Abraham Model Descriptors for Vitamin K4: Prediction of Solution, Biological and Thermodynamic Properties

Abstract

Spectrophotometric measurements were used to determine the mole fraction solubilities of vitamin K4 dissolved in cyclohexane, methylcyclohexane, 1-heptanol, 2-butanol, 2-pentanol, 2-methyl-1-butanol, 4-methyl-2-pentanol, and cyclopentanol at 298.15 K. Results from our experimental measurements, combined with the published solubility data, are used to calculate the solute descriptors of the vitamin K4 solute. The calculated solute descriptors describe the observed solubility data to within an overall standard deviation of 0.110 log units. The calculated solute descriptors were also used to estimate the several blood-to-rat tissue partition coefficients of vitamin K4, as well as the equilibrium vapor pressure above the solid vitamin at 298 K, and the vitamin’s enthalpy of solvation in both water and in 1,4-dioxane organic mono-solvent.

1. Introduction

The pharmaceutical industry faces many challenges in its efforts to bring potential drug candidates successfully through the drug discovery process. Only a small percentage of drug candidates make their way to the market for human consumption. Candidates often fail in the later stages of the discovery process because of poor aqueous solubility and slow dissolution kinetics, which lead to low drug concentration in the gastrointestinal tract and in blood circulation. Low bioavailability adversely affects drug efficacy because higher dosages are needed to provide a sufficient quantity of drug at the target site in order to achieve the desired therapeutic effect. Numerous methods have been suggested to overcome low aqueous solubility, including pH manipulation, addition of organic solvents and complexing agents, nanosuspension modes of delivery, co-crystal formation, and hydrotrope addition. Determining which approach is best for a specific drug molecule is both time-consuming and expensive in terms of employee labor and chemical resources. Several computation methods are available for estimating the solubility of drug candidates [1,2,3,4,5,6,7,8,9] and to assist researchers in selecting an appropriate organic solvent and mixture composition if co-solvency is needed to enhance a low aqueous solubility [4,6,7,9,10,11,12,13].

Our recent efforts pertaining to solubility have focused on experimental measurements for crystalline nonelectrolyte solutes dissolved in select organic mono-solvents [14,15,16,17] and in binary aqueous-organic solvent mixtures [18,19,20,21,22]. The measured solubility data has been used in the calculation of the Abraham model solute descriptors [14,15,16,17] and in developing the Abraham model correlations for predicting the solubilities of pharmaceutical compounds in organic solvents used in recrystallization purifications and in co-solvency solubility enhancements [23,24,25,26,27]. Transcutol was one the recent organic solvents for which the predictive Abraham model expressions were reported [23]. We note that the Abraham model enables estimation of many other important pharmaceutical properties besides solubility. Expressions have been reported for human skin permeations from aqueous solutions [28,29], human and animal air-to-blood partition coefficients [30], air-to-lung and blood-to-lung partition coefficients [31], air-to-muscle and blood-to-muscle distribution coefficients [32], Draize rabbit eye test compatibility and eye irritation thresholds in humans [33], human intestinal absorption of neutral molecules and ionic species [34,35]; air-to-fat and blood-to-fat distribution coefficients of drugs and volatile organic compounds [36], in vivo blood-to-rat brain distribution coefficients [37], in vitro air-to-rat/human brain partition coefficients of volatile organic compounds [38], solute permeabilities from select parallel artificial membrane permeability assay (PAMPA) models [39,40], and water-to-muscle protein partition coefficients [41,42]. The Abraham model correlations have also been used to assist researchers identify organic solvents that can mimic blood [43] and fatty tissue [44] for extraction and leaching studies to test the safety of medical devices that come into direct contact with a patient’s body fluids and tissues.

Prediction of each of the fore-mentioned properties using published Abraham model expressions requires a priori knowledge of the descriptor values of the desired solute molecule. Experimental-based solute descriptors are currently available for over 8000 different organic compounds [45]. Easy-to-use software programs [45,46,47] provide a convenient means to estimate descriptor values for those compounds whose experimental-based values have not been determined. The software programs estimate the Abraham solute descriptors from the molecule’s canonical SMILES code. Our experience in using the internet software programs is that the programs provide reasonably good estimations of the Abraham solute descriptors for molecules containing only a few functional groups. Estimated values do differ rather significantly from experimental-based descriptor values as the number of functional groups increase. The programs often fail to properly account for intramolecular hydrogen-bond in that the A and B solute descriptors are often overestimated. The predictive methods can be no better than the data sets used to train the models. Our comments are not intended to criticize the software programs, but rather to suggest that the best way to improve the predictive capabilities is to increase the chemical diversity of the molecules within the training data sets. We have recently reported experiment-based solute descriptors for four molecules that exhibit intramolecular hydrogen-bond formation [48,49,50], and for one molecule that contains the N-hydroxyl (>N-OH) functional group [15].

In the current communication we continue our efforts to provide the scientific community with experiment-based solute descriptors for additional organic compounds. The compound that we selected to study is vitamin K4 (2-methyl-1,4-napthodiol diacetate; C₁₅H₁₄O₄; Chemical Abstracts Registry Number 573-20-6) which a synthetic hydrophylic menadione compound that is clinically used in the treatment of blood clotting disorders. The chemical compound is also reported to exhibit anticancer activity in human prostate carcinoma PC-3 cells [51], and to both inhibit proliferation and induce apoptosis of U20S osteosarcoma cells [52]. The recently published mole fraction solubility data of Lu and coworkers [53] for vitamin K4 in methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 2-methyl-1-propanol, 1-pentanol, 3-methyl-1-butanol, methyl acetate, ethyl acetate, propyl acetate, butyl acetate, and pentyl acetate provides 13 experimental values for use in the solute descriptor calculations. The published solubility data in cyclohexanol was excluded from the calculations because there is no Abraham model correlation for this organic mono-solvent. We augmented the published data by performing additional solubility measurements in cyclohexane, methylcyclohexane, 1-heptanol, 2-butanol, 2-pentanol, 2-methyl-1-butanol, 4-methyl-2-pentanol, and cyclopentanol at 298.15 K. Cyclohexane and methylcyclohexane were specifically selected because of their non-polar nature and inability to engage in hydrogen-bond formation. Lu and coworkers did not perform solubility measurements in any saturated hydrocarbon solvents. The alcohol solvents were selected for their hydrogen-bonding ability and for the various placements of the -OH and methyl-functional groups within the molecule.

2. Materials and Methods

Vitamin K4 was purchased from a commercial source (TCI America, Portland, OR, USA, 0.98 mass fraction) and recrystallized two times from anhydrous methanol prior to use. The purity of the recrystallized sample of vitamin K4 was 0.997 mass fraction as determined by gas-chromatographic analyses (with thermal conductivity detection). The eight organic solvents were purchased from commercial sources as follows: cyclohexane (Sigma-Aldrich Chemical Company, St. Louis, MO, USA, 0.995 mass fraction, anhydrous), methylcyclohexane (Aldrich Chemical Company, Milwaukee, WI, USA, 0.99+ mass fraction, anhydrous), 2-butanol (Aldrich Chemical Company, 0.995 mass fraction, anhydrous), 2-pentanol (Thermo Scientific, Ward Hill, MA, USA, 0.99 mass fraction), 2-methyl-1-butanol (Sigma-Aldrich Chemical Company, 0.99 mass fraction), 4-methyl-2-pentanol (Acros Organics, Morris Plains, NJ, USA, 0.99+ mass fraction), 1-heptanol (Alfa Aesar, Heysham, UK, 0.99 mass fraction), and cyclopentanol (Sigma-Aldrich Chemical Company, 0.995 mass fraction, anhydrous). All eight solvents were stored over activated molecular sieves shortly before use to remove trace moisture. Gas chromatographic analysis (with thermal conductivity detection) indicated the organic solvent purities to be at least 0.997 mass fraction.

The solubility of vitamin K4 in the eight organic solvents were measured utilizing a static method of equilibration. Mass fraction concentrations in the saturated solutions were calculated from spectroscopic absorbance measurements. The experimental methodology employed in the current communication has been described in earlier publications [54,55] and to conserve journal space we provide only an abbreviated version. Aliquots of the clear saturated solutions were transferred using a heated glass syringe into weighed volumetric flasks after the samples had equilibrated in a constant temperature water bath at 298.15 ± 0.05 K for at least three days with periodic agitation. The transferred aliquot was weighed on a Mettler Toledo ME104E electronic analytical balance (Mettler Toledo, Columbus, OH, USA) and then diluted quantitatively with 2-propanol. Absorbances of the diluted solutions were recorded at an analysis wavelength of 301 nm on a Milton Roy Spectronic 1000 Plus single-beam spectrophotometer (Milton Roy Company, Rochester, NY, USA). The concentration of each diluted solution was computed from a Beer-Lambert law absorbance versus concentration calibration curve generated from the measured absorbances of nine carefully prepared standard solutions of known vitamin K4 concentration. The calculated molar absorptivity, ε ≈ 6450/(mol⁻¹ cm), was constant over the concentration range of 7.83 × 10⁻⁵ Molar to 2.61 × 10⁻⁴ Molar used in the molar absorptivity determination. Molar concentrations determined from the absorbance measurements were first converted to mass fraction solubilities and then to mole fraction solubilities using the mass of the sample analyzed, molar masses of vitamin K4 and the respective organic mono-solvents, volume of the volumetric flasks, and any dilutions needed to get the measured absorbances to fall on the calibration curve. No evidence of solvent formation or solid phase transformation was observed. Melting point temperatures of the solid samples recovered from the saturated solutions within experimental uncertainty of the melting point temperature of the purified, recrystallized sample.

3. Results and Discussion

The experimental mole fraction solubilities, x_S,organic, of vitamin K4 dissolved in eight different organic mono-solvents are reported in

Table 1. Experimental mole fraction solubilities of vitamin K4, x_S,organic, in select organic mono-solvents at 298.15 K.

Organic Mono-Solvent	xS,organic
Cyclohexane	0.001713
Methylcyclohexane	0.002215
1-Heptanol	0.006035
2-Butanol	0.005822
2-Pentanol	0.006857
2-Methyl-1-butanol	0.004897
4-Methyl-2-pentanol	0.005729
Cyclopentanol	0.008789

. The tabulated numerical values represent the average of between four and eight independent determinations and were reproducible to within +2.5%. We were not able to find published solubility data in the chemical literature to compare our measured x_S,organic against. In fact, the only published solubility data that we found in our search of the published literature was the mole fraction solubilities of vitamin K4 contained in the paper by Lu and coworkers [53]. Experimental values reported in the current study significantly increased the available solubility data for vitamin K4.

The calculation of the Abraham model solute descriptors is relatively straightforward, provided that one has a sufficient experimental partition coefficient and molar solubility data to construct the needed mathematical equations. To aid in these calculations, the Abraham model correlations were developed for more than 130 different water-to-organic solvent:

log P or log (C_S,organic/C_S,water) = c_p + e_p × E + s_p × S + a_p × A + b_p × B + v_p × V

(1)

and more than 130 different gas-to-organic solvent solute transfer process:

log K or log (C_S,organic/C_S,gas) = c_k + e_k × E + s_k × S + a_k × A + b_k × B + l_k × L

(2)

Each solute transfer process describes either the logarithm of a water-to-organic solvent partition coefficient, log P, the logarithm of a gas-to-organic solvent partition coefficient, log K, or the logarithm of molar solubility ratios, log (C_S,organic/C_S,water) and log (C_S,organic/C_S,gas), in terms of the product of solute descriptor values (E, S, A, B, V and L) multiplied by the numerical values of complementary solvent/process equation coefficients (c_p, e_p, s_p, a_p, b_p, v_p, c_k, e_k, s_k, a_k, b_k and l_k). Solute descriptors are described as follows: V represents the characteristic McGowan molar volume, L is the logarithm of the solute’s measured gas-to-hexadecane partition coefficient determined at 298 K, E is the solute excess molar refractivity relative to that of a linear alkane of comparable molecular volume, A and B denote the solute’s overall hydrogen bond acidity and basicity, respectively, and S refers to the solute’s dipolarity/polarizability character.

Each product represents a specific type of solute–solvent interaction that is believed to govern the solute transfer process. The sign and magnitude of each product determines whether or not the molecular interaction favors or hinders solute transfer into the organic solvent. For example, in the case of the two hydrogen-bonding terms a positive numerical value of a_k × A and/or b_k × B corresponds to an increase in the given partition coefficient or increase in the solute’s molar concentration in the organic phase, C_S,organic. Conversely, a negative value of either term results in smaller solute partition coefficients or greater relative solute molar solubility in the aqueous, C_S,water, and relative molar gas phase concentration, C_S,gas. Partition coefficients and molar solubility ratios are similarly affected by excess polarizability portion of solute-solvent interactions resulting from the n- and π-electrons, e × E, the dipolarity/polarizability term, s × S, and the cavity formation terms, v_p × V and l_k × L, in the two Abraham model expressions.

When interpreting how the various types of molecular interactions affect solute transfer remember that the coefficients represent the difference in the properties of the destination phase minus those in the origination phase. This is the reason why many of the a_p coefficients and all of the b_p coefficients are negative for the water-to-organic solvent transfer processes listed in Table 2. Even for those organic solvents that can engage in H-bond formation water still possess more H-bond donor character than the organic solvent. Hydrogen-bonding solutes prefer to remain in the aqueous phase if not for the positive contributions from the v_p × V cavity formation term. These are considerations that one considers when designing a biphasic aqueous-organic extraction system to remove organic solutes from aqueous solutions.

The equation coefficients for the various Abraham model solute transfer processes used in the current study are tabulated in the last seven columns of Table 2. Each individual transfer process is designated as either “wet”, “dry” or “both”, depending on whether the organic solvent is in direct contact with the aqueous phase as would be the case for a direct, practical partitioning system. The equation coefficients for the “wet” water-to-organic solvent solute transfer process pertains to the solute partitioning between an aqueous phase saturated with the organic solvent and an organic phase saturated with water. In the case of the “dry” solute transfer processes, the aqueous and the organic phase are not in direct contact with one another, and molar solubility ratios are used to quantify the extent of solute transfer. The “dry” solute transfer correlations can be used to predict the solubility of the solute in additional organic solvents as might be needed in selecting an organic solvent to use in a chemical synthesis or for purifying synthesized compound by recrystallization. For solvents that are almost completely immiscible with water, such as cyclohexane and methylcyclohexane, the designation of “both” is used in Table 2 because the coefficients can be used to describe log C_S,organic/C_S,water, as well as the logarithm of the practical water-to-organic solvent partition coefficient. The presence of small amounts of water in the organic solvent, and small amounts of organic solvent in water, does not affect the solubilizing properties of either water or the organic mono-solvent. Interested readers can find a more detailed discussion of the Abraham model in several informative review articles and book chapters [56,57,58,59,60,61].

Table 2. Abraham Model Equation Coefficients for Various Water-to-Organic Solvent, Equation (1), and Gas-to-Organic Solvent, Equation (2), Solute Transfer Process ^a.

Solvent	c	e	s	a	b	l	v
Equation (1) Coefficients
1-Octanol (wet)	0.088	0.562	−1.054	0.034	−3.460	0.000	3.814
Cyclohexane (both)	0.159	0.784	−1.678	−3.740	−4.929	0.000	4.577
Methylcyclohexane (both)	0.246	0.782	−1.982	−3.517	−4.293	0.000	4.528
Methanol (dry)	0.276	0.334	−0.714	0.243	−3.320	0.000	3.549
Ethanol (dry)	0.222	0.471	−1.035	0.326	−3.596	0.000	3.857
1-Propanol (dry)	0.139	0.405	−1.029	0.247	−3.767	0.000	3.986
1-Butanol (dry)	0.165	0.401	−1.011	0.056	−3.958	0.000	4.044
1-Pentanol (dry)	0.150	0.536	−1.229	0.141	−3.864	0.000	4.077
1-Heptanol (dry)	0.035	0.398	−1.063	0.002	−4.342	0.000	4.317
2-Propanol (dry)	0.099	0.344	−1.049	0.406	−3.827	0.000	4.033
2-Butanol (dry)	0.127	0.253	−0.976	0.158	−3.882	0.000	4.114
2-Methyl-1-propanol (dry)	0.188	0.354	−1.127	0.016	−3.568	0.000	3.986
3-Methyl-1-butanol (dry)	0.073	0.360	−1.273	0.090	−3.770	0.000	4.273
2-Pentanol (dry)	0.115	0.455	−1.331	0.206	−3.745	0.000	4.201
2-Methyl-1-butanol (dry)	0.143	0.388	−1.173	−0.024	−3.817	0.000	4.129
4-Methyl-2-pentanol (dry)	0.096	0.301	−1.100	0.039	−4.081	0.000	4.242
Cyclopentanol (dry)	0.332	0.522	−1.034	−0.106	−3.756	0.000	3.892
Methyl acetate (dry)	0.351	0.223	−0.150	−1.035	−4.527	0.000	3.972
Ethyl acetate (dry)	0.328	0.314	−0.348	−0.847	−4.899	0.000	4.142
Propyl acetate (dry)	0.362	0.280	−0.390	−0.975	−4.928	0.000	4.183
Butyl acetate (dry)	0.289	0.336	−0.501	−0.913	−4.964	0.000	4.262
Pentyl acetate (dry)	0.182	0.261	−0.474	−1.017	−4.952	0.000	4.388
Gas-to-water	−0.994	0.577	2.549	3.813	4.841	0.000	−0.869
Equation (2) Coefficients
1-Octanol (wet)	−0.198	0.002	0.709	3.519	1.429	0.858	0.000
Cyclohexane (both)	0.163	−0.110	0.000	0.000	0.000	1.013	0.000
Methylcyclohexane (both)	0.318	−0.215	0.000	0.000	0.000	1.012	0.000
Methanol (dry)	−0.039	−0.338	1.317	3.826	1.396	0.773	0.000
Ethanol (dry)	0.017	−0.232	0.867	3.894	1.192	0.846	0.000
1-Propanol (dry)	−0.042	−0.246	0.749	3.888	1.076	0.874	0.000
1-Butanol (dry)	−0.004	−0.285	0.768	3.705	0.879	0.890	0.000
1-Pentanol (dry)	−0.002	−0.161	0.535	3.778	0.960	0.900	0.000
1-Heptanol (dry)	−0.056	−0.216	0.554	3.596	0.803	0.933	0.000
2-Propanol (dry)	−0.048	−0.324	0.713	4.036	1.055	0.884	0.000
2-Butanol (dry)	−0.034	−0.387	0.719	3.736	1.088	0.905	0.000
2-Methyl-1-propanol (dry)	−0.003	−0.357	0.699	3.595	1.247	0.881	0.000
3-Methyl-1-butanol (dry)	−0.052	−0.430	0.628	3.661	0.932	0.937	0.000
2-Pentanol (dry)	−0.031	−0.325	0.496	4.792	1.024	0.934	0.000
2-Methyl-1-butanol (dry)	−0.055	−0.348	0.601	3.565	0.996	0.925	0.000
4-Methyl-2-pentanol (dry)	−0.013	−0.606	0.687	3.622	0.436	0.985	0.000
Cyclopentanol (dry)	−0.151	−0.314	0.693	3.549	0.914	0.956	0.000
Methyl acetate (dry)	0.134	−0.477	1.749	2.678	0.000	0.876	0.000
Ethyl acetate (dry)	0.171	−0.403	1.428	2.726	0.000	0.914	0.000
Propyl acetate (dry)	0.246	−0.346	1.318	2.537	0.000	0.916	0.000
Butyl acetate (dry)	0.154	−0.439	1.223	2.586	0.000	0.953	0.000
Pentyl acetate (dry)	0.154	−0.424	1.172	2.506	0.000	0.962	0.000
Gas-to-water	−1.271	0.822	2.743	3.904	4.814	−0.213	0.000

^a Equation coefficients for additional organic mono-solvents can be found in cited reference [49]. Coefficients for ionic liquid solvents can be found in a compilation by Jiang and coworkers [62].

Table 2. Abraham Model Equation Coefficients for Various Water-to-Organic Solvent, Equation (1), and Gas-to-Organic Solvent, Equation (2), Solute Transfer Process ^a.

Solvent	c	e	s	a	b	l	v
Equation (1) Coefficients
1-Octanol (wet)	0.088	0.562	−1.054	0.034	−3.460	0.000	3.814
Cyclohexane (both)	0.159	0.784	−1.678	−3.740	−4.929	0.000	4.577
Methylcyclohexane (both)	0.246	0.782	−1.982	−3.517	−4.293	0.000	4.528
Methanol (dry)	0.276	0.334	−0.714	0.243	−3.320	0.000	3.549
Ethanol (dry)	0.222	0.471	−1.035	0.326	−3.596	0.000	3.857
1-Propanol (dry)	0.139	0.405	−1.029	0.247	−3.767	0.000	3.986
1-Butanol (dry)	0.165	0.401	−1.011	0.056	−3.958	0.000	4.044
1-Pentanol (dry)	0.150	0.536	−1.229	0.141	−3.864	0.000	4.077
1-Heptanol (dry)	0.035	0.398	−1.063	0.002	−4.342	0.000	4.317
2-Propanol (dry)	0.099	0.344	−1.049	0.406	−3.827	0.000	4.033
2-Butanol (dry)	0.127	0.253	−0.976	0.158	−3.882	0.000	4.114
2-Methyl-1-propanol (dry)	0.188	0.354	−1.127	0.016	−3.568	0.000	3.986
3-Methyl-1-butanol (dry)	0.073	0.360	−1.273	0.090	−3.770	0.000	4.273
2-Pentanol (dry)	0.115	0.455	−1.331	0.206	−3.745	0.000	4.201
2-Methyl-1-butanol (dry)	0.143	0.388	−1.173	−0.024	−3.817	0.000	4.129
4-Methyl-2-pentanol (dry)	0.096	0.301	−1.100	0.039	−4.081	0.000	4.242
Cyclopentanol (dry)	0.332	0.522	−1.034	−0.106	−3.756	0.000	3.892
Methyl acetate (dry)	0.351	0.223	−0.150	−1.035	−4.527	0.000	3.972
Ethyl acetate (dry)	0.328	0.314	−0.348	−0.847	−4.899	0.000	4.142
Propyl acetate (dry)	0.362	0.280	−0.390	−0.975	−4.928	0.000	4.183
Butyl acetate (dry)	0.289	0.336	−0.501	−0.913	−4.964	0.000	4.262
Pentyl acetate (dry)	0.182	0.261	−0.474	−1.017	−4.952	0.000	4.388
Gas-to-water	−0.994	0.577	2.549	3.813	4.841	0.000	−0.869
Equation (2) Coefficients
1-Octanol (wet)	−0.198	0.002	0.709	3.519	1.429	0.858	0.000
Cyclohexane (both)	0.163	−0.110	0.000	0.000	0.000	1.013	0.000
Methylcyclohexane (both)	0.318	−0.215	0.000	0.000	0.000	1.012	0.000
Methanol (dry)	−0.039	−0.338	1.317	3.826	1.396	0.773	0.000
Ethanol (dry)	0.017	−0.232	0.867	3.894	1.192	0.846	0.000
1-Propanol (dry)	−0.042	−0.246	0.749	3.888	1.076	0.874	0.000
1-Butanol (dry)	−0.004	−0.285	0.768	3.705	0.879	0.890	0.000
1-Pentanol (dry)	−0.002	−0.161	0.535	3.778	0.960	0.900	0.000
1-Heptanol (dry)	−0.056	−0.216	0.554	3.596	0.803	0.933	0.000
2-Propanol (dry)	−0.048	−0.324	0.713	4.036	1.055	0.884	0.000
2-Butanol (dry)	−0.034	−0.387	0.719	3.736	1.088	0.905	0.000
2-Methyl-1-propanol (dry)	−0.003	−0.357	0.699	3.595	1.247	0.881	0.000
3-Methyl-1-butanol (dry)	−0.052	−0.430	0.628	3.661	0.932	0.937	0.000
2-Pentanol (dry)	−0.031	−0.325	0.496	4.792	1.024	0.934	0.000
2-Methyl-1-butanol (dry)	−0.055	−0.348	0.601	3.565	0.996	0.925	0.000
4-Methyl-2-pentanol (dry)	−0.013	−0.606	0.687	3.622	0.436	0.985	0.000
Cyclopentanol (dry)	−0.151	−0.314	0.693	3.549	0.914	0.956	0.000
Methyl acetate (dry)	0.134	−0.477	1.749	2.678	0.000	0.876	0.000
Ethyl acetate (dry)	0.171	−0.403	1.428	2.726	0.000	0.914	0.000
Propyl acetate (dry)	0.246	−0.346	1.318	2.537	0.000	0.916	0.000
Butyl acetate (dry)	0.154	−0.439	1.223	2.586	0.000	0.953	0.000
Pentyl acetate (dry)	0.154	−0.424	1.172	2.506	0.000	0.962	0.000
Gas-to-water	−1.271	0.822	2.743	3.904	4.814	−0.213	0.000

^a Equation coefficients for additional organic mono-solvents can be found in cited reference [49]. Coefficients for ionic liquid solvents can be found in a compilation by Jiang and coworkers [62].

All experimental solubility data for vitamin K4, including the measured values given in Table 1 of this study, is reported as mole fraction concentrations. The Abraham model correlations that are available for solute descriptor determinations pertain to molar solubility ratios. The conversion of mole fraction solubilities to molar solubilities is achieved by:

C_S,organic ≈ x_S,organic/[x_S,organic V_Solute + (1 − x_S,organic) V_Solvent]

(3)

Dividing the measured x_S,organic values by the ideal molar volume of the saturated solution solution. A value of V_solute = 0.2205 L mol⁻¹ was used for the molar volume of vitamin K4. The solubility of vitamin K4 is sufficiently small in each of the organic mono-solvents considered so that the first term in the denominator, e.g., x_S,organic V_Solute, makes an insignificant contribution in the calculation. The molar solubilities and respective solvent equation coefficients are now substituted into Equations (1) and (2). Based on measured solubility data we now have 21 log (C_S,organic/C_S,water) equations and 21 log (C_S,organic/C_S,gas) equations to use in the solute descriptor computations.

There are two additional Abraham model log (C_S,water/C_S,gas) equations to use in our solute descriptor computations that describe the gas-to-water solute transfer process (coefficients given in the last row of the first and second section of entries in Table 2 entries):

log (C_S,water/C_S,gas) = −0.994 + 0.577 E + 2.549 S + 3.813 A + 4.841 B − 0.869 V

(4)

log (C_S,water/C_S,gas) = −1.271 + 0.822 E + 2.743 S + 3.904 A + 4.814 B − 0.213 L

(5)

plus the two equations based on the water-to-wet 1-octanol transfer process. We use an estimated value of log P = 3.590 [63] for the practical water-to-1-octanol partition coefficient of vitamin K4. In total there are 46 mathematical expressions that can be used in the regression analysis for determining vitamin K4′s six solute descriptors (E, S, A, B, V and L) plus the numerical values of log C_S,water and log C_S,gas needed to calculate the molar solubility ratios. In our search of the published chemical literature, we did not find an experimental value for the solubility of vitamin K4 in water.

The number of equations is more than sufficient to obtain a set of numerical values having predictive capabilities. Fortunately, three of the six solute descriptors can be calculated solely from molecular structure considerations. The E solute descriptor was taken as E = 1.500 [45], the A solute descriptor was set equal to zero because vitamin K4 does not possess a hydrogen atom that is capable of acting as an H-bond donor, and the McGowan molecular volume descriptor, V = 1.9387, was calculated from the number of chemical bonds, as well as the same number and atomic volumes of carbon, hydrogen, oxygen, nitrogen and fluorine atoms [64]. The 46 Abraham model expressions were then solved simultaneously using the built-in Solver add-in feature on Microsoft Excel to give: numerical values of the remaining three solute descriptors: S = 2.143; B = 0.760; and L = 9.931, plus the molar concentrations of log C_S,water = −4.560 and log C_S,gas = −11.891, needed for the molar solubility calculations. The overall standard deviation associated with the regression analysis was SD = 0.110 log units. Individual standard deviations were SD = 0.118 log units and SD = 0.105 log units for the 23 calculated and observed log (C_S,organic/C_S,water) values and the 23 calculated and observed log (C_S,organic/C_S,gas) values, respectively. Deviations between the observed and back-calculated values of log P and log K for solute transfer into wet 1-octanol are included in the respective standard errors for the log (C_S,organic/C_S,water) and log (C_S,organic/C_S,gas) results. Compared to vitamin K3 (menadione) whose experiment-based solute descriptors of: E = 1.250; S = 1.480; A = 0.000; B = 0.540; V = 1.2007; and L = 6.766, which were previously determined by Liu et al. [65] using the published solubility data in several organic mono-solvents and binary aqueous-alcohol mixtures [66,67,68], vitamin K4 exhibits much greater polarity/polarizability and slightly more H-bond basicity than its parent menadione. The increased H-bond basicity likely results from the two lone electron pairs on each of the two additional oxygen atoms (see Figure 1 for the molecular structure of both vitamins).

There is very little published information regarding the physical, chemical, thermodynamic and pharmacokinetic properties of vitamin K4. The experiment-based solute descriptors that were just determined for vitamin K4 can now be used in conjunction with previously published Abraham model correlations to predict the vitamin’s molar solubility in more than 100 different dry organic mono-solvents [23,24,25,26,27,49] and in more than 90 different ionic liquids [62], and to predict practical partition coefficients for many different biphasic aqueous-organic solvent extraction systems [49,69,70,71,72]. The Abraham model correlations have also been developed for predicting the vapor pressure [73], standard molar enthalpies of vaporization [74] and sublimation [75] of organic compounds at 298.15 K, enthalpies of solvation of organic compounds dissolved in both water [76] and in more than 30 organic solvents of varying polarity and hydrogen-bonding character [77,78,79,80], as well as a compound’s blood-to-body fluid/tissue and air-to-body fluid/partition coefficients at 310 K [30,31,32,36,37,38,81,82]. The predicted values are obtained by simply substituting the numerical values of the compound’s solute descriptors into previously published Abraham model correlations. For example, we calculate numerical values of −116 kJ mol⁻¹ and −80 kJ mol⁻¹ for the standard molar enthalpies of solvation of vitamin K4 dissolved in 1,4-dioxane and water at 298 K [77], respectively, and the estimated equilibrium vapor pressure, VP, above the solid vitamin is VP = 6.2 × 10⁻¹² atm at 298 K [73]. In

Table 3. Equations for predicting the logarithm of in vivo blood-to-rat tissue partition coefficients and the calculated log P values for vitamin K4.

System	cp	ep	sp	ap	bp	vp	Calculated
Blood-to-brain	0.547	0.221	−0.604	−0.641	−0.681	0.635	0.298
Blood-to-muscle	0.082	−0.059	0.010	−0.248	0.028	0.110	0.249
Blood-to-liver	0.292	0.000	−0.296	−0.334	0.181	0.337	0.022
Blood-to-lung	0.269	0.000	−0.523	−0.723	0.000	0.720	0.544
Blood-to-kidney	0.494	−0.067	−0.426	−0.367	0.232	0.410	0.452
Blood-to-heart	0.132	−0.039	−0.394	−0.376	0.009	0.527	0.258
Blood-to-skin	−0.105	−0.117	0.034	0.000	−0.681	0.756	0.756
Blood-to-fat	0.077	0.249	−0.215	−0.902	−1.523	1.234	1.225

we list the coefficients for several in vivo blood-to-rat tissue partitioning process at 310 K [81], along with the respective predicted log P value for vitamin K4. Calculations indicate that vitamin K4 is distributed primarily to the fat tissues of the rat, followed by the skin, lung, kidney and brain. All calculated blood-to-tissue partition coefficients exceed unity, with the blood-to-liver value being the smallest value at P = 1.05.

4. Summary

The Abraham general solvation parameter model has been shown to provide a reasonably accurate mathematical description of the observed solubility behavior of vitamin K4 dissolved in 2 cyclic hydrocarbon solvents (cyclohexane, methylcyclohexane), 14 alcohol solvents (methanol, ethanol, 1-propanol, 1-butanol, 1-pentanol, 1-heptanol, 2-propanol, 2-butanol, 2-methyl-1-propanol, 3-methyl-1-butanol, 2-pentanol, 2-methyl-1-butanol, 4-methyl-2-pentanol, cyclopentanol), and five alkyl acetate solvents (methyl acetate, ethyl acetate, propyl acetate, butyl acetate, pentyl acetate) at 298.15 K. The back-calculated molar solubility ratios based on our derived solute descriptors differ from the experimental values by an approximate overall standard deviation of 0.110 log units. The small difference between the observed and back-calculated values suggests that the calculated descriptor values reported in the present communication will enable one to successfully estimate the solubility of vitamin K4 in the additional 130 or so organic mono-solvents and binary aqueous-organic solvent mixtures for which the Abraham model correlations have been determined. The calculated solute descriptors further indicate that vitamin K4 exhibits much greater polarity/polarizability and slightly more H-bond basicity than its parent menadione. The increased H-bond basicity likely results from the two lone electron pairs on each of the two additional oxygen atoms.

The solute descriptors reported in the current study can be used to predict the vapor pressure of vitamin K4 at 298 K, as well as the compound’s standard molar enthalpies of vaporization and sublimation. Important pharmaceutical properties that can be predicted include the logarithm of the in vivo blood-to-rat tissue partition coefficients. Calculations indicate that vitamin K4 is distributed primarily to the fat tissues of the rat, followed by the skin, lung, kidney, and brain. All predicted blood-to-tissue partition coefficients exceed unity, with the blood-to-liver value being the smallest value at P = 1.05.