C. Amorphous solubility and LLPS

How it was shown

Ilevbare and Taylor (2013) pumped a 4 mg/mL methanol stock into stirred medium at 37 °C and followed extinction at non-absorbing wavelengths (280–450 nm) for eight drugs, each in a medium where it is un-ionized; the drugs had Log P 2.2–5.6 and melting points 121–187 °C ilevbare2013a P5, M1, M2, M3, M5, M6, M7, M8, M9, R8. For ritonavir in water they added DLS count rate in a flow cell and a pyrene I1/I3 titration, and they isolated the phase by ultracentrifugation at 274,356 × g from 0.5 mL of a 60 mg/mL stock in 20 mL water at 10, 37 and 50 °C, then measured pellet water by drying and Tg by DSC at 20 °C/min ilevbare2013a P7, P8, P9, P10, P11. Almeida e Sousa and co-workers (2015) used six fast-crystallizing drugs at 25 °C in 50 mM phosphate buffer, pH 2 for the three acids and pH 6.8 for the others: extinction at 400 or 450 nm during continuous pumping, and, in stepwise additions, a polarity-sensitive probe (1-pyrenecarboxaldehyde, pyrene, 4-Di-2-ASP) or phenylbutazone's own fluorescence almeidaesousa2015 P1, P2. Indulkar and co-workers (2015) used antisolvent addition, DLS, nanoparticle tracking with Nile red, XRPD, polarized microscopy and DSC for three weak bases at 37 °C indulkar2015 P8, P11, P12, P14, P15, P16. Mosquera-Giraldo and Taylor (2015) combined extinction at 370 nm, 4-Di-2-ASP fluorescence, ¹H NMR of dissolved drug and DSC of an ultracentrifuged pellet for telaprevir in pH 6.8 phosphate buffer at 37 °C mosqueragiraldo2015 P4, P7, P9, P10. Trasi and Taylor (2015) located the onset of ritonavir, lopinavir and paclitaxel at 25 °C in 100 mM phosphate buffer by extinction, pyrene, ultracentrifugation and flux trasi2015a P1, P2, P3, P4, P6, and Almeida e Sousa and co-workers (2016) that of the CRH1 free base by extinction at 450 nm and pyrene almeidaesousa2016 P3, P4. Sugihara and Taylor (2018) shifted pazopanib HCl from simulated gastric fluid (pH 1.2) to pH 6.5 phosphate buffer at 37 °C and looked at the solution by eye, by DLS (backscatter at 173°), by nanoparticle tracking, by polarized light microscopy at 20× and by DSC of the air-dried precipitate sugihara2018 P6, P7, P8, P9, P10. Ueda and Taylor (2020) formed ibuprofen droplets by adding a methanol-d4 stock (2% v/v) to pH 2.0 phosphate buffer in D2O at 37 °C, with or without 300 µg/mL polymer, and sized them fresh by DLS ueda2020b P6.

What the data show

Ritonavir. In water the solution turned cloudy with a bluish tint, and polarized light showed small, non-birefringent, spherical droplets ilevbare2013a R1. Extinction and DLS count rate rose above 37.2 ± 0.9 and 39.8 ± 0.3 µg/mL ilevbare2013a R8. The pyrene I1/I3 ratio stayed at about 1.67–1.69 up to 39.0 µg/mL and fell to about 1.39 at 120 µg/mL ilevbare2013a R4. The droplets grew from about 345 nm at 1 min to about 842 nm at 64 min ilevbare2013a R3. The isolated phase held 94.85–95.97% drug and 4.03–5.15% water between 10 and 50 °C, with a Tg of 0.4 °C against 50.4 °C for dry amorphous ritonavir ilevbare2013a R6, R5, C23. DLS in the second paper gave about 360 nm at 30 µg/mL ritonavir trasi2015a R15. Eight drugs. UV onsets ran from 5.2 µg/mL (clotrimazole) to 135.8 µg/mL (clozapine), and DLS onsets lay close to them (synthesis: the largest UV–DLS difference is 39.8 − 37.2 = 2.6 µg/mL) ilevbare2013a R8. Six fast crystallizers. Extinction and probe signal changed at about the same concentration, and the turbidity lasted only a few minutes before visible particles formed almeidaesousa2015 R1, R3, C17. The probe read-outs were a shift of the 1-pyrenecarboxaldehyde peak from about 468.5–470.5 to about 446–448 nm (ibuprofen), a rise of phenylbutazone's autofluorescence from about 0 to about 980, a rise of the diclofenac pyrene I3/I1 from about 0.52 to about 0.75, and 4–5 nm shifts of the 4-Di-2-ASP peak, to shorter wavelength for tolnaftate and to longer wavelength for sorafenib almeidaesousa2015 R3, X3. Fluorescence onsets were 125.74, 53.61, 0.558, 50.14, 0.532 and 355.96 µg/mL for ibuprofen, phenylbutazone, tolnaftate, diclofenac, sorafenib and griseofulvin almeidaesousa2015 R5, R6. Griseofulvin at pH 6.8 without polymer showed extinction rising near 150 µg/mL with no probe change and small solid particles; with 100 µg/mL HPMCAS-HF in the buffer, extinction and probe intensity rose together near 350 µg/mL and the solution stayed uniformly turbid without visible particles almeidaesousa2015 R4. Weak bases. All six precipitates (clotrimazole from pH 4 and 8, atazanavir from pH 3.5 and 9, nicardipine from pH 6 and 9) showed only broad halos by XRPD and no birefringence indulkar2015 R19, R20, R21; DLS Z-averages were about 116–387 nm indulkar2015 R14; with Nile red in the buffer a population of lipophilic droplets (about 100–450 nm) appeared only above the onset indulkar2015 R16; the DLS polydispersity index of clotrimazole at pH 8 fell from about 0.76 at 6.1 µg/mL to about 0.27 at 6.65 µg/mL at the onset indulkar2015 R11. Telaprevir. Solutions were clear up to about 90 µg/mL and turbid above mosqueragiraldo2015 R1; extinction at 370 nm stayed near zero up to about 93 µg/mL and rose to about 0.048 at about 129 µg/mL mosqueragiraldo2015 R4; the species measured about 300 nm mosqueragiraldo2015 R5; the pelleted material had a Tg of 52 °C mosqueragiraldo2015 R2. CRH1. Solvent shift gave turbidity without visible particles, later clearing as visible particles appeared, with onsets of 14.15 ± 0.87 (extinction) and 11.31 ± 0.56 µg/mL (pyrene) at pH 6.8 and 13.73 ± 0.47 and 12.49 ± 0.08 µg/mL at pH 4.6 with 20 µg/mL HPMC almeidaesousa2016 R5, R3, R4. Pazopanib. Above the amorphous solubility the solution turned milky with a bluish tinge at once sugihara2018 R14; DLS gave Z-averages of 495.9 nm (PDI 0.027) without and 145.4 nm (PDI 0.140) with 50 µg/mL HPMC, NTA found particles of about 50–650 nm sugihara2018 R15, R16, and at the dose below the amorphous solubility no scattering species appeared sugihara2018 R20. Polarized microscopy showed non-birefringent aggregates beside salt crystals at 2 min, and free-base crystals taking over by 10 min without HPMC and by 60 min with it sugihara2018 R11. The dried precipitate had a Tg of about 83 °C sugihara2018 R17. Ibuprofen. In buffer the drug-rich phase was micron-sized (3182–5313 nm at a PDI of 1.00, the top of the scale), with 300 µg/mL HPMC or PVP-VA it was 268–345 nm and with EUD-E 472–808 nm ueda2020b R5, X10; at 400 µg/mL ibuprofen the buffer and all six polymer solutions of the 2021 paper were milky ueda2021 R15.

Figure

Both signals are flat at first and then rise steeply; the dashed lines cross near 38 µg/mL (UV) and 41 µg/mL (DLS) ilevbare2013a R2.

Panels a–f are ibuprofen, phenylbutazone, tolnaftate, diclofenac, sorafenib and griseofulvin; the concentration axes differ about a thousandfold. In panel f the squares (no polymer) show extinction without a probe change, the triangles (HPMCAS-HF) both signals near 350 µg/mL almeidaesousa2015 R3, R4.

Why

Ilevbare and Taylor read the colloids as liquid–liquid phase separation: the four features reported for drug "aggregation" (no second phase below a threshold, non-crystalline aggregates, constant monomer concentration above the threshold, a reproducible threshold in a given buffer) all appeared in their data, and the solution and the droplets are in equilibrium, so the drug has the same activity in both ilevbare2013a C29, C36. They call a hydrophobic-interaction driving force "intuitive" from the Log P range but did not test it ilevbare2013a C37. The probe logic is that a polarity-sensitive dye partitions into a disordered drug-rich phase and changes its emission, whereas crystallization leaves the dye in solution and the emission unchanged, which is why extinction alone cannot tell the two apart almeidaesousa2015 C12, C16; Indulkar and co-workers add that dyes cannot partition into a crystalline phase and confirm lipophilic droplets with Nile red indulkar2015 C29, C37. NMR sees only dissolved telaprevir because molecules in large, viscous glassy aggregates relax too fast to give a resolved signal ("anticipated") mosqueragiraldo2015 C18. (synthesis: the scattering onset, the probe change, the isolated amorphous pellet and the composition of the pellet are four independent observations, and they agree wherever more than one was made, as for ritonavir and telaprevir.) Sugihara and Taylor read the immediate turbidity, the particle sizes and the Tg together as glass–liquid phase separation into a water-saturated colloidal drug-rich phase ("indicate") sugihara2018 C27, C28, and Ueda and Taylor treat the separated ibuprofen phase as a supercooled-liquid emulsion because its Tg is −45 °C ueda2020b C38, M1. Indulkar and co-workers (2017) show the limit of a spectral marker: the 369 nm emission band that marks the atazanavir drug-rich phase in buffer also appeared in 3.6 mg/mL SDS from about 200 µg/mL, far below the amorphous solubility of about 1450 µg/mL, where they read it as drug–drug contact inside micelles and not as a drug-rich phase indulkar2017a R17, R18, C27. (synthesis: a spectral signature of the drug-rich phase must be checked against the amorphous solubility measured in the same micellar medium.)

Conditions and limits

Temperatures were 25 °C almeidaesousa2015 P1; trasi2015a P1; almeidaesousa2016 P3 or 37 °C ilevbare2013a P5; indulkar2015 P8; mosqueragiraldo2015 P7. Supersaturation came mainly from a solvent shift that left about 1–2% v/v organic solvent almeidaesousa2015 X9; mosqueragiraldo2015 X7. The onset did not depend on the stock solvent or addition rate for ritonavir (methanol, ethanol, THF, or a drug dissolved in 0.1 M HCl) ilevbare2013a C28, R9, R10, and the solvent-free pH titration of the weak bases agreed with the antisolvent runs above the lowest pH indulkar2015 C33, X7. Every compound was poorly soluble and lipophilic, with Log P 2.17–5.75 and melting points 75.9–238 °C almeidaesousa2015 M1, M3, M6; mosqueragiraldo2015 M1. The phase was isolated and examined only for ritonavir, clotrimazole, atazanavir, nicardipine and telaprevir, and the ritonavir DSC trace shown stops at 20 °C, so the absence of melting is stated, not shown ilevbare2013a X8. (synthesis: all 21 compounds are lipophilic and poorly soluble, so hydrophilic drugs were not tested.) The 2017–2022 papers add a pH 1.2 to 6.5 transfer (pazopanib) and a dissolved polymer at 300 µg/mL as ways to generate the phase; the ibuprofen work is in D2O buffer with 2% methanol-d4 sugihara2018 P6; ueda2020b M7, P6.

Counterexamples and conflicts

Griseofulvin without polymer crystallized before it could phase separate, or its probe could not enter a glassy phase; the authors cannot tell which almeidaesousa2015 C18. Crystalline nicardipine lost its birefringence in water within 2 h, so it has no crystal–water equilibrium to compare with indulkar2015 R1. The library finds the same split between slow and fast crystallizers in the fluorescence-probe work: see Solution methods F8. Pazopanib at the dose below its amorphous solubility formed no drug-rich phase at all, as expected sugihara2018 R20, C31.

Transfer check

Raise the concentration slowly in the medium of interest while recording extinction at a wavelength where the drug does not absorb, and repeat with a polarity probe or the drug's own fluorescence almeidaesousa2015 P1, P2. Isolate the pellet by ultracentrifugation and check birefringence, XRPD and a DSC Tg indulkar2015 P14, P15, P16; ilevbare2013a P7, P9. A rise of extinction with a probe change and a birefringence-free, halo-only pellet puts the drug inside this evidence; a rise of extinction without a probe change and with birefringent particles, as for griseofulvin without polymer, means crystallization almeidaesousa2015 R4. In a micellar medium also locate the amorphous solubility in that medium, since a spectral band can appear well before it indulkar2017a R18.

History

Before this subcategory the library's background was "promiscuous aggregation": close to 20% of nearly 300 screened compounds formed aggregates of 50 to more than 1000 nm on dilution into water, and LLPS was proposed as the cause ilevbare2013a C7, C9. Ilevbare and Taylor (2013) were the first in the library to isolate and characterize the phase (ritonavir) and to show the onset for eight drugs ilevbare2013a C2, C56. Almeida e Sousa and co-workers (2015) extended it to six fast crystallizers and argued that a fluorescence check is "clearly essential" for them almeidaesousa2015 C21. Indulkar and co-workers (2015) added XRPD and microscopy for three weak bases, Mosquera-Giraldo and Taylor (2015) a glassy telaprevir phase, and the 2015 and 2016 papers by Trasi and Taylor and by Almeida e Sousa and co-workers used the same onset for ritonavir, lopinavir, paclitaxel and CRH1 indulkar2015 C6; mosqueragiraldo2015 C21; trasi2015a C12; almeidaesousa2016 C17. Sugihara and Taylor (2018) extended the phase to a weak base shifted from stomach to intestinal pH, Ueda and Taylor (2020) to a low-Tg drug whose droplets are stabilized by dissolved polymers, and Indulkar and co-workers (2017) showed where the 369 nm marker fails sugihara2018 C27; ueda2020b C13; indulkar2017a C27.

How it was shown

For ritonavir the pellet from 0.5 mL of a 60 mg/mL stock in 20 mL water at 10, 37 or 50 °C was dried at below 5% RH to constant weight for its water content, and a DSC trace was run at 20 °C/min; the water sorption isotherm of amorphous ritonavir was measured at 37 °C from 0 to 95% RH and extrapolated linearly to 100% RH ilevbare2013a P7, P8, P9, P15. For clotrimazole and atazanavir the precipitates formed in water at 37 °C were ultracentrifuged, equilibrated at −30 °C and heated at 5 °C/min indulkar2015 P16; the nicardipine phase was removed at 37 °C as a supercooled liquid and refrigerated at 2 °C for 4 h indulkar2015 P14, R23. For telaprevir, material pelleted from turbid solutions was scanned at 10 °C/min, the dry Tg came from melt-quenched drug, and the water uptake of the glass from gravimetric sorption mosqueragiraldo2015 P2, P4, P5. For griseofulvin and sorafenib, the water taken up at water activity 1 was estimated, amorphous samples were stored at 97% RH and their Tg measured, and the free energy to the supercooled liquid and to the glass were computed by the heat-capacity method almeidaesousa2015 P7, P8, P9, R8, R9. Sugihara and Taylor filtered and air-dried the pazopanib precipitate and ran DSC, with instrument, heating rate, dose and polymer level not stated sugihara2018 P10. The ibuprofen Tg of −45 °C was taken from earlier work, and Ueda and co-workers (2022) measured it by DSC of melt-quenched ibuprofen–polymer mixtures ueda2020b M1; ueda2022 R26.

What the data show

The ritonavir pellet showed a glass transition centred at 0.42 °C (onset −2.74 °C, end 4.29 °C) and, by the text, no crystallization or melting peak ilevbare2013a R5. Amorphous ritonavir sorbs 5.21% water by extrapolation to 100% RH, matching the 4.03–5.15% found in the pellet, and the Tg drop fits the rule of about 10 °C per 1% water (50.0 °C over 4.87% is 10.3 °C per 1%) ilevbare2013a E14, R6, E15. Clotrimazole and atazanavir precipitates had Tg of about 14 and about 51 °C, below the dry values of 28 and 104 °C, after absorbing 1.4 and 5.2% water as amorphous solids indulkar2015 R22, M1, M2; (synthesis: 14 °C over 1.4% is 10 °C per 1%, and 53 °C over 5.2% is 10.2 °C per 1%). Nicardipine, with a dry Tg of 15 °C, separated as a supercooled liquid at 37 °C that became more viscous only after 4 h at 2 °C indulkar2015 M3, R23. The telaprevir pellet had a Tg of 52 °C against 103 °C for dry melt-quenched drug, and the glass takes up 4.31% water at 95% RH (about 4.6% at 100% RH by the authors' estimate) mosqueragiraldo2015 R2, M1; (synthesis: 51 °C over about 4.6% is about 11 °C per 1%, but the water content of the pellet itself was not measured mosqueragiraldo2015 X3). Griseofulvin and sorafenib take up an estimated 2.7 and 1.8% water at water activity 1, and amorphous samples stored at 97% RH still had Tg of 88 °C (griseofulvin) and 49 °C (sorafenib), above 25 °C almeidaesousa2015 R8, R9. Even so their onsets, 355.96 and 0.532 µg/mL, matched the supercooled-liquid predictions (356.44 and 0.479 µg/mL) and not the glass predictions (531.41 and 0.725 µg/mL) almeidaesousa2015 R5, R6, R11. The dried pazopanib precipitate gave a Tg of about 83 °C, which the authors say "confirms" a glass at 37 °C; the drug-rich phase in the solution was not isolated and its water content was not measured sugihara2018 R17, C28, X5. (synthesis: with the rule of about 10 °C per 1% water ilevbare2013a E15, an uptake of about 4.6% water, the estimate for the telaprevir glass mosqueragiraldo2015 M1, would lower 83 °C to about 37 °C, so whether the wet pazopanib phase is a glass at 37 °C is not settled by the dry value.) Ibuprofen's Tg of −45 °C makes its drug-rich phase a supercooled liquid, and solution NMR sees its ibuprofen because the molecules still tumble fast, whereas the drug-rich phases of higher-Tg drugs (nifedipine 42 °C, glibenclamide 59 °C) are invisible to it ueda2020b C38, C7, R16.

Figure

A glass transition centred at 0.42 °C (onset −2.74 °C, end 4.29 °C); the plotted range is −15 to 20 °C only ilevbare2013a R5, X8.

For sorafenib and griseofulvin the supercooled-liquid prediction sits at the onset and the glass prediction above it almeidaesousa2015 R6, R11.

Why

All four groups give the same first explanation: absorbed water plasticizes the amorphous drug and lowers its Tg by about 10 °C per 1% water, so the drug-rich phase is a supercooled liquid when its wet Tg lies below the test temperature and a glass when it lies above ("consistent with") ilevbare2013a C23; indulkar2015 C39, C38; mosqueragiraldo2015 C13. For the two drugs that stay glassy, Almeida e Sousa and co-workers reason that the new phase forms in water, where molecular mobility is "likely" higher and relaxation rapid, so it holds less enthalpy and entropy than a melt-quenched glass and behaves thermodynamically like the supercooled liquid ("This suggests"; "unsurprising") almeidaesousa2015 C37, C29. Mosquera-Giraldo and Taylor make the same assumption for telaprevir, whose glass formed in water they take to be "highly relaxed" mosqueragiraldo2015 C26. (synthesis: two independent sets of data, drugs with a low or moderate wet Tg and drugs with a high one, support a picture in which the phase is liquid-like in the tests used here whatever its nominal state; neither paper measured the enthalpy relaxation or heat capacity of the new phase, so "relaxed" remains an inference.) Sugihara and Taylor use the Tg to name the process glass–liquid phase separation, which by their definition needs a wet Tg above the test temperature, but measured it on the dried material sugihara2018 C2, C28, X5.

Conditions and limits

The state depends on temperature as well as on the drug: the nicardipine phase thickens at 2 °C indulkar2015 R23, and a later paper reports loratadine droplets that are glassy at 5 °C and supercooled liquid at 37 °C yang2021 C16. The telaprevir pellet's water content and crystallinity were not measured, and whether HPMCAS was present in the pellet and how long it stood before analysis are not reported mosqueragiraldo2015 X3, X12. The 97% RH Tg of griseofulvin falls by only 2.8 °C although it absorbs an estimated 2.7% water, against 39 °C for sorafenib with 1.8%, and storage time and water content are not given almeidaesousa2015 X11. No telaprevir ASD was dissolved; all phases came from solvent shift or antisolvent addition mosqueragiraldo2015 X7. Pazopanib adds a case in which the dry Tg is far above 37 °C and the wet Tg is unknown sugihara2018 X5.

Counterexamples and conflicts

None found in the library for the water-plasticization explanation. A low Tg alone does not make a phase last: the ritonavir and clotrimazole liquids and the atazanavir and telaprevir glasses all behaved alike in onset and probe response indulkar2015 C51; mosqueragiraldo2015 C30.

Transfer check

Measure the water taken up by the amorphous drug at water activity 1 (vapor sorption), estimate the wet Tg as the dry Tg minus about 10 °C per 1% water, and compare it with the test temperature ilevbare2013a E15, P15; indulkar2015 C39. A wet Tg above the test temperature means glass–liquid phase separation; test it by dilution (a probe signal that returns within minutes means a rapidly redissolving phase) mosqueragiraldo2015 R8. If the pellet can be isolated, run DSC on it; if not, treat "relaxed" as unproven almeidaesousa2015 C38. For a drug whose dry precipitate has a Tg above 37 °C, as pazopanib's about 83 °C, measure the water uptake of the amorphous drug before calling the phase glassy sugihara2018 X5.

History

Ilevbare and Taylor (2013) characterized a supercooled-liquid drug-rich phase for low-Tg drugs; the introduction of Mosquera-Giraldo and Taylor (2015) states that high-Tg drugs, whose second phase would be a glass, had not been explored ilevbare2013a C22; mosqueragiraldo2015 C5. Almeida e Sousa and co-workers (2015) tested two high-Tg drugs and Mosquera-Giraldo and Taylor (2015) a third, and both groups concluded that the glassy phases behave "phenomenologically the same" or "very similar" to liquid ones almeidaesousa2015 C40; mosqueragiraldo2015 C30. Indulkar and co-workers (2015) proposed "glass–liquid phase separation" as the more appropriate term for atazanavir indulkar2015 C38. Sugihara and Taylor (2018) called pazopanib a glass–liquid case from a dry Tg, and Ueda and Taylor (2020) used a drug whose Tg (−45 °C) makes the drug-rich phase a supercooled liquid that NMR can quantify sugihara2018 C28; ueda2020b C38.

How it was shown

For ritonavir, Ilevbare and Taylor (2013) found the cloud point on cooling at 0.5 °C/min and the clear point on heating, took their mean as the binodal temperature, and extrapolated the reciprocal 90° scattering intensity (474 nm LED) linearly against temperature to zero for the spinodal; they added the titration onsets from 10 to 65 °C and the shake-flask crystal solubility at 25, 37 and 50 °C ilevbare2013a P12, E13, P5, P1. The nucleation induction time came from a "similar experimental" run at 34 µg/mL with no stated method ilevbare2013a R12, X11. For lifetimes, Mosquera-Giraldo and Taylor watched telaprevir solutions with and without 5 µg/mL HPMCAS-MF by eye and polarized light mosqueragiraldo2015 P12, R1; Almeida e Sousa and co-workers recorded how long the turbidity of the six drugs lasted, and tested griseofulvin with 100 µg/mL HPMCAS-HF almeidaesousa2015 R1, R4; Trasi and Taylor ran the ultracentrifuge method with and without 10 µg/mL HPMC trasi2015a P2, R4; Almeida e Sousa and co-workers (2016) followed CRH1 turbidity and later particles, and preliminary studies without polymer showed that it "crystallized readily" almeidaesousa2016 R5, R13. Sugihara and Taylor followed the apparent concentration of pazopanib after the shift with an in-situ UV probe, took the time over which it stayed near its starting level as the supersaturation maintenance at 0, 5, 20 and 50 µg/mL HPMC (intermediate dose, pH 6.5, 37 °C), and watched the precipitate by polarized microscopy at 2, 10 and 60 min sugihara2018 P6, P7, R22, R11. Ueda and Taylor stirred the ibuprofen dispersion at 300 rpm for 3 h and followed it by DLS, NMR and photographs, and checked the polymer-free dispersion for crystals after 24 h ueda2020b P5, P7, P8.

What the data show

Ritonavir crystal solubility was about 1.2, 2.5 and 6.4 µg/mL at 25, 37 and 50 °C, far below the LLPS concentrations of about 37–58 µg/mL ilevbare2013a R11. Binodal temperatures were about 11.4, 15.7, 20.0 and 24.7 °C at about 58, 50, 46.5 and 42.7 µg/mL, and spinodal temperatures about 17.6, 23.4, 25.2 and 29.2 °C; (synthesis: gaps of 17.6 − 11.4 = 6.2, 23.4 − 15.7 = 7.7, 25.2 − 20.0 = 5.2 and 29.2 − 24.7 = 4.5 K) ilevbare2013a R11. At 34 µg/mL the induction time for crystal nucleation was more than 2 h, and the less stable polymorph, Form I, is only about 5 times as soluble as Form II, so the LLPS solution is supersaturated with respect to every crystal form ilevbare2013a R12, C21. Ilevbare and Taylor distinguish region I (between crystal solubility and the binodal, only crystals can form), IIa (beyond the binodal, LLPS and crystal nucleation can both occur, LLPS only if nucleation is slower) and IIb (beyond the spinodal, spontaneous demixing) ilevbare2013a C32. For faster crystallizers, the telaprevir solution was turbid without visible particles above about 90 µg/mL, birefringent particles appeared after about 15 min, and 5 µg/mL HPMCAS kept the turbid solution for hours mosqueragiraldo2015 R1. The six drugs of the second paper showed turbidity that "lasted only a few minutes" before rapid formation of visible particles almeidaesousa2015 R1. Without HPMCAS-HF griseofulvin crystallized at about 150 µg/mL and showed no probe change almeidaesousa2015 R4. Paclitaxel crystallized during ultracentrifugation without HPMC, whereas ritonavir and lopinavir gave unchanged LLPS concentrations trasi2015a R4. CRH1 turbid solutions cleared as visible particles appeared almeidaesousa2016 R5. Pazopanib. The maintenance time was 5 ± 2, about 8.5, 12.5 and 39 min at 0, 5, 20 and 50 µg/mL HPMC sugihara2018 R22; without HPMC the apparent concentration of the intermediate dose fell from about 18–20 to about 1 µg/mL within about 5 min of the shift (synthesis: total time of about 25 min minus the 20-min shift) sugihara2018 R13, X4. Microscopy at the high dose showed non-birefringent aggregates beside salt crystals at 2 min, free-base crystals only by 60 min without HPMC, and a large non-crystalline aggregate still present at 10 min with it sugihara2018 R11. Ibuprofen. In buffer the droplet-phase ibuprofen fell from about 300 µg/mL to about 52 at 30 min and to zero by 2 h, the vial ended clear with a floating drug-rich droplet, no crystals were present at 3 h and crystals appeared after 24 h (data not shown) ueda2020b R15, R8.

Figure

Crystal solubility (lowest curve) lies far below the LLPS concentration; binodal and spinodal points are a few degrees apart on the left branch; regions I, IIa and IIb are marked ilevbare2013a R11, C32.

Why

Ilevbare and Taylor conclude that ritonavir has a submerged liquid–liquid boundary inside the metastable zone for crystallization ("oiling out"), that the small distance between binodal and spinodal agrees with a low energy barrier to LLPS, and that LLPS is "kinetically favored" because crystal nucleation is slow ilevbare2013a C35, C31, C34, C33. They add that the eight drugs of the first paper are all slow crystallizers by the melt classification, which "may explain" why they phase separate first ilevbare2013a C27. Mosquera-Giraldo and Taylor call the amorphous precipitate a precursor phase, because the split occurs in a metastable region where the solution can crystallize at any time ("Hence"), and note that when high supersaturation is generated rapidly the disordered phase is short-lived mosqueragiraldo2015 C36, C37. Raina and co-workers earlier stated the same competition, that LLPS will not be seen if crystal nucleation is fast relative to it raina2014a C30. (synthesis: the rate at which supersaturation is generated, a solvent shift in seconds, is part of the picture, since efavirenz, which the melt classes call a slow crystallizer, crystallized readily from solution in other work; see Crystallization tendency F18.) Sugihara and Taylor attribute the HPMC effect to delayed crystallization of the free base, with or without amorphous particles present ("clearly extends") sugihara2018 C39, C26. Ueda and Taylor read the loss of droplet-phase ibuprofen as coarsening and creaming, not crystallization ueda2020b C27, C28.

Conditions and limits

The binodal and spinodal exist only for four concentrations (42.7–58 µg/mL) between about 11 and 29 °C; at 37 °C only titration onsets exist; the binodal method names a pH 6.8 buffer while the plotted points coincide with the water values, and the scan direction on this branch is unresolved ilevbare2013a X17, X6, X7. The 2 h induction time has no method, replicates or source ilevbare2013a X11. HPMC at 10 µg/mL was stated for the single-drug ultracentrifuge and flux runs but not for every fluorescence, extinction and binary run, and equilibration time before centrifugation is not given trasi2015a X11. Inhibitors differ between papers (HPMCAS-MF, HPMCAS-HF, HPMC), so lifetimes are not comparable across drugs mosqueragiraldo2015 M2; almeidaesousa2015 M7; trasi2015a M6. The pazopanib maintenance time has no stated end-point criterion or replicate number, and its crystals were identified by appearance sugihara2018 X9, X6. The ibuprofen time course is for stirred dispersions with dissolved polymer, not for dissolving solid ueda2020b P5.

Counterexamples and conflicts

The "slow crystallizer" explanation is only a "may explain" and was not tested here ilevbare2013a C27. Griseofulvin, a fast crystallizer, did not phase separate at all without polymer almeidaesousa2015 C18. None found in the library of LLPS lying outside the metastable zone. In the ibuprofen buffer dispersion the drug-rich phase disappeared long before any crystal appeared, so a drug-rich phase that vanishes within the first hour is not necessarily a precursor of crystals ueda2020b R8, R15.

Transfer check

Record the time course of extinction and the polarized-light appearance after generating a supersaturation above the onset, with and without the intended polymer, and note when birefringent particles first appear mosqueragiraldo2015 P12, R1. A drug whose particles appear within minutes without polymer is inside the fast-crystallizer evidence (telaprevir, griseofulvin, paclitaxel, CRH1) and needs an inhibitor in the medium before its onset can be measured almeidaesousa2015 R4; trasi2015a R4. Measure the induction time at a concentration just below and just above the onset if a claim about kinetic favouring is wanted ilevbare2013a R12. For a weak base shifted from low to neutral pH, record the apparent concentration with and without the intended polymer and read the maintenance time at the polymer concentration expected in vivo sugihara2018 R22.

History

The submerged-boundary idea and the region diagram come from Ilevbare and Taylor (2013), who cite earlier crystallization work on oiling out ilevbare2013a C32, C35. Raina and co-workers (2014) had named the same competition for felodipine and nifedipine raina2014a C30. Mosquera-Giraldo and Taylor (2015) described a glassy precursor that crystallizes in about 15 min without an inhibitor, and Almeida e Sousa and co-workers (2015, 2016) showed it for fast crystallizers and a salt-forming weak base mosqueragiraldo2015 C31, C36; almeidaesousa2015 C15; almeidaesousa2016 R5. Sugihara and Taylor (2018) gave the first dependence of the maintenance time on polymer concentration in the chapter, and Ueda and Taylor (2020) showed a drug-rich phase that is lost by creaming sugihara2018 C32; ueda2020b C27.

How it was shown

Ueda and Taylor (2020) formed ibuprofen droplets by adding a methanol stock (2% v/v) to pH 2.0 phosphate buffer in D2O at 37 °C containing no polymer or 300 µg/mL HPMC, PVP-VA or EUD-E, at the amorphous solubility plus 100, 200 and 300 µg/mL, and measured size and PDI by DLS (backscatter at 173°) and, at plus 300 µg/mL, the zeta potential, all fresh and in triplicate ueda2020b P6. The dispersions at plus 300 µg/mL were then stirred at 300 rpm and 37 °C; DLS (size and derived count rate), NMR of the drug in each phase and photographs were taken after 0.5, 1, 2 and 3 h, and the buffer dispersion was stirred 24 h to check for crystals ueda2020b P5, P7, P8.

What the data show

Fresh droplets. In buffer 3182, 4678 and 5313 nm (PDI 1.00); with HPMC 272, 330 and 345 nm (PDI 0.04); with PVP-VA 268, 296 and 320 nm (PDI 0.07–0.22); with EUD-E 472, 709 and 808 nm (PDI 0.43–0.55) at plus 100, 200 and 300 µg/mL ueda2020b R5. The zeta potential was −1.92 mV (buffer), 0.12 (HPMC), 1.45 (PVP-VA) and 64.85 mV (EUD-E) ueda2020b R6. Stirring. Buffer droplets fell below the DLS detection limit by 30 min; with HPMC the size stayed at about 345–385 nm and the count rate at about 275–320 × 10⁶ cps; with PVP-VA the size stayed at about 320–440 nm while the count rate fell from about 470 to 95 (1 h) and 35 × 10⁶ cps (3 h); with EUD-E the droplets grew from about 810 to 1110 nm while the count rate fell from about 200 to 120 × 10⁶ cps ueda2020b R7. By NMR the dissolved ibuprofen stayed constant for 3 h in every medium; the droplet-phase ibuprofen was about 52 µg/mL at 30 min and zero from 2 h in buffer, 120 (2 h) and 67 µg/mL (3 h) with PVP-VA, about 202 and 183 with HPMC and about 191 and 173 µg/mL with EUD-E ueda2020b R15. After 3 h the buffer vial was clear with a floating drug-rich droplet, the PVP-VA vial less turbid, and HPMC and EUD-E vials little changed; no crystals were present in buffer at 3 h but crystals appeared by 24 h ueda2020b R8. By NMR HPMC and EUD-E lost a similar share, about 30%, of their droplet-phase drug, so HPMC's advantage rests on size and count rate ueda2020b X12.

Figure

Open symbols are dissolved drug (constant), filled symbols drug in the droplets: lost within 1–2 h in buffer, steadily with PVP-VA, slowly with HPMC and EUD-E ueda2020b R15.

HPMC is steady, the PVP-VA count rate falls steeply at constant size, the EUD-E droplets grow to about 1.1 µm ueda2020b R7.

Why

Ueda and Taylor treat the separated phase as an emulsion (Tg −45 °C, supercooled liquid) that can coarsen by Ostwald ripening, flocculation and coalescence and then cream ueda2020b C38, C39. The ripening rate scales with the aqueous solubility of the dispersed drug: HPMC and PVP-VA lower the amorphous solubility only about twofold (177.1 to 96.9 and 90.7 µg/mL), yet the fresh droplets are about 12–15 times smaller than in buffer (synthesis: 3182 ÷ 272 and 5313 ÷ 345), so the polymers "appear to" slow ripening more than the solubility alone explains ueda2020b E1, C42, R1, R5. They propose two further mechanisms that slow ripening, a transient polymer excess in small droplets because polymers diffuse about ten times slower than a drug ("could be envisaged"), and a lower interfacial tension ("would reduce") ueda2020b C44, C45, R17. The small fresh droplets "may reflect" this inhibition in the discussion and "contributed" in the conclusion ueda2020b C47. With EUD-E the higher amorphous solubility and the low mixing "most likely" speed ripening and give larger droplets ueda2020b C48. For the later period they say that adsorbed EUD-E charges the droplets positively and the charge repulsion "would lead to" less flocculation and coalescence, that uncharged PVP-VA "clearly" gives too little steric repulsion, and that HPMC, reported to stabilize emulsions sterically, is the "most effective" through the combination ueda2020b C49, C50, C51. The trade-off is that polymer mixing "provides inter-related benefits and disadvantages", better size stability against a lower maximum supersaturation (F19) ueda2020b C52. The extractor notes that no ripening rate was measured, HPMC adsorption and steric repulsion were not measured (the zeta potential with HPMC was near zero), the droplet composition and location of the polymer are unresolved, and flocculation was not measured apart from creaming ueda2020b X2, X3, X4, X11.

Conditions and limits

One drug at pH 2.0 in D2O with 2% methanol, one polymer concentration (300 µg/mL) for the stability tests, dissolved polymers rather than dissolving ASDs, and no membrane-transport or crystallization data beyond 24 h ueda2020b X6, P1. The age of "fresh" droplets is not stated ueda2020b X7. The buffer droplet sizes sit at the DLS PDI ceiling and are only "micron-sized" ueda2020b X10. The charge stabilization by EUD-E belongs to the pH at which it is ionized ueda2020b X6.

Counterexamples and conflicts

Initial size does not predict stability: PVP-VA gave droplets as small as HPMC but lost most of them in 3 h ueda2020b R5, R15. EUD-E stayed in the water with ibuprofen at pH 2.0 (about 289 of 300 µg/mL), but at pH 6.5 it associated with enzalutamide droplets more than HPMCAS did, and the authors name its ionization as the reason, tested at one pH only ueda2020b R13, C35, X6; wilson2021 R7, C19.

Transfer check

Measure droplet size at a few minutes and the dispersion at 1–3 h with the dissolved polymer at the concentration expected in the ASD, using size, count rate (or turbidity) and, for a low-Tg drug, NMR of the droplet-phase drug; add the zeta potential; do not rank polymers on initial size alone ueda2020b P6, P7, R5, R15.

History

The library's background was that HPMC improves the size stability of the drug-rich phase and suppresses crystallization, and that polymer mixing and adsorption at the amorphous interface might matter but were unclear ueda2020b C3. Ueda and Taylor (2020) compared three polymers with NMR evidence of where each sits and measured dispersion stability for 3 h; Wilson and co-workers (2021) found polymer associated with enzalutamide droplets ueda2020b C13, C18; wilson2021 C16.

How it was shown

Extinction at a wavelength where the drug does not absorb (280–450 nm; dip probe, 5 or 10 mm path) during continuous addition from a syringe pump was used in all six papers of 2013–2016 ilevbare2013a P5; almeidaesousa2015 P1; indulkar2015 P8; mosqueragiraldo2015 P7; trasi2015a P4; almeidaesousa2016 P3. DLS count rate was recorded in a flow cell simultaneously with the UV probe (ritonavir) ilevbare2013a P11. Polarity probes were pyrene (I1/I3 or I3/I1), 1-pyrenecarboxaldehyde and 4-Di-2-ASP, or a drug's own fluorescence, added stepwise ilevbare2013a P10; almeidaesousa2015 P2; indulkar2015 P9; mosqueragiraldo2015 P9. Ultracentrifugation (274,356 × g in the first paper) separated the phases and the supernatant was assayed by HPLC or UV ilevbare2013a P7; trasi2015a P2; almeidaesousa2016 P7. ¹H NMR peaks integrated against water gave dissolved telaprevir mosqueragiraldo2015 P10, and a side-by-side diffusion cell gave a flux break point trasi2015a P6. Indulkar and co-workers added a pH-shift potentiometric titration in 0.15 M NaCl with no organic solvent indulkar2015 P10. Lu and co-workers (2017) used extinction at 370 nm while a 12 mg/mL methanol stock was pumped at 0.05 mL/min, ultracentrifugation of turbid solutions for 40 min, and the flux plateau in a side-by-side cell lu2017b P2, P3, P4. Indulkar and co-workers (2017) used extinction at 350 nm for the solvent-shift onset of atazanavir, flux in a side-by-side cell and atazanavir's own emission (excitation 250 nm) indulkar2017a P6, P7, P10. Sugihara and Taylor (2018) used extinction at 400 nm with 5 µg/mL HPMC and, at pH 8.4, ultracentrifugation with HPLC sugihara2018 P4, P5. Ueda and co-workers used scattering at 400 nm and quantitative ¹H NMR with a 12–15 s delay and a TSP reference ueda2020b P3, P4; ueda2021 P3; ueda2022 P4.

What the data show

For ritonavir in water at 37 °C: 37.2 ± 0.9 (UV), 39.8 ± 0.3 (DLS), 39.0 (pyrene) and 36.0 ± 0.49 µg/mL (supernatant), a spread that the extractor summarizes as 34–40 µg/mL depending on method ilevbare2013a R8, R4, R6, X14. The onset did not depend on the stock: ethanol at 0.01, 0.02 and 0.05 mL/min gave 35.0, 34.0 and 35.0 µg/mL and THF 37.5 µg/mL, and drug dissolved in 0.1 M HCl and pumped into pH 6.8 buffer gave 18.8, 16.9 and 7.6 µg/mL for ritonavir, efavirenz and loratadine against 18.8, 18.4 and 7.6 from methanol stocks ilevbare2013a R10, R9, R8. In the six-drug study, extinction read above fluorescence for every drug (136.12 against 125.74, 63.02 against 53.61, 0.59 against 0.56, 52.15 against 50.14, 0.64 against 0.53 and 357.61 against 355.96 µg/mL for ibuprofen to griseofulvin), which the authors attribute to a higher detection limit or to continuous pumping against static additions; 2 µM pyrene did not move the phenylbutazone onset (53.2 against 53.6 µg/mL by autofluorescence) almeidaesousa2015 R5, C24, R7, C25. For the three weak bases, UV, fluorescence and titration overlapped over most of the pH range: clotrimazole at pH 8 gave 7.7 µg/mL by UV and about 7 by pyrene, nicardipine at pH 6.3 gave 55 µg/mL by UV and a 4-Di-2-ASP blue shift from 55 µg/mL; the titration read lower for clotrimazole at pH 4 (about 320 against about 490 µg/mL) and higher for nicardipine near pH 5.85 (about 163 against about 107) indulkar2015 R5, R13, R12, R8, R7, X7. Trasi and Taylor's four methods gave ritonavir 28, 26, 26.2 and 27.3, lopinavir 17.3, 16, 17.1 and 16.1, and paclitaxel 27.2, 28, 29 and 30.1 µg/mL trasi2015a R1. For CRH1 extinction and pyrene gave 14.15 and 11.31 µg/mL at pH 6.8 and 13.73 and 12.49 µg/mL at pH 4.6 with HPMC; NAP autofluorescence of the hemi-NAP salt gave about 9.8 µg/mL in the same medium almeidaesousa2016 R3, R4, R17, X13. The pitfalls: during dissolution the in-situ UV maximum of the hemi-NAP salt was about 19.1 µg/mL against about 11.4 µg/mL after centrifugation and HPLC, while all other forms agreed within about 1.3 µg/mL almeidaesousa2016 R12; the counterion absorbs between about 250 and 320 nm like the drug almeidaesousa2016 R15. Telaprevir in bile salts. In buffer the three methods agreed (146, 149, 148 µg/mL), and in 1.86 and 12 mM trihydroxy bile salts they agreed within about 15% (for example 170, 182 and 162 µg/mL in 12 mM SGC); micellar STDC and SGDC gave 311–360 µg/mL by all three. Extinction rose on the first addition of drug to STCDC and SGCDC without visible turbidity, so only ultracentrifugation (133 and 290 µg/mL at 1.86 and 12 mM STCDC) and flux (110, 238, 169 and 177 µg/mL) were reported, 12 mM STCDC being the most divergent pair (238 against 290 µg/mL, ultracentrifuge error bar about ±25) lu2017b R2, R4, X4. The buffer series gave 149/146 µg/mL in 50 mM pH 6.5, 153/148 in 50 mM pH 6.8 and 104/102 in 100 mM pH 6.8 lu2017b R9. Atazanavir. In buffer the onset was 70 µg/mL in the text and 67.5 in Table II, and the flux calibration plateaued at the same concentration, with an arrow labelled 70 µg/mL drawn to the line break near the 80 µg/mL point indulkar2017a R6, R10, X2. Pazopanib. At pH 8.4 extinction and HPLC gave 9.2 ± 0.7 and 11.4 ± 0.1 µg/mL, which the authors call good agreement sugihara2018 R1.g, R1.h, C12. Ibuprofen. The NMR-detected dissolved drug during stirring was about 168 (buffer), 117–121 (HPMC), 90–97 (PVP-VA) and 273–278 µg/mL (EUD-E) against UV-extinction values of 177.1, 96.9, 90.7 and 256.3 µg/mL, differing by up to about 20 µg/mL and not compared in the paper ueda2020b R15, R1, X1; in buffer at 37 °C the NMR value of 154 µg/mL agreed with the calculated 151 µg/mL ueda2021 R5, R6.

Figure

UV extinction, fluorescence and pH-shift titration overlap over most of the pH range for clotrimazole (A), atazanavir (B) and nicardipine (C); the y-axes differ, and the legend colours in panel C differ from A and B indulkar2015 R5, R6, R7, R8.

Only the hemi-NAP salt differs between the two methods, about 19 against about 11 µg/mL almeidaesousa2016 R12.

White bars are UV extinction, hatched bars ultracentrifugation, black bars the flux plateau; STCDC and SGCDC have no UV bars lu2017b R2, R4.

Why

The authors' logic is that a second phase scatters light, so extinction at a non-absorbing wavelength rises when it forms, while a polarity probe changes its emission only when it partitions into a disordered drug-rich phase and not when crystals form, so the two together separate LLPS from crystallization almeidaesousa2015 C16, C12. Extinction onsets sit slightly above fluorescence onsets because, they say, of a higher detection limit or of continuous pumping against static additions ("can be explained either by") almeidaesousa2015 C24. The solvent-free titration detects the pH at which the neutral drug reaches its amorphous solubility, and its agreement with the antisolvent routes was taken to show that the organic solvent did not matter ("it can be inferred") indulkar2015 C19, C33. NMR reports only dissolved telaprevir, and flux follows activity, so both stop rising at the boundary mosqueragiraldo2015 C18, C10. In-situ UV overestimates the concentration because nanometre-sized drug-rich species absorb like dissolved drug ("highly probable") almeidaesousa2016 C20, C30. (synthesis: methods that read different physical quantities, scattered light, probe polarity, concentration in the continuous phase, activity, agree within about 25% in most comparisons here, which supports the existence of one boundary; the exceptions arise where a method is used outside its range, at the lowest pH or with a UV-active counterion.) Lu and co-workers read ultracentrifugation and flux as giving the coexistence (binodal) concentration of the continuous phase and continual addition with extinction as "might approach" a spinodal point, and their agreement as showing that telaprevir does not supersaturate substantially with respect to GLPS lu2017b C15, C16. For STCDC and SGCDC they "suspect complex formation between drug and bile salt" as the cause of the scattering below the visible onset lu2017b C23. Sugihara and Taylor take the rise of extinction at 400 nm as the amorphous solubility and attribute the amorphous points under the predicted curve below pH 6 to crystallization before the limit was reached sugihara2018 C8, C11.

Conditions and limits

The onset is method-dependent at the level of a few µg/mL: ritonavir in water at 37 °C spans 34–40 µg/mL and Trasi and Taylor's value in 100 mM buffer at 25 °C (26–28 µg/mL) lies above the value of about 22.6 µg/mL read from Fig. 8 of the first paper at the same temperature and ionic strength (synthesis: 3.4–5.4 µg/mL higher; the second study had 10 µg/mL HPMC in some runs) ilevbare2013a X14, R14; trasi2015a R1, X11. The onset criterion is not defined and fluorescence and extinction used different addition modes and volumes almeidaesousa2015 X12. Fluorescence detection near the ng/mL range (tolnaftate, sorafenib) is uncertain almeidaesousa2015 C33. Replicates are not stated for fluorescence, titration, DLS, NTA or dissolution in the weak-base paper, and several SDs are printed as ± 0.0 indulkar2015 X17. During CRH1 salt dissolution the pyrene ratio crossed the LLPS threshold at an apparent concentration of about 9 µg/mL, below the 11.2–12.5 µg/mL estimate and the solvent-shift onsets, and this offset is not discussed almeidaesousa2016 X3. The telaprevir onset depends on the buffer: 100 mM against 50 mM phosphate lowers it by about 30% (104/102 against 149/146 µg/mL), so compare only matched buffers lu2017b R9, X11. The extinction method needs the scattering species to be absent below the onset, which failed with STCDC and SGCDC lu2017b R4. In SDS the 369 nm emission band of atazanavir is present from about 200 µg/mL, so a fluorescence marker of the drug-rich phase is not unique in micellar media indulkar2017a R18. The NMR work is in D2O buffer with 2% methanol-d4 and counts only dissolved drug ueda2021 X10; ueda2022 X6.

Counterexamples and conflicts

The titration departs from the other routes at the lowest pH indulkar2015 X7. The 4-Di-2-ASP peak shifted to shorter wavelength for tolnaftate and to longer wavelength for sorafenib, although the dye is described as blue-shifting in a more hydrophobic environment, and the paper does not comment almeidaesousa2015 C13, X3. Nanosized particles below the expected onset at pH values where the drug is mostly ionized were called an artefact of the antisolvent addition indulkar2015 C36. Extinction was unusable for STCDC and SGCDC, and in micellar atazanavir the 369 nm band appeared far below the onset lu2017b R4; indulkar2017a R18, C27.

Transfer check

Use at least two principles in the real medium (extinction plus a probe or the drug's own fluorescence), run a no-probe control, and define the onset criterion before the run almeidaesousa2015 R7, X12. For every maximum above the expected limit, centrifuge or ultracentrifuge and assay the supernatant by HPLC; a UV value that exceeds the supernatant value points to drug-rich droplets or a UV-active additive almeidaesousa2016 C15, R12. Expect a spread of up to about 25% between methods (synthesis: from the ratios of the onsets above) and do not compare onsets from different media ilevbare2013a X14, X16. In a micellar or bile-salt medium use ultracentrifugation or flux next to extinction, and confirm any fluorescence marker against the amorphous solubility measured in that medium lu2017b R4; indulkar2017a R18. For a drug with a Tg far below the test temperature quantitative NMR gives the dissolved drug directly (F15).

History

Ilevbare and Taylor (2013) combined UV extinction, DLS and pyrene for ritonavir and showed that the stock solvent and the route did not change the onset ilevbare2013a R10, R9. Almeida e Sousa and co-workers (2015) made the fluorescence check standard for fast crystallizers and added a probe control almeidaesousa2015 C21, R7. Indulkar and co-workers (2015) added a solvent-free titration, Mosquera-Giraldo and Taylor (2015) NMR and flux, Trasi and Taylor (2015) four methods including the flux break, and Almeida e Sousa and co-workers (2016) a counterion fluorescence read-out and centrifugation with HPLC indulkar2015 C49; mosqueragiraldo2015 C19, C20; trasi2015a C12, C13; almeidaesousa2016 C50, C42. Probes and UV artefacts are treated in the Solution methods chapter (Related). Lu and co-workers (2017) gave the first comparison of three onset methods in bile-salt media, with extinction unusable for two of the six bile salts; Sugihara and Taylor (2018) repeated the extinction-against-HPLC check for a weak base at pH 8.4; Ueda and co-workers (2020, 2021) moved to NMR of the dissolved drug lu2017b C14, C23; sugihara2018 C12; ueda2021 C10.

How it was shown

Each paper measured the shake-flask crystalline solubility of the free drug in the test medium (48 h; HPLC) ilevbare2013a P1; almeidaesousa2015 P3; indulkar2015 P3; mosqueragiraldo2015 P1; almeidaesousa2016 P5, Tm and ΔHf by DSC (and, for the Cp method, modulated DSC heat capacities) almeidaesousa2015 P5, P6; mosqueragiraldo2015 P2, P3; almeidaesousa2016 P9, and the activity of the water-saturated amorphous drug, exp[−I(a2)], by gravimetric water sorption and Gibbs–Duhem integration (the Murdande method) ilevbare2013a P2, P15; almeidaesousa2015 P7; indulkar2015 P7; mosqueragiraldo2015 P5; almeidaesousa2016 P8. The crystal-to-amorphous free energy came from the Hoffman equation or the heat-capacity method (F7). Ilevbare and Taylor also dissolved a spin-coated amorphous ritonavir film at 37 °C to get an experimental amorphous solubility ilevbare2013a P3, P4. Indulkar and co-workers (2017) measured the shake-flask solubility of atazanavir free base in 50 mM pH 6.8 phosphate at 37 °C and the solvent-shift onset by extinction at 350 nm, and calculated the ceiling with the Hoffman free energy and a water term from a vapor-sorption isotherm (5.2% water, I(a) 0.22) indulkar2017a P3, P5, P6, M2. Ueda and co-workers (2021, 2022) took the free-energy difference and the water-saturated activity of ibuprofen from the 2015 six-drug paper, measured the crystalline solubility and read the dissolved drug by NMR ueda2021 P7, R6; ueda2022 P11.

What the data show

The table gives, for each compound and medium, the onset, the predicted value (the heat-capacity supercooled-liquid value where there is one) and their ratio (synthesis: onset ÷ prediction). Water-saturated amorphous activities exp[−I(a2)] were 0.27 (ritonavir), 0.44 (efavirenz), 0.60 (loratadine), 0.28 (ketoconazole), 0.73 (indomethacin), 0.85 (felodipine), 0.83 (clotrimazole), 0.80 (clozapine) ilevbare2013a M1, M2, M3, M8, M7, M9, M5, M6, 0.999 to 0.699 from ibuprofen to griseofulvin almeidaesousa2015 M1, M2, M3, M4, M5, M6, 0.82 and 0.22 (clotrimazole and atazanavir) indulkar2015 M1, M2, 0.30 (telaprevir) mosqueragiraldo2015 M1 and 0.755 (CRH1) almeidaesousa2016 M1. Ritonavir's ionic-strength series moved the onset, the film value and the prediction together (F9), and the pH-shift and stock-solvent controls left it unchanged (F4) ilevbare2013a R13, R9, R10. Atazanavir in buffer: crystalline 1.1 µg/mL, predicted ratio 62 and amorphous solubility 68 µg/mL, onset 70 µg/mL indulkar2017a R4, R5, R6. Ibuprofen at pH 2.0 and 37 °C: calculated 151, measured 154 µg/mL ueda2021 R6, R5; in the temperature series the experimental S0a/S0c in buffer was about 5.8, 3.6 and 2.5 at 25, 40 and 60 °C against calculated values of about 3.7, 2.5 and 1.5 ueda2022 R16. The ratio of ibuprofen is not the same from paper to paper: 5.94 in 2020 (177.1 over 29.8 µg/mL) and 3.57 in 2021 (154 over 43.1 µg/mL, synthesis), both at pH 2.0 and 37 °C ueda2020b R1; ueda2021 R5, R2.

Figure

The last three columns sit close together for every drug; the medium pH differs between drugs ilevbare2013a R8, X16.

The crystalline-solubility bar is the pH 6.8 value, the others are pH 4.6 values with 20 µg/mL HPMC almeidaesousa2016 R7, X1.

Why

Ilevbare and Taylor argue from phase equilibrium: the drug has the same activity in the water phase and in the drug-rich droplets, the droplets are water-saturated amorphous drug, so the drug concentration in the water at LLPS is the amorphous solubility, and that follows from the crystalline solubility by the free-energy gap and the lowering of activity by absorbed water ilevbare2013a C36, C61, C60. Eq. 1 assumes that the activity coefficient in the dilute solution is the same at the crystalline and the amorphous concentration and treats the amorphous drug as a supercooled liquid ilevbare2013a C55, C58. Indulkar and co-workers state the same for weak bases ("Assuming") and Almeida e Sousa and co-workers put it as "the extent of supersaturation is limited by the 'solubility' of the amorphous drug, more rigorously of the water-saturated supercooled liquid or glass" indulkar2015 C43; almeidaesousa2015 C39. Water uptake lowers the ceiling: for ritonavir exp[−I(a2)] is 0.27 and for ketoconazole 0.28, so the water-corrected value is about a quarter of the value without the water term (synthesis: 0.27 and 0.28) ilevbare2013a M1, M8. See the amorphous solubility advantage formula with water-sorption correction. Indulkar and co-workers (2017) call the agreement of 70 with 68 µg/mL "excellent" and conclude that atazanavir reaches its full amorphous solubility advantage without crystallizing first, which "may not" hold for all compounds because of crystallization indulkar2017a C12. Ueda and co-workers use the match of 154 with 151 µg/mL to read the NMR-detected dissolved drug as the amorphous solubility ueda2021 C10; ueda2022 C15.

Conditions and limits

The predictions can be reproduced from the printed inputs for ketoconazole, clotrimazole, felodipine and clozapine but not for ritonavir (32.9 against 39.2 µg/mL in water), loratadine, efavirenz (10.3 against 19.8) or indomethacin (42.9 against 26.7), and the inputs actually used are not stated ilevbare2013a X1. Indulkar and co-workers print Eq. 2 with a minus sign that gives a ratio of 0.072 for clotrimazole, and only the plus sign reproduces their ratios of about 9.2 and 62 indulkar2015 X1, E2, R3. The 2015 Almeida e Sousa predictions recompute from the printed inputs for the Hoffman route but the heat-capacity parameters are not printed almeidaesousa2015 E1, E2, E9. Media differ between drugs (pH 2, 6.8, 10 and water), so compare drugs only with their media ilevbare2013a X16. All comparisons were made for un-ionized drug, at 25 or 37 °C, without polymer or surfactant except griseofulvin (HPMCAS-HF) and CRH1 at pH 4.6 (HPMC). The atazanavir ratio of 62 follows from the Table I inputs only with 0.22 read as exp[−I(a)], as Indulkar 2015 lists it, although Table I heads the column I(a): ΔHf 90.1 J/g, Tm 208 °C and T 310 K give ΔG 14.55 kJ/mol and exp(5.65) × 0.22 ≈ 62, against about 228 with 0.22 read as I(a) (synthesis) indulkar2015 M2; the sign of Eq. 3 is printed inverted, and Table II swaps the text values (67.5 and 70 µg/mL) indulkar2017a X1, X2. The ibuprofen calculation uses inputs of the 2015 six-drug paper (exp[−I(a2)] determined in H2O) while the measurements are in D2O with 2% methanol-d4 ueda2022 X6; ueda2021 X10. Pazopanib melts with degradation at about 300 °C, so neither ΔHf nor the water content of the amorphous drug could be determined sugihara2018 C17, M1.

Counterexamples and conflicts

Tolnaftate's onset is 1.72 times its prediction and sorafenib's 1.11 times, which the authors attribute to the fluorescence detection limit and the uncertainty of crystalline solubilities in the ng/mL range ("may be explained") almeidaesousa2015 X5, C33. Clotrimazole's un-ionized onset (7.7 µg/mL) is about 1.6 times the prediction (4.9 µg/mL) indulkar2015 X6. The Hoffman prediction for telaprevir is 2.7 times the onset (F7) mosqueragiraldo2015 C23. The correlation of the 2016 CRH1 data is called "strong" although the pH 6.8 onsets are up to 1.57 times the estimates (synthesis: 14.15 ÷ 9.0) almeidaesousa2016 C19, R2, R3. Ibuprofen's measured ratio lies 1.4–1.7 times above its calculation at all temperatures (synthesis), and its crystalline solubility at 37 °C differs between papers (29.8 and 43.1 µg/mL), so the same drug gives ratios of 5.94 and 3.57 (synthesis: 154 ÷ 43.1) ueda2022 R16; ueda2020b R1; ueda2021 R2.

Transfer check

Measure the crystalline solubility in the test medium, Tm and ΔHf by DSC, and a vapor sorption isotherm of the amorphous drug to get exp[−I(a2)] by Gibbs–Duhem; calculate the ceiling with the Hoffman and the heat-capacity free energies (F7); then measure the onset in the same medium ilevbare2013a P1, P2; almeidaesousa2015 P5, P6, P7. A drug is inside the evidence if its onset falls within about 0.9–1.15 times the calculation, as 17 of the 24 rows here did; a ratio near 1.3–1.7 or 0.4 means check the free-energy method, the crystal form and the detection limit first (synthesis: from the table above) almeidaesousa2015 X5; mosqueragiraldo2015 C23. For ionizable drugs work in the un-ionized form (F10). For a high-melting drug that degrades on melting, as pazopanib, the calculation is not available; measure the onset and the crystalline solubility directly sugihara2018 C17.

History

The estimate of the amorphous solubility from crystalline solubility, the free-energy gap and a water term is the Murdande method, used earlier in the library (2012) for pH-shift precipitation of weak bases hsieh2012 E5, E6. Ilevbare and Taylor (2013) were the first in the library to show, for eight drugs, that the LLPS onset matches it ilevbare2013a C3. Almeida e Sousa and co-workers (2015) extended this to six fast-crystallizing and glassy drugs, Indulkar and co-workers (2015) to weak bases across pH, Mosquera-Giraldo and Taylor (2015) to telaprevir, and Almeida e Sousa and co-workers (2016) to a salt-forming base almeidaesousa2015 C42; indulkar2015 C3; mosqueragiraldo2015 C27; almeidaesousa2016 C19. Later papers of the library use the onset as the experimental amorphous solubility, for example for tacrolimus (ratio about 35) trasi2016 R16. Indulkar and co-workers (2017) and Ueda and Taylor (2021) used the calculation again to validate a measurement, atazanavir in buffer and ibuprofen by NMR indulkar2017a C12; ueda2021 C10; Sugihara and Taylor (2018) report the first drug of the chapter for which it could not be made sugihara2018 C17.

How it was shown

The ratio is the LLPS onset divided by the crystalline solubility of the same drug in the same medium (F4, F5) almeidaesousa2015 E15; ilevbare2013a P1, P5; indulkar2015 P3, P8; trasi2015a P1, P2. Almeida e Sousa and co-workers (2015) tabulated it for six drugs by fluorescence and by extinction; the other papers allow it to be computed from their printed solubilities. Sugihara and Taylor measured the amorphous solubility of pazopanib by extinction at 400 nm with 5 µg/mL HPMC at seven pH values between 5 and 8.4, and the crystalline free-base solubility by a 72-h shake flask at 37 °C sugihara2018 P2, P4. Lu and co-workers took the maximum supersaturation as the amorphous over the crystalline solubility, \ln\delta_{max} = \ln(C_{amorphous}/C_{crystalline}), for a simple dilute solution lu2017b E5.

What the data show

The six-drug set has crystalline solubilities from 29.29 µg/mL (ibuprofen) to 9.86 × 10⁻³ µg/mL (sorafenib), onsets by fluorescence of 125.74, 53.61, 0.558, 50.14, 0.532 and 355.96 µg/mL, fluorescence ratios 4.3, 6.4, 20.0, 52.2, 54.0 and 30.8, and extinction ratios 4.7, 7.5, 21.1, 51.6, 64.5 and 31.0 almeidaesousa2015 R5, R6. The printed extinction ratio of diclofenac (51.6) does not follow from its inputs (52.15 ÷ 0.96 = 54.3) almeidaesousa2015 X1. For the 2013 drugs (onset ÷ equilibrium solubility, synthesis): ritonavir 14.5 in pH 6.8 buffer (printed ratio of LLPS to crystalline solubility 14.5) and 15.5 in water, efavirenz 2.2, loratadine 4.8, ketoconazole 14.7, indomethacin 10.1, felodipine 10.4, clotrimazole 13.0 and clozapine 15.4 ilevbare2013a R8, X5. Clotrimazole and atazanavir were about 13 and about 63 times more soluble when amorphous than as crystals at all pH values, with the exceptions in F10 indulkar2015 R10, C26. Telaprevir was "approximately 18" times the crystalline solubility by the authors' rounding to 5 µg/mL and 19.6 by the measured 4.6 µg/mL (synthesis: 90 ÷ 4.6) mosqueragiraldo2015 R14, R13, X4. CRH1 gave "approximately 10-fold" almeidaesousa2016 R24. In the second-drug paper the ratios from the printed solubilities (extractor arithmetic) are 85–94 for paclitaxel (a hydrate), 15–16 for ritonavir, 6–6.4 for lopinavir and 7–8 for felodipine, outside the "5–50-fold" range the authors cite trasi2015a X12, R18. Outside this chapter, tacrolimus gave about 35 and a literature range of 5–65 is cited for other drugs trasi2016 R16, R17, and felodipine about 9.2 in buffer raina2015c C29. Pazopanib. The amorphous solubility fell from 60 ± 1 µg/mL at pH 5 to 9.2 ± 0.7 µg/mL at pH 8.4 and the crystalline from about 0.60 to 0.018 µg/mL, giving ratios of about 103 (pH 5.0), 304, 379, 505, 603 (pH 6.5), 528 and 501 (pH 8.4) sugihara2018 R1, R2, R4; the authors call the near-neutral range "approximately 500-600" and the highest ratio they have measured, other compounds of their group having ranged from 4 to 80 sugihara2018 C14, C15, R5. Other drugs. Atazanavir in buffer: crystalline 1.1 and amorphous 70 µg/mL (ratio 63.6, synthesis: 70 ÷ 1.1), predicted 62, flux-derived 65 indulkar2017a R4, R6, R5, R13. Telaprevir at 37 °C in 50 mM pH 6.5 buffer: 148 over 5.2 µg/mL is 28.5 (synthesis; the authors state about 28), whereas the 2015 paper's 90 over 4.6 µg/mL in 100 mM pH 6.8 buffer is 19.6 (synthesis: 90 ÷ 4.6); the difference follows the onset, 148 against 90–92 µg/mL, which the stronger buffer lowers (F9) lu2017b R2.a, R1.g, E5; mosqueragiraldo2015 R13, R14. Ibuprofen at pH 2.0 and 37 °C: 5.94 (177.1 over 29.8 µg/mL) in the 2020 paper and 3.57 (154 over 43.1 µg/mL, synthesis) in the 2021 paper; the temperature series falls from about 5.8 at 25 °C to about 3.6 at 40 °C and 2.5 at 60 °C ueda2020b R1; ueda2021 R5, R2; ueda2022 R16.

Figure

Ratios run from 4.3 (ibuprofen) to 54.0 (sorafenib) by fluorescence almeidaesousa2015 R5.

Why

By Eq. 1 the ratio is e^{\Delta G/RT}\,e^{-I(a_2)}, so it grows with the free-energy gap (high melting point and heat of fusion) and shrinks with the water the amorphous drug takes up ilevbare2013a C26; almeidaesousa2015 C27. Indulkar and co-workers attribute atazanavir's larger ratio to its higher melting point (207.8 against 144.4 °C for clotrimazole) ("can be attributed") indulkar2015 C44, M1, M2. In the six-drug set the heat-capacity free energies were 3.56, 4.59, 6.13, 9.99, 10.34 and 9.39 kJ/mol for ibuprofen, phenylbutazone, tolnaftate, diclofenac, sorafenib and griseofulvin, with water activities 0.999, 0.997, 0.985, 0.932, 0.750 and 0.699 almeidaesousa2015 R11, M1, M2, M3, M4, M5, M6. (synthesis: for sorafenib e^{10340/(8.314 \times 298.15)} = 64.8 and 64.8 × 0.750 = 48.6, and for diclofenac e^{9990/(8.314 \times 298.15)} = 56.3 and 56.3 × 0.932 = 52.5, which are the printed Cp-method ratios 48.6 and 52.5; sorafenib's larger free energy is cancelled by a larger water term almeidaesousa2015 R13.) Efavirenz, with the lowest heat of fusion (14.5 kJ/mol), has the lowest ratio ilevbare2013a M2, R8. The authors suggest that the uncertainty of very low crystalline solubilities contributes to the tolnaftate and sorafenib gaps ("may be explained") almeidaesousa2015 C33. Sugihara and Taylor note that the melting point of pazopanib, about 300 °C with degradation, "lends support" to a high ratio because the melting point is a key determinant, and that the ratio is much lower at pH 5 because the system probably crystallizes before the amorphous limit is reached ("likely") sugihara2018 C18, C11. (synthesis: the ibuprofen ratio falls with temperature because the crystalline solubility rises about fivefold, 23.5 to 116 µg/mL between 25 and 60 °C, and the amorphous solubility only about 2.1-fold, 135 to 284 µg/mL, which the authors relate to entropy-driven dissolution of the crystal and a shrinking miscibility gap ueda2022 R1, R9, C6, C16.)

Conditions and limits

Ratios are medium-dependent: the crystalline form matters (paclitaxel is a non-stoichiometric hydrate with around 3.6% water), the temperature differs (25 or 37 °C), and griseofulvin's onset was measured with polymer in the buffer while its crystalline solubility is not stated for that medium trasi2015a M3, X12; almeidaesousa2015 X6. The ratio for an ionizable drug is a property of the un-ionized form (F10). The pazopanib ratio is for simple buffers with 5 µg/mL HPMC and below pH 6 the measured amorphous solubility lies under its model curve sugihara2018 C16, X10, X12. The telaprevir ratio depends on the medium through the onset (F9) lu2017b R9; and one drug can differ by a factor of about 1.7 between two papers of one group (ibuprofen, 5.94 and 3.57, synthesis: 5.94 ÷ 3.57) ueda2020b R1; ueda2021 R2.

Counterexamples and conflicts

The "5–50-fold" range quoted from earlier work is exceeded by paclitaxel hydrate (85–94) and, in the six-drug set, by diclofenac and sorafenib trasi2015a X12, R18; almeidaesousa2015 R5. Efavirenz's ratio (2.2, synthesis: 18.4 ÷ 8.2) is below it ilevbare2013a R8. Pazopanib's 500–600 exceeds the 5–50-fold range quoted from earlier work and the 4–80 range of the authors' own group sugihara2018 R5, C15; trasi2015a X12.

Transfer check

Before choosing an amorphous formulation for a new drug, compute e^{\Delta G/RT}\,e^{-I(a_2)} from Tm, ΔHf and a sorption isotherm; a ratio near 2 (low heat of fusion, as for efavirenz) means little supersaturation potential, a ratio above 30 means large potential but also strong driving force for the drug-rich phase to crystallize ilevbare2013a M2; almeidaesousa2015 R5. Confirm by measuring the onset (F4) and compare with the crystalline form that will actually be present. For a very high-melting, poorly soluble weak base expect a ratio in the hundreds only at pH values where the drug is un-ionized and crystallization is slowed by a polymer; at lower pH measure it, since pazopanib's ratio fell to about 100 at pH 5 sugihara2018 R4, C11.

History

Ilevbare and Taylor (2013) gave the first set of eight, with the printed LLPS-to-crystal ratio of ritonavir; Almeida e Sousa and co-workers (2015) gave the first explicit table of ratios with the 4.3–54.0 range; Indulkar and co-workers (2015) showed the weak-base ratios of 13 and 63; Mosquera-Giraldo and Taylor (2015) the 18-fold of telaprevir and its flux gain; Trasi and Taylor (2015) found that paclitaxel's ratio exceeded the range cited ilevbare2013a X5; almeidaesousa2015 C23, R5; indulkar2015 C26; mosqueragiraldo2015 C34; trasi2015a X12. Sugihara and Taylor (2018) set the upper end at about 600; Indulkar and co-workers (2017) and Lu and co-workers (2017) added atazanavir and telaprevir ratios measured with micelles present, where the concentration ratio and the activity ratio differ (F16); Ueda and co-workers (2020–2022) added ibuprofen and its temperature dependence sugihara2018 C14, C15; indulkar2017a C21; lu2017b C39; ueda2022 R16. The title was widened from "about 2 to about 90" to "about 2 to about 600" because of the pazopanib data sugihara2018 R4.

How it was shown

Almeida e Sousa and co-workers (2015) measured Tm and ΔfusH by DSC at 10 °C/min and the reversing heat capacity of crystal, glass and supercooled liquid by modulated DSC (±1 °C every 60 s, 2 °C/min), fitted each as a straight line in temperature, and computed the crystal-to-supercooled-liquid enthalpy and entropy by a cycle (heat the crystal to Tm, melt, cool the liquid back), and for sorafenib and griseofulvin also the crystal-to-glass change; they plotted measured ratios against estimated ratios for each method almeidaesousa2015 P5, P6, P9, P10, E9, E10, E11, E12, E13, E14. Mosquera-Giraldo and Taylor (2015) computed the telaprevir amorphous solubility with Eq. 1 for three free energies mosqueragiraldo2015 P6. Almeida e Sousa and co-workers (2016) gave Hoffman and Cp values for CRH1 almeidaesousa2016 P10. Indulkar and co-workers (2017) used the Hoffman equation with DSC Tm and ΔHf (TA Q2000, 5 °C/min) and a water term from a vapor-sorption isotherm indulkar2017a P5, E2, E3. Ueda and co-workers (2021, 2022) took the free-energy difference and water-saturated activity of ibuprofen from the 2015 six-drug paper; ibuprofen melts at 75.9 °C almeidaesousa2015 M1, an undercooling of about 39 K at 37 °C (synthesis: 75.9 − 37) ueda2021 P7, E1.

What the data show

Predicted ratios (Hoffman / Cp supercooled liquid) against measured fluorescence ratios were: ibuprofen 3.7 / 4.2 against 4.3; phenylbutazone 7.0 / 6.4 against 6.4; tolnaftate 9.8 / 11.7 against 20.0; diclofenac 28.9 / 52.5 against 52.2; sorafenib 48.4 / 48.6 against 54.0; griseofulvin 34.8 / 30.9 against 30.8 almeidaesousa2015 R13, R5. Free energies to the supercooled liquid (Hoffman / Cp) were 3.24 / 3.56, 4.84 / 4.59, 5.69 / 6.13, 8.51 / 9.99, 10.33 / 10.34 and 9.69 / 9.39 kJ/mol for the same six drugs, and crystal-to-glass values 11.37 (sorafenib) and 10.38 kJ/mol (griseofulvin) almeidaesousa2015 R11. The through-origin fits gave slope 1.06, R² 0.989 (Cp) and slope 1.20, R² 0.926 (Hoffman); the ordinary R² of the same fits are 0.966 and 0.782 (extractor recomputation) almeidaesousa2015 R14, R15, X7. Tg − T∞ was 81, 42, 143, 145, 107 and 122 K and Tm − T 51, 82, 86, 147, 180 and 192 K for ibuprofen to griseofulvin almeidaesousa2015 R16. For the glassy drugs the glass-based predictions overshoot (sorafenib 0.725 against 0.532; griseofulvin 531.41 against 355.96 µg/mL) while the supercooled-liquid predictions (0.479; 356.44) match (F2) almeidaesousa2015 R5, R6, R11. Telaprevir: 245 ± 5 (Hoffman, 2.7 times the onset), 189 ± 16 (crystal-to-glass Cp, 2.1 times) and 97 ± 8 µg/mL (crystal-to-supercooled-liquid Cp) against 92 ± 2 (UV), 92 ± 2 (NMR) and 90 ± 1 (fluorescence) mosqueragiraldo2015 R13, R17. CRH1: ΔG 5.68 (Hoffman), 5.85 (Cp, supercooled liquid) and 5.95 kJ/mol (Cp, glass), giving 11.2, 12.0 and 12.5 µg/mL at pH 4.6 almeidaesousa2016 M1, R2. For atazanavir the printed Table I inputs (ΔHf 90.1 J/g × 704.86 g/mol = 63.5 kJ/mol, Tm 481.15 K, T 310 K) give ΔG 14.55 kJ/mol and exp(5.65) ≈ 284. With 0.22 read as exp[−I(a)], as in Table 2 of Indulkar 2015 for the same drug, the product is about 62, the ratio the paper prints; with 0.22 read as I(a), as Table I heads the column, it is about 228 (synthesis). The paper prints a predicted amorphous solubility of 68 µg/mL and a measured onset of 70 µg/mL indulkar2017a X1, E2, E3, R5, R6; indulkar2015 M2. For ibuprofen, with a small undercooling, the calculated 151 µg/mL matched the measured 154 µg/mL ueda2021 R6, R5.

Figure

Points lie near the line (slope 1.06, R² 0.989); tolnaftate sits well above it almeidaesousa2015 R14.

Wider scatter (slope 1.20, R² 0.926); diclofenac lies far above the line and griseofulvin below it. The text assigns these deviations to its "Figure 6" by mistake almeidaesousa2015 R15, X2.

Why

Almeida e Sousa and co-workers explain the Hoffman errors by its assumptions, a constant, temperature-independent ΔCp, small undercooling and Tg close to T∞ (the temperature at which crystal and supercooled liquid would have equal enthalpy), and expect them largest for diclofenac, which is high on both (undercooling 147 K, Tg − T∞ 145 K) ("should be expected") almeidaesousa2015 C8, C34, C35, R16. The glass carries more enthalpy and entropy than the extrapolated supercooled liquid below Tg, with the enthalpy term dominating, which explains the higher glass estimates almeidaesousa2015 C29, R12. Mosquera-Giraldo and Taylor explain the 2.7-fold Hoffman overestimate for telaprevir "most likely" by its high melting point, a gap of about 200 K between 238 and 37 °C (synthesis: 238 − 37 = 201 K), since the equation assumes a temperature close to Tm, and the remaining 2.1-fold crystal-to-glass discrepancy "most likely" by the quench-cooled DSC glass having more excess enthalpy and entropy than the glass formed in water mosqueragiraldo2015 C23, C24, C25, M1. Both groups close the argument by assuming that the phase formed in water is relaxed and behaves as a supercooled liquid (F2) almeidaesousa2015 C37; mosqueragiraldo2015 C26. (synthesis: that the two independent groups choose the same reference state is encouraging, but in both papers the reference state is justified by the agreement it produces, so the free-energy ranking and the "onset equals amorphous solubility" claim support each other without independent proof almeidaesousa2015 X4; mosqueragiraldo2015 X11.) (synthesis: for atazanavir the Hoffman route at an undercooling of 171 K (481 − 310 K) gives about 62 against a measured 64 once 0.22 is read as exp[−I(a)], so the 2.7-fold overshoot for telaprevir at 201 K is not repeated at 171 K indulkar2017a X1; indulkar2015 M2; mosqueragiraldo2015 R13.)

Conditions and limits

The Cp method needs an un-ionized, non-solvated crystal that melts without degradation, and assumes a Cp linear in temperature, which was shown only for tolnaftate (Fig. 3 of the 2015 paper); for much larger undercooling an expression with a 1/T² term is better almeidaesousa2015 C11, C10, R10. Parameters a–f are not printed in either paper, so Cp-method free energies cannot be recomputed almeidaesousa2015 E9, E10; mosqueragiraldo2015 E3. Tm − T was 51–192 K at 25 °C for the six drugs and 201 K for telaprevir at 37 °C (synthesis: 238 − 37) almeidaesousa2015 R16; mosqueragiraldo2015 M1. For CRH1, with a smaller gap (Tm − T = 116.7 K, synthesis: 141.7 − 25), all three routes agree within 1.3 µg/mL almeidaesousa2016 M1, R2. Pazopanib melts with degradation near 300 °C, so ΔHf and the water content of the amorphous drug could not be determined and neither the Hoffman nor the Cp route could be applied sugihara2018 C17.

Counterexamples and conflicts

Sorafenib has 180 K of undercooling but equal Hoffman and Cp ratios (48.4 and 48.6), so Table 10 does not predict the size of the Hoffman error drug by drug almeidaesousa2015 X8. Tolnaftate deviates by a similar amount under both methods (9.8 and 11.7 against 20.0) almeidaesousa2015 X8. For low-Tg drugs that phase separate as supercooled liquids the Hoffman equation "generally" gave reasonable agreement in earlier work mosqueragiraldo2015 C22. Predictions of the 2013 paper that use the Hoffman route could not all be reproduced (F5) ilevbare2013a X1. Atazanavir (Tm − T = 171 K, synthesis: 481 − 310 K) is a counterexample: with 0.22 read as exp[−I(a)] the Hoffman route gives 62 against a measured 64, so a large undercooling alone did not give an overshoot (synthesis) indulkar2017a X1, R5; indulkar2015 M2.

Transfer check

Compute both free energies. If Tm − T exceeds about 100–150 K or the Hoffman and Cp results differ by more than the measurement spread, trust neither and measure the onset (F4); the differences between the two were small for ibuprofen (Tm − T = 51 K), phenylbutazone and CRH1 and large for diclofenac and telaprevir almeidaesousa2015 R16, R13; mosqueragiraldo2015 R13; almeidaesousa2016 R2. For a drug that is a glass at the test temperature, compute both the supercooled-liquid and the glass value and expect the first to match almeidaesousa2015 R11, C36. Check that the printed Tm, ΔHf and T reproduce the ratio you intend to use, and whether a table lists I(a) or exp[−I(a)], before comparing it with a measurement indulkar2017a X1; indulkar2015 M2.

History

The Hoffman equation was the library's standard route in the 2013 paper ilevbare2013a E2. Almeida e Sousa and co-workers (2015) introduced the heat-capacity method into the library, citing earlier use for alloys, and found it closer to the onsets almeidaesousa2015 C10, C43. Mosquera-Giraldo and Taylor (2015) showed the Hoffman failure for a very high-melting drug and used the same reference state; Almeida e Sousa and co-workers (2016) reported all three routes for CRH1 mosqueragiraldo2015 C27; almeidaesousa2016 R2. Indulkar and co-workers (2017) used the Hoffman equation for atazanavir, where recomputation gives the printed ratio with the water term of the 2015 paper, so this high-melting drug does not show the issue of F7; Sugihara and Taylor (2018) could use neither route for pazopanib indulkar2017a X1; sugihara2018 C17.

How it was shown

Mosquera-Giraldo and Taylor (2015) followed dissolved telaprevir by ¹H NMR (peaks at 9.13, 8.81 and 8.74 ppm integrated against water, 40–200 µg/mL added) and diffusive flux in a side-by-side cell (32 mL per side, regenerated cellulose 6–8 kDa, 37 °C, 5–130 µg/mL donors, both sides with 5 µg/mL HPMCAS-MF), taking flux from the receiver slope over 60 min mosqueragiraldo2015 P10, P11. Trasi and Taylor (2015) ran the same kind of cell at 25 °C with 10 µg/mL HPMC for ritonavir, lopinavir and paclitaxel (4–60 µg/mL), found the break point by intersecting linear fits of the two regions, and compared it with extinction, fluorescence and ultracentrifugation trasi2015a P6, R1. Ilevbare and Taylor (2013) added about 3 mg portions of a 1:9 ritonavir–PVP dispersion to 50 mL of 100 mM phosphate buffer every 10 min at 37 °C and followed concentration and extinction with an in-situ UV probe ilevbare2013a P13, P14. Indulkar and co-workers (2017) followed atazanavir flux in a side-by-side cell (6–8 kDa regenerated cellulose, 7.07 cm², 34 mL per side, 37 °C) from 1 to 200 µg/mL in buffer indulkar2017a P7, R10. Lu and co-workers (2017) did the same for telaprevir with 5 µg/mL HPMCAS-MF in both chambers lu2017b P4. Ueda and co-workers (2021) added ibuprofen from a methanol-d4 stock to give 50–600 µg/mL and read the dissolved drug from the aqueous-phase peaks ueda2021 P2, P3. Sugihara and Taylor shifted pazopanib HCl from simulated gastric fluid to pH 6.5 and recorded the apparent concentration by UV for 180 min sugihara2018 P6.

What the data show

Telaprevir NMR concentration rose from about 33 µg/mL at about 40 µg/mL added to about 77 at about 87, then stayed at about 81–90 µg/mL up to 200 µg/mL added, the two drawn lines crossing at about 92 µg/mL mosqueragiraldo2015 R10. Flux rose from about 0.068 µg/min at 5 µg/mL to about 0.97 at 90 µg/mL (linear) and then stayed at about 0.97–1.07 µg/min up to 130 µg/mL; the receiver never exceeded about 1.95 µg/mL, under half the crystalline solubility mosqueragiraldo2015 R12, R11. The authors state the gain as approximately 18-fold in concentration and in flux; with the measured 4.6 µg/mL the concentration ratio is 19.6 and the flux ratio from the plotted points about 14–16 (synthesis: 90 ÷ 4.6; 0.97 ÷ 0.068 = 14.3 and 1.07 ÷ 0.068 = 15.7) mosqueragiraldo2015 R14, R15, X4, X5. Both plateaus still rise slightly (flux by about 10% from 90 to 130 µg/mL) mosqueragiraldo2015 X6. In the second-drug paper the plateaus were about 0.21–0.22 µg/min (ritonavir, from 30 µg/mL), 0.13–0.14 (lopinavir, from 20 µg/mL) and 0.25–0.27 (paclitaxel, from 30 µg/mL) trasi2015a R2. The ritonavir supernatant held 57.1, 36.0 and 38.9 µg/mL at 10, 37 and 50 °C, following the LLPS concentrations in water ilevbare2013a R6, R14. In the dispersion experiment measured concentrations were 6.4, 11.8, 18.3 and 18.0 µg/mL against 6.0, 12.0, 18.0 and 24.0 µg/mL expected from the mass added, extinction rose from about 0.042–0.046 after the third portion to about 0.117–0.123 after the fourth, and the colloids measured about 325 nm ilevbare2013a R17, R18. Atazanavir. Flux in buffer was about 0.003, 0.028, 0.064, 0.106 and 0.121 µg/min·cm² at 1, 20, 40, 70 and 100 µg/mL (linear fit up to the amorphous solubility, R² 0.998, slope 0.0017 mL/min·cm²) and about 0.114 at 150 and 200 µg/mL indulkar2017a R10. Telaprevir. Flux was linear (0.009 µg/min per µg/mL, R² 0.9932) to about 1.32 µg/min at 150 µg/mL and about 1.25–1.27 at 160–170 µg/mL, with the receiver below half the crystalline solubility lu2017b R6, R25. Ibuprofen. In buffer the dissolved concentration followed the dose to 100 µg/mL, reached about 150 µg/mL at 200 and stayed at about 154–157 µg/mL at 400 and 600 µg/mL added, so the free concentration did not depend on the amount of drug-rich phase ueda2021 R10.a, C23. Pazopanib. After the shift at the high dose (54 times the amorphous solubility) the apparent concentration held about 17 µg/mL for several minutes with 50 µg/mL HPMC; at the intermediate dose (5.4 times) it held about 22–23 µg/mL for the first 20 min after the shift, rose to about 30 µg/mL at about 32 min and fell sharply at about 35 min, against an amorphous solubility of 19.8 µg/mL; a filtered HPLC sample late in the run gave 0.18 ± 0.03 µg/mL sugihara2018 R8, R12, R1.e, R10, X4, X9.

Figure

Flux rises linearly up to about 90 µg/mL and then levels off mosqueragiraldo2015 R12.

Each drug shows a linear rise and a plateau starting at its LLPS concentration trasi2015a R2, C13.

(a) Linear receiver profiles over 60 min; (b) flux rising linearly to the onset at 148 µg/mL and flat above it lu2017b R5, R6.

Why

In an equilibrated two-phase system the chemical potential of the drug is the same in the water-rich and the drug-rich phase, so added drug increases the amount of drug-rich phase while the concentration of the drug-lean phase and its activity stay fixed mosqueragiraldo2015 C35; trasi2015a C5. With D, S, h and γm constant in the cell, flux J = DSa/(h\gamma_m) follows the donor activity a = C\gamma, so a flux plateau means an activity plateau mosqueragiraldo2015 C10, E4, E5. Trasi and Taylor read the linear part as rising activity and the plateau as constant activity once a drug-rich phase forms ("indicates") trasi2015a C14. Ilevbare and Taylor extend this to dosage forms: a supersaturating formulation that exceeds the LLPS concentration "should undergo LLPS" ilevbare2013a C48. Mosquera-Giraldo and Taylor add that the precipitate "will also provide a reservoir" at the maximum, a point taken from earlier work and not tested in their paper mosqueragiraldo2015 C39. See the nanodroplet reservoir effect. Indulkar and co-workers say the plateau occurs because flux is proportional to activity and, above the amorphous solubility, "the excess drug forms a second amorphous phase" whose activity stays at its value at the amorphous solubility, as for felodipine and nifedipine indulkar2017a C17. Ueda and co-workers add that in buffer the dissolved concentration does not depend on the amount of drug-rich phase, a read-out of the same equilibrium without a membrane ueda2021 C23. Sugihara and Taylor read the limited apparent concentration after the pH shift as the amorphous solubility setting the maximum free drug concentration ("clearly demonstrate") sugihara2018 C22, C36.

Conditions and limits

The ASD experiment used one drug, one polymer ratio and total drug by UV; free drug and activity were not measured, and extinction already rose at 18.3 µg/mL, equal to the with-PVP LLPS concentration ilevbare2013a X15, R16. The NMR concentrations lie below the added amount even below the boundary, and the plateau level (81–90 µg/mL) lies below the 92 µg/mL break; the break, not the plateau, should be read as the boundary mosqueragiraldo2015 X10. The low-donor flux traces are curved and the slope window is not stated; replicates are not stated for the flux data mosqueragiraldo2015 X13; trasi2015a X10. The membranes and the polymer additives differ (5 µg/mL HPMCAS-MF against 10 µg/mL HPMC) mosqueragiraldo2015 P11; trasi2015a P6. The pazopanib plateau is an apparent UV concentration in a suspension with colloids, up to about 50% above the amorphous solubility, one trace per condition sugihara2018 X7, X15. With a dissolved polymer the plateau depends on the polymer (F19, F20).

Counterexamples and conflicts

None found in the library for the plateau itself; a later membrane-transport study also found the maximum flux rising linearly up to about 8 µg/mL, close to the amorphous solubility, and not above indulkar2016 R9. Two qualifications: a second drug lowers the plateau (F11), and a solubilizer raises total concentration without raising activity (F12) trasi2015a C28; ilevbare2013a C54. One qualification sits inside the plateau finding: with 500 µg/mL HPMC 603 the dissolved ibuprofen rose from about 88 to 123 µg/mL as the total drug rose from 100 to 600 µg/mL, because the polymer content of the drug-rich phase changed (F20) ueda2021 R10.d, C24.

Transfer check

Measure flux across a membrane against donor concentration in the medium of interest and look for the break; it should coincide with the onset from F4 and the calculation of F5 trasi2015a R1, R2. For an ASD, dissolve it under non-sink conditions and check that the free (ultracentrifuged or filtered) concentration stops at the limit, and that the total UV concentration is not mistaken for free drug (F4) ilevbare2013a X15; almeidaesousa2016 R12. In a micellar medium the plateau of total concentration is not the plateau of free drug; use flux or another free-drug read-out indulkar2017a R11, R13 (F16).

History

Raina and co-workers (2014) had shown that flux peaks at the LLPS concentration for felodipine and nifedipine and proposed a reservoir effect mosqueragiraldo2015 C4, R16; raina2014a C38, C40. Ilevbare and Taylor (2013) were the first in the library to state the limit for a dosage form; Mosquera-Giraldo and Taylor (2015) showed it for a glassy phase with NMR and flux, and Trasi and Taylor (2015) for three more drugs ilevbare2013a C5; mosqueragiraldo2015 C34; trasi2015a C34. Later papers measured the effect for ASDs of telaprevir and for weak bases mosqueragiraldo2018b C4; indulkar2016 R9. Indulkar and co-workers (2017), Lu and co-workers (2017) and Sugihara and Taylor (2018) confirmed the plateau for atazanavir, telaprevir and pazopanib, and Ueda and co-workers (2021) read it without a membrane for ibuprofen indulkar2017a C17; lu2017b C18; sugihara2018 C22; ueda2021 C23.

How it was shown

Ueda and Taylor (2020) used an 800 MHz spectrometer with a cryoprobe at 37 °C, added an ibuprofen stock in methanol-d4 to give 2% v/v, used trimethylsilyl propionate (TSP) as shift and quantification reference with a repetition delay of 12–15 s (five times T1), split the ibuprofen peaks into an aqueous-phase and a drug-rich-phase set by Lorentzian fitting, and worked at half of the amorphous solubility (no droplets) and at the amorphous solubility plus 100, 200 and 300 µg/mL ueda2020b P4. Ueda and co-workers (2021) read dissolved ibuprofen from the aqueous H4 and H5 peaks through a calibration line and dissolved polymer from its peak near 2.0 ppm (PVP, PVP-VA) or 3.1 ppm (HPMC) against its own calibration; the polymer inside the drug-rich phase gives no solution signal, so the missing part gives the polymer:drug weight ratio in that phase (formula not printed), in three series that varied polymer species at 1000 µg/mL, polymer concentration at 400 µg/mL ibuprofen, and total ibuprofen at 500 µg/mL polymer, with n = 3 ueda2021 P2, P3, P4, A1, A2, A3, X3. Ueda and co-workers (2022) repeated the measurement at 15–60 °C in 5 °C steps at 600 MHz with a 15 s delay, HPMC solutions to 55 °C only ueda2022 P4.

What the data show

In ibuprofen-saturated solutions with 1000 µg/mL of any of six polymers all ibuprofen peaks stayed at their buffer positions and the crystalline solubility stayed at 43.1 µg/mL, so the polymers neither solubilize nor change the molecular state of dissolved drug ueda2021 R1, R2. Above the amorphous solubility two sets of drug peaks appeared; in buffer the drug-rich peaks kept their shape and position however much drug-rich phase formed ueda2021 R3, R9. With polymer, the drug-rich H4* peak moved downfield from about 6.95 ppm in buffer to about 7.04 (PVP K12), 7.05 (K25, K90), 7.08 (HPMC 603), 7.09 (HPMC 615) and 7.11 ppm (PVP-VA) and broadened, and the polymer peaks lost intensity ueda2021 R3, R4. The dissolved-drug peaks stayed in place but broadened, only when polymer and drug-rich phase were both present ueda2021 R1, R7, R9. In buffer the dissolved ibuprofen was about 150, 154 and 157 µg/mL at 200, 400 and 600 µg/mL added ueda2021 R10.a. The NMR calibrations were linear (R² 0.9996–0.9999) and the total drug signal of both phases lay on the single-phase line ueda2021 R16. Between 15 and 60 °C the drug-rich H6* peak in buffer stayed at about 6.77 ppm, whereas with polymer it moved downfield with temperature, for HPMC from about 6.835 to about 6.933 ppm between 15 and 55 °C ueda2022 R6. Against UV extinction, the NMR dissolved ibuprofen of the stirring tests differed by up to about 20 µg/mL (about 168 against 177.1 µg/mL in buffer, 117–121 against 96.9 with HPMC) and the paper does not compare the two ueda2020b R15, R1, X1.

Figure

The aqueous-phase peaks (H4, H5) stay at the buffer position, the drug-rich peaks (H4*, H5*) move left and broaden with polymer, most with PVP-VA and HPMC; dashed lines mark the buffer positions ueda2021 R3.

Why

The authors give four explanations, all inferred. The drug-rich peaks lie at higher field because that phase is less polar than water ueda2022 C8. Polymer inside the droplets reorients too slowly (a long re-orientational correlation time) to give a solution-state signal, so its loss measures how much is associated with the droplets ueda2021 C7; ueda2020b C22; ueda2022 C10. The downfield shift and broadening of the drug-rich peaks arise from suppressed ibuprofen mobility together with a changed environment, both from polymer in the phase ueda2021 C6. The broadening of the dissolved-drug peaks is not a viscosity effect, because it is absent in saturated polymer solutions, but exchange of drug between the two phases (lifetime broadening) ueda2021 C9; ueda2022 C14. The reason for choosing ibuprofen is that its Tg of −45 °C keeps the droplet molecules mobile at 37 °C ueda2021 C4; ueda2020b C7.

Conditions and limits

All data are for one drug in 50 mM pH 2.0 phosphate in D2O with 2% methanol-d4; the time from mixing to acquisition is not printed ueda2021 X10, X9. The polymer weight ratio counts all NMR-invisible polymer, so polymer adsorbed at droplet surfaces would be counted the same way, and the composition of the droplets was not determined ueda2021 X4; ueda2020b X4. At 100 µg/mL total drug the ratios rest on only about 12–19 µg/mL of phase-separated drug ueda2021 X3. The method needs a drug-rich phase whose molecules still tumble quickly, which excludes higher-Tg drugs at 37 °C ueda2020b R16.

Counterexamples and conflicts

None found in the library against the read-out. The NMR and extinction values of dissolved ibuprofen differ by up to about 20 µg/mL within one paper ueda2020b X1.

Transfer check

Check that the drug's Tg lies well below the test temperature (or raise the temperature), spike the sample to identify the two peak sets, calibrate drug and polymer against an internal reference with a delay of five times T1, run three replicates and compare the dissolved concentration with the extinction onset and with the calculated amorphous solubility ueda2021 P3, R16, R6. Read the polymer ratio as NMR-invisible polymer per phase-separated drug ueda2021 X4.

History

Ueda and Taylor (2020) applied the method to three ASD polymers with ibuprofen, Ueda and co-workers (2021) quantified dissolved drug and polymer in three series, and Ueda and co-workers (2022) made it variable-temperature ueda2020b C22; ueda2021 C11; ueda2022 C64. Later papers used the same approach for other drugs and additives: Wilson and co-workers (2021) measured the polymer left in solution and the polymer content of enzalutamide droplets by NMR, and Yang and co-workers (2022) used it to predict the distribution of a surfactant in a hydrated ritonavir ASD wilson2021 R7; yang2022b C44.

How it was shown

Indulkar and co-workers (2017) measured atazanavir flux in buffer from 1 to 200 µg/mL in a side-by-side cell (6–8 kDa regenerated cellulose, 37 °C) and used the linear part up to the amorphous solubility as a calibration with unit activity coefficient; flux in 1, 2 and 3.6 mg/mL SDS then gave the free drug, micelle-bound drug followed by mass balance, the concentration-based ratio was total drug over the crystalline solubility in the same medium, and the activity-based ratio was the flux-derived activity over that at the crystalline solubility (Eq. 9) indulkar2017a P7, P8, P9, E9. Lu and co-workers (2017) defined the fundamental supersaturation as δ = F/F°, with F° the flux of a crystal-saturated solution obtained by extrapolating each flux line to the crystalline solubility of the medium, used the buffer series as calibration and plotted δ against C/C° in 13 media lu2017b P4, P6, E6.

What the data show

Atazanavir. In buffer flux rose linearly up to 70 µg/mL (R² 0.998) and was flat above indulkar2017a R10. The slope of free against total drug was 0.998, 0.181, 0.086 and 0.047 at 0, 1, 2 and 3.6 mg/mL SDS, and all four curves ended at about 65–70 µg/mL free drug at totals of about 65, 340, 800 and 1450 µg/mL indulkar2017a R11. In 3.6 mg/mL SDS the activity ratio was about 4.8, 10.3, 18.8, 31.5 and 65 at 100, 200, 400, 800 and 1450 µg/mL total, against concentration ratios of about 2.2, 4.4, 8.9, 17.8 and 31 indulkar2017a R12. At the amorphous solubility the two ratios were 64.8 and 66.0 (0 mg/mL), 37.5 and 70.0 (1), 35.9 and 66.1 (2) and 33.0 and 65.0 (3.6 mg/mL SDS) indulkar2017a R13. The apparent permeability in 3.6 mg/mL SDS equalled that in buffer on an activity basis and was 20-fold lower on a total-concentration basis, which is the inverse of the free fraction (1/0.047 = 21, synthesis) indulkar2017a R27, E12. Telaprevir. In buffer δ matched C/C° (slope 1.011, R² 0.9928); with bile salts most points lay on or above the 1:1 line, for example 12 mM STC at C/C° of about 25.4 gave δ of about 29.0, 12 mM SGCDC about 12.5 gave about 17.1, while the 1.86 mM SGCDC points lay below it (about 28.9 gave about 20.2) lu2017b R16, R17. Later media. In five simulated media and pooled human fluid the flux-based maximum supersaturation stayed at about 50–56 for atazanavir and 29–37 for posaconazole, whereas the concentration-based ratio fell from about 55 and 36 in buffers to about 14 and 5 in human fluid elkhabaz2019b R8, R9, C21. For posaconazole in intestinal media a concentration ratio of about 20 corresponds to a thermodynamic ratio of about 34 duong2022b C9.

Figure

(a) In 3.6 mg/mL SDS the activity ratio (red) runs to 65 while the concentration ratio (black) reaches about 31; (b) at the amorphous solubility both are about 65 in buffer but the concentration ratio drops to 33–38 with SDS indulkar2017a R12, R13.

Buffer points lie on the 1:1 line, most bile-salt points above it, and the 1.86 mM SGCDC points below it lu2017b R17.

Why

Indulkar and co-workers write supersaturation as the activity ratio SR = \gamma c/(\gamma^{*} c^{*}) and note that the concentration ratio equals it only if the activity coefficients at the two concentrations are equal, which micelles break; the activity of the drug in equilibrium with the crystal or with the water-saturated amorphous drug does not depend on the medium, so the activity ratio at the amorphous solubility is the same with and without SDS, and SDS is therefore said not to mix into the drug-rich phase, an inference from the unchanged maximum flux indulkar2017a E9, C20, C43, C40, X11. They conclude that "concentration ratios cannot give an accurate estimate" of supersaturation in this system indulkar2017a C42. Lu and co-workers use the same logic, with flux proportional to activity, and state that the two measures are equal only when the activity-coefficient ratio is one lu2017b E6, C39. The extractor adds that wherever flux is proportional to concentration, the ratio (a/a^{\circ})/(C/C^{\circ}) equals F^{\circ}(\mathrm{medium})/F^{\circ}(\mathrm{buffer}), so part of the bile-salt offsets reflects how the measured crystalline solubility of a medium compares with its flux slope lu2017b X7. (synthesis: the twofold SDS gap is far outside that uncertainty, the 3.9–6.7 spread of the bile-salt data is partly inside it.)

Conditions and limits

One drug and one anionic surfactant at 37 °C for the large effect, one drug and single bile salts without lecithin for the second indulkar2017a X5; lu2017b A1. The free-drug calibration assumes that only dissolved drug crosses the membrane and that SDS does not change the membrane; neither was tested, and the number of flux replicates is not stated indulkar2017a X9, X12; lu2017b X8. The authors expect the ratio of the two supersaturations to approach one below the CMC or with a weak solubilizer indulkar2017a C55.

Counterexamples and conflicts

For 1.86 and 12 mM SGCDC the flux lines coincide while the crystalline solubilities differ (5.7 and 9.6 µg/mL), so a common F° cannot hold and the SGCDC points are internally inconsistent lu2017b X7. The size of the discrepancy is "most likely" drug and surfactant dependent: the authors cite about 2:1 for felodipine and smaller deviations for telaprevir in bile salts indulkar2017a C54, R28, R29.

Transfer check

Measure flux against donor concentration in buffer first, check that it is linear up to the amorphous solubility, repeat in the micellar medium, convert to activity through the buffer line and compare the activity ratio with the concentration ratio at the amorphous solubility and at one lower point; test whether micelles cross the membrane indulkar2017a P7, P8, X9. If the surfactant is below its CMC or solubilizes little, the two ratios may agree indulkar2017a C55.

History

Raina and co-workers (2015) had found about 2:1 for felodipine with solubilizers; Indulkar and co-workers (2017) and Lu and co-workers (2017) measured the gap for atazanavir with SDS and telaprevir with bile salts; Elkhabaz and co-workers (2019) extended it to biorelevant media and Duong and co-workers (2022) to posaconazole indulkar2017a R28; elkhabaz2019b C21; duong2022b C9.

How it was shown

Ilevbare and Taylor (2013) pumped a 4 mg/mL methanol stock into water or 100, 200 or 500 mM sodium phosphate buffer (pH 6.8) at 37 °C and took the onset from extinction at 280–450 nm; they measured crystalline solubility by ultracentrifugation and HPLC after 48 h, and the experimental amorphous solubility by dissolving a spin-coated amorphous ritonavir film (second-derivative UV) in each medium ilevbare2013a P1, P3, P4, P5, R13. The theoretical value came from Eq. 1 with the crystalline solubility of the same medium ilevbare2013a P2. The temperature series ran the same titration at 10–65 °C in water and in 100 mM buffer for ritonavir and at 10–50 °C in water for efavirenz ilevbare2013a P5, R14, R15; crystalline ritonavir solubility was measured at 25 and 50 °C in water too ilevbare2013a P1, R11. Lu and co-workers (2017) measured the telaprevir onset by UV extinction and ultracentrifugation in 50 mM pH 6.5, 50 mM pH 6.8 and 100 mM pH 6.8 phosphate at 37 °C lu2017b P7, R9. Ueda and co-workers (2022) measured dissolved ibuprofen by NMR at 15–60 °C in 5 °C steps (400 µg/mL added, pH 2.0 phosphate in D2O) and the crystalline solubility by shake flask at 25–60 °C ueda2022 P1, P4.

What the data show

In water, 100, 200 and 500 mM buffer, the turbidity onset was 37.2 ± 0.90, 18.8 ± 0.07, 14.3 ± 0.10 and 4.7 ± 0.50 µg/mL, the film amorphous solubility 34.2 ± 2.88, 21.8 ± 1.86, 13.8 ± 1.97 and 5.2 ± 0.65 µg/mL, the crystalline solubility 2.4 ± 0.03, 1.3 ± 0.20 and 0.95 ± 0.11 µg/mL (not given at 500 mM), and the theoretical amorphous solubility 39.2, 20.6 and 15.3 µg/mL ilevbare2013a R13. The onset was thus 0.95, 0.91 and 0.93 times the theoretical value in the three media where both exist (synthesis: 37.2 ÷ 39.2, 18.8 ÷ 20.6, 14.3 ÷ 15.3). Across 10–65 °C the onset in water stayed about twofold above that in 100 mM buffer ilevbare2013a C40, R14. From the plotted points the ratio of the water to the buffer value was 2.1 at 10 °C, 2.1 at 35 °C and 2.7 at 65 °C (synthesis: 58.0 ÷ 27.3, 37.3 ÷ 18.0, 53.0 ÷ 20.0) ilevbare2013a R14. In water the ritonavir onset read about 58.0, 50.3, 46.2, 42.4, 39.4, 37.3, 38.7, 39.5, 40.6, 41.5, 49.0 and 53.0 µg/mL at 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 and 65 °C, and in 100 mM buffer about 27.3, 25.4, 24.6, 22.6, 20.0, 18.0, 18.0, 18.0, 18.6, 18.0, 18.6 and 20.0 µg/mL ilevbare2013a R14. Efavirenz read about 19.3 µg/mL at 10 °C and about 44.6 µg/mL at 50 °C ilevbare2013a R15. The authors state that all compounds except ritonavir rose with temperature but show only efavirenz ilevbare2013a C45, X10. The crystal solubility of ritonavir rose from about 1.2 to about 2.5 and 6.4 µg/mL between 25, 37 and 50 °C ilevbare2013a R11. Telaprevir. The onset was 149/146 µg/mL (ultracentrifugation/extinction) in 50 mM pH 6.5, 153/148 in 50 mM pH 6.8 and 104/102 in 100 mM pH 6.8 buffer, about 0.70 of the 50 mM value in the stronger buffer (synthesis: 104 ÷ 149, 102 ÷ 146); the 50 mM pH 6.8 value shows ionic strength, not pH, to be the variable lu2017b R9, C19, C20. Ibuprofen. In buffer the dissolved ibuprofen was about 127, 135, 179, 250 and 284 µg/mL at 15, 25, 40, 55 and 60 °C, a rise of 2.2 times (synthesis: 284 ÷ 127), while the crystalline solubility rose from about 23.5 to about 116 µg/mL between 25 and 60 °C (4.9 times, synthesis) ueda2022 R9, R1. The measured S0a/S0c therefore fell from about 5.8 at 25 °C to 3.6 at 40 °C and 2.5 at 60 °C ueda2022 R16. The authors read the rising amorphous solubility as a miscibility gap between water and liquid ibuprofen that shrinks with temperature, and the rising crystalline solubility as entropy-driven dissolution ueda2022 C16, C6.

Figure

A U-shape in water with a minimum near 35 °C, and about half the values and a flatter profile in 100 mM buffer ilevbare2013a R14.

Why

Ilevbare and Taylor expect salt to promote aggregation through hydrophobic interactions and lower the coexistence concentration, and they attribute the fall to salting-out: salt makes the solvent more polar and strengthens hydrophobic association, so both the crystalline solubility and the coexistence concentration fall ("may be attributed"); the solvent polarity and drug association were not measured ilevbare2013a C38, C41. They note that the measured amorphous solubility and the theoretical value (which uses the crystalline solubility of the same medium) track the onset, and conclude that experiments need a consistent medium ilevbare2013a C42, C43. For the temperature trend they note that miscibility of partially miscible liquids usually rises with temperature, which efavirenz follows, and "surmise" that the ritonavir minimum near 35 °C resembles the U-shaped temperature dependence of surfactant CMCs (minimum typically at 20–40 °C), a micelle-like origin that had no structural test ilevbare2013a C44, C47, R14. (synthesis: the 37 °C comparison with Eq. 1 is the only one made, so the shape of Fig. 8 is not tested against the thermodynamic estimate; the crystalline solubility rising with temperature and the free-energy term falling with it are both in the equation.) Lu and co-workers give the reason Ilevbare and Taylor gave for the buffer effect, that higher ionic strength "promotes the aggregation of molecules via hydrophobic interactions", without a test lu2017b C20. Ueda and co-workers expect the miscibility of partially miscible liquids to rise with temperature ueda2022 C16. (synthesis: in this chapter the amorphous solubility rises with temperature in two of the three drugs with a temperature series and has a minimum in one; the ratio to the crystalline solubility falls with temperature for ibuprofen.)

Conditions and limits

Salt data are for one drug, in phosphate buffer, at 37 °C; the temperature series covers two drugs (ritonavir, efavirenz) in water and one in 100 mM buffer ilevbare2013a P5, A3, A4. The 500 mM value has no matching crystalline or theoretical value ilevbare2013a R13. Other papers of this chapter used 50 mM phosphate (ionic strength 155 mM), 50 mM citrate (98 mM) or 100 mM phosphate, so onsets from different papers cannot be set side by side almeidaesousa2015 M15, M16; almeidaesousa2016 M16, M17; trasi2015a M8. The ritonavir "boundary at 37 °C" is a range of 34–40 µg/mL in water depending on method (F4) ilevbare2013a X14. The ibuprofen series stops at 55 °C for the HPMC solution because of gelation, and all its data are in D2O buffer with 2% CD3OD at pH 2.0 ueda2022 R3, X6. The telaprevir comparison is for two buffer strengths at one pH lu2017b X11.

Counterexamples and conflicts

None found in the library for the salt effect. For the temperature trend, only efavirenz is shown to rise, and the claim for the other compounds has no data ilevbare2013a X10. The ritonavir value in 100 mM buffer at 25 °C reported by the second-drug paper (26–28 µg/mL) is higher than the plotted value here (F4) trasi2015a R1; ilevbare2013a R14. The ibuprofen trend is monotonic and does not show the ritonavir minimum ueda2022 R9.

Transfer check

Measure the onset in the medium and at the temperature of use (for an oral drug, 37 °C and a buffer of intestinal ionic strength), not in water at room temperature; expect a twofold lower limit in 100 mM phosphate than in water for a lipophilic, un-ionized drug, as for ritonavir ilevbare2013a R13, C40. Check at least two temperatures, since a minimum is possible ilevbare2013a R14. Where a temperature series is available, take the fall of the ibuprofen ratio with temperature as a warning that data at room temperature overstate the ratio at 37 °C (synthesis: 5.8 at 25 °C against about 3.6 at 40 °C) ueda2022 R16.

History

Ilevbare and Taylor (2013) varied ionic strength and temperature ilevbare2013a A3, A4; the other five papers of 2013–2016 each held the temperature fixed, at 25 or 37 °C almeidaesousa2015 A1; indulkar2015 A1; mosqueragiraldo2015 A1; trasi2015a A1; almeidaesousa2016 A3. Lu and co-workers (2017) found the buffer-strength effect for telaprevir after a published value in 100 mM buffer had been about 1.5 times lower than theirs; Ueda and co-workers (2022) added the first full temperature series for a low-melting, low-Tg drug lu2017b C19, R19; ueda2022 R9.

How it was shown

Indulkar and co-workers (2015) studied clotrimazole (pKa 5.89), atazanavir (pKa 4.52, measured by UV-metric titration) and nicardipine (pKa 7.2) at 37 °C over about ±2 pH units around the pKa, in 0.1 M citrate (pH 3–6), phosphate (6.5–8) or carbonate (9) buffers indulkar2015 M1, M2, M3, M10, P2, P3. The crystalline free base was shaken 48 h and assayed by HPLC; the onset came from antisolvent addition with extinction at 400 nm, from pyrene or 4-Di-2-ASP fluorescence, and from a pH-shift titration of drug dissolved at pH 2.0 in 0.15 M NaCl indulkar2015 P3, P8, P9, P10. The un-ionized concentration was obtained as C_U = X\,C_T/(1+X) with X = 10^{\mathrm{pH}-\mathrm{p}K_a}, and the predicted amorphous solubility at each pH was one amorphous-to-crystalline ratio times the crystalline solubility at that pH indulkar2015 E6, P5. Crystalline nicardipine lost its birefringence in water within 2 h, so only its amorphous solubility could be measured indulkar2015 R1. Almeida e Sousa and co-workers (2016) ran CRH1 at pH 6.8 (fully un-ionized) and at pH 4.6, equal to the pKa, and report all concentrations as un-ionized drug through the Henderson–Hasselbalch equation almeidaesousa2016 M16, M17, C13, E3. Sugihara and Taylor (2018) built the pazopanib diagram with the crystalline free base shaken 72 h at 37 °C in 0.1 M citrate (pH 5–6) and phosphate (pH 6.4–8.4) buffers and the amorphous solubility by extinction at 400 nm with 5 µg/mL HPMC; the Henderson–Hasselbalch curves took the pH 8.4 values as the un-ionized solubility and a pKa of 6.4 sugihara2018 P2, P4, P11, E1, C9.

What the data show

Clotrimazole's LLPS concentration was about 145 µg/mL at pH 4.5, 65 at pH 5, 28 at pH 5.5, 14 at pH 6 and 7.7 at pH 8, and 490.2 at pH 4; atazanavir's about 680 at pH 3.5, 265 at pH 4, 125 at pH 4.5, 95 at pH 5 and 72 near pH 6; nicardipine's about 107 near pH 5.85, 57 at pH 6.3, 36 at pH 6.5 and 13–15 near pH 7 indulkar2015 R5, R6, R7, R18. Crystalline solubility rose in parallel: clotrimazole 0.55 µg/mL at pH 8 (the text says pH 9) and about 62 at pH 4, with 1.08 at pH 6, near its pKa; atazanavir 1.04 at pH 9, 2.12 at pH 4.5 (near pKa 4.52) and about 40 µg/mL at pH 3, the doubling at pH = pKa being the expected one indulkar2015 R2, R4, R5, R6, C23, X2, M2. The un-ionized LLPS concentration was about 5.6–8.4 µg/mL for clotrimazole, 57.5–70.4 for atazanavir and 4.5–6.75 for nicardipine, against about 0.5–0.78 and 0.85–1.2 µg/mL for the crystalline un-ionized clotrimazole and atazanavir indulkar2015 R9. The measured ratios were about 13 and about 63, but the clotrimazole ratio is about 8 at pH 4–4.5 and 12–17 from pH 5, and atazanavir's runs from about 48 to 75 (extractor calculation) indulkar2015 R10, X5. The predicted un-ionized amorphous solubilities were 4.9 and 64.9 µg/mL against 7.7 and 65.1 measured, and at pH 3 the measured atazanavir value (about 1970) is about 27% below the prediction (about 2700) indulkar2015 R2, R6, X6. Titration, UV and fluorescence agreed over most of the range (F4), the maxima on dissolving amorphous films were 69–110% of the onsets (F13), and the precipitates were amorphous at both ends of each pH range (F1) indulkar2015 R8, R18, F6, R19, R20, R21. For CRH1, the onsets were 14.15 and 11.31 µg/mL at pH 6.8 and 13.73 and 12.49 µg/mL at pH 4.6 with 20 µg/mL HPMC, "about 12" at both, with crystalline solubility 1.20 and 1.50 µg/mL almeidaesousa2016 R3, R4, R24, R2. Pazopanib. The crystalline free base was about 0.60, 0.13, 0.079, 0.042, 0.033, 0.025 and 0.018 µg/mL at pH 5, 5.5, 6, 6.4, 6.5, 7 and 8.4, and 0.571 mg/mL in gastric fluid (pH about 1.2) sugihara2018 R2. The amorphous solubility was 60 ± 1, 40 ± 2, 29.4 ± 0.9, 20.9 ± 0.9, 19.8 ± 0.5, 13.5 ± 0.4 and 9.2 ± 0.7 µg/mL at the same pH values sugihara2018 R1, following its curve from pH 6 upward but lying below it at pH 5 and 5.5 (curve values about 250 and 80 µg/mL), which the authors attribute to crystallization before the limit was reached ("likely") sugihara2018 R3, C10, C11. The ratio of the two was therefore about 103 at pH 5.0, 304 at 5.5, 379 at 6.0 and 500–600 from pH 6.4 up, not constant across pH sugihara2018 R4. From pH 1.2 to pH 6.5 the crystalline solubility falls by more than four orders of magnitude sugihara2018 C19.

Figure

Flat lines for both forms, about 8- to 17-fold apart for clotrimazole and about 48- to 75-fold for atazanavir (extractor calculation); nicardipine is shown for the amorphous form only indulkar2015 R9, X5.

Why

Indulkar and co-workers reason that a weak base dissolves as neutral free base plus ionized drug, the ionized part growing tenfold per pH unit below the pKa: C_T = C_B\,(1 + 10^{\mathrm{p}K_a-\mathrm{pH}}). Putting the amorphous free-base solubility in for C_B predicts that the amorphous solubility rises below the pKa with a constant ratio to the crystalline one wherever the solution is in equilibrium with the neutral form and not the salt indulkar2015 E8, C42, C46. LLPS "cannot occur until the concentration of the un-ionized form is exceeded", so at low pH the total concentration needed is higher indulkar2015 C47. The ratio is set by the free-energy gap and the absorbed water (F5), and the larger ratio of atazanavir "can be attributed" to its higher melting point indulkar2015 C43, C44. The authors take the supersaturation ratio as "not affected by degree of ionization" indulkar2015 C46. A caution from the extractor: the predicted curve is the crystalline curve times one factor, so its overlap with the LLPS data is not an independent test; the test is the constancy of the un-ionized concentration indulkar2015 X14. Sugihara and Taylor divide the pH-solubility diagram into three regions: below the crystalline curve (subsaturated), between the curves (crystallization the only possible change) and above the amorphous curve (amorphous precipitation); below pH 6 they draw the third boundary along the model curve, so there it is a model and not a measurement sugihara2018 C33, C34, C36, X12.

Conditions and limits

37 °C, 0.1 M buffers, no polymer or surfactant, free-base forms (the sulfate and hydrochloride salts were converted) indulkar2015 P1, M2, M3. The model holds where the solution is in equilibrium with the neutral form; whether a salt limits solubility at the lowest pH was not examined, and the titration departs there indulkar2015 E8, X7. Several plotted pH values differ from the nominal buffer pH indulkar2015 X12. The intestinal conclusions (an atazanavir dose of 400 mg giving about 1.29 mg/mL in the intestine against an onset of 73 µg/mL at pH 6; nicardipine 20 mg giving 64 µg/mL against above 106) are calculations, the two doses differ 20-fold as well as in pKa, and no bile salt or biorelevant medium was used indulkar2015 R26, R27, X15, C58. The pazopanib diagram is for simple buffers with 5 µg/mL HPMC; intestinal solubilizers would raise both boundaries sugihara2018 C37, X10.

Counterexamples and conflicts

The clotrimazole ratio is not constant across pH (about 8 at pH 4–4.5) and its un-ionized amorphous value is about 1.6 times the prediction indulkar2015 X5, X6. The atazanavir onset at pH 3 is below the prediction indulkar2015 X6. None found in the library against the Henderson–Hasselbalch shape of the onset. Pazopanib's amorphous solubility at pH 5 and 5.5 lies below the curve and its ratio falls to about 103, the pattern of clotrimazole's ratio of about 8 at pH 4–4.5 sugihara2018 R3, R4; indulkar2015 X5; the statement of a constant ratio holds only while crystallization is slower than the measurement.

Transfer check

Determine the pKa, measure one onset at a pH at least two units above the pKa (the drug is then un-ionized), compute the total onset at other pH with C_T = C_B(1 + 10^{\mathrm{p}K_a-\mathrm{pH}}), and verify it at one pH near the pKa; do not rely on the constancy of the ratio below pH 5 for a base with a pKa near 6 indulkar2015 E8, X5. In a pH-shift test, a drug-rich phase forms only if the un-ionized concentration after the shift exceeds the free-base amorphous solubility, which depends on dose and volume as much as on pKa indulkar2015 C47, X15. Measure the onset at the intestinal pH of interest with a polymer in the medium; if it falls below the curve drawn from the pH 8 value, suspect crystallization before the limit sugihara2018 C11.

History

The reasoning that weak bases supersaturate on leaving the stomach and that those precipitating as amorphous solids reach the amorphous solubility came from pH-shift work in 2012 indulkar2015 C12; hsieh2012 E5, E6. Indulkar and co-workers (2015) were the first in the library to test the Henderson–Hasselbalch dependence of the LLPS onset across pH, noting that the influence of pH on LLPS had not been widely investigated indulkar2015 C12, C14; Almeida e Sousa and co-workers (2016) used un-ionized concentrations for CRH1 at pH 4.6 almeidaesousa2016 C13; a later paper fitted the same relation to the crystalline and amorphous solubility of posaconazole and another measured both for delamanid across pH duong2022b R1, R2, R3; duong2022a R1, C2. Sugihara and Taylor (2018) built a second pH-solubility diagram with both boundaries after Indulkar and co-workers (2015) and used it to explain a pH-shift test (F23) sugihara2018 C6, C45.

How it was shown

Trasi and Taylor (2015) worked in 100 mM phosphate pH 6.8 at 25 °C with ritonavir as anchor. They pinned each drug's amorphous solubility by four methods (F4); added ritonavir stepwise to pyrene solutions containing 5–20 µM lopinavir, paclitaxel or felodipine and read the onset from the pyrene I3/I1 ratio (or felodipine's emission at 430 nm); added lopinavir (5–30 µg/mL) to 5, 10, 15 and 30 µg/mL ritonavir, ultracentrifuged and analysed both drugs in the supernatant; added lopinavir (4–16 µg/mL) or diclofenac (4–20 µg/mL) to a pre-formed ritonavir-rich phase (45–85 µg/mL ritonavir); and measured flux of 30 + 30 µg/mL binary donors, converting it to free concentration with the single-drug flux curves and checking by ultracentrifugation trasi2015a P3, P7, P8, P9, P10, P6. The ideal-mixing prediction used the single-drug amorphous solubilities and the ritonavir mole fraction in the drug-rich phase from mass balance trasi2015a P12.

What the data show

With 5, 10, 15 and 20 µM of the second drug the ritonavir onset was about 22, 14, 10 and 6 µg/mL (lopinavir), 16, 10, 4 and 2 (paclitaxel) and 18, 12 and 4 µg/mL (felodipine, to 15 µM) against about 26 alone trasi2015a R8, R7. With 5 µg/mL ritonavir, all lopinavir stayed in the supernatant up to 10 µg/mL and then levelled at about 15 µg/mL while ritonavir fell from about 5 to about 3.2 µg/mL; with 30 µg/mL ritonavir both drugs were depleted from the first addition (ritonavir about 21 to about 13 µg/mL as lopinavir rose from 5 to 30 µg/mL) trasi2015a R5. Adding 4, 8, 12 and 16 µg/mL lopinavir to a 45 µg/mL ritonavir solution left about 1.36, 2.76, 4.2 and 5.3 µg/mL lopinavir in bulk and lowered bulk ritonavir from about 23.6 to about 19.8 µg/mL; raising ritonavir to 65 and 85 µg/mL with 16 µg/mL lopinavir lowered bulk lopinavir to about 4.15 and 3.4 µg/mL trasi2015a R9, R10. Diclofenac at 4–20 µg/mL was recovered in full and ritonavir stayed at about 25.1–25.7 µg/mL trasi2015a R11. In the diffusion cell the receiver reached about 0.24–0.26 µg/mL ritonavir at 75 min from the mixtures against 0.48 from ritonavir alone, 0.33 against 0.49 for paclitaxel and 0.21 against 0.41 for lopinavir trasi2015a R12. Free concentrations from flux and ultracentrifugation are in the table trasi2015a R13. As fractions of the single-drug amorphous solubilities (26.2, 17.1 and 29 µg/mL by ultracentrifugation), the ultracentrifuge values are about 0.31 and 0.51 for ritonavir, 0.53 for lopinavir and 0.52 for paclitaxel (synthesis: 8.2 ÷ 26.2, 13.3 ÷ 26.2, 9.1 ÷ 17.1, 15 ÷ 29) trasi2015a R13, R1. Thirty-four ritonavir and 32 lopinavir points, plotted against the ritonavir mole fraction (0.06–0.70), run from about 3.0 to 21.4 µg/mL (ritonavir) and from about 15.2 to 4.6 µg/mL (lopinavir), with ritonavir mostly 2–4 µg/mL above the line S_a^\circ x and lopinavir below it at mid-range; an extractor check at 30 + 30 µg/mL gives x ≈ 0.41 and predicted bulk values of about 10.8 (ritonavir) and 10.1 µg/mL (lopinavir) against 13.3 and 9.1 measured trasi2015a R16, X9. A bile salt or a polymer takes the place of the second drug in the later papers: with 12 mM STCDC or SGCDC the highest telaprevir flux was about 1.06–1.11 µg/min against 1.25–1.32 in buffer (t test p << 0.05), and 1000 µg/mL of six polymers lowered the dissolved ibuprofen from 154 to about 89–131 µg/mL lu2017b R13, R6; ueda2021 R5.

Figure

(a) The pyrene ratio rises at much lower ritonavir concentration with 10 µM second drug, arrows marking the single-drug limits; (b) the onset falls as the second-drug concentration rises trasi2015a R7, R8.

Ritonavir rises and lopinavir falls with the ritonavir mole fraction; the lines are Eq. 4 without fitting trasi2015a R16.

Why

Amorphous drugs are liquid-like and can mix; with two unionized drugs the drug-rich phase holds both, and with ideal mixing each chemical potential falls by RT\ln x, so S_a(x_1) = S_a^\circ x_1 ("suggests"; "readily explained" by conventional thermodynamics) trasi2015a C37, E2, E3, E4, C35. The authors call the lowering of the onset "likely a colligative effect" and compare second drugs at equal molar concentration, which was not tested trasi2015a C19. Lopinavir mixes "fairly ideally" with ritonavir, perhaps from structural similarity, and other pairs "are likely" to be non-ideal, up to immiscibility trasi2015a C43. Ionic charge keeps diclofenac anions out of the hydrophobic phase ("readily explained") trasi2015a C44. (synthesis: this is the same principle as for the solubilizer in F12 read the other way round: what enters the drug-rich phase lowers the amorphous solubility, what stays out leaves it unchanged; the diclofenac result supports the second half, although only one ionized compound was tested.) (synthesis: Lu and co-workers invoke the same principle for bile salts, that a second solute miscible with the drug in the liquid state lowers the maximum supersaturation lu2017b C31, and Ueda and co-workers for polymers, which lower the chemical potential of ibuprofen in the droplets by dilution and, for PVP-VA, by interaction ueda2021 C11, C42; the mechanism is tested directly only for polymers, by NMR of the polymer that disappears from solution (F19).)

Conditions and limits

Solutions were made from methanolic stocks at 25 °C in one buffer; no co-amorphous or co-formulated ASD product was dissolved trasi2015a C45. The second-drug concentration in the fluorescence runs is given as 4 µg/mL in the methods and 10 µM in the results; HPMC (10 µg/mL) is stated only for some runs; equilibration time and the absence of crystals in the binary runs are not documented trasi2015a X3, X11. Paclitaxel at 30 µg/mL is at, not below, its limit, and the reductions are stated against different baselines (30 µg/mL added, or the single-drug amorphous solubility) trasi2015a X7, X8. The below-limit control (no interaction in the bulk) is described inconsistently and its data are not shown trasi2015a X6. The ritonavir curves in the diffusion cell figure are closer to each other than Table 2 suggests trasi2015a X10.

Counterexamples and conflicts

Ritonavir's points lie above the ideal-mixing line, so mixing is only approximately ideal for the one modelled pair trasi2015a X9. The effect is absent for the one ionized second drug trasi2015a R11. None found in the library against the lowering itself; later work reports about half of each drug's flux in a two-drug ASD and mutual reduction in co-amorphous drugs alhalaweh2016 R17; alkalla2026 C7, C12.

Transfer check

For a combination product or co-dosed drugs, test whether both are unionized and lipophilic at the intestinal pH; ultracentrifuge a mixture at the intended ratio and compare the bulk concentrations with each drug's single-drug amorphous solubility times its mole fraction in the drug-rich phase trasi2015a P8, P12, E4. The bulk concentration of the drug present in the smaller molar amount is lowered most, so dose ratio matters as much as drug choice (synthesis: from S_a = S_a^\circ x) trasi2015a C50.

History

Raina and co-workers (2014) had shown flux plateaus for single drugs trasi2015a C5. Trasi and Taylor (2015) were the first in this chapter to show the mutual lowering with a second drug and to propose an ideal-mixing model; later work applied it to binary ASDs and to other combinations trasi2015b C36, C49; trasi2019 C5, C38; alkalla2026 C7. Later the lowering was extended to polymer mixing into the drug-rich phase ueda2020a C36. Lu and co-workers (2017) and Ueda and co-workers (2020–2022) extended the mutual lowering to bile salts and polymers, and the polymer case was measured directly by NMR lu2017b C31; ueda2020b C32; ueda2021 C11.

How it was shown

Ilevbare and Taylor (2013) dissolved PVP K29/32 at 25 µg/mL or Tween 80 at 500 µg/mL in 100 mM phosphate buffer pH 6.8 at 37 °C, measured the crystalline solubility of ritonavir by ultracentrifugation and HPLC after 48 h, and the LLPS concentration by the UV titration (n = 3) ilevbare2013a P1, P5, A6, M10, M11. PVP does not solubilize ritonavir, and Tween 80 was above its CMC ilevbare2013a M10, M11. Lu and co-workers (2017) measured the crystalline solubility (48 h), the GLPS onset by three methods and the flux of telaprevir with six bile salts at 1.86 and 12 mM in 50 mM pH 6.5 buffer at 37 °C lu2017b P1, P2, P3, P4. Indulkar and co-workers (2017) did so for atazanavir with 0.05–3.6 mg/mL SDS in 50 mM pH 6.8 buffer at 37 °C and added flux in 1, 2 and 3.6 mg/mL SDS indulkar2017a P3, P6, P7.

What the data show

Without additive the crystalline solubility was 1.3 ± 0.20 and the LLPS concentration 18.8 ± 0.07 µg/mL; with PVP 1.3 ± 0.03 and 18.3 ± 0.26; with Tween 80 3.5 ± 0.10 (ratio 2.7) and 46.1 ± 0.92 µg/mL (printed ratio 2.6) ilevbare2013a R16. The table values give 46.1 ÷ 18.8 = 2.45 (or 2.52 against the with-PVP value), and LLPS-to-crystal ratios of 14.5 without and 13.2 with Tween 80 (synthesis from R16: 18.8 ÷ 1.3 and 46.1 ÷ 3.5) ilevbare2013a X5. The text calls both increases "~3 times" ilevbare2013a C53. In the outside comparison, felodipine with additives showed crystalline solubility up almost 40-fold while the concentration-based amorphous-to-crystalline ratio fell from about 9.2 in buffer to 4.5 (0.04% vitamin E TPGS) and 5.0 (FaSSIF) raina2015c R11, C29. Bile salts. The crystalline solubility of telaprevir was 5.2 µg/mL in buffer and 11.1, 12.2, 12.4 and 9.6 µg/mL with 12 mM STDC, STCDC, SGDC and SGCDC but 6.7 and 6.2 with STC and SGC (1.8–2.4 times for the dihydroxy salts, synthesis) lu2017b R1, X9. SDS. Atazanavir's crystalline solubility stayed near 1 µg/mL up to the CMC of 0.5 mg/mL and rose linearly above it to about 45.6 µg/mL at 3.6 mg/mL, a molar solubilization ratio κ of 0.006; the LLPS concentration rose too (κ 0.183) but only half as fast as the 0.351 predicted from 62 times the crystalline value indulkar2017a R7, R1, R9. At the amorphous solubility the concentration ratio was about 64.8, 37.5, 35.9 and 33.0 and the flux-derived ratio about 66.0, 70.0, 66.1 and 65.0 at 0, 1, 2 and 3.6 mg/mL SDS indulkar2017a R13. The authors read this as a solubilizer that adds drug to micelles without changing the activity at either boundary indulkar2017a C43, C45.

Figure

PVP changes nothing; Tween 80 raises both values by a similar factor ilevbare2013a R16.

Why

The authors argue that the crystal-saturated solution has the same drug activity with or without additive (same solid), and the drug-rich liquid keeps its activity as long as the additive does not enter it; with a constant activity coefficient in the dilute bulk phase the ratio of the two concentrations "must therefore" stay the same, so a solubilizer puts more drug in solution without raising the maximum supersaturation or the upper limit of activity ilevbare2013a C49, C54, C55. PVP at 25 µg/mL leaving both values unchanged is read as "suggesting" that it does not enter the ritonavir-rich phase ilevbare2013a C51. (synthesis: Raina and co-workers' fall of the concentration-based ratio fits this argument only if the extra solubilized drug is in micelles, which raises crystalline solubility more than the LLPS concentration; the activity argument and the concentration ratio are different measures.) Indulkar and co-workers explain the halving of the concentration ratio by a micelle/water partition coefficient that falls as drug concentration rises: the crystal-based prediction assumes that the adsorption mode seen near saturation also holds at supersaturation (F18) indulkar2017a C46, C48. Lu and co-workers read the bile-salt result as incorporation of telaprevir into micelles with unchanged activity at both boundaries (F17) lu2017b C28, C49. (synthesis: the two cases separate a solubilizer that raises both solubilities in proportion (ritonavir with Tween 80, telaprevir with STDC or SGDC) from one whose partition coefficient depends on the drug concentration (atazanavir with SDS); in both the activity-based boundary is unchanged.)

Conditions and limits

One drug, one buffer, 37 °C, one concentration of each additive; incorporation of Tween 80 or PVP into the drug-rich phase and drug activity in the micellar solution were not measured ilevbare2013a R16, C54. The printed Tween 80 ratio (2.6) does not match the table values ilevbare2013a X5. Single drugs at 37 °C, buffer media with one surfactant or one bile salt at a time and no lecithin indulkar2017a X5; lu2017b A1. In the terms of Ueda 2024 this is Case I, where the maximum supersaturation ratio is unchanged; where the surfactant also mixes into the drug-rich phase (Case III; clopidogrel with TPGS or Tween, ketoprofen with Tween, Table 1, p. 29) it falls.

Counterexamples and conflicts

The felodipine additive results show a falling concentration ratio raina2015c C29. A second drug, unlike Tween 80, does enter the drug-rich phase and lowers the limit (F11) trasi2015a C42. The concentration ratio of atazanavir with SDS falls to about half (65 → 33) and that of felodipine with Vitamin E TPGS to about half (9.2 → 4.5), whereas that of telaprevir with STDC or SGDC is kept, so "the supersaturation ratio does not rise" holds for all, but "stays constant" does not indulkar2017a R13; raina2015c C29; lu2017b R2.

Transfer check

Measure both the crystalline solubility and the LLPS concentration with and without the additive at the concentration to be used, and compare the ratio; also measure flux to get an activity-based ratio, since total concentration in micelles overstates the driving force ilevbare2013a C54, C43; trasi2015a P6. Determine the amorphous solubility in the solubilizing medium itself and the activity-based ratio by flux (F16); multiplying the crystalline solubility in the medium by the buffer ratio overshot twofold for atazanavir in SDS indulkar2017a C16, C64.

History

Ilevbare and Taylor (2013) stated the argument and the Tween 80 result; Raina and co-workers later reported the fall in the concentration-based ratio for felodipine ilevbare2013a C53; raina2015c C29. Lu and co-workers (2017) and Indulkar and co-workers (2017) tested the argument of Ilevbare and Taylor with six bile salts and SDS, and found that it holds for the activity ratio but not always for the concentration ratio lu2017b C49; indulkar2017a C21, C64.

How it was shown

Lu and co-workers (2017) dissolved each of STC, SGC (trihydroxy), STDC, SGDC (hydroxyl missing at R3) and STCDC, SGCDC (missing at R4) at 1.86 mM (called the monomer level) or 12 mM (the micellar level) in 50 mM pH 6.5 phosphate at 37 °C, assuming that telaprevir does not shift the CMC (reported CMCs of these salts are 2–12 mM) lu2017b P1, A1, A2, C42, M3. They measured the crystalline solubility (48 h, ultracentrifugation, UV at 270 nm), the GLPS onset by extinction at 370 nm, ultracentrifugation (40 min) and the flux plateau, and flux across a 6–8 kDa membrane (32 mL per side, 5 µg/mL HPMCAS-MF in both chambers, bile salt at the same concentration on both sides) up to 360 µg/mL lu2017b P1, P2, P3, P4.

What the data show

Solubility. With 12 mM bile salt the crystalline solubility was 6.7 ± 1.2 (STC), 11.1 ± 0.6 (STDC), 12.2 ± 0.6 (STCDC), 6.2 ± 0.4 (SGC), 12.4 ± 1.3 (SGDC) and 9.6 ± 0.6 µg/mL (SGCDC) against 5.2 ± 0.1 in buffer; at 1.86 mM it was 5.5–6.9 lu2017b R1. Onset. By UV, ultracentrifugation and flux the onset was 146, 149, 148 µg/mL in buffer; 174, 178, 181 with 12 mM STC; 335, 353, 311 with 12 mM STDC; 347, 360, 345 with 12 mM SGDC; ultracentrifugation 133 and 290 and flux 110 and 238 with 1.86 and 12 mM STCDC; flux 169 and 177 with 1.86 and 12 mM SGCDC lu2017b R2. The onset-to-solubility ratio therefore stays at about 28–32 for STDC and SGDC (for example 335 ÷ 11.1 = 30.2, 347 ÷ 12.4 = 28.0, synthesis) and is lower, about 16–24, for STCDC (290 ÷ 12.2 = 23.8, 110 ÷ 6.8 = 16.2, synthesis) lu2017b R1, R2. Flux. Trihydroxy salts at 1.86 and 12 mM (and 18 mM STC) left the flux on the buffer curve; 1.86 mM STDC, SGDC and STCDC did as well; 12 mM STDC gave about 0.43 µg/min at 90 µg/mL against about 0.83 in buffer, and 12 mM STDC and SGDC kept rising to about 1.34–1.38 µg/min at 330–360 µg/mL; SGCDC lowered the flux equally at both levels (about 0.58 µg/min at 90 µg/mL) lu2017b R10, R11, R12, R13. The plateau flux with STCDC and SGCDC was about 1.06–1.11 µg/min against 1.25–1.32 in buffer (t test p << 0.05), a ratio of about 0.8–0.9 (X6), and the Fig. 11 schematic draws the GLPS boundary at δ ≈28 without bile salt and with trihydroxy salts and micellar STDC and SGDC, and at ≈22 with monomeric and micellar STCDC and with SGCDC lu2017b R13, R6, R18, X6. Extinction rose on the first addition of drug to STCDC or SGCDC without visible turbidity or birefringence lu2017b R4.

Figure

(a) STC and SGC lie on the buffer curve; (b) 12 mM STDC and SGDC are shifted right, with about half the slope; (c) STCDC and SGCDC level off near 1.1 µg/min, below buffer lu2017b R10, R12, R13.

A schematic drawn from the measured boundaries; the red axes give the matching concentrations lu2017b R18.

Why

The authors say that trihydroxy bile salts are less hydrophobic with higher CMCs and so micellize less at 12 mM ("One possible explanation"), that STDC and SGDC solubilize telaprevir by incorporating it into micelles ("it is apparent"), the drug-loaded aggregates diffusing more slowly by the Stokes–Einstein relation, and that STCDC and SGCDC mix into the telaprevir drug-rich phase on GLPS and lower telaprevir's activity there ("presumably"), as a miscible second solute does lu2017b C45, C28, C46, C50, C51, C31. They suspect drug–bile salt complexes behind the scattering with STCDC and SGCDC lu2017b C23. Because the activity-based boundaries (δ = 1 and ≈28) of micellar STDC and SGDC stay where they are in buffer, they conclude that the composition of the crystalline and amorphous phases is unchanged lu2017b C49. The companion paper measured the surface-tension CMC with 60 µg/mL telaprevir as about 4 mM for STC and about 2 mM for SGDC, in line with 1.86 mM being monomeric for these two lu2017 R5, C14.

Conditions and limits

One drug, 50 mM pH 6.5 phosphate, 37 °C, single bile salts without lecithin or cholesterol, and one concentration each below and above the reported CMCs; the CMC of each salt in this medium was not measured here and crystallization was not measured lu2017b C42, C59. Whether micelles or drug–bile salt aggregates cross the membrane is not stated, so lower activity and slower diffusion of drug-loaded aggregates are not separated lu2017b X8. Replicate numbers and the statistic behind the ± values are not printed, and "no significant" is stated without a test lu2017b X4.

Counterexamples and conflicts

The statement that the taurine or glycine conjugate has no impact holds for STC/SGC and STDC/SGDC but not for STCDC/SGCDC: 1.86 mM STCDC overlies buffer up to 90 µg/mL, 1.86 mM SGCDC lies below it lu2017b X5, R13. Monomeric STCDC lowered the onset (133 and 110 µg/mL) and the boundary (δ ≈22) although monomeric salts are said to interact minimally lu2017b X10, R2, R18. By the concentration ratio alone, 1.86 mM SGCDC shows no fall (169 ÷ 5.7 = 29.6, synthesis), the fall being visible only in the flux plateau lu2017b R1, R2, R13. For SGCDC the flux is the same at the two levels although the crystalline solubility differs lu2017b X7.

Transfer check

Measure the crystalline solubility, the amorphous solubility by two methods and the flux plateau of the drug in single bile salts of both hydroxylation classes and in the mixture at the intended ratio; a lowered plateau flux without a lowered crystalline solubility points to a bile salt mixing into the drug-rich phase; use the activity scale before comparing crystallization in different bile-salt media lu2017b P1, P4, C38. A medium with taurocholate only may mispredict, since SGC, STCDC and SGCDC are more abundant in human intestinal fluid lu2017b C53, R22.

History

Chen and co-workers had shown that STC inhibits crystallization of supersaturated celecoxib, which motivated the study lu2017b R24. Lu and co-workers (2017) gave the first comparison of the six bile salts on the thermodynamics of a supersaturated drug; the companion crystallization paper redraws the STCDC and SGCDC curves, from the same flux data, as levelling off at δ ≈21–24 lu2017 R1. Elkhabaz and co-workers (2019) cite the fall of the telaprevir maximum supersaturation with STCDC and SGCDC to about three-fourths elkhabaz2019b R18. Elkhabaz and co-workers (2017) restate that with solubilizing additives the activity boundaries stay the same while the amorphous solubility increases elkhabaz2017 C40.

How it was shown

Indulkar and co-workers (2017) measured the CMC of SDS in 50 mM pH 6.8 phosphate at 37 °C by the emission maximum of pyrene carboxaldehyde and the aggregation number by static quenching of pyrene (drug-free solutions), the crystalline solubility (48 h, ultracentrifugation at 209,490 g) and the LLPS concentration (solvent-shift extinction at 350 nm) at 0.05–3.6 mg/mL SDS, and fitted S = S_0 + \kappa\,(C_{SDS} - CMC) to the crystalline, measured amorphous and predicted amorphous solubility indulkar2017a P1, P2, P3, P6, E6. Km/w came from the flux-derived free drug and mass balance, and Region I was fitted to a Langmuir isotherm indulkar2017a P8, P14, E10, E11. Drug emission (excitation 250 nm), energy transfer to 0.5 µg/mL curcumin (45–1200 µg/mL drug in 3.6 mg/mL SDS) and pulsed-gradient spin-echo NMR of 40 mg/mL SDS in D2O at 25 °C with 0, 0.4 and 7 mg/mL drug probed where the drug sits indulkar2017a P10, P11, P12.

What the data show

The CMC in buffer was 0.5 mg/mL against 2.4 mg/mL in water, and the aggregation number 60, 58 and 65 at 1, 2 and 3.6 mg/mL SDS indulkar2017a R1, R2, R3. Crystalline solubility stayed near 1 µg/mL to the CMC and reached about 45.6 µg/mL at 3.6 mg/mL; the measured amorphous solubility rose from about 0.16 to about 2.09 mM over the micellized-SDS range, the prediction to about 4.02 mM, and the fitted κ were 0.006, 0.183 and 0.351 with intercepts of 1.04, 64.6 and 69.2 µg/mL indulkar2017a R7, R8, R9. In 3.6 mg/mL SDS Km/w was about 327 at 20 µg/mL, 226 at 45, 99 at 100, 93 at 200, 100 at 400, 114 at 800 and 90 at 1450 µg/mL; Region I fitted a Langmuir isotherm (R² 0.998, K_ad 3.86 × 10⁵ L/mol, saturation mole fraction 0.017 equal to 130 µg/mL total drug), and the slope change occurred at the same activity at 1 and 2 mg/mL SDS (data not shown) indulkar2017a R14, R16, R15. Drug emission showed a single band at 328 nm at 45 µg/mL and a second at 369 nm, the band of the amorphous film, only above 200 µg/mL, with I369/I328 rising from about 0.41 at 500 to about 0.80 at 1400 µg/mL indulkar2017a R17, R18, R19. Energy transfer to curcumin (a new band at about 490–500 nm) appeared at 1200 but not at 45 µg/mL indulkar2017a R22. Drug molecules per micelle rose from 0.4 at the crystalline solubility to 11.5 at the amorphous solubility, similar at the three SDS levels indulkar2017a R24. In the NMR samples the micelle radius was 4.61 nm with 0 and 0.4 mg/mL drug and 6.08 nm with 7 mg/mL, a volume increase of 130% against a theoretical 100% indulkar2017a R25, R26.

Figure

The predicted line (blue) rises about twice as steeply as the measured onset (red); κ values are printed on the lines indulkar2017a R8, R9.

A steep fall in Region I and a plateau in Region II; the y-axis carries no unit indulkar2017a R14, X3.

Why

The authors "speculate" that atazanavir is solubilized by adsorption at the micelle/water interface near saturation, by analogy with nystatin in other micelles, and by association with the dodecyl core once the surface sites are full, by analogy with pyrene; the Langmuir limit corresponding to 130 µg/mL total drug lies "very close" to where Km/w stops falling indulkar2017a C24, C25, R31. The 369 nm band is read as drug–drug contact in the core, because it appears well below the amorphous solubility and so is not a drug-rich phase, the energy transfer as colocalization of drug and curcumin in a non-polar environment, and the swelling as about 7 core-dissolved molecules per micelle against about 0.4 adsorbed ones indulkar2017a C27, C28, C31, C34. A crystal-based prediction assumes the saturated (adsorption) mode throughout, so it overestimates the amorphous solubility; the abstract and conclusions call the change of mechanism "the mechanistic basis" indulkar2017a C46, C48, C65. Raina and co-workers had argued that micelles have a variable capacity that depends on how many solute molecules they already hold raina2015c C37. The extractor cautions that every probe is indirect; Km/w has no stated basis (the plotted values match mole-fraction ratios divided by 1000); the NMR samples differ in temperature, solvent and SDS concentration from the flux conditions; the aggregation number was measured without drug; and Region I ends at about three times the crystalline solubility, not slightly above it indulkar2017a X3, X4, X5, X6, X7.

Conditions and limits

One weak base and one anionic surfactant, always above the CMC, at 37 °C in 50 mM pH 6.8 phosphate (NMR at 25 °C in D2O) indulkar2017a A1, X7. Whether micelles cross the 6–8 kDa membrane was not tested, although the free-drug calibration assumes they do not indulkar2017a X9. The authors expect the twofold error to be "most likely" compound- and surfactant-dependent, and no formulation or in vivo data were measured indulkar2017a C54, C57.

Counterexamples and conflicts

The same 369 nm band marks nanodroplets of atazanavir in buffer in a later study, whereas here it appears in micelles far below the amorphous solubility, so its meaning depends on the medium hate2018 P6; indulkar2017a C27, R18. Whether the plateau of Km/w in Region II reflects core dissolution or another mode is "speculation", supported by the Langmuir fit but not by a direct location of the drug indulkar2017a C24.

Transfer check

Do not multiply the crystalline solubility in a micellar medium by the buffer ratio; measure the amorphous solubility in the medium and the free drug by flux at the crystalline and at the amorphous solubility, then compute Km/w at both points; a lower value at the amorphous solubility, as for atazanavir in SDS and posaconazole in intestinal media, means the prediction overshoots indulkar2017a C16, C48; duong2022b C45.

History

Raina and co-workers (2015) had found about 2:1 for felodipine with solubilizers and suggested a variable micelle capacity; Indulkar and co-workers (2017) proposed the surface-to-core change for atazanavir in SDS; Duong and co-workers (2022) and Elkhabaz and co-workers (2019) used a concentration-dependent Km/w to convert concentration ratios to thermodynamic ratios in biorelevant media raina2015c C37; indulkar2017a C24; duong2022b C8, C9; elkhabaz2019b C33.