How far can a fast-crystallizing drug supersaturate? Measuring the amorphous "solubility" and predicting it from heat capacities

Background and motivation

Most new drug candidates dissolve poorly in water, which limits their bioavailability (Introduction, ref 1). Solubilizing additives such as cosolvents, surfactants and cyclodextrins raise the amount of drug in solution, but often lower its permeability at the same time, so that plasma levels do not rise (Williams et al., ref 2). Formulations that generate supersaturation (salts, cocrystals and amorphous solid dispersions) raise the concentration of free drug instead and leave permeation untouched. C1 The catch is that a supersaturated solution is metastable: sooner or later the drug crystallizes and the concentration falls back to the crystalline solubility. How far, and for how long, a formulation can supersaturate is still poorly understood, and the authors argue that knowing the maximum attainable supersaturation "may in fact be useful" for estimating the gain in exposure. C2

Earlier work had proposed that this maximum is set by the amorphous "solubility" (Ilevbare and Taylor, ref 3; Murdande et al., ref 4). The quotation marks matter: the amorphous form is not in true equilibrium with the solution, only in apparent, metastable equilibrium, so its "solubility" is not a thermodynamic solubility in the strict sense. As with metastable polymorphs, whose apparent solubility differences reflect their free-energy differences (refs 5, 6), it can still be measured if the amorphous phase survives long enough. C3

That condition is the problem. Two methods are commonly used to probe supersaturation (ref 8). Dissolving an amorphous solid depends on the race between dissolution and crystallization, and fast crystallizers never reach their ceiling this way: for griseofulvin, the predicted amorphous "solubility" had never been reached on dissolving the amorphous form (refs 4, 7, 9, 10), and undissolved particles can seed crystals. The cosolvent quench, in which a concentrated solution in a water-miscible solvent is added to water, removes most of these problems and is "very useful" for probing the upper limit, although it does not mimic a dosage form dissolving in the gut. C4·C5

Ilevbare and Taylor (ref 3) had used the quench method on eight poorly soluble drugs. When the concentration became high enough, a new drug-rich liquid phase separated from solution (liquid–liquid phase separation, LLPS), seen as colloids that scatter light. The onset concentration followed the amorphous "solubility" calculated from the crystalline solubility, the free-energy difference between crystal and amorphous form (estimated with the Hoffman equation, ref 11) and the activity of the amorphous drug saturated with water. This "suggested" that the ceiling of supersaturation is fixed by the properties of the amorphous and crystalline states. C6 Those drugs, however, crystallized relatively slowly and had relatively low glass transition temperatures (Tg). C7

This paper extends the test in two directions. Experimentally, it uses six drugs that crystallize rapidly from supersaturated solution and that span a wide range of melting points and Tg: ibuprofen, phenylbutazone, tolnaftate, diclofenac, sorafenib and griseofulvin. Two of them, sorafenib and griseofulvin, are glasses at 25 °C. C7·M1–M6 Theoretically, it replaces the Hoffman approximation with a free-energy calculation built on measured heat capacities, the "Cp method", and asks which estimate predicts the measured onset better. C10

Background concepts

Supersaturation ratio. The supersaturation ratio is the dissolved concentration divided by the crystalline (equilibrium) solubility. In this paper it is evaluated at the onset of phase separation: the onset concentration divided by the crystalline solubility. E15

Amorphous "solubility" from three measurable terms. The prediction used throughout the paper is eq 1:

Here $C_{\mathrm{e}\mathrm{q}}$ is the crystalline solubility, $e^{-I(a_2)}$ is the activity of the amorphous drug when it has absorbed water to saturation, $\Delta G_{c\to a}$ is the free energy of converting the crystal into the amorphous form, $R$ is the gas constant and $T$ the temperature (25 °C here). In plain terms: an amorphous drug is more soluble than its crystal by the Boltzmann factor of the extra free energy it carries, but it is diluted by the water it takes up, which lowers its activity below 1. Dividing both sides by $C_{\mathrm{e}\mathrm{q}}$ gives the predicted enhancement ratio $f·e^{\Delta G/RT}$ of Table 9, with $f=e^{-I(a_2)}$. E1·R13

Activity of the water-saturated amorphous drug. Following Murdande et al. (ref 4), the authors measured how much water each amorphous drug absorbs as relative humidity rises, and integrated the Gibbs–Duhem equation to turn the water activity into the activity of the drug. P7·E16 The values in Table 4 run from 0.999 for ibuprofen to 0.699 for griseofulvin; the two glassy drugs, sorafenib (0.750) and griseofulvin, have the lowest. M1–M6

The Hoffman equation. The usual estimate of the free energy between crystal and supercooled liquid is the Hoffman equation (eq 2):

Here ${\Delta }_{\mathrm{f}\mathrm{u}\mathrm{s}}H$ is the enthalpy of fusion and $T_m$ the melting temperature. In plain terms: the further below its melting point the drug is used, the larger the free-energy gain of the amorphous form, scaled by the heat of fusion. It needs only two DSC numbers. The authors point out that it "can lack accuracy": it assumes that the heat-capacity difference between liquid and crystal is constant and independent of temperature, uses approximations that hold only when the undercooling $T_m-T$ is small and when Tg lies close to $T_{\infty }$, and says nothing about a glass. E2·C8 Other estimates set the heat-capacity difference to zero, take it as constant, or use further approximations. C9

Supercooled liquid, glass and T∞. Above Tg an amorphous drug is a supercooled liquid in internal equilibrium; below Tg it falls out of equilibrium as a glass. Figure 5 sketches enthalpy or entropy against temperature: the supercooled-liquid line falls steeply on cooling from Tm, the glass branches off at Tg and runs above the extrapolated supercooled liquid, and the extrapolated liquid line would meet the crystal line at $T_{\infty }$, the temperature at which crystal and supercooled liquid have the same enthalpy. Below Tg a glass therefore holds more enthalpy and entropy than a hypothetical supercooled liquid at the same temperature. C29 At 25 °C, ibuprofen (Tg −44.5 °C), phenylbutazone (−2.7 °C), tolnaftate (7.5 °C) and diclofenac (18.1 °C) are supercooled liquids, while sorafenib (88.2 °C) and griseofulvin (90.8 °C) are glasses. M1–M6·C28

LLPS and GLPS. When a supersaturated solution passes the amorphous "solubility", the excess drug can separate as a drug-rich liquid (LLPS) or, if that phase is below its Tg, as a drug-rich glass (glass–liquid phase separation, GLPS). Either one scatters light, but so do crystals. To tell them apart the authors use environment-sensitive fluorescence probes: a probe partitions into a disordered drug-rich phase and changes its emission, but it cannot enter a crystal lattice, so crystallization leaves the spectrum unchanged. Pyrene reports polarity through the ratio of its third to first emission peak (I3/I1), which rises in a less polar environment; 1-pyrenecarboxaldehyde shifts to shorter wavelength and loses quantum yield in non-polar media (480 nm in water, 392 nm in cyclohexane); for 4-di-2-ASP, a cationic mitochondrial dye, no literature data were found, and "preliminary studies in our lab indicate" a blue shift in a more hydrophobic environment (ref 15). C12·C13·C16·P2

Methods in detail

The paper has three parts: an experimental route to the onset of phase separation for fast crystallizers, a new way to compute the free energy between crystal and amorphous form, and a comparison of the two. The method flow is:

Contribution 1: catching the onset of phase separation before the crystals

The problem. For a fast crystallizer, any method that waits for an equilibrium with the amorphous solid fails: crystals appear first. The onset of a new disordered phase during a rapid titration is the only window in which the amorphous "solubility" can be seen. C4·C44

What they did. Each drug was dissolved in an organic solvent (methanol for ibuprofen, phenylbutazone and diclofenac; acetone for tolnaftate; DMF for sorafenib and griseofulvin, at 0.03 to 25 mg/mL) and pumped by syringe pump at 50–80 μL/min into a temperature-controlled flask with 13 mL of 50 mM phosphate buffer (ionic strength 155 mM) at 25 °C, stirred at 300 rpm (Table 1). The buffer pH was chosen so that each drug was un-ionized: pH 2 for the three acids, pH 6.8 for tolnaftate, sorafenib and griseofulvin. A fiber-optic dip probe (10 mm path) recorded the extinction at a wavelength where the drug does not absorb, 400 or 450 nm, so that any rise means scattering by a new phase. P1 The total organic solvent was "approximately 1% v/v in all cases", and control experiments with pH solubilization showed that it did not change the concentration at which turbidity appeared. P1·R2 From the stock concentrations and the onsets, the solvent fraction at the onset works out to about 1.0% (diclofenac) to 1.9% (tolnaftate) v/v (extractor calculation). X9

In a second, static experiment, drug stock was added in steps to 10 mL of buffer containing a probe (Table 2): 1 μM 1-pyrenecarboxaldehyde for ibuprofen, 2 μM pyrene for diclofenac, and 1.25 or 5 μM 4-di-2-ASP for tolnaftate, sorafenib and griseofulvin, with a spectrum taken after each addition. Phenylbutazone fluoresces on its own (emission peak at 370 nm, excitation 240 nm) and served as its own probe. The onset was taken as the concentration at which the emission spectrum showed "substantial changes"; no numerical criterion is given. P2·X12 A control with 2 μM pyrene tested whether a probe shifts the phenylbutazone onset. R7

Griseofulvin was run twice: in plain pH 6.8 buffer, and in buffer containing 100 μg/mL HPMCAS-HF, a cellulosic polymer added to slow crystallization over the time of the experiment. P1·M7·A3

Crystalline solubility. Excess crystals were shaken for about 48 h at 25 °C in buffer to which a small amount of the organic solvent had been added to mimic the composition at the onset, then separated by ultracentrifugation (40000 rpm, 15 min) and assayed by HPLC in triplicate (Table 3; calibration down to 10 ng/mL for tolnaftate and 2.5 ng/mL for sorafenib). P3·P4 Whether the griseofulvin solubility was measured with HPMCAS-HF present is not stated. X6

Contribution 2: the Cp method for the free energy of crystallization

The problem. The Hoffman equation is an approximation to the free energy, built on a constant heat-capacity difference and valid for small undercooling. Sorafenib and griseofulvin melt above 200 °C, so at 25 °C they are undercooled by about 180–190 K, and they are glasses, which the Hoffman equation does not cover. C8·M5·M6·R16

What they did. The free energy is built from a thermodynamic cycle (Figures 1 and 2): heat the crystal from $T$ = 25 °C to $T_m$, melt it, then cool the liquid back to 25 °C (for a supercooled liquid) or cool it to Tg and continue along the glass to 25 °C (for a glass). Each leg contributes an enthalpy $\int C_{p}dT$ and an entropy $\int (C_p/T)dT$, and the melting step contributes ${\Delta }_{\mathrm{f}\mathrm{u}\mathrm{s}}H$ and ${\Delta }_{\mathrm{f}\mathrm{u}\mathrm{s}}H/T_m$. E3 The sums give the free energy through eq 3:

In plain terms: the free-energy difference is the enthalpy difference minus temperature times the entropy difference, each summed around the cycle. E4

The new element is the heat capacity. Instead of assuming that the crystal and amorphous heat capacities are equal or differ by a constant, the authors fit each phase's reversing heat capacity, measured by modulated DSC, to a straight line in temperature (eqs 5–7): E6·E7·E8

In plain terms: the heat capacity of the supercooled liquid (sl), the crystal (c) and the glass (g) each rises linearly with temperature, with its own slope and intercept. Figure 3 shows the data behind this for tolnaftate: the glass from about 255 to 275 K (about 330 to 365 J/mol·K), a step at Tg, the supercooled liquid from about 285 to 308 K (about 484 to 504 J/mol·K) and the crystal from about 318 to 356 K (about 395 to 437 J/mol·K), each segment following a straight line (read from Figure 3, approximate). R10 The authors state that the linear form held for all six drugs, while noting that over a much wider undercooling the heat capacity is better described by an expression with a $1/T^2$ term (eq 4), so "it is possible that small errors will result". C10·E5 Only the reversing signal is fitted, because entropy is the heat exchanged reversibly divided by temperature ($\Delta S={\Delta }_{\mathrm{r}\mathrm{e}\mathrm{v}}H/T$). C11·E17

Integrating the lines around the cycle gives, for the crystal-to-supercooled-liquid conversion (eqs 8–10): E9·E10·E11

In plain terms: the heat of fusion is corrected for warming the crystal from 25 °C to its melting point and cooling the liquid back down, each with its own temperature-dependent heat capacity; the entropy is corrected the same way. Setting $\Delta H_{c\to sl}$ to zero and solving for the temperature gives $T_{\infty }$. E9·R16

For the crystal-to-glass conversion the liquid is followed only down to Tg, and the glass line takes over from Tg to $T$ (eqs 11–13): E12·E13·E14

In plain terms: the same bookkeeping, with one extra leg along the glass. The authors describe the approach as using no mathematical approximations and "practically free from assumptions"; it applies only to an un-ionized, nonsolvated crystal that melts without degrading, so salts and hydrates are excluded. C10·C11 Miura had used a similar approach for amorphous alloys (ref 24).

How the inputs were measured. Melting temperature and heat of fusion came from DSC at 10 °C/min (TA Q2000, about 5 mg in Tzero pans, n = 3). Heat capacities came from a modulated program (±1 °C every 60 s, 2 °C/min): first through the crystal and its melting, then, after cooling to at least 20 °C below Tg, through the glass and into the supercooled liquid. Griseofulvin crystallized on cooling in the DSC, so its glass was made by melting about 5 mg on a heating plate and quenching it in liquid nitrogen. P5·P6 The activity of the water-saturated amorphous drug came from water sorption at 25 °C (0 to 95% RH): ibuprofen, phenylbutazone, tolnaftate and diclofenac were melted and quenched in the sample holder and measured as supercooled liquids; griseofulvin and sorafenib were melt-quenched, ground and dried at 50 °C for 1 h. P7

Contribution 3: testing the predictions against the onset

For each drug the authors computed the free energy by Hoffman and by the Cp method (and, for sorafenib and griseofulvin, also to the glass), inserted it into eq 1 with the measured crystalline solubility and amorphous activity, and compared the predicted amorphous "solubility" and enhancement ratio with the measured onset and supersaturation ratio (Tables 6, 8 and 9). P9 They then plotted measured against predicted ratios for each method and fitted a straight line, which in Figures 6 and 7 passes through the origin, reporting its slope and $R^2$. P10·R14·R15 Finally, to judge where the Hoffman approximation should fail, they tabulated $T_m-T$ and $T_g-T_{\infty }$ for each drug (Table 10). R16

Experimental analysis

A disordered phase forms first, then crystals

Once the concentration rose far above the crystalline solubility, every solution turned cloudy with a slight bluish tint. The degree of turbidity differed between drugs and with the level of supersaturation, and it lasted only a few minutes before visible particles formed, as expected for drugs known to crystallize quickly. R1·C15

Figure 4 shows, for each drug, the extinction and the probe signal against the added concentration. In every panel except griseofulvin without polymer, the two start to change at about the same concentration (all values read from Figure 4, approximate): R3

• Ibuprofen (Figure 4a): the 1-pyrenecarboxaldehyde peak stays near 469–471 nm up to about 129 μg/mL, then falls to about 446–448 nm by 161–173 μg/mL, while the extinction rises from about 0.015 at 129 to about 0.29 at 177 μg/mL. R3.a
• Phenylbutazone (Figure 4b): its own fluorescence and the extinction stay at zero up to about 50 μg/mL; the fluorescence then climbs to about 980 by 78 μg/mL. R3.b
• Tolnaftate (Figure 4c): the 4-di-2-ASP peak drifts from about 591.5 to about 587.2 nm above about 0.55 μg/mL, while the extinction starts to rise near 0.6 μg/mL. R3.c
• Diclofenac (Figure 4d): the pyrene I3/I1 ratio jumps from about 0.52–0.53 to about 0.75 between 45 and 57 μg/mL, and the extinction rises from about 0.004 at 52 to about 0.046 at 100 μg/mL. R3.d
• Sorafenib (Figure 4e): the extinction rises above about 0.5 μg/mL, and the 4-di-2-ASP peak moves from about 588.8 to about 593 nm. R3.e

The authors read this coincidence as phase separation to a disordered drug-rich phase ("suggesting") and describe the agreement between the two techniques as "very good". C17 Note by the AI writer (not discussed in the paper): for sorafenib the 4-di-2-ASP peak moves to longer wavelength, the opposite direction to the blue shift reported for this probe in a hydrophobic environment and seen with tolnaftate. The change still marks an onset, but its direction is not explained. X3

Griseofulvin needed a polymer. In plain pH 6.8 buffer, the extinction rose "around 150 μg/mL" without any change in the 4-di-2-ASP intensity, and small solid particles were visible at about that concentration (in Figure 4f the first clear rise is at about 178 μg/mL). R4·R3.f·X10 The authors give two readings: griseofulvin "crystallized before reaching sufficiently high concentrations" to separate as a disordered phase, or the probe "could not penetrate a glassy disordered phase"; crystallization was confirmed visually. C18 With 100 μg/mL HPMCAS-HF in the buffer, the extinction stayed flat to about 350 μg/mL and rose from about 375 μg/mL, and the probe intensity rose at the same concentration; the solutions were uniformly turbid and free of visible particles. R3.g·R4 The authors take the higher onset to mean that the polymer delays crystallization ("This result by itself suggests") and the coincident probe signal to mean that glass–liquid phase separation had occurred. C19·C20 From these cases they conclude that a fluorescence check is "clearly essential" for fast crystallizers, so that turbidity from crystals is not mistaken for phase separation. C21

The onset concentrations and the ceiling of supersaturation

Table A collects the crystalline solubility, the two onset concentrations and the resulting supersaturation ratios.

Table A. Crystalline solubility and onset of phase separation at 25 °C (from Tables 5 and 8; onsets of tolnaftate and sorafenib to three decimals from Table 8, where Table 5 gives 0.59 ± 0.09 and 0.56 ± 0.03, and 0.64 ± 0.11 and 0.53 ± 0.05). Solubility in triplicate; n for the onsets is not stated. R5·R6·X1·X6

Crystalline solubilities span more than three orders of magnitude, from 29.29 μg/mL for ibuprofen to about 10 ng/mL for sorafenib, and tolnaftate and sorafenib separate at concentrations in the ng/mL range. R5·C22 The maximum supersaturation by fluorescence runs from 4.3 for ibuprofen to 54.0 for sorafenib, the least soluble drug; the abstract calls it "highly compound dependent". R5·C23 Extinction onsets lie slightly above the fluorescence onsets for every drug, which the authors attribute "either" to a higher detection limit of the extinction method or to its continuous, dynamic addition, compared with the static stepwise additions of the fluorescence assay. R5·R6·C24 The pyrene control gave 53.2 μg/mL for phenylbutazone, against 53.6 μg/mL from its own fluorescence, so the probe does not shift the onset. R7·C25

Two values in Table 5 need care. The extinction ratio printed for diclofenac, 51.6, does not follow from its own inputs: 52.15 / 0.96 = 54.3, while every other ratio reproduces. X1·E15 And the sorafenib ratio is given as 54 in the abstract and "between 50 and 60" in the text, which is the fluorescence value; by extinction it is 64.5. X14

The two free-energy estimates

Table B. Free energies of crystallization at 25 °C and the amorphous "solubilities" they predict through eq 1 (from Table 6). The glass branch was computed only for the two drugs with Tg above 25 °C. R11

Leaving diclofenac aside, the two methods rank the drugs the same way, ibuprofen < phenylbutazone < tolnaftate < griseofulvin < sorafenib, and differ by less than 0.5 kJ/mol. For diclofenac they part ways: 8.51 kJ/mol by Hoffman, 9.99 by the Cp method. R11·C26 The predicted concentrations differ markedly for ibuprofen (about 15 μg/mL), griseofulvin (about 50 μg/mL) and diclofenac (about 25 μg/mL). For ibuprofen and griseofulvin the authors explain this by their high crystalline solubility, which turns a small free-energy difference into a large concentration difference; for diclofenac the free-energy difference itself is large. C27

The Hoffman values can be checked from Table 4. For ibuprofen, $26.1\times (349.05-298.15)\times 298.15/{349.05}^2$ = 3.25 kJ/mol against the printed 3.24, and inserting 3.24 kJ/mol into eq 1 with $C_{\mathrm{e}\mathrm{q}}$ = 29.29 μg/mL and an activity of 0.999 returns 108.1 μg/mL against the printed 108.16. E2·E1 The Cp-method values cannot be checked in the same way, because the fitted slopes and intercepts $a$ to $f$ are not printed. E9·E10

For sorafenib and griseofulvin, Table 7 splits the free energy into enthalpy and entropy:

Table C. Thermodynamic differences between crystal and the two amorphous forms at 25 °C (from Table 7). R12

The glass carries about twice the enthalpy of the supercooled liquid and far more entropy. The entropy partly cancels the enthalpy, but the enthalpy term wins, so the crystal-to-glass free energy is about 1 kJ/mol higher. The authors explain this with Figure 5: below Tg the glass holds more enthalpy and entropy than the hypothetical supercooled liquid at the same temperature. R12·C29 They also computed the supercooled-liquid branch for these two drugs, to allow for water lowering Tg below 25 °C or for a highly relaxed glass forming in water. C28

Which prediction matches the onset?

Table D. Predicted solubility enhancement $f·e^{\Delta G/RT}$ against the measured supersaturation ratio at the onset (from Table 9). R13·R5

The authors find "in general" good agreement between predicted amorphous "solubility" and the onsets, and state that the supercooled-liquid values from the Cp method are "clearly" closer to the measured onsets than the Hoffman values. C30·C31 In concentration terms the Cp-method predictions for ibuprofen, phenylbutazone, diclofenac and griseofulvin (123.06, 53.15, 50.35 and 356.44 μg/mL) lie within about 3% of the fluorescence onsets (125.74, 53.61, 50.14 and 355.96 μg/mL). R11·R5·X5 Diclofenac is the decisive case: Hoffman predicts a ratio of 28.9, the Cp method 52.5, and the measured value is 52.2. R13.d

Plotting measured against predicted ratios (Figures 6 and 7), the line through the origin has a slope of 1.06 and $R^2$ = 0.989 with the Cp method, against 1.20 and 0.926 with Hoffman; the authors conclude that the correlation is "much better" with the Cp method. R14·R15·C32 Note by the AI writer (not discussed in the paper): both $R^2$ values reproduce from Table 9 as the uncentered $R^2$ of a fit forced through the origin; the ordinary (centered) values for the same fits would be about 0.966 and 0.782. The ranking of the two methods does not change, but the absolute values look better than an ordinary regression would make them. X7

Two drugs remain off the line with the Cp method. Tolnaftate's measured ratio is 20.0 against 11.7 predicted (its onset 0.558 μg/mL is 1.7 times the predicted 0.325 μg/mL), and sorafenib's is 54.0 against 48.6. The authors offer an explanation ("may be explained"): at ng/mL concentrations the fluorescence method approaches its detection limit, and the crystalline solubilities themselves are uncertain. R13.c·R13.e·C33·X5

Why the Hoffman estimate fails where it does

Table E. The two quantities that limit the Hoffman approximation, T = 25 °C (from Table 10; T∞ from eq 8 with zero enthalpy difference). R16

The Hoffman equation's validity "depends critically" on small undercooling and on Tg lying close to $T_{\infty }$. C34 Tg − T∞ is largest for diclofenac (145 K) and tolnaftate (143 K); undercooling is largest for griseofulvin (192 K), sorafenib (180 K) and diclofenac (147 K). The authors therefore expect errors "for diclofenac and, to a smaller extent, griseofulvin, sorafenib, and tolnaftate", report the largest deviations from the fitted line for these four, and conclude that the discrepancies come from the Hoffman approximation; diclofenac, the only drug high on both measures, deviates most. R16·C35

Note by the AI writer (not discussed in the paper): the text cites Figure 6 for these deviations, but in Figure 6 (Cp method) diclofenac and griseofulvin sit on the line; the pattern described is that of Figure 7 (Hoffman). X2 And for sorafenib, despite 180 K of undercooling, the Hoffman and Cp ratios are almost the same (48.4 and 48.6), so Table 10 flags where the Hoffman equation can fail rather than how much it fails for a given drug. X8

The glassy drugs separate at the supercooled-liquid value

Griseofulvin and sorafenib are glasses at 25 °C when dry. They absorb only an estimated 2.7% and 1.8% water at a water activity of 1, and amorphous samples stored at 97% RH still had Tg of 88 °C (griseofulvin) and 49 °C (sorafenib), above the experimental temperature. The authors therefore anticipated that any new noncrystalline phase would be a glass. R8·R9·C14 Note by the AI writer (not discussed in the paper): the storage time, water content and number of these Tg measurements are not given, and sorafenib loses 39 °C of Tg with less water than griseofulvin, which loses about 3 °C; the values are best treated as indicative. X11

The measured onsets, however, match the supercooled-liquid predictions: sorafenib 0.532 μg/mL against 0.479 (supercooled liquid) and 0.725 (glass); griseofulvin 355.96 μg/mL against 356.44 and 531.41. Good agreement was found "only" when the supercooled liquid was assumed. R6·R5·R11·C36 The authors' reading: "This suggests that the new phase is much more relaxed than the glass formed by quenching of the melt", with a heat capacity closer to that of the supercooled liquid, which they find "unsurprising" because the phase forms in water, where mobility "is likely to be higher" and relaxation fast. They could not isolate the phase to test this, because crystallization followed too quickly; its disordered structure rests on the fluorescence response. C37·C38 Both drugs therefore undergo glass–liquid phase separation, yet behave "phenomenologically the same" as the drugs with Tg below 25 °C. C40

Discussion

The authors' picture. The paper's argument can be laid out as a chain:

"All of these results suggest", in the authors' words, that the extent of supersaturation is limited by the "solubility" of the amorphous drug, or more precisely of the water-saturated supercooled liquid or glass; above it, added drug converts to a supercooled liquid or a glass, depending on its Tg, the temperature and plasticization by absorbed water. C39 The conclusions state that all six drugs undergo liquid–liquid or glass–liquid phase separation before crystallizing, that the onset "was found to correlate in all instances" with the predicted amorphous "solubility", and that the Cp method gives "much closer values" than Hoffman. C41·C42·C43

A conclusion that rests on a hypothesis. The authors are explicit about the logic of their last step: "Based on the hypothesis that the LLPS concentration is governed by the 'solubility' of the amorphous supercooled liquid, one can conclude that the Cp method estimates more accurately free energies of crystallization compared to the Hoffman equation." C43 The comparison therefore tests the two free-energy methods only if the onset is accepted as the amorphous "solubility", and the evidence that the onset equals the amorphous "solubility" is mainly the agreement obtained with the Cp method. The probe experiments show that the new phase is disordered, but they do not measure its free energy. Note by the AI writer (not discussed in the paper): the two claims support each other and are consistent, but neither is established independently of the other. X4

Where the approach applies. The authors present the titration with two read-outs as an alternative experimental route to the amorphous "solubility", "particularly useful" for drugs that crystallize too fast for a dissolution measurement; griseofulvin, whose amorphous "solubility" had never been determined experimentally, is their example. C44 They suggest that the resulting ceiling "may be useful to predict improvements in biological exposure" for amorphous solid dispersions and other supersaturating formulations; no formulation or exposure data were collected here. C45 For fast crystallizers, the fluorescence check is "clearly essential". C21

What it suggests for formulation. Note by the AI writer (not discussed in the paper): the ceiling differs more than tenfold between drugs, from about 4 for ibuprofen to 54 for sorafenib, so the gain from making a drug amorphous is worth estimating drug by drug. The inputs are modest: a crystalline solubility, DSC runs on crystal and amorphous material, and one water-sorption isotherm. For high-melting drugs the Hoffman shortcut can be badly off (diclofenac 27.71 μg/mL predicted against 50.14 measured), and for drugs that are glasses at the use temperature the glass estimate overshoots by about 40–50%, so the supercooled-liquid estimate is the one to use. The ceiling says nothing about how long supersaturation lasts, which for these drugs was minutes, and it was measured in buffer containing about 1–2% organic solvent and, for griseofulvin, dissolved polymer. R5·R11·X6·X9

Limitations

Limits the authors state.

• The cosolvent quench probes the upper limit of supersaturation, but "cannot be applied" to the supersaturation generated in the gut by a dissolving dosage form. C5
• For much larger undercooling, a linear heat capacity may introduce errors: "it is possible that small errors will result". C10
• The Cp method requires an un-ionized, nonsolvated crystalline form that melts without degrading; salts, hydrates and other solid forms cannot be used. C11
• The remaining gaps for tolnaftate and sorafenib "may be explained" by the detection limit of the fluorescence method at ng/mL and the uncertainty of very low crystalline solubilities. C33
• The glassy phase of griseofulvin and sorafenib could not be isolated because crystallization followed too quickly. C38

The AI writer's own judgement.

• The heat-capacity parameters $a$ to $f$ are not printed for any drug, and raw heat-capacity data are shown only for tolnaftate (Figure 3). The Cp-method free energies, the $T_{\infty }$ values and therefore the central comparison cannot be recomputed by a reader. E9·E10·R10
• The superiority of the Cp method is concluded on the hypothesis that the onset equals the supercooled liquid's "solubility", while that hypothesis is supported mainly by the Cp-method agreement. X4
• "Correlate in all instances" and "excellent agreement" cover a tolnaftate onset 1.7 times its prediction. X5
• The griseofulvin onset was obtained in buffer with 100 μg/mL HPMCAS-HF, which Tables 5 and 8 do not mention, while the crystalline solubility method does not mention polymer. X6
• The onset criterion is not defined, the probe read-out differs between drugs, and the sorafenib probe shifts in the opposite direction to tolnaftate's. X12·X3
• The $R^2$ values in Figures 6 and 7 are those of fits through the origin; the diclofenac extinction ratio is misprinted (51.6 for 54.3); the text cites Figure 6 where it means Figure 7; and eq 4 uses symbols that clash with eqs 5–7. None of these changes the ranking of the methods. X7·X1·X2·X13
• All work is at 25 °C in buffer with about 1–2% organic solvent; whether the same ceiling is reached when an amorphous solid or an amorphous solid dispersion dissolves was not tested. X9

Conclusion

The paper makes two contributions. Experimentally, it shows that a fast-paced solvent-shift titration, read by both light extinction and an environment-sensitive fluorescence probe, catches a short-lived disordered drug-rich phase for six drugs that crystallize too quickly for a dissolution test, giving supersaturation ceilings from 4.3 (ibuprofen) to 54.0 (sorafenib); griseofulvin needed 100 μg/mL HPMCAS-HF to show it. R3·R5·C41 Theoretically, it computes the free energy between crystal and amorphous form from measured, linear-in-temperature heat capacities. With the new phase treated as a supercooled liquid, the predicted amorphous "solubility" lies within a few per cent of the onset for four drugs, closes the large Hoffman error for diclofenac, and fits the measured ratios with slope 1.06 and $R^2$ 0.989, against 1.20 and 0.926 for Hoffman. R11·R13·R14·R15·C43 For the two drugs that are glasses at 25 °C, the onset follows the supercooled-liquid rather than the glass prediction, which the authors read as a new phase "much more relaxed" than a quenched glass. C36·C37 The conclusion that the Cp method is the more accurate one is drawn, in the authors' words, "based on the hypothesis" that the onset is the supercooled liquid's "solubility", and tolnaftate remains off the line. C43·X4·X5