Nanodroplets as a drug reservoir: how liquid–liquid phase separation keeps membrane flux at its maximum

Background and motivation

Many new drugs dissolve so poorly in water that formulation has to raise their concentration in the gut. There are two broad routes. One is solubilization: surfactants, cyclodextrins and similar additives hold more drug in solution by taking it into micelles or complexes. The other is supersaturation: amorphous solid dispersions (ASDs), pH shifts or antisolvent addition produce a solution that holds more drug than the crystal would allow (Introduction, p.3). The two routes are not equivalent for absorption. A solution is supersaturated only when the thermodynamic activity of the drug is higher than that of a solution in equilibrium with the crystal, and passive flux across a membrane depends on that activity, not on the total amount of drug present. A solubilizer raises the total concentration but lowers the activity, so at the same total concentration an additive-free supersaturated solution can give a higher flux (Higuchi 1960; Barry 1985; Miller 2012; Raina 2015). C2

Supersaturation has an upper limit. If a highly supersaturated solution of a lipophilic drug does not crystallize, it undergoes liquid–liquid phase separation (LLPS) once the concentration passes the solubility of the amorphous drug: the water-rich continuous phase stays at the amorphous solubility, and the excess drug forms a dispersed, colloidal drug-rich phase, the two in metastable equilibrium (Ilevbare and Taylor 2013). Raina et al. (2014) showed with felodipine and nifedipine that activity and flux rise linearly with concentration up to the amorphous solubility and then stop rising. The amorphous solubility is therefore the highest supersaturation that can be sustained. C1·C3·R13 The dispersed phase generally consists of nanodroplets, which are small and amorphous and so can dissolve fast compared with crystalline material. They form when an ASD dissolves fast enough to overshoot the amorphous solubility (Jackson 2016; Alonzo 2011), and weakly basic drugs dissolved in the stomach can undergo LLPS when the pH rises in the intestine (Hsieh 2012; Psachoulias 2011; Indulkar 2015). C1·C4

From these facts a hypothesis followed. In a solution that has undergone LLPS, the continuous phase gives the maximum possible flux, while the droplets, by dissolving quickly, replace drug as it is absorbed and so act as a reservoir; the more drug sits in droplets, the longer the maximum should last. The authors write that these advantages "seem plausible" but "have not been demonstrated experimentally to date". C5·C6 The reason was practical: earlier mass-transport studies of LLPS systems used set-ups in which drug accumulated on the receiver side, so the receiver stopped being a sink as transport proceeded, and a sustained maximum could not be distinguished from a slowing transfer (Raina 2014, 2015). C6

The paper sets out to show the reservoir effect directly. It uses a flow-through diffusion cell, a device used before for skin permeation, in which fresh fluid is pumped continuously through the receiver chamber. The authors argue that this keeps sink conditions for the whole experiment and is closer to absorption in vivo, where blood flow clears absorbed drug, than a static side-by-side cell. Their expectation was stated in advance: below the LLPS concentration, flux should fall as the donor empties; above it, a steady flux should persist until the droplet phase is used up. Clotrimazole is the model drug. C7

Background concepts

Crystalline and amorphous solubility. The crystalline solubility is the concentration of drug in equilibrium with the crystal. The amorphous solubility is higher, because the amorphous form has a higher free energy. The authors give the relation they rely on, taken from Hancock and Parks (2000) and Murdande et al. (2010) (eq 15, p.13): E15·C23

Here $C_{\mathrm{a}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}}$ and $C_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}}$ are the amorphous and crystalline solubilities, $\Delta G$ is the free-energy difference between the two forms, $R$ the gas constant, $T$ the temperature, and $\exp (-I(a))$ a correction for the lower activity of the amorphous drug once it has absorbed water. In plain terms: the amorphous solubility is the crystalline one multiplied by a free-energy factor, reduced somewhat because the amorphous phase takes up water. The paper prints the exponentials with square brackets and does not say which way $\Delta G$ is defined; the factor $\exp (-\Delta G/RT)$ exceeds 1 only if $\Delta G$ is taken as crystalline minus amorphous free energy. E15·X11 The paper uses the equation only to frame the amorphous solubility; no value of $\Delta G$ or $I(a)$ is given, and the amorphous solubility of clotrimazole was measured directly. E15·R2

Liquid–liquid phase separation and the LLPS concentration. When a supersaturated solution passes the amorphous solubility without crystallizing, it splits into a water-rich phase whose drug concentration stays at the amorphous solubility, and a drug-rich phase dispersed as droplets. The paper uses "amorphous solubility", "LLPS concentration" and "LLPS onset concentration" ($C_{\mathrm{L}\mathrm{L}\mathrm{P}\mathrm{S}}$) for the same quantity. Because the two phases are in metastable equilibrium, the activity of the drug in the water-rich phase stays constant however much drug is added above the onset. C1·C24 The onset is commonly detected by light scattering: a clear solution scatters little, and scattering rises once droplets appear.

Flux, activity and sink conditions. Flux $J$ is the mass of drug crossing unit area of membrane per unit time. For a membrane that only molecularly dissolved drug can cross, flux is driven by the free-drug concentration, or more precisely its activity, on the donor side, minus that on the receiver side. If the receiver concentration is kept near zero (sink conditions), flux is set by the donor alone. Two quantities recur below: $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the highest receiver concentration reached in a run, and $J_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the highest flux. C24·P10

The reservoir effect. In a donor that contains droplets, every molecule that crosses the membrane lowers the free-drug concentration slightly; if the droplets dissolve fast enough, they restore it at once, so the donor stays at the amorphous solubility and the flux stays at its maximum until the droplets are gone. Whether this happens depends on how fast the dispersed phase dissolves compared with how fast drug leaves through the membrane. C25·C29

Dose number. The dose number is the dose divided by the amount that can dissolve in the available fluid volume, that is, by the product of solubility and volume. A dose number above 1 means the dose cannot all be in solution at once; the authors use it in a worked example in the Discussion. R17

Methods in detail

The study has three parts that build on one another: fixing the two solubilities and characterizing the droplets; measuring transport across a membrane from donors below and above the amorphous solubility, with a control in which the dispersed drug is coarse amorphous solid instead of droplets; and a first-principles transport model fitted to every receiver profile. All work used clotrimazole, a weak base with a pKa of about 6, in 50 mM phosphate buffer at pH 8 and 37 °C. At pH 8 the drug is about 99% unionized, which the authors chose so that the measured solubilities are those of the unionized form. M1·M4·C8

Solubilities and droplets

Problem. The whole argument turns on one number, the concentration at which clotrimazole separates into two phases, and on the claim that what forms above it are nanodroplets. Both had to be measured under the conditions of the transport experiments.

What they did. The crystalline solubility was measured by equilibrating excess drug in the pH 8 buffer for 48 h at 37 °C, removing the solid by ultracentrifugation at 40000 rpm and assaying the supernatant by HPLC with UV detection at 210 nm (C18 column, 60:40 0.1% phosphoric acid:acetonitrile, 0.5 mL/min). P1·P2 The amorphous solubility was measured by the solvent-shift method: a 5 mg/mL methanolic solution of clotrimazole was pumped at 10 µL/min into 15 mL of buffer at 37 °C, stirred at 300 rpm, while the extinction at 400 nm was recorded; the concentration at which scattering began to rise was taken as the amorphous solubility (Figure 2). P3 Solutions of 0.5 to 20 µg/mL prepared at 37 °C were then examined by nanoparticle tracking analysis (NTA, Nanosight LM10, 532 nm laser). NTA follows the Brownian motion of individual scattering particles under a microscope and converts it to a size with the Stokes–Einstein equation; it also counts them, giving particles per mL. P4·E18

Flow-through diffusion experiments

Problem. To see a reservoir, the receiver must not fill up. In a closed receiver, drug accumulates and slows transport, which can hide or mimic a plateau.

What they did. A Permegear ILC07 automated system of seven in-line Kel-F cells was held at 37 °C (Figure 1). In each cell a 2 mL donor chamber sits over a 0.708 mL receiver chamber, separated by a regenerated cellulose membrane with a molecular weight cut-off of 12–14 kD, which the authors equate to pores of 3–4 nm; the diffusional area is 1.77 cm². Fresh buffer is pumped through each receiver at 1.8 mL/h by a multichannel peristaltic pump, and the outflow is collected at set intervals into vials and assayed by HPLC. Each sample therefore gives the drug concentration of the fluid that left the receiver in that interval, not a cumulative amount. Judging from the figures, samples were taken every 0.5 h in the 5 h runs and every hour in the 16 h runs; the number of replicate cells per condition is not stated. P5·M5·X8 Because the droplets formed above the onset are much larger than the pores, the authors take it that "it can be safely assumed" that only molecularly dissolved drug crosses. C11

Two sets of runs were made. In the first, 11 donor concentrations from 0.5 to 15 µg/mL, spanning the amorphous solubility, were followed for 5 h. In the second, three donors above the amorphous solubility (10, 15 and 20 µg/mL) were followed for 16 h; their solutions were prepared by titrating a methanol stock into buffer, and 10 µg/mL of HPMC-606 was added to the buffer to stop clotrimazole from crystallizing during the long run. The polymer did not change the LLPS onset, and the authors add that "it can be expected that the polymer may prevent aggregation of the nanodroplets". P6·P7·R12·C12 The authors stress that 16 h is not physiologically relevant; it was needed because the small membrane area slows depletion, and in vivo the effect "would be expected to manifest over much shorter time frames". C15·C16

A third comparison asked whether any amorphous drug would do. Clotrimazole was melted at 150 °C (above its 142 °C melting point), quench-cooled in liquid nitrogen and ground to particles of 150–250 µm. Three donors were compared over 5 h: a one-phase solution at 7.5 µg/mL; the same solution with a further 7.5 µg/mL as nanodroplets (15 µg/mL in total); and the same solution with 7.5 mg/mL of the ground amorphous particles, 1000 times more dispersed drug than in the droplet donor (Figure 5). P8·P9·M6

Flux was calculated from the receiver data with eq 11, the definition of flux as mass transferred per area and time: E11·P10

In plain terms: the drug mass collected from the receiver in each interval, divided by the 1.77 cm² membrane area and the interval length, is the flux for that interval. $J_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the highest value in a run. Note by the AI writer (not discussed in the paper): taking the mass per hour as receiver concentration times the 1.8 mL/h flow reproduces the plotted fluxes, for example 0.720 µg/mL × 1.8 / 1.77 ≈ 0.73 µg/(h·cm²) for 15 µg/mL at 3.5 h. E11·R8

The transport model

Problem. A plateau in the receiver could in principle come from droplets crossing the membrane as well as from droplets refilling the donor. The authors built a model from mass balances to describe both regimes with the same physical constants.

What they did. They assumed a pseudo-steady state in the membrane: the membrane holds little drug compared with the chambers, so its concentration profile settles before the chamber concentrations change. Flux then follows Fick's first law (eq 1): C9·E1

Here $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is the effective diffusion coefficient of the solute in the membrane, $l$ the diffusion path length, and $C_d$ and $C_r$ the drug concentrations in the donor and the receiver. In plain terms: flux is proportional to the difference in dissolved-drug concentration across the membrane. The mass balances of the two chambers are (eqs 2 and 3): E2·E3

with $V_d$ and $V_r$ the chamber volumes, $A$ the diffusional area and $F_r$ the flow rate through the receiver. In plain terms: the donor loses what crosses the membrane; the receiver gains what crosses and loses what the flow carries away. Because the receiver is continuously washed and drug is assumed not to accumulate near the membrane, $C_r$ is dropped from eq 1, giving $J=(D_{\mathrm{e}\mathrm{f}\mathrm{f}}/l)C_d$ (eq 4). Combining eqs 2 and 4 gives first-order depletion of the donor (eqs 5–7): C10·E4·E5·E6·E7

where $C_0$ is the initial donor concentration. In plain terms: without droplets, the donor empties exponentially at a rate set by membrane permeability, area and donor volume. Putting this into the receiver balance gives (eqs 8 and 9): E8·E9

which, with $C_r=0$ at $t=0$, integrates to (eq 10): E10

In plain terms: the receiver fills from a donor that is slowly emptying and is washed out at the much faster rate $F_r/V_r$, so its concentration rises to a peak and then falls in step with the donor. Concentrations are in µg/mL, time in h, $K$ and $K^{\prime }$ in h⁻¹, $F_r$ in mL/h and $V_r$ in mL; the authors note that the form resembles one used by Córdoba-Díaz et al. (2000). E10

For donors above the amorphous solubility, eq 10 was modified. While droplets are present ($C_d>C_{\mathrm{L}\mathrm{L}\mathrm{P}\mathrm{S}}$) the donor is held at the amorphous solubility (eq 12); after they are consumed ($C_d\le C_{\mathrm{L}\mathrm{L}\mathrm{P}\mathrm{S}}$) the donor decays again from that value (eq 13): E12·E13

Here $t^{\prime }$ is the time of interest minus the time for which the droplet phase lasts. In plain terms: while the droplets last, the receiver climbs to a plateau set by the amorphous solubility; afterwards it falls as an ordinary one-phase profile would. Note by the AI writer (not discussed in the paper): if the donor is held constant at $C_{\mathrm{a}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}}$, eq 8 integrates to the same expression with $F_r/V_r$ instead of $F_r/V_r-K$ in the denominator; with the printed constants the two differ by about 4% in the plateau, so the printed eq 12 is best read as the authors' approximation. X10

Finally, since $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ depends only on temperature, viscosity and the size of the diffusing species (Stokes–Einstein), all held constant, and $A$ and $l$ are fixed, eqs 6 and 9 give (eq 14): C21·E14

In plain terms: the receiver constant follows from the donor constant and the ratio of the chamber volumes (2 mL over 0.708 mL), so each profile has a single adjustable constant, $K$. E14·P5

Why the design is informative. The flow-through receiver removes the main confounder of earlier work, receiver accumulation, so a flat receiver concentration can only reflect a constant donor activity. The concentration series is dense (11 levels in the 5 h set) and straddles the measured onset, so the break in flux can be located against an independently measured amorphous solubility. And the particle control keeps the continuous phase identical while changing only the form of the extra drug.

Experimental analysis

The two solubilities

The crystalline solubility of clotrimazole in pH 8 buffer at 37 °C was 0.5 µg/mL. R1 In the solvent-shift experiment the extinction at 400 nm stayed close to zero (about 0.001–0.006) up to about 6.5 µg/mL and then rose linearly, to about 0.099 at about 12.2 µg/mL (Figure 2). The two straight lines through the flat and rising parts cross at the marked 7.5 µg/mL, which the authors take as the amorphous solubility, or LLPS concentration. R2 The amorphous-to-crystalline solubility ratio is therefore 15. R3 HPMC-606 at 10 µg/mL did not shift the onset; these data are not shown. R12

What forms above the onset

NTA detected scattering particles only above the amorphous solubility. Just above it, at 8 µg/mL, the droplets were about 110 nm across; at 10 µg/mL about 220 nm; at 15 and 20 µg/mL the size hardly changed further (the text says "fairly constant at 250 nm"). The number of droplets, in contrast, rose about tenfold, from 1.2 × 10⁸ to 12.5 × 10⁸ particles/mL between 8 and 20 µg/mL (Figure 3, Table A). R4

Table A. Nanodroplet size and number above the LLPS onset, pH 8, 37 °C (read from Figure 3, approximate, except where the text gives the value). Error bars are shown in the figure, but n and their meaning are not stated. R4·X8

The authors use these numbers in two ways. Because the size barely changes between 10 and 20 µg/mL, they assume ("it can be assumed") that the droplets at these concentrations dissolve alike; because the number keeps rising, more drug means a larger reservoir. C30·C31 The droplet size is quoted in several ways across the paper: about 220 nm in the Results and in the dissolution estimate, "fairly constant at 250 nm" for higher concentrations, and "approximately 200nm" in the Conclusions. These are roundings of the same data. X9

The flux ceiling

In the 5 h runs (Figure 4A) every receiver profile rose to a peak at about 1.5 h. The authors attribute the rise to two things: the lag before drug crosses the membrane, and dilution of the first samples by drug-free fluid already in the tubing between receiver and collector. R5·C13 The peak receiver concentration, $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$, increased with initial donor concentration up to 8 µg/mL and not beyond. The flux calculated from the same data behaves identically (Figure 6A), and its maximum, $J_{\mathrm{m}\mathrm{a}\mathrm{x}}$, rises linearly up to 8 µg/mL and then stays level (Figure 6B, Table B). R5·R8·R9·C19

Table B. Peak receiver concentration and maximum flux in the 5 h runs (read from Figures 4A and 6B, approximate). R5·R9

The break falls where the independently measured onset lies: the last one-phase donor (7 µg/mL) is still on the rising line and 8 µg/mL is already level with 10 and 15 µg/mL. The authors call 8 µg/mL "close to the amorphous solubility". R9·R2·C19 Their reading is the activity argument: above the amorphous solubility the two phases are in metastable equilibrium, the activity of the solution phase cannot rise further, and "the excess drug in the drug-rich phase is not expected to permeate through the membrane"; flux, $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and supersaturation therefore all level off at the amorphous solubility, a pattern "clearly observed" for clotrimazole and matching their earlier felodipine and nifedipine data. C24·R13 The authors summarise this for three generic cases, starting below, at and above the amorphous solubility (Figure 8, unnumbered relations on p.19): E16

In plain terms: a donor below the amorphous solubility gives a lower peak; at or above it the peak is the same, the maximum attainable. The same ordering is written for $J_{\mathrm{m}\mathrm{a}\mathrm{x}}$. E16

The plateau and its length

The difference between donors shows after the peak. Every one-phase donor gave a receiver concentration that fell after its peak, more steeply at higher concentration: at 7 µg/mL, from about 0.602 µg/mL at 1.5 h to about 0.464 µg/mL at 5 h. The authors attribute the fall to depletion of the donor, "in line with the expected behavior" of flow-through cells. R5·C14 At 8 µg/mL, 0.5 µg/mL above the onset, the profile fell too. At 10 µg/mL the receiver concentration stayed at $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for about 1.5 h and then declined (to about 0.598 µg/mL at 5 h); at 15 µg/mL it stayed at about 0.68–0.72 µg/mL for the whole 5 h. R5·C32

Because 15 µg/mL did not run down within 5 h, the authors ran 10, 15 and 20 µg/mL for 16 h (Figure 4B). $C_{\mathrm{m}\mathrm{a}\mathrm{x}}$ was almost the same for the three donors, but the steady period lengthened with concentration (Table C). R6·C17

Table C. Length of the steady receiver concentration (durations from the text; plateau heights and 16 h values read from Figure 4B, approximate). R5·R6·C17

Two features of Table C deserve care. First, at 20 µg/mL the receiver concentration was still on its plateau when the experiment ended, so the stated 13 h is a lower bound rather than the time the droplets ran out. X5 Second, the plateau in the 16 h runs (about 0.94 µg/mL) is about 35% higher than in the 5 h runs (about 0.69 µg/mL). The 16 h runs differ in sampling interval and in containing HPMC-606, and the paper does not discuss the difference; the reported finding that HPMC-606 leaves the onset unchanged does not address the plateau height. X6·R12

To show when the droplets run out, the authors back-calculated the donor concentration from the initial concentration and the cumulative receiver concentration (Figure 7). The 8 µg/mL donor reaches 7.5 µg/mL at about the time of the peak (about 7.51 µg/mL at 1.5 h); the 10 µg/mL donor between 2 and 2.5 h (about 7.41 µg/mL at 2.5 h); the 15 µg/mL donor is still at about 8.97 µg/mL at 5 h. From this the authors conclude that at 8 µg/mL "no reservoir effect was observed since the droplet phase was depleted before the C_max was reached", that at 10 µg/mL the droplets were consumed in 3 h, which is when the receiver concentration began to drop, and that at 15 µg/mL they lasted beyond 5 h. "The flux decreases only when the drug in the droplet phase is completely consumed." R10·C31·C32

Note by the AI writer (not discussed in the paper): the plotted donor values match the initial concentration minus the running sum of the receiver concentrations; for 10 µg/mL at 3 h, 10 minus about 0.06, 0.44, 0.68, 0.69, 0.69 and 0.70 gives about 6.7, as plotted. A mass balance with the stated volumes would scale each receiver concentration by the volume collected per interval (0.9 mL in 0.5 h at 1.8 mL/h) over the 2 mL donor, a factor of about 0.45; the 10 µg/mL donor would then still be at about 8.5 µg/mL at 3 h. The Figure 7 depletion times are therefore not supported by the stated volumes, although the qualitative order (8 before 10 before 15 µg/mL) does not depend on this. X3

The authors' reading of the plateau is the reservoir model: the observation "is readily explained by a model where the drug-rich nanodroplets replenish drug molecules that have permeated through the membrane", keeping the donor at the amorphous solubility and the flux at its maximum. C25 They contrast this with micelles, which have also been described as reservoirs but lower the activity of the drug, so that "a sustained plateau will not be seen". C26

Large amorphous particles

Figure 5 holds the continuous phase constant at 7.5 µg/mL and changes only the dispersed drug (Table D). With 7.5 µg/mL of extra drug as nanodroplets, the receiver concentration stayed at about 0.68–0.72 µg/mL for the 5 h of the run. With 7.5 mg/mL of 150–250 µm amorphous particles, 1000 times more dispersed drug, it peaked at about 0.63 µg/mL and then fell to about 0.42 µg/mL, much as the one-phase control did (about 0.63 to 0.49 µg/mL). R7·C18

Table D. Receiver concentration (µg/mL) for three donors with the same 7.5 µg/mL continuous phase (read from Figure 5, approximate). R7

The authors call this "absolutely no reservoir effect" and read it ("This suggests") as the coarse particles failing to dissolve and re-equilibrate with the solution as fast as the droplets. C18·C27 They offer two possible reasons. "One possible explanation" is a difference in surface energy, through curvature and wetting, in line with the common view that freshly precipitated material redissolves fast. "Another explanation" is surface area: the particles are about 1000 times larger than 220 nm droplets, and on a previously reported calculation the droplets are expected to dissolve at least 10⁶ times faster, so that if the particles took 20 min to dissolve, the droplets would take 1.2 ms, and "replacement of drug removed by permeation by dissolution of the LLPS nanoparticles can take place instantaneously". C28·C29·R18 Neither dissolution rate was measured. Two details of Figure 5 are worth knowing: after about 3.5 h the particle donor ran below the control (about 0.46 against 0.52 µg/mL at 4 h), so "similar" means no gain rather than identity; and the nanodroplet curve is, point for point, the 15 µg/mL run of Figure 4A. X13·X7

One rate constant for all profiles

Fitting eq 10 to the one-phase profiles and eqs 12–13 to the two-phase profiles, with $K^{\prime }$ tied to $K$ by eq 14, gave "slightly different" values of $K$ for the different runs, with a mean of 0.102 ± 0.012 h⁻¹; the fitted lines in Figure 4 follow the data in both regimes, and the authors conclude that $K$ "is essentially a constant in this study". R11·C20·C22 They then argue that if the droplets crossed the membrane, their much larger size would give a different $D_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and therefore a different $K$; one $K$ for donors with and without droplets "further supports the supposition that only the drug in the continuous phase diffuses through the membrane". C33

Note by the AI writer (not discussed in the paper): only the mean $K$ is printed. With the mean $K$, $K^{\prime }=0.288$ h⁻¹ (eq 14) and $F_r/V_r=2.54$ h⁻¹, eq 12 gives a plateau of about 0.885 µg/mL, between the plotted plateaus of about 0.69 (5 h runs) and about 0.94 µg/mL (16 h runs), which correspond to $K$ of about 0.080 and 0.108 h⁻¹. Eq 10 with the mean $K$ likewise gives 0.354 µg/mL for 5 µg/mL at 5 h, against about 0.29 plotted. The fits therefore depend on per-run values that are not reported. X2 Also, the membrane was chosen precisely so that droplets cannot pass, so a single $K$ is consistent with, rather than an independent test of, non-permeation. X4

Discussion

The authors' causal chain

Put together, the authors' account runs as follows. Above the amorphous solubility the solution separates; the continuous phase stays at 7.5 µg/mL, so the activity of free drug, and with it the flux, cannot rise further. Only dissolved drug crosses the membrane. What happens next depends on what the extra drug is held in. C24·C11·C33

On this picture the length of the plateau is set by how much drug sits in the droplets, which grows with concentration above the onset as the number of droplets grows, and the existence of a plateau is set by how fast the dispersed drug dissolves, which the authors tie to droplet size. C31·C30·C29 The Conclusions state the result as "solutions that contain clotrimazole nanodroplets result in a membrane transport rate that is both maximized and sustained for a period of time that depends on the amount of nanodroplets in the solution", and attribute the failure of the particles "most likely" to their slow dissolution. C45·C46 The authors also add a summary relation for the three cases of Figure 8: because the maximum lasts longer when droplets are present, E17

In plain terms: the area under the receiver curve grows from a donor below the amorphous solubility, to one at it, to one above it with droplets. They write that sustaining the maximum activity "can potentially translate into high bioavailability" and that, "unlike crystallization", LLPS "can be advantageous". C43 No area under the curve was calculated from the measured profiles.

From the cell to the gut

The Discussion moves from clotrimazole to oral formulation in several steps, each hedged by the authors. First, solubilizers: surfactants and cyclodextrins raise solution concentration and often bioavailability, but only free drug permeates, so permeation "may be slower" than from an additive-free supersaturated solution at the same concentration, and "consideration should be given to the free drug concentration". Earlier work showed TPGS and albumin lowering felodipine flux at equal concentration. C34·R15 Second, supersaturating forms: if the solution from a supersaturating dosage form undergoes LLPS, the solution phase "will result in the maximum achievable flux", with the droplets as reservoir. C35 Whether an ASD produces LLPS depends on how fast it dissolves, which depends on composition and particle surface area; Alonzo et al. found that felodipine–HPMC dispersions at 10% drug loading dissolved fast and formed nanodroplets, while those at 50% reached the amorphous solubility without them. C36·R14 Third, the authors cite Frank et al., who found that ABT-102 dispersions with and without micelles gave the same Caco-2 permeation, and that permeation exceeded what free drug alone could explain, which Frank et al. assigned to redissolving amorphous microparticles; the authors say this is consistent with their results and "establishes the role of the droplets formed upon LLPS in enhancing permeation". R16·C37 Fourth, a speculation: small droplets might also be taken up directly by M cells over Peyer's patches ("it can be speculated"). C38

The authors then work an example with ledipasvir, given as an ASD at 90 mg, with an amorphous solubility of 0.025 mg/mL in fasted simulated intestinal fluid and about 100 mL of intestinal fluid after a 240 mL drink; this gives a dose number of 36. They state that the amorphous solubility is reached once 2.25 mg has dissolved, 2.5% of the dose; without LLPS the remaining 97.5% stays as amorphous solid and, depending on how fast it dissolves, absorption may become dissolution-rate limited, as with the bulk amorphous clotrimazole; with LLPS, the rest would sit in droplets acting as a depot. R17·C39 Note by the AI writer (not discussed in the paper): 0.025 mg/mL × 100 mL is 2.5 mg, which matches the dose number of 36 but not the printed 2.25 mg; 2.5 mg would be about 2.8% of the dose. The point of the example does not change. X1

The authors draw two general consequences. LLPS "has two potential consequences for absorption": the number of molecules available for permeation is maximized, with an advantage over the crystal set by the amorphous-to-crystalline solubility ratio (15 for clotrimazole), and dissolution-rate limits disappear because droplets equilibrate instantaneously. C40·R3 And "simply formulating as an amorphous dispersion would not necessarily be expected to eliminate dissolution rate limited absorption", since amorphous formulations can dissolve slowly or slow down with time, which "may be the case" for dispersions with high drug loading. C41 They expect the advantage to be "likely" greatest for poorly soluble drugs with high dose numbers, note that some formulations of a drug give LLPS and others do not, and call this "an area that needs more investigation". C42

Note by the AI writer (not discussed in the paper): for formulation, the study separates two things that are often merged. Raising the total drug in solution above the amorphous solubility does not raise flux; what extra drug can do is extend the time at the maximum, and only if it is held in a form that dissolves fast enough to keep up with absorption. In this study the same amorphous clotrimazole helped as nanodroplets and did nothing as coarse ground glass. The particle sizes tested, about 220 nm and 150–250 µm, are far apart; how fine an amorphous solid must be to act as a reservoir, and how an ASD's own particles behave, were not tested.

Limitations

Limits the authors state. The 16 h runs are "not physiologically relevant"; they were needed because the membrane area is small, and in vivo the effect would show over much shorter times. C15·C16 The reservoir advantage rests on systems that undergo LLPS "without crystallizing"; HPMC-606 was added in the long runs to prevent crystallization. C47·P7 The authors frame their in vivo implications with hedges: LLPS "may prove advantageous for enhancing bioavailability of some poorly water soluble compounds", and the role of nanodroplets in biological absorption needs more study. C47·C42

The AI writer's own judgement.

• The core observations are clear in the raw data: the flux ceiling lies at the independently measured LLPS onset, and above it the plateau lasts longer at higher concentration in both run sets. R9·R2·R5·R6
• Replicate numbers, the meaning of the error bars and the meaning of the ± on $K$ are not given anywhere, so the variability of the plateau lengths cannot be judged. X8
• The two run sets plateau at different heights (about 0.69 and about 0.94 µg/mL) without comment, and the mean $K$ reproduces neither; per-run constants are not printed. X6·X2
• The donor depletion times of Figure 7 come from subtracting summed receiver concentrations without the volume ratio of collected fluid to donor; a mass balance with the stated volumes gives a decline about 0.45 times as fast. X3
• Non-permeation of droplets is built into the choice of a 3–4 nm membrane; the single $K$ does not test it independently, and no droplets were looked for in the receiver. X4·C11
• The explanation for the particle result, slow dissolution, is inferred; droplet and particle dissolution rates were estimated from size, not measured, and only one particle size was tested. C46·R18
• One drug, one pH, one cellulose membrane and no dosage form were studied; the donor solutions were made from a methanol stock whose final content is not stated. M1·M5·X12
• The ledipasvir example contains an arithmetic slip (2.25 against 2.5 mg). X1

Conclusion

The paper makes three contributions. It measures, for clotrimazole at pH 8 and 37 °C, a crystalline solubility of 0.5 µg/mL and an amorphous solubility of 7.5 µg/mL, and shows that above the latter the drug forms droplets of about 110–260 nm whose number grows with concentration. R1·R2·R4 It shows in a flow-through cell under sink conditions that maximum flux rises linearly up to the amorphous solubility and then stays level, and that donors holding nanodroplets keep the receiver concentration at its maximum for a time that grows with the amount of droplet drug, while 150–250 µm amorphous particles at 1000 times the drug do not. R9·R5·R6·R7 And it gives a compartment model that fits the profiles with one rate constant, which the authors read as support that only dissolved drug crosses the membrane. R11·C33 In the authors' words, "solutions that contain clotrimazole nanodroplets result in a membrane transport rate that is both maximized and sustained". C45 The quantitative details (the depletion times, the model constants and the difference between the two run sets) need the caution noted above; the qualitative result, a ceiling plus a reservoir whose usefulness depends on how fast the dispersed drug dissolves, is the part to carry into formulation thinking. X3·X2·X6