Determination and Data Correlation of Solubility of Sofosbuvir Polymorphs in Ethyl Acetate + Toluene and Methyl tert-Butyl Ether Binary Solvents at the Temperature Range from 268.15 to 308.15 K

Abstract

A gravimetric method was used to experimentally determine the (solid + liquid) equilibrium of sofosbuvir of crystalline forms A and B in both ethyl acetate + toluene and methyl tert-butyl ether (MTBE) + toluene binary solvents systems at atmosphere pressure. Experiments were carried out at a temperature range of 268.15−308.15 K. Results show that the solubility of sofosbuvir increases with temperature, and the solubility of form B was higher than that of form A. The modified Apelblat model, the CNIBS/Redlich–Kister model, and the combined version of Jouyban–Acree model were employed to correlate the measured solubility data, respectively. Furthermore, an examination of the solid-state stability of the two polymorphs was conducted, finding that form A and form B exhibit good solid-state stability under high temperature, high humidity, and strong light exposure conditions.

1. Introduction

Sofosbuvir (C₂₂H₂₉FN₃O₉P, CAS No.: 1190307-88-0, presented in Figure 1) is a novel anti-HCV agent, offering a more satisfactory sustained virologic response rate than those that have ever been used in similar cases before [1,2]. It is indicated by our previous studies that ethyl acetate + toluene and MTBE + toluene binary systems can be used to crystallize sofosbuvir of form A or B [3]. However, there is no literature that has reported the solution thermodynamic data of sofosbuvir in these two systems. These are important for the optimization of the crystallization process, especially for the controlling of polymorphs, since the various solid forms generally present different physicochemical properties.

In this study, a gravimetric method [4,5,6] was used to acquire the solubility data of sofosbuvir of forms A and B in both ethyl acetate + toluene binary solvents with ethyl acetate mole fraction ranging from 0.32 to 0.53, and MTBE + toluene binary solvents with MTBE mole fraction ranging from 0.40 to 0.77, at atmosphere pressure, with a temperature range of 268.15−308.15 K. The experimental data were correlated using the modified Apelblat model, the CNIBS (combined nearly ideal binary solvent)/Redlich–Kister model, and the combined version of Jouyban–Acree model. A stability test according to International Conference for Harmonization Quality Guidelines was conducted as well, to disclose the solid-state stability of sofosbuvir polymorphs under the influence of a variety of environmental factors such as temperature, humidity, and light. This can provide experimental data for establishing a re-test period for active pharmaceutical ingredients or a shelf life for the drug product and recommended storage conditions.

2. Modeling

Correlation of the experimental solubility data with different models is helpful for further understanding the thermodynamic properties in the measurement range. With this consideration, three models were used in this study. The modified Apelblat equation describes the dependence of solubility on temperature, the CNIBS/ Redlich–Kister model describes the dependence of solubility on solvent composition, while the Jouyban–Acree model describes the dependence of solubility on both parameters.

2.1. Modified Apelblat Equation

The Apelblat equation, as a well-known equation used to correlate solubility, was originally proposed by Apelblat and Manzurola; in it, the temperature dependence of the mole fraction solubility in different solvents is described as follows [7,8,9,10,11]:

$$\ln x_A=A+\frac{B}{T}+C\ln T$$

(1)

where x_A is the mole fraction solubility and T represents the corresponding absolute temperature. A, B and C are semi-empirical parameters of the model.

2.2. CNIBS/Redlich–Kister Model

The governing equation of CNIBS/Redlich–Kister model [12,13] is defined as follows:

$$\ln x_A=x_B^0\ln {\left(x_A\right)}_B+x_C^0\ln {\left(x_A\right)}_C+x_B^0{x}_C^0{\displaystyle \sum }_{i=0}^N{S}_i{\left(x_B^0-x_C^0\right)}^i$$

(2)

where x_B⁰ and x_C⁰ are the mole fraction of each solvent in a binary solvent system, respectively; N is the number of ‘‘curve-fit” parameters and S_i is the model constant, which is two for a binary solvent system in this case; thus, x_C⁰ can be represented as (1−x_B⁰). Substituting x_C⁰, Equation (2) can be expressed as:

$$\begin{gathered} \ln x_A=(\ln {\left(x_A\right)}_B-\ln {\left(x_A\right)}_C+S_0-S_1+S_2)x_B^0 \\ +\left(-S_0+3S_1-5S_2\right){\left(x_B^0\right)}^2+\left(-2S_1+8S_2\right){\left(x_B^0\right)}^3 \\ +\left(-4S_2\right){\left(x_B^0\right)}^4+\ln {\left(x_A\right)}_C \end{gathered}$$

(3)

Changing constants in Equation (3) with a constant term A_i (A₀, A₁, A₂, A₃, A₄), the CNIBS/R–K model can be simplified as follows [14,15]:

$$\ln x_A=A_0+A_1{x}_B^0+A_2{(x_B^0)}^2+A_3{(x_B^0)}^3+A_4{(x_B^0)}^4$$

(4)

2.3. Jouyban–Acree Model

The original function of Jouyban–Acree model, which was obtained by modifying the CNIBS/R–K model, is shown below [16]:

$$\ln x_A=x_B^0\ln {\left(x_A\right)}_B+x_C^0\ln {\left(x_A\right)}_C+x_B^0{x}_C^0{\displaystyle \sum }_{i=0}^N\frac{J_i}{T}{\left(x_B^0-x_C^0\right)}^i$$

(5)

in which J_i is a constant of the function.

In order to enlarge the applicability of solution behavior of non-ideal systems, the modified Apelblat equation can be used to substitute ln(x_A)_i; then Equation (6) is obtained, as in the CNIBS/R–K model:

$$\begin{gathered} \ln x_A=a_B+\frac{b_B}{T}+c_C\ln T+\left(a_B-a_C\right)x_B^0+\left(b_1-b_2+J_0-J_1+J_2\right)\frac{x_B^0}{T} \\ +\left(3J_1-J_0-5J_2\right)\frac{{(x_B^0)}^2}{T}+\left(8J_2-2J_1\right)\frac{{(x_B^0)}^3}{T} \\ +\left(-4J_2\right)\frac{{(x_B^0)}^4}{T}+\left(c_B-c_C\right)x_B^0\ln T \end{gathered}$$

(6)

A final equation of the Jouyban–Acree model can be obtained by simplifying equation 6 as well, with a constant term B_i (containing B₀, B₁, B₂, B₃, B₄, B₅, B₆, B₇, B₈) [17]:

$$\begin{gathered} \ln x_A=B_0+\frac{B_1}{T}+B_2\ln T+B_3{x}_B^0+B_4\frac{x_B^0}{T}+B_5\frac{{(x_B^0)}^2}{T}+B_6\frac{{(x_B^0)}^3}{T} \\ +B_7\frac{{(x_B^0)}^4}{T}+B_8{x}_B^0\ln T \end{gathered}$$

(7)

3. Experimental Section

3.1. Materials.

Sofosbuvir of forms A and B was prepared and identified using the methods published in previous studies [3]. Ethyl acetate, MTBE and toluene were of analytical grade and used without further purification. Detailed information on the above-mentioned materials is listed in

Table 1. Sources and purity of the materials.

Chemical Name	Source	Purification Method	Mass Fraction Purity	Analysis Method
Sofosbuvir (form A)	synthesis	crystallization	>0.99	HPLC a
Sofosbuvir (form B)	synthesis	crystallization	>0.99	HPLC
Ethyl acetate	Sinopharm	none	0.995	GC b
Toluene	Sinopharm	none	0.995	GC
MTBE	Sinopharm	none	0.995	GC

^a High performance liquid chromatography. ^b Gas chromatography.

.

3.2. Characterization of Sofosbuvir Polymorphs

Powder X-ray diffraction (PXRD) was used to identify the crystalline form of sofosbuvir, and the patterns were recorded using a Rigaku Ultima IV diffractometer with Cu Kα radiation (40 kV, 40 mA) scanned at 20°/min over an angular range of 5–45° of 2θ. The patterns are shown in Figure 2, and the main PXRD data are listed in Table S1, Supplementary Information.

3.3. Stability Test

3.3.1. Temperature

The two polymorphs of sofosbuvir were kept for 10 days at a temperature of 60 ± 2 °C and sampled on the 5th and 10th day, respectively. PXRD analysis shows no obvious changes, which means that the solid-state stability of the two polymorphs is good at high temperature conditions. The patterns are given in Figures S1 and S2, Supplementary Information.

3.3.2. Humidity

The two polymorphs of sofosbuvir were kept for 10 days at a humidity of RH 92.5 ± 5% and sampled on the 5th and 10th day, respectively. PXRD analysis shows no obvious changes, which means that the solid-state stability of the two polymorphs is good at high humidity conditions. The patterns are given in Figures S3 and S4, Supplementary Information.

3.3.3. Light

The two polymorphs of sofosbuvir were kept for 10 days under a light illumination of 4500 ± 500 lux and sampled on the 5th and 10th day, respectively. PXRD analysis shows no obvious changes, which means that the solid-state stability of the two polymorphs is good at strong light exposure conditions. The patterns are given in Figures S5 and S6, Supplementary Information.

3.4. Solubility Measurement

Sufficient amounts of sofosbuvir polymorphs were added to 25 ml solvent in a 50 ml jacket glass vessel with a magnetic stir bar and a thermometer, to form a saturated solution. A thermostatic bath was employed to control the temperature with an accuracy of ±0.1 K. The solution was agitated for 2 h to ensure that the system had reached the state of solid-liquid equilibrium. Then the stirring was stopped and the solution was kept stationary for 10 min. Subsequently, the upper clear solution (about 5 mL) was transferred into a small beaker using a membrane filter (0.45 μm). The beaker was dried in a vacuum oven at 333.15 K for about 48 h to make sure that the weight of the beakers became constant. Both of the polymorphs were proved by PXRD to be maintained over the temperature range. Measurements were conducted in triplicate at each temperature to minimize the relative deviation. All of the masses were measured using an analytical balance (Mettler Toledo XS105, Switzerland) with an accuracy of ±0.01 mg.

The solubility of the sofosbuvir polymorphs, described in mole fraction x in different systems, was calculated by the following equation [18]:

$$x=\frac{\frac{m}{M}}{\frac{m}{M}+\sum \left(\frac{m_s}{M_s}\right)}$$

(8)

in which m and m_s represent the mass of sofosbuvir and the solvent, respectively; M and M_s are the molecular mass of sofosbuvir and the solvent, respectively.

4. Results and Discussion

4.1. Solubility Data in Binary Solvents

Experimental data on solubility measurement are presented in Tables S2−S5, Supplementary Information, and graphically illustrated in Figure 3; Figure 4. It is obvious that the solubility of sofosbuvir increases with the temperature and the mole fraction of ethyl acetate or MTBE. Solubility of form B is higher than that of form A under all of the experimental conditions, which means that form A is more stable than form B at the temperature range of measurement. The relative thermodynamic stability is in line with our previous research [3].

4.2. Data Correlation

The Apelblat model, the CNIBS/Redlich-Kister model, and the Jouyban–Acree model were used to correlate the solubility data, based on the average relative deviation (ARD) defined by Equation (9), which was used to compare among three different models [19].

$$ARD\%=\frac{100}{N}{\displaystyle \sum }_{i=1}^N\left|\frac{x_{A,i}-x_{A,i}^{cal}}{x_{A,i}}\right|$$

(9)

in which N is the number of the experimental measurement, x_A,i is each measured solubility and x_A,i^cal is each calculated value. The 1stOpt program was applied to calculate the data using Equations (1), (4), and (7).

Parameters of data correlation and values of ARD% for each individual modeling system are listed in

Table 2. Fitting parameters of modified Apelblat model for ethyl acetate + toluene system.

xB0	A	B	C	ARD%
Form A
0.32	−19.11	−1655.76	4.55	2.13
0.36	55.30	−5021.69	−6.51	1.58
0.40	−70.83	416.91	12.45	2.41
0.46	70.70	−5406.75	−8.91	0.90
0.53	−101.85	2463.61	16.78	0.88
Form B
0.32	−118.97	3120.84	19.29	1.07
0.36	−187.47	5626.11	29.90	0.62
0.40	−73.51	508.07	12.96	1.10
0.46	−36.90	−873.12	7.39	1.31
0.53	30.75	−4517.73	−2.22	1.86

,

Table 3. Fitting parameters of modified Apelblat model for MTBE + toluene system.

xB0	A	B	C	ARD%
Form A
0.40	−48.64	794.58	8.30	0.55
0.46	1.45	−1661.09	1.00	1.24
0.53	76.70	−5256.26	−10.05	0.73
0.63	−43.24	−54.52	7.98	1.26
0.77	−47.35	302.27	8.53	1.43
Form B
0.40	−27.26	−937.26	5.66	1.46
0.46	97.35	−6501.56	−12.89	0.72
0.53	−60.70	323.62	10.85	1.08
0.63	1.13	−2362.87	1.61	1.18
0.77	−97.89	1952.94	16.48	1.65

,

Table 4. Parameters of CNIBS/R–K model for ethyl acetate + toluene system.

T/K	A0	A1	A2	A3	A4	ARD%
Form A
268.15	−44.17	469.34	−1833.07	3119.79	−1938.72	0.31
273.15	−19.22	221.48	−922.75	1664.43	−1082.88	0.42
278.15	−3.58	59.73	−298.58	616.49	−437.45	0.06
283.15	−64.11	669.20	−2567.66	4325.46	−2682.28	0.06
288.15	−38.48	415.32	−1626.87	2795.33	−1762.46	0.06
293.15	−21.05	226.97	−879.77	1515.09	−960.63	0.01
298.15	−66.28	682.86	−2568.42	4252.38	-2600.93	0.05
303.15	−42.04	436.51	−1636.54	2708.64	-1656.26	3.35 × 10−3
308.15	24.61	−221.89	775.88	-1167.56	648.80	5.15 × 10−3
Form B
268.15	−20.13	205.25	−772.52	1297.59	−803.01	5.14×10-3
273.15	−10.78	111.46	−419.60	714.80	−444.44	3.83×10−3
278.15	12.13	−133.23	543.43	−933.40	593.65	0.15
283.15	−20.09	193.32	−676.07	1065.87	−619.34	1.29×10−3
288.15	53.46	−550.38	2113.54	−3523.38	2175.54	0.46
293.15	10.07	−138.44	669.97	−1302.14	909.16	0.46
298.15	17.10	−210.49	948.41	−1771.84	1201.58	0.88
303.15	−1.99	−33.95	349.46	−879.34	708.39	0.66
308.15	12.11	−181.10	921.67	−1851.30	1318.01	0.43

,

Table 5. Parameters of CNIBS/R–K model for MTBE + toluene system.

T/K	A0	A1	A2	A3	A4	ARD%
Form A
268.15	2.93	−14.50	31.95	−25.98	7.38	3.33 × 10−5
273.15	−2.85	24.15	−62.23	76.30	−34.29	1.67 × 10−4
278.15	−5.23	40.73	−103.47	122.26	−53.62	4.93 × 10−4
283.15	13.09	−95.73	270.85	−323.18	140.77	1.73 × 10−3
288.15	0.51	−7.34	44.02	−68.49	35.06	1.30 × 10-4
293.15	14.54	−111.81	331.94	−413.21	186.39	2.04 × 10−3
298.15	36.57	−272.47	763.73	−917.64	402.86	0.03
303.15	17.44	−130.78	378.15	−458.88	201.78	3.52×10−3
308.15	14.07	−107.14	318.83	−393.48	175.10	6.76×10−3
Form B
268.15	7.40	-53.96	155.74	−186.41	80.93	2.03 × 10−3
273.15	5.30	-36.30	107.39	−129.93	56.98	1.16 × 10−3
278.15	−1.11	9.07	−7.47	−3.26	5.59	1.57 × 10−6
283.15	−6.73	52.11	−125.01	136.72	−55.91	1.22 × 10−4
288.15	−8.90	63.65	−142.97	144.93	−54.73	1.45 × 10−3
293.15	−29.22	214.75	−553.63	631.87	−267.26	0.02
298.15	−18.94	144.72	−375.83	435.24	−187.22	2.16 × 10−3
303.15	−7.52	59.50	−139.44	150.07	−60.82	1.45 × 10−4
308.15	0.23	0.16	30.20	−60.21	34.51	9.61 × 10−3

,

Table 6. Parameters of modified Jouyban–Acree model for ethyl acetate + toluene system.

Parameters	Form A	Form B
B0	−54.84	78.01
B1	−9575.17	1486.82
B2	482.53	−525.11
B3	−254.27	123.06
B4	27916.88	1322.24
B5	−71,896.69	8370.77
B6	150,328.07	−93,564.23
B7	−108,692.03	102,694.79
B8	36.72	−14.50
ARD%	2.96	4.06

and

Table 7. Parameters of modified Jouyban–Acree model for MTBE + toluene system.

Parameters	Form A	Form B
B0	−9.03	−32.63
B1	−1580.68	−7699.82
B2	107.52	292.16
B3	−69.37	−142.54
B4	−8506.99	21944.77
B5	34761.70	−37826.14
B6	−44,739.66	39619.22
B7	20376.34	−15596.72
B8	11.05	21.83
ARD%	3.51	1.82

, with which the calculated solubility values were obtained and listed in Tables S2–S5, Supplementary Information. It was observed that experimental solubility data are satisfactory, fitting with the calculated solubility values. Values of ARD% calculated by the modified Apelblat model, the CNIBS/R–K model, and the Jouyban–Acree model are less than 2.41%, 0.88%, and 4.06%, respectively, indicating that the three models all correlate well in this system. After comparing calculated stabilities with corresponding experimental ones, the CNIBS/R–K model stood out to be more suitable with a higher accuracy than the other two models.

5. Conclusions

In this paper, the solubility data of sofosbuvir of forms A and B in ethyl acetate + toluene and MTBE + toluene binary solvent systems were measured at atmosphere pressure with T = 268.15−308.15 K by a gravimetric method. Three models were adopted to correlate the experimental solubility data and all of them manifested satisfactory consistency, especially for CNIBS/R–K, with a higher accuracy. It was found that the solubility of sofosbuvir increases with temperature and the mole fraction of ethyl acetate or MTBE. Solubility of form B was higher than that of form A under all of the experimental conditions, which means form A was more stable than form B in the measurement temperature range. A stability test was conducted as well, finding that the two polymorphs show good solid-state stability under high temperature, high humidity, and strong light exposure conditions. All of the experimental data may provide valuable guidance for the crystallization and purification process of sofosbuvir.