INTRODUCTION:

Predictive models for drug solubility in solvent blends are essential for drug formulation, which has practical implications across a range of pharmacy domains. Solvent mixtures find extensive utilization in formulation, extraction, or analysis within the pharmaceutical field¹.

The implementation of theoretical and semiempirical approaches allows for significant savings in cost and time, reducing reliance on expensive and time-consuming experimental methods. Due to the marginal solubility of drugs in water and other pharmaceutically interesting solvents, understanding the solubility behavior of drugs becomes particularly crucial for liquid dosage forms².Appropriate qualitative standards for choosing a solvent are provided by the Solubility Parameter (SP) and the theory of regular solutions^3,4,5,6. Under these situations, it is useful to postulate the behavior of a solution based on certain assumptions. If the postulates are reasonably accurate, the observed behavior should match the theoretical conclusions. In this investigation, efforts are concentrated to arrive at such conclusions.

Ideal Solutions:

A dispersion in which unlike molecules have the same affinities as they do for their own kind is the ideal solution of solid in liquid mixture. The magnitude of the attractions present is the same in ideal solutions^7,8. Energy is required to liquefy the solute when a perfect solution of a solid in a liquid form can be explained as a liquid-solid equilibrium process. The solution is anticipated to function as an ideal liquid-liquid solution upon liquefaction. The heat absorbed when one mole of solid melts is known as the molar heat of fusion (DH_f)^9,10. Molar heat of solution is equal to DH_f in ideal solutions¹¹.An ideal solution is formed when a solute's Mole Fraction Solubility (MFS) in a solvent meets the Ideal Solubility (IS), represented as X₂ⁱ¹².

(1)

where, T_o =melting point of solid solute in absolute scale (K), T = absolute temperature of the solution (K), R = Ideal gas constant (cal/mol. K), Δ H_f= molar heat of fusion (cal/mol). The ideal MFS may otherwise be represented for solubility of solute in polar and nonpolar liquids as:

(2)

where, Entropy of fusion, (3)

Nonideal Solutions ¹¹

The IS Eq. (1) serves as the foundation for the progress of solution theories. However, there are numerous scenarios that cannot be explained accurately by Eq. (1). Therefore, IS expression needs further improvement. The word "effective concentration" (activity) of the solute might be used in place of "concentration". When the concentration is provided on the MFS scale , activity (₂) is stated as the concentration multiplied by the rational activity coefficient ( .

(4)

Log on both sides of Eq. (4) result in:

(5)

g₂ represents the total work needed to extract a solute from its nearby solute molecules (W₂₂), to create a solvent cavity big enough to hold a solute, (W₁₁), and to place a solute into a cavity, the effort that is obtained is (W₁₂). It can be shown mathematically that

(6)

where ϕ₁ is the solvent volume fraction and V₂is the solute molar volume (V_m). Regular solutions are those in which there is little variation in the size and polarity of the solute as well as the solvent. For these regular solutions, geometric mean of W₁₁ followed by W₂₂ can be used to accurately estimate the term W₁₂, which is hard to evaluate.

(7)

Eq. (6) becomes

(8)

The solubility parameter of the solvent blend (δ₁) and the solute (δ₂) are the square roots of the work terms. Hence, the nature of the solute (V_m and SP), solvent (volume fraction and SP), in addition to solution temperature, determines the rational activity coefficient, g₂. The solute-solvent interaction term is d₁d₂.

Substituting δ₁ and δ₂ values in Eq. (8) gives:

(9)

Substituting Eq. (9) in Eq. (5) gives the MFS as:

(10)

A regular solution refers to a nonideal solution to which the Scatchard-Hildebrand (SH) Eq. (10) applies. Irregular solutions are those that exhibit a divergence from regular solutions. The geometric mean assumption was conveniently proposed in the process of deriving SH expression¹³. The justifications for the assumptions are; it would be anticipated that intermolecular forces would function similarly to electrostatic and gravitational forces between point masses. The geometric mean relationship was confirmed experimentally. Unlike pair potential, where the arithmetic mean of two like pair potentials, leads to the cancellation of terms, resulting in zero energy of mixing.

The solubilities of steroidal esters in hydrocarbon solvents have been predicted with some degree of effectiveness using Hildebrand regular solutions theory. Most of the cyclo and aromatic hydrocarbons, as well as halogen derivatives were examined for testosterone esters' solubilities. A d value of 19.5 to 20.5 Mpa^½ (9.5 to 10.0 cal^½/(cm³)^½) is assigned for testosterone propionate¹⁴.An important contribution of the SH Eq. (10) is the introduction of V_m and SP in the solubility predictions for regular solutions.

Extended Hildebrand Solubility Approach (EHS)

Predictions regarding the solubility of drugs in polar liquids are not very accurate when using the regular solution theory. Irregularities in polar liquids include complexation of more than two solids in solution, solvation of the solute by the solvent, and solute and solvent self-association¹⁵.The solubility expression utilised in the EHS approach is;

(11)

where the solubility of pharmaceutically significant crystalline solids in polar and non-polar solvents⁷ can alternatively be represented, (log X₂ⁱ), as follows:

(2)

ΔS_fstands for the entropy of fusion. The logarithm of the activity coefficient γ₂ (in Eq.10) is divided into two terms according to the EHS approach. represents primarily physical or van der Waals forces. represents residual presumably greater forces.

(12)

in which

(13)

and

(14)

In terms of Eq.s (13) and (14), the Eq. (2) is expressed as follows:

(15)

where, (16)

In regular solution theory, the W term, an interaction term, is thought to be equivalent to a geometric mean in which δ₁ and δ₂ represent the SPs of solvent and solute respectively. The expression, is denoted as A. Both polar and nonpolar solutes in single solvent systems and mixed solvent systems can be treated with the EHS approach. Various drugs such as testosterone propionate¹⁴, caffeine¹⁵,testosterone and testosterone propionate¹⁶,theophylline¹⁷, p-hydroxy benzoic acid¹⁸ and paracetamol¹⁹ were investigated by the EHS approach. Since it offers a large variety of solvent combinations with varying polarities that can be used in industrial processes, the dioxane-water system is a great choice of cosolvents to test solubility models. The HS equation has been modified by the EHS approach. This method of back calculating drug solubility in polar and nonpolar solvents is not a novel physical theory; rather, it is a methodology based in part on polynomial regression analysis of experimental findings.

The solubility behaviour of Aripiprazole (ARP) is described in the current report. The discussion begins with the ideal solution, followed by regular solutions exhibiting moderately polar characteristics, and concludes with irregular solutions.

MATERIALS AND METHODS:

Materials

Gift sample of ARP was provided by Hetero Labs, Jadcherla, Telangana, India. SD Fine Chemicals provided research-grade dioxane, which was utilized as such. The entire work was conducted using double-distilled water that had been prepared in the lab. Other chemicals used were of research grade. To calculate solubility, GW BASIC an in-house developed programme was utilized. The statistical functions of the MS-Excel programme were used to do a regression analysis on the experimental data.

Analytical method

Evaluation of solubility of ARP was done using UV spectrophotometric method (UV-1700, Shimadzu, Japan) at an analytical wavelength of 248.6 nm within a concentration range of 5 to 35mg·mL^–1.

Determination of ∆H_f

Using a Differential Scanning Calorimeter (DSC 8500, Perkin Elmer, USA), ∆H_f was calculated calorimetrically. Under a nitrogen atmosphere, the heating rate was 5°C per minute (flow rate 50–60 mL/min). The energy needed to keep the samples at necessary temperature was noted on a chart adjusted for temperature rise^20,21,22.

Calculation of SP of solvent blends

The following equation was used for calculating the SPs for a mixture of solvents using SPs of pure solvents^23,24.

(17)

where, δ₁ and δ are SPs of mixed solvents and pure solvents A and B, respectively and f is the volume fraction.

Calculation of V_m of solvent blends

The V_m of solvent blends were calculated using following equation²³.

(18)

where the V_m of solvents A and B are denoted by V_A and V_B, respectively.

Determination of V_m of ARP

Literature rarely provides molar quantities of chemicals that are generally present in the solid state^25,26. ARP was submerged in n-hexane, and the density of the displaced liquid was determined²⁷. The V_m of a compound is showcased as:

(19)

Determination of experimental solubility

ARP's solubility was assessed in eleven dioxane-water blends with varying polarity (10.04 H to 23.4 H). The solvents were mixed with excess ARP. To achieve equilibrium, the flasks were shaken for 72 hours at a constant temperature (25±1°C) in a cryostatic constant temperature reciprocating shaker bath (Research and Test equipment, Bangalore, India). After 72 hours, the samples were taken out, filtered using 0.22, appropriately diluted, and examined using UV-Visible spectrophotometer at 248.6 nm (n=3).

RESULTS AND DISCUSSION:

Determination of ΔH_f

ΔH_fdetermined based on the data obtained from DSC thermogram²⁴ was 2983.0307 cal/mol.

Calculation of solubility parameter (SP) and molar volume (V_m) for mixed solvents

The SP and V_m for dioxane-water mixtures in the current study were determined using Eq.s (17) and (18), respectively and are projected in Table 1. The density and molar volume of (V_m) of ARP were determined to be 1.2626 g/cc and 355.124 cm³/mol, respectively.

Table 1. SP and V_m of dioxane-water mixtures

Sl No	Dioxane-water Ratio	SP (H)	*V_m* (cm³/mol)
1	100:0	10.04	85.7
2	90:10	11.37	78.93
3	80:20	12.71	72.16
4	70:30	14.04	65.39
5	60:40	15.98	58.62
6	50:50	16.72	51.85
7	40:60	18.05	45.08
8	30:70	19.39	38.31
9	20:80	20.72	31.54
10	10:90	22.06	24.77
11	0:100	23.40	18.00

Ideal solubility

At the temperature of fusion T_O (139.25^OC), D was calculated. Further the value obtained (2983.0307 cal/mol) was used for IS determination. The ideal MFS () was determined using the data and Eq. (1), yielding a value of 0.2476. However, Eq. (2) indicated that 0.3067 was the . The solubility of caffeine¹⁵ and theophylline¹⁷ in polar and non-polar solvents were calculated using Eq. (2). The current study utilized the ideal MFS () of ARP, 0.3067, for additional computations. The ARP solutions exhibited significant deviation from ideal behavior.

Determination of experimental solubility

ARP's MFS in each system determined are shown in Table 2. Figure 1 displays a plot of the mole fraction solubilities of ARP vs the solvent blend SP. The ideal solution behavior is insufficient to explain the experimental data, according to a comparison of the calculated value with the experimental mole fraction solubilities. In comparison to the , the observed solubilities are lower. As a result, ARP solutions do not function like ideal solutions.

Figure 1. MFS of ARP in dioxane- water system at 25^OC

Regular solutions

Scatchard Hildebrand presented the idea of a regular solution, which is not perfect but does not need a strong solute-solvent interaction. A solution cannot incorporate hydrogen bonding or other types of chemical complexation to be considered "regular." Since ARP solutions did not meet the IS expression, they were categorized as regular solutions for the purpose of the present discussion. Consequently, all the data was examined to confirm the SH equation Eq. (10), which assumes the SP (δ₂) as 10.03 H²⁴. Table 2 provides the estimated MFS values. There were inaccuracies more than 100% between the calculated solubility values using Eq. (10) and the experimental values. ARP forms solutions in mixtures of dioxane and water. The solubility trend was almost linear, suggesting that the interactions between the solute and the solvent were similar. In dioxane containing water up to 20%, calculated solubilities exceeded experimental solubilities; however, the trend reversed in dioxane including water above 20%. This might be because of the nonpolar nature of drug that dissolves least readily in polar solvents like water.

Table 2: Data of experimental and calculated MFS [using Eq. (10) for regular solutions] of ARP in dioxane: water system, at 25°C

Ratio	V₁	δ₁	φ₁	A	X₂, experimental^c	X₂, calculated
100:0	85.7	10.04	0.3551	0.2537	0.0031	0.3048
90:10	78.93	11.37	0.3345	0.2575	0.0012	0.3067
80:20	72.16	12.71	0.9915	0.2577	0.0010	1.433E-03
70:30	65.39	14.04	0.9999	0.2586	0.0006	4.7649E-06
60:40	58.62	15.38	1	0.2586	0.0005	1.6747E-09
50:50	51.85	16.72	1	0.2595	0.0002	6.927E-14
40:60	45.08	18.05	1	0.2601	0.00005	3.3727E-19
30:70	38.31	19.39	1	0.2602	0.00003	1.9324E-25
20:80	31.54	20.72	1	0.2602	0.000008	1.3031E-32
10:90	24.77	22.06	1	0.2603	0.000002	0
00:100	18.0	23.4	1	0.2602	0.0000023	0
^c àAverage of three determinations

Irregular solutions:

Evidently, regular solution theory is not very useful for forecasting nonelectrolyte solubilities in polar and aqueous solvents. However, it provides a useful basis for explaining the behavior of nonpolar solutes in nonpolar or moderately polar solvents. Similar internal pressures in the solvent and solute are the fundamental presumption for regular solution behavior. Experience, however, has demonstrated that this presumption is not necessarily true. Alcohol and water are two examples of intermediate systems whose properties resemble regular solutions, despite the fact that their properties drastically diverge from regular solution theory. The solubilities of piroxicam did not fall within the definition of a regular solution in a mixture of polar and highly hydrogen bonded solvents such as ethanol-water mixture²⁸. They formed highly irregular mixtures and Eq. (10) gave inaccurate predictions in ethanol-water mixtures. Extensive attempts were made to modify the existing SH equation.

EHS Approach - solubility predictions using regression of “W” values

Examining Figure 1 revealed that, across the whole range of the dioxane-water solvent blends, the observed solubilities of ARP significantly differed from those predicted for a regular solution. The results deviated from SH equation. Therefore, the results were treated for EHS approach. The EHS approach was proposed for different drugs like sulfonamides,²⁹ salmeterol xinafoate,³⁰ methylxanthines,³¹tolbutamide, acetohexamide, sulpha-somidine combination,³² and itraconazole³³. Here is the modification of the current regular solution expression.

(20)

where (16)

The ϕ₁ values used in Eq. (16) were obtained directly using experimentally determined MFS.

₁ (21)

Assuming that Eq. (21) is more accurate, the “W” values (also (log ₂)/A values) were calculated by including ϕ values in Eq. (20). In regular solution theory, the interaction energy term denoted by "W" was assumed to be equivalent to the "geometric mean." Currently, the evaluation of "W" is based on knowledge of other terms in Eq. (20) that have been discovered experimentally or found in the literature. Table 3 provides "W" values. Once the “W” value is established, precise solubility predictions can be derived by utilising regression of “W” versus δ₁ (solvent mixture SP). Even if "W" cannot yet be determined using the solute and solvent's fundamental physicochemical properties, Eq. (20) can be used to get "W" from the experimental values for each solvent blend. It is possible to regress these “W” values against a polynomial in δ₁. Table 3 provides the experimental "W" values for blends of dioxane and water. The regression of the experimental values of “W” for dioxane-water mixtures yielded the following quadratic, cubic, and quartic equations.

Quadratic equation :

(22)

n = 11 ; s = 0.365995 ; R² = 0.9999 ; F = 39996 ; F (2, 8, 0.01) = 8.65

Cubic equation:

(23)

n = 11 ; s = 0.35572 ; R² = 0.9999 ; F =23331; F (3, 7, 0.01) = 8.45

Quartic equation :

(24)

n = 11; s = 0.23771; R² = 0.9999; F = 14998.5; F (4, 6, 0.01) = 9.15

where n- number of cases, R²- squared correlation coefficient, s- standard deviation, F-Fisher F ratio. The R² and residual analysis were used to determine equation of the best choice according to Bustamante and Rellio³⁴. Table 3 presents the data. The randomly distributed SD units demonstrate the applicability of Eq. (24).

Table 3: Residuals derived from the 'W' based quartic regression Eq. (24) for the solubility of ARP in a dioxane-water blend at 25°C

SI No	W(exp)^b	W(cal)^a	Residuals^c	SD units
1	94.0000	93.9763	0.0237	0.1264
2	107.5635	107.651	-0.0875	-0.4659
3	123.5051	123.4407	0.0644	0.3430
4	140.9679	141.0725	-0.1046	-0.5572
5	160.5094	160.3611	0.14832	0.7898
6	181.2224	181.2084	0.0140	0.0745
7	203.1723	203.6041	-0.4318	-2.2993
8	227.793	227.6251	0.1679	0.8941
9	253.5504	253.4359	0.1145	0.6096
10	280.9801	281.2885	-0.3084	-1.6425
11	311.4664	311.5223	-0.0559	-0.2979
^a– Equation (24); ^b – the mean of 3 determinations; ^cW(exp) – W(cal)

The quartic equation (Eq. 24) selected for this study is based on the values of "s" and "R²." Table 4 displays the calculated values of “W” derived from the quartic equation. The calculated "W" values match the experimental "W" values. To back compute the solubility of ARP X₂_(calc), the W calculated values are inserted in Eq. (20). Table 4 records these data together with the % error. Table 4 indicates an excellent agreement between the anticipated solubilities from the SH approach Eq. (20) and the actual results, with the majority of the values falling within an 18% error range. Note that reporting the residuals as a percentage tends to overstate an error. It is possible to regard the outcome of X₂_exp = 0.003061 versus X₂_calc = 0.003015 as reasonably excellent.

Solubility predictions by regression of activity coefficient, (log g₂)/A versus d₁

Relationship between the activity coefficient, ideal MFS and MFS for a nonideal solution can be written as follows:

(25)

where the solute's rational activity coefficient in the solution is denoted by g₂. Eq. (25) can be expressed as follows:

(26)

where (16)

Eq. (26) may be written as

(27)

By passing 'W' and eliminating the necessity for d₂ in the calculations, (log g₂)/A can be directly regressed against d₁. Using Eq. (26) for each solvent blend, (log g₂)/A values were computed from the experimental data. In Table 5, these values are provided. Regressing this data against a polynomial in d₁ of the solvent mixture is feasible. Table 6 provides experimental (log ₂)/A values for the dioxane-water series. By regressing the experimental (log ₂)/A values for the dioxane–water series, following quadratic, cubic, and quartic equations were formed.

Quadratic equation :

(28)

n = 11 ; s = 0.7319 ; R² = 0.9767 ; F = 167.673 ; F (2, 8, 0.01) = 8.65

Cubic equation :

(29)

n = 11 ; s = 0.7114 ; R² = 0.9808 ; F = 119.194 ; F (3, 7, 0.01) = 8.45

Quartic equation :

(30)

n = 11; s = 0.4754: R² = 0.9926; F = 201.202; F (4, 6, 0.01) = 9.15

The quartic equation is chosen based on the residual plot and R² value. Figure 2 displays the scattergram of the quartic equation (Eq. 30) for ARP in the dioxane-water series. The information is presented in Table 5. Figure 2 illustrates the degree of randomness in the point scattering, supporting the applicability of Eq. (30).

Table 4. MFS of ARP in dioxane-water blend based on “W” at 25^oC*

Ratio	δ₁	A	W(exp)^c	W(cal)^a	X₂(exp)^c	X₂(cal)^b	Percent error
100:0	10.04	0.2537	94.00001	93.9763	0.0031	0.0030	1.5224
90:10	11.376	0.2575	107.5635	107.651	0.0012	0.0013	-10.3822
80:20	12.712	0.2577	123.5051	123.4407	0.0010	0.0009	10.8831
70:30	14.048	0.2586	140.9679	141.0725	0.0007	0.0007	-4.6786
60:40	15.384	0.2586	160.5094	160.3611	0.0005	0.0004	16.8632
50:50	16.72	0.2595	181.2224	181.2084	0.0002	0.0002	1.6688
40:60	18.056	0.2601	203.1723	203.6041	4.51E-05	7.54E-05	-67.2756
30:70	19.392	0.2601	227.793	227.6251	2.8E-05	2.29E-05	18.3418
20:80	20.728	0.2602	253.5504	253.4359	7.55E-06	6.97E-06	7.6785
10:90	22.064	0.2603	280.9801	281.2885	2E-06	2.89E-06	-44.6795
00:100	23.4	0.2602	311.4664	311.5223	2.28E-06	2.46E-06	-7.8914
àΔH_f₌ 2983.0307 cal/mol, T_o* = 412 K, δ₂= 10.03 (cal/cm³)^1/2, X₂ⁱ = 0.3067, log X₂ⁱ = -0.5133, Molar volume = 355.124 cm³/mol; ^a^{, b}àaccording to Eqs. (24) and (20), respectively, c àmean of three determinations.

Table 5. Residuals found from quartic regression Eq. (26) based on log γ₂/A for ARP solubility in dioxane and water at 25^oC

SI. No.	log γ₂/A (exp)	log γ₂/A (cal)^a	Residuals^b	SD units
1	7.8640	7.8894	-0.0253	-0.0547
2	9.3489	9.1375	0.2114	0.3229
3	9.6473	9.7196	-0.0723	-0.1105
4	10.4731	10.1798	0.2933	0.4480
5	10.7111	10.8872	-0.1761	-0.2690
6	12.1761	12.0360	0.1401	0.2134
7	14.7371	13.6451	1.0920	1.6680
8	15.5263	15.5583	-0.0312	-0.0489
9	17.6115	17.4444	0.1670	0.2552
10	19.9223	18.7971	1.1252	1.7186
11	19.6898	18.9345	0.7548	1.1523
^aaccording to equation (30); ^blog γ₂/A(exp) – log γ₂/A(calc)

Figure 2. Scattergram obtained from quartic regression of Eq. (30) for ARP in dioxane-water solvent system

Table 6 reports the estimated values of (log g₂)/A from the quartic equation. The experimental results and the calculated (log g₂)/A values observed were comparable (Table 6). To compute the MFS X₂(calc) of ARP, the (log g₂)/A computed values are substituted in Eq. (26) using back calculation. These values are listed in Table 6 along with the error percentage. The values are shown in Figure 3. Table 6 makes it very evident that the percentage error is negligible. Most of the points can be superimposed. Analogous outcomes were also obtained using calculated "W" values. These findings support the idea that, in the case of a single drug, the volume fraction of the solvent combination and IS can be ignored when determining the solubility model while dealing with a single drug. Eq. (30) allows X₂ to be calculated directly on different solvent compositions without requiring an iterative procedure, which is required in the EHS method to estimate the two unknowns, X₂ and ϕ₁. The model also produced similar results for pimozide³⁵.

Figure 3. Experimental MFS (♦) in dioxane-water solvent systems with calculated MFS (■) using logγ₂/A of ARP at 25^oC

Table 6. Solubility of ARP on dioxane -water solvent system based on log γ₂/A at 25^oC

Ratio	δ₁	A	log γ₂/A (exp)^c	log γ₂/A (cal)^a	X₂(exp)^c	X₂(cal)^b	Percent error
100:0	10.04	0.2537	7.8640	7.8894	0.0030	0.0031	0.2186
90:10	11.37	0.2574	9.3489	9.1375	0.0012	0.0014	-12.8115
80:20	12.71	0.2577	9.6472	9.7195	0.0010	0.0001	7.8263
70:30	14.04	0.2586	10.4731	10.1797	0.0007	0.0007	-10.0704
60:40	15.38	0.2587	10.7110	10.8872	0.0005	0.0005	10.6539
50:50	16.72	0.2595	12.1761	12.0360	0.0002	0.0002	-8.7432
40:60	18.056	0.2600	14.7371	13.6451	4.52E-05	8.66E-05	-91.8365
30:70	19.392	0.2601	15.5263	15.5583	2.80E-05	2.75E-05	2.0285
20:80	20.728	0.2602	17.6115	17.4444	7.55E-06	8.84E-06	-17.0881
10:90	22.064	0.2602	19.9223	18.7971	2.00E-06	3.93E-06	-96.2616
0:100	23.4	0.2602	19.6897	18.9349	2.28E-06	3.62E-06	-58.6372
* à ΔH_f= 2983.0307 cal/mol, T_o= 412 K, δ₂ = 10.03 (cal/cm³)^1/2, X₂ⁱ = 0.3067, log X₂ⁱ = 0.51329, Molar volume = 355.124 cm³/mol; ^a àaccording to Eq. (30); ^b àaccording to Eq. (26); ^c àmean of three determinations

CONCLUSIONS:

The success of the EHS approach in predicting ARP solubility highlights its potential for broader applications in pharmaceutical formulation. By bypassing the need for complex calculations involving solute-solvent interactions, the EHS approach offers a more efficient and accurate method for predicting drug solubility in various solvent blends. Furthermore, the findings suggest that similar models could be applied to other drugs exhibiting similar solute-solvent interactions, potentially streamlining the drug development process, and reducing the need for extensive experimental trials. The case study of ARP solubility in dioxane-water mixtures showcases the effectiveness of the Extended Hildebrand Solubility approach in accurately predicting drug solubility. By leveraging advancements in computational modeling and regression analysis, pharmaceutical scientists can enhance their understanding of solubility behavior and expedite the drug development process. This study underscores the importance of adopting innovative approaches to address the challenges associated with drug solubility prediction in complex solvent systems

CONFLICT OF INTEREST:

The authors do not have any conflict of interest.

ACKNOWLEDGEMENTS:

The authors would like to express their gratitude to Dr. A P Basavarajappa, Principal, Bapuji Pharmacy College, for providing the facilities needed to conduct the research, and to Dr. C.V.S. Subrahmanyam, Emeritus Professor for sharing the concept of the project.

REFERENCES:

1. Nozomu S. Regular solution theory for nonlinear composition dependency of enantioselectivity by mixed micelle. J. Mol. Liq. 2022; 367: 120597. https://doi.org/10.1016/j.molliq.2022.120597

2. Yanmin S. Yu B. Peixia Z. Xiaolong Y. Zheng Z. Dan D. Han W. Wenju L. Equilibrium solubility of 6-propyl-2-thiouracil in nine pure solvents: Determination, correlation, Hansen solubility parameter and thermodynamic properties. J. Indian. Chem. Soc. 2023; 100: 100934

3. Hildebrand JH. Scott RL. Regular solutions. Inorg. Chem. 1963; 2: 431–432. https://doi.org/10.1021/ic50006a060

4. María MM. Darío A. Tinjacá. Jouyban A. Martínez F. William E., et al. Volumetric properties of {PEG 200 (or 300) (1) + water (2)} mixtures at several temperatures and correlation with the Jouyban–Acree model. Phys. Chem. Liq. 2017: 100- 109. https://doi.org/10.1080/00319104.2017.1303700

5. Kolar P. Shen JW. Tsuboi. Akio T. Takeshi I. Solvent selection for pharmaceuticals. Fluid. Ph. Equilibria. 2002; 194: 771-782. http://dx.doi.org/10.1016/S03783812(01)00716-6

6. Li A. Yalkowsky SH. Solubility of organic solutes in ethanol/water mixtures. J. Pharm. Sci. 1994;83: 1735-1740.

7. Smith PE. Mazo RM. On the theory of solute solubility in mixed solvents. J. Phys. Chem. B. 2008; 112: 7875-7884. https://doi.org/10.1021/jp712179w

8. Stephen WT. Ideal solution laws: Apparatus and experiment. J. Chemi. Edu. 1962; 39: 258 https://doi.org/10.1021/ed039p258

9. Estanislao S. Arturo A. Tuñón I. Fundamental Principles Governing Solvents Use. Hand book of solvents. 2nd Ed, 2014. Chap (2) PP.11-72. https://doi.org/10.1016/B978-1-895198-64-5.50004-0

10. Bustamante P. Escalera B. Enthalpy and entropy contributions to the solubility of sulphamethoxypyridazine in solvent mixtures showing two solubility maxima. J. Pharm. Pharmacol. 1995; 47: 550-555. https://doi.org/10.1111/j.2042-7158.1995.tb06712.x.

11. Rathi PB. Determination and evaluation of solubility parameter of satranidazole using dioxane-water system. Ind. J. Pharm. Sci. 2010; 72: 671-674. https://doi.org/10.4103/0250474x.78546

12. Martin AN. Swarbrick J. Cammarata. Physical Pharmacy 3rd Ed, Philadelphia, Lee and Febiger. 1983:371-374.

13. Hildebrand JH. An improvement in the theory of regular solutions. Proc. Natl. Acad. Sci. 1979;76: 6040-6041.

https://doi.org/10.1073/pnas.76.12.6040

14. James KC. Ng CT. Noyce PR. Solubilities of testosterone propionate and related esters in organic solvents. J. Pharm. Sci. 1976; 65: 656-659. https://doi.org/10.1002/jps.2600650506

15. Adjei A. Newburger J. Martin A. Extended Hildebrand approach: solubility of caffeine in dioxane-water mixtures. J. Pharm. Sci. 1980; 69: 659-661. https://doi.org/10.1002/jps.2600690613

16. Martin A. Wu PL. Adjei A. Mehdizadeh M. James KC. Metzler C. Extended Hildebrand solubility ap roach: testosterone and testosterone propionate in the binary solvents. J. Pharm. Sci. 1982; 71: 1334-1340. https://doi.org/10.1002/jps.2600711207

17. Martin A. Newburger J. Adjei A. Extended Hildebrand solubility approach: Solubility of theophylline in polar binary solvents. J. Pharm. Sci. 1980; 69: 487-491. https://doi.org/10.1002/jps.2600690503

18. Wu PL. Martin A. Extended Hildebrand solubility approach: p-hydroxy benzoic acid in mixtures of dioxane and water. J. Pharm. Sci. 1983; 72: 587-592. https://doi.org/10.1002/jps.2600720603

19. Subrahmanyam CVS. Sreenivasareddy M. Venkatarao J. Gundurao P. Irregular solution behaviour of paracetamol in binary mixtures. Int. J. Pharm. 78 1992 17-24. https://doi.org/10.1016/0378-5173(92)90350-B

20. Peggy C. David T. John M. Benjamin PP. David LK. Rufina GA. et al. Heat of fusion of polymer crystals by fast scanning calorimetry. Polymer. 2017; 126: 240-247. https://doi.org/10.1016/j.polymer.2017.08.042

21. Franz JL. Nicolai W. Joachim K. Dirk WS. On the Determination of the Enthalpy of Fusion of α-Crystalline Isotactic Polypropylene Using Differential Scanning Calorimetry, X-Ray Diffraction, and Fourier-Transform Infrared Spectroscopy: An Old Story Revisited. Adv. Eng. Mater. 2019; 22: 1900796. DOI: 10.1002/adem.201900796

22. Oseph WH. Emily AL. Michael LC. Andrew DF. Nicole LQ. Deborah MR. et al. Melting Point, Enthalpy of Fusion, and Heat Capacity Measurements of Several Polyfunctional, Industrially Important Compounds by Differential Scanning Calorimetry. Chem. Eng. Data. 2018; 63: 2500–2511. https://doi.org/10.1021/acs.jced.7b01026

23. Barton AFM. Handbook of Solubility Parameters and other cohesion parameters, 2nd Ed, CRC press, Florida (1991) PP.167-168.

24. Shwetha SKK. Sharda S. Thimmasetty J. Shashank NN. Jayesh SP. Introduction of new method for prediction of solubility parameter using aripiprazole as a model drug. Chem. Data Collect. 2023; 44: 100995. https://doi.org/10.1016/j.cdc.2023.100995

25. Khalil SA. Abdallah OA. Moustafa MA. Absorption of some barbiturates by gambusia fish and its correlation to solubility parameter. Can. J. Pharm. Sci. 1976; 11: 126-130.

26. Kenneth CJ. Solubility and related properties, Vol.28., Marcel Dekker Inc., New York (1986).

27. Beckett AM. Stenlake JB. Practical pharmaceutical chemistry, 4th Ed, CBS Publishers., New Delhi (1986) pp 10.

28. Reinaldo GS. Holguín RA. Cristancho MD. Delgado RD. Martínez F. Extended Hildebrand Solubility Approach applied to piroxicam in ethanol + water mixtures. J. Mol. Liq. 2013; 180: 34-38. https://doi.org/10.1016/j.molliq.2012.12.028

29. Martin A. Wu P L. Velasquez T. Extended Hildebrand solubility approach: sulfonamides in binary and ternary solvents. J. Pharm. Sci. 1985; 74: 277-282. https://doi.org/10.1002/jps.2600740311.

30. Jouyban-Gharamaleki A. York P. Hanna M. Clark BJ. Solubility prediction of salmeterol xinafoate in water--dioxane mixtures. Int. J. Pharm. 200; 216: 33-41. https://doi.org/10.1016/s0378-5173(00)00694-3

31. Martin A. Paruta AN. Adjei A. Extended Hildebrand Solubility Approach: methylxanthines in mixed solvents. J. Pharm. Sci. 1981; 70: 1115-1120. https://doi.org/10.1002/jps.2600701007

32. Martin A. Miralles MJ. Extended Hildebrand solubility approach. Solubility of tolbutamide, acetohexamide and sulpha-somidine in binary solvent mixtures. J. Pharm. Sci. 1982; 71: 439-442. https://doi.org/10.1002/jps.2600710416

33. Jagdale S. Nawale RB. Extended Hildebrand Solubility Approach: Prediction and Correlation of the Solubility of Itraconazole in Triacetin: Water Mixtures at 298.15°K. Turk. J. Pharm. Sci. 2020; 17: 228-234. https://doi.org/10.4274/tjps.galenos.2019.20438

34. Reillo A. Bustamante P. Escalera B. Jimenez MM. Selle E. Solubility parameter-based methods for predicting the solubility of sulfapyridine in solvent mixtures. Drug. Dev. Ind. Pharm. 1995; 21: 2073-2084. http://dx.doi.org/10.3109/03639049509065891

35. Thimmasetty J. Subrahmanyam CVS. Satheshbabu PR. Maulik MA. Viswanath BA. Solubility behavior of pimozide in polar and nonpolar solvents: Partial solubility parameters approach. J. Sol. Chem. 2008; 37: 1365-1378. https://doi.org/10.1007/s10953-008-9317-