Following Ueda K., Moseson D.E., Taylor L.S. (2024) Amorphous Solubility Advantage: Theoretical Considerations, Experimental Methods, and Contemporary Relevance. Journal of Pharmaceutical Sciences.
Equations and their derivation: the amorphous solubility advantage
How to read this page. Each section takes one equation of Ueda, Moseson and Taylor (2024, J. Pharm. Sci., pre-proof) as printed, with the review page ([review p. N] is the PDF page, one above the printed "Page N of 70"), and gives the algebra that connects it to the equations before it. The review derives almost nothing line by line, so intermediate lines are marked (added), and numbers from the library and not from the review are marked with their source. Sections W1–W14 treat quantitative relations that the review states only in words or in a figure. See also the check and the storyline.
The derivation map
Eq. 1 — S_a^0/S_c^0 from the crystal-to-amorphous free energy — from outside (equal chemical potential at saturation; refs 20, 79) + \gamma equal at both solubilities
Eq. 2 — the same ratio with the water-saturated amorphous drug — from Eq. 1 + Gibbs–Duhem on the water sorption isotherm (ref 27)
Eq. 3 — the same ratio for an ionizing drug without buffer — from Eq. 2 + total solubility = un-ionized part / (1-\bar\alpha) (ref 27)
Eq. 4 — supersaturation ratio as an activity ratio — definition (ref 85), from \mu=\mu^\circ+RT\ln a
Eq. 5 — Case I: S_a as free plus complexed drug, free part fixed at S_a^0 — mass balance + the additive stays out of the drug-rich phase
Eq. 6 — Case I: S_a from S_a^0 and the complexed fraction — from Eq. 5 + f_{free}+f_{com}=1
Eq. 7 — Case I: S_a/S_c — from Eq. 6, the same balance at the crystal (added line), and Eq. 2
Eq. 8 — Case II, ideal mixing: S_a from the drug mole fraction — from equal chemical potential + a_{drug}=x_{drug} (ref 90)
Eq. 9 — Case II, regular solution: adds \chi — from Bragg–Williams (ref 99); reduces to Eq. 8 at \chi=0
Eq. 10 — Case II, polymer: volume fraction and molar volume ratio — from Flory–Huggins (ref 100); reduces to Eq. 9 at r=1
Eq. 11 — Case II: S_a/S_c with \chi — from Eq. 9 and Eq. 2 + S_c=S_c^0
Eq. 12 — Case II: S_a/S_c for a polymer — from Eq. 10 and Eq. 2 + S_c=S_c^0
Eq. 13 — Case III: S_a — from Eq. 6 and Eq. 9 combined
Eq. 14 — Case III: S_a/S_c — from Eq. 13, the crystal balance of Eq. 7, and Eq. 2
Eq. 15 — sink index against the crystalline solubility — definition (ref 125)
Eq. 16 — sink index against the amorphous solubility — definition, same form as Eq. 15
Eq. 17 — dose number — definition (ref 131); the reciprocal of Eq. 15 or 16 at V=Vo (added)
Relations stated in words or in a figure:
W1 — \Delta G_{c\to a} at temperature T from T_m, melting enthalpy and \Delta C_p (p. 26) — from outside (refs 1, 31, 81–83); three approximations
W2 — \Delta G_{c\to a}\lesssim 10 kJ/mol, ratio mostly below 100-fold, predicted ranges 12–1652 and 1.1–3.6 (pp. 8, 27) — from Eq. 1
W3 — ratio falls with temperature and reaches 1 at the melting point in water; glass above the supercooled liquid (pp. 21–24, Fig. 5) — from Eq. 2 + Gibbs–Helmholtz
W4 — solubility ratio equals dissolution rate ratio (pp. 12–13), and its limit (p. 44) — from Noyes–Whitney (not named in the review)
W5 — amorphous solubility follows Henderson–Hasselbalch in buffer (p. 17) — from outside (ref 18); consistent with Eq. 3 at fixed \bar\alpha
W6 — flux proportional to activity; amorphous-to-crystalline flux ratio reads SR_{max} (pp. 17–19, 51–52, Figs. 4, 13a) — from outside (refs 50, 59); with Eq. 4
W7 — how S_c, S_a, free drug and SR_{max} move in each case (pp. 30–40, Figs. 6, 7) — from Eqs. 5–14
W8 — dissolved drug against total drug and the amount of drug-rich phase (pp. 37, 39–40, 49, Fig. 8) — from Eqs. 8–10, 13 + mass balance (added)
W9 — S_a first falling and then rising with additive in Case III (pp. 38–39, Fig. 7c) — from Eq. 13 + additive mass balance (added)
W10 — D_{obs} in PGSE-NMR (pp. 51–52, Fig. 13b) — from outside (ref 118), fast exchange
W11 — drug-rich droplets and the aqueous boundary layer (pp. 15–17, 53) — no equation in the review; film model (added)
Symbols: S_a^0, amorphous solubility without additive; S_c^0, crystalline solubility without additive (same concentration unit, µg/mL in practice); \Delta G_{c\to a}, Gibbs free energy difference between crystalline and amorphous drug (J/mol, positive for the amorphous form above the crystal); R, gas constant (8.314 J mol⁻¹ K⁻¹); T, temperature (K).
Where it comes from: the review gives the equation with refs 20 and 79 (Parks and co-workers, glucose, 1934 and 1928). Steps (added):
At saturation the solid and the dissolved drug have the same chemical potential: \mu_i^{solid}=\mu^{\circ}+RT\ln a_i, for i = amorphous or crystal, with the same solution standard state \mu^{\circ}.
Subtracting the two: \mu_a^{solid}-\mu_c^{solid}=\Delta G_{c\to a}=RT\ln(a_a/a_c).
With a=\gamma c: a_a/a_c=\gamma_aS_a^0/(\gamma_cS_c^0).
With \gamma_a=\gamma_c: \ln(S_a^0/S_c^0)=\Delta G_{c\to a}/RT, which is Eq. 1.
Assumptions and valid range: stated: unionized drug, aqueous solution without additives, both forms single-component [review pp. 25–26]. The equation also serves for a metastable polymorph with the free energy difference between the polymorphs (refs 77, 80). Needed and not stated at this point: \gamma_a=\gamma_c (step 4), which the review states only later for Eq. 4 [review p. 28]; and a value of \Delta G_{c\to a} for one amorphous state, supercooled liquid or glass (W3).
What it means for a formulation: the ratio rises exponentially with free energy, so each RT\ln 10=5.71 kJ/mol at 25 °C (5.94 kJ/mol at 37 °C) is one factor of ten (added). The review prints no worked value here; W2 applies it to its ~10 kJ/mol limit.
In the wiki: advantage from free energy difference. The wiki holds the same relation in chemical potentials (S_A/S_C=\exp[(\mu_A-\mu_C)/RT], li2018), in mole fractions with \Delta G_{a\to c} (ilevbare2013a) and with a negative sign (\Delta G=-RT\ln(C_a/C_s), duong2022a), so the sign depends on the direction written. The polymorph use is not on that page. Used in Amorphous solubility F6.
Symbols: as Eq. 1; e^{-I(a_2)}, activity of the amorphous drug saturated with water, referred to the dry amorphous drug (dimensionless, ≤ 1); I(a_2), the integral below (dimensionless). The review prints the factor without defining I.
Where it comes from: Eq. 1 applies to a one-component amorphous form; a real amorphous drug absorbs water up to its miscibility limit, which lowers its chemical potential [review p. 26]. The review names Gibbs–Duhem and cites ref 27 (Murdande 2010). Steps (added):
In the wet amorphous phase, \mu_a^{wet}=\mu_a^{dry}+RT\ln a_2, with a_2 the drug activity on the dry-drug scale (index 1 = water, 2 = drug).
Replacing \mu_a^{solid} in step 2 of Eq. 1 by \mu_a^{wet}: RT\ln(S_a^0/S_c^0)=\Delta G_{c\to a}+RT\ln a_2. The crystal term is unchanged because a non-hydrate crystal takes up negligible water [review p. 21].
So S_a^0/S_c^0=a_2\,e^{\Delta G_{c\to a}/RT}, and Eq. 2 is this line with a_2\equiv e^{-I(a_2)}.
Gibbs–Duhem at constant T and P: n_1\,\mathrm{d}\ln a_1+n_2\,\mathrm{d}\ln a_2=0, so \mathrm{d}\ln a_2=-(n_1/n_2)\,\mathrm{d}\ln a_1.
Integrating from dry solid (a_1\to 0, a_2=1) to water saturation (a_1=a_1^S\approx 1): I(a_2)=-\ln a_2=\int_0^{a_1^S}(n_1/n_2)\,\mathrm{d}\ln a_1.
Where the isotherm is linear, n_1/n_2=ka_1, so \int_0^{a_1^H}ka_1\,\mathrm{d}a_1/a_1=(n_1/n_2)_H. This is the first term of the form in the wiki. For an isotherm that curves upward, I(a_2) is below the water content at saturation expressed as n_1/n_2.
Assumptions and valid range: stated: unionized drug, no additives; the crystal takes up negligible water unless it forms a hydrate [review pp. 21, 27]. Needed and not stated: \gamma equal at both solubilities; a sorption isotherm that reaches equilibrium and can be extrapolated to water activity near 1 (library measurements stop at 95% RH); and a hydrate crystal needs its own \Delta G, because S_c^0 then belongs to the hydrate.
What it means for a formulation: drugs that take up little water (ibuprofen, tolnaftate) give nearly the same value from Eqs. 1 and 2; for telaprevir and griseofulvin Eq. 2 is 30–70% lower [review pp. 26–27]. The range is reproduced by the library's activities (added): griseofulvin e^{-I}=0.699 is 1 − 0.699 = 30% (I=0.358), telaprevir 0.30 is 70% (I=1.20). Worked for griseofulvin at 25 °C with library inputs (added): \Delta G=9.39 kJ/mol gives Eq. 1 = exp(9390/2478.8) = 44.2, and Eq. 2 = 44.2 × 0.699 = 30.9. Atazanavir at 0.22 (I=1.51) lies beyond the review's upper figure, at 78%. Ibuprofen at 25 °C, as a closure test: 29.29 × 0.999 × exp(3240/2478.8) = 108.1 µg/mL against a printed 108.16 [almeidaesousa2015].
In the wiki: advantage with water-sorption correction, identical in the form of Ueda 2022; other papers print \exp[-\Delta G/RT] with \Delta G taken as negative or with an unstated sign (indulkar2015, indulkar2017a, duong2022b), and the wiki records that several printed inputs do not reproduce. The integral is on Gibbs–Duhem equation, with separate Henry and high-water parts (hsieh2012). The concept page water sorption correction tabulates e^{-I}. Tested in Amorphous solubility F5 and F7.
Symbols: as Eq. 2, with S_a^0 and S_c^0 now total solubilities (ionized plus un-ionized); \bar\alpha^a, \bar\alpha^c, ionized fractions of the drug at the amorphous and crystalline solubilities (dimensionless).
Where it comes from: cited from ref 27; the review states the reason in words, that the ionized fraction falls as drug concentration rises. Steps (added):
Only the un-ionized drug is in equilibrium with the solid, so Eq. 2 holds for the un-ionized parts S_{a,u} and S_{c,u}.
Total = un-ionized /(1-\bar\alpha), so S_a^0=S_{a,u}/(1-\bar\alpha^a) and S_c^0=S_{c,u}/(1-\bar\alpha^c).
Dividing gives Eq. 3. The form is the same as Eq. 7, with ionization in place of complexation.
Why \bar\alpha falls without a buffer. For a weak acid HA with no other proton source, [\mathrm{H}^+]=[\mathrm{A}^-], so [\mathrm{A}^-]=\sqrt{K_aS_u} and \bar\alpha=[\mathrm{A}^-]/(S_u+[\mathrm{A}^-]). Then \bar\alpha^a<\bar\alpha^c whenever S_{a,u}>S_{c,u}.
With a buffer, \bar\alpha=K_a/(K_a+[\mathrm{H}^+]) does not depend on concentration, so \bar\alpha^a=\bar\alpha^c and Eq. 3 returns Eq. 2 at every pH. This is the content of "only relevant for unbuffered conditions".
Assumptions and valid range: stated: unbuffered conditions; Eq. 3 becomes Eq. 2 if the change in ionized fraction is minimal [review p. 27]. Needed and not stated: the ionized species do not associate with the solid or the drug-rich phase; no salt limits the crystalline solubility.
What it means for a formulation: the review gives no number. A hypothetical case (added): K_a=10^{-5} M, S_{c,u}=10^{-5} M and un-ionized ratio 10. Then [\mathrm{A}^-]_c=10^{-5} M, \bar\alpha^c=0.50; [\mathrm{A}^-]_a=3.16\times10^{-5} M, \bar\alpha^a=0.24. The prefactor is 0.50/0.76 = 0.658, so the total ratio is 6.58 and not 10, and direct totals give 1.316\times10^{-4}/2\times10^{-5}=6.58. For formulation media, which are buffered, step 5 applies, and the additive sections assume this [review p. 29].
In the wiki: the check (R35) calls Eq. 3 absent. It is present once, as a general form printed by raina2014a on the water-sorption formula page with (\mu_2^a-\mu_2^x)/RT in the exponent, and with the term set to 1 at pH 6.8 (ionization); it is never evaluated. Step 5 gives the pH-independent ratio that the wiki reports at about 13 and 63 for clotrimazole and atazanavir, and it does not account for the moving ratios of clotrimazole and pazopanib in buffer (Amorphous solubility F10, supersaturation ratio). The review gives no term that would explain those.
Symbols: SR, supersaturation ratio (dimensionless); a, solute activity; \gamma, solute activity coefficient; c, solute concentration (µg/mL); the asterisk marks the same property in a saturated solution in equilibrium with the drug crystal. SR_{max} is SR at the highest attainable activity, in equilibrium with the drug-rich phase (Fig. 6).
Where it comes from: stated with ref 85 (Mullin). Steps (added):
\mu=\mu^\circ+RT\ln a and \mu^*=\mu^\circ+RT\ln a^*, so \Delta\mu=RT\ln(a/a^*)=RT\ln SR: SR is the chemical potential excess in exponential form.
a=\gamma c gives the second equality.
Consequence for an additive-free solution: with \gamma/\gamma^*\sim 1 and c=S_a^0, c^*=S_c^0, SR_{max}=S_a^0/S_c^0, which is why Eqs. 1–3 are also statements about SR_{max} [review p. 28].
Consequence for a complexing additive: with c total drug and \gamma the apparent coefficient on the total, activity is fixed by the free drug, \gamma c=\gamma_{free}c_{free}=\gamma_{free}f_{free}c. Hence \gamma=\gamma_{free}f_{free}, and if \gamma_{free} is constant, \gamma/\gamma^*=f^a_{free}/f^c_{free}. This line explains the review's sentence that \gamma/\gamma^*\sim 1 "is not correct" with a complexing agent [review p. 33], and it joins Eq. 4 to Eq. 7.
Assumptions and valid range: stated: \gamma/\gamma^*\sim 1 is reasonable for dilute additive-free solution and may fail with micelles and solubilizers [review p. 28]. Needed: both states at the same temperature and on the same crystal reference form; if the crystal form changes, a^* changes.
What it means for a formulation: the review relates supersaturation directly to the expected improvement of solubility-limited passive absorption [review p. 28]. From Fig. 4 and step 4 (added, values read from the plot): \gamma/\gamma^* = flux ratio / concentration ratio is 54.8/56.5 = 0.97 in Buffer-V1 but 52.6/14 = 3.8 in FaHIF. A concentration ratio of 14 therefore reads a driving force of about 53.
In the wiki: activity-based supersaturation ratio, same form [indulkar2017a E9]; written also as \ln\delta and \ln S with the same meaning. Bioperformance F29 rests the proportionality of absorption to supersaturation on one model.
Symbols: S_a, apparent amorphous solubility with additive (total dissolved drug, µg/mL); f^a_{free}, f^a_{com}, fractions of free and complexed drug when total drug is at the amorphous solubility (dimensionless). "Complex" is an umbrella term for any additive–drug species that solubilizes drug, including micelles and cyclodextrin complexes.
Where it comes from: mass balance plus one thermodynamic step. The first two equalities are the definition of the fractions. The third uses S_af^a_{free}=S_a^0. Steps for that one (added):
If the additive shows minimal partitioning into the drug-rich phase, the drug chemical potential in that phase is the same as without additive [review p. 31].
At equilibrium, free drug has the same chemical potential as the drug-rich phase.
With \gamma_{free} unchanged by the additive, free drug is at the additive-free value, S_a^0.
Assumptions and valid range: stated: minimal partitioning of the additive into the drug-rich phase; a water-soluble complex [review pp. 31, 33]; independent of stoichiometry or binding model [review p. 32]. Needed and not stated: the additive does not change \gamma_{free} in the water. A cosolvent that does so does not fit the free-plus-complex picture.
What it means for a formulation: total concentration above S_a^0 does not mean phase separation; LLPS waits until the free drug reaches S_a^0. Eq. 6 gives the number.
Where it comes from: Eq. 5 rearranged (added): S_a-S_af^a_{com}=S_a^0, so S_a(1-f^a_{com})=S_a^0; the same as S_a=S_a^0/f^a_{free}.
Assumptions and valid range: as Eq. 5. f^a_{com} is an output of a binding or partition model at the drug concentration S_a, not a constant of the additive.
What it means for a formulation: the review gives no number. Library example (added): atazanavir in 3.6 mg/mL SDS, slope of free against total drug 0.047, so f^a_{free}=0.047; with S_a^0 = 65–70 µg/mL, Eq. 6 gives 65/0.047 = 1383 to 70/0.047 = 1489 µg/mL against about 1450 measured [indulkar2017a R11]. So 95% of the drug at this limit is in micelles and does not add driving force.
Symbols: as Eqs. 2 and 5; S_c, apparent crystalline solubility with additive; f^c_{free}, f^c_{com}, fractions at the crystalline solubility. The review prints f^{C}_{free} with a capital C once at this point, a typographic slip for f^c_{free}.
Where it comes from: Steps (added; the review prints only the result):
At the crystal the free drug is at S_c^0 [review p. 32], so by the argument of Eqs. 5–6, S_c=S_c^0/(1-f^c_{com}).
Dividing Eq. 6 by this: S_a/S_c=[(1-f^c_{com})/(1-f^a_{com})]\,(S_a^0/S_c^0).
Inserting Eq. 2 gives the second equality.
Why f_{com} generally falls with drug concentration. With a fixed additive total A_t and 1:1 binding, [\mathrm{DA}]=K[\mathrm{D}]A_t/(1+K[\mathrm{D}]), so f_{com}=KA_t/(1+K[\mathrm{D}]+KA_t), which decreases as free drug [\mathrm{D}] rises. Since [\mathrm{D}] is S_a^0 at one limit and S_c^0 at the other, f^a_{com}<f^c_{com} and the prefactor is below 1. This is the review's statement in algebra, that binding "becomes limited by the amount of the additive" [review p. 32].
When is the prefactor 1? A constant partition, [\mathrm{complex}]=K_mA[\mathrm{free}], gives f_{com}=K_mA/(1+K_mA), independent of drug. Then Eq. 7 returns Eq. 2.
From steps 2 and 4 of Eq. 4: SR_{max}=(S_a/S_c)(\gamma/\gamma^*)=(S_a/S_c)\,(1-f^a_{com})/(1-f^c_{com})=S_a^0/S_c^0. So SR_{max} stays at the additive-free value (SR^0_{max}) [review p. 33].
Assumptions and valid range: as Eq. 5; the same crystal form with and without additive. The review separates the case of a poorly soluble complex: ibuprofen/β-CD shows a B_S-type diagram, S_c plateaus because the complex crystal limits it, S_a keeps rising and S_a/S_c becomes much larger while SR_{max} stays constant [review p. 33]. Step 1 does not hold there, since a new solid sets the limit at the crystal side, and the prefactor can exceed 1. "SR_{max} larger than S_a/S_c" therefore holds only in the general soluble-complex case.
What it means for a formulation: a concentration ratio under a solubilizer does not measure driving force for crystallization or transport [review p. 33]. Library examples (added):
Atazanavir in 3.6 mg/mL SDS: S_c 45.6, S_a about 1450, S_c^0 1.1, S_a^0 70 µg/mL. 1-f^c_{com} = 1.1/45.6 = 0.0241, 1-f^a_{com} = 70/1450 = 0.0483, prefactor 0.50. Then 0.50 × 63.6 = 31.8 = 1450/45.6, against a flux-based ratio of about 65 [indulkar2017a R13]. f_{com} only moves from 0.976 to 0.952, yet the prefactor is one half, because the free fractions matter.
Ibuprofen with EUD-E at pH 2.0: S_c 29.8 → 42.1, S_a 177.1 → 256.3 µg/mL. f^c_{com} = 0.292, f^a_{com} = 0.309, prefactor 1.02, ratio 5.94 → 6.09, which is the review's constant-f_{com} example [ueda2020b R1].
In the wiki: not in the wiki as an equation. Outcomes appear as examples in Amorphous solubility F12 (ratio constant or falling), F16, F18, F19, and Bioperformance F18. The title of F12 ("the supersaturation ratio does not rise") holds for Case I. The B_S case is not in the wiki.
Eq. 8. Case II: ideal mixing in the drug-rich phase
S_a=S_a^0\cdot x_{\mathrm{drug}}\tag{8}
[review p. 35]
Symbols: x_{\mathrm{drug}}, mole fraction of drug in the drug-rich phase when the additive is present (dimensionless); others as Eq. 5. Here S_a is free drug, since no complex forms.
Where it comes from: cited to ref 90 (Trasi and Taylor 2015). Steps (added):
The aqueous free drug has the activity of the drug in the drug-rich phase: a_{aq}=a_{rich}.
With \gamma in the water unchanged, S_a/S_a^0=a_{rich}/a^0_{rich}, where a^0_{rich} refers to the phase without additive.
Assumptions and valid range: stated: ideal mixing [review p. 35]; unchanged water [review p. 36]. Needed: the water in the phase is left out of the composition, so the phase is treated as pseudo-binary; x_{\mathrm{drug}} is an input that must be measured or found from a partition and mass balance, which the review does not give.
What it means for a formulation: two drugs given together lower each other's limit by composition [review pp. 41–42]. Library example (added): a 7:3 ritonavir:lopinavir film by weight gives x = (7/720.9)/(7/720.9 + 3/628.8) = 0.67 (molar masses from outside the review), so 25 × 0.67 = 16.8 µg/mL against the wiki's calculated 16.7 and measured 15.2 µg/mL [trasi2015b R5]. For a polymer the mole fraction fails: per chain it stays at 0.97 or more while S_a/S_a^0 reaches 0.52–0.54 [ueda2022 X3], which is why Eq. 10 uses volume fractions.
Symbols: \chi, drug–additive interaction parameter (dimensionless); others as Eq. 8.
Where it comes from: named as the Bragg–Williams model (ref 99). Steps (added):
G_{mix}/RT=n_1\ln x_1+n_2\ln x_2+\chi n_1n_2/(n_1+n_2), with 1 = drug and 2 = additive.
\partial/\partial n_1 of the first two terms gives \ln x_1, since n_1\,\partial\ln x_1/\partial n_1+n_2\,\partial\ln x_2/\partial n_1=0.
\partial/\partial n_1 of the third gives \chi(x_2-x_1x_2)=\chi x_2^2.
So \ln a_1=\ln x_1+\chi(1-x_1)^2. With steps 1–2 of Eq. 8 this is Eq. 9; \chi=0 returns Eq. 8.
Assumptions and valid range: stated: molecular volume of additive similar to that of the drug; a more negative \chi means a stronger interaction and a greater reduction relative to ideal mixing [review pp. 35–36]; unchanged water [review p. 36]. Needed: one phase. For \chi>2 the regular solution separates, and the equation does not apply across the gap.
What it means for a formulation: strong drug–additive interaction lowers the limit further. Hypothetical values (added): at \chi=-1, x_{\mathrm{drug}} = 0.9 gives 0.9 × e^(−0.01) = 0.891, and 0.5 gives 0.5 × e^(−0.25) = 0.389 instead of 0.5. The interaction term therefore matters mostly at high additive content. The review prints no example.
Symbols: \Phi_{\mathrm{drug}}, volume fraction of drug in the drug-rich phase; r, molar volume ratio of polymeric additive and drug (both dimensionless); \chi as Eq. 9, on a lattice site of the drug's volume.
Where it comes from: Flory–Huggins (ref 100). Steps (added), with \Phi_1=n_1/(n_1+rn_2) and \Phi_2=rn_2/(n_1+rn_2):
Sum: \ln a_1=\ln\Phi_1+(1-1/r)\Phi_2+\chi\Phi_2^2, with \Phi_2=1-\Phi_{\mathrm{drug}}. Steps 1–2 of Eq. 8 give Eq. 10. At r=1, \Phi=x and this is Eq. 9.
Small polymer content with large r: \ln a_1\approx(\chi-\tfrac12)\Phi_2^2, second order in \Phi_2. At \Phi_2=0.1 and \chi=0, a_1=0.9\,e^{0.1}=0.995. Only a strongly negative \chi or a large \Phi_2 then lowers the limit substantially.
Assumptions and valid range: stated: polymer additive with molar volume much larger than that of the drug; unchanged water [review p. 36]. Needed: one mixed phase. With \chi positive the expression can exceed 1 (r\to\infty, \chi=1, \Phi_2=0.5: a_1=0.5\,e^{0.75}=1.06), which signals a miscibility gap rather than a real activity. The review gives no correction term for water; it only reports that one has been proposed (ref 70), so that form cannot be written from the review.
What it means for a formulation: a hydrophobic or strongly interacting polymer that enters the phase lowers the ceiling. Library example (added): ibuprofen, S_a^0 = 154 µg/mL, polymer:drug about 1 by weight, so \Phi_{\mathrm{drug}}\approx 0.5. With \chi=0 and large r: 154 × 0.5 × e^0.5 = 127 µg/mL, the wiki's "about 127–129". With \chi=-1: 154 × 0.5 × e^(0.5−0.25) = 98.9 µg/mL, against the wiki's "about 100" to "about 102" and about 90 measured with PVP-VA [ueda2021 R12, R13].
In Case II the additive does not mix with the crystal or complex the drug, so S_c=S_c^0 ("minimally changed") [review p. 36].
Dividing Eq. 9 by S_c^0 gives the first equality.
Inserting Eq. 2 gives the second.
Both S_a and S_c are free drug, so \gamma/\gamma^*\sim 1 and S_a/S_c=SR_{max} [review p. 37].
Assumptions and valid range: as Eq. 9, and no solubilization of the crystal. It has an additional implicit assumption, namely that e^{-I(a_2)} keeps its value for the phase containing additive.
What it means for a formulation: here a concentration ratio can be read as the driving force. Library example (added): ibuprofen with 300 µg/mL PVP-VA, S_c 29.8 → 30.1, S_a 177.1 → 90.7 µg/mL, ratio 5.94 → 3.01 [ueda2020b R1]. Then 5.94 × 90.7/177.1 = 3.04, which differs from 3.01 only through the small change in S_c.
Where it comes from: as Eq. 11, with Eq. 10 in place of Eq. 9 (added).
Assumptions and valid range: as Eqs. 10 and 11. The review adds the caution that Eqs. 9–12 assume mixing does not change the water in the phase [review p. 36].
What it means for a formulation: the fall of SR_{max} follows the polymer content of the phase. Library example (added): ibuprofen with PVP-VA at a polymer:drug weight ratio of 0.58 in the phase, so \Phi_{\mathrm{drug}}\approx 1/1.58=0.633 if densities are taken as equal, with large r. The factor is 0.633 × e^0.367 = 0.914 with \chi=0, and 0.633 × e^(0.367−0.135) = 0.798 with \chi=-1, against a measured 101/154 = 0.656 [ueda2021 R5, R11]. The ideal term accounts for about a quarter of the measured loss (0.086 of 0.344). The review gives no number.
In the wiki: not written as a ratio; see Eq. 10. The water hypothesis was examined on solid dispersions in Amorphous solubility F20.
Where it comes from: the review says "combining Eq. 6 and 9". Steps (added):
Free drug is in equilibrium with the mixed phase, so by Eq. 9, free drug =S_a^0X with X=x_{\mathrm{drug}}e^{\chi(1-x_{\mathrm{drug}})^2}.
Mass balance as Eq. 5: S_a=S_a^0X+S_af^a_{com}.
Solving: S_a=S_a^0X/(1-f^a_{com}), which is Eq. 13.
For a polymer the same steps with Eq. 10 give S_a=\dfrac{S_a^0}{1-f^a_{com}}\Phi_{\mathrm{drug}}e^{\{(1-1/r)(1-\Phi_{\mathrm{drug}})+\chi(1-\Phi_{\mathrm{drug}})^2\}}. The review does not print this form, although Table 1 lists a polymer (TPV/CAS) under Case III [review p. 29].
Assumptions and valid range: stated: molar volumes comparable; water sorption of the amorphous drug unchanged by mixing [review p. 38]. Needed: Eq. 13 is implicit. f^a_{com} here is not the Case I value, because the free drug and the additive left in the water are both lower, and x_{\mathrm{drug}} depends on how much additive leaves the water [review pp. 38–39]. It organizes data, but cannot predict S_a without an additive partition and binding model, which the review does not give.
What it means for a formulation: an additive can raise the measured amorphous solubility and lower the driving force at the same time, as for ketoprofen with CTAB [review p. 41]. S_a lies below the value expected from the crystalline solubility enhancement [review p. 39]; W9 gives the algebra.
At the crystal the additive stays in the water because it does not mix with drug crystals, and the free drug is at S_c^0 [review p. 38, Fig. 9a], so S_c=S_c^0/(1-f^c_{com}), a single value at each additive concentration whatever the amount of crystal.
Dividing Eq. 13 by this gives the first equality; inserting Eq. 2 gives the second.
SR_{max} is free drug at the limit over free drug at the crystal: SR_{max}=(S_a^0/S_c^0)X. The review states this only in words, as continuous reduction of SR_{max} [review p. 39].
Therefore S_a/S_c=[(1-f^c_{com})/(1-f^a_{com})]\,SR_{max}: the ratio does not reflect SR_{max} unless the prefactor is 1 [review p. 39].
Assumptions and valid range: as Eq. 13.
What it means for a formulation: the concentration ratio and the driving force separate here more than in any other case. Library example (added): telaprevir with 12 mM STCDC [lu2017b R1, R2], the wiki reading the fall in flux as \delta ≈28 to ≈22. Then X\approx 22/28=0.786. S_c 5.2 → 12.2 µg/mL gives 1-f^c_{com}=0.426. With S_a about 264 µg/mL (the middle of 238–290) and free drug 150 × 0.786 = 118 µg/mL, 1-f^a_{com}=0.447. Then 0.953 × 28.8 × 0.786 = 21.6 = 264/12.2, so here the ratio lies close to the flux-based value only because the two free fractions happen to be similar. The mixing itself is presumed and not measured [lu2017b C31].
Eq. 15. Sink index against the crystalline solubility
SI_{cr}=\frac{S_c}{Dose/V}\tag{15}
[review p. 54]
Symbols: SI_{cr}, sink index for the crystal (dimensionless); S_c, crystalline solubility in the medium (µg/mL); Dose, drug amount added to the dissolution experiment (µg); V, medium volume (mL).
Where it comes from: definition (ref 125, Sun, Ju and Lee 2012). Rearranged (added): V_{sat}=Dose/S_c is the volume needed to form a saturated solution, so SI_{cr}=V/V_{sat}; this is why SI_{cr}\geq 3 reads as three times that volume.
Assumptions and valid range: a dimensionless description of the test; it says nothing of release kinetics. The review adds that it may not represent the driving force for release, which for many ASDs relates to the polymer, but is important for the driving force for crystallization [review p. 55].
What it means for a formulation: see W12 for the regimes. Library example (added): tacrolimus 5 mg, S_c 1.8 µg/mL, 900 mL: Dose/V = 5.56 µg/mL and SI_{cr} = 1.8/5.56 = 0.32, nonsink for the crystal even in a compendial volume [purohit2018 R4].
In the wiki: sink index (formula) writes one index, SI=C_sV/Dose, with C_s either solubility; same content, and sink index (concept). Bioperformance F16.
Eq. 16. Sink index against the amorphous solubility
SI_{am}=\frac{S_a}{Dose/V}\tag{16}
[review p. 54]
Symbols: SI_{am}, sink index for the amorphous form; S_a, amorphous solubility in the medium; others as Eq. 15.
Where it comes from: same form as Eq. 15 with S_a. Dividing the two (added): SI_{am}/SI_{cr}=S_a/S_c.
Assumptions and valid range: as Eq. 15. Added: in Cases I and III S_a is a total concentration, so SI_{am}=1 marks the onset of the drug-rich phase in that medium, not a free drug value. In Cases II and III S_a can depend on the amount of drug-rich phase (W8), so the index at one dose does not transfer exactly to another.
What it means for a formulation: library example (added), tacrolimus 5 mg with S_a 50 µg/mL: 900 mL gives 50/5.56 = 9; 100 mL gives 50/50 = 1; 40 mL gives 50/125 = 0.4; the crystalline indices are 0.32, 0.036 and 0.014. The two scales are 28-fold apart.
Symbols: Mo, administered dose (mg); Vo, volume of media (mL; 250 mL in the review's example); S, solubility of the compound in intestinal media, generally reflecting S_c (mg/mL). Dimensionless.
Where it comes from: definition, cited to ref 131 (Wuelfing 2022), which the review uses as a summary. Link to Eqs. 15–16 (added): with V=Vo and Dose=Mo, Dose number =1/SI. So a dose number of 10 is SI=0.1 and a dose number of 1 is SI=1.
Assumptions and valid range: stated: for poorly soluble, highly permeable compounds; ideally biorelevant solubility; no crystallization for the amorphous form [review pp. 55–56]. Not stated, and in tension with the review's central theme: a solubility that includes solubilized drug lowers the dose number in Cases I and III even where free drug does not rise (Eqs. 5–7, W6). The equation therefore reads capacity to hold the dose in solution, not driving force.
What it means for a formulation: the review's worked examples, with arithmetic added:
Ritonavir 100 mg: dose numbers ~60 and ~7 [review p. 56]; the review prints neither volume nor solubilities. With 250 mL, Mo/Vo = 400 µg/mL, and the FaSSIF-V1 values of its source (6.7 and 56 µg/mL) give 400/6.7 = 59.7 and 400/56 = 7.1 [indulkar2018 R8], so these are the inputs.
The ratio of the two dose numbers is S_a/S_c: 11.1/0.4 = 28; 59.7/7.1 = 8.4.
In the wiki: no formula page. One use on dissolution-rate-limited absorption: ledipasvir 90 mg, 0.025 mg/mL, about 100 mL gives 90/(100 × 0.025) = 36 [indulkar2016 R17], at another volume convention than the review's.
W1. \Delta G_{c\to a} at the temperature of interest
[review p. 26; no display equation]
Symbols: T_m, melting temperature (K); \Delta H_f, melting enthalpy (J/mol), equal to \Delta H_{c\to a} at T_m; \Delta S_{c\to a}, entropy difference, equal to \Delta H_f/T_m at T_m; \Delta C_p, heat capacity difference, amorphous minus crystal (J mol⁻¹ K⁻¹).
Where it comes from: the review describes the route in words (refs 1, 31, 81–83). The algebra, all (added):
\Delta C_p constant at its value at T_m: \Delta G_{c\to a}=\Delta H_f(T_m-T)/T_m-\Delta C_p[(T_m-T)-T\ln(T_m/T)]. The bracket is positive, so \Delta C_p>0 lowers the estimate.
The Hoffman form used through the library is not named in the review: \Delta G=\Delta H_f(T_m-T)T/T_m^2. Expanding step 3 to second order in T_m-T, it corresponds to \Delta C_p\approx 2\Delta S_fT/T_m.
Assumptions and valid range: stated: the C_p method with measured temperature dependence gives improved predictions, and can also be used if the solubility follows the GLPS boundary [review p. 26]. Needed: a crystal that melts without degrading, which pazopanib does not [sugihara2018 C17].
What it means for a formulation: the route changes the prediction by a large factor for a high-melting drug. Library inputs for enzalutamide (added): \Delta H_f 33,820 J/mol, T_m 470 K, T 310 K, RT = 8.314 × 310 = 2577 J/mol ([wilson2018 E2] on the formula page):
Hoffman: 33,820 × 160 × 310/470² = 7.59 kJ/mol, ratio 19.0, the value printed by that paper (7594 J/mol). The wiki records that its printed 39 µg/mL does not follow; 19.0 × 0.502 × 2.9 = 27.7 µg/mL [wilson2018 X1].
Statements: \Delta G_{c\to a} of most drugs is typically below ~10 kJ/mol at ambient conditions; measured ratios of most drugs are below 100-fold (refs 31, 37, 65). Hancock and Parks predicted 12–1652-fold for amorphous forms and 1.1–3.6-fold for crystalline polymorphs (ref 1).
Where it comes from: Eq. 1 applied to the numbers (added):
10 kJ/mol: e^(10000/2478.8) = 56 at 25 °C; e^(10000/2578.6) = 48 at 37 °C.
100-fold corresponds to RT\ln 100 = 11.4 kJ/mol at 25 °C and 11.9 kJ/mol at 37 °C. The two statements of the review are therefore consistent.
The Hancock and Parks ranges at 25 °C (temperature assumed; the review does not give it): 12-fold = 6.2 kJ/mol, 1652-fold = 18.4 kJ/mol; polymorphs 1.1-fold = 0.24 and 3.6-fold = 3.2 kJ/mol.
Assumptions and valid range: "most" and "typically" are the review's limits. The range uses Eq. 1, without water; Eq. 2 can only lower it.
What it means for a formulation: the review reads this as a constraint on the achievable gain before any formulation work. Above about 12 kJ/mol the ratio would exceed 100-fold before the water correction. The polymorph range sits well below the amorphous range.
In the wiki: Amorphous solubility F6 holds about 2 to about 600. Hoffman values above 10 kJ/mol exist for telaprevir (13.31) and atazanavir (14.55; check R34). For atazanavir, e^(14550/2578.6) = 283 before water; with the water activity 0.22 [indulkar2015 M2] this gives 62, the ratio printed in 2015 and 2017; Table I of the 2017 paper heads the same 0.22 as I(a), which read literally gives about 228 [indulkar2017a X1] (advantage with water-sorption correction). The Hancock and Parks ranges are not in the wiki.
W3. Temperature: the ratio falls and reaches 1 at the melting point in water
[review pp. 21–24, Fig. 5]
Symbols: T_{m\,(\text{in water})}, melting point of the crystal in contact with water; wet-T_g, glass transition of the water-saturated amorphous drug; others as Eq. 2 and W1.
Where it comes from: the review argues from Fig. 5. The algebra, all (added):
From Eq. 2, the free energy difference between crystal and wet amorphous drug is \Delta G_{wet}(T)=\Delta G_{c\to a}(T)-RT\,I(a_2), and \ln(S_a^0/S_c^0)=\Delta G_{wet}/RT.
Gibbs–Helmholtz on the dry term: \mathrm{d}(\Delta G_{c\to a}/RT)/\mathrm{d}T=-\Delta H_{c\to a}/RT^2. Since \Delta H_{c\to a}>0, the ratio falls as T rises, if I(a_2) changes little.
The ratio equals 1 where \Delta G_{wet}=0, that is \Delta G_{c\to a}(T_{m\,(\text{in water})})=RT\,I(a_2). At the dry T_m, \Delta G_{c\to a}=0, so \Delta G_{wet}=-RT\,I<0; hence the crossing lies below the dry melting point. This is the melting point depression of the review's text, and the intersection of the crystal solubility curve with the LLPS boundary in Fig. 5c.
Above T_{m\,(\text{in water})} no crystal can exist, and any measured solubility lies on the LLPS boundary. Extending that boundary below it gives the amorphous solubility in the supercooled, metastable region [review p. 22].
Below the wet-T_g the glass has a higher free energy, so Eq. 1 with \Delta G_{c\to g} gives a higher solubility (the dotted GLPS branch of Fig. 5b, c). The review calls it controversial whether this branch is the one measured [review pp. 22–23].
Assumptions and valid range: stated: "generally" [review p. 22]; a non-hydrate crystal. Needed: the same phase composition across the range; polymorphs each have their own T_m and T_{m\,(\text{in water})} (crystals A and B in Fig. 5).
What it means for a formulation: a low-melting drug can have no crystalline solubility at 37 °C at all; a shake-flask measurement then gives the amorphous solubility [review p. 48]. Library numbers (added): ibuprofen's measured ratio falls 5.8, 3.6, 2.5 at 25, 40, 60 °C [ueda2022 R16]; through step 2, the first and last give an apparent enthalpy of 8.314 × ln(5.8/2.5)/(1/298.15 − 1/333.15) = 19.9 kJ/mol. On step 5: griseofulvin at 25 °C has \Delta G 10.38 (glass) and 9.39 kJ/mol (supercooled liquid), so the predicted solubilities differ by e^(990/2478.8) = 1.49, and 531.41/356.44 = 1.49; the measured 355.96 µg/mL follows the supercooled liquid [almeidaesousa2015 R5, R11].
W4. Solubility ratio equals dissolution rate ratio
[review pp. 12–13, with the limit at p. 44]
Statements: sulfathiazole: solubility ratio 3.8 at 37 °C, initial dissolution rate ratio 3.7 for high-loading coprecipitates. Reserpine–PVP 1:5: surface-normalized rate factor ~2.5, solubility ratio ~3, absorption in rats ~2.7 (rate) and ~3.2 (extent). Stagner and Guillory (1979) linked intrinsic dissolution rate ratios, solubility ratios and free energies for iopanoic acid.
Where it comes from: the review does not write the equation; it is the film form of Noyes–Whitney (added). \mathrm{d}M/\mathrm{d}t=(DA/h)(S-C); at C\approx 0 the initial rate is proportional to S, so with the same D, A, h, rate ratio = solubility ratio. From the ratio to free energy by Eq. 1, as those authors did: RT\ln 3.8 = 3.44 kJ/mol at 37 °C.
Assumptions and valid range: same hydrodynamics and area; solid unchanged during the measurement. The review excludes these methods from its measurement section, because the crystalline rate of a poorly soluble drug is typically inaccessible and multicomponent systems undergo phase separation at the interface [review p. 44].
What it means for a formulation: historically this was the indirect route to the ratio while the amorphous solid recrystallized. Today it does not give S_a for a dispersion.
In the wiki: Noyes–Whitney equation; intrinsic dissolution rate carries the Bhardwaj equality without this limit (itraconazole 51 against 2.8 µg/min/cm², ratio 18; check R52). The historical ratios are not in the wiki.
W5. Henderson–Hasselbalch in a simple buffer
[review p. 17]
Statement: amorphous solubility concentrations appear to follow Henderson–Hasselbalch for weakly basic drugs in simple buffer, provided the amorphous solubility of the free form is known (ref 18).
Where it comes from: (added) for a base, total =C_B(1+10^{pK_a-pH}). Only the un-ionized part is limited by the drug-rich phase, so C_B is the free-base amorphous solubility, and the crystal has the same factor. The ratio of totals then does not depend on pH, which is Eq. 3 at fixed \bar\alpha.
Assumptions and valid range: "simple" buffer; no salt or complex formation.
What it means for a formulation: a pH-dependent limit is predictable from one measured value. Library: clotrimazole's total rose from 7.7 µg/mL at pH 8 to 490.2 at pH 4, a factor of 64, with the un-ionized part staying within 5.6–8.4 µg/mL [indulkar2015 R9].
Symbols: J, flux per unit area (µg min⁻¹ cm⁻²); D, diffusion coefficient in the membrane; h, membrane thickness; \gamma_m, activity coefficient in the membrane; J^*, flux from a solution saturated with crystal.
Where it comes from: the review states it in words, citing Higuchi 1960 (ref 59), transdermal work (refs 60, 61) and Raina 2014 (ref 50). Steps (added):
Fick across the membrane, sink receiver: J=DC_{m}/h, with C_m the drug concentration in the donor-side face.
Equilibrium at that face: a=\gamma_mC_m. Hence J=Da/(h\gamma_m).
With D, h, \gamma_m the same, J/J^*=a/a^*=SR, by Eq. 4. At the drug-rich limit, J_{max}/J^*=SR_{max}.
Assumptions and valid range: stated: the membrane limits permeation of droplets and complex, and is poorly permeable to the drug; controls at known supersaturation or at the crystalline solubility [review p. 51]. Needed: the membrane, not the boundary layer, is rate limiting, otherwise step 1 fails.
What it means for a formulation: read from Fig. 4 (added; values approximate). By steps of Eqs. 4 and 7, the prefactor is the concentration ratio over the flux ratio:
medium
concentration ratio
flux ratio
prefactor
Buffer-V1
56.5
54.8
1.03
Buffer-V2
54.8
53.6
1.02
FaSSIF-V2
38.9
54.2
0.72
FaSSIF-V1
37.8
56.0
0.68
Composite FaSSIF
32.2
50.3
0.64
FaHIF
14.0
52.6
0.27
So a constant driving force is measured while solubilization differs fourfold. The caption does not name the drug; the values are those of atazanavir in the library.
Where it comes from: Eqs. 5–14 and the review's text; the formula column for Case III is (added), step 3 of Eq. 14. X=x_{\mathrm{drug}}e^{\chi(1-x_{\mathrm{drug}})^2} or the Flory–Huggins equivalent.
S_c
S_a
free drug at crystal
free drug at S_a
SR_{max}
S_a/S_c reads SR_{max}?
No additive
S_c^0
S_a^0
S_c^0
S_a^0
S_a^0/S_c^0
yes
Case I
rises
rises
S_c^0
S_a^0
S_a^0/S_c^0
no; generally smaller
Case II
\approx S_c^0
falls
S_c^0
S_a^0X
(S_a^0/S_c^0)X
yes
Case III
rises
falls then rises, or rises
S_c^0
S_a^0X
(S_a^0/S_c^0)X
no
Assumptions and valid range: unionized or constant ionized fraction under buffer [review p. 29]. Table 1 assigns a case to a drug/additive pair, not an additive. Fig. 7 labels the crystalline line S_b in all three panels in its legend; the caption and text have S_c. This is a misprint.
What it means for a formulation: the assignment needs two measurements (added): (i) S_c with additive, and (ii) maximum flux or free drug with additive. S_c up and maximum flux unchanged is Case I; S_c unchanged and S_a, flux down is Case II; S_c up and flux down is Case III. Ketoprofen gives one drug in all three: EUD-E (I), PVP (II), CTAB (III) [review pp. 40–41, Fig. 10]. For the formulator, an inhibitor that mixes with the phase lowers X, hence the trade-off the review names [review p. 41].
In the wiki: no classification page. Dissolution F12, Overview F2. The library shows the pair and the medium matter: EUD-E stays in water with ibuprofen at pH 2.0 and associates with enzalutamide droplets at pH 6.5 (Amorphous solubility F19), and atazanavir in 3 mM STC alone has a flux-based maximum of about 40 against 63 in buffer [indulkar2018 R10], which is not the review's Case I for STC (Crystallization from solution F19, check R49).
W8. Dissolved against total drug: the amount of drug-rich phase
[review pp. 37, 39–40, 49; Fig. 8]
Symbols: C_{tot}, total drug per volume; D=C_{tot}-S_a, drug in the drug-rich phase per volume; A_r, additive in the drug-rich phase per volume (molar basis in step 2).
Where it comes from: the review reasons in words; Fig. 8 is a schematic. The algebra, all (added):
Without additive and in Case I, the phase composition does not depend on D, so dissolved drug stays constant above the onset.
Case II, ideal mixing, A_r fixed: x_{\mathrm{drug}}=D/(D+A_r), so S_a=S_a^0D/(D+A_r), which rises with D toward S_a^0. The self-consistent solution needs D=C_{tot}-S_a.
Case III: with the partition form of W9, S_a=S_a^0X(1+KA_{aq}). A larger phase removes additive from the water, lowering A_{aq} and the complexed drug, which is the review's explanation of the fall [review pp. 39–40].
The same larger phase dilutes the additive within it, which raises X as in step 2. The review's statement that S_a decreases therefore requires that the loss of complex dominates; it does not follow from Eq. 13 alone.
Assumptions and valid range: the onset itself needs an additive partition coefficient, which the review does not give. Fig. 8 draws the Case II onset below and the Case I and III onsets above the additive-free value.
What it means for a formulation: in Cases II and III a measured S_a refers to a stated total drug. NMR reads this directly: linear rise of the dissolved peak to the onset, then constant, slower rise, or fall [review p. 49]. The sentence there writes "additive-free solution (Case I)", and Fig. 8 shows the two behave alike above the onset. In the library at 500 µg/mL polymer, dissolved ibuprofen levels at about 120 and 90 µg/mL with PVP K25 and PVP-VA, and rises from about 88 to 123 with HPMC 603 [ueda2021 R10]: the schematic rise holds only where the additive-to-drug ratio in the phase keeps falling.
Statement: if complexation dominates, S_a rises with additive as in Case I. If partitioning is stronger, a small amount of additive nearly all enters the drug-rich phase, little complex forms and free drug falls, so S_a falls; further additive stays in the water and S_a rises. The supersaturation region is narrow at low additive and wider at high additive.
Where it comes from: Eq. 13 with a partition form for the complex and an additive balance, all (added):
[\mathrm{complex}]/[\mathrm{free}]=KA_{aq}, so 1/(1-f^a_{com})=1+KA_{aq}, where A_{aq}=A_t-A_r.
Eq. 13: S_a/S_a^0=X(A_r)\,(1+K(A_t-A_r)).
At the crystal, A_{aq}=A_t, so S_c/S_c^0=1+KA_t. This gives the review's statement that S_a lies below what the crystalline enhancement predicts: it would be S_a^0(1+KA_t) [review p. 39].
S_a/S_c=SR_{max}\,(1+KA_{aq})/(1+KA_t), so with constant K the ratio is below SR_{max} and approaches it at large A_t. The same constant-K form applied to Case I gives a prefactor of 1, so the Case I shortfall needs a K that falls as free drug rises, which is the wiki's reading for atazanavir in SDS (Amorphous solubility F18).
Assumptions and valid range: K constant, and X depends on A_r alone. Both are simplifications for the algebra, not from the review.
What it means for a formulation: hypothetical numbers (K=0.002 mL/µg): A_t = 50, A_r = 45, X = 0.8 gives S_a/S_a^0 = 0.8 × 1.01 = 0.81, against crystal factor 1.10. A_t = 500, A_r = 150, X = 0.7 gives 0.7 × 1.70 = 1.19, against 2.0. A_t = 2000, A_r = 200, X = 0.7 gives 0.7 × 4.60 = 3.22, against 5.0. A low surfactant level can therefore reduce a measured limit that a higher level raises, while free drug remains below S_a^0 at every level.
Symbols: D_{\mathrm{obs}}, observed diffusion coefficient of the dissolved drug peak (m²/s); D_{\mathrm{free}}, D_{\mathrm{complex}}, those of free and complexed drug; f_{\mathrm{free}}, f_{\mathrm{complex}}, fractions "per total dissolved components".
Where it comes from: PGSE method (ref 118, Stejskal and Tanner). Under fast exchange on the NMR time scale, the dissolved peak is the population average; phase-separated drug has its own peak and a lower D, so it is excluded [review p. 52]. Steps (added):
Free drug =f_{\mathrm{free}}\times dissolved drug by peak area; at the limit, this is S_af^a_{free} of Eq. 5.
SR_{max} = free drug at the limit over free drug at the crystal.
Assumptions and valid range: stated: exchange faster than the NMR time scale; reference values from each component in the absence of complex formation [review p. 52]. Not stated: viscosity and obstruction changes, and a complex with the same D as its additive. D_{\mathrm{free}} and D_{\mathrm{complex}} must differ well.
What it means for a formulation: gives free drug with droplets and complex present, and with no membrane. Hypothetical (added): D_{\mathrm{free}}=6, D_{\mathrm{complex}}=1, D_{\mathrm{obs}}=3 (×10⁻¹⁰ m²/s) gives f_{\mathrm{free}}=2/5=0.40. The review gives no number.
Statement: drug-rich species may raise boundary layer permeability of highly membrane-permeable drugs by particle drifting, reducing the concentration gradient across the layer, so that the concentration at the membrane surface approaches the value set by the amorphous solubility (refs 45–49). Size is called "of paramount importance" [review p. 17].
Where it comes from: the review prints no equation. A film model that gives the statement form, all (added):
Across a film of thickness h with free drug at S_a in the bulk and droplet concentration C_p falling to zero at the membrane: J\leq (D_fS_a+D_pC_p)/h, with D_f, D_p the diffusion coefficients of free drug and droplet.
Relative to droplet-free drug at S_a, the supply rises by 1+(D_p/D_f)(C_p/S_a).
Stokes–Einstein: D_p=k_BT/(6\pi\eta r), so the term scales as 1/r.
Assumptions and valid range: boundary-layer-limited transport; droplets stable across the layer and dissolving at the surface; an assumed D_f.
What it means for a formulation: an order-of-magnitude check (added) with T = 310 K, \eta = 0.69 mPa·s, D_f=5\times10^{-10} m²/s (assumed), C_{tot} = 1000 and S_a = 80 µg/mL (C_p/S_a = 11.5): diameter 283 nm gives D_p=2.3\times10^{-12} m²/s and factor 1.05; 100 nm gives 1.15; 50 nm gives 1.30. The measured rise for atazanavir in PAMPA with a sink receiver is about 8.8/5.0 = 1.76 between 80 and 1000 µg/mL [hate2022 R19]. The measured effect exceeds this estimate, and droplets of about 1000 nm and 283 nm gave the same mass flow at 150 µg/mL [hate2022 R17], whereas step 3 predicts a size dependence; those authors reject size-dependent shuttling on that result. This simple film estimate therefore does not account for those data, and the conflict with the review's size statement is the "differs" verdict at check R12, with different systems on the two sides.
Statements: SI_{cr}\geq 3 is sink, SI_{cr}<3 nonsink. At SI_{am}=1 the dose confers maximum activity, and all amorphous material can in theory dissolve molecularly. At SI_{am}<1 a colloidal phase can form and concentrations exceed S_a, with a risk of crystallization and coarsening.
Where it comes from: Eqs. 15–16, with mass balance (added). If all drug is released, C_0=Dose/V=S_a/SI_{am}. For SI_{am}<1, free or dissolved =S_a, colloid =C_0-S_a, and the colloid share is 1-SI_{am}. For SI_{am}>1 the maximum activity relative to the limit is 1/SI_{am} (additive-free or Case II).
Assumptions and valid range: complete release and no crystallization. The review adds "if enabled by the formulation design" [review p. 54]. Added: "highest driving force" holds for activity; below SI_{am}=1 the activity is unchanged, not lower.
What it means for a formulation: tacrolimus at SI_{am}=0.4 (5 mg in 40 mL, library): C_0 = 125 µg/mL, 50 free, 75 in colloid = 60% = 1 − 0.4, if all is released. The review's interpretation is that the index speaks to crystallization and not release in polymer-controlled ASDs [review p. 55].
In the wiki: sink index, sink conditions, non-sink dissolution. Bioperformance F16, F28 (seed-driven loss of dissolved drug at 40 mL, SI_{am}=0.4, and not at larger volumes), F15 (transfer keeps rising above S_a, that is below SI_{am}=1). The colloid share is not on those pages as an equation.
W13. Dose number threshold
[review pp. 55–56]
Statement: enabling formulations might be needed when the aqueous dose number is above 10, and for below 10 conventional formulations may suffice (Van den Bergh, ref 132; also ref 133); favourable values from other studies are summarized in ref 131.
Where it comes from: an empirical rule from outside; the review does not derive it. By Eq. 17, a dose number of 10 means the dose in 250 mL is ten times the solubility.
Assumptions and valid range: poorly soluble, highly permeable compounds; the review applies it with amorphous solubility as well, under "achieve and maintain" [review p. 56].
What it means for a formulation: tacrolimus goes from 11 to 0.4; ritonavir from ~60 to ~7. Added: the crystalline value for tacrolimus crosses the threshold with the input: 20/1.4 = 14, 20/1.8 = 11, 20/2.5 = 8 (check R69). The threshold cannot be read sharply for this drug.
In the wiki: no threshold in the wiki. Simulation evidence on tacrolimus is in Bioperformance F17.
W14. Stated numerical limits on methods and particle size
Below 1% organic solvent in the final aqueous solution, stated as ideal for the titration method [review p. 45]. No derivation. It keeps the medium the one being studied.
Droplets below several tens of nanometres may not separate even by ultracentrifugation, so the measured value can be too high [review p. 49]. A sedimentation argument (added): velocity scales as r^2\Delta\rho, so small droplets with a small density difference settle slowly. A drug-rich phase with lower density than water forms the upper layer.
Aggregates below 100 nm were linked to good absorption in a survey of non-nucleoside reverse transcriptase inhibitors, called "speculation" [review p. 15]. The review prints the author as Frankel; its reference 41 is Frenkel.
Dates and attributions before 2013 are as the review cites them; the originals are not in the library unless a card is linked.
1. The ratio follows the free energy of the solid forms (1928 to 2000; Eq. 1, W1). Parks, Huffman and Cattoir (1928, ref 79) studied the glass transition of glucose, and Parks, Snyder and Cattoir (1934, ref 20) compared its glass and crystal in organic solvents and found agreement with calorimetric free energies [review p. 10]. W. I. Higuchi and co-workers applied the relation to polymorphs (1963, ref 80). Hancock and Parks (2000, ref 1) brought it to drugs, with indomethacin and a survey of other compounds, using heat capacity, enthalpy and entropy of fusion and melting temperature. Recrystallization kept their measured values below the true advantage [review p. 8].
2. First measurements on drugs, with the solid crystallizing as the obstacle (1960 to 1979; W4). Mullins and Macek (1960, ref 21) found about tenfold for novobiocin, with absorption in dogs only from the amorphous free acid; the review reads their spheroid precipitate as LLPS in retrospect. Simonelli and co-workers (1969, 1976) obtained 3.8 for sulfathiazole by equilibrating coprecipitates. Stupak (1974) linked rate, solubility and absorption for reserpine–PVP, and Stagner and Guillory (1979) used intrinsic dissolution rates to go around crystallization [review pp. 11–13].
3. Activity, not concentration, drives transport (1960 onward; Eq. 4, W6). T. Higuchi (1960, ref 59) stated the thermodynamic basis, developed further in transdermal work (refs 60, 61) [review p. 17]. Taylor group: Raina 2014 found the flux plateau at the amorphous solubility; Raina 2015 separated solubilizers from supersaturation; Elkhabaz 2019 carried this into biorelevant media and human intestinal fluid.
4. Colloids and a liquid second phase (1965 to 2013; W11). Tachibana and Nakamura (1965, ref 22) saw particles of about 100 nm from a β-carotene–PVP dispersion. Bonnett and co-workers (2003, ref 36) described oiling out as a precursor to crystallization. Shoichet's aggregation work (refs 38–40) and Frenkel and co-workers' link of aggregate size to absorption (2005, ref 41, printed as Frankel) came from other fields. Lindfors and co-workers (2006, ref 30) gave the scattering method. Taylor group: Ilevbare 2013 established the same onset by several routes at the predicted amorphous solubility for ritonavir and others, and called it the maximum achievable supersaturation (ref 37); Raina 2015 added fluorescent probes.
5. The water in the amorphous phase and ionization (2010 to 2011; Eqs. 2, 3). Murdande and co-workers (refs 27–29) added the water sorbed by the amorphous solid, an ionization correction and tighter heat capacity data; predictions still exceeded kinetic measurements until a polymer delayed crystallization [review p. 13]. Paus and co-workers (2015, ref 35) applied PC-SAFT. Taylor group: Almeida e Sousa 2015 compared Hoffman and heat capacity routes across six drugs and glass against supercooled liquid; Mosquera-Giraldo 2015 did so for telaprevir, whose phase is a glass; Wilson 2018 used the route on enzalutamide. For ionized drugs: Indulkar 2015 and Sugihara 2018 (W5).
6. Complexation and solubilization (Case I; Eqs. 5–7). Higuchi and Connors (1965, ref 98) gave the phase solubility types that frame the complex. Taylor group: Indulkar 2017 showed that SDS raised the amorphous solubility of atazanavir about half as much as predicted from the crystal, and explained it by a partition coefficient that falls with drug concentration (ref 58); Lu 2017 did the same comparison across six bile salts (ref 57); Indulkar 2018 measured intestinal-fluid analogues (ref 135).
7. Mixing into the drug-rich phase (Case II; Eqs. 8–12). Flory (1953, ref 100) and the Bragg–Williams model (ref 99) supply the expressions. Taylor group: Trasi 2015 and Trasi 2015 showed ideal mixing between two drugs (ref 90), carried to dispersions by Alhalaweh 2016. For polymers: Li 2018 with an insoluble polymer in the phase; Ueda 2020 on polymer type (ref 64); Ueda 2020 on HPMCAS grade with flux (ref 65); Ueda 2021 on partitioning by NMR with a Flory–Huggins calculation (ref 93); Ueda 2022 on partition against temperature (ref 92). Wilson 2021 tied polymer association to induction times (ref 108), and Duong 2022 examined the phase behaviour of posaconazole across media (ref 103). In this line the trade-off between inhibition and SR_{max} took form.
8. The three-case scheme (2024; Eqs. 13–14, W7). The separate lines are put together here as Cases I, II and III [review pp. 28–40]. The ketoprofen series behind it comes from refs 66, 67, 95 and 96, which the library does not hold. Case III in the library is Lu 2017 with STCDC and Correa-Soto 2022 for clopidogrel with TPGS and Tween 80 (ref 86; Table 1 places clopidogrel with CTAB under Case I). The review prints no earlier source for the case names, so the numbering appears to be this review's.
9. Droplet formation and transfer from the dispersion (2016 onward; W8, W11). Taylor group: Indulkar 2016 on droplets as a reservoir for flux; Indulkar 2017 on how they form (ref 127); Hate 2018 on transfer above S_a in a hollow-fibre module; and Hate 2022 on PAMPA (ref 46). Particle drifting comes from refs 45, 47 and 48, outside the library.
10. Sink index and dose number (2012 to 2022; Eqs. 15–17, W12, W13). Sun, Ju and Lee (2012, ref 125) defined the sink index; Van den Bergh and co-workers (2018, ref 132) gave the threshold of 10; Wuelfing and co-workers (2022, ref 131) summarize dose numbers. Taylor group: Trasi 2016 measured tacrolimus (ref 134); Purohit 2018, Hate 2019 and Purohit 2019 tested it by volume, with absorption and in vivo; Deac 2023 (ref 73) is the release side of the same limit.
After the review, in the library.Deac 2024 extends the release side to polymer molecular weight. Shah 2025 measures polymer in posaconazole droplets (about 31 wt% for HPMCAS grade H, about 5 for L, about 14 for M), and the grade that entered most delayed nucleation longest. Beside enzalutamide with Eudragit E PO, which entered most and inhibited worst, the wiki reads this as polymer uptake not by itself ranking inhibitors (Crystallization from solution F37, check R47). Alkalla 2026 brings a crystalline second drug into the co-dosing line, with the atazanavir flux plateau falling from 9.5 × 10⁻⁵ to 2.9 × 10⁻⁵ µg·min⁻¹·cm⁻². Earlier framing for comparison: Taylor and Zhang 2016.
Open points. These follow the storyline's list, with the derivation gap each one leaves.
Whether a glass below its wet T_g sets the same limit as a liquid: W3 step 5 has two branches and the review calls the choice controversial; griseofulvin follows the supercooled liquid in one comparison (Amorphous solubility F2).
Whether pH-dependent ratios in buffer are real: Eq. 3 with a constant ionized fraction gives a ratio independent of pH, measured ratios move (Amorphous solubility F10).
The composition of the phase with its water: Eqs. 8–14 use a pseudo-binary phase, and the water-corrected form (ref 70) is not printed (F17, F20).
No equation predicts x_{\mathrm{drug}} or \Phi_{\mathrm{drug}}: Eqs. 8–14 organize data, and the additive partition must be measured (W8, W9).
Which case applies to a real mixed medium: one bile salt is Case II alone and behaves otherwise with lecithin (Crystallization from solution F19).
PGSE-NMR for free fractions in formulated systems: W10 has no worked number in the review, and the library uses the method only for micelle radius (Amorphous solubility F18).
What transfers drug above S_a: W11 does not reach the measured rise and predicts a size effect that one PAMPA comparison did not find (Bioperformance F15, F22).
S_a as a model input: Eq. 17 with a solubilized S reads capacity, not driving force, and the library's one model assumes no precipitation (Bioperformance F17).