Prediction of Paracetamol Solubility in Binary Solvents Using Reichardt’s Polarity Parameter Combined Model

: The objective of this research is to propose a general model utilizing the solvatochromic polarity of electronic transition energy (ET) of the Reichardt indicator to predict paracetamol solubility in the solvent mixtures. In order to model validation, the available ET (30) values of nine aqueous mixtures obtained from existing literature sources were utilized. The trained model yielded a relatively accurate estimation of paracetamol solubility in the investigated systems.


Introduction
Paracetamol, known as N-acetyl-p-aminophenol, is highly valued for its analgesic and antipyretic properties in the treatment of various conditions such as fever, headache, arthritis, neuralgia, post-surgical pain, and providing palliative care to advanced cancer patients [1].While it is mostly administered as a tablet, other forms such as intravenous preparations, suppositories, and solutions are also available in the market [2].For efficient drug absorption, it must be in an aqueous solution form at the absorption site.The improved aqueous solubility of drugs or drug candidates can increase their bioavailability, reduce their dosage, and ultimately enhance their efficacy.Therefore, the aqueous solubility of any drug candidate is a crucial physicochemical property essential for its successful development.This aspect of drug development is often limited by poor solubility, and, as a result, it is crucial to determine drug candidate solubility as early as possible.There is considerable interest in the development of models that accurately predict aqueous solubility directly from a chemical structure [3].In the case of the low aqueous solubility of a drug, addition of a permissible organic solvent, cosolvency, is an appropriate solution.Cosolvency helps the formulation scientists to dissolve the desired amount of the drug in a given volume of the liquid formulation.In some cases, i.e., in injectable solution, there is a volume restriction problem too.More solubilizing cosolvent with a lower toxicity and less side effects is more favorable.Desolubilization of a drug is also required where recrystalization is the aim of the experiments.In these cases, the drug and the related compounds are dissolved in a good solvent; usually an organic solvent and an anti-solvent is added to the mixtures to induce crystallization process.These practical applications reveal the importance of solubility data in binary solvent mixtures.Despite the experimental determination of the solubility in cosolvent + water mixtures, there are some models to calculate the solubility in mixed solvent systems.These models facilitate the process of data usage in industrial applications.The extended Hildebrand solubility approach of Martin [4], mixture response surface [5], the combined nearly ideal binary solvent/Redlich-Kister equation [6], the log-linear model of Yalkowsky [7], the modified Wilson model [8], phenomenological model [9], fluctuation theory [10], the excess free energy approach [11], the Jouyban-Acree model [12], and Kamlet-Abboud-Taft-linear solvation energy relationship [13,14] were the well-known reported mathematical models for solubility prediction in cosolvency mixtures.One of the commonly used models that have demonstrated accurate predictions of solubility is the Jouyban-Acree model, which is dependent on both temperature and solvent compositions [12].Beyond their general forms, these models can be customized by introducing the chemical and physical properties of solvent and solute into their parameters.Some of these parameters that exhibit quantitative structure-property relationships (QSPRs) are the Hansen [15] and Catalan [16] solubility parameters, Abraham solvation parameters and solvatochromic polarity parameters (e.g., electronic transition energy (ET) or Reichardt's polarity).In continuation of our previous works in combining the QSPR parameters with the Jouyban-Acree model, this study seeks to suggest a combined Jouyban-Acree model with Reichardt's polarity parameter that can predict and correlate paracetamol solubility in the cosolvency systems.To achieve this, data on paracetamol solubility along with ET 30 values in different cosolvency systems were gathered from the literature and utilized to develop a comprehensive model capable of predicting paracetamol solubility accurately.We used paracetamol as a model drug in this work, since a very wide range of solubility data in cosolvent + water mixtures are available for this drug.
The investigated model in this work was the Jouyban-Acree model as the most precise cosolvency model available, it depicts the correlation between the solubility of a solute and both the temperature and the solvent composition.In binary cosolvency systems at different temperatures, the Jouyban-Acree model can be expressed in a general form as follows [12]: where x m,T , x c,T and x w,T denote the solubility of the solute in the solvent mixtures, cosolvent and water at temperature T/K; w c , and w w are the mass fractions of mono solvents  d-w-c) interactions in the solute saturated mixture solution [12].One can integrate the Jouyban-Acree model with certain parameters such as Reichardt's polarity parameter to analyze the characteristics of solvents with regard to their physicochemical properties.
By including these values in Equation ( 1) for a given solute, the combined model can be obtained as where J terms are the model parameters and E N m,T is the ET (30) values for the desired binary mixtures.The symbols used in prior models remain unchanged in this case.The model constants in Equation ( 2) are determined through a no-intercept least square analysis.
To investigate the capability of Reichardt's polarity parameter for improving the solubility prediction power of the Jouyban-Acree model, the obtained results were compared with the Jouyban-Acree model combined with Abraham solvation parameters.For this purpose, Equation ( 3) as a simplified model for one solute was used: Solvent coefficients, namely c, e, s, a, b and v, exhibit variation based on the type of solvent being analyzed.The phase's affinity to interact with solutes via polarizability-based interactions is expressed as e, whereas s quantifies the dipolarity/polarity of the solvent phase.Hydrogen-bond acidity and basicity of the solvent phase are designated as a and b coefficients.Additionally, v represents the overall dispersion interaction energy between the solvent phase and the solute.Also, J terms are the model parameters.
To determine accuracy, the mean relative deviation (MRD) is employed and computed via the following formula: The formula involves NDP, which represents the quantity of data points in every set.The definition of the MRD is very similar to that of the relative standard deviation (RSD) for the repeated experiments.One could directly compare the numerical values of the MRDs with the RSD values for experimental measurements, where the ideal model should provide MRD% close to RSD values.The RSD for repeated paracetamol solubility data using the same chemicals and the same instruments and procedures varied from 3.3% to 17.0% and as a general rule; with a lower solubility, a larger RSD is obtained [34].Concerning the paracetamol solubility data reported from different laboratories, the overall RSD varied from 17.6% to 21.1% [35].In order to demonstrate the predictive ability of the models in question, a leave-one-solvent-system-out method was utilized for cross-validation.During each analysis, one data set was omitted from the training process and the trained model was then used to predict its corresponding solubility.

Results and Discussion
The experimental paracetamol solubility data in binary aqueous mixtures of ethanol, methanol, PG, 2-propanol, 1-propanol, acetonitrile, DMF, DMSO, and 1,4-dioxane were used to train Equations ( 1)-( 3).In the first step, the Jouyban-Acree model and its combined form with Reichardt's polarity parameter were used for each binary system data correlating, individually.The MRD% values for these computations are given in Table 1.As can be seen, MRD% values for all solubility systems were less than 15%, showing the reliability of data for fitting to the mathematical model.Furthermore, the low MRD% values being obtained separately for each cosolvency system is an initial criterion for including them in the generation of a general model.Another point in Table 1 was the low value of MRD% for the combined form of the Jouyban-Acree model with Reichardt's polarity parameters compared with the Jouyban-Acree model.The Jouyban-Acree model, in its general form, is not influenced by the characteristics and properties of either the solute or solvent.Despite this, factors such as solute ionization in solvent mixtures, solubilization/desolubilization capacity, density, dielectric constant, and physical/chemical stability can impact solubility.These parameters can be described in ET (30) values reported for the solvent mixtures.
The next step was the correlation of all data for the generation of a general model for solubility prediction.The trained version of the combined form of the Jouyban-Acree model with Reichardt's polarity parameters for the paracetamol solubility prediction in aqueous solvent mixtures was as: It is important to highlight that the statistical significance of all the model constants was confirmed through t-test analysis at a probability level of <0.1.The back-calculated solubility data, comprising 422 data points, showed an overall MRD% of 37.6%.Table 2 displays the MRD% values calculated for paracetamol solubility data in different solvent mixtures at varying temperatures, using Equation (5).For the trained model, the lowest predicted solubility data deviation (MRD = 4.3%) can be observed for a solvent mixture of 2-propanol and water at a temperature of 303.2 K. Conversely, the highest deviation (MRD = 139.0%)occurs for a solvent mixture of PG and water at a temperature of 293.2 K.
One can remove J 1 , J 3 , and J 5 from Equation ( 2) to reach below model with The overall MRD% for back-calculated data with this trained equation is 46.9% which does not have significant difference with Equation ( 5) demonstrating Equation ( 6) with three parameters can be used instead of Equation ( 5) with six parameters.
The effectiveness of Reichardt's polarity parameter in improving the solubility prediction accuracy of the Jouyban-Acree model was examined by comparing the results with those obtained from the Jouyban-Acree model that was combined with Abraham solvation parameters.Abraham solvation parameters are a set of empirical coefficients that include multiple parameters that represent different molecular interactions such as polarizability, dipolarity/polarity, hydrogen-bond acidity, basicity, and dispersion.Each parameter contributes to a different aspect of solvation, creating a more accurate representation of the overall behavior of the solvent.The use of multiple parameters in Abraham solvation parameters, as well as their flexibility and applicability to a wider range of solvents, offers advantages in predicting solvation behavior over other solubility parameters.
The trained form of Equation ( 3) for the paracetamol solubility in nine included aqueous binary systems is The overall MRD% is 12.4% (Table 2).As can be seen, a relatively high difference was observed for back-calculated MRD% of Equation ( 7) with 10.0% and Equation ( 5) with 37.6 for the similar data.A similar trained model was proposed for the solubility of paracetamol in various cosolvent + water mixtures with an overall MRD% of 19.6%, employing the Hansen solubility parameters [35].However, these differences are normal and the possible reason for this difference in accuracy is the number and nature of parameters used in each model.The Abraham solubility parameter model incorporates multiple parameters that represent different types of molecular interactions, whereas Reichardt's polarity parameter represents only the solvents' relative polarities, which can be less specific to the solute.Another possible reason could be the variant data set used for model validation.The models' performances heavily depend upon the data set used for validation, and any biases in the data set can affect the predictive capability of a model.For example, this difference between the MRD% values of the two models decreased when excluding the PG+ water system with MRD% 10.2% for Equation (7) and 23.4% for Equation (5).Therefore, it is essential to use a diverse data set for validation, including compounds with different chemical structures and properties, to ensure accurate predictions.
Table 2. MRDs% for solubility of paracetamol in the aqueous binary systems at various temperatures for Equations ( 5) and (7).

No. Solvent Mixtures T (K)
MRDs (±SD)% Equation ( 5) Equation ( 7) It should be noted that even though Reichardt-polarity-parameter combined model gave a higher error percentage compared to the Abraham-solubility-parameter combined model, the error range was still acceptable.As mentioned above, the RSD values for repeated paracetamol solubility determination varied from 17.6 to 21.1% [34].These observations suggest that Reichardt's polarity parameter can potentially be used as an alternative to Abraham solubility parameters if a less complex model is desired, although it may not provide the same level of accuracy as Abraham solubility parameters.
Cross-validation was employed using the leave-one-solvent system-out method to assess the prediction capabilities of the trained models.A comprehensive report of the cross-validation process for the analyzed models is presented in Table 3.The tabulated results show that the overall MRDs% increased from 15.3 to 20.2 for the ethanol + water mixture, 127.8 to 177.8 for PG + water, 60.9 to 56.4 for methanol + water, 41.2 to 12.6 for 1,4-dioxane +water, 25.7 to 21.5 for 1-propanol + water, 41.3 to 47.6 for acetonitrile +water, 35.2 to 36.3 for DMSO + water, 35.5 to 37.0 for DMF + water, and 7.3 to 7.8 for the 2-propanol + water system.It can be concluded that the combined form of the Jouyban-Acree model with Reichardt's polarity parameters has an acceptable reliability to predict the paracetamol solubility data in the investigated mixtures.A cross-validation process was also employed for the Jouyban-Acree model combined with Abraham solvation parameters, and the overall MRD% value increased from 10.0% to 1.1 × 10 7 .As can be seen, the Reichardt-polarity-parameter combined model showed better results compared to the Abraham-solvation-parameters combined model.A possible reason for it can be this fact that the Reichardt-polarity-parameter combined model is relatively simple, requiring only one parameter to predict solubility (the solvent polarity parameter) whereas, the Abraham-solvation-parameters combined model requires multiple parameters, including the hydrogen-bond acidity and basicity, polarizability, and volume parameters.In some cases, having fewer model parameters can make a model less prone to overfitting and better suited to predict solubility, especially if the data set is limited.The performance of the models also depends on the quality and diversity of the training data used to optimize the parameters.Therefore, a more detailed investigation is needed to determine the performance differences between the models.

Conclusions
This research involved the development of a trained model based on Reichardt's polarity parameter to predict paracetamol solubility in cosolvency systems.The use of the Jouyban-Acree model was examined, as well as its combined version with Reichardt's polarity parameter.The effectiveness of Reichardt's polarity parameter in improving the solubility prediction accuracy of the Jouyban-Acree model was examined by comparing the results with those obtained from the Jouyban-Acree model combined with Abraham solvation parameters.Upon analysis, the model was deemed to have a satisfactory level of accuracy in predicting solubilities, as evidenced by the overall MRDs% of 37.6.

1 (
the cosolvents (c) in this work), and 2 (water (w) in this work) in the absence of the solute; and J i terms are the model coefficients representing the two-body (d-d, d-c, c-c, d-w, w-w (d = drug)) and three-body (d-d-d, d-d-c, d-c-c, d-c-d, c-c-c, d-d-w, d-w-w, d-w-d, w-w-w,

2 T
ln x m,T = w c • ln x c,T + w w • ln x w,T + w c w w T 5922.694− 75.549EN m,T + w c w w (w c −w w ) T 6900.277− 135.008EN m,T + w c w w (w c −w w ) 8395.463 − 156.285EN m,T

Table 1 .
MRDs% for solubility of paracetamol in the aqueous binary systems at various temperatures for Equations (1) and (2).

Table 3 .
Leave-solvent-system-out cross-validation for the proposed models.