The ceiling on supersaturation: liquid–liquid phase separation caps passive membrane flux

Background and motivation

Many new drug candidates dissolve poorly in water, either because they are lipophilic, because they have a high melting point, or both, and formulations that generate supersaturated solutions, such as amorphous solid dispersions (ASDs), are one answer (Augustijns 2012, ref 1; Brouwers 2009, ref 2). A supersaturated solution is one whose solute chemical potential exceeds that of a solution saturated with the crystal. When lipophilic drugs are pushed to high supersaturation, they form colloidal aggregates, a phenomenon familiar from enzyme screening, where aggregates formed by adding a concentrated organic stock to water ("solvent shifting") give false positives; such aggregates appear only above a critical concentration C1. Colloidal species also appear when ASDs dissolve, again only once a critical dissolved concentration is reached, and they have been "postulated" to help oral delivery C2. It had been suggested that the common cause is liquid–liquid phase separation (LLPS): above a threshold, the supersaturated solution splits into a drug-rich, hydrophobic phase and a water-rich phase that holds little drug, the two being in dynamic equilibrium while the whole system stays supersaturated C3·C4.

The formulation question behind the paper is what such high concentrations buy in terms of absorption. Measuring the total drug in solution does not tell whether a formulation has raised the drug's chemical potential (true supersaturation) or merely its solubility. Low concentrations of polymer do not raise the crystalline solubility but help keep a supersaturated solution from crystallizing, whereas micelles, cyclodextrins, cosolvents and pH adjustment raise the crystalline solubility and therefore increase concentration without raising the chemical potential C5. Because membrane transport is driven by the chemical potential gradient (Higuchi 1960, ref 32), it was "well established" that surfactants and cyclodextrins can lower membrane transport, whereas supersaturated solutions raise it as long as crystallization is prevented; most of that evidence came from transdermal work, with more recent intestinal perfusion studies C6.

Where the paper starts is an unexplained observation from the same group. Diffusion data of Alonzo et al. (2010, ref 46) "suggested" that flux reached a maximum once concentrations at or above the "amorphous solubility" were generated, so "there may be a breakdown" in the relation between concentration and diffusion rate at very high supersaturation R18·C7. Separately, Ilevbare and Taylor (2013, ref 17) had shown that the LLPS concentration of several hydrophobic drugs lies very close to their calculated amorphous solubility R19. The present paper connects these two lines. Its stated goal is to show that small organic molecules at and above their amorphous solubility undergo LLPS, and that the LLPS concentration corresponds to the maximum achievable diffusive flux. Felodipine and nifedipine were the model drugs, and, following Corrigan et al. (1980, ref 35), the crystallization inhibitor hydroxypropyl methylcellulose (HPMC) was dissolved in the donor so that supersaturation would last long enough to measure diffusion C8.

Background concepts

Amorphous solubility and how it is estimated

The amorphous solubility is the highest concentration that the amorphous (supercooled liquid) form of a drug can generate in water before a drug-rich phase appears. It exceeds the crystalline solubility because the amorphous form has a higher free energy. The paper starts from the general expression for the mole-fraction solubility of a crystal (Eq. 1) E1:

In plain terms: the solubility of a crystal falls with its heat of fusion $\Delta H_f$ and with how far the experiment's temperature $T$ lies below the melting temperature $T_m$ (the first two terms are the cost of melting the crystal), and falls further if the drug interacts unfavourably with water, expressed by the activity coefficient $\gamma$. $R$ is the gas constant and $\Delta C_p$ the heat-capacity difference between crystal and liquid. For an amorphous solid there is no crystal lattice to break, so the melting terms drop out and only $\ln (x{)}_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}}=-\ln \gamma$ remains (Eq. 2) E2. The ratio of amorphous to crystalline solubility, ${\sigma }_a/{\sigma }_c$, is therefore set by the free-energy difference $\Delta G$ between the two solid forms, which the paper first estimates with the Hoffman equation (Eq. 3) E3:

In plain terms: the amorphous form is more soluble than the crystal by a factor that grows with the heat of fusion and with the undercooling $\Delta T=T_m-T$. The Hoffman equation is a "reasonable first approximation", but it ignores two effects: an ionizable drug may be ionized to different degrees in solutions equilibrated with the amorphous and the crystalline form, and the amorphous solid absorbs water, which lowers its activity C10. The paper adopts the correction of Murdande et al. (2010, ref 50), Eq. 4 E4:

In plain terms: the solubility ratio is the product of an ionization term (with ${\alpha }^a$ and ${\alpha }^x$ the fractions ionized in the amorphous and crystalline cases), a water-sorption term $e^{-I(a_2)}$ obtained by integrating the Gibbs–Duhem equation over the moisture sorption isotherm of the amorphous drug up to a water activity of 1, and the free-energy term (the standard-state chemical potential difference between dry amorphous and crystalline drug). Felodipine (pKa below 2) and nifedipine (pKa below 3.5) are unionized at pH 6.8, so the ionization term is 1 C9·M1·M2. With the free-energy difference again taken from the Hoffman estimate, Eq. 4 reduces to Eq. 5 E5:

In plain terms: the predicted amorphous solubility is the crystalline solubility multiplied by the Hoffman ratio and then by a moisture factor smaller than 1. For these drugs the moisture factor was 0.74–0.83 R7.

Liquid–liquid phase separation and the binodal

Why a drug solution can hold only up to a certain concentration before a liquid drug-rich phase appears is explained in the Discussion with a free-energy-of-mixing diagram for drug (A) and buffer (B), adapted from Deneau and Steele (2005, ref 23) (Figure 7). The free energy of mixing is written as (Eq. 8) E8:

In plain terms: the free energy of mixing is the sum, over the two components, of each mole fraction $x$ times how far that component's chemical potential $\mu$ lies below its pure (standard, *) state. For a partially miscible pair this curve has two minima. The mixture can lower its free energy by splitting into two phases of compositions $x_{AB}$ and $x_{BA}$, the binodal points, which mark the theoretical onset of LLPS C27. $x_{BA}$, on the water side, is the highest bulk concentration an amorphous drug can generate, the amorphous solubility; $x_{AB}$ is the most water the drug-rich phase can hold C27. Adding drug to a dilute solution raises its concentration until $x_{BA}$ is reached; beyond that, more drug forms more of a drug-rich phase of composition $x_{AB}$, and the two compositions stay fixed while only the relative amounts change C27. Approaching from the drug side, an amorphous drug above its glass transition takes up water until it reaches $x_{AB}$ C27. Inside the binodal points lie the spinodal points, where the solution becomes unstable and separates without any free-energy barrier C30. Because the drug-rich phase contains water, both felodipine and nifedipine are taken to be supercooled liquids in these experiments, a point the authors assert but did not measure C28.

Metastable zones: why LLPS is not always seen

Figure 8 places the binodal on a concentration–temperature diagram. The coexistence curve (the amorphous solubility) lies above the equilibrium crystalline solubility, within the metastable zone for crystal nucleation C30. Between the crystalline solubility and the coexistence curve only crystallization can occur; above the coexistence curve either LLPS or crystallization can occur, and LLPS will be seen only if crystal nucleation is slow compared with the formation of the drug-rich phase C30. The width of a metastable zone reflects the free-energy barrier to forming a new phase, which falls as supersaturation rises; temperature, viscosity and the time scale of the experiment also affect it C30. For organic molecules the spinodal and the binodal have been found to lie close together (Ilevbare 2013, ref 17) R19.

Flux, activity and supersaturation

Passive diffusion across a membrane with sink conditions on the receiver side and a nearly constant donor concentration follows Eq. 6 E6:

In plain terms: the flux $J$, the mass $M$ crossing per unit time $t$, is proportional to the drug's thermodynamic activity $a$ in the donor, with the diffusion coefficient $D$, the membrane area $S$, the membrane thickness $h$ and the drug's activity coefficient in the membrane ${\gamma }_m$ as constants of the set-up C11. The activity in solution is the product of an activity coefficient and the concentration, $a=\gamma C$ (Eq. 7) E7. When a drug-rich and a drug-lean phase coexist, the drug in both has the same chemical potential and therefore the same activity (Eq. 9) E9·C33:

In plain terms: the drug-rich (DR) phase holds far more drug than the drug-lean (DL) phase, but each drug molecule "feels" the same driving force in both, because the activity coefficients differ by the same factor as the concentrations. Taking the activity coefficient as constant between the crystalline and the amorphous solubility, "a reasonable assumption for these highly dilute solutions", the activity relative to the crystal is simply the supersaturation ratio (Eq. 10) E10·C34:

In plain terms: $C_i$ is the concentration in the drug-lean phase and $C_S$ the equilibrium crystalline solubility, so the activity is 1 for a saturated solution and rises in proportion to concentration up to the amorphous solubility $C_{DL}$, the highest drug-lean concentration possible E10. Note that the paper uses $\alpha$ here for activity, the same symbol it uses for the fraction ionized in Eq. 4, and that the text refers to this relation as "Eq. (12)" and "Eq. (9)" although it is printed as Eq. 10 X2.

Methods in detail

The study has two parts that are run separately and then compared: locating the phase boundaries of each drug in buffer (crystalline solubility, onset of a new phase, predicted amorphous solubility), and measuring flux across a membrane over a range of donor concentrations that brackets those boundaries. All experiments used 50 mM pH 6.8 phosphate buffer (ionic strength 0.155 M) M4.

Contribution 1: locating the phase boundaries in buffer

Crystalline solubility. Excess crystalline felodipine (Attix Pharmaceuticals) or nifedipine (Euroasia) was shaken in buffer in foil-wrapped vials for 48 h at 20, 25, 30, 37 and 45 °C; preliminary experiments had shown that equilibrium was reached by 48 h P1·M1·M2. The suspensions were ultracentrifuged at 274,356 g for 15 min, and the supernatant was diluted and assayed by reversed-phase HPLC with UV detection at 360 nm against standards of 0.1–50 µg/mL (r² above 0.998), in triplicate P1·P2. The same measurement was repeated with HPMC dissolved in the buffer, at 100 µg/mL for felodipine and 1 mg/mL for nifedipine P1·M3.

Onset of a new phase by UV extinction. A 5 mg/mL methanol stock of each drug was pumped with a syringe pump into stirred buffer held at 20–45 °C, at 20 µL/min for felodipine and 40 µL/min for nifedipine, with or without HPMC P7. Extinction was recorded every 10 s at 455 nm, a wavelength at which neither drug absorbs, so any rise in signal comes from light scattered by a new phase P7·C12. The onset concentration is the intersection of the flat baseline and the rising branch of extinction versus added concentration (Figure 3) C12·R3. Stirring rates of 120–300 rpm were used, slower for nifedipine because faster stirring led to crystallization; Table 1 lists 300 rpm and 20 µL/min for its felodipine and nifedipine-in-buffer columns and 120 rpm and 40 µL/min for nifedipine with HPMC, which does not fully match the methods text for nifedipine P7·X16. The turbid solutions (50 and 100 µg/mL, made the same way) were examined between slide and cover slip under polarized light, to tell an amorphous liquid phase (no birefringence) from crystals (birefringent) P6.

Predicted amorphous solubility. Melting temperature and enthalpy of fusion were measured by DSC (147 °C and 28.67 kJ/mol for felodipine; 173 °C and 32.5 kJ/mol for nifedipine), and moisture sorption of the amorphous drugs was measured gravimetrically P3·P4·M1·M2. These gave the Hoffman ratio (Eq. 3) and the moisture factor of Eq. 5, which were multiplied by the measured crystalline solubility P5·R7.

Contribution 2: flux across a membrane, from below to far above the amorphous solubility

The cell. A side-by-side diffusion cell (Figure 2) had two chambers of 34 mL capacity, each filled with 30 mL of buffer and stirred, separated by a regenerated cellulose membrane with a molecular weight cutoff of 6–8 kDa over an orifice of 30 mm diameter (7.065 cm²), and held at 25 or 37 °C P8·M6. HPMC was dissolved in the donor only: 100 µg/mL for felodipine and 1 mg/mL for nifedipine, the higher level being needed to stop nifedipine crystallizing P8·C8. Supersaturated donor solutions were made by adding methanol stock (5 or 25 mg/mL felodipine; 25 or 50 mg/mL nifedipine) in portions that introduced less than 100 µL of methanol at a time into 30 mL; donor concentrations were chosen below, at and above the predicted amorphous solubility P8. UV dip probes in both chambers followed concentrations in situ without sampling, and the receiver concentration was kept below one-third of the crystalline solubility, so the receiver acted as a sink P8.

From concentration to flux. The linear part of the receiver concentration versus time was fitted by linear regression, and the slope was multiplied by the cell volume to give the flux in µg/min P9·E11. The text cites this step as "Eq. (8)", but the printed Eq. 8 is the free energy of mixing; the flux relation itself is given only in words X2. Each condition was run in triplicate P9. With Eq. 6, a flux measured this way is directly proportional to the donor activity, because $D$, $S$, ${\gamma }_m$ and $h$ do not change between runs C11.

Statistics. One-way ANOVA at $\alpha$ = 0.05 with a Bonferroni post hoc test and a test for equal variances compared the fluxes within the presumed plateau: felodipine at 5, 10, 25, 50 and 100 µg/mL (25 °C) and at 10, 25, 50 and 100 µg/mL (37 °C), and nifedipine at 72, 108 and 145 µg/mL (37 °C) P10·R15.

How this differs from earlier work. The earlier diffusion study by Alonzo et al. had hinted at a flux maximum near the amorphous solubility R18. This paper adds an independent location of the LLPS onset (UV extinction), a thermodynamic prediction of the amorphous solubility corrected for water uptake, two drugs with different crystallization tendency, two temperatures for felodipine, and a statistical test of the plateau R1·R7·R15.

Experimental analysis

Felodipine: a liquid phase at the predicted amorphous solubility

When felodipine stock was added to buffer at 37 °C, extinction stayed close to zero up to about 7.5 µg/mL and then rose steadily; the two fitted lines meet at 8.7 µg/mL (Figure 3) R3. At that concentration the solution was visibly turbid but contained no discernible particles, and no birefringent particles were seen under polarized light, from which the authors conclude that felodipine undergoes LLPS above about 9 µg/mL ("It is thus apparent") R4·C13. The text describes 8.7 µg/mL as "approximately 11 times" the crystalline solubility; with the crystalline solubility at 37 °C of 1.21 µg/mL the ratio is about 7.2, and about 11.8 would follow only from the 25 °C value of 0.74 µg/mL R2·X1.

Table A collects the phase-behaviour data for felodipine. The onset rose with temperature, from 5.70 µg/mL at 20 °C to 13.4 µg/mL at 45 °C in buffer and from 6.53 to 14.56 µg/mL with 100 µg/mL HPMC, and the crystalline solubility rose from 0.55 to 1.63 µg/mL R1·R2·C15. The authors describe the onset as "not notably influenced" by HPMC C16. The HPMC values are, however, higher at every temperature, by 0.83–1.73 µg/mL (9–22%); at 25 °C the two ranges (6.01 ± 0.54 and 7.34 ± 0.38 µg/mL) do not overlap R1·X15. According to the text, HPMC did not raise the crystalline solubility of either drug at the concentrations used, but no values are printed R6·C19.

Table A. Felodipine in 50 mM pH 6.8 phosphate buffer (µg/mL, mean ± 1 SD, n = 3), from Tables 1, 2 and 3 of the paper R1·R2·R7.

The predicted amorphous solubility matches the UV onset in buffer almost exactly: 5.99 against 6.01 µg/mL at 25 °C and 8.70 against 8.70 µg/mL at 37 °C, a correlation the authors call "extremely well" R7·R16·C17. The prediction itself reproduces from the printed inputs: at 25 °C the Hoffman equation with the printed melting data gives a ratio of about 10.84 (printed 10.92), and 0.74 × 10.92 × 0.74 gives 5.98 µg/mL E3·E5. One discrepancy concerns the plot: Figure 4a draws the predicted values at about 6.8 µg/mL (25 °C) and about 7.9 µg/mL (37 °C), not at the tabulated 5.99 and 8.70; the reason is not stated R8·X5.

Note by the AI writer (not discussed in the paper). The Figure 4a points are close to what one obtains by exchanging the two moisture factors between the temperatures (0.74 × 10.92 × 0.83 ≈ 6.7 and 1.21 × 8.64 × 0.74 ≈ 7.7 µg/mL), which suggests a plotting slip rather than a different calculation; the tabulated values are the ones the text and Table 6 use X5.

Nifedipine: crystals first, unless HPMC is present

In buffer alone, nifedipine did not show LLPS. The onset recorded was spontaneous nucleation followed by crystal growth, and polarized light showed crystalline particles R5. With 1 mg/mL HPMC dissolved in the buffer, the scattering phase appeared as a diffuse phase that showed no birefringence when the plane of polarized light was rotated, which the authors read as LLPS R5·C14. Table B shows that between 25 and 45 °C the crystallization onset in buffer (37.90–62.30 µg/mL) lay below the LLPS onset with HPMC (49.38–88.89 µg/mL) R1·C18. At 20 °C the two were similar, 32.30 ± 3.0 against 30.86 ± 2.83 µg/mL, so the statement that nifedipine crystallizes before reaching the LLPS concentration holds clearly only from 25 °C upward R1·X14.

Table B. Nifedipine in 50 mM pH 6.8 phosphate buffer (µg/mL, mean ± 1 SD, n = 3), from Tables 1 and 2 of the paper R1·R2.

The authors explain the contrast between the two drugs by kinetics. Felodipine, a relatively slow crystallizer (Baird 2010, ref 54), nucleates slowly compared with the rate at which supersaturation is generated, so "it is clear" that it can exceed its amorphous solubility and undergo LLPS; nifedipine, a faster crystallizer, crystallizes just below the coexistence curve unless HPMC, acting as a nucleation inhibitor, holds crystallization off R20·C31. The crystallization rates themselves were cited, not measured here R20.

At 37 °C the predicted amorphous solubility of nifedipine (72.09 µg/mL in Table 6) agrees with the UV onset with HPMC (72.84–72.85 µg/mL), which the authors call "excellent" R16·R1·C17. Two cautions apply to this number. Table 3 prints the prediction as 72.85 ± 5.66 µg/mL, the same figures as the UV onset in Table 6, whereas Eq. 5 with the Table 3 inputs (10.00 × 8.78 × 0.82) gives 72.0 µg/mL, matching Table 6 R7·X4. More important, the printed Hoffman ratio of 8.78 cannot be reproduced from the printed melting point (173 °C) and enthalpy of fusion (32.5 kJ/mol), which give a ratio of about 14.4 at 37 °C and would put the moisture-corrected prediction near 119 µg/mL E3·X3. The paper gives no other input that would resolve this, so the excellent agreement for nifedipine rests on a ratio that cannot be checked from the printed data X3.

Flux versus donor concentration

Figure 5 shows a felodipine run at 25 °C. In the donor, the apparent concentration at each level from 2 to 100 µg/mL stayed flat for the 60 min shown (the 50 µg/mL trace drifted from about 53 to about 48 µg/mL), and in the receiver the concentration rose steadily but stayed small: at 60 min about 0.024 µg/mL for a 2 µg/mL donor and at most about 0.155 µg/mL R9·R10. The receiver never approached the crystalline solubility, so sink conditions held R10·C20. The receiver traces fall into two groups, a low one for donors of 2–4 µg/mL and a high one for 5–100 µg/mL (Figure 5b) R10.

Table C gives the fluxes. At 25 °C, flux rose from 12 to 24 × 10⁻³ µg/min between 2 and 4 µg/mL, then jumped to 63 × 10⁻³ µg/min at 5 µg/mL and stayed at 63–71 × 10⁻³ µg/min up to 100 µg/mL R11. The authors describe a linear rise over 2–5 µg/mL, which corresponds to supersaturation ratios of 2.7–6.8, and no statistically significant increase above 5 µg/mL; "It thus appears" that once a threshold supersaturation is passed the flux approaches a constant, and this threshold "correlates closely" with the predicted amorphous solubility at 25 °C R17·C21. In the data, flux divided by concentration is about 6–7 × 10⁻³ mL/min at 2–4 µg/mL and about 12.6 × 10⁻³ mL/min at 5 µg/mL, so the 5 µg/mL point already belongs to the plateau; the change happens somewhere between 4 and 5 µg/mL, where no measurement was made X8.

Table C. Flux versus donor concentration (mean ± 1 SD, n = 3). Felodipine from Table 4 R11·R12; nifedipine read from Figure 6c, approximate, because the paper prints no table for it R14. Note the different flux units.

At 37 °C felodipine flux rose from 6 × 10⁻³ µg/min at 1 µg/mL through 24, 52 and 72 at 2.5, 5 and 7.5 µg/mL to 120 × 10⁻³ µg/min at 10 µg/mL, and then levelled at 136–149 × 10⁻³ µg/min from 25 to 100 µg/mL R12·C22. Figure 6b plots the 10 µg/mL point at about 104 rather than 120 × 10⁻³ µg/min, and Figure 6a plots the 25 µg/mL point at 25 °C at about 75 rather than 71; all other plotted points agree with Table 4 R13·X6. Nifedipine flux at 37 °C, shown only in Figure 6c, rose from about 0 at 10 µg/mL through about 0.30, 0.63 and 1.15 µg/min at 20, 40 and 60 µg/mL to about 1.74 µg/min at 72 µg/mL, and then stayed at about 1.73 and 1.93 µg/min at 108 and 145 µg/mL R14·C22. In each panel the authors drew a steep line and a nearly flat one and marked the break at about 5, 11 and 72 µg/mL R13·R14.

One-way ANOVA found no significant difference among the plateau fluxes: P = 0.37 for felodipine at 25 °C, 0.90 for felodipine at 37 °C and 0.75 for nifedipine R15·C23. Two details of Table 5 do not fit the groups listed in its footnotes. The between-groups degrees of freedom are 2 for felodipine at both temperatures, which corresponds to three groups, not the five (25 °C) and four (37 °C) named; the total degrees of freedom (14 and 11) do match five and four groups of three runs R15·X7. The 37 °C group also includes 10 µg/mL, which lies below the flux onset of 11.07 µg/mL and on the rising line of Figure 6b R15·R16·X10.

Note by the AI writer (not discussed in the paper). The nifedipine flux at about 10 µg/mL, essentially the crystalline solubility at 37 °C (10.03 µg/mL), is about zero, and the authors' rising line crosses zero flux at a positive concentration. If flux were strictly proportional to activity, a saturated solution (activity 1) should still give a measurable flux, as felodipine does at 1 µg/mL and 37 °C (6 × 10⁻³ µg/min) R14·R12·X11. The paper does not comment, so the nifedipine rising branch is better not used to estimate flux at low supersaturation.

Three routes to the same threshold

Table 6 of the paper sets the three thresholds side by side, reproduced here as Table D R16. For felodipine at 25 °C, the flux onset (5.12 µg/mL), the predicted amorphous solubility (5.99) and the UV onset (6.01) lie within 0.9 µg/mL of each other; at 37 °C the flux onset (11.07) lies 2.4 µg/mL above the other two (8.70) R16. For nifedipine all three are close to 72 µg/mL R16. The authors add that, if the drug-rich phase is hydrated and in equilibrium with water, the LLPS onset is the amorphous solubility ("Assuming") C24.

Table D. LLPS thresholds by three routes (µg/mL, ± 1 SD, n = 3), from Table 6 of the paper R16. The nifedipine crystalline solubility is 10.03 ± 0.00 µg/mL in Table 2 R2·X4.

Two points qualify the comparison. For felodipine, Table 6 uses UV onsets measured in buffer without HPMC, whereas the diffusion donor contained 100 µg/mL HPMC; the UV onsets with HPMC were 7.34 µg/mL (25 °C) and 10.43 µg/mL (37 °C), and the latter is closer to the 37 °C flux onset R1·X9. And the paper does not say how the flux onset was obtained (presumably from the drawn lines in Figure 6), while two of the three values carry an SD of 0.00 despite n = 3 R16·X16.

Discussion

How the authors connect phase behaviour and flux

The argument runs from thermodynamics to transport. Below the amorphous solubility the solution is a single phase and, with a constant activity coefficient, its activity relative to the crystal equals the supersaturation ratio (Eq. 10) C34. For felodipine at 25 °C the amorphous solubility corresponds to a supersaturation ratio of about 8, so the activity rises from 1 at the crystalline solubility to about 8 at the onset of LLPS R17. Above that point extra drug only forms more drug-rich phase, which has the same activity as the drug-lean phase (Eq. 9), so the activity stays at about 8 "irrespective of total solution concentration" C33·C35. The usual assumption that concentration can stand in for activity therefore "clearly breaks down when LLPS occurs", and so does the link between flux and concentration C35. Since the flux depends on the activity alone (Eq. 6), Figures 6a–c "clearly demonstrate" a constant flux at and above the amorphous solubility, "indicating" a constant activity C11·C36.

The lower half of this chain is the authors' interpretation, drawn as a schematic in Figure 9 and not tested in the paper. Once activity and flux reach a constant, "the upper limit of drug absorption potential based on a passive absorption mechanism will be reached", and concentrations at and above the amorphous solubility "will not yield a higher drug transport rate across biological membranes" C37. An LLPS system "will serve as a drug reservoir": as drug crosses the membrane from the drug-lean phase, the drug-rich phase replenishes it and keeps the activity at its maximum, a process "likely to be rapid relative to the dissolution of a crystalline material" (Hsieh 2012, ref 19) C38. A one-phase supersaturated solution below the amorphous solubility still has an activity above 1 and gives more uptake than a crystalline material, but "it will never attain the maximum flux", and its activity falls as absorption depletes it C39. Overall, transport from an LLPS system "would be expected to be constant and much higher" than from a crystallizing system or a depleting one-phase solution, and crystallization-inhibiting polymers are "expected" to convert a crystallizing system into an LLPS system and so promote uptake C40·C41. The authors close the discussion by limiting it to passive diffusion: "real life in vivo situations are much more complex" C42.

Where the threshold for LLPS comes from

The second strand concerns whether LLPS is seen at all. LLPS is a transition inside the metastable zone for crystal nucleation (Figure 8), so a drug that nucleates quickly crystallizes before reaching it C30. This is the authors' reading of the felodipine–nifedipine contrast: felodipine, a slow crystallizer, reaches LLPS without help, whereas nifedipine crystallizes just below the coexistence curve unless HPMC inhibits nucleation C31. The close match between predicted amorphous solubility and LLPS onset "suggests" that the metastable zone for LLPS itself is small, so that LLPS follows almost as soon as the binodal is crossed, in line with earlier work (Ilevbare 2013, ref 17) C32·R19.

What it suggests for formulation

Stated by the authors: supersaturated solutions differ in kind from solutions enhanced by cyclodextrins, surfactants, cosolvents or pH, which raise the crystalline solubility but not the chemical potential and therefore "may not enhance passive absorption" C25. Concentrations above the amorphous solubility do not raise passive transport further C37, and crystallization inhibitors are "expected" to help by converting crystallizing systems into LLPS systems C41. In the conclusions the authors state the result in its most general form: the activities and passive transport rates that can be generated in aqueous media "are limited by the amorphous activity, regardless of the mechanism by which supersaturation is generated" C44. The abstract words the same point more cautiously: "This suggests that there is an upper limit in passive membrane transport that is dictated by the LLPS concentration" C44.

Note by the AI writer (not discussed in the paper). For dissolution testing, the practical consequence is that above the amorphous solubility the total drug in a sample (dissolved plus drug-rich droplets) overstates the absorbable driving force; what matters is whether the amorphous solubility is reached and held. The ceiling can be estimated before formulation work from melting data and moisture sorption, or measured directly by UV extinction, and in this paper the two agreed well for felodipine R16. Because HPMC raised the felodipine LLPS onset modestly and was needed for nifedipine to reach LLPS at all, the ceiling is best measured with the formulation's own polymer present R1·X15.

Limitations

Limitations stated by the authors. The authors restrict their conclusions to uptake by passive diffusion and state that "real life in vivo situations are much more complex" C42. They note that, for most amorphous compounds, equilibrium between the amorphous solid and buffer cannot be reached by adding the solid, because crystallization intervenes (Murdande 2010, ref 50) C29. Their equivalence between the LLPS onset and the amorphous solubility rests on the assumption that the drug-rich phase is hydrated and in equilibrium with water ("Assuming"), and their conversion of concentration to activity on a constant activity coefficient, "a reasonable assumption for these highly dilute solutions" C24·C34. The Hoffman equation is described as a "reasonable first approximation" that needs correction for water sorption and ionization C10.

The AI writer's own judgement (1): what the data show and what they do not. The flux plateau is measured directly in three data sets and passes a statistical test, and for felodipine three independent routes place the threshold within 0.9 µg/mL of each other at 25 °C and within 2.4 µg/mL at 37 °C R11·R12·R14·R15·R16. This is the solid core of the paper. The interpretation built on it goes further than the experiments in three ways. First, the activity plateau is inferred from flux only: the donor probes record the apparent total concentration, and the drug-lean concentration or activity above LLPS was not measured separately R9·X12. Second, every supersaturated solution was made by solvent shift from a methanol stock with HPMC present; no ASD was dissolved, and no pH-shift or lipolysis route was tested, so "regardless of the mechanism by which supersaturation is generated" is an extrapolation C44·X13. Third, the reservoir effect and the advantage over crystallizing or one-phase systems (Figure 9) were not tested: the runs lasted about 60 min with a nearly constant donor, and no condition depleted the drug-lean phase C38·C40·X17.

The AI writer's own judgement (2): numbers that do not fit together. Several printed values are inconsistent. The nifedipine Hoffman ratio (8.78) cannot be reproduced from the printed melting data, which give about 14.4, and the nifedipine prediction is printed as 72.85 ± 5.66 in Table 3 but 72.09 ± 1.76 in Table 6 X3·X4. Figure 4a plots the felodipine predictions at about 6.8 and 7.9 µg/mL instead of the tabulated 5.99 and 8.70 X5. Two flux points in Figure 6 differ from Table 4, the "11 times" supersaturation for 8.7 µg/mL at 37 °C is about 7.2 times, and the text's equation numbers do not match the printed ones X6·X1·X2. The felodipine ANOVA degrees of freedom correspond to three groups rather than the five and four listed, and the 37 °C plateau group includes a point below the flux onset X7·X10. At 25 °C the claimed linear rise to 5 µg/mL is in fact a step between 4 and 5 µg/mL with no point in between, so the onset there is bracketed rather than located X8. Finally, the felodipine diffusion onsets (HPMC in the donor) are compared with UV onsets measured without HPMC, which were 17–18% lower than the HPMC values at 25 and 37 °C X9. None of these overturns the plateau, but they mean that the close numerical agreements in Table 6 should be read as approximate.

Conclusion

The paper makes two linked observations on felodipine and nifedipine in pH 6.8 buffer. First, highly supersaturated solutions of both drugs form a non-crystalline drug-rich phase at a concentration that matches the predicted, moisture-corrected amorphous solubility, nifedipine only when HPMC holds off its faster crystallization R1·R5·R16·C17·C18. Second, in a side-by-side diffusion cell the flux rises with donor concentration up to about that concentration and then stays flat, at about 5 µg/mL for felodipine at 25 °C, about 11 µg/mL at 37 °C and about 72 µg/mL for nifedipine at 37 °C R11·R12·R14·R16. The authors interpret the plateau as the solution activity reaching its maximum at the amorphous solubility and conclude that passive transport from aqueous media is "limited by the amorphous activity, regardless of the mechanism by which supersaturation is generated"; the reservoir role of the drug-rich phase and the advantage of LLPS systems over crystallizing ones are proposed rather than tested C36·C44·C38·C40.