4.1. PSP of Solvents
A first step to apply the PSP model is to obtain the LSER descriptors of the compounds in given mixtures. Since LSER descriptors have been determined for many common solvents and other additives, it is often possible to use tabulated values from the literature. Some example values are listed in
Table 2, which are widely used as solvents or probe gases in measurements of inverse gas chromatography (IGC).
The PSP approach was used to characterize the studied drugs, that is, to determine their LSER descriptors from the experimental IGC data. The McGowan volume,
Vx, is an atom-specific quantity and, thus, it is obtained either from the ACD software or from the freely accessible database [
7] via their SMILES form. The
E and
S descriptors are first obtained from IGC data (
Ω1 or
χ12 parameter data) with alkane and aromatic hydrocarbon probes correlated with Equation (24), which, in the infinite dilution limit (
φ2 → 1,
x2 → 1 of IGC measurements, takes the following analytical form:
where:
A and B descriptors were then obtained from IGC data with hydrogen-bonding probes (acidic, like chloroform, basic, like acetone, and homosolvated, like ethanol) correlated with Equation (34).
4.2. Applications to Pharmaceutical Drugs
Six model drugs were selected that represent rather poorly water-soluble compounds due to their importance in pharmaceutical sciences.
The LSER descriptors for the drugs studied in this work may be estimated using the ACD software (ACD Percepta, Absolv. V.2016) and obtained values are reported in
Table 3. These values were compared with determined LSER descriptors using the PSP approach based on the experimental IGC measurements.
In
Figure 1, the experimental activity coefficients at infinite dilution are compared to calculated estimates for cyclosporine A. As observed, there is a rather significant discrepancy between experimental data and calculations when using the ACD/LSER descriptors. The picture is even worse for the other five studied drugs. Indeed, due to the rather complex structure of these drugs, it was not expected to have very precise estimations of their LSER descriptors by such a pure in silico approach as used by the ACD software.
Figure 2 displays the molecular structures and screening charge densities of the different model compounds. The compounds are typically larger and more complex than most solvent molecules; they also exhibit dipolar and hydrogen-bonding capabilities. Larger molecules have, compared to rather small solvents, more options for specific effects of the three-dimensional (3D) conformation, such as shielding of moieties that are relevant for interaction. This is neglected by the 2D input structure of the ACD software. Supramolecular effects and self-association can further complicate the situation, thereby leading to poor estimates of the LSER descriptors of drugs using the 2D in silico method.
As observed in
Figure 1, the PSP calculations are made much closer to the diagonal (equal experimental and calculated data) compared to the ACD values, which are significantly off the diagonal.
Relevant deviations are observed with the IGC data for all probe gases studied.
Figure 3 displays the experimental and calculated activity coefficients at infinite dilution of methanol (the most hydrophilic solute/probe) for the studied drugs. In this case, the PSP calculations practically coincide with the diagonal, while the calculations with ACD/LSER descriptors are often significantly off the diagonal. However, with the exception of methanol, n-alkanes, and a couple of other solutes/probes, the IGC data could not be reconciled satisfactorily, either by the ACD or by the PSP/LSER descriptors. Thus, the reported LSER descriptors in
Table 3 should be considered as initial or tentative estimates. It should be kept in mind that there can be issues of individual probe gases on an experimental level, such as association in the gas phase and deviation from the infinite dilution conditions. Analysis of further probe gases can further improve estimates of PSP/LSER descriptors determined by IGC.
Having determined the LSER descriptors of drugs, Equation (25) can be used for the prediction of their solubilities in the various solvents. The required melting points and heats of fusion of the drugs are reported in
Table 4. The predictions with both sets of the LSER descriptors (ACD and PSP) are shown in
Figure 4 and
Figure 5, along with the experimental solubilities. Both sets of descriptors were used with the PSP thermodynamic framework of
Section 2.
The approach based on ACD estimations is depicted in
Figure 4 as a regression line together with 95% confidence and prediction limits. The regression coefficient was
r = 0.942 (
p < 0.0001; slope of 0.676 with an intercept of −0.717) and the mean absolute error was 0.537. Such errors of logarithmic solubility around 0.5 are often obtained with drug solubility predictions of different methods. Limited prediction accuracy was already expected for the ACD in silico estimation of LSER parameters based on the previous comparison with IGC data. However, there can also be experimental factors contributing to limiting prediction accuracy by any theoretical prediction. It is, here, useful to consider the results of the residual solid form analysis that was determined as part of the solubility experiments. For example, the x-ray diffraction analysis of the residual solid revealed, for zafirlukast, at least one different solid form following equilibration, so we omitted the value with ethanol for the statistical evaluation due to a likely solvate formation. Moreover, the value of loratadine in heptane was omitted, and there were further cases in which likely the experimental complexity of the residual solid form was given. The residual solid XRPD revealed, for example, in the case of carvedilol, a changed solid form in dichloromethane after equilibration, which likely suggested a solvate formation.
Figure 5 shows the corresponding regression analysis of estimated values by the PSP approach, and the regression was slightly better with
r = 0.958 (
p < 0.0001; slope of 0.753 with an intercept of −0.634) and the mean absolute error yielded 0.481. Overall, the solubility predictions were reasonable, and it is well possible that higher precision would have been achieved by omission of all solubility data where solid form changes were detected during solubility equilibration. However, most solubility studies in the literature were not analyzing the residual solid form, so we decided to only omit clear outliers. As mentioned before, it is well possible that further precision can be achieved using more probe gases in IGC to determine PSPs, but the present approach (with a minimal dataset) was supposed to reflect a typical situation for practical solubility prediction, as it is feasible in the pharmaceutical industry.
The PSP framework can also be used for the estimation of the surface energy components of the studied drugs. However, there are practically no data on the total surface energies of drugs available in the literature. Instead, the dispersion components,
γd, were determined by the Dorris–Gray method [
50]. This dispersion component is practically equivalent to our nonhydrogen-bonding component,
γVES. Thus, we may use Equation (29) in the equivalent form of Equation (36) and obtain the total surface energy of the drugs,
γtot:
In a similar manner, the acidity and basicity components of the surface energy are obtained from Equation (27). The obtained surface energy components of the studied drugs are reported in
Table 5.
Any reported value of surface energy contribution by IGC also involves a theoretical framework and, hence, the results in
Table 5 are obtained surface estimates using IGC with the PSP approach. Due to the importance of surface energy contributions for pharmaceutics, one would expect that many poorly water-soluble drugs have been thoroughly characterized. However, values of surface energy contributions obtained by either contact angle measurements or IGC are only occasionally reported in the literature. Dispersive energy contribution was reported, for example, in the case of ketoconazole and values between about 40 and 50 mJ/m
2 were obtained (based on IGC) depending on the surface disorder that was introduced by a milling process [
51]. This result is in good agreement with our finding of
γd for ketoconazole.
The surface energy estimations by the PSP approach can be further harnessed in future pharmaceutical studies on, for example, drug wettability. The PSP approach allows further theoretical options, like the estimation of cohesive energy density or equivalently, the solubility parameter and its components using Equations (1)–(4) and (13). These alternative characteristic descriptors of the drugs are reported in
Table 6. As observed, the main component of the cohesive energy density in all studied drugs is the one reflecting dispersion forces, although the other components are by no means negligible. It is certainly an advantage of the PSP approach that a hydrogen-bonding contribution to cohesive energy density is differentiated according to acidity and basicity, which is missing in classical solubility parameter concepts.