Water uptake of amorphous drug–polymer blends: why the blend takes up less than its parts, and what Flory–Huggins theory can and cannot extract from it

Background and motivation

Many poorly water-soluble drugs are formulated as amorphous solid dispersions (ASDs): the drug is mixed with a polymer at the molecular level so that no crystal lattice has to be broken on dissolution, and the polymer slows crystallization of the drug back to its less soluble crystalline form (Introduction, refs 1–6). C1 The polymers used for this, such as poly(vinylpyrrolidone) (PVP), hypromellose (HPMC) and hypromellose acetate succinate (HPMCAS), are somewhat hydrophilic and highly disordered, so they are hygroscopic. A blend therefore takes up more water than the amorphous drug alone, the water lowers the glass transition temperature ($T_g$), and this "may lead" to crystallization of the drug. Water uptake matters for physical stability and also for processing. C2

The obvious way to estimate how much water a blend takes up is to add up the water taken up by its parts, weighted by their proportions. Shamblin and Zografi (ref 7) tested this "additivity" approach on colyophilized sucrose–PVP and sucrose–PVP/VA. It worked for mixtures with 50 wt% PVP, but not for 20 wt% PVP or for PVP/VA above 32% relative humidity (RH), where sucrose crystallized. R18 Zhang and Zografi (ref 9) found the same for sucrose–PVP and trehalose–PVP as long as the sugar stayed amorphous, and went further: they fitted the three-component (ternary) Flory–Huggins equation to the sorption data and extracted interaction parameters for sugar–water, PVP–water and sugar–PVP that were all consistent with miscible pairs. R19·C3

The problem that motivates this paper came when Crowley and Zografi (ref 10) applied the same approach to blends of hydrophobic drugs (indomethacin, ursodeoxycholic acid, indapamide) with PVP. Additivity agreed with indomethacin–PVP at 60–90% PVP, but the drug–polymer interaction parameters they extracted from the ternary Flory–Huggins fits were 1.27–1.49, which indicates limited miscibility. R16 They themselves called this counterintuitive: indomethacin–PVP shows a single $T_g$, infrared evidence of drug–polymer hydrogen bonding and an amorphous X-ray pattern, all signs of a miscible blend, and later work estimated a much lower interaction parameter (Marsac et al., ref 14). R20·C4

The paper therefore asks two linked questions. Can the water uptake of a one-phase blend of a hydrophobic drug and a hydrophilic polymer be predicted from its components? And is the ternary Flory–Huggins equation a valid tool for these blends, in particular for extracting the drug–polymer interaction parameter? The authors evaluate several models of increasing complexity on the same set of isotherms to answer them (Introduction). A4

Background concepts

Moisture sorption isotherm. An isotherm is the amount of water a solid takes up at equilibrium as a function of relative humidity, at fixed temperature. Here it is expressed as grams of water per 100 g of dry solid, at 25 °C, from 5 to 95% RH. P2 The relative humidity divided by 100 is the water activity $p/p_0$: the partial pressure of water vapour $p$ divided by the saturated vapour pressure $p_0$.

Additivity. The additivity model predicts the uptake of a blend as the weight-averaged sum of the uptakes of the pure components at the same RH. The authors point out what this assumes at the molecular level: the blend must behave like a physical mixture, so every functional group that binds water in the pure amorphous drug or pure polymer must be just as available to water in the blend. E5·C9 The deviation of a blend from additivity is expressed in Figures 6–8 as a percentage of the additivity prediction; positive values mean that the blend took up less than predicted. E6

Flory–Huggins theory for one solid and water. Flory–Huggins (FH) theory describes mixing in polymer solutions with two parts: an entropic part that depends only on the volume fractions and the sizes of the molecules, and an enthalpic part summarized in one interaction parameter, $\chi$. Treating the absorption of water into an amorphous solid as a dissolution process, the water activity follows (Eq. 1): C16

Here ${\varphi }_1$ is the volume fraction of water, ${\varphi }_2$ the volume fraction of the amorphous solid, $x_{12}$ the size of a solid molecule relative to a water molecule, and ${\chi }_{12}$ the water–solid interaction parameter. In plain terms: once the volume and size of each component are fixed, a single number ${\chi }_{12}$ sets how much water the solid holds at each humidity; a large positive ${\chi }_{12}$ means water and solid dislike each other and little water is absorbed. E1 A negative $\chi$ means net attraction and a positive one net repulsion. For a small molecule mixing with a polymer, a $\chi$ above a critical value of about 0.5 means the two cannot mix in all proportions; for two small molecules the critical value depends on their size ratio. C17 Literature values of $\chi$ for water–PVP, taking the size ratio as infinite, are around 0.5. R17

Flory–Huggins theory for water, drug and polymer. With three components (1 = water, 2 = drug, 3 = polymer) there are three pair parameters: ${\chi }_{12}$ (water–drug), ${\chi }_{13}$ (water–polymer) and ${\chi }_{23}$ (drug–polymer). The paper uses the ternary form (Eq. 2): E2

In plain terms: the first four terms are entropic and depend only on how much of each component there is and how big its molecules are; the next term adds the two water–component interactions; the last term carries the drug–polymer interaction, scaled by the drug's size relative to water. The paper prints the fourth term as ${\varphi }_3/13$; by the symbol definitions it is ${\varphi }_3/x_{13}$. E2·X1 If all three $\chi$ values are known, Eq. 2 predicts the blend isotherm; if the isotherm is measured and two $\chi$ values are known, the third, usually ${\chi }_{23}$, can in principle be fitted. P5

Interaction parameter for drug and polymer. ${\chi }_{23}$ is the parameter that speaks to drug–polymer miscibility: a negative value means net attraction between drug and polymer, and values such as the 1.27–1.49 of ref 10 indicate limited miscibility. C17·R16 It can also be estimated without water, from the melting point depression of the crystalline drug in the polymer, which is how ref 14 obtained the indomethacin–PVP value used in this paper. P5

Mean interaction parameter. For solutions of a solvent in a blend of two polymers, Panayiotou and Vera (ref 29) introduced a mean, or mixed, parameter ${\chi }_{1,23}$ that describes the interaction of the solvent with the blend as a whole; it was used by Sabzi and Boushehri, Eliassi and Modarress, and Csaki et al. (refs 26–28) because the polymer–polymer parameter measured in a ternary system depends strongly on the solvent (ref 25). C33 The pseudobinary equation (Eq. 4) has the same form as Eq. 1: E4

Here $a_1$ is the water activity, ${\varphi }_{23}$ the combined volume fraction of drug and polymer and $x_{23}$ their size relative to water; the paper does not define the last two explicitly. In plain terms: the blend is treated as a single new solid, and one number ${\chi }_{1,23}$ says how much water likes it. E4·C34

Methods in detail

The paper has one experimental data set, equilibrium water vapour isotherms, and four ways of analysing it. The study design is simple: two hydrophobic drugs (felodipine and indomethacin) and three polymers of very different hygroscopicity (PVP K29/32, HPMC grade Pharmacoat 606, HPMCAS grade AQOAT AS-MF), combined into four blend series, felodipine–PVP, indomethacin–PVP, felodipine–HPMC and felodipine–HPMCAS, each across the full composition range, all measured at 25 °C. A1·A2·A3·M1·M2·M3·M4·M5

Contribution 1: a complete set of isotherms for drugs, polymers and blends

① The problem. Testing any model of blend sorption needs isotherms of the pure components and of blends across the whole composition range, all made the same way and all free of crystallization.

② What they did. Each polymer was dried over phosphorous pentoxide for at least one week, mixed with the drug in a glove-box flushed with dry air (RH below 15%), and dissolved in pure ethanol (indomethacin–PVP) or in 1:1 (w/w) dichloromethane–ethanol (the felodipine blends). The solvent was removed on a rotary evaporator, the solid was kept under vacuum for at least 12 h, heated to 5 °C above the melting temperature of the pure drug for about a minute to remove any remaining crystals, cooled and gently ground in the glove-box. P1 Blends made by this method had been shown before to be miscible, with a single $T_g$ (refs 8, 10, 15–17); miscibility was not re-measured in this paper. C5 About 10 mg of sample was dried in the analyzer at below 2% RH for 3 h at 25 °C, then held at one RH (5, 15, 25 … 95%) until its weight levelled off: about 600 min for the pure amorphous drugs and about 1000 min for the blends. Each RH was a separate experiment, and only plateau values without evidence of crystallization were reported. P2 PVP and indomethacin–PVP blends were made at 0, 10, 25, 50, 75, 90 and 100% polymer; the HPMC series is listed with the same levels in Figure 3, and the HPMCAS series at 0, 15, 25, 50, 70, 85 and 100% in Figure 4. A2

③ Why it works. Measuring every RH as a separate experiment from the dry state avoids carrying sorption history from one step to the next, and the melt step makes sure the starting material is fully amorphous. Excluding points with crystallization keeps the data about the amorphous state only. P1·P2

④ Difference from earlier work. Earlier studies of this kind used sugars with PVP (refs 7, 9) or several drugs with PVP only (ref 10). Here the same drug, felodipine, is combined with three polymers that span a wide range of hygroscopicity, so the effect of polymer type can be separated from that of the drug. A1·R16·R18·R19

Contribution 2: testing additivity

① The problem. If additivity held, no interaction parameter would be needed to predict blend uptake. Its failure, and where it fails most, is the first sign that something in the blend differs from a physical mixture.

② What they did. For every blend and every RH from 15 to 85%, the predicted uptake was the weight-averaged sum of the drug and polymer uptakes, and the difference between measured and predicted uptake was normalized against the prediction. These deviations are plotted against polymer content in Figure 6 (felodipine–PVP), Figure 7 (felodipine–HPMC) and Figure 8 (felodipine–HPMCAS); the indomethacin–PVP deviations were computed but not shown. P3·E5·E6·R6

③ Why it works. The comparison needs no model parameters at all, so it isolates the effect of mixing. The authors then compare where the deviation peaks with the hypothesis of Crowley and Zografi that the deviation reflects specific drug–polymer hydrogen bonds and should be largest near one drug hydrogen-bond donor per PVP repeat unit. C11

Contribution 3: testing the ternary Flory–Huggins equation

① The problem. The ternary equation is the natural next step: it adds the interaction terms that additivity ignores. But it needs reliable binary parameters, and the question is whether ${\chi }_{23}$ can be obtained from sorption data at all.

② What they did. First, Eq. 1 was fitted to each pure-component isotherm with the constants of Table I. The size ratios were the molecular-volume ratios to water, 16.7 for felodipine and 14.9 for indomethacin; the ratios used for the polymers are not stated. The fit used $p/p_0$ from 0.15 to 0.85 or 0.95 and minimized the squared error between predicted and measured $\ln (p/p_0)$ at each measured amount of water; one average $\chi$ was reported per isotherm. P4·M1·M2·M3·M4·M5 Table I lists density, molecular weight and the derived molar and molecular volumes: for example 1.28 g/cm³, 384.3 g/mol and 300.2 cm³/mol for amorphous felodipine, and 1.11 g/cm³, 40,000 g/mol and 36,036 cm³/mol for PVP. M1·M3 The molecular volumes are given in the table with the unit cm³/mol, but their values are per molecule (molar volume divided by Avogadro's number). X3

Second, for indomethacin–PVP all three parameters were available, with ${\chi }_{23}$ taken from melting point depression in ref 14 (the value is not printed). The isotherms of the 25% and 75% PVP blends were then predicted with Eq. 2 and compared with the measurements and with additivity (Figure 9). P5·R12

Third, to see which term matters, Eq. 2 was rewritten for the water volume fraction (Eq. 3), shown as printed except that $x_{13}$ is written where the paper prints 13:

In plain terms: the water volume fraction taken up is set by the water activity and the entropic terms (the first five on the right), and is then adjusted up or down by the three interaction terms. A positive ${\chi }_{12}$ or ${\chi }_{13}$ (water dislikes the drug or the polymer) lowers the predicted water; a negative ${\chi }_{23}$ (drug and polymer attract each other) also lowers it. E3·C22 Two misprints should be noted: the polymer size term is again printed as ${\varphi }_3/13$, and moving $-{\varphi }_2/x_{12}-{\varphi }_3/x_{13}$ from the right of Eq. 2 to the right of Eq. 3 should change their signs to plus, while the paper keeps them negative. The interaction terms, which are the ones analysed, change sign correctly. X1·X2 Table II gives, for indomethacin–PVP and felodipine–PVP at 10–90% PVP, the share of each of the three interaction terms in the total interaction adjustment. P6·R13

Fourth, ${\chi }_{23}$ was fitted to the felodipine–PVP and indomethacin–PVP isotherms with Eq. 2, using the binary ${\chi }_{12}$ and ${\chi }_{13}$ from Eq. 1. Only the ranges of the fitted values are reported. P5·R14

③ Why it works. The term analysis does not depend on fitting: it simply asks how much each term changes the predicted water uptake. If the drug–polymer term is tiny compared with the others, then any error in ${\chi }_{12}$ or ${\chi }_{13}$ will dominate a fitted ${\chi }_{23}$. C25·C29

④ Difference from earlier work. Crowley and Zografi reported counterintuitive ${\chi }_{23}$ values without a term-by-term analysis; Zhang and Zografi noted the insensitivity for sugar–PVP. This paper quantifies the insensitivity for two drug–PVP systems and names the likely cause. R16·R19·C25·C31

Contribution 4: a mean water–blend parameter

① The problem. If the binary water–drug and water–polymer interactions change once drug and polymer are mixed, as the authors argue, any approach that imports binary values into the blend is built on a wrong assumption. C26·C27

② What they did. Eq. 4 was fitted to the isotherms of each blend composition in all four series to obtain an average ${\chi }_{1,23}$. For felodipine–PVP only data from 15 to 65% RH were used, because this blend separates into drug-rich and polymer-rich phases at high RH (ref 30). The pure-component values from Eq. 1 are included as the 0 and 100% polymer points of Figure 10. P7·C35·R21

③ Why it works. Treating drug and polymer as one solid removes the need to know, or to assume constant, the binary parameters. The authors argue that this is "more appropriate" for blends in which drug and polymer are mixed at the molecular level, as shown for these systems in ref 16, and "particularly useful" because no binary parameters are needed in advance. C34·C38·C39

④ Difference from earlier work. The mean parameter had been used for solvent–polymer–polymer systems (refs 26–28); this paper applies it to water with drug–polymer blends. C33

Experimental analysis

The pure components: PVP takes up far more water than anything else

Figure 1 shows the isotherms of the five pure materials. PVP is by far the most hygroscopic: about 33 g water per 100 g at 75% RH and about 75 g/100 g at 95% RH. HPMC reaches about 22 and HPMCAS about 12 g/100 g at 95% RH, so HPMCAS is the least hygroscopic polymer. The amorphous drugs take up very little, about 2.1 (indomethacin) and 1.3 g/100 g (felodipine) at 85% RH; indomethacin is slightly less hydrophobic than felodipine. Neither drug has a 95% RH point in the figure; the paper reports only data without evidence of crystallization. R1·C6·X12

Table A. Water sorbed at 25 °C (g/100 g dry solid), read from Figure 1 (approximate), and the χ values with water stated in the text. R1·R11

The binary Flory–Huggins fits had correlation coefficients better than 0.985 for every component, which the authors take as a good fit. R11·C18 The fitted values follow the ranking of the isotherms: 3.27 for felodipine and 2.86 for indomethacin, large positive values that "indicate" unfavourable mixing with water, as expected for hydrophobic drugs; 1.33 for HPMC and 1.62 for HPMCAS, both above 0.5 and in line with these polymers being relatively insoluble in water; and 0.36 for PVP. R11·C19·C20 For comparison, Crowley and Zografi had reported 3.13 for water–indomethacin from a single point at $p/p_0$ = 0.8 with a size ratio of 2, and other authors around 0.5 for water–PVP; the authors call their PVP value similar to, though slightly smaller than, the latter. R16·R17·C20

Note by the AI writer (not discussed in the paper): solving Eq. 1 for $\chi$ at each plotted point of Figure 1, with the densities of Table I, gives point values that bracket the stated averages for both drugs, HPMC and HPMCAS, but about 0.47–0.75 for PVP across 15–95% RH, all above the stated 0.36. A least-squares fit of the kind described cannot fall below every point value, so the PVP value cannot be recovered from the plotted isotherm; the polymer size ratio and the points used are not stated. The plotted isotherm corresponds to a $\chi$ of about 0.5–0.6, close to the literature value. E1·X5

The blends: uptake rises with polymer content, steeply only with PVP

Figures 2–5 show the blend isotherms. In every series, uptake at a given RH rose with polymer content, and the rise was much larger with PVP than with the cellulose polymers at the same polymer content. The authors attribute this to the higher polarity, and hence hygroscopicity, of PVP ("presumably"). R2·R3·R4·R5·C7·C8

Table B. Water sorbed at 85% RH and 25 °C (g/100 g dry solid), read from Figures 2–5 (approximate); the indomethacin–PVP values at 0 and 10% PVP are from overlapping symbols. R2·R3·R4·R5

All blends take up less than additivity predicts

Additivity failed for every system: the measured uptake was "much" lower than the weight-averaged sum. C10·C13 For felodipine–PVP every plotted deviation is positive. It is about 45–57% at 10% PVP and 43–64% at 25% PVP across the eight RH series, falls to about 32–59% at 50% PVP, and to about 9–22% and 2.4–21% at 75 and 90% PVP (Figure 6). R7 Indomethacin–PVP showed "similar trends", but the data are not shown. R6 Figure 9 gives an example: at 25% PVP and 85% RH the indomethacin–PVP blend took up about 3.6 g/100 g, whereas additivity predicts about 14.0. R12

Table C. Deviation of measured uptake from the additivity prediction, as % of the prediction (positive = less water than predicted), read from Figures 6–8 (approximate). R7·R8·R9

Where the shortfall is largest. The authors place the largest felodipine–PVP discrepancies "around 25% polymer". Crowley and Zografi had hypothesized that the shortfall should be largest at about one drug hydrogen-bond donor per PVP repeat-unit acceptor, which corresponds to 76:24 (indomethacin–PVP) and 78:22 (felodipine–PVP) by weight, and the authors write that this "is indeed around" where they see the maximum. C10·C11·C12·R10 The figure supports this for most RH series: the 15, 35 and 85% RH series peak at 25% PVP (about 64, 56 and 52.5%). But the 25% RH series is higher at 10% PVP (about 56.5%) than at 25% (about 43%), so the maximum lies somewhere between 10 and 25% PVP, and the indomethacin–PVP data behind the 76:24 match are not shown. R7·X11

For the cellulose blends the picture differs. The deviations are smaller than with PVP, "much more dependent on RH", and largest between 50 and 70% polymer (Figures 7 and 8). C13 With HPMCAS the RH dependence is clear: at 50% HPMCAS the deviation falls from about 55% at 15% RH through about 47.6 and 43.8% at 25 and 35% RH to about 25% at 85% RH, and at 15% HPMCAS and 85% RH it is about −4.5%, the only case in which a blend took up slightly more water than predicted. R9 The authors' explanation: unlike PVP, HPMC and HPMCAS carry several hydrogen-bond donors and acceptors per repeat unit, so water keeps sites to bind to, and they "have been suggested" to interact more weakly with felodipine (ref 17). If the reduced uptake comes from specific drug–polymer interaction, "it is intuitive" that these blends are less affected, and it is "reasonable to speculate" that the size of the shortfall follows the nature of the drug–polymer interactions and how they change water's access to each component. C14·C15 No measurement of these interactions was made in this paper. X14 Note also that the composition levels of the cellulose blends are not consistent across figures: Figure 7 plots HPMC at 15, 25, 50, 70 and 85%, whereas Figure 3 lists 10, 25, 50, 75 and 90%. X4

The ternary prediction for indomethacin–PVP

Figure 9 compares measured indomethacin–PVP isotherms with the Eq. 2 prediction and with additivity. At 25% PVP, Eq. 2 tracks the data closely (about 3.8 against 3.6 g/100 g at 85% RH), while additivity overpredicts strongly (about 14.0). At 75% PVP, additivity stays within about 2 g/100 g of the data (about 38.0 against 36.0 at 85% RH), while Eq. 2 falls below the data at every RH (about 26.9 at 85% RH). R12 The authors summarize that the additivity model underpredicts at 75% polymer and overpredicts at 25%, that the 75% data are "reasonably similar" to additivity, and that at 25% both the data and the FH prediction are much lower than additivity. C21 Note by the AI writer: in the figure it is the ternary prediction, not additivity, that lies below the 75% PVP data at all humidities; additivity is below the data only at 25–55% RH and above it at 65 and 85% RH. X8

The authors also report that the five entropic terms of Eq. 3 on their own overpredicted uptake for both PVP systems, "suggesting" unfavourable enthalpic contributions, and that including the positive ${\chi }_{12}$ and ${\chi }_{13}$ and the negative ${\chi }_{23}$ brought the prediction closer to the data. C23

The drug–polymer term barely matters

Table II is the central piece of evidence. It splits the interaction adjustment of Eq. 3 into its three parts. R13

Table D. Contribution of each interaction term of Eq. 3, as % of the overall adjustment (from Table II, as printed). R13

Up to 75% PVP, most of the reduction in water uptake is attributed to the more hydrophobic component, the drug; only at 90% PVP does the polymer term carry more. The drug–polymer term never exceeds 0.160%. The authors conclude that this "highlights the insensitivity" of the ternary equation to ${\chi }_{23}$, as Zhang and Zografi had also found for sugar–PVP: drug–polymer interactions do affect blend uptake, but mathematically their contribution is "dwarfed" by the hygroscopicity of the components. C24·C25·R19

Note by the AI writer (not discussed in the paper): the ratio of the water–drug to the water–polymer share in Table II depends only on ${\chi }_{12}{\varphi }_2/({\chi }_{13}{\varphi }_3)$, not on the water content. With the stated values (2.86 and 0.36) and the Table I densities, the indomethacin–PVP ratios are reproduced within 1.5% at every composition, but the felodipine–PVP ratios are about 11% lower than those computed with the stated water–felodipine $\chi$ of 3.27 (6.99 against 7.88 at 50% PVP). Both rows behave as if computed with a water–drug $\chi$ near 2.9. The paper also does not state the RH or water content, or the ${\chi }_{23}$ values, behind Table II. None of this changes the authors' point, because a larger water–drug term would make the drug–polymer share smaller still. E3·X6·X7

Fitted drug–polymer parameters scatter widely

The consequence shows in the fitted values. Fitting Eq. 2 to the blend isotherms gave ${\chi }_{23}$ from −9.6 to 26.9 for felodipine–PVP and from −20.4 to 22.0 for indomethacin–PVP; the paper reports only these ranges, not the individual values, compositions or RH. R14 Because the ${\chi }_{23}$ term is so small, small changes in ${\chi }_{12}$ or ${\chi }_{13}$ caused by the other component produce large changes in ${\chi }_{23}$, so the fitted values "may not be physically meaningful or even show the correct trend", as Crowley and Zografi had inferred. C29 The authors argue that the binary parameters do change in the blend: it is "unlikely" that water–drug, water–polymer and drug–polymer interactions are the same in the ternary system as in the binaries, because drug–polymer interactions "would most likely" change how water interacts with each component, and water will change the drug–polymer interactions. They cite solvent sorption studies in which copolymers took up more solvent than either homopolymer, a miscible PVP–poly(acrylic acid) blend took up less water than expected (attributed to hydrogen bonds between the polymers), and butanone lowered the solubility of toluene in poly(methyl methacrylate). C26·C27·C28·R22 Their conclusion is that the method "cannot be used to unambiguously determine drug–polymer interaction parameters in solid dispersions"; the abstract says it "led to ambiguous conclusions", the conclusions section that it "led to erroneous values", and the cause is "thought" (abstract) or "hypothesized" (conclusions) to be the wrong assumption of unchanged binary interactions combined with the small ${\chi }_{23}$ term. C30·C31

The mean parameter falls smoothly with polymer content

Figure 10 shows ${\chi }_{1,23}$ from Eq. 4 against polymer content. In all four series it decreases as polymer is added, which the authors read as water liking the solid mixture more as the hydrophilic component increases ("can be interpreted"). R15·C36 The fall is steepest with PVP, which they explain by PVP being more hygroscopic than HPMC or HPMCAS ("can be explained"). C37

Table E. Mean water–blend parameter ${\chi }_{1,23}$, read from Figure 10 (approximate). The 0 and 100% points are the pure-component values from Eq. 1. Figure 10 plots the HPMCAS series at 10, 25, 50, 75 and 90%, while Figure 4 lists 15, 25, 50, 70 and 85%. R15·X4·X13

Two features stand out in the table. The two PVP series nearly coincide from 50% PVP upward (about 1.71 and 1.58 at 50%, about 0.37 for both at 100%), although the drugs differ at 0% (about 2.87 and 3.27). And with HPMC and HPMCAS the parameter stays above 1.3 at every composition. R15 The authors present the approach as one that "was found to better describe the changing affinity of water" (abstract) and that "clearly shows" the affinity rising with polymer proportion and hygroscopicity (conclusions). C40 No goodness of fit, no isotherm recalculated from ${\chi }_{1,23}$ and no side-by-side comparison with Eq. 2 is reported. X10

Discussion

How the authors connect the results. The argument runs in four steps. The polymer brings the water-binding groups and the drug is hydrophobic, so blend uptake rises with polymer content. C2·C6·C7 If every component kept its water-binding ability, the blend would take up the weight-averaged sum; it takes up much less, so something in the blend removes water-binding capacity. C9·C10 With PVP, the drug's hydrogen-bond donors bind the single acceptor of the PVP repeat unit, which is then unavailable to water, so the loss is largest near one donor per repeat unit; with the cellulose polymers, which have many sites and bind felodipine more weakly, the loss is smaller. C11·C12·C14 Finally, the ternary equation that should capture this through ${\chi }_{23}$ is insensitive to it, so any change in water–component interactions inside the blend ends up in the fitted ${\chi }_{23}$. C25·C27·C29

The two cases in the middle are different regimes, not a good and a bad outcome, so they are shown side by side. The weakest link in the chain is step 2 to the branches: the hydrogen-bond picture fits the pattern of the shortfall, but no measurement in the paper tests it, and the authors themselves offer it as speculation ("reasonable to speculate"). C15·X14

Where the conclusions apply. All data are for 25 °C, for miscible blends prepared by solvent evaporation with a brief melt, and for points without detectable crystallization. P1·P2 For felodipine–PVP, an earlier paper of the group (ref 30) showed that the blend separates into two amorphous phases at high RH; the authors therefore fitted ${\chi }_{1,23}$ for this blend only up to 65% RH. C35·R21 Note by the AI writer: the same high-RH felodipine–PVP points are used in the additivity comparison (Figure 6 goes to 85% RH) and, as far as the paper says, in the ${\chi }_{23}$ fits, so some of those values may describe a phase-separated blend rather than a one-phase blend. X9

What it means for formulation, as stated by the authors. Moisture uptake of a dispersion matters for its physical stability and for processing, because water lowers $T_g$ and "may lead" to crystallization. C2 "Care must be taken" when applying the FH model, which works for single amorphous components, to drug–polymer blends, and the ternary approach "cannot be used to unambiguously determine" drug–polymer interaction parameters. C30·C32 The mean-parameter approach is "particularly useful" because it needs no binary parameters in advance. C39

Note by the AI writer (not discussed in the paper). Three practical points follow from the data. First, adding up the component isotherms overestimates the water uptake of these blends, by up to about two-thirds at 10–25% PVP and about half at 50% HPMCAS; it is at best an upper bound for a stability assessment, and the blend's own isotherm is the safer basis. R7·R9 Second, at equal polymer content the HPMC and HPMCAS blends took up several times less water than the PVP blends at high RH (about 4.6 and 3.5 against 15.7 g/100 g at 50% polymer and 85% RH for felodipine), which is consistent with choosing a less hygroscopic polymer when moisture is the main risk. R2·R3·R4 Third, a drug–polymer interaction parameter for miscibility or drug-loading decisions is better obtained by a method that does not involve water, such as melting point depression (the source of the literature value used in Figure 9), than from sorption isotherms. P5·R14 The mean parameter ${\chi }_{1,23}$ is fitted separately for each composition, so it summarizes measured isotherms; it is not shown to predict compositions or humidities it was not fitted to. X10

Limitations

Limitations stated by the authors. The authors state the central limitation themselves: Eq. 2 assumes that water–drug, water–polymer and drug–polymer interactions in the blend equal those of the binaries, and they consider this "unlikely". C26·C27 They note that the felodipine–PVP blend separates at high RH and restrict its mean-parameter fit to 15–65% RH. C35 They report only data without evidence of crystallization. P2 And they present the link between the shortfall and drug–polymer interactions as speculation ("it is intuitive", "reasonable to speculate"). C14·C15

The AI writer's own judgement (1): what is solid. The two main results are robust. The shortfall against additivity is large, appears in all four series, and the plotted additivity predictions can be recomputed from the pure-component isotherms (for indomethacin–PVP at 85% RH, 0.75 × about 49.9 + 0.25 × about 2.1 gives about 37.9 g/100 g against about 38.0 plotted). E5·R1·R12 The insensitivity of the ternary equation to ${\chi }_{23}$ follows from the arithmetic of the model itself, and the indomethacin–PVP split of Table II can be reproduced from the stated parameters. R13·E3 The wide ranges of fitted ${\chi }_{23}$, which disagree even in sign, demonstrate the ambiguity directly. R14

The AI writer's own judgement (2): what does not reconcile. Several reported values do not reconcile on recalculation. The stated water–PVP $\chi$ of 0.36 lies below every point value implied by the plotted PVP isotherm (about 0.47–0.75). X5 The felodipine–PVP row of Table II matches a water–felodipine $\chi$ near 2.9 rather than the stated 3.27, and neither the RH nor the ${\chi }_{23}$ values behind Table II are given. X6·X7 The HPMC and HPMCAS composition levels differ between Figures 3, 4, 7, 8 and 10. X4 Eqs. 2 and 3 print the polymer size term as ${\varphi }_3/13$, and Eq. 3 keeps the wrong sign on the size terms. X1·X2 Table I gives molecular volumes with the unit cm³/mol although the values are per molecule. X3 In the discussion of Figure 9, the text names the additivity model as underpredicting at 75% PVP, while the figure shows this for the ternary prediction. X8 None of these changes the main conclusions, but they make the paper's numbers unsafe to reuse without checking.

The AI writer's own judgement (3): what is not tested. The mechanism, drug–polymer hydrogen bonds taking water-binding sites away and binary parameters changing inside the blend, is argued from the pattern of the data and from studies of other systems, and is not measured; spectroscopy of the blends at controlled humidity would be the direct test. X14·C27 The location of the felodipine–PVP maximum lies between 10 and 25% PVP depending on RH, and the indomethacin–PVP data supporting the 76:24 match are not shown. X11 The isotherms appear to be single runs, the way crystallization was excluded is not described, and the neat drugs have no 95% RH point. X12 The claim that the mean parameter "better" describes the data rests on fitted values without fit statistics. X10 The study covers one temperature, two drugs and three polymers. A1·A3

Conclusion

This paper tested four ways of describing the water uptake of amorphous drug–polymer blends on one consistent data set: felodipine and indomethacin with PVP, HPMC and HPMCAS at 25 °C. A1·A4 The simplest, additivity, fails: the blends take up much less water than the sum of their parts, by up to about 64% for felodipine–PVP, with the largest gap near 25% PVP in the PVP blends and at 50–70% polymer in the cellulose blends. R7·R8·R9·C10·C13 The binary Flory–Huggins equation describes the single components and ranks them sensibly, from 0.36 for PVP to 3.27 for felodipine. R11 The ternary equation, however, is almost blind to the drug–polymer interaction, whose term carries at most 0.160% of the interaction adjustment, so ${\chi }_{23}$ fitted to sorption data scatters between −20.4 and 26.9 and "cannot be used to unambiguously determine" drug–polymer interaction. R13·R14·C30 The authors attribute the shortfall and the scatter to drug–polymer interactions that change how water interacts with each component, a hypothesis the paper does not test. C15·C31·X14 A mean water–blend parameter ${\chi }_{1,23}$ describes the trend instead, falling with polymer content and most steeply with PVP. R15·C36·C37