To probe the characteristics of solubility of neutral-form drugs added in excess to pure water, the measured S_w values of thirty-two weakly-ionizable druglike molecules were taken from published sources [16-27]. A systematic series of benzene carboxylic acids (benzoic, phthalic, trimellitic, hemimellitic, pyromellitic, mellophanic, benzene pentacarboxylic, and mellitic acid) and eight other weak acids covering a wide range of solubility values (acetaminophen, crocetin, glibenclamide, indomethacin, isotretinoin, naproxen, phenytoin, and quercetin) were selected. The ten selected weak bases include atenolol, bedaquiline, brigatinib, carvedilol, emtricitabine, imatinib, imiquimod, ketoconazole, lamotrigine, and lumefanatrine. The six ampholytes considered are celecoxib, enrofloxacin, L-lysine, mycophenolate mofetil, piroxicam, and sulfamethoxazole.

Values of pK_a were taken from standard compilations, matching the temperature of the solubility measurements and harmonizing on ionic strengths, as described below [28-34]. For newer drugs, relevant literature sources were consulted, including patents and public regulatory agency documents.

The solubility analysis, refinement, and simulation computer program, pDISOL-X™ (in-ADME Research), was used in this study. The mathematical approach based on a mass action equilibrium model continues to evolve [35-39]. The program has been effectively applied in several studies involving multifactorial equilibria: self-aggregation [40-45], cocrystals [46-48], complexation [49,50], and salt disproportionation [51,52]. Recently, salting-out activity corrections have been added [15].

The data analysis method uses log S (molarity basis) as measured input data, as a function of pH. The analytical concentrations of all added reagents are specified. The mass action algorithm considers the contribution of all species proposed to be present in the solution. The algorithm derives its own implicit polynomial equations internally, given a practical number of equilibrium reactions and the corresponding roughly estimated constants. The program refines the estimated constants and calculates the distribution of species and reactants consequent to a simulated sequence of additions of titrant (e.g. HCl or NaOH, or a few program-recognized ionizable titrants: phosphoric acid, maleic acid, lysine, etc.), to simulate the suspension pH speciation from pH 0 to 13. The ionic strength, I, is iteratively calculated at each pH. Values of pK_a, intrinsic solubility, along with pH electrode standardization constants, are accordingly adjusted at each pH point for activity deviations from the benchmark level of I_ref = 0.15 M, selected as the basis of the ‘constant ionic medium’ thermodynamic state [12].

All the equilibrium constants reported here are based on the concentration scale, i.e. the ‘constant ionic medium’ thermodynamic standard state, without loss of thermodynamic rigor [12] (pp. 43-47). Since the measured pH is based on the ‘operational’ activity scale (p_aH), such values need to be converted to the concentration scale, p_cH (= -log c_H+), as described elsewhere [12] (pp. 55-66; cf. Appendix B). In such a system, the electrode is first ‘calibrated’ using NIST standard pH buffers, which establishes the relationship between meter voltage (mV) readings and the p_aH scale. These values are then converted into the p_cH scale.

Since I at any given pH point in an acid-base titration is likely to be different from I_ref, all ionization constants are locally transformed (from I_ref to local I) for the calculation of local point concentrations (as detailed in Supplementary material, Appendix A). It is a reasonable practice to designate 0.15 M as the benchmark ‘reference’ ionic strength, I_ref (‘physiological’ level), to which all equilibrium constants are harmonized in the data analysis. The procedure uses activity corrections based on the hydration theory proposed by Stokes and Robinson [13,14,53-55], as described by examples in the next four sections and in further detail in Appendix A.

Activity corrections using the Stokes-Robinson hydration theory model (SRHT) [13] have seldom been applied in solution speciation studies involving multiprotic drugs in media containing numerous charged and neutral species. One exception appears to be that of Wang et al. [14].

We adapted the version detailed by Wang et al. into pDISOL-X in 2013 [40] and continue to make slight improvements to the treatment. In many of the classical papers embracing the SRHT method, most of the emphasis has been placed on correcting the activity of charged species, since long-range ion-ion interactions play a dominant role in solutions with I < 0.5 M, in comparison to the effect of short-range ion-solvent interactions. Activity corrections for neutral species are seldom discussed in the context of SRHT model. There are three components to the model (Eq. A4 in Appendix A). The first term in Eq. A4 addresses the dominant ion-ion electrostatic interactions. The impact of the SRHT model on uncharged species arises from the second and third ‘solvation’ terms in Eq. A4 (ion-solvent interactions), although the calculated activity coefficients are practically unit value in the model, unless I > 1 M.

As ions are added to pure water, some of the water molecules are removed from the initial pure water and taken up by the ions as part of their primary solvation shells. For example, hydrogen ions are thought to sequester seven water molecules into the solvation shell, but chloride ions only hold one water molecule as such [14]. More water is sequestered if the ions are multiply charged and arise from relatively soluble electrolytes (e.g. mellitic acid, lysine, etc.). So, the presence of charged species results in less free water to dissolve the neutral molecule, because some of the water is removed from the bulk solvent into the solvation shells of the ions. In terms of the total volume of water, the solubility of uncharged species appears to be reduced in the presence of a large concentration of solvated ions, since it is the free water volume that dictates the solubility equilibrium. The phenomenon is termed ‘salting out’ [15] (and references therein).

In our recent study of the impact of salting-out on equilibria involving self-association of an ionizable solute [15], a model to predict salting-out constants was introduced, as further described by Eq. A5 in Appendix A. In the present study, we compared the predicted activity coefficients of neutral molecules using the SRHT model (Eq. A4) with those arising from the salting-out equation (Eq. A5). In all the cases we considered, the legacy SRHT appears to underestimate the effect of ion-water interactions on the activity of uncharged species, in comparison to that of the salting-out effect.

In the present commentary, we propose a modified SRHT model, which incorporates predicted salting-out constants to estimate the activity coefficients of neutral molecules. The modified SRHT model is a tentative proposal, awaiting more direct experimental confirmation. Since salting-out studies have been performed far more often than applications of SRHT and since there are many measurements of the effect of salt on the solubility of uncharged species, our approach in this commentary may be an opportunity to improve the SRHT model for future applications involving druglike molecules.

Consider the case of a monoprotic weak acid, acetaminophen, whose ionization constant at I_ref = 0.15 M and 37 °C is pK_a^ref = 9.41. When 20 mg of pure acetaminophen are added to 1.0 mL of distilled water, the equilibrated pH = 5.29 and ionic strength I = 5.7 μM at the 132 mM subsaturation concentration. To perform mass action calculations in the near-zero ionic strength, the pK_a^ref needs to be adjusted for the changes in activities between the two ionic strengths (5.7 μM and 0.15 M). The general Eq. A3 in Appendix A reduces to Eq. (1):

where K_a^I is the ionization constant at ionic strength, I; f_A, f_H, and f_HA are the activity coefficients of A^-, H⁺, and HA. In the subsaturated solution, the activity coefficient of the neutral species, HA, is expected to be near unit value, independent of ionic strength. The activity coefficients of the charged species may be calculated using Eq. A4 (SRHT). Since both ionic strengths are relatively low, the first term of Eq. A4 (Debye-Hückel term) dominates. At I = 5.7 μM, the calculated activity coefficients are: f_A^I = f_H^I = 0.997, f_HA^I ≡ 1. At I_ref = 0.15 M, the corresponding coefficients change to f_A^ref = 0.769, f_H^ref = 0.828, f_HA^ref ≡ 1. Inserting these values into Eq. 1 produces Eqs (2a)

In negative log form, Eq. (2b)

Next, consider the case of a saturated solution of the drug: when 25 mg of pure acetaminophen are added to 1.0 mL of distilled water, the calculated saturation pH 5.28 at the added total concentration of 164 mM, with I = 6.6 μM. The two equilibrium equations (in cumulative form, Eq. A2) are A + H (HA and A + H ( HA(s). The cumulative constant for the second equation, β_HA(s)^ref = pK_a^ref + pS₀^ref = 9.41 +0.87 = 10.28.

The activity coefficient of the solid species, f_HA(s)^I and f_HA(s)^ref are generally defined as unit value. The salting-out constant of acetaminophen, K_salt = 0.117 M^-1, is used to determine the activity of HA: log f_HA^I = 0.117×6.66×0^-6 = 7.72×10^-7 or f_HA^I = 1.000002; log f_HA^ref = 0.117×0.15 = 0.0176 or f_HA^ref = 1.04124. The activity coefficients of the ions are practically unchanged from the prior example: f_A^I = f_H^I = 0.997, f_A^ref = 0.769, f_H^ref = 0.828.

The pK_a equation is treated similarly as in the prior example: pK_a^I = 9.62 (slightly different from 9.60 due to the use of salting-out constant). The solubility equation in cumulative form (cf. Eq. A2) at the local ionic strength is determined (Eq. A3) as log β_HA(s)^I = log β_HA(s)^ref + log (f_A^I/f_A^ref) + log (f_H^I/f_H^ref). With the above activity coefficients, log β_HA(s)^I = 10.28 + log (0.997/0.769) + log (0.997/0.828) = 10.47. From this, pS₀^I = 10.47-9.62 = 0.85 (140 mM), which is only slightly lower than pS₀^ref by 0.02 (=-K_salt(I-I_ref)).

As a further example drawing on analytic continuation of the above case, consider that 0.5 M HCl is added so that the pH is lowered to 0.05 (p_aH). The activity coefficients of the various species need to be adjusted accordingly. So, at the target (local) pH, the ionic strength increases to I = 0.5 M. The second two terms in Eq. A4 take on more significance in the high ionic strength case. Using the approach presented above, f_A^I = 0.743, f_A^ref = 0.822, f_H^I = 0.888, f_H^ref = 0.885, and f_HA^I = 1.144, f_HA^ref = 1.041. These values substituted into Eq. 1 indicate that the pK_a value at I = 0.50 M becomes 9.33; that is, higher salt concentration makes the weak acid slightly stronger. The pS₀^I at pH 0.05 (Eq. (3)) now increases from the value of 0.87 at pH_w 5.28 to

Thus, at pH 0.05, the solubility (solid curve in Fig. 1c) crosses the HH curve (140 mM: dashed curve in Fig. 1c) as the solubility in the higher ionic strength suspension drops to 122 mM.

In this manner, at every pH point, a different set of pK_a^I and pS₀^I need to be determined, from which the local concentrations of reactants and product species are then calculated.

Table 1 lists pairs of pK_a values (pK_a^ref and pK_a^I) of the 32 weakly ionizable molecules considered in the study. The listed temperature is that of the corresponding solubility study. In the solubility measurements in pure water, the benchmark reference ionic strength was selected as I_ref = 0.15 M. The local values of ionic strength, I_w (calculated at pH_w ), are listed. Values of pK_a^ref are taken from published sources [28-34]; the corresponding pK_a^I (at pH_w) were calculated using pDISOL-X (cf. Eqs. (1) and (2).

Tables 2-4 summarize the details of the analyses of the published weak acid, base, and ampholyte solubility measurements in pure water. The literature values of log S_w, along with the corresponding pK_a values (Table 1), were the starting points in the determination of pH_w and log S₀^ref. Activity corrections (Eqs. (1) to (3), based on the general Eq. A3 in Appendix A) were applied to ionic species. In addition, K_salt salting-out constants (Eq. (3) based on Eq. (A5)) were used to adjust the activities of uncharged species. In most cases in the tables, the I_w value at pH_w is near zero, particularly in the cases of weak bases, but the values increase significantly with the benzene carboxylic acid series, reaching a high value of 0.49 M for mellitic acid (2-fold excess added solid) or 0.38 M for mellitic acid (10-fold excess). Higher yet, lysine at pH_w indicates I_w = 0.61 M (1.1-fold excess) or I_w = 0.70 M (2-fold excess).

In almost all the simple acids, bases, and ampholytes, the measured solubility was slightly higher than the value harmonized to I_ref, i.e. the presence of salt lowers water solubility at I_ref due to activity effects. In the case of lysine, the presence of salt elevates the solubility in water. These trends will be graphically illustrated below for a few of the substances studied.

The average values (± standard deviations) of salting-out coefficients, K_salt (Tables 2 to 4, estimated using Eq. (A5b)), are 0.28 ± 0.14 (ampholytes), 0.29 ± 0.13 (acids, first 8 in Table 2), and 0.40 ± 0.17 M^-1 (bases). The average values are notably lower at 0.06 ± 0.06 for the benzene polycarboxylic acid series, particularly in the cases of three or more carboxylic acid substituents.

At pH_w, salting-out accounts for 92 to 100 % of the total activity correction for intrinsic solubility (i.e. log S₀^ref - log S₀^I ≈ K_salt(I_ref - I)) and Eq. A3 does not contribute to the shift in log S₀ between the measurement ionic strength and the reference 0.15 M value, since the second and third terms in Eq. (A3) are not significant at these levels of ionic strength.

Figure 1 illustrates the log S-pH relationships in three selected examples of acids (shown in increasing order of solubility): crocetin, glibenclamide and acetaminophen. At pH_w, the ionic strength and buffer capacity are at their minimum values (with I_w ~0 M, buffer capacity < 1 mM/pH) for the three acids. As pH is altered by the addition of titrant, both properties increase, particularly below pH 3 and above pH 8.

In all cases in Figure 1, the equilibrium pH_w is less than that of water (pH ~7). Both the water solubility (S_w) and the acid strength (pK_a) of the molecule play a critical role, as detailed in the examples in Appendix C. Consider the weak acids in the figure. To better understand the resulting shifts from pH ~7 to pH_w, it is helpful to think of the weak acid as a titrant added to pure water that produces a certain shift in water pH. For the most soluble of the acids in Figure 1, acetaminophen, the pH_w is shifted below the pK_a by 4.3 units. Since acetaminophen is quite soluble, enough of the free acid dissolves and sufficiently ionizes, to overcome the buffering of pure water in the neutral pH domain. In simulation calculations, acetaminophen, when added to water, releases ~0.08 μmol of H⁺ per mL of solution, which results in the pH shift from ~7 to ~75, well inside the minimum solubility region (pH <7, as dictated by the acetaminophen pK_a) in the log S-pH profile (Fig. 1c). The lower the solubility of the acid, the less the acid impacts on the shift in pH_w from pH ~ 7 of pure water. For example, crocetin would need to release ~0.73 μmol/mL of H⁺ to effect the pH shift from ~7 to ~3, where the molecule would be in the intrinsic (minimum) solubility part of the log S-pH curve (Fig. 1a). However, due to its very low solubility, only about 0.06 μmol/mL of H⁺ are released, so crocetin cannot reach the intrinsic solubility region in the log S-pH profile (pH <3, as defined by its two pK_a values). The measured pH_w values are apt to be in the diagonal region of the profile, as in the case of the poorly soluble crocetin, where pH_w is above pK_a1 by 0.94 log unit. By comparison, the pH_w of the more soluble glibenclamide is below its pK_a by 0.43 log unit.

In Figure 1c, the solid curve has a ‘rotated sigmoidal’ distortion, compared to the benchmark HH hyperbolic (dashed) curve. The solubility difference between the two curves maximizes at pH_w (where I and I_ref are most different). The solid and dashed curves cross at pH near 0.9 and 8.5, as the ionic strengths become equal at those points.

Figure 2 shows log S-pH plots for the benzene polycarboxylic acid series. For benzoic, phthalic, trimellitic, and pyromellitic acids, the hyperbolic profiles take on typical activity-distorted shapes. As the number of carboxylic groups (with overlapping pK_a) increases, so does the solubility, with mellitic acid reaching the high value of 988 mg/mL (Table 2). Due to the increasing ionic strength with solubility, especially as the charges increase on the polycarboxylates as pH increases, the distortions from the Henderson-Hasselbalch benchmarks (dashed curves in Fig. 2) become most pronounced. As pH increases, the increasing ionic strength exceeds 7 M past pH 2.3, 3.7, 4.1, and 7.0 for mellitic, benzene pentacarboxylic, mellophanic, and hemimellitic acids, respectively. The other four (mostly less soluble) carboxylic acids have ionic strengths that level off at pH 7.4 at I = 4.5 (trimellitic), 3.6 (pyromellitic), 1.2 (phthalic), and 0.3 M (benzoic).

The extreme distortions in Figure 2 related to activity corrections appear not to have been previously reported. In the case of mellitic acid, I_w exceed I_ref, and depends on the excess quantity of neutral acid solid added to distilled water (cf. Table 1). This concentration dependence is also evident in the case for L-lysine. From the perspective of solubility measurement, when I_w exceeds I_ref, it may be of limited use to adjust the water solubility observed at I_w to the expected value at the physiologically relevant I_ref, since at that reference ionic strength, both mellitic acid and L-lysine are apt to dissolve completely at practical concentrations.

Figure 3 illustrates the log S-pH relationships of three selected bases (arranged in increasing order of intrinsic solubility). The ionic strength values are at their minimum at pH_w. As pH is adjusted by additions of HCl/NaOH in the simulated analytic continuation procedure, the ionic strength increases, just as in the cases of simple weak acids. The characteristic distortions of the solid (green) curves are the consequence of activity corrections, which depend on the differences, I - I_ref. A partial ‘rotated sigmoidal’ shape distortion is evident for bedaquiline and brigatinib, compared to the HH hyperbolic dashed curve at I_ref = 0.15 M.

The position of pH_w in relation to the pK_a mirrors the trends illustrated in Figure 1 and can be explained by the effects of solubility and how far the pK_a values are from pH ~7 of pure water. The less soluble the base, the more is the pH_w apt to be in the diagonal region of the log S-pH profile.

Figure 4 shows the log S-pH profiles of three selected ampholytes, arranged in increasing order of intrinsic solubility. The differences in the curves between the low ionic strengths (solid green) and the reference level (dashed) curves in the first two examples may be anticipated based on the preceding acid-base discussions.

Except for lysine, the ampholytes (Table 4) show positive log S₀^I - log S₀^ref differences of 0.029 to 0.051 (less soluble with increased salt). Lysine appears to show negative (salting-in) differences of -0.048 (1.1-fold excess solid added) or -0.056 (2-fold excess) - more soluble with increased salt. This salting-in value appears to be entirely K_salt driven, with no contribution from the traditional SRHT Eq. A3. The dash-dot (red) curve in Figure 4c shows the asymmetric lysine ionic strength profile as a function of pH. Ionic strength values exceeding 7 M are reached as pH is altered from the pH_w.

In the absence of high quality S_w solubility and pK_a measurement, the additional ‘fine tuning’ corrections for the ionic strength effect would hardly be worthwhile. The uncertainty in the conversion of S_w (typically at I_w ~ 0 M) to S₀ at I_ref = 0.15 M and consequently extending that into a log S-pH reference-state profile probably has as much to do with the reliability of the measured S_w and the provided I_ref -based pK_a values, as with the modified SRHT model described here.

The following shortfalls underlie the challenge in validating the conversion procedure we discussed:

As part of the S_w assay, measurement of the pH_w would be helpful, as this would allow for recognition/correction of the effects of impurities (e.g. CO₂) and could lead to more certain S_w measurement.