Supersaturation in surfactant micelles: why concentration ratios mislead, shown with atazanavir and SDS

Background and motivation

Poorly soluble drugs are formulated in two broad ways. One raises the crystalline solubility, for example with surfactant micelles or cyclodextrins. The other creates supersaturation, that is, a dissolved concentration above the crystalline solubility, for example with amorphous solid dispersions (ASDs), salts, cocrystals, lipid systems that are digested, or weak bases that precipitate on moving from stomach to intestine. The paper starts from a contrast between the two. Miller, Dahan and co-workers measured the apparent permeability of progesterone while raising its apparent concentration: cyclodextrins, surfactants and cosolvents raised the concentration but lowered membrane transport, whereas an ASD raised the concentration without that cost R30·C1. The authors explain this with the strict thermodynamic definition of supersaturation: what drives diffusion across a membrane is the activity of the solute, and in a surfactant solution only the free (molecularly dissolved) drug contributes to activity, not drug held in micelles C1·C2.

A second line of earlier work, much of it from the same group, concerns what happens at very high supersaturation. When the concentration exceeds the amorphous solubility and the drug does not crystallize, the solution separates into a drug-lean continuous phase that stays at the amorphous solubility and a drug-rich phase of nanodroplets. This liquid–liquid phase separation (LLPS; glass–liquid phase separation, GLPS, when the drug-rich phase is glassy) caps the activity, so flux across a membrane reaches its maximum at the amorphous solubility, and the droplets act as a reservoir that keeps the flux there (Ilevbare 2013; Raina 2014; Indulkar 2016) C3.

Surfactants are common in ASD products (the paper names Span 20 in Kaletra and SDS in Incivek and Kalydeco), and the intestine adds bile-salt and phospholipid micelles. The direct precedent is Raina et al. (2015): at equal total drug concentration at and below the LLPS concentration, solutions without solubilizing additives gave higher flux than solutions with them; LLPS still occurred with the additives, but at a higher total concentration, and the ratio of amorphous to crystalline solubility differed from that in plain buffer R28·C4. How much supersaturation such solutions contain was therefore not easy to say. The authors add a second point from colloid science: the textbook picture of a drug dissolving in the hydrophobic micelle core is an oversimplification. Feng found that many drugs sit at the micelle/water interface at their equilibrium solubility, and Eriksson and Gillberg found that cetyltrimethylammonium bromide micelles hold benzene at different sites depending on its concentration R31·C5.

The paper sets out to answer three questions for one model pair, atazanavir (ATZ, a poorly soluble weak base, molecular weight 704.86 g/mol) and SDS (288.37 g/mol): how micelles change the "amorphous solubility", what the thermodynamic supersaturation is when micelles are present, and how micelles solubilize the drug in supersaturated solutions M2·M3·C6. The working hypothesis is that the micelle/water partition coefficient $K_{m/w}$ changes with drug concentration, which would make the relation between supersaturation and concentration non-trivial C6.

Background concepts

Supersaturation as an activity ratio. The supersaturation ratio (SR) is defined by the activity $a$ of the solute in the solution relative to its activity $a^{*}$ in a solution saturated with the crystal (Eq. 9):

In plain terms: supersaturation compares how "eager" the dissolved drug is to leave solution with how eager it is at saturation; $c$ is the concentration, $\gamma$ the activity coefficient, and the asterisk marks the saturated solution E9. Activity is related to concentration by $a=\gamma c$ (Eq. 8), and in a dilute aqueous solution of a drug up to its amorphous solubility the authors take $\gamma =1$ E8·C18. The everyday practice is to replace the activity ratio by the concentration ratio $c/c^{*}$, which is exact only if $\gamma /{\gamma }^{*}=1$; the authors stress that in micellar solutions this requires the micelle/water partition coefficient to be constant over the whole concentration range C20·C37.

Flux measures activity. In a side-by-side diffusion cell, drug moves from a donor compartment through a membrane into a receiver. The flux $J$, mass transferred per unit area and time ($J=dm/(Adt)$, Eq. 4), is proportional to the activity of the solute (Eq. 7) E4·E7:

In plain terms: for a given membrane (diffusion coefficient $D$ in the membrane, thickness $h$, activity coefficient of the solute in the membrane ${\gamma }_m$), flux rises in proportion to activity, so a flux measurement can be read as an activity measurement E7·C17. Above the amorphous solubility the activity no longer rises, because the excess drug forms a second amorphous phase, so the flux plateaus C17.

Micelles, CMC and solubilization. SDS forms micelles above its critical micelle concentration (CMC); the aggregation number $N_{agg}$ is the number of SDS molecules in one micelle. Only surfactant above the CMC solubilizes drug, and the total solubility $S$ often rises linearly with the micellized surfactant (Eq. 6) E6:

In plain terms: the solubility without surfactant, $S_0$, is increased by an amount proportional to the concentration of surfactant in micelles, $C_{SDS}-CMC$; the slope $\kappa$ (the molar solubilization ratio) is the number of drug molecules solubilized per micellized surfactant molecule E6. The micelle/water partition coefficient is the ratio of micelle-bound to free drug (Eq. 10) E10:

In plain terms: $K_{m/w}$ says how strongly the micelles pull drug out of water; if it is the same at every drug concentration, the free drug is always the same fraction of the total and concentration ratios still track activity ratios E10·C37. The paper does not state the concentration basis of $K_{m/w}$ (see the AI writer's note in the analysis section) X3.

Amorphous solubility and its prediction. The amorphous solubility is the highest concentration the solution can reach before LLPS; the authors measure it as the LLPS onset and also predict it from the crystalline solubility. The free-energy difference between amorphous and crystalline drug is estimated with the Hoffman equation (Eq. 2), and the solubility ratio is corrected for the activity lowering caused by water that the amorphous form absorbs (Eq. 3) E2·E3:

In plain terms: the more heat of fusion $\Delta H_f$ the crystal has and the further the temperature $T$ lies below the melting point $T_m$, the larger the solubility advantage of the amorphous form; the factor $\exp (-I(a))$ reduces it for absorbed water, and $S_{xtal}$ is the crystalline solubility in the medium of interest E2·E3. Note by the AI writer: as printed, a positive $\Delta G$ from Eq. 2 in the first factor of Eq. 3 would make the prediction smaller than the crystalline solubility; the sign convention is evidently the reverse of what is printed X1.

Langmuir adsorption. If drug adsorbs to a limited number of sites on the micelle surface, its mole fraction in the micelle $x$ should follow a Langmuir isotherm (Eq. 11) E11:

In plain terms: plotting $C/x$ against the free drug concentration $C$ gives a straight line whose slope is $1/x_m$, the inverse of the maximum mole fraction that can be adsorbed, and whose intercept gives the adsorption constant $K_{ad}$ E11. Once the surface sites are full, extra drug must go somewhere else, here into the core C24.

Spectroscopic and NMR probes. Atazanavir is fluorescent: dissolved in water it emits at 328 nm, and as an amorphous film, where the molecules touch each other, at 369 nm R17·C26. Förster resonance energy transfer (FRET) passes energy from an excited donor (here atazanavir) to a nearby acceptor dye (curcumin) only when the two are within a few nanometres, so the appearance of the dye's emission reports that they share the same small region P11·C30. Pulsed-gradient spin-echo NMR (PGSE-NMR) measures the self-diffusion coefficient $D_{NMR}$ of the micelles, which the Stokes–Einstein equation converts into a hydrodynamic radius $r$ (Eq. 5) E5:

In plain terms: larger particles diffuse more slowly; with the Boltzmann constant $k_B$, the temperature $T$ and the solvent viscosity $\eta$ known, a slower diffusion means a larger micelle E5.

Methods in detail

All solution work used 50 mM phosphate buffer at pH 6.8 and 37 °C, with SDS from 0.05 to 3.6 mg/mL; only the NMR measurements departed from these conditions P1·P3·P6·P7·P12. The study has four connected parts: characterize the micelles, measure crystalline and amorphous solubility across SDS levels, convert membrane flux into free drug and supersaturation, and probe where the drug sits in the micelle. The figure below shows the flow.

Part 1: Characterizing the SDS micelles

Problem. The CMC of an ionic surfactant depends on salt and ionic strength, so the literature value in water cannot be used in a 50 mM buffer C10. What they did. Pyrene carboxaldehyde (0.5 µg/mL) was added to SDS solutions of 0.05–2 mg/mL; its emission maximum shifts to shorter wavelength when it moves from water into the non-polar micelle interior, and the CMC was taken where the maximum changed (excitation 360 nm) P1. The aggregation number was measured by static fluorescence quenching: pyrene (0.4 µg/mL) is the probe, dodecyl pyridinium chloride the quencher, and SDS was 1, 2 or 3.6 mg/mL P2. The relation used is Eq. 1 E1:

In plain terms: $I_p$ and $I_q$ are the pyrene intensities without and with quencher at concentration $c_Q$ (written [Q] in the paper); because a micelle that holds a quencher does not emit, the log intensity ratio rises in proportion to the number of quenchers per micelle, and the slope against $c_Q$ gives $N_{agg}$ once the concentration of micellized SDS is known E1. Probe and quencher are assumed to be distributed among micelles by Poisson statistics E1. All $N_{agg}$ values were measured in drug-free solutions P2·X5.

Part 2: Crystalline and amorphous solubility with SDS

Crystalline solubility. Excess crystalline ATZ (free base prepared from the sulfate salt by precipitation and recrystallization from water:ethanol) was shaken for 48 h at 37 °C in buffer with 0.05–3.6 mg/mL SDS, separated by ultracentrifugation at 209,490 g and assayed by HPLC at 210 nm M2·P3·P4.

Predicted amorphous solubility. The melting point (208 °C) and heat of fusion (90.1 J/g) came from DSC, and the water taken up by melt-quenched amorphous ATZ (5.2%) from vapor sorption at 37 °C, giving $I(a)=0.22$ (Table I) M2·P5. With Eqs. 2 and 3 these give an amorphous-to-crystalline solubility ratio, which the authors then multiply by the crystalline solubility measured at each SDS level P5·R5.

Experimental amorphous solubility. The LLPS onset was found by the solvent-shift method: a 30 mg/mL methanol solution of ATZ was added by syringe pump to the stirred medium (300 rpm, 37 °C) within 10 min, and extinction at a non-absorbing wavelength (350 nm) was followed; the concentration at which the scattering slope changed sharply was taken as the amorphous solubility P6. The authors note that GLPS would be the strict term for ATZ, because its water-equilibrated amorphous precipitate has a glass transition above 37 °C, but they keep the more common term LLPS C7·R32.

Part 3: From flux to free drug and supersaturation

Problem. In a micellar solution the total concentration is easy to measure, but supersaturation depends on the free drug, which is not C2. What they did. A side-by-side diffusion cell with a regenerated cellulose membrane (molecular weight cut-off 6–8 kD, 7.07 cm²) held 34 mL of the same medium on both sides at 37 °C; ATZ was added to the donor from a methanol stock, and the receiver was sampled every 10 min for 60 min P7. In plain buffer, where activity equals concentration, flux rose linearly with concentration up to the amorphous solubility (slope 0.0017 mL/min·cm², R² 0.998) R10. This line was then used as a calibration: a flux measured in an SDS solution was converted into the free-drug concentration that would give the same flux in buffer, and the micelle-bound drug followed by mass balance from the known total P8. The approach is the flux equivalent of dialysis methods used for the same purpose C18.

Two supersaturation ratios were then calculated: $S{R}_{conc}$, the total concentration divided by the crystalline solubility in the same medium, and $S{R}_{activity}$, the flux-derived activity divided by the activity at the crystalline solubility P9. The drug molecules per micelle were obtained by dividing the micelle-bound drug by the number of micelles, itself the micellized SDS divided by $N_{agg}$ P13. Region I of the partition-coefficient plot was fitted to the Langmuir isotherm P14. Finally, the apparent permeability was computed from the same flux data with Eq. 12, once with the free and once with the total donor concentration P15·E12:

In plain terms: the apparent permeability is the flux divided by the donor concentration $C_0$; it gives a property of the membrane only if $C_0$ is a concentration that reflects activity, which in a micellar solution means the free concentration E12·C58.

Part 4: Locating the drug in the micelle

ATZ emission spectra (excitation 250 nm) were recorded in buffer and in 1, 2 and 3.6 mg/mL SDS from below the crystalline solubility up to the LLPS concentration, and for a spin-coated amorphous ATZ film as a reference for drug in contact with drug P10. For FRET, curcumin (0.5 µg/mL) was dissolved in 3.6 mg/mL SDS; on its own it emits strongly only when excited at 369 nm, near the ATZ emission at 369 nm, so curcumin emission under 250 nm excitation of ATZ signals energy transfer from nearby ATZ P11·R20. Curcumin absorption spectra were also compared with curcumin in dodecane (data not shown) P11·R21. PGSE-NMR was run on 40 mg/mL SDS in D₂O with phosphate salts at 25 °C, with 0, 0.4 and 7 mg/mL ATZ; the higher SDS level keeps monomers to 1.25% of the surfactant, and the lower temperature was chosen to slow crystallization P12·C9. Radii were calculated with the viscosity of water at 25 °C, assuming that neither surfactant nor drug changes it P12·C9.

Experimental analysis

The micelles: a CMC lowered by buffer, a nearly constant aggregation number

The emission maximum of pyrene carboxaldehyde stayed at about 471 nm up to 0.4 mg/mL SDS, dropped to about 465 nm at 0.5 mg/mL and levelled at about 458 nm from 1 mg/mL (Figure 2a, approximate); the CMC was taken as 0.5 mg/mL R1. This is well below the 2.4 mg/mL reported for SDS in water, which the authors attribute to the higher ionic strength of the buffer, and close to a value reported in 50 mM electrolyte R2·C10. The aggregation number was 60, 58 and 65 at 1, 2 and 3.6 mg/mL SDS; the authors read this as micelles that grow in number rather than size as SDS is added, as reported before for SDS R3·C11. Note by the AI writer: recomputing $N_{agg}$ from readings of Figure 2b (slope about $6.0\times {10}^3$ M⁻¹ times 10.75 mM micellized SDS) gives about 64.5, consistent with the printed 65 E1.

Solubility: SDS lifts the amorphous solubility only half as much as predicted

In plain buffer the crystalline solubility was 1.1 µg/mL, the predicted amorphous-to-crystalline ratio 62, the predicted amorphous solubility 68 µg/mL and the measured LLPS onset 70 µg/mL, which the authors call "excellent agreement": atazanavir reaches its full amorphous solubility advantage before it crystallizes R4·R5·R6·C12. Table A shows that Table II of the paper lists the same quantities with slightly different values R9.

Table A. Reference solubilities in buffer as given in the text and in Table II of the paper R4·R5·R6·R9.

Note by the AI writer (from the card's extractor notes): the "experimental" and "predicted" entries of Table II look swapped relative to the text, so the amorphous solubility in buffer is best read as about 65–70 µg/mL X2. More importantly, the ratio of 62 cannot be reproduced from the Table I inputs: with $\Delta H_f$ = 90.1 J/g × 704.86 g/mol, $T_m$ = 481.15 K and $T$ = 310 K, Eq. 2 gives $\Delta G$ = 14.55 kJ/mol and Eqs. 2–3 a ratio of about 228; a ratio of 62 would require a heat of fusion near 69 J/g X1·E3. The printed ratio agrees with the measured LLPS onset and with the flux results below, so the problem lies in the printed inputs or the calculation route, not in the value used for the conclusions X1.

With SDS, the crystalline solubility stayed near 1 µg/mL up to the CMC and then rose linearly to about 45.6 µg/mL at 3.6 mg/mL SDS (Figure 3, approximate) R7·C13. The LLPS onset also rose, to about 2.09 mM (about 1.47 mg/mL) at the highest SDS level, but the prediction, 62 times the crystalline solubility at each SDS level, rose to about 4.02 mM (Figure 4, approximate) R8. Fitting Eq. 6 to the three profiles gave the values in Table B R9.

Table B. Molar solubilization ratio and intercept for the three solubility profiles (from Table II; fits of Fig. 4) R9.

The intercepts match the buffer values, so the linear micellar model describes both forms; but SDS solubilizes the amorphous form to a different extent than the crystalline-based prediction implies, and Eq. 3 "fails to predict" the amorphous solubility in SDS C15·C16. As a check, the fitted line for the experimental amorphous form, $64.6\mu \text{g/mL}+0.183\times 10.75\text{mM}$ of micellized SDS, gives about 1451 µg/mL at 3.6 mg/mL SDS, in line with the 1450 µg/mL used as the amorphous solubility in that medium in Figure 7a E6·R12. The concentration ratio of amorphous to crystalline solubility is therefore about 31–33 in 3.6 mg/mL SDS, against about 65 in buffer R12·R13.

Flux: the true supersaturation is unchanged

In buffer the flux rose from about 0.003 µg/min·cm² at 1 µg/mL to about 0.12 µg/min·cm² near the amorphous solubility and stayed at about 0.114–0.121 µg/min·cm² up to 200 µg/mL (Figure 5, approximate) R10. The authors explain the plateau by activity: above the amorphous solubility the extra drug forms a second phase and the activity of the dissolved drug stops rising, as seen earlier for felodipine and nifedipine C17.

Read through the buffer calibration, the free-drug concentration in SDS solutions ran from about 1 µg/mL at the crystalline solubility to about 65–70 µg/mL at the amorphous solubility, while the total concentration at that point was about 340, 800 and 1450 µg/mL at 1, 2 and 3.6 mg/mL SDS (Figure 6, approximate) R11·C19. The slope of free against total drug, which the authors label $\gamma$ and estimate with the ideal dilute solution as standard state, fell from 0.998 in buffer to 0.181, 0.086 and 0.047 R11. Table C collects the supersaturation ratios at the amorphous solubility.

Table C. Supersaturation at the amorphous solubility (read from Figs. 6 and 7b, approximate) R11·R13.

In buffer the two ratios agree, as they must; with SDS the concentration ratio is about half the activity ratio, and the activity ratio stays at about 65 in every medium, close to the predicted amorphous-to-crystalline ratio of 62 R13·R5. The authors conclude that $S{R}_{conc}$ "clearly underpredicts" the fundamental supersaturation, by "a factor of nearly two", and that the agreement of 65 with 62 confirms that flux gives a good estimate of the true supersaturation C21·C22·C41. The gap is not limited to the amorphous solubility: in 3.6 mg/mL SDS the activity ratio was about 4.8, 10.3, 18.8 and 31.5 at 100, 200, 400 and 800 µg/mL total, against concentration ratios of about 2.2, 4.4, 8.9 and 17.8 (Figure 7a, approximate; Table D) R12. The authors also conclude that SDS does not mix into the drug-rich phase formed above the amorphous solubility, because such mixing would lower the activity and the maximum flux ("This observation confirms") C40. Note by the AI writer: this is an inference from the unchanged maximum activity; the drug-rich phase itself was not analysed X11.

The partition coefficient falls, then levels off

From free and micelle-bound drug the authors calculated $K_{m/w}$ in 3.6 mg/mL SDS (Figure 8) R14. It fell from about 327 at about 20 µg/mL total to about 226 at the crystalline solubility (about 45 µg/mL) and about 99 at 100 µg/mL (Region I), then stayed at about 90–114 up to the amorphous solubility at 1450 µg/mL (Region II) R14. At 1 and 2 mg/mL SDS the change of slope occurred at the same activity (data not shown) R15.

Table D. Supersaturation and partition coefficient in 3.6 mg/mL SDS (read from Figs. 7a and 8, approximate) R12·R14.

The Region I data fitted a Langmuir isotherm (Figure 9; R² 0.998 in the text, 0.9977 on the figure), with $K_{ad}=3.86\times {10}^5$ L/mol and a maximum adsorbed mole fraction $x_m$ = 0.017, which corresponds to 130 µg/mL total drug, "very close" to the point where $K_{m/w}$ stops falling R16·C25. Comparing the shape with literature on nystatin (falling $K_{m/w}$, interface-bound) and pyrene (constant $K_{m/w}$, core-bound) in other surfactants, the authors "speculate" that atazanavir is adsorbed at the micelle interface near saturation and associates with the hydrophobic core once the surface sites are full R31·C24. Note by the AI writer: a least-squares fit of readings of Figure 9 gives $x_m$ ≈ 0.0169 and $K_{ad}$ ≈ 3.85 × 10⁵ L/mol, reproducing the printed values E11. The paper gives no unit or basis for $K_{m/w}$; the plotted numbers are about five times the simple ratio of micelle-bound to free mass concentration but match the ratio of the drug mole fraction in micelles to that in water divided by 1000, so the values are best compared only within this paper X3·E10. The boundary is drawn at about 150 µg/mL, about three times the crystalline solubility in this medium, rather than "slightly above" it as the text says X6.

Spectroscopy: drug meets drug, and a dye, at high concentration

In buffer, 50 µg/mL ATZ emitted at 328 nm, the amorphous film at 369 nm, and a buffer solution at 100 µg/mL, above the LLPS onset, at both wavelengths (Figure 10a) R17. The authors assign 328 nm to drug in a dilute polar environment and 369 nm to drug in a less polar, aggregated environment C26. In 3.6 mg/mL SDS, 45 µg/mL ATZ (Region I) gave only the 328 nm peak, while 1200 µg/mL (Region II, still below the amorphous solubility of about 1450 µg/mL) gave both; across the concentrations studied the second peak appeared only above 200 µg/mL, and $I_{369}/I_{328}$ rose from about 0.41 at 500 µg/mL to about 0.80 at 1400 µg/mL (Figure 10b, approximate) R17·R18·R19. Because this happens below the amorphous solubility, the authors exclude a drug-rich phase and read the 369 nm band as drug-drug contact inside the micelle core ("suggests"); in Region I the adsorbed drug still senses water and emits only at 328 nm C27·C28.

The FRET experiment points the same way. Curcumin alone in 3.6 mg/mL SDS emitted only weakly at 524 nm under 250 or 328 nm excitation but strongly under 369 nm excitation R20. With 45 µg/mL ATZ, excitation at 250 nm gave no curcumin band beyond that of the drug alone; with 1200 µg/mL ATZ a new band appeared near 490–500 nm (Figure 11; the text gives 500 nm) R22. Other concentrations followed the Region I / Region II split (data not shown), and curcumin absorption with 1200 µg/mL ATZ shifted to shorter wavelength with "some spectral resemblance" to curcumin in dodecane R21·R23. The authors conclude that ATZ and curcumin are colocalized in a non-polar environment, which "further confirm[s]" the mechanism C29·C31. Note by the AI writer: Figure 11 does not state the excitation wavelength of the curcumin-alone curve, and both 45 µg/mL curves show a small band near 520 nm (about 80 and 110 a.u.), so "no FRET" at 45 µg/mL rests on a visual comparison of small bands X10.

How much drug one micelle holds, and when it swells

Dividing micelle-bound drug by the number of micelles gave the loads in Table E (Figure 12) R24.

Table E. ATZ molecules per SDS micelle (bars read from Fig. 12, approximate; means as printed) R24.

At a given activity each micelle holds about the same number of drug molecules at all three SDS levels, so more SDS means more micelles, not differently loaded ones; at the amorphous solubility a micelle holds about 11 molecules, with about 70 µg/mL of drug free in water R24·C32·C51. The authors take this finite capacity as the point at which LLPS begins: beyond it, drug exists free, micelle-bound and in a water-saturated drug-rich phase C51. As a check by the AI writer, 1450 µg/mL total with 70 µg/mL free in 3.6 mg/mL SDS (10.75 mM micellized SDS, $N_{agg}$ = 65) gives about 11.8 molecules per micelle, matching the 3.6 mg/mL bar; the counts use $N_{agg}$ measured without drug R24·R3·X5.

Table F. Self-diffusion coefficient and radius of SDS micelles in 40 mg/mL SDS, D₂O with 50 mM phosphate salts, 25 °C (Table III) R25.

PGSE-NMR showed no change in micelle size with 0.4 mg/mL ATZ and an increase from 4.61 to 6.08 nm with 7 mg/mL, a 130% increase in volume against a theoretical 100% estimated from the size of an atazanavir molecule in a close-packed crystal; ATZ diffused at the same rate as the micelles R25·R26. The authors explain the difference by about 0.4 adsorbed molecules per micelle at the low concentration and about 7 core-dissolved molecules at the high one C34. Note by the AI writer: these NMR samples differ from the flux experiments in SDS concentration, solvent and temperature, and the stated loadings do not follow from the stated concentrations with $N_{agg}$ = 65 (about 0.27 and 4.7 molecules per micelle if all drug is bound); the abstract's statement that the micelle grows "when the drug concentration reaches the amorphous solubility" refers to conditions where the amorphous solubility was not measured X4·X7.

Permeability: total concentration gives a 20-fold error

Calculated with activity as the donor concentration, the apparent permeability in 3.6 mg/mL SDS equalled that in buffer; calculated with the total donor concentration, it was 20-fold lower (values not shown) R27. The authors call the lower value implausible, since permeability is a property of the molecule for a given membrane, temperature and hydrodynamics as long as the additive does not change the membrane, and they link it to the reduced apparent permeability that Miller et al. reported with solubilizers C58·R30. Note by the AI writer: the 20-fold factor is the inverse of the free fraction in this medium ($1/0.047\approx 21$), so it follows directly from the flux calibration rather than adding independent evidence E12.

Discussion

The authors' explanation, step by step

The discussion ties the results into one chain. The flow below follows the authors' own order; the two branches are two regimes of the same micelle, not a good and a bad outcome.

The authors place this within an older idea from colloid science: Mukerjee and Cardinal's "two-state" model, in which benzene is "adsorbed" at the micelle surface at low concentration and "dissolved" in the core at high concentration, and the solubilization-site change seen by NMR for benzene in cetyltrimethylammonium bromide; "This dissolution-adsorption model for solubilization by micelles is also observed in this study" C47·R31. The strength of their wording grows along the paper, from "we can speculate" when the shape of $K_{m/w}$ is first described, through "suggests" for the emission and "further confirm" for FRET, to "The mechanistic basis for these observations" in the Conclusions C24·C27·C31·C65.

What the results imply for estimating supersaturation and amorphous solubility

The authors draw a practical rule from Eq. 3: it can still predict the supersaturation ratio at the amorphous solubility (about 62 here), because the activities of the crystalline and amorphous reference phases do not depend on the medium, but it cannot predict the amorphous-to-crystalline solubility ratio in surfactant solutions, because that needs equal activity coefficients at both solubilities, which "clearly breaks down" for atazanavir in SDS C44. They expect the same assumption to be "a problematic assumption for other compounds", so that neither the amorphous solubility nor the supersaturation ratio can be estimated in surfactant solutions from the rise of the crystalline solubility C48. They do see a possible shortcut: because the load per micelle at a given activity is the same at all SDS levels, the amorphous solubility measured at one SDS concentration "may" be extrapolated to others, provided $N_{agg}$ stays constant and the mechanism does not change again C53.

On generality they are cautious. The two-fold ratios found here are "most likely both compound and surfactant dependent"; Raina et al. found about 2:1 for felodipine in micellar surfactants, Lu et al. smaller deviations for telaprevir in bile-salt micelles, and the ratio should approach 1 when the surfactant is below its CMC or solubilizes little C54·C55·R28·R29.

Implications for formulation, as the authors state them

The authors argue from the results to formulation design. Because permeation and desupersaturation both depend on supersaturation, formulations must be compared by accurately determined supersaturation C56. They describe two hypothetical ASDs: formulation 1 contains enough surfactant to pass its CMC on dissolution and dissolves quickly to a high total concentration; formulation 2 has no surfactant and dissolves more slowly. A scientist who looks only at dissolution rate and total concentration may choose formulation 1, yet it "may exhibit slower rate of permeation in vivo", because its free drug is lower; in vitro dissolution rankings of surfactant-containing formulations "may lead to erroneous in vivo predictions" C57. Micelle-bound drug can replenish free drug, but an LLPS system holds the flux at its maximum, while a micellar solution of the same total concentration gives lower flux, a difference that "may play a more significant role" for drugs with a narrow absorption window C61·C62. Their recommendation is to choose additive levels that improve wetting and dissolution without forming solubilizing structures, to measure the amorphous solubility in relevant media including bile salts and phospholipids, and to design the formulation "to undergo LLPS if possible" C63. Permeability models that assume a constant micelle-drug association, such as the reciprocal permeability approach, "must be used with caution" C59.

Note by the AI writer (not discussed in the paper): the most transferable lesson is about measurement. A dissolution test of a surfactant-containing formulation reports total drug; for atazanavir in SDS that number overstated the gain in driving force at the amorphous solubility, which did not change at all R13. A free-drug measure (flux or dialysis) and an LLPS onset measured in the actual medium would have avoided both errors. These are shown here for one weak base and one anionic surfactant.

Limitations

Limitations the authors state. The authors say that the 2:1 ratios are "most likely" compound- and surfactant-dependent, that concentration-based supersaturation can be "safely employed" only when the surfactant is below its CMC or solubilizes little, and that extrapolation across SDS levels holds only while $N_{agg}$ and the mechanism stay unchanged C53·C54·C55. They note that the maximum solubility advantage "may not be achieved by all compounds due to crystallization" C12. The NMR temperature of 25 °C was chosen to slow crystallization, and the viscosity of water was assumed for the radius calculation C9.

The AI writer's own judgement (1): the central result is solid. The activity-based supersaturation at the amorphous solubility is about 65–70 in buffer and at three SDS levels, the concentration-based value about half of it with SDS, and the flux calibration in buffer is linear up to the measured LLPS onset R10·R13. The intercepts and slopes of Table II and the Langmuir parameters can be reproduced from the plotted data E6·E11. The free-drug method rests on one assumption the paper does not test: that only molecularly dissolved drug crosses the 6–8 kD membrane and that SDS, present on both sides, does not change the membrane X9. Replicate numbers are not stated for the flux points, and intermediate free-drug values carry wide error bars X12.

The AI writer's own judgement (2): the mechanism is plausible but indirect. No measurement locates the drug in the micelle directly. The partition-coefficient shape, the Langmuir fit, the 369 nm band, the FRET signal and the micelle swelling are all consistent with surface adsorption giving way to core solubilization, but each could have other readings, and the NMR, which carries the size argument, was done at 40 mg/mL SDS in D₂O at 25 °C with loadings that do not follow from the stated concentrations X4·X7. $N_{agg}$ was measured only without drug, although the per-micelle counts and the statement that size grows "although not aggregation number" rely on it X5. The Region I/II change lies at about three times the crystalline solubility, not "slightly above" it X6.

The AI writer's own judgement (3): several printed numbers do not reconcile. The amorphous-to-crystalline ratio of 62 does not follow from the Table I inputs (about 228 by Eqs. 2–3), and Eq. 3 is printed with an inverted sign X1. Buffer values in the text and Table II differ (70 versus 67.5 µg/mL for the LLPS onset, 68 versus 70 µg/mL for the prediction) X2. $K_{m/w}$ has no stated basis; its plotted values match mole-fraction ratios divided by 1000 X3. Two references are mis-cited (refs 71 and 72 concern nanoparticles in the gastrointestinal tract, not the statements they support), and the discussion cites Figure 7 for data shown in Figure 6 X8. None of these changes the main conclusion, because the ratio of 62 is consistent with the measured LLPS onset and the activity ratio of 65.

Conclusion

The paper makes three contributions. First, it shows with flux measurements that SDS micelles leave the activity of atazanavir at the crystalline and amorphous solubilities unchanged, so the supersaturation at the amorphous solubility is about 65 with or without SDS, while the usual concentration ratio falls to about 33–38 R13·C43. Second, it shows that the amorphous solubility in SDS cannot be predicted by scaling the crystalline solubility, because SDS solubilizes the drug about half as effectively near the amorphous solubility as the crystalline data imply (κ 0.183 against 0.351 predicted) R9·C16. Third, it proposes, from a falling-then-flat partition coefficient, emission spectra, FRET and micelle size, that atazanavir adsorbs at the micelle surface near saturation and dissolves in the core when supersaturated, a concentration-dependent mechanism that the Conclusions call "the mechanistic basis" for these observations R14·R19·R22·R25·C65. The authors' broader message is that "rigorous methodology needs to be employed when evaluating supersaturation in complex media where solubilizing additives are present" C66.