Solubility Prediction of Different Forms of Pharmaceuticals in Single

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Thermodynamics, Transport, and Fluid Mechanics

Solubility Prediction of Different Forms of Pharmaceuticals in Single and Mixed Solvents Using Symmetric eNRTL-SAC Model Getachew S. Molla, Michael Frederick Freitag, Stuart Michael Stocks, Kim Troensegaard Nielsen, and Gurkan Sin Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04268 • Publication Date (Web): 19 Feb 2019 Downloaded from http://pubs.acs.org on February 24, 2019

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Industrial & Engineering Chemistry Research

Solubility Prediction of Different Forms of Pharmaceuticals in Single and Mixed Solvents Using Symmetric eNRTL-SAC Model Getachew S. Molla†*, Michael Frederick Freitag‡, Stuart Michael Stocks‡, Kim Troensegaard Nielsen‡, Gürkan Sin† †Process

and Systems Engineering Center (PROSYS), Department of Chemical and Biochemical

Engineering, Technical University of Denmark, Building 229, DK-2800 Kgs. Lyngby, Denmark ‡LEO-Pharma

A/S, Industriparken 55 DK-2750 Ballerup, Denmark

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Abstract An improved framework has been developed for solubility prediction of different forms of a medium-sized antibiotic (i.e. nonelectrolyte, electrolyte and solvate) in single and mixed solvents using a symmetrically reformulated eNRTL-SAC model. The methodology incorporates key features of the symmetric eNRTL-SAC model structure to reduce number of parameters and using a hybrid of global search algorithm for parameter estimation. Moreover, a design of experiments is included in the methodology in order to generate and use experimental data appropriately for model parameter regression and model validation. Due to the semi-predictive nature of the symmetric eNRTL-SAC model, the segment parameter regression is a critical step for solubility prediction accuracy. A particle swarm optimization (PSO) algorithm is incorporated to pre-regress conceptual segment parameters of solutes. The pre-regressed segment parameters were used as initial guesses for further segment parameter estimation. In this way, consistent segment parameters that reflect the characteristics of solutes in solution were estimated. The methodology application is demonstrated by predicting the solubility of fusidic acid, sodium fusidate and fusidic acid acetone solvate in single and mixed solvents as well as at different temperature. The solubility predictions of fusidic acid, fusidic acid acetone solvate and sodium fusidate in various single solvents show good agreement with experimental solubility with average squared relative errors (ASREs) of 0.055, 0.079 and 0.084 in logarithmic mole fraction scale, respectively. The model moreover predicts solubilities in binary solvents mixture and as a function of temperature in a satisfactory agreement with experimental solubility.

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1. INTRODUCTION The use of solvent is most often an essential requirement in the reaction, the separation and the formulation sections of pharmaceutical manufacturing processes. Reaction rates, separation efficiency, product quality, safety and manufacturing economy are affected by the choice of solvent or solvents mixture

1 2.

Therefore, screening of an optimal solvent or solvents mixture is a critical

step to design an efficient pharmaceutical process. Even though the safety, health and environmental (SHE) impacts of solvents can be used to quickly shortlist the number of solvent candidates, the pharmaceutical industry still must put a major effort to screen an optimum solvent or solvents mixture among hundreds of solvents. The solubility of active pharmaceutical ingredients (APIs) is a key thermodynamic property that reflects the effect of solvents on reaction rates and separation efficiency of various downstream processing (DSP) unit operations such as crystallization, extraction and adsorption. Consequently, solubility is most often used as a criterion to screen an optimum solvent or solvents mixture. Solubility can be determined by either performing experiment or applying thermodynamic models. Experimental solvent screening is an expensive and laborious task or sometimes not practically possible due to the lack of sufficient amount of an API despite the availability of high-throughput technologies 3 4. Moreover, designing of DSP unit operations such as crystallization further requires constructing a solubility phase diagram of an API as a function of temperature, solvent composition or combination thereof. Hence, experimental solvent screening becomes practically limited to identify an optimal mixture of binary or ternary solvents. A reliable computational tool to estimate solubility for systematic solvent screening and guiding design of experiments is therefore useful to support pharmaceutical process designing. There are several existing thermodynamic activity coefficient models for estimating solubility in single and mixed solvents. Among the most commonly used models are universal functional activity coefficient (UNIFAC) group contribution 5, conductor-like screening model for real solvent (COSMO-RS) 6, conductor-like screening model-segment activity coefficient (COSMO-SAC) 7 and nonrandom two-liquid segment activity coefficient (NRTL-SAC) 1. Each of these models shows different accuracy that sometimes is related to their inherently different basis of assumptions 8. For instance, as a preliminary model assessment UNIFAC, COSMO-SAC, COSMO-RS and NRTLSAC were used for the solubility prediction of medium-sized steroid-like structure antibiotic where NRTL-SAC showed better prediction accuracy 9. NRTL-SAC is a semi-predictive model that

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computes activity coefficient by identifying solute and solvent molecules in terms of predefined conceptual segments and regressed binary segment interaction parameters 1. The conceptual segments are identified as hydrophobic (X), polar attractive (Y-), polar repulsive (Y+) and hydrophilic (Z). The aforementioned activity coefficient models are applicable to estimate the liquid-phase nonideality of organic nonelectrolytes. However, most APIs possess acidic and/or basic functional group that nearly half of all APIs are prepared as their salt form 12. Preparing APIs in their salt form is often preferred to improve stability and aqueous solubility

13.

Therefore, determining the

solubility of salt is useful in the separation and the formulation sections of pharmaceutical manufacturing. The NRTL-SAC model has been extended to electrolyte nonrandom two-liquid segment activity coefficient (eNRTL-SAC) model for solubility estimation of organic electrolytes 14.

In addition to the conceptual molecular segments identified in the NRTL-SAC model, the

eNRTL-SAC model has introduced an electrolyte conceptual segment (E). The electrolyte segment is presumed to completely dissociate to a cationic (c) and an anionic (a) segments for calculating both local-composition and long-range interactions contributions 14. The original eNRTL-SAC was formulated as unsymmetric activity coefficient model by using aqueous phase infinite dilution as a reference state 14 15. In this model, the long-range interaction term is calculated using Pitzer-DebyeHückel equation that uses an infinite dilution reference state in the actual solvent

14 16.

Therefore,

calculating Born correction term is required for non-aqueous system to transfer the reference state of the long-range interaction term. However, using aqueous phase infinite dilution reference state is not straightforward for non-aqueous systems and includes uncertainty related to the Born term calculation like the choice of ionic radii

17 18.

reformulated to use symmetric reference state

17.

The original

eNRTL-SAC model has been

Therefore, implementation of the symmetric

eNRTL-SAC model makes it a simplistic thermodynamic framework to estimate the solubility of various forms of APIs such as nonelectrolytes, electrolytes and solvates. Furthermore, unlike other activity coefficient models, neither the original NRTL-SAC nor eNRTL-SAC accounts the effect of temperature on binary segment interaction parameters. It has been described that the prediction accuracy of NRTL-SAC model reduces for the solubility prediction as a function of temperature 20.

8

An improvement of prediction accuracy has been described by extending the original NRTL-

SAC model using temperature dependent binary segment interaction parameters

11.

Prediction

accuracy as a function of temperature is a requirement; for example, for solvent screening and designing of cooling crystallization. 2 ACS Paragon Plus Environment

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Parameter estimation, typically done, by nonlinear regression is very critical in a general property modeling

21,22.

Regressing the conceptual segment parameters of a solute using experimental

solubility data is a precondition for the activity coefficient calculation using eNRTL-SAC model. The segment parameters of most commonly used solvents in pharma industry are provided elsewhere 1. Moreover, due to the semi-predictive nature of eNRTL-SAC model, the segment parameter regression step is critical for prediction accuracy 8. For estimating the segment parameters of APIs, therefore, the regression step should provide consistent segment parameters reflecting the characteristics of a solute in solution of all types of solvents. This step includes using reliable experimental solubility data and an appropriate regression algorithm such as maximum likelihood

estimation

(MLE).

Furthermore,

API

purification

using

crystallization

and

recrystallization in different solvent systems can lead to the formation of undesired intermediate solvate. Solvates possess different thermophysical properties compare to pure component crystals that influence design variables of the next purification unit operation in a DSP flowsheet. Therefore, for predicting the solubility of solvates factors such as solvate stoichiometry, purity of solid solvate and the amount of solid solvate used to prepare slurry in solubility experiments as well as the solidstate properties of residual solid at equilibrium must be determined for accurate model parametrization and validation. This paper describes an improved symmetric eNRTL-SAC model based framework to predict the solubility of different forms of pharmaceuticals. Main objectives of this work were defined as implementation of symmetrically reformulated eNRTL-SAC model incorporating a design of experiments to generate reliable solubility data in appropriate solvents and using a hybrid global search algorithm to provide better initial guess for another nonlinear segment parameters regression method. A particle swarm optimization (PSO) algorithm was incorporated to pre-regress initial guess segment parameters. In this way, consistent solute segment parameters that reflect the characteristics of a solute in solution can be regressed. The symmetric eNRTL-SAC model based framework utility was demonstrated by predicting the solubility of nonelectrolyte (i.e. fusidic acid), electrolyte (i.e. sodium fusidate) and solvate (i.e. fusidic acid acetone solvate) in single and mixed solvents as well as at different temperature.



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2. THE eNRTL-SAC MODEL FRAMEWORK The equilibrium solubility of a solute in nonideal solutions is commonly calculated using equation (1). This equation is derived from the thermodynamic fugacity equilibrium of a component in solid and liquid phases taking two major assumptions such as neglecting the change in heat capacity and using melting point instead of triple point temperature 23. 𝑙𝑛𝑥𝑠𝑎𝑡 𝐼 =

∆𝐻𝑓𝑢𝑠 1 1 ― ― 𝑙𝑛𝛾𝑠𝑎𝑡 𝐼 𝑅 𝑇𝑚 𝑇

(

)

}

𝑓𝑜𝑟 𝑇 ≤ 𝑇𝑚

(1)

𝑠𝑎𝑡 where 𝑥𝑠𝑎𝑡 𝐼 and 𝛾𝐼 are the saturated mole fraction and the saturated activity coefficient of a solute 𝐼

dissolved in a solvent at a system temperature (𝑇), respectively. ∆𝐻𝑓𝑢𝑠 and 𝑇𝑚 are the enthalpy of fusion and melting point of a solute, respectively, and 𝑅 is the gas constant. The pure component thermal properties of a solute crystal can be obtained experimentally using differential scanning calorimetry (DSC). These properties can as well be predicted using group contribution (GC+) model though this model cannot distinguish between different crystal morphology

24.

Using the thermal

properties of a solute crystal, its solubility can be predicted by calculating the activity coefficient iteratively using an appropriate thermodynamic model. The original eNRTL-SAC model was formulated for calculating unsymmetric activity coefficient using aqueous phase infinite dilution as a reference state 25 14. In this work, the eNRTL-SAC model was implemented using pure component reference state for symmetric activity coefficient calculation

17.

A symmetric expression of Pitzer-Debye-Hückel (PDH) was used for long-range

interactions contributions

18 19 26 27.

Likewise the original eNRTL-SAC model, retaining the two

central assumptions (i.e. like-ion repulsion and local electroneutrality), the symmetric eNRTL-SAC model basis on three types of interactions contributions. Local-composition (lc) interactions contributions that exist at the immediate neighborhood of a species calculated using a segment based symmetric electrolyte NRTL equation. Long-range interactions contributions that exist beyond the immediate neighborhood of a species calculated using symmetric PDH equation. An entropic contribution that takes into account a molecular size and shape calculated using FloryHuggins equation. Equation (2) shows the logarithm of symmetric activity coefficient of component 𝐼 (𝑙𝑛𝛾𝐼) calculated as the sum of the aforementioned three terms using pure component reference state.

𝑙𝑛𝛾𝐼 = 𝑙𝑛𝛾𝐼𝐶 + 𝑙𝑛𝛾𝐼𝑙𝑐 + 𝑙𝑛𝛾𝐼PDH

(2) 4

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The 𝑙𝑛𝛾𝐶𝐼 in equation (2) is the combinatorial term calculated using Flory-Huggins approximation for the entropy of mixing in equation (3). Φ𝐼 𝑙𝑛𝛾𝐶𝐼 = 𝑙𝑛 + 1 ― 𝑟𝐼 x𝐼

Φ𝐽

∑𝑟

(3)

𝐽

𝐽

where 𝑟𝐼 is the total segment number and Φ𝐼 is the segment mole fraction of component 𝐼. The 𝑙𝑛𝛾𝑙𝑐 𝐼 in equation (2) is to the local-composition term that represents the contribution of intermolecular and intramolecular interactions (i.e. enthalpy of interaction). This term is calculated as the sum of local-composition interactions contributions by each segment as shown in equation (4). ln𝛾𝑙𝑐 𝐼 =

∑𝑟

[ln Γlc𝑖 ―

𝑖,𝐼

lnΓlc,𝐼 𝑖 ]

𝑖 = 𝑚,𝑐,𝑎

𝑖

=

∑𝑟

𝑚,𝐼

[ln Γlc𝑚 ―

lc lc,𝐼 lc lc,𝐼 lnΓlc,𝐼 𝑚 ] + 𝑟𝑐,𝐼[ln Γ𝑐 ― lnΓ𝑐 ] + 𝑟𝑎,𝐼[ln Γ𝑎 ― lnΓ𝑎 ]

(4)

𝑚

where r is the segment number, m is the index of conceptual molecular segments (i.e. x, y-, y+ and z), c is the cationic segment, a is the anionic segment, Γlc 𝑖 is the activity coefficient of segment species 𝑖 in a solution and Γlc,𝐼 𝑖 is the activity coefficient of segment species 𝑖 in the pure component 𝑃𝐷𝐻 lc,𝐼 14 in equation (2) 𝐼. Details on the calculations of Γlc 𝑖 and Γ𝑖 are provided elsewhere . The 𝑙𝑛𝛾𝐼

is the symmetric Pitzer-Debye-Hückel term accounting for the long-range interaction calculated as the sum of contributions of each segment as shown in equations (5), (6) and (7) where cationic and anionic segments are considered to be univalent. 𝑙𝑛𝛾𝐼𝑃𝐷𝐻 =

∑𝑟

ΓPDH , 𝑖

𝑖,𝐼ln

𝑖 = 𝑚,𝑐,𝑎

(5)

𝑖

1

ln

ΓPDH 𝑚

( )

=2

1000 𝑀𝑠

2

3

𝐴𝜑𝐼𝑥2

(6)

1

1 + 𝜌𝐼𝑥2



1

ln

ΓPDH 𝑖

= ―

{ [ ( )]

( ) () 1000 𝑀𝑠

2

𝐴𝜑

2 ln 𝜌

1 + 𝜌𝐼𝑥

1

1

3

2

𝐼2𝑥

2𝐼2𝑥

1

1 + 𝜌(𝐼0𝑥)2

+

―

1

1 + 𝜌𝐼𝑥2

= 𝑐,𝑎

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( )

―

2𝐼𝑥 𝐼0𝑥

―

1 2

∂𝐼0𝑥

( ) ( ) 1

1 + 𝜌 𝐼0𝑥

2

}

𝑛 , 𝑖 ∂𝑛𝑖

(7)

where 𝐴𝜑 is the Debye-Hückel parameter and 𝐼𝑥 is ionic strength calculated based on segment based mole fraction. 1

3

)( )

(

2

1 2𝜋𝑁𝐴𝑑𝑠 𝐴𝜑 = 3 1000

𝑄𝑒

2

2

(8)

𝜖𝑠𝑘𝐵𝑇

1 𝐼𝑥 = (𝑥𝑐 + 𝑥𝑎) 2

(9)

𝐼0𝑥 = 𝐼𝑥(𝑥𝑠𝑜𝑙𝑣𝑒𝑛𝑡𝑠 = 0) ∂𝐼0𝑥

1 𝑛 = ∂𝑛𝑖 2

∑

∂𝑥0𝑗 𝑛 , ∂𝑛𝑖

( )

𝑍2𝑗

𝑗

(10)

𝑖,𝑗 = 𝑐,𝑎

∂𝑥0𝑗 𝛿𝑖𝑗 ― 𝑥0𝑗 𝑛 = , ∂𝑛𝑖 ∑𝑐𝑥𝑐 + ∑𝑎𝑥𝑎

(11)

𝑖,𝑗 = 𝑐,𝑎

(12)

𝜌 is the closest approach parameter (i.e. 14.9) which is fixed for all components. 𝑁𝐴 is the Avogadro’s number (i.e. 6.0225×1023 mol-1). 𝑄𝑒 is the electron charge (i.e. -1.602 ×10-19 C). 𝑘𝐵 is the Boltzmann constant (i.e. 1.38×10-23 J/K). 𝑀𝑠 is an average molecular weight of mixed solvents.

𝑑𝑠 is an average density of mixed solvents. 𝜖𝑠 is an average dielectric constant of mixed solvents. The average molecular quantities (i.e. 𝑀𝑠, 𝑑𝑠 and 𝜖𝑠) of mixed solvents are calculated using composition-average mixing rules 14.

2.1.

Segment parameters estimation

Estimating the conceptual segment parameters of a solute using minimum number of experimental data is a precondition for the activity coefficient calculation using eNRTL-SAC model. The segment parameters of a solute can be estimated using a nonlinear regression algorithm such as MLE method

28.

An appropriate numerical solver is typically used to iteratively minimize an 6 ACS Paragon Plus Environment

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objective function such as in this case the root-mean squared error (RMSE) between experimental and calculated solubility data as shown in equation (12). The RMSE was used as an objective function for the sake of comparison as it is commonly used in literature about eNRTL-SAC 14.

(∑ 𝑁

𝑅𝑀𝑆𝐸 =

(

𝑖=1

𝑙𝑛𝑥𝑒𝑥𝑝 𝐼

― 𝑁

1 𝑐𝑎𝑙 2 2 𝑙𝑛𝑥𝐼

)

)

(12)

where 𝑥𝑒𝑥𝑝 and 𝑥𝑐𝑎𝑙 𝐼 𝐼 are experimental and calculated equilibrium mole fractions of component 𝐼, respectively, and 𝑁 is the number of experimental data. Estimation of segment parameters depend on various factors such as quality and number of experimental data, property of solvents used to generate experimental data, uncertainty of pure component thermal properties and regression algorithm as well as initial guesses of regression. Figure 1 shows a comprehensive schematic workflow of the solubility prediction framework. Step 1 is generating input experimental thermophysical data. While experimental data uncertainty propagation to segment parameters can be handled by advanced statistical analysis, the main issue in the quality of experimental data is an appropriate solid-state characterization of the residual solid at equilibrium. Phenomena such as polymorphic transformation, solvate formation or chemical transformation can occur in dissolution unless otherwise a solute crystal is stable in a solution. The activity coefficient calculation cannot consider any of those phenomena nor the segment parameters regression step. Such transformations can so far only be revealed by experiment though recently COSMO-RS based and statistical models have been described to assess solvents ability to form solvate with an API

29 30 31.

Therefore, it is a requirement to experimentally characterize the solid-

state properties of the wet and dried solid residue at the equilibrium. Only characterizing the solidstate properties of dried solid residue using DSC is a common

32

but misleading procedure as it

cannot guarantee the crystal stability nor polymorphic transformation. Polymorphic transformation including solvation and desolvation could occur during sample drying. An appropriate solid-state characterization is for example X-ray powder diffraction (XRPD) measurement of wet as well as dried solid residue. Therefore, in this work, the crystal structures of solutes were characterized before and after dissolution using XRPD. Moreover, the thermal properties of solute crystals and dried solid residue at equilibrium were characterized using DSC.



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While the aim is to use minimum number of experimental data for segment parameters estimation, using a large number of experimental data can be advantageous. However, it is necessary to generated experimental solubility data in a set of solvents covering a wide property spectrum from hydrophobic (e.g. n-hexane), polar (e.g. acetonitrile) to hydrophilic (e.g. water). As shown in the step 2 and step 3 of the schematic workflow of the solubility prediction framework, finding a global minimum of the objective function for parameter estimation is difficult for most optimization algorithms as they have to deal with nonlinear function. Moreover, results of most nonlinear regression algorithms are influenced by initial guesses that typically provided based on researcher’s prior knowledge. Therefore, in this work, hybrid PSO algorithm was implemented to estimate segment parameters 33. The pre-regressed segment parameters by PSO algorithm were used as initial guesses for further segments parameter estimation using trust-region-reflective or interiorpoint algorithms. In this way, consistent segment parameters with minimum influence of initial guesses and the most minimum objective function value can be estimated. Once reliable solute segment parameter are regressed and can be used for solubility prediction of the solute including thermal properties of the solute as well as segment parameters of a solvent or mixture of solvents at step 4. The final step is evaluating solubility prediction accuracy by comparing predicted solubility with experimental solubility generated in the same conditions. Figure 1 shows the schematic workflow of the symmetric eNRTL-SAC model based framework for the solubility prediction of pharmaceuticals.


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Figure 1: The schematic workflow of a symmetric eNRTL-SAC model based framework for solubility prediction of pharmaceuticals; the segment parameters of a solvent can be obtained from literature or can be regressed from VLE and LLE data

3. MATERIALS AND METHODS 3.1.

Materials

Fusidic acid hemihydrate, sodium fusidate and fusidic acid acetone solvate samples were kindly supplied by LEO Pharma A/S (Ballerup, Denmark). All organic solvents used in this study were analytical grade purchased from Merck KGaA and were used without further purification.



3.2.

Page 12 of 25

Methods

3.2.1. Solubility Measurements Solubility experiments of fusidic acid hemihydrate, sodium fusidate and fusidic acid acetone solvate were performed in 4 ml glass vials in a thermostated STEM RS1000 reaction station from Electrothermal. Saturated solutions were prepared using an excess amount of solid. Analytical balance AX205 from Mettler Toledo was used to weigh samples to prepare slurry of solubility experiments as well as for DSC measurements. Slurry samples were incubated for 24 h and HPLC samples were collected by using pre-warmed syringe at the temperature of solubility experiments equipped with 2 µm filter. Another set of HPLC samples were collected and analyzed after 48 h in order to check whether the solubility reached equilibrium. The dissolved concentration of fusidic acid was analyzed by using UHPLC (Ultimate 3000 from Dionex) equipped with a 2.1 x 50 mm Agilent eclipse+ C18RRHD 1.8 µm column using 0.25 mL/min of acetonitrile:1% formic acid in water (90/10, v/v) as an eluent and DAD at 235 nm. The typical retention time of fusidic acid was about 9.2 mins. Wet residual solid samples at the equilibrium were collected for the XRPD analysis. The rest of solid residue samples were dried in desiccator under vacuum. The dried solid residue samples were further characterized by XRPD in order to check solid-state transformation during drying. Moreover, the thermal properties of dried solid residue were determined by using DSC. 3.2.2. Solid-state Characterizations using XRPD, DSC and TGA-IR XRPD patterns were collected with a PANalytical X’pert PRO MPD diffractometer using an incident Cu Kα radiation and operating at 45 kV and 40 mA. The XRPD patterns were collected in the 2 theta range from 3 to 45 degrees with a step size of 0.007°, counting time of 148.93 s and in transmission geometry. In the incident beam path an elliptically graded multilayer mirror together with a 4 mm fixed mask, fixed anti-scatter slit 1° and fixed divergence slits of ½° were placed to line focus of the Cu Kα X-rays through the sample and onto the detector. At the diffracted beam path a long antiscatter extension was placed to minimize the background generated by air. Furthermore, Soller slits of 0.02 rad were placed on both the incident and diffracted beam paths to minimize broadening from axial diver-gence. Samples were placed on a 3 μm thick foil impregnated in a 96 wells high-throughput plate and oscillated in the X direction for better particle statistics. The diffraction patterns were collected using a PIXel RTMS detector with active length of 3.347° and located at 240 mm from the sample. XRPD data were collected and analyzed using Data Viewer from PANalytical B.V.


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The thermal properties (i.e melting points and enthalpy of fusion) of solutes and dried solid residue samples of solubility experiments were analyzed by using DSC on a DSC8500 (Perkin Elmer, Co. Norwalk, USA) equipped with a cooling device Intracooler 2. A purge by dry nitrogen gas (20 mL/min) was used for all measurements. About 1.5 to 3 mg of sample was used for DSC analysis. The DSC profiles of fusidic acid and sodium fusidate as well as all dried solid residue samples were generated at heating rate of 20 °C/min from 25 °C to 500 °C. In addition to the XRPD and the DSC analyses, fusidic acid acetone solvate sample was analyzed by using thermal gravimetric analysis (TGA) Pyris 1 coupled with an infrared (IR) spectroscopy Spectrum100. 2.31 mg solid sample of fusidic acid acetone solvate was heated from 25 °C to 500 °C at a rate of 10 °C/min under nitrogen gas purge. The TGA-IR analysis was performed for weight loss during heating in order to investigate the stoichiometry of fusidic acid acetone solvate and to identify material evaporated during heating. Moreover, combining the results of TGA-IR and XRPD of fusidic acid acetone solvate raw material leads to determine the raw material sample composition. In-house IR spectral library was used in order to identify the material evaporated during heating.

4. RESULTS AND DISCUSSION The solubility of fusidic acid, fusidic acid acetone solvate and sodium fusidate were experimentally determined in order to generate data for segment parameters estimation and model validation. The solubility data were determined in a set of solvents representative of a wide solvent property. The segment parameters of fusidic acid were estimated using solubility data in methanol, acetonitrile, nheptane, water and ethanol-water mixture. The segment parameters of sodium fusidate were estimated using solubility data in acetone, acetonitrile, n-heptane, ethyl acetate and 1,4-dioxane. Segment parameters regression analysis revealed that multiple sets of parameters with sufficiently small RMSEs can be estimated for the same set of experimental data but different initial guesses. Predicted solubility data with the set parameters regressed from different initial guesses showed no plausible range comparison. This demonstrates the sensitivity of eNRTL-SAC model accuracy on the solute segment parameters. Therefore, in this work initial guesses for segment parameters estimation were pre-regressed using PSO algorithm. In PSO, the RMSE in equation (12) is iteratively solved by improving the location of candidate solution. Each iteration solves the RMSE, which in turn determines the direction of the next iteration. Moreover, the PSO algorithm was



Page 14 of 25

hybridized with another local optimizer in order to combine the power of PSO in exploring the global minimum region and the speed of other local optimizer. In the hybrid PSO, the PSO locates the optimum region and the other local optimizer take over to find the global minimum. For examples, the RMSE of segment parameter estimation of fusidic acid reduced from 2.0×10-3 in the pre-regression step by PSO to 7.1×10-7 by trust-region-reflective method in mole fraction scale. Table 1 shows the estimated eNRTL-SAC segment parameters of fusidic acid and sodium fusidate and the RMSEs of estimation in mole fraction scale. For estimating the segment parameters of fusidic acid the electrolyte segment was set to be zero. The estimated segment parameters of fusidic acid and sodium fusidate indicate the change of interaction properties of fusidic acid with solvents due to the replacement of its acidic proton by sodium in sodium fusidate. As depicted in Table 1 the property

of

fusidic

acid

shifts

from

hydrophobic/hydrophilic/polar

combination

to

hydrophilic/polar/hydrophobic combination for sodium fusidate. Table 1: eNRTL-SAC segment parameters (i.e. hydrophobic (X), polar attractive (Y-), polar repulsive (Y+), hydrophilic (Z), and electrolyte (E)) of fusidic acid, sodium fusidate and the RMSEs of regression

Solute

X

Y-

Y+

Z

E

RMSE

Fusidic acid

1.92

0.19

0.21

0.96

0

7.1×10-7

Sodium fusidate

0.43

0.78

1.42

2.71

0.19

9.4×10-8

4.1.

Solubility prediction of fusidic acid in single and mixed solvents

The solubility of fusidic acid was predicted in various single solvents, binary solvents mixtures and as a function of temperature using symmetric eNRTL-SAC model. The prediction capability of the model was evaluated by calculating the average squared relative error (ASRE) between experimental and predicted solubility data as shown in equation (13). The ASRE was calculated after excluding data used for model parameter estimation. 1 𝐴𝑆𝑅𝐸 = 𝑁

(∑ 𝑁

2 (𝑙𝑛𝑥𝑒𝑥𝑝 ― 𝑙𝑛𝑥𝑐𝑎𝑙 𝑖 𝑖 )

𝑖=1

𝑙𝑛𝑥𝑒𝑥𝑝 𝑖

)

(13)

where 𝑥𝑒𝑥𝑝 and 𝑥𝑐𝑎𝑙 𝑖 𝑖 are experimental and calculated equilibrium mole fractions, respectively, and 𝑁 is the number of experimental data. Figure 2 shows comparison of predicted vs. experimental solubility of fusidic acid in water, 2-propanol, methanol, acetonitrile, acetone, toluene, heptane, ethyl acetate, 1-butanol, 1,4-dioxane, ethanol, iso-butanol, methyl-tert-butyl ether, methyl-isobutyl


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ketone at 298.15 °K. The prediction shows good agreement with the experimental data with ASRE of 0.055 in logarithmic mole fraction scale.

Figure 2: Predicted vs. experimental solubility of fusidic acid with average squared relative error (ASRE) of 0.055 at 298.15 °K

The solubility of fusidic acid was predicted in binary solvents mixtures of methanol:water and ethanol:water. Figure 3 and Figure 4 show the solubility of fusidic acid as a function of volume percentage of water in methanol:water and ethanol:water mixture, respectively. The prediction in methanol:water mixture shows good agreement with experimental data until about 10% (v/v) of water (i.e. ASRE of 4.8×10-5 in logarithmic mole fraction scale). However, the deviation increases with increasing the water content (i.e. ASRE of 6.9×10-3 in logarithmic mole fraction scale). The thermal properties of fusidic acid anhydrous III were used to predict solubility in the entire curve of Figure 3 because the solid residue in methanol was anhydrous III. However, solid-state characterization of the solid residue at the equilibrium by XRPD in methanol:water mixtures of 75:25 and 50:50 v/v revealed that the residue solid is not anhydrous III. Figure 5 shows the XRPD pattern of fusidic acid anhydrous III, solid residue in methanol, solid residue in methanol:water mixtures of 75:25 and 50:50, v/v and methanol adduct. Therefore, the increase of deviation after 10% of water evidently attributes to the formation of methanol adduct. As shown in Figure 4, the model captures the solubility trend of fusidic acid in ethanol:water mixture and predicts with ASRE of 0.07 in logarithmic mole fraction scale.



Page 16 of 25

Figure 3: Predicted and experimental solubility of fusidic acid in a mixture of methanol (MeOH) and water at 296.15 °K

Figure 4: Predicted and experimental solubility of fusidic acid in a mixture of ethanol (EtOH) and water at 294.15 °K

Figure 5: XRPD pattern of fusidic acid anhydrous III, solid residue in methanol, solid residues in methanol:water mixtures of 75:25 and 50:50, v/v and methanol adduct


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The solubility of fusidic acid was moreover predicted as a function of temperature in methanol and in methanol:water mixture of 90:10, v/v. Figure 6 and Figure 7 show the solubility of fusidic acid as a function of temperature in methanol and a mixture of methanol:water (90:10, v/v), respectively. As graphically depicted, the predictions show good agreement with experimental data with ASREs of 4.1×10-4 and 5.0×10-4 in logarithmic mole fraction scale in methanol and methanol:water mixture, respectively. Generating a solubility phase diagram as a function of temperature in a mixture of solvents is useful for example to design a cascade of antisolvent and cooling crystallization in order to improve process yield 34.

Figure 6: Predicted and experimental solubility of fusidic acid as a function of temperature in methanol

Figure 7: Predicted and experimental solubility of fusidic acid as a function of temperature in methanol:water mixture of 90:10, v/v



4.2.

Page 18 of 25

Solubility prediction of fusidic acid acetone solvate

An accurate solubility prediction of API solvates is useful for example to design and optimize crystallization. The qualitative and quantitative composition of fusidic acid acetone solvate sample was analyzed by using XRPD and TGA-IR measurements. Figure 8 shows that the raw material contains fusidic acid acetone solvate and fusidic acid anhydrous II polymorphic form. Since the stoichiometry of fusidic acid and acetone in fusidic acid acetone solvate is 1:1 mole/mole, the raw material quantitative composition can be determined from the TGA curve. Therefore, from the TGA curve the raw material contains 59.4% of fusidic acid anhydrous II and 40.6% of fusidic acid acetone solvate. Moreover, the IR spectrum in the TGA-IR measurement confirmed that the mass loss at about 370 °K is due to the evaporation of acetone. The solubility of fusidic acid acetone solvate in solvents other than acetone was predicted in binary solvents mixture by including the amount of acetone released into the solution. The amount of acetone released into the solution was calculated using the amount of solid used to prepare slurry. Figure 9 shows comparison of predicted vs. experimental solubility of fusidic acid acetone solvate in water, methanol, ethanol, acetonitrile, acetone, toluene, 1-butanol and 1,4-dioxane at 298.15 °K. The prediction shows good agreement with experimental data with ASRE of 0.079 in logarithmic mole fraction scale.

Figure 8: XRPD pattern of fusidic acid anhydrous II, fusidic acid acetone solvate and raw material of fusidic acid acetone solvate used for experimental solubility data generation


Page 19 of 25 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


Figure 9: Predicted vs. experimental solubility of fusidic acid acetone solvate with average squared relative error (ASRE) of 0.079 at 298.15 °K

4.3.

Solubility prediction of sodium fusidate

The utility of symmetric eNRTL-SAC model for predicting the solubility of salts is demonstrated by predicting the solubility of sodium fusidate in various single solvents and binary solvents mixture. Figure 10 shows comparison of predicted vs. experimental solubility data of sodium fusidate in methanol, acetonitrile, acetone, n-heptane, ethyl acetate, 1-butanol, 1,4-dioxane and in binary solvents mixture of methanol:water (75:25, v/v) and acetone:water (95:5, v/v) at 298.15 °K. The prediction shows satisfactory agreement with the experimental solubility data with ASRE of 0.084 in logarithmic mole fraction scale.

Figure 10: Predicted vs. experimental solubility of sodium fusidate average squared relative error (ASRE) of 0.084 at 298.15 °K



Page 20 of 25

Figure 11 shows predicted and experimental solubilities of sodium fusidate as a function of volume percentage of water in acetone:water mixture at 298.15 °K. The prediction shows good agreement with experimental data until 5% of water with ASRE of 0.02 in logarithmic mole fraction scale. However, with increasing the amount of water higher than 5% the prediction shows a linear offset. Solid-state characterization of the residue solid revealed that the residual solid at the amount of water higher than 5% was not the same as the initial sodium fusidate solid crystal. On the other hand, the thermal properties of sodium fusidate were used for solubility prediction in the entire region of the curve in Figure 11. The linear offset after 5% of water therefore attributes to the solidstate transformation of sodium fusidate crystal that may be due to the formation of solvate. Moreover, the model captures the solubility trend of sodium fusidate as a function of percentage of water in acetone:water mixture.

Figure 11: Predicted and experimental solubility of sodium fusidate in a binary acetone:water mixture at 298.15 °K

Thus, the methodology is applicable for a reliable solubility prediction of different forms of medium-sized APIs. The solubility prediction was performed in various single and mixed solvents as well as a function of temperature. Moreover, the methodology performance for solubility prediction was evaluated by comparing the ASREs obtained in this study with described model accuracy in literature. For a fair evaluation, the comparison was limited with models of the same domain (i.e. NRTL-SAC or eNRTL-SAC) but different model compounds. As shown in Table 2, the methodology in this study predicts solubility with low ASREs of an order of magnitude. However, the large prediction uncertainty in the described literature could also attribute to the quality of experimental data with respect to unrevealed residual solid polymorphs and corresponding thermal properties used for solubility calculation as well as number of equilibrium data and uncertainty associated with experimental data.


Page 21 of 25 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


Table 2: Methodology performance evaluation by comparing the average squared relative errors (ASREs) of solubility prediction in logarithmic mole fraction scale of different compounds

Model compound

Model

ASRE

Number of

Source

equilibrium data Fusidic acid

symmetric eNRTL-SAC

0.055

19

This study

Sodium fusidate

symmetric eNRTL-SAC

0.084

10

This study

Fusidic acid acetone symmetric eNRTL-SAC

0.079

8

This study

solvate Ibuprofen

NRTL-SAC

0.165

9

8

Paracetamol

NRTL-SAC

0.250

12

8

Salicylic acid

NRTL-SAC

0.436

12

8

Benzoic acid

NRTL-SAC

0.114

13

8

4-Aminobenzoic acid

NRTL-SAC

0.672

n/a

8

Anthracene

NRTL-SAC

3.631

n/a

8

5. CONCLUSIONS The solubility property of APIs is a key thermodynamic property for screening of an optimum solvent or solvents mixture to design an efficient pharmaceutical manufacturing process. The symmetrically reformulated eNRTL-SAC model was implemented for symmetric activity coefficient calculation. The symmetric eNRTL-SAC model eliminates the need to compute the Born correction term for non-aqueous systems. This provides computationally simple and straight forward as well as circumvents uncertainty associated with the Born correction term calculation. Moreover, due to its semi-predictive nature, the eNRTL-SAC model accuracy highly depends on the estimated segment parameters of a solute. Therefore, design of experiments is also considered in the framework in order to generate and use experimental data appropriately for model parameter regression and model validation. To improve estimation of the eNRTL-SAC model parameters, a particle swarm optimization (PSO) algorithm is incorporated to pre-regress conceptual segment parameters of solutes. The pre-regressed segment parameters were used as initial guesses for further segment parameter estimation using another nonlinear regression algorithms (e.g. trust-regionreflective). This hybrid PSO approach leads to regress consistent segment parameters with much low RMSE. The application of symmetric eNRTL-SAC model was demonstrated by predicting the solubility phase properties of fusidic acid, fusidic acid acetone solvate and sodium fusidate in single 19 ACS Paragon Plus Environment


Page 22 of 25

and mixed solvents representative of a wide solvent property spectrum. Moreover, using temperature dependant binary segment interaction parameters, the solubility of fusidic acid was accurately predicted as a function of temperature in single and binary solvents. The solubility predictions of fusidic acid, fusidic acid acetone solvate and sodium fusidate in various single solvents showed good agreement with experimental solubility data with ASREs of 0.055, 0.079 and 0.084 in logarithmic mole fraction scale, respectively. Likewise, the model predicted the solubility of fusidic acid and sodium fusidate in binary solvents mixtures and fusidic acid as a function of temperature in a satisfactory agreement with the experimental solubility data. AUTHOR INFROMATION Corresponding Author *(Getachew S. Molla) E-mail: [email protected] Notes The authors declare no competing financial interest.

Acknowledgment: We would like to thank the Danish Council for Independent Research (DFF) for financing the project under the grant ID: DFF-6111600077B. We would like to thank Heidi Lopez de Diego for fruitful discussions during our project meetings. Supporting Information: Experimentally determined thermal properties of fusidic acid polymorphs and sodium fusidate (i.e. ∆𝐻𝑓𝑢𝑠 and 𝑇𝑚) as well all experimental solubility data used in the manuscript are provided as supporting information.

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Solubility Prediction of Different Forms of Pharmaceuticals in Single

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