Dissolved polymers and supersaturated danazol: the thermodynamic ceiling stays, crystallization slows

Background and motivation

Amorphous solid dispersions (ASDs) raise the apparent solubility and the bioavailability of poorly water-soluble drugs because, when they dissolve, they generate solutions that are supersaturated with respect to the crystal (refs 3–9). Such a solution has a driving force for crystallization. In the last decade the Taylor group and others showed that a highly supersaturated solution can undergo a second transition before it crystallizes: liquid–liquid phase separation (LLPS). Past a threshold concentration, the drug separates into a colloid of non-crystalline, water-saturated drug-rich droplets, and the surrounding aqueous phase is left at a concentration equal to the amorphous solubility (refs 9–12). Both phases remain supersaturated with respect to the crystal, so the equilibrium between them is metastable and crystallization can still follow. C1·C2

LLPS is not a rarity. It has been seen after a pH shift, on ASD dissolution, on dilution of organic solvent solutions and on cooling, and drug-rich colloids are a known nuisance in high-throughput screening (refs 12, 14–19). It has been suggested that the drug-rich phase could influence absorption (refs 9, 13), but the mechanisms and formulation factors that control it are not well understood. C3·C4

Formulations always contain excipients, and those excipients are dissolved together with the drug. The introduction lists two ways in which they can act. Surfactants and polymers change kinetic properties, the coalescence of droplets and the crystallization of the drug (refs 11, 20, 21). Surfactants can also change the thermodynamics: they can raise the crystalline solubility and shift the concentration at which LLPS appears (ref 22). Both matter for formulation. A shift in thermodynamics changes how much free drug the formulation can deliver; a shift in kinetics changes how long it can deliver it. C5·C6

For danazol, the group's own earlier work (Jackson et al. 2014, ref 20) had left an open question. That study had estimated the amorphous solubility of danazol at 16 μg/mL, read the LLPS onset with a pyrene fluorescence probe at 13 μg/mL, and found lower onsets when 10 μg/mL PVP, HPMC or HPMC-AS was present in the buffer. R4·R12 If polymers really lower the LLPS onset, they lower the maximum free-drug concentration and, with it, the maximum rate of membrane transport. The present paper set out (1) to test whether these three common ASD polymers change the thermodynamic properties of danazol solutions, through the crystalline solubility, the LLPS onset and the flux across a membrane, and (2) to measure how they change the kinetics of the phase transformations, through crystallization induction times and the properties of the droplets.

Background concepts

Crystalline solubility, amorphous solubility and supersaturation. The crystalline solubility is the equilibrium concentration over the stable crystal; for danazol in the 10 mM pH 6.8 phosphate buffer used throughout, it was 0.9 ± 0.1 μg/mL at 25 °C, within the 0.58–1.05 μg/mL reported elsewhere. R1.a·R2 Any concentration above it is supersaturated. The amorphous solubility is the much higher concentration in equilibrium with the amorphous (liquid-like) drug; above it the solution cannot stay single-phase and separates by LLPS. The paper calculated it at 7.8 μg/mL for danazol, about nine times the crystalline solubility. R3 The Discussion separates supersaturation from solubilization. Solubilization (by micelles, complexation or co-solvents) raises the equilibrium solubility but not the chemical potential of the drug, so a solubilized solution does not deliver more free drug to a membrane. Supersaturation raises the chemical potential itself, which both speeds membrane transport and creates the driving force for crystallization (refs 31–33). C31

Two routes out of supersaturation. Figure 10 sketches the two transitions open to a supersaturated solution. It can crystallize directly, once the concentration exceeds the threshold for primary nucleation, and crystallization pulls the concentration back down to the crystalline solubility. Or, above a certain supersaturation, it can first undergo LLPS into a water-rich phase and drug-rich droplets; the system is then still supersaturated, and crystals can form later from either phase. C32 The difference matters for a formulation: LLPS keeps the supersaturation, crystallization destroys it.

The free-energy picture. Figure 11 (adapted from Steele and Deneau, ref 40) shows the Gibbs free energy of mixing of drug and water against the mole fraction of drug. For a system with a miscibility gap, a common tangent touches the curve at two binodal points. The left one, $x_{AB}$, is the highest drug concentration the aqueous phase can hold (the LLPS concentration, equal to the amorphous solubility); the right one, $x_{BA}$, is the composition of the drug-rich droplets. Between them lie two spinodal points, $x_{AB}^{*}$ and $x_{BA}^{*}$. Between a binodal and the neighbouring spinodal the solution is metastable: a new phase can form only after an activation barrier is overcome. Beyond the spinodal the solution is unstable, the barrier vanishes and demixing is spontaneous (spinodal decomposition). If the drug's melting point lies above the experimental temperature, the two liquid phases coexist only in metastable equilibrium, and crystallization can occur at any time. C33 For danazol at 25 °C the paper places $x_{AB}$ at 4.27 × 10⁻⁷ mole fraction, from the measured LLPS concentration, and $x_{BA}$ at 0.92, from moisture-sorption data on amorphous danazol extrapolated to a water activity of 1; the spinodal points are drawn without values and the axis is "not to scale". R22 The text credits the diagram to ref 36 while the caption credits ref 40. X15

Thermodynamic activity and flux. The activity $a$ of a dissolved drug is its effective concentration, the concentration $C$ multiplied by an activity coefficient $\gamma$:

In plain terms: two solutions with the same total drug concentration can have different activities if something in one of them binds or solubilizes the drug, which lowers $\gamma$. E2 Diffusion across a membrane is driven by activity, not by total concentration, so a flux measurement reads the activity directly. This is why the paper treats flux as its most direct thermodynamic test. C9·C41

Induction time. The crystallization induction time is the time between creating supersaturation and the first detectable crystallization. Here it was taken from the apparent drug concentration measured by UV at 288 nm: the moment the concentration begins an abrupt and sustained fall marks the end of the induction period. P10

Zeta potential. The zeta potential is the electrical potential at the slipping plane of a particle in solution. A change in its value when a polymer is added shows that something has adsorbed at the particle surface; the direction of the change depends on the charge of the adsorbing species (ref 29). R18·C25

Methods in detail

The design is simple and deliberate: one drug, danazol (99% purity), three polymers that are common ASD carriers, PVP K29/32, HPMC (606 grade) and HPMC-AS (MF grade), each pre-dissolved at 1, 10, 100 and 1000 μg/mL (and 500 μg/mL for two of the measurements), one medium, 10 mM pH 6.8 phosphate buffer with sodium chloride for constant ionic strength, and one temperature, 25 °C. M1·M2·M3·M4·M9·A1·A2 Supersaturation was always created by the solvent-shift method, adding small aliquots of a 5 mg/mL methanolic danazol stock to the polymer-containing buffer. P4·P5·P7·P8·P9 PVP and HPMC are neutral; HPMC-AS carries negative charge at pH 6.8. M2·M3·M4 The work has three parts: the same thermodynamic limit measured in several independent ways, flux as a direct read-out of activity, and kinetic measurements on the same kind of solutions.

Contribution 1: The LLPS onset, measured five ways and calculated once

① The problem. The earlier fluorescence onset for danazol (13 μg/mL) did not match either a careful amorphous solubility estimate or other methods, and polymer seemed to lower it. R12 A single method cannot tell whether a polymer really changes the thermodynamic limit or only changes how fast the droplets become detectable.

② What they did. The amorphous solubility was calculated by the method of Almeida e Sousa et al. (ref 24) from the crystalline solubility in buffer and the melting temperature, heat of fusion, glass transition temperature and heat capacity of danazol measured by DSC (TA Q2000), with a correction for the activity lowering caused by sorbed water taken from a dynamic vapour sorption profile at 25 °C. HPLC showed no degradation of danazol after DSC heating. P3·R5·E3 The crystalline solubility itself was measured by shaking excess crystalline danazol for 48 h in 50 mL buffer with each polymer level, ultracentrifuging (40,000 rpm, 15 min) and assaying the supernatant by HPLC at 288 nm, n = 3. P1·P2 The LLPS onset was then measured four ways in the presence and absence of polymer:

• UV extinction: danazol was pumped into 50 mL stirred buffer at 1 μg/mL/min while the extinction at 450 nm, a wavelength danazol does not absorb, was followed; the onset is where scattering from the new phase appears. Without polymer the run started from 5 μg/mL so that LLPS could be reached before crystallization. n = 3. P4
• Fluorescent probe: 2 μM pyrene, whose ratio of emission peaks I₁/I₃ falls when it moves into a hydrophobic environment such as a drug-rich droplet; danazol was added stepwise. P5·C8
• Ultracentrifugation with HPLC: 15 μg/mL danazol, spun at 35,000 rpm for 30 min (polymer below 100 μg/mL) or 60 min (100 μg/mL and above); the supernatant is the continuous water-rich phase, so its concentration is one binodal point, a direct measurement of the amorphous solubility. n = 3. P7·C7
• Nanoparticle tracking analysis (NTA): a NanoSight LM10 with a 532 nm laser counts and tracks individual scattering particles; the onset is where the particle count rises above background. Triplicate. P8

The fifth estimate comes from the flux plateau (Contribution 2). R10

③ Why it works. The methods detect different things: a turbidity signal, a change in a probe's environment, the concentration left in the continuous phase, and countable particles. If all of them give the same onset, with and without polymer, the onset is a property of the solution and not an artefact of one detection method. C14

Contribution 2: Flux as a direct read-out of activity

① The problem. A concentration measurement cannot tell free drug from drug that is bound to or solubilized by a polymer. The quantity that matters for absorption is the activity.

② What they did. A side-by-side diffusion cell (PermaGear) with a regenerated cellulose membrane (molecular weight cut-off 6–8 kDa, orifice area 7.065 cm²) separated two 34 mL chambers that both contained the same pre-dissolved polymer. Danazol was added to the donor; the receiver concentration was followed by a UV dip probe at 288 nm, and the slope of concentration against time gave the flux. The receiver never exceeded 0.3 μg/mL, so it stayed at sink with respect to the crystalline solubility. A crystalline danazol suspension served as reference. P9·M10 The flux is given by Equation 1:

In plain terms: the flux $J$, the mass $M$ crossing a membrane of area $S$ per unit time $t$, equals the diffusion coefficient $D$ times the drug activity $a$ in the donor, divided by the membrane thickness $h$ and the drug's activity coefficient in the membrane ${\gamma }_m$. With $S$, $D$, $h$ and ${\gamma }_m$ fixed by the set-up, the flux is directly proportional to the activity in the donor. E1·C9 Combined with $a=C\gamma$ (Equation 2), this gives the test the paper relies on: if a polymer lowered danazol's activity, by binding it or by mixing into the drug-rich phase, the flux at a given total concentration would fall. E2·C15

③ Why it works. Once LLPS has occurred, the activity of the drug in the water-rich phase is fixed at its maximum; adding more drug only makes more droplets, which do not add to the flux. The flux therefore rises with donor concentration up to the LLPS onset and then levels off, and the plateau height is the maximum activity the system can reach (Raina et al. 2014, ref 26). C16 ④ Difference from the earlier work. The earlier danazol paper probed the solution optically; the flux experiment adds a measurement of activity itself, which the authors call their most definitive evidence because flux depends on the thermodynamic activity of the solute rather than on its concentration (ref 31). C41

Contribution 3: The kinetics, measured on the same solutions

① Time-resolved fluorescence. To test whether the earlier high onset came from slow droplet formation, pyrene spectra were recorded 15 s (the original method of ref 20), 1 min and 5 min after each danazol addition, without polymer and, in additional runs, with each polymer (the polymer concentration of these runs is not stated). P6·C19 ② Induction times. Solutions of 15 μg/mL danazol, almost twice the onset and therefore already two-phase, were prepared in 50 mL buffer with 0–1000 μg/mL polymer, and the apparent concentration was followed by UV at 288 nm until it dropped. P10·R7.j ③ Droplet surface and size. Zeta potential and z-average diameter were measured by dynamic light scattering (Malvern Nano-ZS, backscatter at 173°) on 0.5 mL samples from 10 mL stirred solutions, and the mean size was also measured by NTA. The danazol concentration of these samples is given only as "as described above", and the time between supersaturation and measurement is not stated. P11·P12·X12

Experimental analysis

The onset of LLPS is about 8 μg/mL, and it equals the amorphous solubility

NTA gives the most direct picture (Figures 1 and 2). At 3 μg/mL danazol the frames show no scattering centres, at 8 μg/mL a few, at 11 μg/mL many. R6 The particle count stays at about 2–6 × 10⁷ per mL up to 7 μg/mL, reaches about 1.8 × 10⁸ at 8 μg/mL, and then rises steeply, to about 3.2 × 10⁹ per mL at 15 μg/mL. R7 For danazol alone, the UV extinction onset was 8.3 ± 1.0 μg/mL, the fluorescent probe gave 8 and NTA 8 μg/mL; the HPLC value could not be determined because danazol crystallized during the 30 min spin. R8.a·C56 The flux rose linearly with donor concentration up to 8 μg/mL and was constant within experimental error above it, so the flux-based onset is also 8 μg/mL (Figure 3). R10·C17 All of these lie close to the calculated amorphous solubility of 7.8 μg/mL, which the authors call "good agreement". R3·C50

The calculated value is half the 16 μg/mL estimated in ref 20. The authors explain the difference by the more rigorous calculation of ref 24, which uses heat capacity data and has given predictions closer to experiment ("This can be rationalized"). R4·C11 The inputs of the new calculation are not printed, so the 7.8 μg/mL cannot be recomputed from the paper. E3

Note by the AI writer (not discussed in the paper): the DNZ Only row of Table II is printed with one entry too many. The value 8.3 ± 1.0 sits in the column where the polymer rows give the polymer amount, and the row then continues 8, 8, n/d, 8 under the four technique headings. Because footnote a compares every UV extinction value with the DNZ only system by t-test, 8.3 ± 1.0 is most likely the UV extinction value; the fluorescent probe, HPLC and NTA entries are unaffected. X1

The polymers do not move the onset, and change the crystalline solubility little

Table II reports the LLPS onset for 15 polymer conditions. Across PVP, HPMC and HPMC-AS at 1–1000 μg/mL, UV extinction gave 7.6–8.7 μg/mL, the fluorescent probe 7–9, HPLC 8.0–9.1 and NTA 7–8 μg/mL. No UV extinction value differed statistically from danazol alone (all p > 0.3). R8·R9 Table A condenses the table.

Table A. Range of LLPS onset concentrations over the five polymer levels, condensed from Table II (n = 3; fluorescent probe and NTA at 1 μg/mL steps). The DNZ-only UV value is read as 8.3 ± 1.0 (see the note above). R8·R9·X1

The authors read Table II as showing "no trend" with polymer type or concentration. C14 Their argument for why this matters is explicit: if a polymer were "substantially incorporated" into the drug-rich phase, it would lower the activity of danazol, and the LLPS onset would fall; "this does not appear to be the case". C15 The HPLC column is the most informative here, because it measures the continuous phase directly: it stays at 8.0–9.1 μg/mL whatever the polymer and its amount, close to the calculated 7.8 μg/mL. R8·C40

The crystalline solubility (Table I) was 0.9 ± 0.1 μg/mL in buffer. With polymer it was 0.8–1.3 μg/mL for PVP, 0.9–2.2 μg/mL for HPMC and 1.0–1.4 μg/mL for HPMC-AS (Table B). R1 The authors call these "relatively minor increases" and, in the Discussion, "no significant changes". C10·C39

Table B. Crystalline solubility of danazol (μg/mL) at 25 °C, mean ± SD, n = 3 (from Table I). R1

Note by the AI writer (not discussed in the paper): no statistical test is reported for Table I, and with HPMC at 10, 100 and 1000 μg/mL the crystalline solubility is about twice the buffer value. In absolute terms the rise is about 1 μg/mL, small against the 7.8 μg/mL amorphous solubility, so it barely changes the attainable supersaturation; but "no significant changes" is a stronger statement than the table supports. X5

The flux ceiling is the same with and without polymer

Without polymer, the flux (plotted in units printed as μg/mL/cm²) rose from about 0.0034 at 2 μg/mL to about 0.0152 at 8 μg/mL and then stayed at about 0.0156–0.0159 up to 12 μg/mL (Figure 3). R10.b–j At 15 μg/mL with 10 μg/mL polymer, the flux was about 0.0149 with PVP, 0.0156 with HPMC and 0.0165 with HPMC-AS. R10.k–m Figure 4 extends this to every polymer level at 15 μg/mL danazol (Table C).

Table C. Maximum flux from 15 μg/mL danazol (axis unit as printed, μg/mL/cm²), read from Figure 4 (approximate). R11

The values lie between about 18% below and 14% above the no-polymer value, with no consistent direction for any polymer, and the authors conclude that "Within experimental error" the polymers do not reduce the flux and so do not change the free-drug concentration; in the Discussion they call this result "Most definitively" supporting their conclusion. R11·C18·C41

Note by the AI writer (not discussed in the paper): three details of the flux data limit how precisely they can be used. The axis unit, μg/mL/cm², has no time unit, although Equation 1 defines flux per unit time; the number of replicates is not stated. X9 The crystalline-suspension reference reads about 0.0011 in Figure 3 but about 0.0008 in Figure 4. X10 And if flux were strictly proportional to activity, the line through the supersaturated points (about 0.0019 per μg/mL) would predict about 0.0017 at the crystalline solubility of 0.9 μg/mL; the measured crystalline flux is lower, so the plateau is about 14–19 times the crystalline flux although the concentration ratio is only about 9. The paper does not comment on this. X11 None of this affects the main comparison, which is between polymer and no polymer at the same donor concentration.

Without polymer, droplets form late at first

Figure 5 explains the old 13 μg/mL reading. Without polymer, the pyrene ratio stayed at the buffer value of about 1.93 up to 6 μg/mL at all three reading times. Read 1 or 5 min after addition, it fell to about 1.87–1.88 at 8 μg/mL and to about 1.57–1.60 at 14 μg/mL. Read after 15 s, it was still about 1.90 at 10 μg/mL and about 1.88 at 12 μg/mL, and fell only to about 1.75 at 14 μg/mL. R13 So the 15 s method detects the drug-rich phase only between 10 and 14 μg/mL, while 1 min later it is detected at about 8 μg/mL; all readings were taken before crystallization began. C20·R13 The authors note that they had analysed more than 20 compounds with this method and never seen such a time effect. R15 With pre-dissolved PVP the delay was the same as without polymer; with HPMC or HPMC-AS the ratio changed at about 7–8 μg/mL at every reading time. These polymer runs are described but not shown, and their polymer concentration is not given. R14

The polymers delay crystallization for hours

Figure 6 shows representative traces at 15 μg/mL danazol. Without polymer the apparent concentration falls within the first minutes, to about 2 μg/mL by about 40 min. With 10 μg/mL polymer the plateau holds for about 100 min with PVP, about 120 min with HPMC and about 240 min with HPMC-AS before the concentration falls. R16 Figure 7 gives the averages (Table D).

Table D. Average crystallization induction time (min) of 15 μg/mL danazol, read from Figure 7 (approximate); HPMC-AS error bars span about ±130–150 min; n not stated. R17

The authors summarise: all three polymers prolong supersaturation, PVP least and HPMC-AS most; at 1 μg/mL the polymers have "equal impact" and extend induction times only marginally; from 10 μg/mL the ranking is clear, HPMC-AS "much more effective" than PVP and HPMC; and raising the polymer from 10 to 1000 μg/mL "did not extend induction times any further" (abstract: "maximum effectiveness being reached at 10 μg/mL"). C21·C22·C23·C24 The figure supports the ranking and the large step between 1 and 10 μg/mL, but it shows HPMC ahead at 1 μg/mL (about 48 against 26–27 min) and modest further rises above 10 μg/mL, about 40–50 min for each polymer, which the Discussion more cautiously calls "fairly minimal increases". X3·X4·C46 The text puts the no-polymer crystallization at 10–15 min, whereas the bar in Figure 7 reads about 5 min and the trace in Figure 6 falls within minutes. X2

Two features make these numbers notable. The solutions were already two-phase: at 15 μg/mL NTA counts more than 3 × 10⁹ droplets per mL, so the polymers act in the presence of a large drug-rich interface. R7.j And the effective polymer concentrations are very low; the authors point out that earlier danazol precipitation studies used much higher polymer levels (refs 16, 50). C45

The polymers sit on the droplet surface

Without polymer the droplets had a zeta potential of −8.1 mV. R18.a Neutral PVP made them steadily less negative, about −7.1, −5.4, −4.7 and −3.5 mV at 1, 10, 100 and 1000 μg/mL; neutral HPMC brought them to about −2.9 mV already at 1 μg/mL and about −1.7 to −2.2 mV above. Anionic HPMC-AS did the opposite: about −8.9, −10.7, −14.6 and −18.6 mV (Figure 8). R18 The authors read the shift with the neutral polymers as "consistent with the polymers interacting with the surface and reducing the negative charge", and the continuing fall with HPMC-AS as more polymer at the surface at higher concentration. C25·C26·C49

Droplet size depends on the method

By DLS, the z-average diameter fell from about 350 nm without polymer to about 172–218 nm at 100 μg/mL and about 103–136 nm at 1000 μg/mL, similarly for all polymers (Figure 9, left). R19·C27 The polydispersity index was 0.7–1 without polymer and with 1 μg/mL PVP or HPMC, and 0.15–0.7 in all other samples. R20 NTA, which sizes droplets one by one, gave means of about 217 nm without polymer and about 181–270 nm with polymer, with no obvious trend (Figure 9, right). R21·C28 The authors explain the difference by the known bias of DLS toward a few large particles (ref 30): polymer "appears" to reduce the formation of some larger droplets, "probably through coalescence", so the DLS average falls while the NTA mean does not. C29·C30 The summary size of the drug-rich phase is given as "a mean size of 250 nm" in the abstract and "around 200–250 nm" in the conclusion. C50·X13

Discussion

The authors' chain of reasoning

The Discussion turns the results into one argument about thermodynamics and one about kinetics. It can be laid out as a chain that ends in two outcomes for the formulation:

The first half of the chain, steps 1 to 3, is the paper's main message: "fortunately, these additives do not appear to impact the bulk thermodynamic properties of the solution", so supersaturated solutions with higher membrane transport than saturated solutions can still be generated. C54 The authors contrast danazol with resveratrol, where cellulose polymers did solubilize the drug and reduced the supersaturation reached (Wegiel et al., ref 45). C43 The second half, the kinetic effect, agrees with earlier work on polymers and crystallization (refs 27, 46–49) and with earlier danazol precipitation studies at much higher polymer concentrations (refs 16, 50). C44·C45

Why do the polymers inhibit so differently?

The authors first exclude two simple causes. The polymer concentrations are too low to change the viscosity of the solution, and they do not change the supersaturation, the thermodynamic driving force for crystallization. C47 Surface adsorption alone cannot explain the ranking either: all three polymers change the zeta potential, yet HPMC-AS is "clearly the most effective crystallization inhibitor" and PVP the least. C48 The authors therefore conclude that the different inhibitory properties "most likely arise from differences in molecular level interactions between the various polymers and danazol nuclei", interactions they describe as very difficult to probe experimentally. C48 One observation sharpens the puzzle: with HPMC-AS the zeta potential kept falling as the concentration rose from 10 to 1000 μg/mL, "suggesting that more polymer was present at the droplet surface", but the induction time did not rise correspondingly, which the authors read as "indicating that additional studies are needed". C49

Why is the onset late without polymer?

The time-dependent onset of Figure 5 is unusual. The authors use the free-energy picture of Figure 11 for a "potential explanation". If the gap between binodal and spinodal is wide and the activation energy for forming the new phase is high, a solution can be pushed past the binodal, the amorphous solubility, for a short time before droplets nucleate. For most systems little such overshoot is seen (refs 12, 24), but for danazol "we speculate" that the barrier is higher than usual, so the amorphous solubility can be exceeded when supersaturation is generated very rapidly. C34·C35 Polymer "seemingly" lowers this barrier: it is "most likely" interacting with the droplets and lowering the surface tension needed to create the new interface, and the zeta potential "provides support for this supposition". C36·C37 Note by the AI writer (not discussed in the paper): the supporting data cover only HPMC and HPMC-AS. With PVP the delay remained, although PVP also changed the zeta potential, so adsorption at the droplet surface is not sufficient to remove the delay. X6·R14·R18

What this means for formulation

The authors draw two practical points. Very small amounts of dissolved polymer give most of the protection against crystallization, "even in the presence of a drug-rich phase": the largest gain came between 1 and 10 μg/mL. C46 And the three polymers do not compromise the higher membrane transport of maximally supersaturated solutions, whose maximum transport rate is limited only by the amorphous solubility. C51 They close by recommending that the solution phase behaviour of supersaturated solutions with polymers be characterised to understand solubility-enhancing formulations such as ASDs. C55

Note by the AI writer (not discussed in the paper): for danazol the choice of polymer can be made on kinetics alone, at least in the range tested, because none of the three lowered the ceiling set by the amorphous solubility. The ceiling itself, about 8 μg/mL here, cannot be raised by adding more drug: above it, extra drug only makes more droplets, not more flux. And an LLPS onset read within seconds of creating supersaturation can overestimate the amorphous solubility for a drug that forms droplets slowly; a reading after a minute or more, or an independent method such as flux or ultracentrifugation, avoids this. R10·R13

Limitations

Limitations stated by the authors. The authors state that the interactions between polymers and danazol nuclei, which they hold responsible for the ranking, are "very difficult to probe experimentally", and that more studies are needed to understand how polymers inhibit crystallization "in these complex two phase solutions". C48·C49 Without polymer, the ultracentrifugation method could not give an onset because danazol crystallized during the spin, and the UV extinction run had to start at 5 μg/mL to reach LLPS before crystallization. C56·P4 The mechanism for the late onset is offered as speculation. C35

The AI writer's own judgement (1): what is measured and what is inferred. The thermodynamic conclusion rests on solid ground: five methods with different physical bases agree on an onset of about 8 μg/mL with and without polymer, and the UV extinction values were tested statistically against danazol alone. R8·R9·R10 What is inferred rather than measured is the step from "the activity is unchanged" to "polymer does not mix with the droplets". The droplet composition was never measured; the zeta potential shows that polymer sits at the droplet surface; and the onset methods resolve about 0.5–1 μg/mL, so a small polymer uptake that shifts the activity by less than that would go unseen. The "confirm" in the Discussion is therefore stronger than the data. X14·C42 The same caution applies to the crystalline solubility: HPMC about doubles it in Table I, and "no significant changes" is stated without a test. X5

The AI writer's own judgement (2): precision of the kinetic and flux data. Flux, induction time, zeta potential and size are available only as plotted bars or points, with replicate numbers and error-bar definitions not stated; the flux axis lacks a time unit, and the crystalline reference flux differs between Figures 3 and 4. X9·X10·X12 The no-polymer induction time is about 5 min in Figure 7 but 10–15 min in the text. X2 The statement that induction times did not increase from 10 to 1000 μg/mL is an approximation: the means rise by about 40–50 min for each polymer, within the large scatter of the HPMC-AS data but not zero. X3 The droplet size and zeta potential were measured at an unstated danazol concentration and time after supersaturation, although droplet properties "as a function of time" were among the stated aims; no time series is shown. X12 The HPLC onsets at 1 μg/mL polymer were measured after a 30 min spin, a time comparable to the induction times at that polymer level (about 26–48 min), although the values agree with the other methods. X16 Finally, there are a few textual inconsistencies: the DNZ Only row of Table II is misaligned, the fluorescence step size is given as 0.5 μg/mL in the methods, 1 μg/mL in the Table II footnote and 2 μg/mL in Figure 5, the spectrofluorometer slit widths are printed in cm, and the source of Figure 11 is credited to ref 36 in the text but ref 40 in the caption. X1·X7·X8·X15 The scope is one drug, one buffer at 25 °C, and polymer up to 1 mg/mL.

Conclusion

The paper separates two questions that are easily confused when a formulation contains polymer: does the polymer change how much free drug the solution can hold, and does it change how long the solution can hold it. For danazol the answers are no and yes. The LLPS onset, about 8 μg/mL, matches the calculated amorphous solubility of 7.8 μg/mL and is the same with PVP, HPMC or HPMC-AS at 1–1000 μg/mL by five independent methods, and the maximum membrane flux is unchanged; in the authors' words, the additives "do not appear to impact the bulk thermodynamic properties of the solution". R3·R8·R11·C54 At the same time, as little as 10 μg/mL polymer extends the life of the two-phase supersaturated solution from minutes to hours, HPMC-AS by far the most. R17·C46 The polymers adsorb at the droplet surface, but surface charge does not decide their effectiveness; the authors place the difference, "most likely", in molecular interactions with danazol nuclei. C48·C49 The paper also corrects the group's earlier onset for danazol: the 13 μg/mL reading was an artefact of measuring 15 s after mixing, for a drug that forms its droplets slowly. R12·R13·C20