent–solvent and solute–solute pairs of molecules, respectively. By defining ∆gsj = gsj −gjj 12


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and ∆gjs = gjs − gss , only two parameters are necessary to describe the binary interactions in the frame of the Wilson model. The key point is that ∆gsj and ∆gjs are properties of the liquid phase only and are thus independent from the polymorphic form in contact with the liquid. They can therefore be estimated from the solubility data of any polymorphic form and subsequently be used to predict the solubility of other forms. Based on the thermodynamic background given earlier, the following method can be used: 1. measure the solubility of the stable form at various temperatures 2. measure the solid-state thermodynamic properties (T f , ∆H f , ∆Cp ) of all the polymorphs under investigation, for instance with differential scanning calorimetry (DSC) 3. compute the ideal solubility of the stable form with Equation 5 at each considered temperature using the solid-state properties of the stable form 4. estimate the activity coefficients at saturation at each considered temperature using Equation 4 and the solubilities (measured and ideal) of the stable form 5. fit the binary interaction coefficients using a thermodynamic model (e.g., fit ∆gsj = gsj − gjj and ∆gps = gjs − gss for the Wilson model given by Equations 6 and 7) 6. estimate the solubility of the metastable form with Equation 4 using the solid-state properties of the metastable form and the binary interaction coefficients of the activity model previously determined. Note that Equation 4 is implicit. In order to obtain accurate solubility predictions with the method described above, it is essential to obtain reliable thermodynamic data in point 2. For illustrative purposes, Figure 3(a) shows DSC thermograms of both forms of paracetamol. 52 The fusion temperature T f is usually estimated as the onset of the temperature endotherm, while the fusion enthalpy ∆H f is determined from the area below the melting endotherm. The results are shown in Figure 3(a). The knowledge of the temperature and enthalpy of fusion of polymorphs can be used to assess the relative thermodynamic stability of the different forms thanks to the so-called “thermodynamic rules”. 53,54 Just to give an example, the heat of fusion rule states

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that polymorphic pairs are enantiotropically related if the low melting form has the higher heat of fusion, otherwise they are monotropically related. Let us now try to quantify the impact of experimental uncertainties in the measured values of T f and ∆H f on the solubility predictions. To do so, we consider a simplified expression of Equation 4:

ln x∗,id j

∆Hjf = R

1 − f T Tj 1

! (8)

This expression neglects ∆Cp , whose impact is discussed later on. The uncertainty in x∗,id j ∗,id and can be estimated from Equation 8 by using partial derivatives: is denoted d xj d x∗,id j x∗,id j

∆Hjf f 1 1 1 = f − d ∆Hjf + d Tj R Tj T R(Tjf )2

(9)

where d ∆Hjf and d Tjf denote the uncertainties in ∆Hjf and Tjf , respectively. To get an order of magnitude of the two terms in Equation 9, we consider the following numerical values: ∆Hjf = 25 kJ/mol, Tjf = 200 ◦ C, T = 20 ◦ C, d ∆Hjf = 1 kJ/mol, d Tjf = 1 ◦ C. ∗,id This leads to a relative uncertainty of d xj /x∗,id = 18.3%, with the first term equal j to 17.0% and the second term to 1.3%. This brief analysis shows that errors in the heat of fusion contribute the most to errors in the ideal solubility, while errors in the melting temperature are less troublesome. The impact of experimental errors on the ratio of ideal solubilities of two polymorphs j and k can be estimated as follows:   ∗,id ∗,id ∗,id ∗,id d x d x x∗,id ± d x j j j x k  ≈ j∗,id 1 + + ∗,id ∗,id ∗,id ∗,id x x x xk ± d xk j k k

(10)

In our example, an error of 4% in ∆Hjf results in an error of 17% in x∗,id j . Considering that d x∗,id /x∗,id /x∗,id ≈ d x∗,id j j , this may lead up to 34% error in the solubility ratio. k k These calculations show that small errors in the determination of the heat of fusion may lead

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to large errors in the prediction of the solubility of the metastable polymorph. Therefore, efforts should be put forth to obtain as accurate values of ∆Hjf as possible. A potential issue that can be encountered while performing DSC experiments is that of polymorphs transforming before melting. 54–57 Indeed, polymorphic transformations do not only occur as a solvent-mediated mechanism, but can also occur in the solid-state, 19,58 in particular when the temperature approaches that of the melting point. To circumvent this issue, one strategy consists in using very high heating rates with a view to kinetically hindering the polymorphic transformation. 56,57 Note that the use of an appropriate DSC set-up is required to perform accurate measurements at high heating rates.

Heat flow [mW]

(a)

40 TIIf =156.4 oC f ΔHII =27.6 kJ/mol

20 0

TIf=168.6 oC f

ΔHI =28.1 kJ/mol

-20 -40 -60 -20 0

20 40 60 80 100 120 140 160 180 200

Temperature [oC]

(b) 400 Heat capacity [J/K/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Liquid

350

ΔCp,I ~ΔCp,II

300 250

Form II

200

Form I

150 100 -20 0

20 40 60 80 100 120 140 160 180 200

Temperature [oC]

Figure 3: (a) Heat flow data from DSC experiments for both forms of paracetamol, allowing the determination of T f and ∆H f . (b) Molar heat capacity of both forms of paracetamol and of the melt as a function of temperature, allowing the determination of ∆Cp . Adapted from Sacchetti 52 with permission from Springer Nature, copyright 2000.

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DSC experiments allow the determination of the heat capacity as a function of temperature. 59 As an illustration, the heat capacity of both forms of paracetamol and of the melt are shown as a function of temperature in Figure 3(b). In Equation 2, ∆Cp is defined as the difference between the heat capacity of the solid and that of the hypothetical supercooled melt at the same temperature. It was necessary to assume that ∆Cp is constant with temperature to integrate Equation 2 and obtain Equation 5. In practice, the value of ∆Cp at the melting point is often considered in calculations due to the difficulty to extrapolate the heat capacity of the melt to low temperatures.

Ideal solubility estimate [-]

(a)

0.1 C p=0

(b)

C p=

0.08

S

f

0.06 0.04 0.02 0

Ideal solubility ratio [-]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0

0.02

0.04

0.06

0.08

0.1

Ideal solubility with ΔCp [-] 4 3.5 3 2.5 2 without ΔCp with ΔCp

1.5 1

1

1.5

2

2.5

3

3.5

4

Measured solubility ratio [-]

Figure 4: (a) Comparison between the ideal solubility computed with Equation 5 and results estimated with two approximations: ∆Cp = 0 (filled symbols) and ∆Cp = ∆S f (closed symbols) for several systems. Data taken from Neau et al. 60 (b) Comparison between solubility ratios measured experimentally and the ratio of ideal solubilities for several systems. The solubility ratios are defined with respect to the most stable form under the investigated temperature, so that they are all larger than 1. Data taken from Mao et al. 61

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The term in ∆Cp in Equation 5 is thus relatively difficult to estimate experimentally with accuracy and is often neglected. In the following, we discuss the impact of neglecting this term. To do so, let us consider the data shown in Figure 4(a), which show the ideal solubility of various systems computed by neglecting the term in ∆Cp in Equation 5 as a function of the ideal solubility computed with the full equation (filled circles). 60 It is observed that neglecting the term in ∆Cp leads to an underestimation of the ideal solubility for the systems under consideration. These data were also analyzed by considering another approximation, namely assuming that ∆Cp,j ≈ ∆Sjf , where ∆Sjf is the molar entropy of fusion of j. This approximation leads to the following simplified expression of the ideal solubility: 60 x∗,id j (T ) = exp −

∆Hjf RTjf

ln

Tjf T

!! (11)

The results of the ideal solubility computed with Equation 11 are shown with open circles in Figure 4(b). It is seen that the approximation ∆Cp ≈ ∆S f leads to estimates of the ideal solubility that are closer to the values obtained with the full expression (Equation 5) than the assumption ∆Cp ≈ 0. Figure 4(b) shows another set of data, where the ideal solubility ratio of various polymorphs is plotted as a function of the experimentally measured solubility ratio. The experimental data were obtained with flufenamic acid, carbamazepine, sulfamerazine, sulfathiazole, mefenamic acid, lifibrol, indomethacin, and MK571 in several separate studies and have been analyzed collectively by Mao and al. 61 The ratios are defined as metastable/stable so that all the values are greater than 1. Filled symbols correspond to results obtained with the full expression of the ideal solubility (Equation 5), while open symbols correspond to the results obtained by neglecting the ∆Cp term. It is observed that similar ratios of the ideal solubility are obtained with or without the ∆Cp term. This is because the heat capacities of polymorphs are in general relatively similar (see Figure 3(b)), so that the errors in the ∆Cp term often compensate when computing solubility ratios. It is interesting to notice in Figure 4(b)

17



that the ratio of ideal solubilities is relatively close to the experimentally measured solubility ratio when the solubility ratio is close to 1. However, larger discrepancies are observed for higher solubility ratios. This is most likely because the ratio between the activity coefficients at saturation deviates from 1 when the solubilities of the two forms are significantly different. ∗,id is valid only for Indeed, for two polymorphs j and k, the assumption that x∗j /x∗k ≈ x∗,id j /xk

γj ≈ γk . This is a reasonable approximation if the concentrations at saturation are relatively close. There is often the tendency in the literature to characterize the solubility of metastable polymorphs in terms of a solubility ratio estimated only at one particular temperature. 3,61,62 This is for example the case of the data shown in Figure 4(b), where each point corresponds to the solubility of one system under specific conditions of temperature (and solvent). However, this approach prevents the determination of the temperature–composition dependency of the activity coefficient. This may lead to discrepancies between predicted and measured solubility ratios at a given temperature if the solubility ratio is high (see Figure 4(b)) and prevents predicting the solubilities at other temperatures. In this context, it is important to stress that the solubility ratio computed at one given temperature cannot be used to predict the solubility of metastable polymorphs at other temperatures, as it is sometimes done in the literature. 30 Indeed, it is likely that the solubility ratio changes with temperature, and it is even possible that the solubility ranking changes (enantiotropic systems). It is therefore always advisable to measure solubility data at different temperatures, as shown in the next section.

Application of the method In the following, we show how to apply the step-by-step methodology described earlier to three systems: paracetamol in ethanol, chloramphenicol palmitate in water and sulfathiazole in 1-propanol. 1. The experimental solubility data of the stable polymorph of each system were already 18


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presented with blue symbols in Figure 2 (in mg of solute per g of solvent). Although it is clear that form I of paracetamol and form β of chloramphenicol palmitate are the most stable in the investigated range of temperature, the solubility values of form III and IV of sulfathiazole are so close that it is difficult to establish a stability ranking. Here, the methodology was applied considering that form IV is the most stable. 2. Table 1 shows the values of the solid-state parameters measured by DSC for the three systems under investigation. The values of T f and ∆H f reported for paracetamol and chloramphenicol palmitate correspond to averages from the references given in Table 1. Measuring the fusion parameters of sulfathiazole has been shown to be challenging due to polymorphic solid-state transformation. This issue was overcome by using a proper DSC set-up that allows accurate measurements of thermal properties at very fast heating rates. 56 The results obtained at 500 K/min, where the polymorphic transformation is kinetically inhibited, are reported in Table 1. In the absence of ∆Cp data for chloramphenicol palmitate and sulfathiazole, its contribution was neglected. Table 1: Solid-state thermodynamic parameters of the systems under investigation Polymorph T f [◦ C] Paracetamol 52,54,60,63 I 168.5 II 156.1 Chloramphenicol palmitate 39,54,64 β 93.0 α 87.5 56 Sulfathiazole III 171 IV 171

∆H f [kJ/mol]

∆Cp [J/K/mol]

28.1 27.0

99.8 104.5

67.2 44.3

n/a n/a

8.83 8.76

n/a n/a

3. The ideal solubility values of the most stable form of each system were then computed at each considered temperature using Equation 5 and the solid-state parameter values reported in Table 1. 4. The activity coefficients at saturation were then estimated at each considered temperature using Equation 4 and the solubilities (measured and ideal) of the stable forms. The 19



so-obtained values of the activity coefficient are shown as black circles in Figure 5 for the three systems under investigation. 5. The surfaces in Figure 5 instead represent the simulated activity coefficients using the Wilson model given in Equation 6 and fitting the values of the binary interaction coefficients (i.e., ∆gsj = gsj − gjj and ∆gjs = gjs − gss ). It is observed in Figure 5 that the activity coefficients of the three systems vastly differ, from around 1.2 for paracetamol up to 3 × 107 for chloramphenicol palmitate, which has a very low solubility. In addition, it is seen that the dependence of the activity coefficient with temperature and composition is strongly system dependent. While the activity coefficient of paracetamol is practically independent from temperature and composition, variations by a factor of 2 are observed for sulfathiazole in the range of investigated conditions. 6. Once the binary interaction coefficients are determined, the solubility of the stable form can be computed at any temperature using Equations 4 to 7. The results are shown with blue solid lines in Figure 2. In addition, the solubility of the metastable forms can be predicted using the corresponding solid-state thermodynamic properties. Those results are shown with red solid lines in Figure 2 for the three systems under investigation. The shaded red areas represent the confidence interval obtained by varying the fusion enthalpy by ±5%, which is within the range of experimental uncertainty. It is seen that the model predictions are in good agreement with the experimental data. Therefore, these results show that it is possible to use a thermodynamic model to predict the solubility of a metastable polymorph based on the solubility data of the stable form and the solid-state thermodynamic properties of the stable and metastable forms.

In silico approaches There is a high interest in developing simulation tools to predict the solubility of pharmaceutically relevant molecules. 65–69 The description of the various modeling techniques is out

20


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(a) Paracetamol 2

γp [-]

1.5 1 0.5 0 60

0.1

40

T [oC]

20 0

0.05 0

xp [-]

(b) Chloramphenicol palmitate

γp [-]

107 3 2.5 2 1.5 60

1

40

T [oC]

20 0

0.5 0

xp [-]

10-9

(c) Sulfathiazole 800

γp [-]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


600 400 200 60

2

40

T [oC]

20 0

1 0

10-3

xp [-]

Figure 5: Activity coefficients of the three systems under investigation. Black circles correspond to experimental data and surfaces to fits with the Wilson model (Equation 6). of the scope of this review and the interested reader is referred to the existing literature on the topic. 70,71 Here, we simply stress that predicting the solubility of various polymorphs requires selecting a model based on the crystal structure (e.g., atomistic models) rather than a model based on the molecular structure (as most quantitative structure–property relationships (QSSR) models). Indeed, models based solely on the molecular structure are not capable of predicting the impact of a change in the crystal structure on the solubility. Only few simulation studies aiming at predicting the solubility of polymorphs are available in the literature. 72,73 Further improvements are necessary to make such tools widely accessible and

21



applicable for the solubility prediction of polymorphic systems.

Conclusion To conclude, three experimental methods have been identified to estimate the solubility of metastable polymorphs: dissolution in an undersaturated solution (M1), crystallization of a supersaturated solution (M2), and the observation of the behavior of few crystals in solutions of increasing concentrations (M3). In method M1, it is essential that the dissolution of metastable crystals occurs faster than the crystallization of the more stable form(s). When the solute concentration goes through a maximum (i.e., the solute concentration does not reach a plateau before decreasing toward the solubility of a more stable form), it is strongly advised to verify the accuracy of the measurements with method M3. In method M2, it is necessary to select conditions where the metastable form crystallizes faster than the more stable form(s) and where the solvent-mediated transformation is limited by the recrystallization of the more stable form (rather than by dissolution of the metastable form). Method M3 is widely applicable but has a relatively low precision (which is given by the accuracy in the concentration of the prepared samples). Alternatively, the solubility of metastable polymorphs can be estimated with a thermodynamic model. The model requires two types of inputs: (i) solubility data of the stable form, and (ii) solid-state properties of the stable and metastable polymorphs. This model requires accurate thermodynamic data and cannot be applied if the considered polymorphs undergo solid-state transformation during DSC experiments.

Acknowledgement Financial support from the Swiss National Science Foundation is gratefully acknowledged (grant P2EZP2_168909).

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Graphical TOC Entry Solute concentration

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


solubility metastable polymorph solubility stable polymorph faster transformation Time

Solvent-mediated transformation complicates the measurement of the solubility of metastable polymorphs.

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Estimation of the Solubility of Metastable Polymorphs: A Critical

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