Following Moseson D.E., Taylor L.S. (2023) Crystallinity: A Complex Critical Quality Attribute of Amorphous Solid Dispersions. Molecular Pharmaceutics.
Equations and their derivation: crystallinity in amorphous solid dispersions
How to read this page. Each section takes one equation of Moseson and Taylor (2023, Mol. Pharmaceutics 20, 4802–4825) as printed, with the review page ([review p. N] is the PDF page; p. 1 is journal page 4802), and gives the algebra that connects it to the equations before it. The review prints the nine equations without derivation, so every intermediate line here is marked (added), and numbers that come from the library and not from the review are marked with their source. Sections W1–W10 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 — rate of crystal dissolution into a liquid (polymer melt) — from outside (Noyes–Whitney; film form), no source cited in the review
Eq. 2 — diffusion coefficient of the dissolving species — from outside (Stokes–Einstein), feeds D of Eq. 1
Eq. 3 — depressed melting point T_c against composition, which fixes C_s of Eq. 1 — from outside (Flory–Huggins lattice, refs 42, 44–48) + solid–liquid equilibrium with \Delta C_p = 0
Eq. 4 — nucleation rate — from outside (classical nucleation theory, ref 79); the barrier follows from Eq. 5; printed without the minus sign
Eq. 5 — supersaturation as chemical potential and activity ratio — definition, from \mu=\mu^\circ+RT\ln a; last equals sign missing in print
Eq. 6 — supersaturation as concentration ratio — from Eq. 5 + \gamma/\gamma^*\approx 1
Eq. 7 — power-law crystal growth rate per unit area — from outside (diffusion–reaction model, ref 63); exact only for first-order integration
Eq. 8 — sink index against the crystalline solubility — definition (refs 80, 137, 138)
Eq. 9 — sink index against the amorphous solubility — definition, same form as Eq. 8
Relations stated in words or in a figure:
W1 — miscibility criterion on \chi (p. 4) — from the Flory–Huggins free energy behind Eq. 3
W2 — phase-diagram regions and processing regimes, and how many crystals must remain below T_c (pp. 4, 11) — from Eqs. 1 and 3 + mass balance
W3 — rule T_g - T > 50 °C (pp. 6, 16) — from outside (Hancock 1995, 2001)
W4 — dose scenarios \mathrm{SI_{am}}=1 and \mathrm{SI_{am}}<1, highest supersaturation and the crystal fraction that costs no plateau (p. 9) — from Eqs. 6, 8, 9
W5 — theoretical maximum concentration against % crystallinity (Fig. 10c, p. 11) — empirical line between two measured end points
W6 — seed effect scales with surface area, not mass (pp. 9–10) — from Eq. 7 + sphere geometry
W7 — stated detection limits of the solid-state methods (pp. 7–8, 11) — from outside (cited values)
W8 — limits of dissolution as a crystallinity test: ±5% and <10%, 6%, 2–20× (pp. 8, 9, 18) — from outside; arithmetic added
W9 — crystal growth stops when the matrix is no longer supersaturated (pp. 13–14, 16) — from Eq. 7 + mass balance; the mass balance does not support the review's reading
W10 — thresholds from case studies: crystallinity against degradation (p. 12), polymer dissolution pH (p. 17) — from outside (Davis 2021; Nguyen 2023)
Eq. 1. Rate of crystal dissolution into the molten polymer
Symbols: C, solute concentration in the liquid phase at time t (mass per volume; the review gives no units, and the population balance work uses g/g); C_{\mathrm{s}}, equilibrium solubility of the crystalline drug in the molten polymer; D, diffusion coefficient (m²/s); A, surface area available for dissolution (m²); h, thickness of the mass transfer boundary layer at the solid–liquid interface (m); V, volume of the liquid phase (m³); t, time (s).
Where it comes from: the review names Noyes–Whitney and cites no origin. The form with D and h is the film model. Steps (added):
Steady diffusion across a film of thickness h, with C_{\mathrm{s}} at the crystal surface and C in the bulk (Fick's first law): J = D(C_{\mathrm{s}}-C)/h.
Mass balance over area A: \mathrm{d}M_{\mathrm{dis}}/\mathrm{d}t = AJ = \frac{DA}{h}(C_{\mathrm{s}}-C).
Constant V, C = M_{\mathrm{dis}}/V: dividing by V gives Eq. 1.
Constant A and C(0)=C_0: C(t)=C_{\mathrm{s}}-(C_{\mathrm{s}}-C_0)\exp\!\left(-\frac{DA}{hV}t\right). In a process A is not constant, as steps 5–6 show.
One spherical crystal of density \rho and radius r: \mathrm{d}M/\mathrm{d}t = 4\pi r^2\rho\,\mathrm{d}r/\mathrm{d}t = -4\pi r^2 J, so \mathrm{d}r/\mathrm{d}t = -\frac{D}{\rho h}(C_{\mathrm{s}}-C). At fixed h the time to dissolve is proportional to the initial size.
A finite amount of drug: with M the undissolved mass and C_{\mathrm{tot}} the drug content per melt volume, C = C_{\mathrm{tot}}-M/V, so C_{\mathrm{s}}-C = (C_{\mathrm{s}}-C_{\mathrm{tot}})+M/V. This has three cases:
C_{\mathrm{s}}>C_{\mathrm{tot}} (T>T_c): the driving force stays positive at M=0, and dissolution ends in finite time.
C_{\mathrm{s}}=C_{\mathrm{tot}} (T=T_c): C_{\mathrm{s}}-C=M/V. With A=aM^{2/3} (fixed number of particles), \mathrm{d}M/\mathrm{d}t=-\frac{Da}{hV}M^{5/3}, which integrates to M^{-2/3}=M_0^{-2/3}+\frac{2Da}{3hV}t. M falls as t^{-3/2} at long times and reaches zero only as t\to\infty.
C_{\mathrm{s}}<C_{\mathrm{tot}} (T<T_c): dissolution stops at M_\infty=V(C_{\mathrm{tot}}-C_{\mathrm{s}})>0.
Assumptions and valid range: the review states none. The derivation needs a stagnant film of fixed h, constant D and V, a surface at C_{\mathrm{s}}, and a shrinking particle with no breakage. The review itself reports that TEM "suggests dissolution mechanisms beyond diffusion", with defects raising A by fragmentation [review p. 5]; fragmentation enters Eq. 1 only through A(t). In the melt, D also changes with composition, because viscosity falls as drug dissolves [moseson2018 E5, as recorded on the wiki's formula page].
What it means for a formulation: the review reads Eq. 1 as a list of levers, namely area, diffusivity, temperature and mixing [review p. 5]. Step 6 adds what the review leaves out: at T=T_c a pure dissolution model does not finish in finite time, so a margin above T_c is needed and not only a longer residence time. This agrees with the crystals of 5–10 nm found by TEM in the sample made at T_c for the longest residence time [review pp. 8, 11], and with the population balance result in which dissolution alone left 26–73% or 3–61% crystals after 20 min at T_c (with C_{\mathrm{sat}} = 0.3 g/g equal to the loading) [moseson2022c R12, E9]. The review gives no numerical example of Eq. 1.
In the wiki: Noyes–Whitney equation, same form [moseson2018 E5]; the wiki also holds the mass-rate form \mathrm{d}M/\mathrm{d}t and the aqueous forms. Used or tested in ASD process F11 (hot-stage times), F12 (fragmentation) and F37 (population balance model; fitted k_d of 0.1002 µm/s). The finite-time argument of step 6 is not in the wiki.
Eq. 2. Diffusion coefficient of the dissolving species
D=\frac{k_{\mathrm{B}}T}{6\pi\eta r}\tag{2}
[review p. 3]
Symbols: D, diffusion coefficient (m²/s); k_{\mathrm{B}}, Boltzmann constant (1.381 × 10⁻²³ J/K); T, absolute temperature (K); \eta, viscosity (Pa·s); r, radius of the diffusing species (m).
Where it comes from: outside result, named by the review without a citation. Steps (added):
Stokes drag on a sphere with no slip at its surface: friction coefficient \zeta = 6\pi\eta r.
Einstein's relation. In equilibrium under a force F, the drift flux cF/\zeta balances the diffusion flux D\,\mathrm{d}c/\mathrm{d}x. The Boltzmann distribution gives \mathrm{d}c/\mathrm{d}x = cF/(k_{\mathrm{B}}T), so D=k_{\mathrm{B}}T/\zeta.
Substituting step 1 in step 2 gives Eq. 2.
Assumptions and valid range: the review states none. The derivation needs a sphere much larger than the molecules of the medium and a Newtonian continuum. A drug molecule in a polymer melt meets neither condition, and a small solute need not feel the bulk zero-shear viscosity of an entangled or high-molecular-weight melt. The wiki records the authors' own caution that the relation is limited at high viscosity [moseson2022c E19].
What it means for a formulation: from 131 to 161 °C the factor T rises only 7% (434.15/404.15 = 1.074), so the rise of D with temperature that the review relies on comes almost entirely through \eta (added). The review gives no numbers. An order-of-magnitude check from library data (added): the gradient of about 8 µm after 1 h at 130 °C in copovidone [moseson2019 R9] gives D\approx x^2/2t = (8\times10^{-6})^2/(2\times3600) = 8.9\times10^{-15} m²/s; with r\approx 1 nm [saboo2020a E6] and k_{\mathrm{B}}T = 5.57\times10^{-21} J, Eq. 2 then requires \eta = 5.57\times10^{-21}/(6\pi\times10^{-9}\times8.9\times10^{-15}) \approx 33 Pa·s. This is an effective value, and the comparison with a measured melt viscosity remains to be done.
In the wiki: Stokes–Einstein equation, same form. The wiki notes that it was not tested against the measured gradient; the estimate above is the first step of that test, and it depends on the assumed radius. Findings: ASD process F11, F36, F37.
Eq. 3. Depressed melting point: the formulation critical temperature
Symbols: T_{\mathrm{c}}, depressed melting point of the drug–polymer system (K); T_{\mathrm{m}}, melting temperature of the pure drug (K); R, gas constant (8.314 J mol⁻¹ K⁻¹); \Delta H, enthalpy of fusion of the drug (J/mol); \phi, volume fraction of the drug; m, ratio of the polymer segment to the drug molecular volume; \chi, Flory–Huggins interaction parameter (the last three dimensionless).
Where it comes from: Flory–Huggins framework of melting point depression (refs 42, 44–48). The review states the origin in words, as a reduced chemical potential of the drug from a favourable entropy of mixing and, for some systems, an exothermic mixing enthalpy. Steps (added):
Free energy of mixing of n_1 moles of drug (one lattice site each) and n_2 moles of polymer (m sites each), with \phi=n_1/(n_1+mn_2): \Delta G_{\mathrm{mix}}=RT\left[n_1\ln\phi+n_2\ln(1-\phi)+\chi n_1(1-\phi)\right].
With N=n_1+mn_2: \partial\phi/\partial n_1=(1-\phi)/N, n_1/N=\phi, n_2/N=(1-\phi)/m. Differentiating at constant n_2: $\frac{\mu_1-\mu_1^{\circ,\mathrm{liq}}}{RT}=\ln\phi+(1-\phi)-\frac{1-\phi}{m}+\chi(1-\phi)-\chi\phi(1-\phi)=\ln\phi+\left(1-\frac{1}{m}\right)(1-\phi)+\chi(1-\phi)^2$
Pure crystal against pure supercooled liquid, with \Delta H constant and \Delta C_p=0, so that \Delta S=\Delta H/T_{\mathrm{m}}: \mu^{\mathrm{cr}}-\mu_1^{\circ,\mathrm{liq}}=-\Delta H\,(1-T/T_{\mathrm{m}}).
Equilibrium of crystal and mixture at T=T_{\mathrm{c}}: set step 2 times RT_{\mathrm{c}} equal to step 3, and divide by RT_{\mathrm{c}}: [\ldots]=-\frac{\Delta H}{R}\left(\frac{1}{T_{\mathrm{c}}}-\frac{1}{T_{\mathrm{m}}}\right).
Multiplying by -R/\Delta H gives Eq. 3. The bracket is negative for a miscible pair, so 1/T_{\mathrm{c}}>1/T_{\mathrm{m}} and T_{\mathrm{c}}<T_{\mathrm{m}}.
Assumptions and valid range: stated by the review: T in kelvin, and limitations of the approach are referred to the literature (refs 49–52). Needed by the derivation and not stated: \Delta C_p=0; one \chi, independent of temperature and composition; a drug-sized lattice site; equilibrium at the offset temperature. Eq. 3 uses the volume fraction, whereas Figs. 3a and 9a plot the drug weight fraction; for indomethacin the wiki finds that using weight fraction in place of \phi gives 133.9 instead of 131.0 °C [moseson2018 X7].
What it means for a formulation: T_{\mathrm{c}} is the lowest product temperature at which all crystals of a given composition can dissolve. Worked example (added) for the review's Case 1, indomethacin–PVPVA at 50% loading, with T_{\mathrm{c}}=131 °C in Fig. 9a [review pp. 10–11]; inputs from the library: \phi=0.4739, m=170.7, \chi=-1.97, \Delta H=37.8 kJ/mol, T_{\mathrm{m}}=433.79 K [moseson2018 E2, M1]:
Bracket: \ln 0.4739=-0.7468; (1-1/170.7)(0.5261)=+0.5230; -1.97\,(0.5261)^2=-0.5453; sum =-0.7690.
R/\Delta H=8.314/37800=2.1996\times10^{-4} K⁻¹, so 1/T_{\mathrm{c}}-1/T_{\mathrm{m}}=+1.6915\times10^{-4} K⁻¹.
1/T_{\mathrm{c}}=2.30526\times10^{-3}+0.16915\times10^{-3}=2.47441\times10^{-3} K⁻¹, so T_{\mathrm{c}}=404.14 K =131.0 °C.
With \chi=0 the bracket is -0.2237 and T_{\mathrm{c}}=151.6 °C. Of the total depression of about 29.7 °C, the entropy terms therefore give 9.1 °C and the \chi term 20.6 °C.
At 10 wt% drug (\phi=0.091) the same inputs give 61 °C, near the low end of the curve printed in Fig. 9a.
The same equation gives \phi\approx0.021 at 25 °C, an extrapolation far below the fitted range (50–90% drug), which illustrates why the review places most compositions in region 2 [review p. 16].
In the wiki: Flory–Huggins melting point depression, same form and same symbols [moseson2018 E2]; the wiki reproduces the indomethacin, bicalutamide and posaconazole values. Used in ASD process F8 and F41. The wiki notes what the review does not: 131 °C is the predicted value and 132.3 ± 0.8 °C the measured one, and the origin of the "131 °C ± 2 °C" printed in Fig. 9a is unexplained (storyline, step 2).
[review p. 6]. Corrected misprint: the review prints the exponent as +\frac{16\pi\gamma^3\upsilon^2}{3(k_{\mathrm{B}}T)^3(\ln S)^2}, with no minus sign (checked on a 300 dpi rendering). The correction is required by the barrier derivation below and by the review's own next sentence, that higher temperatures and supersaturation "will increase nucleation rates".
Symbols: J, nuclei formed per unit time per unit volume (m⁻³ s⁻¹); A, a constant (same units; in fuller treatments a kinetic factor, see below); \gamma, interfacial tension (J/m²); \upsilon, molecular volume (m³); k_{\mathrm{B}}, Boltzmann constant; T, temperature (K); S, supersaturation (dimensionless, Eq. 5). The symbol \gamma means the activity coefficient in the next equation.
Where it comes from: classical nucleation theory, "originally derived for condensation of a vapor into a liquid", cited to Mullin (ref 79). Steps (added):
A spherical cluster of radius r holds (4\pi r^3/3)/\upsilon molecules, each gaining \Delta\mu=k_{\mathrm{B}}T\ln S (Eq. 5 per molecule), and has interface 4\pi r^2: \Delta G(r)=-\frac{4\pi r^3}{3\upsilon}k_{\mathrm{B}}T\ln S+4\pi r^2\gamma.
Maximum at \mathrm{d}\Delta G/\mathrm{d}r=0: r^*=\dfrac{2\gamma\upsilon}{k_{\mathrm{B}}T\ln S}. Below this "critical size" the nuclei redissolve, as the review states in words.
Rate over a barrier: J=A\exp(-\Delta G^*/k_{\mathrm{B}}T), which is Eq. 4 with the minus sign.
Dependence on S and T at constant A, with B=16\pi\gamma^3\upsilon^2/[3(k_{\mathrm{B}}T)^3]: \partial\ln J/\partial\ln S=+2B/(\ln S)^3 and (\partial\ln J/\partial T)_S=+3B/[T(\ln S)^2]. Both are positive with the minus sign, and both become negative with the printed form.
Assumptions and valid range: from the review: the theory is the "most widely used" for crystals from supersaturated solutions. Needed and not stated:
homogeneous nucleation, spherical clusters and a size-independent \gamma;
A constant, which is not so in a viscous phase. The review's statement that in glasses molecular mobility "is often the rate-limiting step" [review p. 7] requires a kinetic term that Eq. 4 lacks, as in the forms \exp[-(\Delta G^*+\Delta G_a)/kT] on the wiki's page [konno2006 E8; baird2012 E1];
"higher temperatures ... will increase nucleation rates" holds at fixed S only. In a solution of fixed c, for a drug whose solubility rises with temperature, a rise in T raises c^* and lowers S, and the rise in T can then lower J;
the review's agitation effect ("faster agitation rates ... will promote nucleation") has no term in Eq. 4.
What it means for a formulation: J depends on (\ln S)^{-2} in an exponent, which is the formal basis of the metastable zone (p. 6): over a small range of S the rate changes by orders of magnitude. The review gives no numbers. Worked example (added) with the fitted set on the wiki's formula page, posaconazole in buffer: \gamma=3.52 mJ/m², \upsilon=920 ų [duong2022b R16], at 37 °C (k_{\mathrm{B}}T=4.282\times10^{-21} J):
At S=33.6: \Delta G^*/k_{\mathrm{B}}T=7.88/(3.515)^2=0.64, so \Delta G^*=2.73\times10^{-21} J, the value on the wiki's page.
S
B/(\ln S)^2
J/A, with minus sign
J/A, as printed
2
16.4
7.6 × 10⁻⁸
1.3 × 10⁷
5
3.04
0.048
20.9
10
1.49
0.23
4.4
33.6
0.64
0.53
1.9
The printed form would make J largest at the lowest supersaturation and diverge as S\to1. The same inputs give r^*=0.43 nm, below the radius of one molecule of that volume (0.60 nm), so at that S the fitted \gamma is an effective parameter and not an interfacial tension of a macroscopic nucleus (added).
[review p. 6]. Corrected misprint: the review prints no equals sign between \ln(a/a^*) and \ln[\gamma c/(\gamma^*c^*)]; only a gap appears there (checked on a 300 dpi rendering).
Symbols: S, supersaturation (dimensionless); \mu, solute chemical potential in the supersaturated solution (J/mol); a, solute activity; \gamma, solute activity coefficient; c, solute concentration; the asterisk marks the property at saturation, for a solute in solution in equilibrium with the crystal; R, gas constant; T, temperature (K).
Where it comes from: a definition, with the steps left to the reader (added):
\mu=\mu^\circ+RT\ln a in the supersaturated solution, and \mu^*=\mu^\circ+RT\ln a^* at saturation, with the same standard state.
Subtracting: \mu-\mu^*=RT\ln(a/a^*), which defines \ln S.
With a=\gamma c, the last form follows.
Division by RT per mole equals division by k_{\mathrm{B}}T per molecule, which is the step used in Eq. 4.
Assumptions and valid range: the crystal form that defines the asterisk must be stated, since a metastable polymorph has a higher a^* and so a lower S [review p. 15]. Eq. 5 itself does not depend on dilution.
What it means for a formulation: Eq. 5 applies to a solid matrix as well as to a solution, and joined with Eq. 3 it gives the "relative supersaturation" that the review invokes at p. 16 but does not write. Combining steps 2–3 of Eq. 3 with Eq. 5 (added), for amorphous drug in polymer against its crystal at temperature T: $\ln S_{\mathrm{matrix}}=\ln\phi+\left(1-\frac{1}{m}\right)(1-\phi)+\chi(1-\phi)^2+\frac{\Delta H}{R}\left(\frac{1}{T}-\frac{1}{T_{\mathrm{m}}}\right) which is zero at T=T_{\mathrm{c}}. For 50% indomethacin–PVPVA, with the inputs of Eq. 3 (\Delta H/R=4546$ K):
25 °C: -0.769+4546\,(3.3540-2.3053)\times10^{-3}=-0.769+4.768=4.00, so S\approx55.
40 °C: \ln S=3.27, S\approx26.
5 °C: \ln S=5.10, S\approx163.
Cooling therefore raises the driving force while it lowers mobility, and the review's suggestion of subambient storage [review p. 16] accepts the first to gain the second. The values are extrapolations (\Delta C_p=0 and constant \chi, far below the fitted range). For scale, the wiki's Hoffman estimate for amorphous nifedipine, −9.0 kJ/mol at 20 °C [marsac2006b R12], corresponds to \ln S=9000/(8.314\times293)=3.7.
Symbols: c, solute concentration; c^*, concentration in equilibrium with the crystal (µg/mL in the case studies); S dimensionless.
Where it comes from: from Eq. 5. Steps (added): \ln S=\ln(\gamma/\gamma^*)+\ln(c/c^*); with \gamma/\gamma^*=1 the first term vanishes.
Assumptions and valid range: stated: dilute aqueous solutions "without solubilizing components". Needed and not stated: c below the amorphous solubility, because above it the free drug stays near C_{\mathrm{am}} and c/c^* with total c overstates S (the wiki's page holds this limit [dening2018 E6]). With a solubilizer the ratio can under-read the activity-based value, as for atazanavir in SDS, 33–38 against about 65–70 [indulkar2017a R12, R13].
What it means for a formulation: Eq. 6 is what links a measured concentration to Eqs. 4 and 7. Examples with the review's systems (added; solubilities from the library): indomethacin test of Case 1, S=50/8=6.25 [moseson2020a R1]; tacrolimus at the amorphous solubility in Fig. 14b, S=50/1.8=27.8 [purohit2018 R4].
Symbols: R_{\mathrm{G}}, overall growth rate per unit area (mass per area per time); K_{\mathrm{G}}, overall crystal growth coefficient (its units depend on g); C_{\mathrm{x}}-C_{\mathrm{eq}}, "the driving force for crystal growth"; g, order of the growth process (dimensionless). The review does not define C_{\mathrm{x}} and C_{\mathrm{eq}} separately; they are read here as the bulk solute concentration and the equilibrium crystalline solubility.
Where it comes from: diffusion–reaction model (ref 63, Ilevbare, Marsac and Mitra 2015). The review says that the two rates are combined "into eq 7" and does not show how. Steps (added):
Diffusion to the surface, with interfacial concentration C_i: R_{\mathrm{G}}=k_d(C_{\mathrm{x}}-C_i).
Integration at the surface, with order r: R_{\mathrm{G}}=k_r(C_i-C_{\mathrm{eq}})^r.
For r=1, eliminating C_i: C_i-C_{\mathrm{eq}}=\frac{k_d}{k_d+k_r}(C_{\mathrm{x}}-C_{\mathrm{eq}}), so R_{\mathrm{G}}=K_{\mathrm{G}}(C_{\mathrm{x}}-C_{\mathrm{eq}}) with \frac{1}{K_{\mathrm{G}}}=\frac{1}{k_d}+\frac{1}{k_r} and g=1. This case is exact.
For r=2: R_{\mathrm{G}}=k_r\left(C_{\mathrm{x}}-C_{\mathrm{eq}}-R_{\mathrm{G}}/k_d\right)^2, a quadratic in R_{\mathrm{G}} that is not a single power law. It tends to g=2 when k_r\ll k_d and to g=1 when k_d\ll k_r.
Eq. 7 with non-integer g is therefore an empirical fit, with g between 1 and r, and not a result of the combination.
Desupersaturation in a closed vessel with crystal area A_c: \mathrm{d}C_{\mathrm{x}}/\mathrm{d}t=-(A_c/V)R_{\mathrm{G}}, so seeds act through their area (see W6).
Assumptions and valid range: from the review: polymers reduce growth by adsorbing on the crystal surface and interfering with integration, and as supersaturation falls the rate falls. Derivation gap: the review then says that with inhibitors growth "may be arrested even though supersaturation still persists". Eq. 7 with constant K_{\mathrm{G}} cannot give this, since R_{\mathrm{G}}>0 for every C_{\mathrm{x}}>C_{\mathrm{eq}}. An arrest needs a coefficient that depends on the additive and on S, as in two models on the wiki that the review does not use:
Kubota–Mullin: R/R_0=1-\alpha\theta_{\mathrm{eq}}, with \alpha=\gamma a/(kT\ln S\,L) [alonzo2012 E5, E7];
Cabrera–Vermilyea: arrest when the spacing of adsorbed molecules equals 2\rho_c [alonzo2012 E8, C9].
Setting \alpha\theta_{\mathrm{eq}}=1 in the first gives a critical value \ln S_{\mathrm{crit}}=\gamma a\theta_{\mathrm{eq}}/(kTL) below which growth stops (added). The wiki's Cabrera–Vermilyea page records that the chapter papers did not test that relation numerically.
What it means for a formulation: the order g sets how strongly volume or dose changes growth. Worked example (added) with Fig. 14 (5 mg tacrolimus, C_{\mathrm{eq}}\approx1.8 µg/mL) and the library's order g=1.677 [hate2019 E5]:
450 mL: all dose dissolved gives 5000/450 = 11.1 µg/mL, driving force 11.1-1.8=9.3.
40 mL: C_{\mathrm{x}} limited at about C_{\mathrm{am}}=50 µg/mL, driving force 50-1.8=48.2.
Ratio: 48.2/9.3=5.18; growth per unit crystal area is higher by 5.18^{1.677}\approx16 at 40 mL. This corresponds to the review's reading of "greater thermodynamic driving force for crystallization" at 40 mL [review p. 13], and it assumes the same K_{\mathrm{G}} in both volumes.
Symbols: \mathrm{SI_{cr}}, sink index (dimensionless); C_{cr}, crystalline solubility (µg/mL); dose, amount of drug added to the dissolution experiment (µg); V, media volume (mL).
Where it comes from: a definition (refs 80, 137, 138). Steps (added): C_{cr}V is the largest mass of crystal that volume V can hold, so \mathrm{SI_{cr}}=C_{cr}V/\mathrm{dose} is that mass as a multiple of the dose. For \mathrm{SI_{cr}}<1, its reciprocal is the nominal supersaturation of Eq. 6 with the whole dose in solution: 1/\mathrm{SI_{cr}}=(\mathrm{dose}/V)/C_{cr}=S_{\mathrm{nominal}}.
Assumptions and valid range: the review states that sink and non-sink conditions "can be defined" by the index and gives no numerical boundary beyond the scenarios of W4. The wiki adds that SI above 1 means enough medium for the dose, and that sink in the compendial sense is three times the solubility needed [purohit2018 E1]. C_{cr} has to be measured in the same medium. The review's phrase that the index represents "the driving force for dissolution and crystallization" needs care, since a low index means a high driving force for crystallization.
What it means for a formulation: a volume or dose change is a change of the test's driving force. Worked example (added) for the review's Fig. 14, 5 mg tacrolimus, C_{cr}\approx1.8 µg/mL (read from the plot; 1.8 in the source [purohit2018 R4]):
Thus the "highest sink index" of Case 6 is itself non-sink for the crystal. Indomethacin Case 1: 8/50 = 0.16 [moseson2020a R1].
In the wiki: sink index, written \mathrm{SI}=C_sV/\mathrm{Dose} with one symbol for either solubility; algebraically identical. The concept page sink index holds the rule on crystal dissolution. Findings: Bioperformance F16, F17, F28, F31.
Eq. 9. Sink index against the amorphous solubility
Symbols: as Eq. 8, with C_{\mathrm{am}} the amorphous solubility (µg/mL), defined by the review as the concentration at which a liquid–liquid phase separation event may be observed, which can change with the drug–polymer composition and media additives [review p. 9].
Where it comes from: a definition, the same form as Eq. 8. Dividing Eq. 9 by Eq. 8 (added): \mathrm{SI_{am}}/\mathrm{SI_{cr}}=C_{\mathrm{am}}/C_{cr}, the amorphous solubility advantage, which does not depend on dose or volume.
Assumptions and valid range: C_{\mathrm{am}} is not a fixed property of the drug, so the index moves when a polymer or medium changes it (a point the review makes in words). The wiki gives the size of that shift, for example ibuprofen from 177.1 to 90.7–96.9 µg/mL with dissolved polymers (LLPS F19).
What it means for a formulation: worked example (added), Fig. 14, C_{\mathrm{am}}\approx50 µg/mL (read from Fig. 14b; about 50 in the library [hate2019 E1]):
450 mL: 50/11.11 = 4.5.
40 mL: 50/125 = 0.40.
Check: 4.5/0.16=0.40/0.0144=27.8=50/1.8.
Other systems: indomethacin Case 1, \mathrm{SI_{am}}=80/50=1.6 [moseson2020a R1], so that case does not reach the \mathrm{SI_{am}}<1 the review calls more likely biorelevant; bicalutamide at 30 µg/mL (Fig. 11a), 150/30=5, and at 150 µg/mL, 150/150=1moseson2021, in [ASD process F27].
Spinodal, second derivative zero: \frac{1}{\phi}+\frac{1}{m(1-\phi)}-2\chi=0.
Critical point, third derivative also zero: \frac{1-\phi}{\phi}=\frac{1}{\sqrt m}, so \phi_c=\frac{\sqrt m}{1+\sqrt m}.
Substituting: \chi_c=\frac12\left(1+\frac{1}{\sqrt m}\right)^2. Below \chi_c the pair mixes at every composition in this model.
Assumptions and valid range: \chi independent of composition and temperature, and no specific association. The critical value depends on how the lattice is defined [marsac2006b C18].
What it means for a formulation: "small positive" means roughly below 0.5–0.7. With library values of m:
m=25.95 (posaconazole–HPMCAS): 0.72, so the measured \chi=+0.43 lies below it and the depression is small, about 7 °C.
In the wiki: Flory–Huggins free energy of mixing holds step 1 and quotes a limit of "about 0.6 or less" from simulated curves for PVP [marsac2006b R3]; the closed form in step 4 is not printed there. Finding: ASD process F8.
W2. Phase-diagram regions, processing regimes and the crystals left below T_c
From Fig. 3 and its text [review p. 4]; Case 1 [review pp. 10–11].
Relations (written out here, added), with T_c(w) the solubility line of Eq. 3 and T_g(w) the formulation T_g line at drug weight fraction w:
Region 1: T<T_g(w) and T>T_c(w), thermodynamic and kinetic stability.
Region 2: T<T_g(w) and T<T_c(w), "thermodynamic metastability and kinetic stability".
Melting regime T>T_{\mathrm{m}}; dissolution regime T_c<T<T_{\mathrm{m}}; suspension regime T<T_c. Polymer T_{\mathrm{min}} is a lower processing limit that the review does not define by an equation.
Where it comes from: from Eq. 3 and the third case of Eq. 1, step 6. Mass balance (added): 1 g of mixture at loading w_0 holding X g crystals leaves a matrix of drug fraction (w_0-X)/(1-X). At equilibrium with the crystal this equals w_s(T), so $X_{\mathrm{eq}}=\frac{w_0-w_s(T)}{1-w_s(T)} which is the lever rule, and X_{\mathrm{eq}}=0 at T=T_c(w_0)$.
Assumptions and valid range: equilibrium with the stable form; the line of Eq. 3 taken as exact. The review states the qualitative result: below T_c residual crystallinity "remained regardless of temperature or residence time" [review p. 10].
What it means for a formulation: the phase diagram predicts a definite amount of crystal below T_c, which can be compared with the extrudates of Fig. 9b. Worked example (added), 50% indomethacin–PVPVA, w_s from Eq. 3 with the inputs of that section; observed values from the library [moseson2018 R10]:
T (°C)
w_s
X_{\mathrm{eq}}, % of mass
X_{\mathrm{eq}}, % of drug
observed by XRPD (%)
117
0.389
18.1
36.3
26.1 (2 min)
121
0.419
14.0
27.9
16.0 (2 min); 1.8 (20 min)
126
0.458
7.7
15.5
7.2 (2 min); 1.5 (10 min)
131
0.500
0
0
2.8 (2 min); ≈0 (10, 20 min)
Derivation gap: at long residence time the XRPD values lie far below X_{\mathrm{eq}} on either basis (the basis of % crystallinity was not checked against the source paper). Dissolution alone cannot go below X_{\mathrm{eq}}, so either the crystals are under-read (nanometre and defective crystals are poor XRPD signals, and the 121 °C, 20 min sample showed dense birefringence by PLM [moseson2018 R11]), or the actual melt temperature or solubility differs from the line. The measured point at 40% drug (129 ± 6 °C against 118.5 °C predicted) would widen this gap, not close it. The review and, as far as the wiki records, the source do not make this comparison.
Stated in words at p. 6 (ref 64, Hancock 1995: storage 50 °C below T_g for a stability period of years) and as an inequality at p. 16 (ref 182, Hancock 2001).
Symbols: T_g, glass transition temperature of the formulation at its water content (°C); T, storage temperature (°C).
Where it comes from: an empirical rule from outside; the review gives no derivation, and one cannot be completed from what it prints. It is a statement about molecular mobility and contains no term for the driving force of Eq. 5.
Assumptions and valid range: the review gives exceptions on the lenient side, ASDs stable for years at "much less stringent" conditions (refs 65, 66, Theil 2017). Water lowers T_g through its uptake, so the value applies at the storage humidity.
What it means for a formulation: at 25 °C the rule asks for T_g\ge75 °C (added). Fig. 17, calculated by PC-SAFT for naproxen at 20 and 40% loading [review p. 16]: dry, T_g-T = 10 to 100 °C; at 60% RH, −5 to 25 °C, so at 25 °C T_g = 20 to 50 °C and no listed composition meets the rule (added); at 75% RH, T_g below ambient for 20% with PVP and PVPVA and for 40% with PVPVA and HPMCAS. This is why the review turns to packaging.
In the wiki: not a formula page. Crystallization tendency F4 records the exception in the other direction (nilutamide nucleating most about 60 °C below T_g); Physical stability F5 records that T_g and T/T_g do not rank polymers or drugs; Physical stability F6 holds measured humid T_g values. Gordon–Taylor equation is the wiki's route to T_g(w) with water; the review's Fig. 17 lines are PC-SAFT results, with a different route.
W4. Dose scenarios, highest supersaturation and tolerated crystal fraction
Stated in words at p. 9: the scenario \mathrm{SI_{cr}}<1, \mathrm{SI_{am}}=1 (Fig. 7), and the scenario \mathrm{SI_{am}}<1, called more likely biorelevant, with colloidal drug-rich aggregates acting as a reservoir.
Symbols: as Eqs. 8–9; X, crystalline fraction of the drug; S_{\max}, highest supersaturation against the crystal.
Where it comes from: from Eqs. 6, 8 and 9. Steps (added):
With no crystals: C_{\max}=\min(\mathrm{dose}/V,\ C_{\mathrm{am}}), so S_{\max}=\min(1/\mathrm{SI_{cr}},\ C_{\mathrm{am}}/C_{cr}).
With inert crystals (the review's first scenario, in which "crystals do not grow"): amorphous supply (1-X)\,\mathrm{dose}/V, so C_{\max}(X)=\min[(1-X)\,\mathrm{dose}/V,\ C_{\mathrm{am}}].
For \mathrm{SI_{am}}\ge1 any X lowers the plateau in proportion to (1-X). This is the "lost solubility advantage".
For \mathrm{SI_{am}}<1 the plateau is unchanged until X=1-\mathrm{SI_{am}}; what falls first is the excess drug available as colloids, C_{\mathrm{res}}=(1-X)\,\mathrm{dose}/V-C_{\mathrm{am}}.
Assumptions and valid range: crystals at C\gg C_{cr} do not dissolve; complete release of the amorphous part; C_{\mathrm{am}} unchanged by X.
What it means for a formulation: step 4 gives a test of whether a loss is potency or growth. Worked example (added), Fig. 14b, 40 mL, \mathrm{SI_{am}}=0.4, so a plateau at 50 µg/mL should survive up to X=0.6 if crystals are inert. Peak release at 10 min, read from the plot (1% release = 1.25 µg/mL):
X
(1-X)\times125 (µg/mL)
C_{\mathrm{res}} (µg/mL)
peak read from plot
0.2
100
50
≈37.5% = 47 µg/mL
0.3
87.5
37.5
≈34% = 42.5 µg/mL
0.5
62.5
12.5
≈30.5% = 38 µg/mL
0.6
50
0
≈24% = 30 µg/mL
All lie below 50 µg/mL and then fall to about 8–11% release (10–14 µg/mL) by 120 min, so the loss is growth and not potency. At 450 mL (\mathrm{SI_{am}}=4.5; 1% release = 0.111 µg/mL, Fig. 14a) step 3 predicts 90% and 80% at X = 0.1 and 0.2, against about 81% and 78% at 120 min, close to the review's "no marked decrease" at 20%; it predicts 70% at X=0.3, whereas the plotted curve stays near 67% to 90 min and falls to about 55% at 120 min, with a large error bar. At high X the inert-crystal step fails in the other direction, since C is then no longer far above C_{cr}: X=0.6 predicts 40% against about 47% observed, and X=1 predicts no amorphous supply against about 33% (3.7 µg/mL), twice the marked crystalline solubility of about 16% (1.8 µg/mL). The review notes that last result and offers three possible reasons (incomplete crystallization, defective crystals, very small crystals) [review p. 14]; Eqs. 6–9 do not discriminate between them. The barrier of Eq. 4 also differs between these volumes: \ln S = 1.82 and 3.32, so (\ln S)^2 differs 3.3-fold (added).
W5. Theoretical maximum concentration against % crystallinity
Fig. 10c [review p. 11]; stated in words as "the theoretical maximum concentration which corresponds to the level of amorphous content".
Relation (written out here, added): C_{720}^{\max}(X)=C_0-(C_0-C_{100})X, with C_0\approx43.2 and C_{100}\approx4.4 µg/mL (100 on the x-axis of Fig. 10c).
Where it comes from: the review prints no formula. The library records that the line joins the 720 min concentrations of the all-amorphous and all-crystalline samples [moseson2020a E4, C34]. It is thus an interpolation between two measured end points, not a calculation from dose: C_0 is below the dose of 50 µg/mL, and C_{100} is below the crystalline solubility of about 8 µg/mL. At C\gg C_{cr} crystals do not contribute (W4, step 2), so the term in C_{100} has no physical basis, though it changes the line by little.
Assumptions and valid range: points on the line are read as crystals that did not grow; one dose, \mathrm{SI_{am}}=1.6.
What it means for a formulation: the arithmetic (added; observed values from the library [moseson2020a R6, R7]):
sample
X
line (µg/mL)
observed
below line
physical mixture
0.10
39.3
36.7
2.6
physical mixture
0.20
35.4
34.9
0.5
physical mixture
0.40
27.7
26.7
1.0
extrudate
0.07
40.5
34.5
6.0
extrudate
0.16
37.0
32.6
4.4
extrudate
0.25
33.5
28.8
4.7
Difference within the review: p. 11 says that for both preparations the loss "fell along" the line, which "indicates that the crystal seeds ... do not grow"; p. 17 reports greater desupersaturation with residual crystals. In the 7% extrudate, seeds of 0.07\times50=3.5 µg/mL stand against a further loss of 6.0 µg/mL (added), which, if the gap is crystallized drug and not undissolved drug or under-read crystallinity, is growth of more than the seeds' own mass.
In the wiki: ASD process F26 (the line and the values) and F29; recorded as a difference at R37 of the check. No formula page.
W6. Seed effect scales with surface area, not with mass
Stated in words at pp. 9–10: the key property of seeds is "directly related to surface area ... but notably not related to their mass amount", and "the same 10% crystallinity" from spiked crystals represents far lower surface area.
Symbols: A_{\mathrm{sp}}, specific surface area (m²/g); \rho, crystal density (g/m³); d, particle diameter (m); m_c, crystal mass.
Where it comes from: from Eq. 7, step 6, \mathrm{d}C/\mathrm{d}t=-(A_c/V)R_{\mathrm{G}}. Sphere geometry (added): area \pi d^2 over mass \rho\pi d^3/6 gives A_{\mathrm{sp}}=6/(\rho d), so A_c=6m_c/(\rho d).
Assumptions and valid range: smooth equal spheres with all surface exposed and K_{\mathrm{G}} independent of origin. Embedded crystals have less exposed area and defective ones may have a higher K_{\mathrm{G}}, neither of which is measured. Wording: A_c is proportional to mass at fixed size, so the review's "not related to their mass" holds only when size varies between samples; the correct form is that mass alone does not fix the area.
What it means for a formulation: at equal mass the area ratio equals the inverse size ratio. With the review's sizes (added): spiked above 10 µm against residual below 1 µm, more than 10-fold; against 5–10 nm domains, 1000- to 2000-fold. Check against a measurement: d = 21.9 µm, \rho\approx1.3 g/cm³ (assumed) gives 6/(1.3\times10^6\times21.9\times10^{-6})=0.21 m²/g, against BET 0.14 m²/g for indomethacin bulk crystals [moseson2020a M1]. Bicalutamide comparison, from the library: 3.5% residual crystals of 140–340 nm against 10% seeds of D50 3.5 µm give an area ratio (3.5/10)\times(3500/340\ \text{to}\ 3500/140) = 3.6 to 8.8, against a ratio of relative growth rates of 0.088/0.044=2.0 [moseson2021 R22, C65, C66]. The direction agrees and the size of the effect is smaller than the geometry gives.
In the wiki: crystal seeding has the same principle, without the geometry. Findings: ASD process F26; Supersaturation F22 (paclitaxel seeds, equal mass, ≈7.9–25.3 µg/mL). The size, position and defects of residual crystals were not measured directly as potency factors [moseson2021 C65, C66, X10].
W7. Stated detection limits
method
limit as stated by the review
page
remark
XRPD
LOD ≈0.4%, LOQ ≈1% in Fig. 9b; "relatively poor sensitivity"
7, 10
limit set by mass fraction, dilution, crystal quality
19F ssNMR
0.04% in a tablet of a fluorinated drug
7
"may not be generalizable"
PLM
diffraction limited
7
the limit is not given; with d=\lambda/(2\mathrm{NA}), 550 nm and NA 0.9 give about 0.3 µm (added)
TEM
crystals of 5–10 nm
8
low throughput
SEM
surface crystals only
8
SHG
"parts per billion range"
8
the wiki's lowest is 10 ppm, calculated
micro-CT
below 1%; crystals of about 10 µm or more
8
Where it comes from: cited values from outside; no calculation is printed.
What it means for a formulation: every limit is a mass fraction of something, and dilution shifts it. Worked example (added) with the wiki's conversion w_{\mathrm{cryst}}=f_{\mathrm{drug,cryst}}\times\mathrm{DL}: 1% of the drug at 30% loading is 0.3% of the ASD powder, and in a tablet holding 30% ASD it is 0.09% of the tablet, below the XRPD LOD above. This is the arithmetic behind the review's warning on dilution (p. 7). The SHG statement differs from the wiki's quoted limit by at least a factor of 10^4 (10 ppm = 10,000 ppb); the cited papers (refs 117–119) are not in the library.
Stated in words: variability of ±5% means that detecting less than 10% crystallinity "might not be possible" in QC (ref 124, p. 8); 6% might be detectable assuming no other variability (ref 140, p. 18); most amorphous compounds have release rate advantages of 2–20× (ref 141, p. 9).
Where it comes from: the review prints no calculation. Arithmetic (added):
In a non-sink test with inert crystals the result falls by the fraction X (W4, step 3). Two results each within ±5% can differ by 10% without any real difference, which gives the stated <10%.
For an initial rate test, with rates additive in mass fraction: R/R_{\mathrm{am}}=(1-X)+X/k=1-X(1-1/k).
Assumptions and valid range: step 1 takes the worst case of two independent results and ignores growth, which makes a test more sensitive in the systems where crystals grow. Step 2 needs equal surface area per unit mass, and release governed by the drug and not by the polymer, the same conditions the review lists as concerns (p. 9). Missing: the statistical model behind 6% is in Hermans 2019, which is not in the library, so that number cannot be reproduced here.
What it means for a formulation: step 2 gives, at X=0.10, a rate lower by 5% at k=2 (1-0.1\times0.5=0.95) and by 9.5% at k=20 (1-0.1\times0.95=0.905), which is at or inside the ±5% band of each result. This is the quantitative content of "sufficient discriminating ability may not be found".
In the wiki: not in the wiki as a formula; Bioperformance F16 holds the tacrolimus data (USP test I, about 93 to 58%). In the check, R25 records that the wiki extends the statement with data, and R31 and R60 record that the 2–20× ratio and the statistical limits of QC dissolution are not in the wiki.
W9. Growth stops when the matrix is no longer supersaturated
Stated in words at p. 16: in a system that is not supersaturated "the crystals will not grow", in a metastable system T_g-T affects the growth rate, and in Case 7 growth ceased "when the solubility of the amorphous drug within the polymeric matrix was presumably reached" (Fig. 15a; nifedipine–PVP 70%, total below 10%). At p. 14 the same plateau is described after Ivanisevic as the amount of drug "not stabilized by the polymer".
Where it comes from: from Eq. 7 with C_{\mathrm{x}}\to C_{\mathrm{eq}} in the matrix, and the mass balance of W2.
Derivation gap: the mass balance (added) does not support a thermodynamic reading.
If crystallinity below 10% means 10% of total mass, the remaining matrix holds (0.70-0.10)/(1-0.10)=0.667 drug by weight; if it means 10% of the drug (7% of total mass), (0.70-0.07)/(1-0.07)=0.677. The plateau would therefore put the solubility limit at or above about 0.67 by weight at ambient temperature.
The library's predicted solubility of crystalline nifedipine in PVP is 0.06–0.07 g/g [marsac2006b R17, R18], that is w_s=0.057–0.065. Reaching that would need X_{\mathrm{eq}}=(0.70-0.06)/(1-0.06)\approx0.68 of total mass, about 97% of the drug.
So the plateau lies far from crystal solubility as predicted. The review's wording "solubility of the amorphous drug" also differs from its Eq. 3 and region 2, which are defined by the crystal.
A kinetic stop is the reading that fits the numbers: as drug leaves the matrix its T_g rises, and with it T_g-T (added, not tested). The predicted solubility also carries the uncertainty of an extrapolated \chi, and the storyline (step 5) records that the XRPD signal in Fig. 15a is small and not monotonic.
What it means for a formulation: a plateau in stored crystallinity should not be taken as a thermodynamic safe limit without a check by this mass balance against a measured solubility.
In the wiki: Physical stability F15 has plateaus for felodipine–PVP without attributing them to solubility; Miscibility F9 holds the predicted solubilities; R53 of the check.
W10. Thresholds from case studies
Crystallinity against degradation [review p. 12, Case 4]. LY3009120 by KinetiSol: degradation 0.7, 2.0, 3.1% with crystallinity 2.3, 0.9, 0.1%; desupersaturation at 2.3% and not at 0.9 or 0.1%. So the performance threshold lies between 0.9 and 2.3% in that test, not located further. Arithmetic (added): slope (0.1-2.3)/(3.1-0.7)=-0.92% crystallinity per % degradation, so on this process line about one is exchanged for the other; the sums are 3.0, 2.9 and 3.2%. Three points only. In the wiki: ASD process F35 for the same trade-off with extrusion; KinetiSol is not in the wiki.
Polymer dissolution pH [review p. 17]. HPMCAS HF "does not ionize and dissolve until ≥ pH 6.8", used to explain a negative food effect. This is a stated threshold with no equation, and the review prints the polymer name as "HMPCAS". The library shows release well below that nominal value in fed medium: about 143 and 128 µg/mL at FeSSIF pH 6.0 and 5.8, and only the crystalline solubility at pH 5.5 and 5.0 (Bioperformance F26) [nguyen2023a R19]. The gastric pH series (1.6, 3, 6.5) is in Supersaturation F25.
How the ideas developed
Years and reference numbers are as the review cites them. The review names the Noyes–Whitney, Stokes–Einstein and Flory–Huggins relations without citing the original papers; the dates given for those are general history and do not come from the review.
Crystal dissolution in a melt (Eqs. 1–2).
Before the review's sources: Noyes and Whitney (1897), with the film form from Nernst and Brunner (1904); Stokes (1851) and Einstein (1905) for the diffusion relation. The review gives none of these references.
Taylor group: the review applies both relations to crystal dissolution in polymer melts through the extrusion work: Moseson and Taylor 2018 (ref 30), Moseson et al. 2019 (ref 31, fragmentation and dissolution in the melt), and Moseson et al. 2022 (ref 41, a population balance on amorphization kinetics).
Melting point depression and \chi (Eq. 3, W1, W2).
Polymer blends: Nishi and Wang 1975 (ref 44), Avella 1988 (ref 46), Greco 1989 (ref 45), Painter 1991 (ref 47), Hoei 1992 (ref 48); the lattice model itself from Flory and Huggins (about 1941–42).
Classical nucleation and supersaturation as used here: Mullin 2001 (ref 79) (Volmer and Becker–Döring are the usual origins, not cited by the review); Rodríguez-Hornedo and Murphy 1999 (ref 83).
Other groups in the same period: Haser 2017 (ref 75), Theil 2018 (ref 10), Hermans 2019 (ref 140), Davis 2021 (ref 11), Chasse 2022 (ref 6), Seiler 2022 (ref 61), Purohit 2023 (ref 124). Nguyen 2023 (refs 184, 186) adds the gastric and fed routes: delamanid, pretomanid.
After the review, in the library.
Shah et al. 2024: posaconazole–HPMCAS extruded below its calculated T_c, with crystals from 0.3 to 19%; SHG was used at the low end, and supersaturation after a pH shift ended earlier at every loading (ASD process F41). This applies Eq. 3 and W2 prospectively.
Shah et al. 2025: HPMCAS grade and pH trade release against inhibition. HF delayed nucleation most (none within 240 min at pH 6.5) and released only about 22–31 µg/mL at that pH; LF released at every pH and lost the drug within 3 h. This extends W10.
Alvarenga et al. 2025: in FeSSIF-V2, praziquantel dispersions released less at the higher dose and crystal peaks appeared in the residues, more clearly with HPMC than with HPMCAS.
Benson et al. 2025: pressing coated ritonavir–PVPVA particles lowered release at pH 6.8 (ASD process F48), which bears on the tablet concerns the review raises and is not a crystallinity effect.
Alkalla et al. 2026: crystalline darunavir hydrate lowered the concentration and flux of atazanavir from its ASD, while darunavir's own flux did not change. This is a loss caused by a crystalline phase of a different drug, which Eqs. 6–9 do not contain.