Measuring how much a drug likes its polymer: interaction parameters from melting point depression and from solubility in a liquid analogue of PVP

Background and motivation

An amorphous drug can dissolve faster and reach higher apparent solubility than its crystal, but it tends to crystallize back over time. Dispersing it at the molecular level in a polymer (an amorphous solid dispersion, which this paper calls an "amorphous molecular level solid dispersion") helps on both counts: the dispersion keeps the drug amorphous, and drug–polymer blends that are miscible resist crystallization better than the drug alone (Andronis and Zografi 2000; Crowley and Zografi 2003, refs 28, 29). The introduction places this in a broad context of drug–polymer products, from contact lenses and stents to oral formulations, and notes the estimate that about half of new molecules have solubility problems (ref 27).

The difficulty the paper addresses is how miscibility is judged. In practice a blend made by melt extrusion, spray drying or freeze drying is called miscible when it shows a single glass transition temperature (Tg) (refs 30–38). The authors point out that a single Tg is "not an infallible indicator of miscibility" and, more importantly, says nothing about the thermodynamics of mixing: it does not tell whether the blend will stay mixed at other temperatures or compositions, nor how much drug the polymer can actually dissolve. Polymer scientists have long used quantitative methods for polymer–polymer blends (refs 30–32, 35, 36, 40–44); for drug–polymer pairs the thermodynamics of mixing was, in the authors' words, "relatively unexplored".

The authors frame the chemistry in terms borrowed from polymer science (Coleman and Painter, refs 32, 41, 45). Mixing is exothermic when strong adhesive interactions (between drug and polymer) form at the expense of weak cohesive ones (within each pure component); if forming the adhesive contacts means breaking strong cohesive ones, the enthalpy of mixing "may not be as favorable". Several authors, including the Taylor group (refs 46–51), had linked specific interactions such as hydrogen bonds to the formation of single-phase dispersions.

A second question is solubility. Even when the amorphous drug mixes with the polymer, the crystalline drug has a finite solubility in the polymer. Above that solubility the dispersion is thermodynamically supersaturated with respect to the crystal. The solubility of a crystal in any liquid follows from two factors: the crystal lattice (heat of fusion, melting point, heat capacities) and the activity coefficient of the drug in the mixture (Sandler 1999; Yalkowsky 1999, refs 52, 53). The lattice term is accessible from thermal data; the activity coefficient of a drug in a polymer is not, because polymers are viscous, have no measurable vapour pressure and are usually out of equilibrium at room temperature.

In an earlier paper (Marsac, Shamblin and Taylor 2006, ref 39) the group had used melting point depression to estimate χ for felodipine and nifedipine with PVP. Here they extend that work: they compare two experimental routes to the activity coefficient of a drug in a polymer, the melting point depression of the drug in the polymer and the solubility of the drug in a low-molecular-weight liquid analogue of the polymer; they discuss the strengths and limits of each; and they use the results to estimate free energy of mixing curves, the solubility of crystalline drugs in PVP and the thermodynamic driving force for crystallization (C45).

Background concepts

The Flory–Huggins free energy of mixing. Flory–Huggins lattice theory describes mixing of molecules of very different size by placing them on a lattice of equal sites. A molecule occupies $m$ sites, its molar volume divided by the volume of one site. The free energy of mixing per site is (Eq. 10):

Here $\Phi$ is a volume fraction, $R$ the gas constant, $T$ the absolute temperature and $\chi$ the Flory–Huggins interaction parameter. In plain terms: the first two terms are the combinatorial entropy of mixing, always negative (favourable), but small for a large molecule because ${\Phi }_{polymer}/m_{polymer}$ is tiny when the chain occupies thousands of sites; the last term is the enthalpy of mixing, set by $\chi$. A negative $\chi$ means that drug–polymer contacts are more favourable than the average of drug–drug and polymer–polymer contacts (exothermic mixing); $\chi =0$ is athermal mixing, driven by entropy alone; a positive $\chi$ opposes mixing. Because drug molecular weights are about 200–600 and those of pharmaceutical polymers 10,000–1,000,000, the entropy term is roughly the same for any drug–polymer pair, so, as the authors put it, any pair "can tolerate a certain maximum (unfavorable) enthalpy of mixing", and predicting miscibility comes down to knowing $\chi$ (C26, R13).

Activity coefficient and ideal solubility. The solubility of a crystalline drug in a liquid, written as a mole fraction $x_{drug}$, obeys (Eq. 1):

Here ${\gamma }_{drug}$ is the activity coefficient of the drug at the solubility limit, $\Delta G_{fus}$ the free energy difference between the supercooled liquid and the crystal, $\Delta H_{fus}$ the heat of fusion, $T_m$ the melting temperature and $\Delta C_P$ the heat capacity difference between liquid and crystal. In plain terms: the right-hand side is the free energy penalty of turning the crystal into liquid at temperature $T$, fixed by the crystal's thermal properties; it sets the ideal solubility (the solubility when $\gamma =1$). A real solvent changes the solubility through $\gamma$: $\gamma <1$ means the drug dissolves better than ideal, $\gamma >1$ worse. If the ideal solubility is known, measuring the real solubility gives $\gamma$ directly.

Combinatorial and residual parts. The activity coefficient can be split into a combinatorial part (non-ideal entropy of mixing, which depends on molecular size) and a residual part (the enthalpy of mixing): exothermic mixing lowers the residual part, endothermic mixing raises it (Eqs. 3 and 4). The authors add that the two parts are linked, since order imposed by strong interactions lowers the entropy of mixing (C10).

Melting point depression. A crystal melts when the chemical potential of the crystalline drug equals that of the molten drug. If the molten drug mixes with a polymer, its chemical potential in the mixture is lower than that of the pure melt, so the crystal melts at a lower temperature. Strongly exothermic mixing should give a large depression; weakly exothermic, athermal and endothermic mixing progressively less; and an immiscible polymer none, because it does not change the chemical potential of the melt (C5). The Flory–Huggins form of this relation is (Eq. 2):

Here $T_M^{pure}$ is the melting point of the pure drug, $T_M^{mix}$ the depressed melting point in the presence of the polymer, $MV$ a molar volume and $M{V}_{lattice}$ the volume of a lattice site, which the authors set equal to the volume of one 1-ethyl-2-pyrrolidone molecule (Table II footnote b). In plain terms: the first two terms in the bracket are the depression that entropy alone would cause (the "athermal" prediction, drawn as hashed lines in Figures 2–4); the $\chi$ term adds or subtracts depression according to the interaction. More depression than the athermal line means negative $\chi$; less means positive $\chi$; none means the components do not mix, so the enthalpy of mixing cannot be measured.

Onset and offset of melting. A DSC melting peak has an onset (where melting starts) and an offset (where it ends). In a physical mixture of crystals and polymer the offset reflects the melting point at the final composition, assuming complete mixing (Nishi and Wang 1975, ref 56), and this is the point the authors use.

Methods in detail

The paper has four connected pieces of work: an estimate of χ from melting point depression, an estimate of the activity coefficient from solubility in a liquid analogue of PVP, the use of these numbers to compute free energy of mixing and crystalline solubility, and a look at the chemistry behind the numbers (FTIR and computed dipole moments). The systems were chosen because their miscibility had already been classified by other means (Table I): five compounds with PVP K12 (felodipine, nifedipine, ketoconazole, sucrose, indomethacin), all reported miscible; three with dextran (sucrose miscible, ibuprofen and indomethacin immiscible); and itraconazole with Eudragit E100, reported partially miscible (R1).

Contribution 1: χ from melting point depression

Problem. To use Eq. 2 the drug and polymer must actually mix while the crystal melts, and the melting point must be read at a well-defined composition. A physical mixture of crystals and polymer powder starts with the two components touching only at particle surfaces, so the local composition near a melting crystal is not the bulk composition, and mixing during heating takes time (C8).

What they did. Crystalline drug and non-crystalline polymer were sieved to 45–90 µm and dried over nitrogen for at least a week, then mixed as powders; one to three samples were prepared at each polymer concentration. Melting was recorded on a TA 2920 modulated DSC at 1 °C/min under nitrogen (45 ml/min), with temperature calibrated against indium and benzophenone. The offset of melting was taken as the extrapolated offset of the bulk melting endotherm. Polymer volume fractions reached about 0.23 with PVP K12 and about 0.54 with Eudragit E100 (Figures 2–4). Eq. 2 was linearized in χ as in the earlier paper and fitted to the offset melting points (P1, P6). The input properties are in Table II: molecular weight, density, molecular volume, heat of fusion and melting point. Densities were measured by helium pycnometry or taken from the literature; PVP K12 was given 2,500 g/mol, dextran 860 g/mol (the supplier's MW; MN = 570) and Eudragit E100 4,840 g/mol, "assuming 20 repeat units in the polymer" (M8, M11, M12).

For felodipine one more test addressed the mixing problem. The crystals were mixed either with pure PVP K12 or with an amorphous felodipine–PVP K12 dispersion made by rotary evaporation from ethanol, at the same overall composition. The χ in Table II for felodipine was measured against a dispersion containing 50 wt% felodipine (P3, R8).

Conditions the method needs. The authors list them (C6, C7): the drug must melt before it decomposes; the polymer must be liquid-like at the drug's melting point, so the method suits polymers whose Tg lies well below the drug's Tm (a miscible higher-Tg polymer might instead be plasticized by the drug); χ refers to temperatures close to the drug's melting point; and Eq. 2 is linear only at relatively low polymer content, "likely a result of the kinetics of mixing", so the χ obtained is "a composite value over this limited concentration range".

How it differs from earlier work. Melting point depression is a standard tool for polymer–polymer blends (refs 34, 54–56). The earlier Taylor-group paper (ref 39) applied it to two drugs with PVP; this paper applies it to nine pairs that include immiscible ones, and compares it with an independent route.

Contribution 2: the activity coefficient from solubility in a liquid analogue of the polymer

Problem. The activity coefficient of a drug in a polymer cannot be measured directly. A small molecule that resembles the polymer's repeat unit can be measured: the drug's solubility in it gives its activity coefficient there. 1-Ethyl-2-pyrrolidone (EP) is such an analogue of the vinylpyrrolidone unit of PVP.

What they did (experiment). Excess crystalline drug was stirred in EP in a capped, jacketed vessel at 25 °C for at least 24 h, protected from light; the liquid was decanted, centrifuged, filtered through a 0.2 µm nylon filter, diluted with ethanol and assayed against a standard curve: by UV-Vis for felodipine, nifedipine, ketoconazole and indomethacin, and by HPLC with refractive index detection for sucrose (P2). The number of replicates is not stated. The ideal solubility came from Eq. 1 with literature values of the heat of fusion, melting point and heat capacities (refs 51, 59). For ketoconazole no heat capacity data were available, so the heat capacity difference between liquid and crystal was approximated by its value at Tg, integrated between Tg and Tm, as done before by Hancock and Parks (ref 60). The activity coefficient in EP is then the ratio of the ideal to the measured solubility (P7).

What they did (model). The step from EP to PVP is a chain of assumptions, written out as Eqs. 3–8. Splitting each activity coefficient into combinatorial and residual parts, the difference between PVP and EP is (Eq. 5):

In plain terms: the change in activity coefficient from the small analogue to the polymer is the change in its entropy part plus the change in its enthalpy part. The authors then assume, first, that the drug mixes with EP with ideal entropy, so the combinatorial part in EP is $\ln (1)=0$ (Eq. 6), and second, that the residual parts are identical, that is, the drug interacts with EP exactly as it does with a PVP repeat unit (Eq. 7):

In plain terms: under these two assumptions, the only difference between the drug in EP and the drug in PVP is the smaller entropy of mixing with a long chain. Inserting the Flory–Huggins combinatorial term gives (Eq. 8):

In plain terms: the activity coefficient in PVP is the one measured in EP, corrected for the reduced entropy of mixing with the polymer. The authors stress that this is the activity coefficient at the solubility limit in EP, and that using it to compute the solubility in PVP through Eq. 1 assumes the same value at the solubility limit in the polymer (C11). To compare with the melting route, the activity coefficient is converted into χ by Eq. 9, which adds the χ term to Eq. 8's bracket:

In plain terms: an activity coefficient measured at one composition can be expressed as the χ that would produce it in Flory–Huggins theory, on the same lattice as Eq. 2. The predicted solubility was computed for PVP K12 (2,500 g/mol), PVP K29/32 (40,000 g/mol) and PVP K90 (molecular weight not printed).

Why it can work, and its limits. The authors state two assumptions plainly: that the drug's solubility in EP "provides an indication of the compatibility of the drug with a polymerized version of the solvent", and that the result is a solubility in an equilibrium liquid, not in a glass (C9). Note by the AI writer (not discussed in the paper): the molar volume of EP, which defines the lattice site in both Eq. 2 and Eqs. 8–9, is not printed, nor are the heat capacities or the ideal solubilities, so none of the χ, γ or predicted solubility values can be recomputed from the paper alone (card section F).

Contribution 3: from χ to free energy of mixing and to the driving force for crystallization

The χ values from both routes were inserted into Eq. 10, with χ assumed independent of composition, to draw ΔG/RT against polymer volume fraction for each PVP pair and for sucrose–dextran (Figure 5). Each panel shows three curves: athermal (χ = 0), melting-route χ and solubility-route χ. The figure does not state which PVP grade (and so which $m_{polymer}$) or which temperature was used, and its y-axes carry no numbers (X12).

The predicted solubility in PVP K29/32 was then compared with the drug loading. Any drug above the solubility is "supersaturated" (the authors' quotation marks) and counted as available for crystallization. Figure 6 plots this fraction against polymer content for the five compounds. The relation used is not printed. Note by the AI writer: at 0.50 polymer the indomethacin curve reads about 0.82, which is higher than the simple ratio (loading minus solubility) divided by loading (0.71–0.73 for a solubility of 0.144–0.134); the curve therefore seems to count the drug that the polymer can hold at its saturation ratio, but the paper does not say (X13).

Contribution 4: the chemistry behind the numbers

Two kinds of data link χ to molecular structure (Table IV). Spin-coated films of the drug–polymer systems on ZnSe, prepared in a glovebox below 10% relative humidity at one hydrogen bond donor group per PVP carbonyl group, were measured by FTIR (128 scans, 4 cm−1 resolution); the shift of the PVP carbonyl band to lower wavenumber indicates hydrogen bonding to the carbonyl (P4). Dipole moments of single molecules were calculated with Gaussian 03 at the B3LYP level (P5). The authors use these, together with the drugs' structures (Figure 1), to discuss which interactions can make mixing exothermic.

Experimental analysis

What the data are. Melting data are shown for three systems (Figure 2, indomethacin–PVP K12; Figure 3, ketoconazole–PVP K12; Figure 4, itraconazole–Eudragit E100); the other six pairs are described in the text with "data not shown" (R5). χ for all nine pairs is in Table II, the solubility data and predictions in Table III, and the FTIR shifts and dipole moments in Table IV. The table below gathers the melting route.

Table A. The melting route for the nine pairs (Tables I and II; melting temperatures read from Figures 2–4, approximate). Card rows R1–R5, R7.

Main result 1: miscible pairs depress the melting point, immiscible pairs do not. For indomethacin with PVP K12 (Figure 2) the offset melting point fell steadily from about 161.5 °C to about 155 °C as the PVP volume fraction rose to about 0.23, more than the athermal line, which predicts only about 158.6 °C even at 0.25. For ketoconazole (Figure 3) the offset fell from about 150.0 to 147.5 °C, slightly more than the athermal prediction. For itraconazole with Eudragit E100 (Figure 4) the offset stayed between about 168.0 and 167.4 °C up to a fraction of about 0.54, although entropy alone predicts a fall to about 156.5 °C at 0.6. The authors conclude that systems identified as miscible "show melting point depression" while those identified as immiscible or partially miscible "show little or no melting point depression" (C12). Note by the AI writer: the x-axis of Figure 4 and the second sentence of its caption name "PVP K12", although the figure shows itraconazole with Eudragit E100, and its first caption sentence speaks of weight fraction rather than volume fraction (X5).

Main result 2: the point of the melting peak matters a great deal. In Figure 2 the onset melting point of indomethacin fell much further than the offset, from about 160 °C to about 135.5 °C. χ from the same data was −6.16 from the onset, −1.94 from the midpoint and −0.82 from the offset (R6). The authors argue that "theoretically" the offset should be used, since it gives the melting point at the final composition assuming complete mixing (ref 56), and all values in Table II are offset values (C16). The eight-fold range for one pair shows how sensitive χ from this route is to how the thermogram is read.

Main result 3: χ sorts the pairs. All five PVP pairs reported miscible gave negative or near-zero χ (−0.82 to 0.02); the immiscible pairs gave larger positive values (C17). Where the melting point did not move, the value is only a lower limit: "in the limit of complete immiscibility, the melting temperature will remain unaffected by the presence of the polymer" (Table II footnote e; C19). How a finite value was obtained in those cases is not explained (X11). The text says the values range "to as high as 0.72 for the ibuprofen–dextran system", but Table II assigns 0.72 to indomethacin–dextran and 0.38 to ibuprofen–dextran (X1). Two further cautions come from the authors: dextran's Tg of around 140 °C means the depression with dextran "may be underestimated" (C20), and sucrose degrades on melting, so its data are "a gross estimation" (C14).

The felodipine test of mixing kinetics gave, in the authors' words, a melting point depression with the premixed dispersion "slightly lower than when in contact with pure PVP K12" (data not shown). From this they conclude that "mixing of the drug and the polymer over the timescale of the experiment does contribute to an underestimation of the true interaction", most noticeably when the drug's melting point is close to the polymer's Tg (C21, C22). Note by the AI writer: if the premixed sample, in which mixing is not limiting, gave a smaller depression, the comparison would point the other way; the conclusion follows if it was the melting point, not the depression, that was lower (X6).

Table B. Solubility in 1-ethyl-2-pyrrolidone at 25 °C as weight fraction w and mole fraction x, activity coefficient γ and χ, and predicted weight-fraction solubility in three PVP grades (from Table III). The predicted activity coefficients in PVP are 3.94, 1.18, 5.04, 1.76 and 0.50 in K12 (same order), slightly higher in K29/32 and K90. Card row R9.

Main result 4: solubility in EP gives more favourable χ. All five compounds dissolved in EP better than their ideal solubility, so every γ is below 1, from 0.89 (felodipine) to 0.15 (indomethacin; given as 0.146 in the discussion) (C23). The χ values from this route are −0.08, −0.81, −0.12, −0.53 and −1.83 for felodipine, nifedipine, ketoconazole, sucrose and indomethacin, against −0.08, 0.00, −0.08, 0.02 and −0.82 from melting. The authors find "a reasonable consistency" between the two sets and note that the solubility values "in general" indicate more favourable interactions, "most probably" because they refer to 25 °C and one composition (the solubility limit) while the melting values refer to the melting point and low polymer fractions, and because mixing may be kinetically hindered in the melting test (C27, C28). Note by the AI writer: the agreement is exact only for felodipine; the differences reach 0.81 (nifedipine), 0.55 (sucrose) and 1.01 (indomethacin), and the two routes rank the compounds differently: by melting, nifedipine and sucrose are the least favourable; by solubility, they are second and third most favourable (X7). The later statement that felodipine, nifedipine, sucrose and ketoconazole "all yield slightly negative or close to zero interaction parameters" fits the melting values, but not the solubility values of nifedipine and sucrose, which are close to the −0.82 that the authors call "much larger in magnitude" for indomethacin (X8).

Main result 5: the free-energy curves. Figure 5 shows that with the melting-route χ, indomethacin–PVP has a ΔG/RT curve clearly below the athermal one, while nifedipine, felodipine, ketoconazole and sucrose with PVP lie about on it. With the solubility-route χ the curves are deeper: indomethacin most, then nifedipine and sucrose, with felodipine and ketoconazole similar to each other (C30). Read from Figure 5e, the minimum of the indomethacin curve is about 2.4 times (melting route) and about 5 times (solubility route) as deep as the athermal minimum; all minima lie at polymer fractions of about 0.5–0.6. For sucrose with dextran the melting-route curve lies above the athermal one, as a positive χ (0.20) implies. Curves for the immiscible pairs are not drawn, because their enthalpy of mixing cannot be accessed. The authors conclude that the simulations "illustrate how the magnitude of the interaction parameter rather than the entropy term leads to differences between the various compounds", since the combinatorial entropy is "virtually constant" across systems (C31).

Main result 6: low crystalline solubility in PVP. Correcting for the smaller entropy of mixing with the polymer lowers the predicted solubility sharply: indomethacin falls from 0.396 weight fraction in EP to 0.144 in PVP K12, "a factor of more than two", and only slightly further to 0.134 in PVP K29/32, with K90 the same. The authors explain the plateau by the entropy term $(1/m_{drug}-1/m_{polymer})$, which "quickly approaches $1/m_{drug}$" as the chain grows (C24). Across the five compounds the predicted solubility in PVP K12 is 0.011–0.144 and in PVP K29/32 0.010–0.134. In Figure 6 the fraction of drug available for crystallization reaches zero only at about 0.86 (indomethacin), 0.93 (felodipine), 0.95 (nifedipine) and 0.99 (sucrose, ketoconazole) weight fraction polymer; at 0.50 polymer it is about 0.82–0.99 (R11). The authors read this as follows: "It is clear that these predictions suggest that there is a thermodynamic driving force for crystallization of the drug at most practical polymer concentrations", so that "thermodynamically stable solid dispersions" are difficult to achieve and any apparent stability "is most likely achieved through modification of the kinetics" (C34). They add that the small dissolved fraction "may help explain why 100% crystallization of the drug is not always observed" (C35). One number in this passage disagrees with the table: the discussion gives 0.144 as the predicted solubility of indomethacin in K29/32, whereas Table III and the results section give 0.144 for K12 and 0.134 for K29/32; the zero point of the indomethacin curve in Figure 6 (about 0.86) cannot tell the two apart (X2).

Table C. Interaction parameters set against chemistry (Tables II, III and IV). Dipole moments were also computed for ibuprofen (1.76 D), alpha-D-glucose (4.22 D) and vinyl pyrrolidone (3.91 D). Card rows R7, R9, R12, R14.

Main result 7: the chemistry behind χ. The authors start from the structures: all the compounds are polar, so they can form interactions beyond the weak van der Waals forces that, alone, would make mixing endothermic (C36, C37). Felodipine, nifedipine and sucrose hydrogen bond to each other in the amorphous state (refs 61, 49), but the authors argue that this self-association is compensated by hydrogen bonds to PVP that are stronger than the drug–drug bonds, because PVP's carbonyl is a better acceptor than any group in the drugs; the downshift of the PVP carbonyl band is their evidence (C39). Ketoconazole has no hydrogen bond donor (ref 63) and causes no carbonyl shift, yet it mixes about athermally; "it seems reasonable to speculate" that it interacts with PVP through dipole–dipole forces, given its large computed dipole moment (6.03 D), and its large size is "believed to contribute considerably" to van der Waals interactions (C40, C41). For indomethacin, the large negative χ (−0.82) and low γ in EP (0.146) "can most likely be rationalized" by the acidity of its carboxyl donor, expected to form stronger hydrogen bonds with PVP than the NH groups of felodipine and nifedipine or the OH groups of sucrose; the larger carbonyl shift (45 cm−1 against 25) supports this, and indomethacin "appears to form the most energetically favorable mixture with PVP" (C42). Note by the AI writer: Table IV gives no carbonyl shift for nifedipine, so its hydrogen bonding to PVP rests on the earlier paper (ref 51), not on this table (X9); and nifedipine, with the largest dipole moment (9.58 D), is not discussed in terms of dipoles.

The immiscible pairs. For dextran with ibuprofen and with indomethacin neither drug is expected to form contacts with dextran stronger than those in the pure components, and "it can be speculated that mixing is not achieved because the mixing entropy is not sufficient to offset the unfavorable adhesive interactions" (C43). Eudragit E100 is a cationic methacrylate polymer with strong self-interactions of ionic origin, and itraconazole interacts mainly by dispersion forces; "it can be predicted that adhesive interactions are not energetically favorable relative to cohesive interactions", consistent with the reported miscibility gap and the absent depression (C44). No interaction measurements were made on these pairs, and the Tg evidence cited as "this study" for indomethacin–dextran is not shown (X10).

Discussion

How the pieces fit together. The authors' argument runs as a chain.

Where the approach applies. The melting route needs a drug that melts without decomposing and a polymer whose Tg lies well below the drug's melting point, so that the polymer is liquid-like when the crystal melts; it gives χ near the drug's melting point and at low polymer fractions (C6, C7). The solubility route needs a liquid analogue of the polymer; the authors used it only for PVP, as their conclusion says ("for PVP systems"). It gives χ at room temperature and at one composition, the solubility limit, and it rests on the assumption that the analogue interacts with the drug exactly as the polymer's repeat unit does (C9, C11, C27). Both routes treat χ as a single number, although the authors note that the activity coefficient varies with temperature and composition, which is why they prefer the solubility route for estimating solubility in the polymer at room temperature (C32).

What remains open. The authors state that mixing in the melting test "may still be kinetically limited" and that the interaction is then underestimated, most for polymers whose Tg is close to the drug's melting point (C8, C22). How χ changes between the melting point and storage temperature, and across composition, is not measured. Nor is the predicted solubility of any crystalline drug in a polymer compared with a measurement, and no stability data test the conclusion that stability is kinetic. Note by the AI writer (not discussed in the paper): the paper's own numbers show how much these routes can differ. For one pair the choice of onset, midpoint or offset changes χ from −6.16 to −0.82; for another, the two routes give 0.00 and −0.81.

What it suggests for formulation. The authors' message is that miscibility "can be explored by coupling solution theories with experimental data", and that the estimates give insight into physical stability and the thermodynamic driving force for crystallization (C47). They also conclude that the solubility of most drugs in polymeric matrices "is likely to be low at room temperature" (C46). Note by the AI writer: on these numbers, a typical drug loading of 20 wt% or more in PVP would be far above the predicted crystalline solubility (1–14 wt%) for all five compounds, so such dispersions rely on kinetic stabilization, for example a high Tg and low molecular mobility. Within this set, the clearest sign of favourable mixing with PVP was a strong hydrogen bond donor that shifts PVP's carbonyl band (indomethacin); a large dipole without donors (ketoconazole) gave only about athermal mixing.

Limitations

Limitations stated by the authors. For the melting route: Flory–Huggins assumptions "may lead to inconsistencies between theoretical predictions and experimental data"; crystals and polymer meet only at particle surfaces and mixing during the experiment is finite, so "mixing may still be kinetically limited" (C8); Eq. 2 is linear only at low polymer content, so χ is "a composite value over this limited concentration range" (C7); the value depends strongly on the point of the melting peak (C16); in the limit of complete immiscibility χ is only a lower limit (C19); depression with dextran "may be underestimated" because of its high Tg (C20); and sucrose degrades on melting (C14). For the solubility route: the estimate assumes that the solubility in EP indicates compatibility with the polymer and refers to an equilibrium liquid rather than a glass (C9), and it assumes ideal entropy with EP, identical drug–EP and drug–PVP contacts and the same activity coefficient at both solubility limits (C11). The conclusion sums up: "Limitations were found with both experimental approaches" (C45).

The AI writer's own judgement (1): what the numbers can bear. The qualitative result is solid: in the three systems whose data are shown, indomethacin and ketoconazole are depressed by PVP K12 and itraconazole is not depressed by Eudragit E100, and the offset χ values sort all nine pairs in line with their known miscibility. The quantitative values are softer. χ for one pair spans −6.16 to −0.82 depending on the point of the peak; the fit range of Eq. 2 is not stated and the fitted lines miss the curvature of the points in Figure 3 and of the onset points in Figure 2 (X15); six of the nine pairs rest on data not shown (R5); and the two routes differ by up to 1.01 and rank the compounds differently, so "reasonable consistency" is generous (X7). The two routes also refer to different conditions, melting temperatures of about 415–449 K against 25 °C, so they are better compared qualitatively (X16).

The AI writer's own judgement (2): the solubility prediction and internal consistency. The headline that crystalline drugs have "a rather low solubility" in PVP is a model prediction for five compounds, built on the assumptions above, without an experimental check in a polymer and without stability data (X17). Several inputs are not printed (the molar volume of EP that defines the lattice, the heat capacities, the ideal solubilities and the relation behind Figure 6), so the χ, γ and Figure 6 values cannot be recomputed from the paper (X4, X13). The paper also contains a handful of internal inconsistencies: the χ of 0.72 attributed to ibuprofen–dextran in the text belongs to indomethacin–dextran in Table II (X1); the predicted solubility of indomethacin in PVP K29/32 is 0.134 in Table III but 0.144 in the discussion (X2); the molecular volumes in Table II follow from molecular weight and density for every row except indomethacin (267.01 calculated, 268.01 printed) and ibuprofen (196.46 calculated from 206.28/1.05, 158.68 printed, which equals 206.28/1.30) (X3); the Table II melting points lie 2–3 °C below the pure-drug offsets plotted in Figures 2–4 (X4); Figure 4 is labelled with PVP K12 (X5); and the closing sentence asks for "extremely favorable cohesive interactions" where, on the paper's own definitions, adhesive (drug–polymer) interactions would be needed to raise solubility (X14). None of these changes the qualitative conclusions, but they limit the use of individual numbers.

Conclusion

The paper makes three contributions. First, it shows that melting point depression of a crystalline drug in a polymer separates pairs known to be miscible (depression, χ from −0.82 to 0.02 with PVP K12) from immiscible or partially miscible ones (no depression, positive lower-limit χ), while warning that the value of χ depends strongly on how the melting peak is read and on mixing kinetics. Second, it introduces solubility in a low-molecular-weight liquid analogue, 1-ethyl-2-pyrrolidone for PVP, as a second route to the activity coefficient, which gives more favourable χ and, after a Flory–Huggins correction for chain length, a predicted crystalline solubility in PVP of about 0.010–0.144 weight fraction. Third, it links the numbers to chemistry: indomethacin's acidic carboxyl, with the largest carbonyl shift, forms the most favourable mixture with PVP, and ketoconazole, without hydrogen bond donors, is thought to interact through its dipole.

Within its scope, five small molecules with PVP, three with dextran and one with Eudragit E100, with χ treated as independent of composition, the paper offers two practical ways to rank drug–polymer pairs by their interaction and a thermodynamic argument that amorphous solid dispersions at normal drug loadings are supersaturated with respect to the crystal. The next steps it points to are measuring χ over temperature and composition, checking predicted crystalline solubilities against measured ones, and testing whether the predicted supersaturation predicts crystallization on storage.