Following Taylor L.S., Zhang G.G.Z. (2016) Physical chemistry of supersaturated solutions and implications for oral absorption. Advanced Drug Delivery Reviews.
Equations and their derivation: supersaturated solutions and oral absorption
How to read this page. Each section takes one display equation of Taylor and Zhang (2016, Adv. Drug Deliv. Rev., uncorrected proof), checked against the page image, with its page ([review p. N] is the PDF page, equal to the printed page and to the numbering of the storyline), and gives the algebra that links it to the equations before it. The review derives little line by line, so intermediate lines are marked (added), and numbers that come from the library and not from the review carry their source; arithmetic on them is also (added). Sections W1–W20 treat relations stated only in words or in a figure. See also the check. The review reuses letters: S is surface area in Eq. 1, supersaturation in Eqs. 3 and 14, and part of a solubility symbol in Eq. 16 (S_w, S_{tot}); D and h belong to the aqueous layer in Eq. 1 and to the membrane in Eq. 14.
The derivation map
Eq. 1 — dissolution rate and its sink limit — from outside (Noyes–Whitney with the Nernst–Brunner layer; no reference printed); Fick's first law across a stagnant layer (added)
Eq. 2 — crystalline solubility from the crystal-to-supercooled-liquid chemical potential — from outside (refs 19, 20); equal chemical potential at saturation, supercooled liquid as standard state
Eq. 3 — supersaturation as chemical potential and activity ratio — from outside (ref 21); same step as Eq. 2 written for a supersaturated solution (added)
Eq. 4 — concentration-based supersaturation — definition; equals Eq. 3 when \gamma_D^{W(S)}/\gamma_D^{W(c)}=1 in a dilute solution
Eq. 5 — amorphous solubility from the activity coefficient alone — from Eq. 2 with the pure supercooled liquid in place of the crystal (added)
Eq. 6 — amorphous from crystalline solubility — from Eqs. 2 and 5 + \gamma_D^{W(c)}\cong\gamma_D^{W(a)}
Eq. 7 — crystal-to-supercooled-liquid chemical potential from melting data — from outside (ref 20); \Delta G=\Delta H-T\Delta S with constant \Delta C_p (added); left side printed without RT (misprint)
Eq. 8 — chemical potential of drug in the water-saturated amorphous phase — definition of activity, pure amorphous drug as standard state
Eq. 9 — activity in that phase from mole fraction and activity coefficient — definition
Eq. 10 — amorphous solubility with water in the amorphous phase — from Eqs. 6, 8, 9 + equal chemical potential between the phases (the step printed later as Eq. 12)
Eq. 11 — Murdande's concentration form — from outside (ref 12); from Eq. 10 by exponentiating + dilute solution + Gibbs–Duhem for the water term (added)
Eq. 12 — equal drug activity in the water-rich and drug-rich phases — from outside (phase equilibrium) + common standard state
Eq. 13 — logarithmic form with mole fractions and activity coefficients — from Eqs. 9 and 12
Eq. 14 — membrane flux proportional to supersaturation — from outside (refs 39, 78, 79); Fick's first law in the membrane + interface equilibrium + Eq. 3 (added)
Eq. 16 — total solubility in micelles — from Eq. 15 at crystal saturation + mass balance (ref 19)
Relations stated in words or in a figure:
W1 — binodal (common tangent) and spinodal (\partial^2G/\partial x^2=0) (p. 4, Fig. 1) — from outside (ref 34); Eq. 12 at the binodal
W2 — order of the phase boundaries and the regions of the phase diagram (pp. 5–6, Figs. 2, 3) — from Eqs. 2 and 10
W3 — LLPS or GLPS: temperature against the wet T_g (pp. 3, 4, 12) — definition (ref 29)
W4 — fast, intermediate and slow crystallizers: 2.5 min and 1 h (p. 6) — from outside (ref 45)
W5 — reading the onset: crossing of two lines, pyrene I_1/I_3 (pp. 6–7, Figs. 5, 6) — from outside (refs 53, 61)
W6 — amorphous-to-crystalline ratio, its range and its errors (p. 8, Figs. 4, 7) — from Eq. 10
W7 — ionic strength lowers the amorphous solubility (p. 8, Fig. 8) — from outside (ref 35); no equation given
W8 — pH: total solubility rises, un-ionized part and ratio constant (pp. 8–9) — from outside (Henderson–Hasselbalch, ref 43)
W9 — a second solute: amorphous solubility proportional to its mole fraction in the drug-rich phase (p. 9, Fig. 9) — from Eq. 13 + ideal mixing (ref 68)
W10 — reciprocal solubility and partition coefficient; constant flux at constant activity (p. 9) — from outside (refs 70, 71, 77, 78); follows from Eq. 14 (added)
W11 — flux linear up to the amorphous solubility, then constant; the cap for any route (pp. 9–10, 13, Figs. 10, 11) — from Eqs. 12 and 14
W12 — supersaturated against solubilized solution (p. 10) — definition from Eq. 3
W13 — when Eq. 4 holds with micelles: constant K_m (p. 11) — from Eqs. 3, 4, 15, 16 (added)
W14 — only free un-ionized drug in the central compartment drives flux (pp. 11–12, Fig. 12) — from outside (ref 89), with Eq. 14
W15 — dissolution-, solubility- and permeability-limited absorption; BCS II and IV (pp. 9, 14) — stated in words; with Eqs. 1 and 14 (added)
W16 — supply against removal in the central compartment (pp. 14, 16–18, Fig. 18) — no equation in the review; balance from Eq. 1 (added)
W17 — reservoir: plateau length and droplet number against excess drug (pp. 14–15, Figs. 13, 14) — mass balance (added)
W18 — activity scale of Section 4.3: amorphous 1, crystal 0.1 (p. 15) — definition; differs from the scale of Eq. 8
W19 — dose number and the percent of dose in the compartment (pp. 13, 15) — not defined in the review; definition (added)
W20 — the simulation: flux, fraction absorbed, Table 1 (pp. 2, 15–18, Figs. 15–19) — equations not printed; reconstructed from Eq. 14 + mass balance (added)
Symbols: M, mass of solute dissolved (g); dM/dt, dissolution rate (g/s); D, diffusion coefficient of the solute in the medium (m²/s); S, surface area of the exposed solid (m²); h, thickness of the diffusion layer (m); C_s, solubility of the solute in the medium (g/m³); C, concentration in the medium (g/m³).
Where it comes from: the review names the Whitney–Noyes equation and the Nernst–Brunner diffusion layer model and prints no reference. Steps (added):
Across a stagnant layer of thickness h, with C_s at the solid surface and C in the bulk, a linear steady profile gives Fick's first law as J=D(C_s-C)/h.
Multiplying the flux by the area gives dM/dt=SJ, the first form of Eq. 1.
Under sink conditions, C\ll C_s, the difference reduces to C_s, the second form.
Assumptions and valid range: stated: C approximately zero for the second form. Needed and not stated: quasi-steady layer, constant S and h, and a solid surface that stays at C_s. The review applies the same equation to neat amorphous drug, with its higher solubility as C_s, and says that for multicomponent solids the driving force is "poorly understood" [review p. 12], so Eq. 1 is not a derived rate law for an ASD.
What it means for a formulation: D and h are set by the molecule and the gut, and S and C_s can be changed [review p. 2]. Worked with the simulation's solubilities (added): at equal S/h and under sink conditions, C_s=10 against 1 µg/mL gives a tenfold rate. The rate also falls as the solution fills: at C=8 µg/mL the amorphous term is 10-8=2, one fifth of its sink value (W16).
Symbols: x_D^{W(c)}, mole fraction solubility of the crystalline drug in water (dimensionless); \gamma_D^{W(c)}, activity coefficient of the drug in the liquid at saturation, relative to the pure supercooled liquid (dimensionless); \mu_D^{(c)}, chemical potential of the crystal (J/mol); \mu_D^{0(a)}, chemical potential of the pure supercooled liquid at T (J/mol); R, 8.314 J mol⁻¹ K⁻¹; T, temperature (K).
Where it comes from: cited from refs 19 and 20. Steps (added):
In solution, with the pure supercooled liquid as standard state: \mu_D^{W}=\mu_D^{0(a)}+RT\ln\left(x_D^{W}\gamma_D^{W}\right).
At saturation the dissolved drug and the crystal have equal chemical potential: \mu_D^{W(c)}=\mu_D^{(c)}.
Combining and dividing by RT gives Eq. 2.
Assumptions and valid range: stated: the solvent is not in the crystal; supercooled liquid as standard state. The right side is negative below T_m, since the crystal lies below the supercooled liquid, so the left side, the crystal's activity on this scale, is below 1 (added).
What it means for a formulation: solubility splits into a solid term (right side) and a solvation term (\gamma). Worked for felodipine at 37 °C with library inputs (added): the Hoffman ratio 8.64 [raina2014a] gives \ln(x\gamma)=-\ln 8.64=-2.16, so the crystal's activity is 0.116. With 1.21 µg/mL and MW 384, x=(1.21\times10^{-6}/384)/0.0551=5.7\times10^{-8} (0.0551 mol/mL for water at 37 °C), so \gamma_D^{W(c)}=0.116/5.7\times10^{-8}=2.0\times10^{6}.
In the wiki: solid–liquid equilibrium solubility equation, printed as \ln x=-\Delta G_{a\to c}/RT-\ln\gamma [ilevbare2013a E3]. This agrees with Eq. 2 only if \Delta G_{a\to c} is read as a positive magnitude, since the review writes the signed \mu_D^{(c)}-\mu_D^{0(a)} (noted in the check). Used in Amorphous solubility F5.
Eq. 3. Supersaturation
\ln S = \frac{\mu_D^{W(S)} - \mu_D^{(c)}}{RT} = \ln\frac{\gamma_D^{W(S)}x_D^{W(S)}}{\gamma_D^{W(c)}x_D^{W(c)}} \tag{3}
[review p. 3]
Symbols: S, supersaturation (dimensionless); \mu_D^{W(S)}, chemical potential of the drug in the supersaturated solution (J/mol); x_D^{W(S)}, \gamma_D^{W(S)}, its mole fraction and activity coefficient; others as Eq. 2.
Where it comes from: cited from ref 21 (Mullin). Steps (added):
From step 1 of Eq. 2 for the supersaturated solution: \mu_D^{W(S)}=\mu_D^{0(a)}+RT\ln\left(\gamma_D^{W(S)}x_D^{W(S)}\right).
From Eq. 2: \mu_D^{(c)}=\mu_D^{0(a)}+RT\ln\left(\gamma_D^{W(c)}x_D^{W(c)}\right).
Subtracting removes \mu_D^{0(a)} and gives Eq. 3, so S does not depend on the standard state.
Assumptions and valid range: none beyond a homogeneous solution with one chemical potential for the drug. S>1 is the review's definition of a supersaturated solution, against the most stable crystal.
What it means for a formulation: S is an activity ratio, so it is the quantity that flux reads (Eq. 14). Worked (added): felodipine with Vit E TPGS at its LLPS onset has a concentration ratio of about 4.5 and an activity ratio of about 9.2 (Eq. 14; W13), so the apparent \gamma_D^{W(S)}/\gamma_D^{W(c)} on total concentration is 9.2/4.5=2.0.
Symbols: S_C, concentration-based supersaturation (dimensionless); C, molar concentration in the supersaturated solution (mol/L); C^{*}, molar concentration of the saturated solution (mol/L). Misprint: the sentence after the equation prints "10C^{*}" (p. 3, confirmed at 300 dpi); the "10" is stray, since the symbol defined in Eq. 4 is C^{*}.
Where it comes from: a definition. The link to Eq. 3 (added):
In a dilute solution, x\approx C\,\bar V with \bar V the solution's molar volume, so x_D^{W(S)}/x_D^{W(c)}=C/C^{*}.
Eq. 3 then reads S=\left(\gamma_D^{W(S)}/\gamma_D^{W(c)}\right)S_C, the link the review states in words.
Assumptions and valid range: stated: good when the activity coefficient ratio is close to 1, likely at very low supersaturation and/or very dilute solutions in simple media. Section 3.8 removes the "simple media" case (W13). Needed and not stated: C^{*} measured in the same medium, with the same additives.
What it means for a formulation: for a drug in plain buffer the ratio can be read from concentrations; with micelles, proteins or cyclodextrins it under-reads the driving force. The felodipine numbers are worked in W13.
In the wiki: supersaturation ratio; [sugihara2018 E2] prints the same symbols. The review treats changes of C^{*} only for surfactants, whereas the library also has polymers raising it (Crystallization from solution F19).
Eq. 5. Amorphous solubility from the activity coefficient
\ln x_D^{W(a)} = -\ln\gamma_D^{W(a)} \tag{5}
[review p. 3]
Symbols: x_D^{W(a)}, mole fraction solubility of the amorphous drug in water (dimensionless); \gamma_D^{W(a)}, activity coefficient of the drug in the solution at that solubility, relative to the pure supercooled liquid (dimensionless).
Where it comes from: stated in words, as the amorphous form lacking the long-range interactions of the crystal. Steps (added):
Replace the crystal in step 2 of Eq. 2 by the pure supercooled liquid: \mu_D^{W(a)}=\mu_D^{0(a)}.
Then RT\ln\left(x_D^{W(a)}\gamma_D^{W(a)}\right)=0, so the activity at saturation is 1, which is Eq. 5.
Assumptions and valid range: the amorphous drug is treated as a supercooled liquid, also below T_g by extrapolation [review p. 3] (W3), and at this point it is pure. Water uptake, removed in Eq. 10, breaks this assumption.
What it means for a formulation: in this limit the amorphous solubility depends only on drug–water interactions. Worked (added): felodipine in buffer at 10.4 µg/mL [raina2015c] gives x=(10.4\times10^{-6}/384)/0.0551=4.9\times10^{-7} and so \gamma=2.0\times10^{6}; with the water term of 0.83 [raina2014a] (Eq. 13) it is 1.7\times10^{6}. This is close to the 2.0\times10^{6} from the crystal under Eq. 2, which is the assumption that Eq. 6 needs.
With \gamma_D^{W(c)}\cong\gamma_D^{W(a)}, and \ln\gamma_D^{W(a)}=-\ln x_D^{W(a)} from Eq. 5: \ln x_D^{W(c)}-\ln x_D^{W(a)}=\left(\mu_D^{(c)}-\mu_D^{0(a)}\right)/RT.
Rearranging gives Eq. 6. Since the fraction is negative, x_D^{W(a)}>x_D^{W(c)}.
Assumptions and valid range: stated: \gamma constant between the two solubilities, reasonable for poorly soluble compounds at very low concentration. Pure amorphous drug.
What it means for a formulation: the amorphous solubility can be predicted from one measured crystalline solubility and calorimetry. Worked for felodipine at 37 °C (added): 1.21\times8.64=10.5 µg/mL, against 10.4 measured [raina2014a; raina2015c].
Misprint. The page image (read at 300 dpi) prints the left side as \mu_D^{(c)}-\mu_D^{0(a)}, without the division by RT. As printed the equation cannot hold: the left side is in J/mol, and every term on the right is dimensionless (\Delta H_f/R in K times a term in K⁻¹; \Delta C_p/R dimensionless). The derivation below gives the right side for (\mu_D^{(c)}-\mu_D^{0(a)})/RT, the group that appears in Eqs. 2, 6 and 10, so the left side is corrected here. The right side is as printed. A reader who took the printed left side and divided by RT again for Eq. 6 would get a wrong solubility.
Symbols: \Delta H_f, enthalpy of fusion at T_m (J/mol); T_m, melting temperature (K); T, temperature of interest (K); \Delta C_p, heat capacity difference between solid and liquid, taken constant (J mol⁻¹ K⁻¹). The review does not state its direction; the signs of Eq. 7 require liquid minus crystal (steps below).
Where it comes from: cited from ref 20 (Sandler) with no steps. Steps (added), writing \Delta X=X_{\text{liquid}}-X_{\text{crystal}}:
At T_m: \Delta G=0, so \Delta S_f=\Delta H_f/T_m.
Kirchhoff, with constant \Delta C_p: \Delta H(T)=\Delta H_f-\Delta C_p(T_m-T).
Changing the sign and dividing by RT gives the three terms of Eq. 7 in order.
Assumptions and valid range: stated: \Delta C_p constant; pure crystal and pure amorphous drug, which holds for non-solvated crystals but not for amorphous drug in contact with water (Eq. 8). Needed and not stated: the supercooled liquid line is extrapolated down to T, so for a drug below its wet T_g the value belongs to the liquid and not to the glass (W3); the crystal melts without degrading, and pazopanib, which degrades, could not be treated (Amorphous solubility F7). Since y-\ln(1+y)>0 with y=(T_m-T)/T, a positive \Delta C_p always lowers the magnitude of the difference (added).
What it means for a formulation: a high T_m and \Delta H_f give a large gap and a large advantage [review p. 8]. Worked with library inputs (added), the first term alone, then the Hoffman form that the library uses, \Delta H_f(T_m-T)T/T_m^2 divided by RT:
Felodipine at 37 °C (\Delta H_f 28.67 kJ/mol, T_m 420.15 K, T 310.15 K [raina2014a M1]): first term 3448\times110/(420.15\times310.15)=2.91, factor e^{2.91}=18.4; Hoffman 2.91\times310.15/420.15=2.15, factor 8.57 (printed 8.64).
Atazanavir at 37 °C (\Delta H_f 63.5 kJ/mol, T_m 481.15 K, T 310 K [indulkar2017a M2]): first term 7638\times171.15/(481.15\times310)=8.76, factor about 6400; Hoffman 8.76\times310/481.15=5.65, factor 283.
The \Delta C_p terms therefore carry a large part of the result. To match Hoffman, Eq. 7 needs \Delta C_p/R=(2.91-2.15)/(0.3547-0.3036)=14.9 for felodipine and (8.76-5.65)/(0.5521-0.4396)=27.7 for atazanavir, or 124 and 230 J mol⁻¹ K⁻¹, about 1.8 times \Delta H_f/T_m in both. The Hoffman form is therefore not the \Delta C_p=0 limit of Eq. 7, and the error of omitting the term grows with undercooling.
Symbols: \mu_D^{L(a)}, chemical potential of the drug in the amorphous phase holding water (J/mol); \mu_D^{0(a)}, chemical potential of pure amorphous drug at T (J/mol); a_D^{L(a)}, activity of the drug in the water-saturated amorphous phase L (dimensionless). The equation is printed run together as "RTlna".
Where it comes from: the definition of activity with pure amorphous drug as standard state, so a_D^{L(a)}=1 for the dry phase and below 1 once water is present.
Assumptions and valid range: a single amorphous phase with one chemical potential at a given water content; standard state at the same T and pressure.
What it means for a formulation: this is the term that makes dry-state calorimetry insufficient: RT\ln a_D^{L(a)} is the free energy lost when the amorphous drug takes up water. Worked (added): a=0.27 for ritonavir [ilevbare2013a M1] is 2.579\times\ln 0.27=-3.4 kJ/mol at 37 °C.
Symbols: x_D^{L(a)}, mole fraction of the drug in the water-saturated amorphous phase; \gamma_D^{L(a)}, its activity coefficient (both dimensionless). The sentence before it gives the concentration as "\ln x_D^{L(a)}", where the concentration is x_D^{L(a)} (a slip in the text; the equation is right).
Where it comes from: the definition of the activity coefficient.
Assumptions and valid range: none; the values have to be found separately, which is what the water sorption isotherm does (Eq. 11).
What it means for a formulation: a few percent of water by weight is a large mole fraction, because the drug is heavy. Worked with library values (added): ritonavir (MW 720.9) with about 5 wt% water has x_D=(0.95/720.9)/[(0.95/720.9)+(0.05/18.0)]=0.32, so with a=0.27, \gamma_D^{L(a)}=0.84. Atazanavir (MW 704.86, 5.2 wt%) gives x_D=0.32, and with a=0.22, \gamma_D^{L(a)}=0.69. The mole fraction alone gives most of the reduction in both.
In the wiki: activity from concentration, the same form as a_{i1}=x_i\gamma_{i1} [ilevbare2013a E7]; most other papers there write a=\gamma C on a concentration scale. Composition of the phase: Amorphous solubility F1.
Eq. 10. Amorphous solubility with water in the amorphous phase
Where it comes from: the review says that Eq. 6 can be modified with Eqs. 8 and 9. The modification needs equal chemical potential between the phases, which is printed only later (Eq. 12). Steps (added):
Equilibrium: \mu_D^{W(a)}=\mu_D^{L(a)}.
With step 1 of Eq. 2 on the left and Eqs. 8–9 on the right, and the same standard state: \ln\left(x_D^{W(a)}\gamma_D^{W(a)}\right)=\ln\left(x_D^{L(a)}\gamma_D^{L(a)}\right). Eq. 5 is this line with a_D^{L(a)}=1.
From Eq. 2 with \gamma_D^{W(c)}\cong\gamma_D^{W(a)}: -\ln\gamma_D^{W(a)}=\ln x_D^{W(c)}-\left(\mu_D^{(c)}-\mu_D^{0(a)}\right)/RT.
Substituting step 3 in step 2 gives Eq. 10.
Assumptions and valid range: as Eq. 6, plus a metastable equilibrium with crystallization absent [review p. 4]; a non-solvated crystal, whose chemical potential water does not change.
What it means for a formulation: three things set the amorphous solubility: the crystalline solubility, the solid-state gap and the water taken up. Worked for ritonavir at 37 °C in water with library inputs (added): Hoffman 60.4\times84\times310.15/394.15^2=10.13 kJ/mol, e^{10.13/2.579}=50.8; with the water term 0.27, 13.7, about the review's "around 15"; and 13.7\times2.4=32.9 µg/mL against 37.2 measured. The wiki records this 32.9 against the paper's printed 39.2 [ilevbare2013a]. Without the water term the estimate would be 122 µg/mL.
Symbols: C_{\text{amorphous}}, amorphous solubility (same unit as C_{\text{eq}}); C_{\text{eq}}, measured solubility of the crystal; \Delta G_{c\to a}, free energy difference from crystal to amorphous (J/mol, positive); I(a_2), the integral below (dimensionless), with 1 = water and 2 = drug. Wording slip: the text calls "-I(a_2)" the activity of the water-saturated drug and equates it with x_D^{L(a)}\gamma_D^{L(a)}. It is e^{-I(a_2)} that equals this activity; -I(a_2) is its logarithm. The equation itself is right.
Where it comes from: quoted from ref 12 and called "essentially equivalent" to Eq. 10, with the isotherm procedure in words and no formula. Steps (added):
With \Delta G_{c\to a}=\mu_D^{0(a)}-\mu_D^{(c)}, the last factor is e^{\Delta G_{c\to a}/RT}.
In a dilute solution x is proportional to C with the same factor for both, so x_D^{W(a)}/x_D^{W(c)}=C_{\text{amorphous}}/C_{\text{eq}}. This is the "essentially".
Set e^{-I(a_2)}\equiv x_D^{L(a)}\gamma_D^{L(a)}=a_2.
Gibbs–Duhem at constant T and P: n_1\,d\ln a_1+n_2\,d\ln a_2=0, so d\ln a_2=-(n_1/n_2)\,d\ln a_1.
Integrating from dry solid (a_1\to0, a_2=1) to saturation (a_1\approx1): I(a_2)=-\ln a_2=\displaystyle\int_0^{a_1^{S}}\frac{n_1}{n_2}\,d\ln a_1.
Assumptions and valid range: as Eq. 10. The isotherm has to be extrapolated to water activity 1 [review p. 4], and library isotherms stop at 95% RH. Ionization is not included here; the library's general form adds a factor (W8).
What it means for a formulation: the review's two end points of the ratio can be reproduced (added). Ibuprofen at 25 °C with \Delta G 3.24 kJ/mol and e^{-I} 0.999 gives 0.999\times e^{3240/2478.8}=3.7, against the review's "only 4". Atazanavir at 37 °C with \Delta G 14.55 kJ/mol and e^{-I} 0.22 gives 0.22\times e^{14550/2577}=0.22\times283=62, against the review's ~65.
In the wiki: water-sorption correction, identical in [almeidaesousa2015 E1; ueda2021 E1]; Gibbs–Duhem equation, with separate Henry and high-water parts [hsieh2012 E5]. Wiki correction applied (2026-11-04): Amorphous solubility F7 and the formula page had recorded atazanavir as a mismatch (ratio about 228 against the printed 62) by reading 0.22 as I(a), as Table I of the 2017 paper heads the column (checked on that table). Table 2 of Indulkar 2015 lists 0.22 under exp[−I(a)] [indulkar2015 M2], and with that reading the printed inputs give 62; only the printed exponent sign of [indulkar2017a E3] stays wrong.
Eq. 12. Equal activity in the two liquid phases
a_D^{W(a)} = a_D^{L(a)} \tag{12}
[review p. 5]
Symbols: a_D^{W(a)}, activity of the drug in the drug-saturated bulk aqueous solution; a_D^{L(a)}, activity of the drug in the water-saturated drug-rich liquid (dimensionless). "Drug-rich liquid" includes stable liquids and metastable supercooled liquids or glasses.
Where it comes from: stated as well established. Steps (added): phase equilibrium is equal chemical potential; activity follows only if both phases share a standard state: \mu_D^{0(a)}+RT\ln a_D^{W(a)}=\mu_D^{0(a)}+RT\ln a_D^{L(a)}. The review's Eqs. 2 and 8 both use the pure supercooled liquid, so the condition is met, although it is not stated.
Assumptions and valid range: stated: for supercooled liquids and glasses the equilibrium is metastable, since the crystal is the stable form. Needed: no change of the coexisting compositions by a third component without allowing for it (W9).
What it means for a formulation: the drug has one activity however much drug-rich phase is present, which gives the flux plateau (W11) and the reservoir (W17).
Symbols: as Eqs. 5 and 9. As printed, the logarithm applies to each product.
Where it comes from: from Eq. 12 with Eq. 9 and its aqueous counterpart a_D^{W(a)}=x_D^{W(a)}\gamma_D^{W(a)} (added). This is step 2 of Eq. 10.
Assumptions and valid range: as Eq. 12.
What it means for a formulation: the amorphous solubility is x_D^{W(a)}=a_D^{L(a)}/\gamma_D^{W(a)} (added): anything that lowers the activity in the drug-rich phase, such as water or a second solute, lowers it, and anything that lowers \gamma in the water raises it. Worked for ritonavir (added): 37.2 µg/mL is x=(37.2\times10^{-6}/720.9)/0.0551=9.4\times10^{-7}, and with a=0.27, \gamma_D^{W(a)}=2.9\times10^{5}.
Symbols: J, membrane flux (mass per unit membrane area per unit time, e.g. µg cm⁻² min⁻¹); S, supersaturation of Eq. 3 (dimensionless), not an area; D, solute diffusion coefficient in the membrane (cm²/min); h, membrane thickness (cm); \gamma_m, solute activity coefficient in the membrane, on a concentration scale so that a/\gamma_m is a concentration (added). "*" is a multiplication sign.
Where it comes from: the review cites Higuchi (ref 78) and refs 39, 79, and gives the result as a proportionality. Steps (added):
Fick's first law in the membrane, with concentrations C_{m,0} and C_{m,h} at its two faces: J=D(C_{m,0}-C_{m,h})/h.
Sink receiver, C_{m,h}\approx0: J=DC_{m,0}/h.
Equilibrium at the donor face: the activity in the membrane equals the activity a in the donor, and C_{m,0}=a/\gamma_m. So J=\dfrac{D\,a}{h\,\gamma_m}.
From Eq. 3, a=S\,a_c, with a_c=\gamma_D^{W(c)}x_D^{W(c)} the activity at crystal saturation. So J=a_c\,S\,\dfrac{D}{h\gamma_m}, and the constant hidden by "\propto" is a_c.
Dividing by the flux from a crystal-saturated solution (S=1): J/J_c=S.
Assumptions and valid range: stated: D, h, \gamma_m constant for a known system, and systems where the membrane is the major barrier [review p. 9]. Needed and not stated: sink receiver and steady state; a donor at constant activity over the measurement; a membrane that passes only dissolved drug; no aqueous boundary layer in series, or one that adds no resistance. The last condition fails in the library's hollow-fibre and PAMPA data above the amorphous solubility (W11).
What it means for a formulation: flux reads activity, so it measures S whatever the medium. Worked from Fig. 10 for danazol (added): the review's values give J_{\max}/J_c=C_{am}/C_c=8/1=8, "about an order of magnitude". Read from the plot, the antisolvent points lie on J\approx0.0019\,C (0.0034 at 2, 0.0075 at 4, 0.0116 at 6 µg/mL), and the plateau of about 0.0156 corresponds to 0.0156/0.0019=8.2 µg/mL, matching the stated amorphous solubility. The crystalline point (about 0.0010 at about 0.9 µg/mL) lies below this line, which predicts 0.0017, so the measured flux ratio is about 15 against a concentration ratio of about 9. The review does not comment on this. The axis of Fig. 10 is labelled "µg/mL/cm²", which has no time in it and is not a flux unit (misprint in the figure).
Symbols: K_m, micellar partition coefficient "for a given concentration of micelle" (dimensionless); C_m, solute concentration in the micelle; C_w, solute concentration in the aqueous phase (same unit, e.g. µg/mL).
Where it comes from: a definition; the review notes that it is commonly measured at the saturation solubility.
Assumptions and valid range: for Eq. 16 to follow, C_m must be micelle-bound drug per volume of whole solution (added). K_m is then proportional to the micelle concentration and is not a material constant, which the phrase "for a given concentration of micelle" concedes. The review states that it depends strongly on solute concentration, normally falling as concentration rises [refs 86–88; review p. 11].
What it means for a formulation: a K_m measured at the crystalline solubility cannot be carried to a supersaturated solution; W13 gives the size of the error.
In the wiki: not on a formula page. The concept pages micellar solubilization and free drug concentration and Amorphous solubility F18 carry the measured K_{m/w}. That value (about 327 falling to about 99 for atazanavir in SDS) is on the paper's own scale; the review's definition applied to the totals of that study gives (45.6-1.1)/1.1=40 at the crystal and (1450-70)/70=20 at the amorphous solubility (added).
Eq. 16. Total solubility with micelles
S_{tot} = S_w + K_m S_w \tag{16}
[review p. 11]
Symbols: S_{tot}, total solubility at a given micellar surfactant concentration; S_w, solubility of the crystalline drug in water (same unit, e.g. µg/mL); K_m as Eq. 15, at crystal saturation.
Where it comes from: cited from ref 19. Steps (added): mass balance, S_{tot}=C_w+C_m; at saturation with the crystal, C_w=S_w; from Eq. 15, C_m=K_mS_w.
Assumptions and valid range: free drug stays at the crystal value, the review's "two populations"; only free drug drives absorption [refs 84, 85]. Monomeric surfactant is neglected. The form holds at saturation; using it above saturation with the same K_m is what fails (W13).
What it means for a formulation: more dissolved drug without more activity. Worked with the review's felodipine values (added): 40=1+K_m\times1 gives K_m=39 in Vit E TPGS, so 39 of every 40 molecules are bound and the driving force is unchanged. Calculating permeability from total concentration gives errors, as noted from ref 85.
Relation: at the binodal compositions x_{DW} and x_{WD} the chemical potential of each species is equal in both phases (common tangent); at the spinodal compositions x^{*}_{DW} and x^{*}_{WD}, \partial^2G/\partial x^2=0 [review p. 4, Fig. 1].
Where it comes from: Fig. 1 is adapted from ref 34. Steps (added):
For \Delta G_{mix}(x), the tangent at composition x meets the pure-component axes at the chemical potentials of the two species. A line tangent to the curve at two points gives equal \mu_D and equal \mu_W in both phases, which is Eq. 12 for the drug.
On the drug axis of Fig. 1 the gap between \mu_D^{*} (pure) and \mu_D (tangent) is -RT\ln a_D^{L(a)}, which is RT\,I(a_2) of Eq. 11: the water term is a length on this diagram.
Where \partial^2G/\partial x^2>0, small fluctuations raise G and a nucleus is needed; where it is negative they lower G and separation is spontaneous.
Assumptions and valid range: a binary drug–water system at fixed T and pressure; the drug treated as a liquid.
What it means for a formulation: the amorphous solubility is x_{WD}, and the drug-rich phase composition is x_{DW}. Between binodal and spinodal a one-phase solution can last for a time (W11).
Relation: at fixed T, x_D^{W(c)}<x_D^{W(a)}<x_D^{L(a)}. Below x_D^{W(c)} the solution is stable; between x_D^{W(c)} and x_D^{W(a)} it is supersaturated, one-phase and metastable, and only crystals can form; above x_D^{W(a)} LLPS or crystallization can occur [review pp. 5–6, Figs. 2, 3].
Where it comes from: the first inequality follows from Eq. 10, with the free energy term larger than the water term (added). The gap between the crystal and amorphous curves narrows on heating because \mu_D^{(c)}-\mu_D^{0(a)}\to0 at T_m in Eq. 7 (added). The review gives no equation for the shape of the LLPS dome in Fig. 2.
Assumptions and valid range: schematic; the upper closure of the dome and whether a drug has one are not given.
What it means for a formulation: every solution above the crystal curve is at risk of crystals, and the LLPS region lies wholly inside the region where crystals are favoured, so the drug-rich phase is itself supersaturated. Fig. 3 places a metastable boundary for fast crystallizers below the binodal.
Relation: the process is glass–liquid phase separation when T is below the T_g of the water-saturated drug-rich phase [review pp. 4, 12]. For drugs that are glasses at T, the supercooled liquid properties are extrapolated to T [review p. 3].
Where it comes from: a definition from ref 29. No equation for the wet T_g is given; [review p. 13] cites ref 95 for water plasticization.
Assumptions and valid range: the thermodynamic treatment of Eqs. 2–13 is kept for glasses by assumption, "for the purposes of understanding" [review p. 3]. A glass lies above the supercooled liquid in free energy, so its solubility should differ, and the review does not derive this.
What it means for a formulation: the review gives no example with numbers. From the library: telaprevir, with a wet T_g of about 52 °C at 37 °C, follows the supercooled-liquid heat capacity route and not the glass one (97 and 189 against 90–92 µg/mL measured) [mosqueragiraldo2015 R2, R13].
Relation: of about 50 compounds, about a third crystallized within 2.5 min of a solvent shift, a third did not crystallize after 1 h although a precipitate formed, and a third fell in between [review p. 6, ref 45].
Where it comes from: an empirical classification; the cut-offs come from the protocol, not from theory.
Assumptions and valid range: the protocol of ref 45 at high supersaturation; additives move a drug between classes [review p. 6].
What it means for a formulation: a slow crystallizer can reach W2's third region and form the reservoir without help, whereas a fast one needs an inhibitor before the amorphous solubility can be reached.
Relation: the phase separation concentration is the crossing of two straight lines through extinction at a non-absorbing wavelength against added concentration (Fig. 5, nisoldipine at 450 nm, 25 °C, about 7.4 µg/mL read from the plot). With pyrene, I_1/I_3 falls at the onset and rises at crystallization (Fig. 6, nifedipine, from about 1.8 to about 0.75, rising again beyond about 200 µg/mL) [review pp. 6–7].
Where it comes from: refs 53 and 61; empirical read-outs without an equation.
Assumptions and valid range: stated limits: overshoot under continuous addition; no separation of crystalline and amorphous phases by scattering; enough scattering near the onset; no autofluorescence or probe effect. Good agreement between methods suggests little overshoot [review p. 7].
What it means for a formulation: the onset is the routine measurement of the ceiling. Both signals are needed to tell droplets from crystals.
Relation: ratio =C_{\text{amorphous}}/C_{\text{eq}}, set by the free energy gap and the water taken up; ibuprofen only 4 (T_m ~77 °C), atazanavir ~65 (T_m ~208 °C), ritonavir about 15, telaprevir about 20 [review p. 8].
Where it comes from: from Eq. 11 divided by C_{\text{eq}} (added): ratio =e^{-I(a_2)}e^{\Delta G_{c\to a}/RT}. The solubility of the crystal cancels, so the predicted ratio does not depend on it.
Assumptions and valid range: as Eqs. 7 and 10; un-ionized drug. The caption of Fig. 7 assigns the extra scatter to errors in the crystalline solubility of compounds below 100 ng/mL, carried into the predicted amorphous solubility. By the cancellation above, the predicted ratio is free of this error, and it enters through the measured ratio on the other axis: \delta(\text{ratio})/\text{ratio}=-\delta C_{\text{eq}}/C_{\text{eq}} (added).
What it means for a formulation: the ratio is the largest gain in passive flux that supersaturation can give. Worked as free energies (added): RT\ln ratio at 37 °C is 3.57 kJ/mol for a ratio of 4, 6.98 for 15, 7.72 for 20 and 10.76 for 65; each factor of ten corresponds to 5.94 kJ/mol (5.71 at 25 °C). The review reads a ratio near 4 as unlikely to bring a substantial gain, and 15–20 as enough for two marketed ASDs.
Relation: higher ionic strength lowers crystalline and amorphous solubility; ritonavir reads about 37, 19, 14 and 5 µg/mL at 0, 100, 200 and 500 mM [review p. 8, Fig. 8].
Where it comes from: an observation from ref 35; the review gives no equation. A test (added) with the Setschenow form \ln(C_0/C)=K_sc, which the review does not cite, and the library's 37.2, 18.8, 14.3 and 4.7 µg/mL: K_s comes out at 6.8, 4.8 and 4.1 M⁻¹ at 0.1, 0.2 and 0.5 M, so one constant does not describe the curve from water. Part of the first step may come from replacing water by buffer rather than from salt, which the review does not separate.
Assumptions and valid range: one drug, 37 °C, phosphate.
What it means for a formulation: the amorphous solubility is quoted for one medium. Comparing values between papers needs the same buffer strength.
In the wiki: Amorphous solubility F9, which adds that the crystalline solubility falls with it (2.4, 1.3, 0.95 µg/mL).
W8. pH
Relation: for a weak base the total concentration rises as pH falls, the un-ionized concentration stays at the amorphous solubility, and the amorphous-to-crystalline ratio stays constant with pH [review pp. 8–9, ref 43].
Where it comes from: Henderson–Hasselbalch, named and not printed. Steps (added):
Total = un-ionized \times\left(1+10^{\mathrm{p}K_a-\mathrm{pH}}\right) for a base.
The un-ionized part is the species in equilibrium with the solid, so it is fixed at C_{\text{eq}} or C_{\text{amorphous}}.
The same factor multiplies both totals and cancels in the ratio, as long as the solid phases do not change.
Assumptions and valid range: stated: until the salt form crystallizes. Needed: no ionized drug or counterion in the drug-rich phase; crystallization slower than the measurement.
What it means for a formulation: pH moves the total concentration, not the ceiling of activity. Worked (added): clotrimazole with pK_a 5.89 at pH 4 has a factor of 1+10^{1.89}=78.6, so 490.2 µg/mL total is 6.2 µg/mL un-ionized, within the 5.6–8.4 µg/mL that the library reports [indulkar2015 M1, R9].
Relation: a second component that mixes into the drug-rich phase lowers the amorphous solubility in proportion to the mole fraction of the drug there; at a mole fraction of 0.5 the solubility of each drug is halved (ritonavir–lopinavir) [review p. 9, Fig. 9]. Polymers below 7 wt.% in the drug-rich phase have no discernible impact.
Where it comes from: stated with Eqs. 11 and 12 and ideal mixing (ref 68). Steps (added):
From Eq. 13, x_D^{W(a)}\propto a_D^{L(a)} at constant \gamma_D^{W(a)}.
Ideal mixing with mole fraction X_D on a water-free basis: a_D^{L(a)}=X_D\,a_{D,\text{pure}}^{L(a)}.
So C_{\text{amorphous}}(X_D)=C_{\text{amorphous}}^{\text{pure}}X_D, the lines of Fig. 9.
Assumptions and valid range: ideal mixing; unchanged water content; the second component stays out of the aqueous activity term. The ionized diclofenac does not mix and has no effect. The basis of X (with or without water) is not stated in the review.
What it means for a formulation: Worked from Fig. 9 (added): the pure values are 26 and 16 µg/mL, so at X_{\text{ritonavir}}=0.5 the lines give 13 and 8 µg/mL; the plotted ritonavir points lie mostly above the line at intermediate X. For a polymer at 7 wt%, the athermal Flory–Huggins activity with a long chain is \ln a=\ln0.93+0.07=-0.003, a 0.3% fall (added), so the "no discernible impact" follows from this composition if \chi\approx0; the same form cannot give the larger falls found later.
W10. Reciprocal solubility and partition coefficient
Relation: solutions saturated with crystal gave nearly constant flux over about two orders of magnitude in concentration; apparent solubility and apparent membrane–solution partition coefficient are reciprocal, so a surfactant, co-solvent or complexing agent does not raise the flux [review p. 9].
Where it comes from: refs 70, 71, 77, 78, in words. Steps (added):
In concentration terms, J=(D/h)KC, with partition coefficient K=C_{m,0}/C.
Equal activity at the interface: \gamma C=\gamma_mC_{m,0}, so K=\gamma/\gamma_m.
For a crystal-saturated vehicle, \gamma C_s=a_c, so KC_s=a_c/\gamma_m, whatever the vehicle. K\propto1/C_s is the reciprocity, and J reduces to Eq. 14 at S=1.
Assumptions and valid range: as Eq. 14; \gamma here is the apparent coefficient on total concentration; the vehicle does not change the membrane.
What it means for a formulation: solubilizing raises concentration, not delivery. Worked from the library (added): felodipine suspensions at about 1 and 40 µg/mL gave about 0.009 and 0.006 µg/min, a 40-fold concentration change with no gain in flux [raina2015c R4]; 5 µg/mL HPMCAS doubled the celecoxib solubility with mass flow unchanged at 0.44 µg/min [li2016a R15].
W11. The flux plateau and the upper limit of supersaturation
Relation: flux rises linearly with concentration up to the amorphous solubility and is constant above it; solutions from ASDs and from antisolvent give a similar flux; the upper limit of supersaturation for "any" formulation is set by the amorphous solubility, "or strictly speaking, the spinodal decomposition point" [review pp. 9–10, Figs. 10, 11].
Where it comes from: from Eqs. 12 and 14. Steps (added):
Below the binodal, one phase, constant \gamma: S=C/C_c, so J=J_c\,C/C_c.
Above it, with two phases at equilibrium, Eq. 12 fixes the activity at the value for C_{am}, whatever the amount of drug-rich phase, so J=J_c\,C_{am}/C_c.
Together: J=J_c\min(C,C_{am})/C_c.
Gap in the derivation: the review uses both limits. At equilibrium the coexisting solution is at the binodal, x_{WD}, so a two-phase solution cannot exceed the amorphous solubility. Only a homogeneous solution held between binodal and spinodal can, and Fig. 1 calls that range metastable. The review's "spinodal" is therefore a transient limit, as Fig. 3 draws it (the slow crystallizer falls from the spinodal to the binodal in time), whereas Eq. 12 and the flux plateau describe the binodal. The text on p. 13 has the solution "undergo spinodal decomposition" as dissolution proceeds, which is not shown by any measurement in the review.
Assumptions and valid range: as Eq. 14, with droplets that do not cross the membrane and do not otherwise add to transport. "Any" is the review's statement ("an important conjecture"), which needs validation with activity-based supersaturation [review p. 10].
What it means for a formulation: the amorphous-to-crystalline ratio bounds the gain in passive flux. Danazol: 8/1 by concentration. Adding drug above about 8 µg/mL enlarges the drug-rich phase and leaves the flux unchanged (Eq. 14).
Relation: in a supersaturated solution the drug's activity exceeds that of the crystal; in a solubilized solution it stays at or below it [review p. 10].
Where it comes from: Eq. 3 read as S>1 or S\le1 (added), where total concentration plays no part.
Assumptions and valid range: a definition. A solution can be both, with a solubilizer present and activity above the crystal, as in Section 3.8.
What it means for a formulation: concentration alone does not show which case applies. Worked (added): felodipine at 40 µg/mL is saturated (S=1) in Vit E TPGS. In buffer the same concentration would be 40-fold on concentration, which a single phase cannot reach because LLPS sets in at about 10 µg/mL.
Relation: Eq. 4 under-predicts supersaturation with micelles, cyclodextrins and proteins; if K_m were constant Eq. 4 would hold; a K_m falling with concentration is "the most plausible explanation"; the amorphous-to-crystalline ratio for felodipine fell from ~9 to ~4.5 while maximum flux was unchanged [review p. 11].
Where it comes from: the algebra is left to the reader. Steps (added):
Total concentration, with Eq. 15 per volume of solution: C_{tot}=C_w(1+K_m).
Saturation with the crystal, Eq. 16: C^{*}=S_w(1+K_m^{*}).
Activity follows free drug: S=C_w/S_w at constant \gamma in the aqueous phase.
So S_C=C_{tot}/C^{*}=S\,(1+K_m)/(1+K_m^{*}).
If K_m=K_m^{*}, then S_C=S, the review's statement. If K_m<K_m^{*}, then S_C<S, the review's under-prediction.
Assumptions and valid range: the free-drug activity coefficient is unchanged by the micelles; the micelles do not change with drug loading; flux reads free drug (Eq. 14).
What it means for a formulation: Worked with the review's felodipine numbers (added). In Vit E TPGS, at the crystal, 1+K_m^{*}=40/1, so K_m^{*}=39. At the amorphous solubility, free drug is at the buffer value of 10 µg/mL and the total is 177 µg/mL, so 1+K_m=17.7, K_m=16.7. Then S_C=177/40=4.4, while S=10, and 4.4=10\times17.7/40. So under Eq. 15 the printed numbers require K_m to fall by a factor of 2.3 between the two solubilities, and the same factor is the under-prediction. A formulator reading 4.4 would conclude that half the benefit had been lost, whereas the flux shows none lost. Ref 79 did not measure K_m, so this is the reading of the review, and the fall was measured later (below).
In the wiki: activity-based supersaturation ratio; Bioperformance F4 (onsets 10.4, 177.3, 214.6 µg/mL; ratios 9.2, 4.5, 5.0); Amorphous solubility F18, which measures the fall of K_m, and F16 (atazanavir in 3.6 mg/mL SDS: 33.0 by concentration, about 65 by flux; with a total of about 1450 µg/mL and free drug of about 65–70 µg/mL, 1+K_m\approx21–22, and step 4 then requires 1+K_m^{*}\approx21.5\times65/33\approx42, added). The ratio does not always fall (F12).
W14. Free un-ionized drug in the central compartment
Relation: only un-ionized drug in the central aqueous compartment partitions into the membrane, and its activity, not the total in all compartments, sets the kinetics of passive absorption [review pp. 11–12, Fig. 12].
Where it comes from: ref 89 (pH-partition) with Eq. 14; as an equation (added): J\propto a_u, with a_u=\gamma_uC_u and C_u=C_{T}/(1+10^{\mathrm{p}K_a-\mathrm{pH}}) for a base in the absence of micelles.
Assumptions and valid range: the review says "thought to", and only passive transcellular absorption is considered. Exchange between compartments is taken as an assumed rate, without an equation.
What it means for a formulation: droplets, micelles, complexes, ionized drug and crystals matter only through how fast they refill this one pool. The review groups droplets and micelles as compartments that "equilibrate rapidly" [review p. 13]; the two differ in activity (W13).
Relation: dissolution-limited bioavailability is met by faster dissolution, and solubility-limited by supersaturation [review p. 9]. For BCS class IV drugs membrane diffusion is likely rate-determining and activity "may not be the most critical parameter"; for class II, supersaturation helps; removal may outpace dissolution, and then absorption is dissolution-rate-limited [review p. 14].
Where it comes from: in words. Steps (added), with V the compartment volume:
Supply by Eq. 1, (DS/h)(C_s-C); removal by Eq. 14, AJ=Pa in the units of the case, with k_a=PA/V.
If removal is much faster than supply, C falls below C_s and rate equals supply: dissolution-limited.
If supply is much faster, C stays at C_s and rate is k_aC_sV: solubility-limited.
Gap in the derivation: the class IV statement does not follow from Eq. 14, in which J is proportional to activity at any fixed permeability; a low P lowers the prefactor, not the proportionality. What a low P changes is step 3 against step 2: with slow removal, supply keeps up, and the fraction absorbed is limited by k_a and residence time. The statement should therefore be read as being about the rate-limiting step and not as a weaker dependence of flux on activity. The review prints no numbers for it.
Assumptions and valid range: as Eqs. 1, 14; a single well-mixed compartment.
What it means for a formulation: the simulation takes the case of step 3 by choosing a high permeability [review p. 15].
Relation: when the dissolution rate is comparable to the absorption rate, activity in the central compartment may be significantly lowered by absorption; Fig. 18 compares activity with and without absorption [review pp. 14, 16–17, 18].
Where it comes from: no equation is printed. A balance (added), with a=C/C_{am}: da/dt=k_d(1-a)-k_aa.
Without absorption (k_a=0): a=1-e^{-k_dt}. The upper curve of Fig. 18 reads about 0.97 at 8 h, giving k_d=0.44 h⁻¹, and then 0.36, 0.58 and 0.83 at 1, 2 and 4 h, as plotted.
With absorption the steady value is a_{ss}=k_d/(k_d+k_a). With k_a=0.8 h⁻¹ (W20): 0.44/1.24=0.35, reached within about 3 h.
The figure gives 0.29 at 8 h and is still rising after 3 h (about 0.22), so a constant k_d does not reproduce the curve. It would need k_d=0.33 h⁻¹ at the end, and a fall in k_d as the solid is used up would act in this direction, but the review does not state its dissolution model.
Assumptions and valid range: added form only; the missing items are the review's rate law and rate constants for ASD dissolution.
What it means for a formulation: the closed-vessel reading overstates the supersaturation of a slow ASD: about 0.97 against 0.29 at 8 h here. A closed-vessel test can show a formulation at the ceiling, while in the gut it never reaches it.
Relation: for clotrimazole (amorphous solubility 7.5 µg/mL), flux maxima are proportional to donor concentration from 0.5 to 7 µg/mL and decline with time; above 7.5 µg/mL, 200–250 nm droplets form, their number rises with the excess "in accordance with mass balance", and the plateau length "roughly correlates" with the excess [review pp. 14–15, Figs. 13, 14].
Where it comes from: stated in words, from ref 107. Steps (added):
Droplet mass per volume is m_d=C_0-C_{am}, so n\,(\pi/6)d^3\rho=C_0-C_{am}.
Removal at constant flux: t_{\text{plateau}}=(C_0-C_{am})V/(JA), which predicts a plateau proportional to the excess. V and A are not printed in the review, so the absolute time cannot be computed.
Assumptions and valid range: fast equilibrium, which the review expects from pre-wetting and surface area; no coarsening or crystallization during the run.
What it means for a formulation: Worked from Fig. 14 (added; density 1.2 g/cm³ assumed, one size per run): at 10, 15 and 20 µg/mL, about 223, 246 and 257 nm and about 3, 5.8 and 12.4 × 10⁸ droplets/mL (printed "x108", a lost superscript) give about 2.1, 5.3 and 13 µg/mL, against excesses of 2.5, 7.5 and 12.5. The mass balance holds in order of size. For length, excesses in the ratio 1:3:5 predict from the 1 h at 10 µg/mL about 3 and 5 h, against about 5 and at least 13 h in the library. From Fig. 13 (added), the plateau of about 0.96 µg/h·cm² exceeds what the proportional range predicts at 7.5 µg/mL (0.62\times7.5/7=0.66). Depletion before the peak in panel A may account for part of this, and the review does not comment on it.
Relation: activity of the amorphous phase 1, of the crystalline phase 0.1 for solubilities of 10 and 1 µg/mL; "1" is "defined as the activity of the water-saturated amorphous material" [review pp. 15].
Where it comes from: a definition: a_{\text{sim}}=C/C_{am} (added).
Gap in the derivation: this is not the scale of Eqs. 2, 8 and 13, on which pure dry supercooled liquid is 1 and the water-saturated phase is a_D^{L(a)}=e^{-I(a_2)}<1. The conversion (added) is a_D=a_{\text{sim}}\,e^{-I(a_2)}. Ratios do not change: S=a_{\text{sim}}/a_{\text{sim},c}=10. The text on p. 14 also calls the highest activity "the amorphous solubility", which is a concentration.
Assumptions and valid range: constant \gamma between 1 and 10 µg/mL.
What it means for a formulation: a value of "activity" must be read with its scale. A statement such as a=0.3 here means 30% of the ceiling and S=3.
W19. Dose number and the percent of dose in the compartment
Relation: "high dose number" is used without a definition [review pp. 13–15]; 20% of the dose is in the aqueous compartment at the amorphous solubility, 2% at the crystalline solubility, ~6% at 0.29 activity.
Where it comes from: definition (added): D_0=(\text{dose}/V)/C_s. Percent of dose dissolved at C: 100\,CV/\text{dose}.
Assumptions and valid range: well-mixed compartment; V of 100 mL from ref 109.
What it means for a formulation: Worked (added): 5\ \text{mg}/100\ \text{mL}=50 µg/mL, so D_0=50 against the crystal and 5 against the amorphous solubility. Percent dissolved: 10\times100/5000=20\%, 1\times100/5000=2\%, 3\times100/5000=6\%, 2.9\times100/5000=5.8\%, matching panels D of Figs. 15–19. With D_0\le1 against the amorphous solubility there would be no droplets, so the reservoir applies only for D_0>1 on this basis.
Relation: for a 5 mg dose, 100 mL, solubilities 1 and 10 µg/mL, and high permeability, the fractions absorbed at 8 and 4 h are those of Table 1 (printed on p. 2, introduced on p. 18) [review pp. 15–18, Figs. 15–19].
Where it comes from: the review prints neither the equations nor the permeability, so this is a reconstruction (added). The model that fits is first-order removal from the compartment, d(\%\,\text{absorbed})/dt=k_a\times(\%\ \text{of dose dissolved}), which is Eq. 14 in dose units, flux tracking activity: 2400, 720 and 240 in Figs. 15A–16A for a of 1, 0.3, 0.1.
From Case 1, 96% in 6 h with 20% dissolved: k_a=16/20=0.8 h⁻¹.
Case 4: 0.8\times2=1.6 %/h, \times8=12.8 (printed 13).
Case 2a: first 0.5 h, 16\times0.5=8; next 0.5 h with mean a=0.55, 0.55\times16\times0.5=4.4; 1 h total 12.4; then 1.6\times7=11.2 ("another 11%"); total 23.6.
Case 2b: 8+0.65\times16\times0.5=13.2; then 4.8\times7=33.6 ("34%"); total 46.8.
Case 2c: 16\times2=32 ("over 32%"); 0.65\times16\times2=20.8; 4 h total 52.8; then 4.8\times4=19.2; total 72.0.
Case 3, integrating the flux curve of Fig. 17A read at each hour (about 0.13, 0.18, 0.23, 0.26, 0.28, 0.30, 0.30, 0.31 of 2400): about 11 at 4 h and 29 at 8 h.
Case
8 h printed
8 h reconstructed
4 h printed
4 h reconstructed
1
100
see below
65
64
2a
23
23.6
17
17.2
2b
46
46.8
27
27.6
2c
71
72.0
52
52.8
3
30
29
11
11
4
13
12.8
6
6.4
Gap: Case 1 does not close its mass balance. The text gives 96% absorbed at 6 h and, for the same 6 h, 20% of the dose in the aqueous compartment, and panels B and D of Fig. 15 show the same. The sum is 116% of the dose. Fig. 15B rises only from about 95 to 100 after 6 h, while panel D falls from 20 to 0, so about 15% of the dose is unaccounted for, and the fall in panels A, C and D (about tenfold per hour) is faster than the k_a of 0.8 h⁻¹ of the plateau would give. A closed balance (added): droplets hold 80%, removed at 16 %/h, so they are gone at 80/16=5 h, when 80% is absorbed. After that, the remaining 20e^{-0.8(t-5)} gives 91% at 6 h and 98% at 8 h. The 4 h value of Table 1 is not affected, and Case 1 still leads by a wide margin (98 against 72 for Case 2c), so the ranking stands, but "constant for 6 h", 96% at 6 h and 100% at 8 h are not what the stated inputs give.
Other inconsistencies: Case 2b says that the first hour is "the same as for Case 1a" (no such case; read 2a) and that at 1 h the drug is distributed "with the same ratios as in Case 2a". Fig. 16D shows 6% against 2% at 1 h, and Fig. 16C shows 0.3 against 0.1, so only the first 30 min coincide. "(See Fig. 17.)" closes the Case 2c paragraph, whereas Case 2c is in Fig. 16. Text activity 0.29 for Case 3 and flux in Fig. 17A (about 750, i.e. 0.31) differ slightly. All are small except the mass balance.
Assumptions and valid range: stated: passive absorption only; 8 h window with constant permeability, which is "somewhat long" and "unlikely to be valid for a majority of compounds" [review p. 18]; recrystallized solids hold the stated solubility. The simulation does "not contain sufficient parameters to represent the complex biological situation" [review p. 15]. Missing: k_a, membrane area and permeability; the dissolution rate law of Case 3; crystallization kinetics, which are imposed as times and not computed, so absorption cannot delay crystallization here.
What it means for a formulation: the 8 h result turns on how long the maximum activity lasts, with each hour at a=1 worth 16% of the dose, and each hour at a=0.1 worth 1.6%. Moving crystallization from 30 min to 2 h (2b to 2c, same final solubility) is worth about 25 points at 8 h. Halving the window hurts the slow cases most in ratio: Case 3 falls from 30 to 11, Case 1 from 100 to 65.
Years are those of the review's reference list; "printed" marks where this differs from the card's year. Attribution is the review's own and has not been checked against the older sources.
1. Dissolution of solid dispersions (Eq. 1). The rate law is named Noyes–Whitney, with the Nernst–Brunner film, and no reference is printed [review p. 2]. For dispersions, Simonelli, Mehta and Higuchi (1969, 1976, refs 9, 10) noted for sulfathiazole–PVP that the polymer dissolved faster than the drug, so that the drug's dissolution rate fell with time, more so at higher drug loading; Corrigan (1985, ref 94) proposed drug-controlled dissolution at moderate to high loadings and polymer-controlled release at low loadings [review p. 13]. Taylor group: Alonzo 2010 and Alonzo 2011 followed dissolution and crystallization from amorphous drugs and dispersions together, and showed that a polymer is needed for the concentration to keep rising [review p. 13]. After the review: dissolution was followed with a sink present, Hate 2017 and Hate 2018, which measure the difference that W16 leaves as a model.
2. Amorphous solubility from thermodynamics (Eqs. 2, 5–7, 10, 11). The heat-capacity route to the free energy of a glass goes back to Parks and co-workers (1934, 1936, refs 23, 24). Hancock and Parks (2000, ref 27) asked what the true solubility advantage is. Murdande and co-workers (2010, refs 12, 26, 50) added water in the amorphous phase, giving Eq. 11, and found that the prediction still often exceeded experiment, in systems that crystallized before reaching the limit. Paus and co-workers (2015, ref 22) treated the amorphous solubility as a liquid–liquid equilibrium [review p. 3]. Taylor group: Almeida e Sousa 2015 (printed 2014) and Mosquera-Giraldo 2015 (printed 2014) compared Eq. 10 with an experimental onset, so that the comparison was no longer made against a crystallizing system [review pp. 4, 8]. After the review: predictions now extend to other drugs and to the choice of heat-capacity route; see the formula pages under Eqs. 7 and 10 and the Ueda 2024 review.
3. The amorphous solubility as a phase boundary (Eq. 12, W1–W5). Miscibility gaps in water were known from phenol (Campbell and Campbell, 1937, ref 33) and described later for local anaesthetic drugs (Ceolin 2010, Rietveld 2013, refs 31, 32) [review p. 4]. Tachibana and Nakamura (1965, ref 42) and Brick and co-workers (2003, ref 25) made colloidal dispersions by solvent shifting, and the link to the amorphous solubility is "inferred" from Brick [review p. 4]. "Oiling out" before crystallization came from process chemistry (Bonnett 2003, Deneau and Steele 2005, refs 36, 34) [review pp. 4–5]. Light scattering for the binodal and spinodal came from proteins (Thomson 1987, ref 54), passed to a drug by Lafferrere (2004, ref 55), and nanosuspensions gave a bulk monomer concentration (Lindfors 2006, ref 52) [review p. 6]. Taylor group: Hsieh 2012 used titration; Ilevbare 2013 made LLPS the subject for pharmaceuticals, including the effect of ionic strength, and Ilevbare 2013 the effect of additives on droplets; Van Eerdenbrugh 2014 (ref 45, no card in the library) classified crystallization behaviour; Raina 2015 introduced fluorescent probes for the onset; Indulkar 2015 showed the pH dependence. After the review: the effect of polymer type and grade on the onset and on droplet stability, Ueda 2020 and Ueda 2020, and the effect of polymer ionization on crystallization, Shah 2025.
4. Flux follows activity (Eq. 14, W10, W11). Higuchi (1960, ref 78) gave the analysis for creams and ointments, Yalkowsky and Flynn (1974, ref 71) the correlation with vehicle polarity, and Twist and Zatz (1986, ref 70) the near-constant flux at constant activity across solubilities differing by orders of magnitude. Supersaturated donors were then tested on skin and silicone: Davis and Hadgraft (1991, ref 108), Megrab (1995, ref 72), Pellett (1997, ref 73), Moser (2001, ref 74) [review p. 9]. For oral delivery, Brouwers and co-workers (2009, ref 76) made supersaturation an absorption strategy, and Miller and co-workers (2012, ref 75) showed that it gives solubility without a loss of permeability [review p. 9]. Taylor group: Raina 2014 established the upper limit of flux at the amorphous solubility; Trasi 2015 lowered the limit with a second drug miscible in the drug-rich phase (ritonavir and lopinavir, W9); Jackson 2015 (printed 2016) gave the same limit from dispersions; Indulkar 2016 added duration through the reservoir [review pp. 9–10, 14]. After the review: Hate 2021 and Hate 2018 reported transport rising above the limit where droplets add to transfer (Bioperformance F15, F22); Almeida e Sousa 2016 tested other solid forms against the limit.
5. Solubilized drug and free drug (Eqs. 15, 16, W12–W14). The pH-partition hypothesis dates from Shore, Brodie and Hogben (1957, ref 89) [review p. 12]. That only free drug drives absorption was recognized by Poelma (1991, ref 84), and Katneni (2006, ref 85) showed that permeability calculated from total drug is wrong with micelles [review p. 11]. Eq. 15 follows Yalkowsky (1999, ref 19), and the fall of K_m with concentration comes from Goto and Endo (1978), Lee (1990) and Croy and Kwon (2005), refs 87, 86, 88 [review p. 11]. Taylor group: Raina 2015 showed that a concentration ratio under-reads activity while the maximum flux is unchanged [review p. 11]. After the review: Indulkar 2017 measured the fall of K_m, and Elkhabaz 2019 extended the flux-based reading to biorelevant media and human fluid (Amorphous solubility F16, F18).
6. From solution to absorption (W15–W20). The "spring and parachute" description is cited to Brouwers (2009, ref 76), and the 100 mL compartment to Mudie and co-workers (2014, ref 109) [review pp. 15–16]. Taylor and Zhang's addition is the simulation and the term "spring and plateau" for Case 1, a design target and not a measurement [review p. 16]. After the review: rats, with Wilson 2018 (ceiling against crystals) and Wilson 2020 (short inhibition was enough for a drug absorbed quickly, Bioperformance F24); and Hate 2019, which couples absorption with growth on seeds (Bioperformance F29).
Open points.
Is the limit the binodal or, as the review also allows, the spinodal (W11)? The library uses the binodal and has only a kinetic overshoot for danazol (Crystallization from solution F22).
Do co-crystals, lipid formulations and prodrugs reach the amorphous solubility? The one co-crystal tested did not, and lipid and prodrug routes are absent (Amorphous solubility F14).
Is the activity at the amorphous solubility kept in every micellar medium, and does K_m fall by the amount W13 requires? Tested for two drugs (Amorphous solubility F16).
Does activity matter less for low-permeability drugs (W15)? Eq. 14 alone does not say so, and all the library's data concern high-permeability drugs.
How do absorption and growth on seeds trade off? The one model has constants that do not reproduce (Bioperformance F29).
Is "spring and plateau" better than "spring and parachute" for a slowly absorbed drug? Short inhibition sufficed for enzalutamide, and no slow-absorption comparison exists (Bioperformance F24).
Do the simulated fractions hold in humans (Overview F14)? In addition, the model has not been published in a form that can be rerun, and Case 1 does not close its mass balance (W20).
Does the plateau length scale with the excess as the mass balance requires (W17)? The library's values are longer than proportional.