Review: storyline · equations and derivations

Following Ueda K., Moseson D.E., Taylor L.S. (2024) Amorphous Solubility Advantage: Theoretical Considerations, Experimental Methods, and Contemporary Relevance. Journal of Pharmaceutical Sciences.

The amorphous solubility as the limit of supersaturation: from the free-energy calculation, through additives and measurement, to the dose number

What this page is. Ueda, Moseson and Taylor (2024) reason from one definition (the concentration in equilibrium with a water-saturated amorphous drug) to its calculation, to what additives do to it, to how it is measured, and to how it should be used in release tests and dose assessment. Each step below gives that reasoning and then adds what the library's papers, with their years, have measured on the same point, including where they qualify the review. Review pages are pages of the journal pre-proof PDF, one above the printed "Page N of 70". The review prints no absorption simulation of its own; its design target is the dose number recalculated with the amorphous solubility, together with the physiologically based models it cites.

The chain at a glance

1. One limit for all routes: the maximum supersaturation from any route → the limit needs a definition that does not depend on the route.
2. What the amorphous solubility is: metastable equilibrium with the water-saturated amorphous drug → a defined equilibrium can be calculated.
3. Calculating the additive-free ratio: S_a^0/S_c^0 from \Delta G_{c\to a}, water uptake and ionization → the calculated value must be set against a measured phase boundary.
4. Phase separation marks the limit: the onset of LLPS, and the type of precipitate → the supersaturation that remains has to be expressed as an activity.
5. Concentration and activity: SR=a/a^*, read by membrane flux → additives separate concentration from activity, so they have to be classified.
6. Case I, complexation in the aqueous phase: f_{com} raises S_a with SRmax unchanged → other additives enter the drug-rich phase instead.
7. Case II, mixing into the drug-rich phase: x_{drug} and \chi lower S_a and SRmax → many additives do both.
8. Case III, both, and the formulation trade-off: the combined equation, and the inhibitor that also lowers the limit → the limit in real media cannot be calculated, so it must be measured.
9. Measuring the limit and the maximum activity: kinetic and static S_a, flux and diffusometry → these values are the reference for release from a formulation.
10. Release under nonsink conditions: drug-rich colloids and the sink index → the measured values can then be applied to the dose.
11. Design target: dose number and absorption modelling: dose number with S_a, reaching and maintaining S_a, and PBBM input [adapted: no absorption simulation printed].

Step 1. One limit for all routes

Question. Why does one quantity bound the supersaturation reached by such different routes?

The review's reasoning (2024). The review starts from Hancock and Parks (2000), who estimated an amorphous advantage of 12–1652 fold against 1.1–3.6 fold for crystalline polymorphs, and found that experiment under-estimated it because the amorphous form recrystallized [review p. 8]. Supersaturation can arise by storage or process disorder, by hydration-induced disorder, by a pH or solvent shift, from salts, cocrystals, polymorphs and prodrugs, from solubilized formulations and from amorphous solids; in all of these the maximum extent is stated to be limited by the amorphous solubility [review pp. 8–9, Fig. 1]. An early solubility screen by anti-solvent addition may therefore report the amorphous value without its being recognized [review p. 9].

The history supplies the link to performance. Amorphous novobiocin gave about ten times the solubility of the crystalline free acid and was absorbed in dogs, whereas the crystal gave undetectable absorption [review p. 11]. Sulfathiazole gave a solubility ratio of 3.8 at 37 °C against a dissolution rate ratio of 3.7, and reserpine–PVP gave a surface-normalized rate factor of ~2.5, a solubility ratio of ~3, and in rats a rate of absorption ~2.7 fold and an extent ~3.2 fold above the crystalline suspension [review pp. 12–13]. These are the review's historical examples; it needs no earlier step here.

What the library shows. The library holds the cap for several of these routes. A dissolving film, a ritonavir–PVP dispersion and salts reach about the same value: amorphous films of clotrimazole, atazanavir and nicardipine reached 69–110% of the LLPS concentration at the same pH [indulkar2015 R18, F6]; a 1:9 ritonavir–PVP dispersion levelled off at 18.3 and 18.0 µg/mL against an onset of 18.8 µg/mL [ilevbare2013a R17, R8, R13]; and pazopanib HCl shifted from pH 1.2 to 6.5 held about 17 and 22–30 µg/mL against 19.8 µg/mL (2018) [sugihara2018 R8, R12] (Amorphous solubility F13). Flux, NMR and ultracentrifugation put the free drug from six marketed tacrolimus capsules at ≈46–50 µg/mL [trasi2016 R10, R4, R15, R14], and in rats the enzalutamide dispersions that reached the limit exceeded crystalline drug while the one that crystallized did not (2018) [wilson2018 R18, R6, C47] (Overview F1).

Corrections and qualifications.

• "All routes" is measured for fewer routes than Fig. 1 draws. The library has solvent shift, pH shift, films, dispersions, salts, a co-crystal and capsules; the CRH1 glass and co-crystal never reached the limit, and the salts reached it only with HPMC in the medium [almeidaesousa2016 R12, C31, R8, R9] (Amorphous solubility F13). The cap holds where crystallization is slower than the route.
• Silica formulations are a route that stays below the limit: ritonavir from SBA-15 dosed at up to ten times the amorphous solubility plateaued at about 24 µg/mL, about 85% of it (2018) [dening2018 R22, C32, R35] (Overview F1).
• The check finds the library silent on the polymorph range of Hancock and Parks, on the screening artefact and on the sulfathiazole and reserpine ratios (check R2, R3, R4); those statements rest on the review here.

Next. The cap is observed for many routes, but the observation alone does not say what state the drug is in at the cap. That needs a definition.

Pages. amorphous solubility, amorphous solubility advantage, supersaturation, supersaturating drug delivery systems.

Step 2. What the amorphous solubility is

Question. With which phase is the aqueous solution in equilibrium when its concentration equals the amorphous solubility?

The review's reasoning (2024). A crystal solubility is a true equilibrium and does not depend on the starting solid given enough time; if transformation is slow or inhibited, the value measured belongs to the metastable form, so an amorphous residue in metastable equilibrium with the solution gives the amorphous solubility [review p. 21]. Free-energy differences between the anhydrous forms do not translate directly into solubility differences, because the amorphous drug absorbs water and its chemical potential falls, whereas a non-hydrate crystal takes up negligible water [review p. 21].

Temperature gives the phase diagram [review p. 24, Fig. 5]. Both solubilities generally rise with temperature and their ratio falls, since the free-energy difference disappears at the melting point; water depresses that melting point, the crystalline and amorphous solubilities converge there, and above it the crystal cannot exist and any solubility follows the LLPS boundary between liquid drug and water [review p. 22]. Extending that boundary below the depressed melting point gives the amorphous solubility curve in the supercooled region. Above the wet-Tg the value refers to the drug-lean phase in equilibrium with the water-saturated supercooled liquid; below the wet-Tg the glass has the higher free energy, and the review calls it controversial whether the solubility reflects the glass or the supercooled liquid [review pp. 22–23]. This step needs the cap of Step 1 only as motivation.

What the library shows. The isolated phase fits the water-saturated description. The ritonavir phase held 94.85–95.97% drug and 4.03–5.15% water with a Tg of 0.4 °C against 50.4 °C dry, and amorphous ritonavir sorbs 5.21% water by extrapolation to 100% RH (2013) [ilevbare2013a R6, R5, E14]. The clotrimazole, atazanavir and telaprevir phases had Tg values of about 14, about 51 and 52 °C, against dry values of 28, 104 and 103 °C, so the last two are glasses at 37 °C (2015) [indulkar2015 R22, M1, M2; mosqueragiraldo2015 R2, M1] (Amorphous solubility F2).

On the review's open question the library has one direct comparison. Griseofulvin and sorafenib stay glassy at 25 °C after 97% RH storage (Tg 88 and 49 °C), yet their onsets of 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) (2015) [almeidaesousa2015 R5, R6, R8, R9, R11] (Amorphous solubility F2). The authors read this as a phase "much more relaxed" than a melt-quenched glass, an inference, since the phase was not isolated [almeidaesousa2015 C36, C37, C38].

Corrections and qualifications.

• "Generally both increase" has an exception in the library. The ritonavir limit in water fell from about 58 µg/mL at 10 °C to about 37 near 35 °C and rose to about 53 at 65 °C, while efavirenz rose from about 19 to about 45 µg/mL and ibuprofen from about 127 to about 284 µg/mL (2013, 2022) [ilevbare2013a R14, R15; ueda2022 R9] (Amorphous solubility F9).
• The falling ratio is measured for ibuprofen: about 5.8 at 25 °C, 3.6 at 40 °C and 2.5 at 60 °C (2022) [ueda2022 R16] (Amorphous solubility F6).
• The check finds the library has the ritonavir binodal but not the construction of Fig. 5c, with the two curves meeting at the depressed melting point (check R28); it rests on the review.

Next. The definition names the phases and hence the terms any calculation needs: the crystal-to-amorphous free energy and the water taken up.

Pages. amorphous solubility, temperature-composition phase diagram, melting point depression, plasticization by water, supercooled liquid, glass-liquid phase separation.

Step 3. Calculating the additive-free ratio

Question. How far can the ratio of amorphous to crystalline solubility be calculated from solid-state properties?

The review's reasoning (2024). For an unionized drug without additives the ratio follows from the free-energy difference, an equation that also applies to metastable polymorphs [review p. 25]:

The free energy comes from the melting temperature and enthalpy, with the heat capacity difference taken as zero, as constant at its value at Tm, or measured against temperature for improved predictions [review pp. 25–26]. Eq. 1 treats the amorphous form as one component. Using the definition of Step 2, the review adds the activity of the water-saturated amorphous drug, obtained by applying Gibbs–Duhem to the sorption isotherm [review p. 26]:

Eqs. 1 and 2 almost coincide for ibuprofen and tolnaftate, while Eq. 2 is 30–70% lower for telaprevir and griseofulvin [review pp. 26–27]. Since \Delta G_{c\to a} of most drugs is typically below ~10 kJ/mol, the ratio is constrained, and measured ratios of most drugs are below 100-fold [review p. 27]. For ionization, which falls at the higher concentration of the amorphous form, the review gives a further correction, relevant only for unbuffered conditions [review p. 27]:

What the library shows. The calculated value has been set against measured onsets for 17 compounds, and 17 of 24 rows lie between 0.91 and 1.15 times the prediction (2013–2022) [ilevbare2013a R8; almeidaesousa2015 R5, R11; indulkar2015 R2; mosqueragiraldo2015 R13; almeidaesousa2016 R2, R4] (Amorphous solubility F5). Water-saturated activities run from 0.999 (ibuprofen) and 0.699 (griseofulvin) [almeidaesousa2015 M1, M6] through 0.30 (telaprevir) [mosqueragiraldo2015 M1] and 0.27 (ritonavir) [ilevbare2013a M1] to 0.22 (atazanavir) [indulkar2015 M2], so the correction reaches about 78% for atazanavir (synthesis: 1 − 0.22), beyond the review's upper figure.

The route to the free energy matters for high-melting drugs. For telaprevir (Tm 238 °C, 37 °C test) the estimates were 245 ± 5 µg/mL by the Hoffman equation, 189 ± 16 by the crystal-to-glass heat capacity method and 97 ± 8 by the crystal-to-supercooled-liquid method, against 90–92 measured (2015) [mosqueragiraldo2015 R13]. For six drugs at 25 °C the fits through the origin gave slope 1.06, R² 0.989 (heat capacity) and slope 1.20, R² 0.926 (Hoffman), with diclofenac predicted at 52.5 and 28.9 against 52.2 measured [almeidaesousa2015 R14, R15, R13, R5] (Amorphous solubility F7). The evidence is circular to a degree: the reference state is justified by the agreement it produces [almeidaesousa2015 X4; mosqueragiraldo2015 X11].

Corrections and qualifications.

• "Below 100-fold" has a marked exception. Pazopanib gave about 500–600 from pH 6.4 to 8.4 (about 603 at pH 6.5: 19.8 against 0.033 µg/mL), with high uncertainty because the crystalline value is so low (2018) [sugihara2018 R4, R1.e, R2.g, C16]; the library's range runs from 2.2 (efavirenz) upward, and paclitaxel hydrate gives 85–94 [ilevbare2013a R8; trasi2015a X12] (Amorphous solubility F6).
• Pazopanib also shows that Eq. 1 is unavailable for a drug that degrades on melting [sugihara2018 C17] (Amorphous solubility F5).
• Eq. 3 is absent from the library, whose ionizable series were all buffered, and the review's constant ionized fraction under buffer implies a ratio independent of pH. Total LLPS concentrations do follow Henderson–Hasselbalch, with the un-ionized part flat at about 5.6–8.4 µg/mL for clotrimazole and 57.5–70.4 µg/mL for atazanavir (2015) [indulkar2015 R9]. But measured ratios move: clotrimazole about 8 at pH 4–4.5 and 12–17 from pH 5 [indulkar2015 R10, X5], pazopanib about 103 at pH 5.0 and 500–600 from pH 6.4, its amorphous values at pH 5 and 5.5 lying under the curve (60 against about 250 µg/mL, 40 against about 80), which the authors attribute to crystallization before the limit ("likely") [sugihara2018 R3, R4, C11] (Amorphous solubility F10). The constant ratio holds only while crystallization is slower than the measurement.
• For atazanavir, Eq. 2 with the printed inputs gives the printed 62 against a measured ratio near 64, provided the 0.22 that the 2017 Table I heads I(a) is read as exp[−I(a)], as in Indulkar 2015 [indulkar2017a X1, E2, E3; indulkar2015 M2] (Amorphous solubility F7).

Next. A calculation needs a measurement to test it, and the amorphous solid recrystallizes when dissolved. The discovery of phase separation supplies that measurement.

Pages. amorphous-crystalline free energy difference, water sorption correction of amorphous solubility, Gibbs-Duhem equation, Hoffman equation, advantage from free energy difference, advantage with water-sorption correction, free energy difference from heat capacity.

Step 4. Phase separation marks the limit

Question. What happens when the limit is exceeded, and why does the type of precipitate decide the outcome?

The review's reasoning (2024). Bonnett and co-workers (2003) showed that crystallization is not the only transition in a supersaturated solution, so a precipitation event should be defined by the structure of the precipitate; the solute has the same chemical potential in the two phases that LLPS forms, though both are metastable with respect to crystal [review p. 14]. Ilevbare and Taylor (2013) generated supersaturated ritonavir by cooling, antisolvent addition and ASD dissolution, and found LLPS at the same concentration, which matched the predicted amorphous solubility and the maximum from dissolving neat amorphous drug [review p. 15]. The review takes this as establishing that the amorphous solubility is the maximum achievable supersaturation, and plots predicted against measured values for 23 compounds around a line of slope 1 [review p. 16, Fig. 3].

The outcome differs by type: after LLPS the system remains highly supersaturated, whereas crystallization depletes supersaturation [review pp. 16–17]. For an amorphous precipitate the review calls size of paramount importance because of redissolution kinetics and particle drifting across the boundary layer, and for a crystalline one lists size, surface area, disorder, polymorph and residual supersaturation [review p. 17]. Step 3 supplies the prediction that the onset is compared with.

What the library shows. Twenty-one poorly soluble compounds in these papers turned turbid and took up polarity probes past a drug-specific concentration, with onsets from 0.53 µg/mL (sorafenib) to 355.96 µg/mL (griseofulvin, with polymer), and for the five drugs examined gave an amorphous pellet or precipitate (2013–2016) [ilevbare2013a C2, R8; almeidaesousa2015 C41, R5, R6; indulkar2015 C6; mosqueragiraldo2015 C21; trasi2015a C12; almeidaesousa2016 C17] (Amorphous solubility F1). The onset did not depend on the stock solvent for ritonavir, and drug in 0.1 M HCl 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 [ilevbare2013a R10, R9, R8] (Amorphous solubility F4).

The phase is a transient precursor. Crystal solubility of ritonavir lies far below it (37.2 against 2.4 µg/mL in water at 37 °C), but telaprevir gave birefringent particles after about 15 min without polymer and for hours not with 5 µg/mL HPMCAS [ilevbare2013a R8, R11; mosqueragiraldo2015 R1], and pazopanib supersaturation after a pH shift lasted 5 ± 2 min without and about 39 min with 50 µg/mL HPMC (2018) [sugihara2018 R22, C32] (Amorphous solubility F3).

Corrections and qualifications.

• Griseofulvin without polymer crystallized near 150 µg/mL before any probe signal, whereas with 100 µg/mL HPMCAS-HF the two signals rose together near 350 µg/mL [almeidaesousa2015 R4] (Amorphous solubility F1). Phase separation at the limit is therefore not inevitable in the sense that it is the first event; fast crystallizers may never show it.
• A drug-rich phase that disappears is not necessarily crystallizing: ibuprofen droplets in buffer were lost within 1–2 h by creaming, with crystals only after 24 h (2020) [ueda2020b R15, R8, C27] (Amorphous solubility F3).
• "Paramount" for size is a statement about kinetics that Step 10 qualifies with library data.

Next. Past the onset the free drug stops rising. The review's next argument is that this is a statement about activity, not concentration.

Pages. liquid-liquid phase separation, drug-rich phase, crystallization inhibition, desupersaturation.

Step 5. Concentration and activity

Question. What does the amorphous solubility limit: the concentration, or the thermodynamic activity?

The review's reasoning (2024). Supersaturation is fundamentally an activity ratio [review pp. 27–28]:

In dilute solution without additives \gamma/\gamma^*\sim 1, so the ratio of Step 3 gives SRmax directly; in media with micelles the assumption may fail and a concentration ratio may mislead [review p. 28]. The evidence is mass transport. The maximum flux across a cellulose acetate membrane coincided with concentrations at or above the amorphous solubility (Raina 2014) [review p. 17], and in Fig. 4 the amorphous-to-crystalline flux ratio stays at about 50–56 across media while the concentration ratio ranges from about 14 to about 57 (values read from the plot) [review p. 19]. The review calls this the central theme: variations in concentration must be separated from their thermodynamic consequences [review pp. 17–18]. Solubility-limited absorption is then related to supersaturation because supersaturation raises the free drug available for diffusion [review p. 28].

What the library shows. Flux against donor concentration has been measured for many systems. Felodipine at 25 °C rose from 12 to 63 × 10⁻³ µg/min between 2 and 5 µg/mL and stayed at 63–71 × 10⁻³ µg/min up to 100 µg/mL (2014) [raina2014a R11, R12, R14]; the same shape followed for clotrimazole, for telaprevir released from an ASD (about 1.0–1.2 µg/min above about 90 µg/mL), for five drugs in a PAMPA cell (2022) and for enzalutamide (2018) [indulkar2016 R9; mosqueragiraldo2018b R1, R2; hate2022 R1; wilson2018 R13] (Bioperformance F2). Crystalline suspensions holding about 1 and about 40 µg/mL dissolved felodipine gave about the same low flux (about 0.009 and 0.006 µg/min), so solubilized drug does not add driving force (2015) [raina2015c R4] (Bioperformance F3).

The Fig. 4 result comes from Elkhabaz (2019): flux-based maxima of about 50–56 for atazanavir and 29–37 for posaconazole across six media, with concentration-based values falling from about 55 and 36 in buffers to about 14 and 5 in pooled human fluid [elkhabaz2019b R8, R9, C21]. For atazanavir in 3.6 mg/mL SDS the free drug was about 65–70 µg/mL at a total of about 1450 µg/mL, and the concentration and activity ratios at the amorphous solubility were 33.0 and 65.0 against 64.8 and 66.0 in buffer (2017) [indulkar2017a R11, R13] (Amorphous solubility F16).

Corrections and qualifications.

• The flux plateau is a property of a membrane-limited device. In a hollow-fibre module and in PAMPA with a sink receiver transfer continued above the amorphous solubility, for atazanavir from about 5.0 to 8.8 µg/min between 80 and 1000 µg/mL (2022) [hate2018 R9; hate2022 R5, R19] (Bioperformance F2). Step 10 returns to this. The activity is still capped; the mass transferred is not.
• The free-drug calibration assumes micelles do not cross the 6–8 kDa membrane, which was not tested [indulkar2017a X9] (Amorphous solubility F16).
• The direct relation of supersaturation to expected absorption gain rests in the library on one model (check R36; Bioperformance F29, Step 11).

Next. Once activity and concentration can differ, an additive has to be described by what it does to each. The review sorts additives into three cases, for an unionized drug or a fixed ionized fraction [review pp. 28–30, Table 1, Fig. 6].

Pages. thermodynamic activity, supersaturation ratio, free drug concentration, membrane flux, activity-based supersaturation ratio, membrane flux in terms of solute activity, supersaturation from membrane flux ratio.

Step 6. Case I, complexation in the aqueous phase

Question. What happens when an additive binds dissolved drug but stays out of the drug-rich phase?

The review's reasoning (2024). The drug-rich phase is unchanged, so the free drug at the limit stays at S_a^0 and complexed drug is added to it [review p. 31]:

The ratio then carries both complexed fractions and the terms of Step 3 [review p. 32]:

A constant complexed fraction returns Eq. 2, as for ibuprofen and ketoprofen in EUD-E. More generally f_{com} falls as drug concentration rises, so S_a/S_c<S_a^0/S_c^0, while SRmax stays at its additive-free value and exceeds the apparent ratio [review pp. 32–33]. Cyclodextrins of BS type are the reverse case: for ibuprofen with β-CD the apparent ratio is larger and SRmax is again constant [review p. 33]. Table 1 lists surfactants, bile salts, cyclodextrins and EUD-E here [review p. 29].

What the library shows. Ritonavir with 500 µg/mL Tween 80 gave crystalline 1.3 to 3.5 µg/mL and LLPS 18.8 to 46.1 µg/mL, ratios of 14.5 and 13.2 (2013) [ilevbare2013a R16, X5] (Amorphous solubility F12). Telaprevir with 12 mM STDC or SGDC went from 5.2 to 11.1 and 12.4 µg/mL and from about 150 to 311–360 µg/mL, with activity boundaries unchanged (2017) [lu2017b R1, R2, C49] (Amorphous solubility F17). Ibuprofen with EUD-E at pH 2.0 went from 29.8 to 42.1 and from 177.1 to 256.3 µg/mL, ratio 6.09 against 5.94, and about 289 of 300 µg/mL polymer stayed in solution (2020) [ueda2020b R1, R13] (Amorphous solubility F19).

The falling f_{com} has measurements. In SDS the molar solubilization ratios of atazanavir were 0.006 (crystal), 0.183 (measured amorphous) and 0.351 (predicted from 62 times the crystal), and Km/w fell from about 327 to about 99 as total drug rose to 100 µg/mL (2017) [indulkar2017a R9, R14]; Km/w was almost halved at the amorphous solubility for posaconazole in four intestinal media (2022) [duong2022b C45] (Amorphous solubility F18). The surface-to-core mechanism proposed for this is "speculation" in the authors' word [indulkar2017a C24].

Corrections and qualifications.

• Table 1 lists the case by drug and additive, and the library shows why it must. EUD-E, which stayed in the water with ibuprofen at pH 2.0, associated with enzalutamide droplets at pH 6.5 (2021) [wilson2021 R7, C19] (Amorphous solubility F19).
• The check records the library's statement that the ratio does not rise as true of Case I only; in the library's data it may stay constant (telaprevir with STDC) or fall (atazanavir with SDS) [lu2017b R2; indulkar2017a R13] (Amorphous solubility F12). Eq. 7 covers both.
• The library has no measurement of the β-CD case with the amorphous limit (check R24); it rests on the review.

Next. Case I leaves the maximum activity alone because nothing enters the drug-rich phase. The next case asks what happens when something does.

Pages. solubilization, micellar solubilization, surfactant partitioning into the drug-rich phase, supersaturation ratio.

Step 7. Case II, mixing into the drug-rich phase

Question. How much does a component that enters the drug-rich phase lower the limit?

The review's reasoning (2024). Distribution is set by relative chemical affinity and by the entropy of mixing, and both hydrophobic additives and hydrophilic ones with strong specific interactions (HPMC, PVPVA, HPMCAS) can enter [review pp. 34–35]. Dilution gives the ideal term, a regular-solution term adds \chi for similar molecular volumes, and Flory–Huggins handles a polymer of relative size r [review pp. 35–36]:

Inserting Eq. 2 gives the ratios (Eq. 12 is the same with \Phi_{drug} and the exponent of Eq. 10) [review p. 36]:

These assume unchanged water [review p. 36]. Crystalline solubility stays, and S_a, free drug, SRmax and maximum flux all fall, so here S_a/S_c does reflect SRmax [review pp. 36–37]. Since x_{drug} rises as more drug-rich phase forms, S_a may rise with the amount of drug added [review p. 37, Fig. 8].

What the library shows. Second drugs follow the ideal term: 10 µM lopinavir, paclitaxel or felodipine moved ritonavir from about 26 to about 14, 10 and 12 µg/mL, and five of six ritonavir–lopinavir mixed values lay within 0.5 µg/mL of C_{max}=S_a\,x_d (2015) [trasi2015a R7, R9, R11; trasi2015b R5, E1]; Caco-2 flux from 1:1 ritonavir–atazanavir solutions was halved (2016) [alhalaweh2016 R12, R13, E7, R17]; and crystalline darunavir hydrate cut the atazanavir flux plateau from about 9.5 × 10⁻⁵ to 2.9 × 10⁻⁵ µg·min⁻¹·cm⁻² (2026) [alkalla2026 R7, R9, R13] (Overview F2).

Polymers need the non-ideal terms. At 300 µg/mL HPMC and PVP-VA lowered ibuprofen from 177.1 to 96.9 and 90.7 µg/mL at unchanged crystalline solubility (2020) [ueda2020b R1] (Amorphous solubility F19). Ideal Flory–Huggins predicted about 127–129 µg/mL at a polymer:drug ratio of 1.0, and a cited \chi of about −1 brought PVP-VA to about 102 against about 90 µg/mL measured (2021) [ueda2021 R12, R13] (Amorphous solubility F20). HPMCAS-HF lowered atazanavir from 81.5 to 35.5 µg/mL while AS-LF moved it by under a tenth (2020) [ueda2020a R1] (Overview F2). Between 15 and 55 °C the polymer ratio in the phase rose for HPMC from about 0.20 to 1.46, and the ranking of the polymer solutions reversed (2022) [ueda2022 R9, R10] (Amorphous solubility F21).

Corrections and qualifications.

• The rise of S_a with drug amount depends on the polymer. With 500 µg/mL polymer dissolved ibuprofen held at about 120 (PVP K25) and 90 µg/mL (PVP-VA) up to 600 µg/mL total, and only HPMC 603 rose, from about 88 to 123 µg/mL [ueda2021 R10] (Amorphous solubility F20).
• Water was tested as an explanation for the PVP-VA effect, and uptake was lower with PVP-VA than with HPMC 603 (13.0 against 13.9% at 1/1), so water does not explain it, though the measurement was on solid ASDs [ueda2021 R17, C35] (Amorphous solubility F20).
• Not every polymer enters. Danazol onsets stayed at 7–9.1 µg/mL with 1–1000 µg/mL PVP, HPMC or HPMC-AS (2016) [jackson2016 R8] (Crystallization from solution F19), and ionized diclofenac did not lower the ritonavir limit [trasi2015a R7, R9, R11] (Overview F2).

Next. Surfactants and bile salts can do both things at once, and neither Eq. 7 nor Eq. 11 alone then applies.

Pages. polymer partitioning into the drug-rich phase, ideal mixing, Flory-Huggins interaction parameter, ideal-mixing reduction of amorphous solubility, Flory-Huggins activity equation.

Step 8. Case III, both, and the formulation trade-off

Question. What follows when an additive both complexes the drug and enters the drug-rich phase, and what does this cost a formulator?

The review's reasoning (2024). The two corrections multiply, assuming comparable molar volumes and unchanged water sorption [review p. 38]:

S_a is smaller than the crystalline enhancement predicts, may rise throughout or fall at low additive and then rise, and free drug and SRmax fall continuously, so S_a/S_c no longer reflects SRmax; S_a falls as drug-rich phase grows [review pp. 38–40, Fig. 9].

Ketoprofen shows the three cases side by side: EUD-E (I), PVP (II), CTAB (III) [review pp. 40–41, Fig. 10]. From this the review draws its formulation warning. Crystallization inhibitors may reduce the maximum activity, and there may be a trade-off, since more hydrophobic polymers that mix more with the drug-rich phase inhibit crystallization from it more effectively [review p. 41]. Drugs given together, as ritonavir and lopinavir, lower each other's limit in a ratio-dependent way, and among bile salts STC is Case I while STCDC and SGCDC lower SRmax for posaconazole but not atazanavir [review pp. 41–42].

What the library shows. Telaprevir gives the bile-salt split directly: STCDC and SGCDC lowered maximum flux to 1.06–1.11 µg/min against 1.25–1.32 in buffer, read as a fall of the maximum from δ ≈28 to ≈22, whereas STDC and SGDC left it unchanged (2017) [lu2017b R13, R6, R18] (Amorphous solubility F17). The mixing is presumed there, not measured [lu2017b C31].

The inhibition side has its own results. Crossing enzalutamide's amorphous solubility of 42 µg/mL cut the induction time with 2.5 µg/mL HPMCAS from about 129 to 43 min (2021) [wilson2021 R3], and for posaconazole the HPMCAS grade with the most polymer in the droplets (about 31 wt% for H, against about 5 for L and 14 for M) inhibited longest (2025) [shah2025 R17, R3] (Crystallization from solution F37). The same dependence reached exposure from another side: the polymer best at delaying crystallization of enzalutamide, CA Sub, gave an AUC0–inf of 45 µg·h/mL at 50:50 in rats, like the crystalline slurry (43.1), whereas CPHPC-106, which delayed it for about 0.7 h, gave 305 and 516 (2020) [wilson2020 R1, R6] (Bioperformance F24). The authors assign that result to poor polymer dissolution, not to a lowered limit.

Corrections and qualifications.

• The trade-off as a simple rule has a counterexample. Eudragit E PO loaded enzalutamide droplets most (about 41 w/w% against 13–25 w/w% HPMCAS by NMR) and inhibited worst, although a colorimetric check gave about 19–23 w/w% for both [wilson2021 R7, R9, X5] (Crystallization from solution F37). How much polymer enters does not by itself rank inhibitors.
• The review's Case I label for STC holds in the micellar biorelevant media it cites. In 3 mM STC alone, atazanavir's amorphous solubility fell from 65 to 38 µg/mL with crystalline solubility about 1–1.8 µg/mL, and lecithin raised it to about 82 µg/mL at 0.75 mM (2018) [indulkar2018 R6, R8] (Crystallization from solution F19). The case depends on the composition of the medium, and that behaviour is Case II without lecithin.
• The check finds the library has the mixing and solubilization separately but not Eqs. 13–14, and no ketoprofen data (check R46); the three-way comparison rests on the review.

Next. Since f_{com}, x_{drug} and \chi are not known for a real medium, the review concludes that the limit is easiest to measure in the medium, and that the maximum activity has to be measured as well [review p. 44].

Pages. drug-rich nanodroplets, crystallization inhibition, polymer adsorption to drug-rich droplets, bile salt hydroxylation pattern.

Step 9. Measuring the limit and the maximum activity

Question. How are the amorphous solubility and the maximum activity measured in a given medium, and what does each read-out leave out?

The review's reasoning (2024). This step needs Steps 5–8: the ratio S_a/S_c may not reflect the maximum activity, especially in Case I and III, so limit and activity are separate measurements [review p. 44]. Intrinsic dissolution and single-particle methods are excluded, because the crystalline rate is often inaccessible and a multicomponent rate does not reflect S_a [review p. 44]. More initial supersaturation and more drug-rich phase raise the crystallization risk, which favours rapid kinetic methods: titrating a stock into the medium (organic solvent ideally below 1%), lowering the temperature, or shifting the pH, with detection by UV at a nonabsorbing wavelength, fluorescence probes (pyrene, PRODAN) or light scattering [review pp. 45–47]. Turbidity alone cannot tell droplets from crystals, the extinction coefficient may change once the phase is present, and predissolved polymer may change S_a in Case II and III, which is also the value relevant to a comparably formulated system [review p. 46].

Static methods work for slow crystallizers, but crystallization is their greatest risk [review pp. 47–48, 50–51]. Centrifugation may not separate droplets below several tens of nm, even in an ultracentrifuge, and a low-density phase forms an upper layer [review p. 49]. In solution NMR the dissolved-drug peak area rises linearly to S_a, then stays constant for an additive-free solution, slows in Case II and falls in Case III [review p. 49]. Activity follows from S_a only additive-free or in Case II, so otherwise it needs flux across a membrane that excludes droplets and complex and limits drug transport, with controls at known supersaturation, or PGSE-NMR for the free fraction [review pp. 51–52].

What the library shows. Where nothing interferes the methods agree: ritonavir in water at 37 °C gave 37.2 (UV), 39.8 (light scattering), 39.0 (pyrene) and 36.0 µg/mL (supernatant) [ilevbare2013a R8, R4, R6], and telaprevir in 50 mM pH 6.5 buffer 146, 149 and 148 µg/mL by extinction, ultracentrifuge and flux (2017) [lu2017b R2.a, R4], although extinction failed in STCDC and SGCDC [lu2017b R4] (Amorphous solubility F4). The review's warning on turbidity has measured cases: the extinction rise marked crystals for nifedipine at 25 and 37 °C and nimodipine at 37 °C, and the authors assigned droplets only where there was haze with no visible particles and no birefringence (2015) [raina2015a R4, C9, P7] (Solution methods F3).

Read-outs diverge most from dispersions. Danazol 90:10 with HPMCAS gave about 16.2 µg/mL by UV against 8.2 ± 1.2 µg/mL by flux (2015) [jackson2015 R2, R9] (Supersaturation from ASD F8). A 10% lopinavir dispersion read about 466 µM by UV against 36 µM in the supernatant (2018) [tres2018 R8, R16, R9, R17], and tacrolimus droplets of about 220 nm were retained at 0.22 µm and passed at 0.7 µm (2016) [trasi2016 R8, R11] (Solution methods F11). The NMR read-out was shown for ibuprofen at pH 2.0 in D₂O at 37 °C: dissolved peaks at 7.29 and 7.25 ppm, drug-rich peaks at 6.95 and 6.77 ppm, 154 against a calculated 151 µg/mL (2021) [ueda2021 R3, R5, R6]; nifedipine (Tg 42 °C) and glibenclamide (59 °C) showed only dissolved drug [ueda2020b R16] (Amorphous solubility F15). Predissolved polymer did not move the onset for five dihydropyridines (largest difference 1.5 µg/mL) [raina2015a R5], while 10 µg/mL polymer lowered danazol from 13 to 11, 9 and 8 µg/mL [jackson2014 R5, R8] (Solution methods F7).

Corrections and qualifications.

• Flux reads activity only where the membrane is the right resistance for the question. In a lipid-impregnated PVDF cell the authors read atazanavir (log P about 4.5) as limited by the unstirred water layer, with filter and lipid changing mass flow at most 1.7-fold at 80 µg/mL (2022) [hate2022 C42, X2, R5] (Bioperformance F21); that reading is an inference, since layer thickness was not varied [hate2022 C32].
• The check says the library has no PGSE-NMR (check R62). It has PGSE for micelle radius, not for free drug: atazanavir in 40 mg/mL SDS in D₂O at 25 °C, 4.61 to 6.08 nm [indulkar2017a P12, R25] (Amorphous solubility F18). The free-fraction use remains untested here.
• The library is silent on shake-flask validity, melting in water, and the ultracentrifuge size limit (check R58, R59), and does not use the slope above S_a to assign a case (check R60).
• The review's exclusion of intrinsic dissolution has a library contrast: itraconazole's 51 against 2.8 µg/min/cm² was equated with a solubility ratio of about 18 that was not measured (2018) [bhardwaj2018 R4, R5, C45] (Release mechanism F1).

Next. With the limit and the activity measured in the medium, these values become the reference for what a formulation releases, which is the next step.

Pages. amorphous solubility, free drug concentration, membrane flux, intrinsic dissolution rate, light scattering by particles, membrane flux in terms of solute activity.

Step 10. Release under nonsink conditions

Question. How does S_a set the conditions of a release test, and what can such a test say about absorption?

The review's reasoning (2024). This step needs the measured values of Step 9. Dissolved ASDs can give colloids, particularly at low drug loading, though not all do at higher loading, and these particles may pass centrifugation and filtration, so such methods and in-situ UV can be ambiguous about dissolved drug and the amount available to permeate [review p. 53]. Their size and structure may affect absorption through particle drifting across the aqueous boundary layer [review pp. 15–17, 53]. Because precipitation is poorly characterized, differences in dissolution profiles do not necessarily correlate with absorption, and sink tests do not discriminate crystallization [review p. 54]. The review therefore writes the sink index for both forms:

SI_{cr}\geq 3 is sink and SI_{cr}<3 nonsink [review p. 54]. At SI_{am}=1 the dose confers the maximum activity with all amorphous material in theory molecularly dissolved; at SI_{am}<1 a colloidal phase and concentrations above S_a are possible, with a risk of crystallization and coarsening, so discriminatory analytics are desirable [review pp. 54–55]. In the review's interpretation the release driving force of many ASDs relates to the polymer, so the index may not represent it, but it does bear on the driving force for crystallization [review p. 55].

What the library shows. Colloid formation is tied to release. Ritonavir–PVPVA tablets of 10–25% passed the amorphous solubility within 5–10 min (218 ± 15 nm droplets at 10%) and those of 35–50% never did (2019) [indulkar2019a R9, R20]; since 2021 partly incongruent release of low-Tg drugs has also passed the limit, for loratadine–PVPVA at 35% [yang2021 R19, R16, C14, C15] (Release mechanism F4). The polymer-control reading has direct tests: nifedipine dispersions kept dissolving into water already at their amorphous solubility of 110 µg/mL, reaching about 214–218 µg/mL (5 °C, D₂O) [indulkar2017b R8, R9] (Release mechanism F1).

The sink index has been applied to one capsule. For 5 mg tacrolimus, indices (crystalline; amorphous) were 0.32 and 9 at 900 mL, 0.04 and 1 at 100 mL, and 0.01 and 0.4 at 40 mL [purohit2018 R4]. In 100 mL peak release fell from about 76.5% (10% crystalline) to 10% (100%) against about 98% fresh [purohit2018 R7] (Bioperformance F16). With an absorptive sink the fresh generic gave receiver AUC0–120 of about 362, 682 and 992 µg/mL·min at 240, 100 and 40 mL, and a 20% crystalline sample lost dissolved drug only at 40 mL (48 to 9 µg/mL) [hate2019 R12, R10] (Bioperformance F28).

Transfer above the limit was measured with sink receivers. In a flow-through cell, clotrimazole donors above the amorphous solubility (7.5 µg/mL) held the peak receiver signal for about 1.5 h (10 µg/mL), 5 h (15) and at least 13 h (20) (2016) [indulkar2016 R5, R6, R7] (Bioperformance F5). In a hollow-fibre module the atazanavir AUC slope was 5.51 below S_a (78 µg/mL) and 1.29 above it, with the count of 204–226 nm droplets rising from 2.3 × 10⁹ to 1.06 × 10¹⁰ per mL between 75 and 150 µg/mL (2018) [hate2018 R9, E3, E4, R6, R7]; free drug was not measured and three explanations stayed untested [hate2018 X11, C56, C57, C59] (Bioperformance F15). Against exposure, mass flow across a dialysis membrane ranked five enzalutamide formulations (0.632 to 0.063 µg/min) apart from one crystallizing ASD, while closed dissolution AUC changed order with the integration window (2018) [wilson2018 R15, R17, R7] (Bioperformance F25).

Corrections and qualifications.

• Size "of paramount importance" (Step 4) is not what the library's direct comparison found. At 150 µg/mL atazanavir, 1000 nm CTAB agglomerates gave the same mass flow as 283 nm HPMCAS droplets, and the authors rejected size-dependent shuttling [hate2022 R17, C37, C39, C40] (Bioperformance F22). The interface concentration there is rescaled from mass flow, not measured [hate2022 X7, X8], and the rat evidence for a colloid benefit (about 30% more AUC) has no test and is confounded with dissolution rate [wilson2018 C46, X6, X14] (Bioperformance F23).
• "Highest driving force" at SI_{am}=1 holds for activity, not for transferred mass (check R66; Step 5). The plateau appears in a side-by-side cell read with a buffer receiver and even there atazanavir kept rising, from about 1.3–2.2 to 3.2–3.7 µg/min [hate2022 X12, R5] (Bioperformance F21).
• The crystallization risk is not confined to SI_{am}<1. In a closed 100 mL vessel (SI_{am}=1) the fresh generic fell from about 98% to about 39.5% at 120 min [purohit2018 R7, R9], and did not crystallize in the absorptive test at any volume [hate2019 C35, R5]; no run without the membrane at the same volume separates absorption from the polymer effect [hate2019 X15] (Bioperformance F28).

Next. The sink index describes a test vessel. The design question is whether the same ratio, with a physiological volume, predicts that absorption is no longer limited by the dose.

Pages. non-sink dissolution, sink conditions, sink index, congruent release, drug-rich nanodroplets, nanodroplet reservoir effect, unstirred water layer, absorptive sink, sink index, resistances-in-series model.

Step 11. Design target: dose number and absorption modelling [adapted: no absorption simulation printed]

Question. At which dose and volume does the amorphous form remove the dose limit, and do absorption models with S_a as input support this?

The review's reasoning (2024). The review prints no simulation of its own; its design target is a dose number written for both forms, with cited models as the support. This step needs Steps 8–10:

S is generally S_c, ideally biorelevant [review p. 55]. By a cited rule, a value above 10 means enabling formulations might be needed, and below 10 conventional ones may suffice [review p. 55]. The worked cases assume no crystallization: tacrolimus at 1.8 µg/mL crystal and 50 µg/mL amorphous, 5 mg in 250 mL, gives 11 and 0.4; ritonavir in fasted-state intestinal fluid at 100 mg gives ~60 and ~7 [review pp. 55–56]. Formulations that "achieve and maintain" S_a should no longer be limited by dose number, with Prograf and Norvir as examples [review p. 56].

The cited physiologically based models carry the same statement, and absorption kinetics are linked to the maximum flux at S_a [review p. 56]. Mitra predicted linear pharmacokinetics with S_a, whereas crystalline solubility under-predicted AUC and Cmax above 50 mg; venetoclax was modelled with S_a and slow crystallization; belinostat with three polymers (likely Case II) agreed well in dogs, with dissolution rate also likely critical; encorafenib with S_a predicted no significant proton-pump inhibitor effect, matching clinical data [review pp. 56–57]. Posaconazole was accurate with molecularly dissolved input and over-predicted with total drug, and the review expects S_a input to become routine [review p. 57]. These are the review's citations, not its calculations and not library measurements.

What the library shows. Tested in vivo. Reaching the limit raised exposure: in rats enzalutamide ASDs reaching 42 µg/mL gave AUC0–48 of about 770 and 575 µg·h/mL against about 91 for crystalline drug, and the 10% HPMCAS ASD that crystallized gave about 61 (2018) [wilson2018 R1, R17, R18] (Bioperformance F23). Having the polymer that inhibits best was not sufficient: the CA Sub ASD gave 45 against 43.1 µg·h/mL crystalline, and CPHPC-106 ASDs 305 and 516 (2020) [wilson2020 R1, R6] (Bioperformance F24). Six marketed tacrolimus capsules gave free drug of about 46–50 µg/mL in vitro [trasi2016 R10, R4, R15, R14] (Overview F1), so the "achieve" half of the target has been measured for that product class. On "maintain", beagles at 1 mg gave AUC0–24 of 42.1 ng·h/mL fresh and 21.6 fully crystalline, yet 5.29 for reference crystals (2019) [purohit2019 R9] (Bioperformance F30); crystallization cost less in vivo here than in the low-volume tests, about 51% against 17% at 100 mL in the absorptive test (a cross-paper comparison of ratios) [purohit2019 R9; hate2019 R13] (Bioperformance F31). The dog dose in about 95 mL gives at most about 10 µg/mL [purohit2019 R12, C40], which with Eq. 17 and 50 µg/mL is a dose number of about 0.2 and with 2.5 µg/mL about 4 (synthesis), a regime the review's rule would call conventional.

Modelled. The only simulation in the library is GastroPlus with tacrolimus dissolution as input and no precipitation assumed: virtual trials of 24 subjects failed from 20% crystallinity with 100 mL input (AUC about 70%) and from 40% with 900 mL input (about 69%) against Prograf, with an over-prediction of about 1.4-fold [purohit2018 R13, R15, R17, R11, X3] (Bioperformance F17). A second model sets absorption, proportional to supersaturation, against growth on seeds of order about 1.7, crossing near S 18.5 [hate2019 R18, R19] (Bioperformance F29); it bears on "maintain", which the worked dose numbers set aside.

Untested in the library. The threshold of 10, the ritonavir values, and the review's PBBM cases: none is held here (check R68, R70), and S_a as model input has no library test against the crystalline input. Aggregates were not tracked in vivo, and their redissolution rate was not measured [wilson2018 C40, C48].

Corrections and qualifications.

• The tacrolimus crystalline dose number depends on the value chosen (check R69). 5 mg in 250 mL is 20 µg/mL, so 1.4 µg/mL (pH 4.5, 37 °C) [trasi2016 R16, R17, X9] (amorphous solubility advantage) gives about 14, 1.8 µg/mL (medium I, with 50 µg/mL HPC) [purohit2018 E1, R4] (Bioperformance F16) about 11, and 2.5 µg/mL (pH 4.5 with HPC) [hate2019 E1, M1] (Bioperformance F28) 8. The spread between closely similar media crosses the threshold, so 10 cannot be read sharply for this drug. The amorphous value of 0.4 is unaffected.
• A single Vo of 250 mL is a convention: fasted human small-intestinal fluid is given as 30–420 mL [hate2019 R21] (Bioperformance F31), and the seed effect in tacrolimus appeared only at the lowest volume (Step 10).
• The direct relation of supersaturation to expected absorption rests on a model whose rate units, growth constant and absorption line cannot be reproduced from the printed values, and whose threshold moves with the seed amount [hate2019 X1, X2, X3, X4, X16] (Bioperformance F29). Its qualitative form (absorption linear in supersaturation, growth faster than linear) applies, not the value of 18.5.
• A dose number with S_c does not describe crystals formed from the formulation: the enzalutamide ASD that crystallized gave less than the crystals, the tacrolimus capsule four times the reference crystals [wilson2018 R17; purohit2019 R9, C52] (Bioperformance F30).

Next. This closes the chain: measure S_a and the maximum activity in the relevant medium, then design to reach and maintain them at the dose and volume.

Pages. dissolution-rate-limited absorption, oral bioavailability, virtual bioequivalence, in vitro–in vivo correlation, residual crystallinity, crystal seeding, power-law crystal growth rate, hollow-fibre donor depletion.

Open questions along the chain

• Does a glass below its wet Tg set the same limit as a liquid? — Step 2 — the near-equivalence rests on inference without isolating the phase (Amorphous solubility F2).
• Are the pH-dependent ratios in buffer real or a crystallization artefact? — Step 3 — Eq. 3 with a constant ionized fraction implies a ratio independent of pH, measured ratios move, and the library cannot separate the two (Amorphous solubility F10).
• What is the composition of the water-containing drug-rich phase? — Steps 7–8 — mixing is inferred from the fall in the onset, not measured with the water present (Amorphous solubility F17, F20).
• Is there a rule for the trade-off between inhibition and SR_{max}? — Step 8 — the polymer with most in the droplets is not always the best inhibitor, and the best inhibitor may not release (Crystallization from solution F37, Bioperformance F24).
• Which case applies to a real mixed medium? — Steps 6 and 8 — one bile salt behaves as Case II alone and differently with lecithin (Crystallization from solution F19).
• Can PGSE-NMR give the free fraction for formulated systems? — Step 9 — the library has the method only for micelle radius (Amorphous solubility F18).
• What transfers drug above S_a, and do droplet size and number matter in vivo? — Steps 4 and 10 — interface free drug was not measured, one PAMPA comparison found no size effect, and in vivo relevance "needs further investigation" [hate2018 C67; hate2022 R17] (Bioperformance F15, F22).
• How long is S_a maintained in vivo, and at which volume? — Step 11 — the dog result fits the large-volume conditions on a cross-paper comparison with weak strength (Bioperformance F31).
• Does S_a as model input hold beyond the cited cases? — Step 11 — the library's one model assumes no precipitation and has its human study pending [purohit2018 C58, C59] (Bioperformance F17, check R70).