Which dissolved polymers delay crystal nucleation from supersaturated drug solutions, and why hydrophobicity matching matters

Background and motivation

Amorphous solids and other high-energy forms are used to raise the dissolved concentration of poorly water-soluble drugs above their crystalline solubility. In this supersaturated state the drug's thermodynamic activity is higher, and so is the driving force for transport into and across the intestinal membrane; but the state is thermodynamically unstable and ends in crystallization, so keeping it in the gastrointestinal tract may be difficult (Gao 2004, ref 3) C6. Polymers added to the solution are known to delay this crystallization (refs 4–12), yet the authors note that very few tools exist to choose the right polymer for a given drug, that little quantitative work had addressed how polymers change nucleation and growth kinetics, and that little was known about which polymer properties matter C7.

Crystallization from solution has two steps that a polymer can act on: forming new crystals (nucleation) and growing existing ones. Growth inhibition is generally explained by additive adsorption on the crystal surfaces (refs 14–16), whereas the way additives slow nucleation is "more speculative", because the molecular events cannot be observed directly C9. Molecular simulations by Anwar and co-workers (ref 17) suggested that the relative strength of the solute–solvent–additive interactions, together with size factors, sets an additive's effect on nucleation C9. Nucleation is also the energetically more demanding step: there are supersaturations at which crystals grow but new ones do not form (refs 14, 18, 19) C10.

The group's earlier work gave the practical reason to look at nucleation. Polymers were not very effective at slowing the crystal growth of ritonavir and felodipine at the high supersaturations expected from dissolving an amorphous formulation, and in the presence of seed crystals the solutions desupersaturated rapidly (Alonzo 2012, ref 20; Ilevbare 2012, ref 21) R19. The authors concluded that "it seems essential to prevent the initial formation of crystals" with effective nucleation inhibitors if supersaturation is to last for biologically relevant times C11. Many of the polymers used here, including a family of newly synthesized cellulose ω-carboxyesters with wider chemical diversity than commercial cellulose derivatives, had already been tested as ritonavir growth inhibitors (Ilevbare 2012, ref 22; Liu 2012, ref 33) R19.

The study therefore set out to quantify how chemically diverse polymers change the nucleation behaviour of three chemically diverse model drugs, and to find which interplay of polymer and drug properties controls it. Its working hypothesis was that a polymer's ability to inhibit nucleation depends on the strength of its interaction with the crystallizing solute ("It is hypothesized") C12.

Background concepts

Supersaturation. For dilute solutions the authors define the supersaturation ratio, the driving force for nucleation, as the dissolved concentration divided by the crystalline solubility (Eq. 1) E1:

In plain terms: $S$ says how many times more drug is dissolved than the crystal allows at equilibrium; $C$ is the concentration of the supersaturated solution and $C_{\mathrm{e}\mathrm{q}}$ the equilibrium solubility of the crystalline drug at the same temperature E1. At pH 6.8 and 37 °C the crystalline solubilities were 1.5 µg/mL for celecoxib, 8.2 µg/mL for efavirenz and 1.3 µg/mL for ritonavir, all three drugs being un-ionized at that pH R1.

Amorphous solubility. The highest supersaturation an amorphous formulation can produce is set by the solubility of the amorphous form. The authors estimate it from the crystalline solubility, the free-energy difference between the amorphous and crystalline forms, and a correction for the water that the amorphous solid absorbs (Eq. 6) E6:

In plain terms: the amorphous form is more soluble than the crystal by the exponential of its excess free energy $\Delta G_C$ (divided by the gas constant $R$ times temperature $T$), and less soluble by the factor $\exp (-I(a_2))$, where $I(a_2)$ accounts for the lowered activity of an amorphous solid saturated with water; $I(a_2)$ comes from a Gibbs–Duhem treatment of the measured water sorption isotherm (ref 36) E6·P6. The free-energy difference is estimated at 37 °C from the heat of fusion $\Delta H_f$ and the melting point $T_m$ with the Hoffman approximation (Eq. 7) E7:

In plain terms: the further below its melting point the drug sits, and the larger its heat of fusion, the larger the free-energy excess of the amorphous form E7. The paper prints neither the $\Delta G_C$ values nor $I(a_2)$; from the printed heats of fusion and melting points the extractor obtains about 10.2, 2.70 and 7.74 kJ/mol for ritonavir, efavirenz and celecoxib at 310.15 K, a calculation not given in the paper E7. The resulting theoretical amorphous solubilities are 20.6, 19.8 and 22.6 µg/mL for ritonavir, efavirenz and celecoxib, and the induction-time experiments start at about these concentrations R2·P9.

Induction time. Nuclei cannot be seen when they form; they are detected only after they grow. The measured induction time is therefore the sum of the true nucleation time and the time to grow to a detectable size (Eq. 2) E2:

In plain terms: $t_{\mathrm{i}\mathrm{n}\mathrm{d}}$ is the time from creating supersaturation to the first observable change, here a rise in light scattering; $t_n$ is the time to form critical nuclei and $t_g$ the time for them to grow large enough to be seen E2. The authors stress that the measured induction time is not a fundamental property and depends on the detection method C13. If steady-state nucleation is reached quickly and growth to detectable size is fast ($t_n\gg t_g$), the induction time is inversely proportional to the nucleation rate $J$ (Eq. 3) E3·C14:

In plain terms: a longer induction time means slower nucleation E3. Classical nucleation theory gives the nucleation rate as (Eq. 4) E4:

In plain terms: nucleation is exponentially slower the higher the free-energy barrier $\Delta G_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ for forming a critical nucleus, compared with the thermal energy $kT$ ($k$ the Boltzmann constant); the pre-exponential factor $A$ reflects how fast molecules attach to the nucleus and so depends on molecular mobility E4. Since the barrier falls as supersaturation rises, nucleation speeds up sharply at high supersaturation; the authors use this to argue later that polymers should work better at lower supersaturation C45.

Two-step nucleation. Classical theory pictures nuclei growing by adding one molecule at a time and says nothing about the structure of the aggregates. The two-step model, first proposed for proteins, has a disordered cluster of solute molecules form first and then reorganize into an ordered crystal structure; it has been shown for small organic molecules, it appears in molecular dynamics simulations, and the cluster's organization time has been proposed as the rate-limiting step (Erdemir 2009, ref 30; Vekilov 2010, ref 31; Anwar 2009, ref 17) C15. This picture is the frame in which the authors later explain how a polymer could delay nucleation C5.

Solubility parameter. The solubility parameter measures cohesive energy density and is used here as a hydrophobicity scale for both drugs and polymers. The authors estimate it with Fedors' group-contribution method (Eq. 5), which needs only the structural formula (ref 34) E5·P1:

In plain terms: $\delta$ is the square root of the energy of vaporization $\Delta E_v$ per molar volume $V$, both built up from tabulated contributions $\Delta e_i$ and $\Delta v_i$ of the atoms and groups in the molecule E5. The higher the solubility parameter, the more hydrophilic the compound; water has 49.01 MPa^1/2 M29·C18. The three drugs lie at 20.03 (ritonavir), 21.89 (efavirenz) and 23.38 MPa^1/2 (celecoxib), and the 21 polymers span 19.62 to 32.24 MPa^1/2 M1·M2·M3·R10. Log P values (5.6, 4.7 and 3.5) give the same order, ritonavir the most hydrophobic and celecoxib the most hydrophilic of the three M1·M2·M3·C18.

Crystallization tendency classes. Earlier work by the group sorted compounds by how readily they crystallize from the undercooled melt or on rapid solvent evaporation: Class I fast, Class II intermediate, Class III slow (refs 44, 45). Ritonavir and efavirenz are Class III and celecoxib Class II M1·M2·M3·C24.

Methods in detail

The study has one central measurement, the unseeded induction time of each drug with and without each polymer, set up by two supporting steps (choosing the starting concentration and ranking the polymers by hydrophobicity) and followed by two targeted tests (polymer ionization and supersaturation level). The flow below summarizes it.

Step 1: the three drugs and where to start them

Problem. To be relevant to amorphous formulations, the induction times must be measured at the supersaturation an amorphous solid would generate, and the three drugs should differ in structure, hydrophobicity and possible interactions with polymers C17·A1.

What they did. Ritonavir (a very weak base with two thiazole groups, pK_a 1.8 and 2.6, 720.9 g/mol), efavirenz (a very weak acid, pK_a 10.2, 315.7 g/mol) and celecoxib (a very weak acid, pK_a 11.1, 381.4 g/mol) were chosen (Figure 1, Table 3) M1·M2·M3. Their melting points (122.7, 139.0 and 163.5 °C) and heats of fusion (60.4, 14.5 and 37.6 kJ/mol) were measured by DSC on a TA Q2000 at 5 °C/min M1·M2·M3·P4. Water sorption of melt-quenched amorphous samples (about 15 mg, 5–95% RH in 10% steps at 37 °C, three samples per drug) supplied the water correction of Eq. 6 P5·P6. The crystalline solubility was measured by a 48 h shake-flask method at 37 °C in 100 mM sodium phosphate buffer pH 6.8, with ultracentrifugation at 40 000 rpm and HPLC analysis; six polymers were also predissolved at 5 µg/mL to check whether they change it P3·R1. The ritonavir values were taken from the group's earlier paper (Table 4, footnote a) R1·X17.

Checks on the amorphous route. The experimental amorphous solubility was measured by dissolving solvent-evaporated amorphous drug in the same buffer at 37 °C and following the concentration with an in situ UV fiber-optic probe and second-derivative spectra P7. Separately, melt-quenched amorphous celecoxib and efavirenz particles were wetted with buffer and watched under crossed polarizers (Figures 4 and 5) P8.

Step 2: the induction-time measurement

What they did. A 4 mg/mL solution of drug in methanol was pumped with a syringe pump into 50–80 mL of buffer at 37 °C, stirred at 300 rpm, at a pH where the drug is un-ionized P9. The starting concentrations were 20 µg/mL ritonavir, 19 µg/mL efavirenz and 22 µg/mL celecoxib, which the authors equate with the theoretical amorphous solubility; they give the corresponding supersaturation ratios as 15.4, 2.4 and 14.7 P9. Polymers were predissolved in the buffer at 5 µg/mL, and every condition was run in triplicate P9. The first crystals were detected by light scattering: the extinction at 280 nm (ritonavir) or 350 nm (celecoxib and efavirenz), wavelengths at which the drugs do not absorb, was recorded every 30 s with a fiber-optic dip probe, while the absorption peaks followed the dissolved concentration P9. The induction time was read as the intersection of two regression lines, one through the flat early part of the extinction curve and one through the rising part (Figure 6) P10·R6.

Why it works. The authors argue that in the relatively large volumes and at the high supersaturations used, the probability of nucleation is high, perhaps because random impurities induce heterogeneous nucleation, so replicate induction times should vary little ("it is reasonable to expect"); the observed variation was below 15% C16·R17. Starting from a clear solution also separates nucleation from the solution from crystallization on a dissolving amorphous particle C24.

Note by the AI writer (not discussed in the paper): two details of this step matter for reading the numbers. First, the efavirenz ratio of 2.4 matches the theoretical amorphous solubility (19.8/8.2), whereas the 19 µg/mL actually used gives 2.32 X4. Second, the methanol fraction in the medium is not stated; adding a 4 mg/mL stock to reach about 20 µg/mL implies roughly 0.5% v/v methanol, while the crystalline solubility used to compute $S$ was measured without methanol X16.

Step 3: ranking 21 polymers by hydrophobicity

The 21 polymers fall into three groups (Tables 1 and 2, Figures 2 and 3) M4·M15. Six are synthetic polymers: PVP K29/32, PVPVA K28, poly(acrylic acid) (PAA), poly(4-vinylpyridine N-oxide) (PVPdn-O), and two flexible polymers with drug-like solubility parameters, poly(allylamine) (PAlAmn, 20.70 MPa^1/2) and poly(N-isopropylacrylamide) (Pn-IPAAmd, 22.64 MPa^1/2) M4·M5·M6·M7·M8·M9. Five are commercial cellulose derivatives: hydroxypropyl cellulose (HPC), cellulose acetate phthalate (CAPh), HPMC 606 grade, HPMCAS AS-MF and carboxymethyl cellulose acetate butyrate (CMCAB) M10·M11·M12·M13·M14. Ten are cellulose ω-carboxyesters synthesized in the Edgar laboratory, which carry adipate, suberate or sebacate side chains with terminal carboxylic acids alongside acetate, propionate or butyrate groups, placed without regioselectivity (Figure 2) M15·M16·M17·M18·M19·M20·M21·M22·M23·M24. Their solubility parameters, from 19.62 (CAB Seb) to 32.24 MPa^1/2 (PVPdn-O), order the polymers from hydrophobic to hydrophilic; Figures 7–9 plot the polymers in order of decreasing solubility parameter, with PAlAmn placed slightly out of order R10·R9·X6.

Note by the AI writer (not discussed in the paper): the paper gives no molecular weights or degrees of substitution for the polymers, and the numbers in names such as CAP Adp 0.85 or CAB Adp 0.25 are not defined here; they are taken from the synthesis papers X15.

Step 4: two follow-up tests

Polymer ionization. The synthesized esters carry carboxylic acids. To test whether their ionization matters, the efavirenz runs with the two most effective esters, CAP Adp 0.85 and CAB Adp 0.81, were repeated at pH 3.8 in 100 mM sodium acetate buffer, where the acid groups are mostly un-ionized, alongside the nonionizable HPMC as a control P11. With the adipate pK_a taken as about 4.43, the substituents are nearly fully ionized at pH 6.8 and less than 25% ionized at pH 3.8 R18.

Supersaturation level. Celecoxib, which nucleates within minutes, was also started at about 11 µg/mL, half its theoretical amorphous solubility ($S$ = 7.3), with and without HPMC P12. To compare how much supersaturation each start delivers over time, the supersaturation-time profiles were integrated to 240 min and divided by the area for a saturated solution (Eq. 8) P13·E8:

In plain terms: the supersaturation factor SF is the area under the supersaturation ratio-time curve of the test solution, ${\mathrm{A}\mathrm{U}\mathrm{C}}_{\mathrm{s}\mathrm{p}}$, divided by that of a solution held at saturation ($S$ = 1), ${\mathrm{A}\mathrm{U}\mathrm{C}}_s$, both to 240 min; an SF of 7 means the solution delivered on average seven times the crystalline solubility over that period E8.

Experimental analysis

The drugs on their own

The amorphous solubility measurements already separate the three drugs (Table A). Ritonavir dissolved to 19.8 µg/mL against a theoretical 20.6 µg/mL, which the authors read as slow crystallization both of the dissolving amorphous solid and of the solution, so that a solution near the amorphous solubility forms and crystallizes only later R2·C21. Efavirenz reached 12.3 against 19.8 µg/mL, and celecoxib only 2.5 against 22.6 µg/mL, a peak supersaturation of 1.7, "indicating that crystallization was rapid" R2·R3·C22. Under the polarized light microscope an amorphous celecoxib particle showed a birefringent spot after about 1 min and was covered in crystals by 6 min (Figure 4), so celecoxib crystallized on contact with the buffer R4·C23. Amorphous efavirenz particles shrank over 3 h without birefringence (Figure 5), "suggesting that crystallization of efavirenz occurs from the supersaturated solution generated after dissolution" R5·C23.

Table A. The three model drugs (from Tables 3 and 4; induction times read from Figures 7–9, approximate; the text gives about 120, 5 and 2 min) M1·M2·M3·R1·R2·R8.

In the induction-time runs, the solutions stayed clear and the dissolved concentration stayed at its starting value until the extinction began to rise; the concentration then fell rapidly toward the crystalline solubility as the crystals grew R7. Without polymer the average induction times were about 2 min for celecoxib, 5 min for efavirenz and 120 min for ritonavir R8. The authors find it "apparent" that ritonavir nucleates much more slowly, which they attribute "most likely" to its higher molecular weight and greater conformational flexibility; neither property was tested C25. Note by the AI writer (not discussed in the paper): the comparison is not at equal driving force, since ritonavir and celecoxib started at a supersaturation of about 15 and efavirenz at only about 2.3 X4.

Ritonavir and efavirenz: the hydrophobic cellulose esters win

Table B lists all the induction times, read from Figures 7–9. For ritonavir, the four most effective polymers were synthesized esters: CAB Adp 0.81 (about 244 min), CAP Adp 0.33 (about 239), CA 320S Seb (about 178) and CAP Adp 0.85 (about 172), against about 120 min without polymer R9. The text gives the best result as a factor of 2, from 120 to about 250 min R12. HPMCAS (about 150 min) and the hydrophobic sebacate esters followed; all synthetic polymers, HPC and CP Adp stayed between about 93 and 122 min, with PAA (about 93 min) even below the drug alone R9.

For efavirenz the same four esters led: CAB Adp 0.81 (about 28.0 min), CAP Adp 0.85 (about 24.9), CA 320S Seb (about 24.1) and CAP Adp 0.33 (about 21.6), against about 5.1 min without polymer R9. The other cellulose polymers gave about 14.9–20.4 min, while the synthetic polymers and HPC stayed at about 4.4–6.6 min R9. The text gives the best result as 5 → 25 min with CAB Adp 0.81, a factor of about 5; the bar for CAB Adp 0.81 reads about 28 min in Figure 8 and in Figure 10 at pH 6.8, and 25 min matches CAP Adp 0.85 instead R12·X5. The abstract summarizes the gains as "up to a factor of 5−6 at the highest supersaturations tested" C2.

The authors call these four esters "the most effective nucleation inhibitors for both efavirenz and ritonavir", increasing the induction time "significantly" relative to the commercial HPMCAS, HPMC and CMCAB; no statistical test is reported C26. "The very hydrophilic polymers had no impact on induction times for either compound" C27. The flexible synthetic polymers with drug-like solubility parameters, PAlAmn (20.70) and Pn-IPAAmd (22.64 MPa^1/2), were also ineffective: about 112 and 119 min for ritonavir, about 4.4 and 5.0 min for efavirenz R9·C38.

Table B. Induction times with polymer at 5 µg/mL, in the order of Figures 7–9 (all values about, read from the bar charts; n = 3, error bars not transcribed). Starting concentrations 20 µg/mL ritonavir, 19 µg/mL efavirenz, 22 µg/mL celecoxib, 37 °C, pH 6.8. SP from Table 5 R9·R10.

The comparison with the group's earlier growth experiments shows that the two kinds of inhibition rank polymers differently. CA 320S Sub, the most effective growth-rate inhibitor for ritonavir in earlier work, gave about 139 min here, against about 244 min for CAB Adp 0.81 R9·R19·C28. Pn-IPAAmd, "a highly effective crystal growth rate inhibitor" of ritonavir, left the induction time at about 119 min R9·R19. The authors conclude that the properties that matter for growth-rate modification are "not necessarily the same" as those for nucleation inhibition ("Clearly") C40.

Celecoxib: the order reverses

For celecoxib a different set of polymers worked (Figure 9). HPMC (about 12.6 min), CA 320S Adp (about 11.2), Pn-IPAAmd (about 10.9) and HPMCAS (about 10.5) were the most effective, against about 2.4 min without polymer R9·C29. The hydrophobic esters that led for ritonavir and efavirenz were nearly inactive: CAB Adp 0.81 and CAB Adp 0.25 about 2.8 min, CAB Seb about 3.0, CAP Seb about 3.1 and CAP Adp 0.33 about 4.0 min R9. The hydrophilic polymers had some effect (about 5.9–8.3 min) R9. The authors find it "very clear" that the more hydrophobic polymers were ineffective, the most hydrophilic had some effect, and the moderately hydrophobic polymers were most effective C29. Even so, the best polymer raised the induction time only from about 2 to 13 min, which they call "rather modest" next to the 3–4 h over which absorption occurs R12·C32.

The authors summarize the three drugs together: "in general", hydrophilic polymers were less effective or ineffective for the more hydrophobic efavirenz and ritonavir, while the more hydrophobic cellulose polymers worked; for the more hydrophilic celecoxib the hydrophilic and some moderately hydrophobic polymers worked and the most hydrophobic did not C30. In the text, 9 of 21 polymers were effective for ritonavir (induction time above 120 min), 14 had an effect for efavirenz and 15 for celecoxib R11.

Solubility parameter matching

Table 5 sorts the polymers by solubility parameter, colour-codes each as ineffective, moderately or very effective for each drug, and boxes the range of effective polymers R10. Table 6 condenses this (Table C): the effective polymers lie at 19.62–22.44 MPa^1/2 for ritonavir, 19.62–23.28 for efavirenz and 21.27–32.24 for celecoxib, and their averages, 20.87, 21.73 and 24.28, lie close to the drugs' own values of 20.03, 21.89 and 23.38 R13. The authors conclude that cellulose polymers with solubility parameters close to the drug's "were effective nucleation inhibitors" and that, "in general", the average SP of the effective polymers is close to the SP of the drug C37.

Table C. Drug solubility parameters against those of the polymers that delayed their nucleation (MPa^1/2; from Table 6) R13.

Note by the AI writer (not discussed in the paper): two checks qualify Tables 5 and 6. The averages in Table 6 are means over every polymer inside each boxed range, including PAlAmn (and, for efavirenz, Pn-IPAAmd), which Table 5 codes as ineffective; over the polymers coded effective only, the means would be 20.89 and 21.74, a negligible difference X3. More important, the colour classes in Table 5 do not follow the time thresholds in its own footnote when compared with the bar heights: for ritonavir, CAP Seb (about 158 min) and HPMCAS (about 150) are coded only moderately effective although above the 145 min threshold, while CMCAB, HPMC and CA 320S Sub (about 129–139 min) are coded ineffective although inside the 120–144 min band; for celecoxib, CA 320S Sub (about 10.0 min) is coded moderate and CA 320S Seb (about 6.0 min) very effective, the reverse of the bars X1. The text's count of 9 effective ritonavir inhibitors matches neither the 8 coloured polymers nor the 11–13 bars above 120 min, but equals the number of polymers inside the boxed ritonavir range X2. The ranking pattern itself is clear in the bar heights; the precise class boundaries are not.

Polymer ionization

At pH 3.8 the efavirenz induction time with CAP Adp 0.85 fell from about 24.9 to about 12.6 min, and with CAB Adp 0.81 from about 27.9 to about 13.6 min, while with the nonionizable HPMC it was about 14.9 min at pH 6.8 and about 19.1 min at pH 3.8 (Figure 10, Table D) R14. The authors define effectiveness as the induction time with polymer divided by that without, and report that the effectiveness of the two esters "decreased by ∼50%" at pH 3.8, whereas that of HPMC is "comparable at pH 3.8 and 6.8, as expected" E9·C31. The text names the first ester "CAP Adp 0.81", a misprint for CAB Adp 0.81 X6.

Table D. Efavirenz (19 µg/mL) induction times at two pH values, polymer 5 µg/mL (read from Figure 10, approximate) R14.

Note by the AI writer (not discussed in the paper): the "about 50%" fits the raw induction times. Because efavirenz alone took longer at pH 3.8 (about 6.6 against 5.1 min), the effectiveness ratio as the paper defines it fell by about 61–62% (from about 4.9 to 1.9 and from about 5.5 to 2.1), and HPMC's ratio stayed at about 2.9 X8. The low-pH runs also used acetate rather than phosphate buffer, so pH and buffer changed together X9.

A lower supersaturation

Starting celecoxib at about 11 µg/mL instead of 22 changed the picture. Without polymer the induction time was 27.5 ± 0.7 min, and HPMC extended it to 855.5 ± 6.0 min, "increasing the induction time by a factor of 31" R15. At 22 µg/mL the same polymer had raised it only from about 2.4 to about 12.6 min R9. Figure 11 shows what this means over time: from 22 µg/mL the HPMC solution stayed at a supersaturation of about 15 for roughly 15 min, fell to about 7.8 by 35 min and levelled near 4.3 by 200–240 min, while from 11 µg/mL it stayed at about 7.3 throughout the 240 min R16. The areas under the curves to 240 min were 1426 (from 22 µg/mL) and 1745 (from 11 µg/mL), against 239 for a saturated solution, giving supersaturation factors of 5.9 and 7.3 R16·E8. The authors summarize that higher levels of inhibition were reached at lower supersaturation, giving "an overall enhancement in the extent of supersaturation" over time C33.

Note by the AI writer (not discussed in the paper): 1426/239.0 is 5.97, which rounds to 6.0 rather than the printed 5.9 X12. The y-axis of Figure 11 is labelled as concentration in µg/mL, but the plotted values are supersaturation ratios (the 22 µg/mL curve starts near 15, and the line labelled "saturated solution (1.5 µg/mL)" sits at 1) X13. Also visible in Figure 11, but not discussed, is that after nucleation the 22 µg/mL solution with HPMC did not fall to saturation within 240 min but levelled at a supersaturation of about 4.3 R16.

Discussion

How the authors explain the pattern

The authors' explanation starts from the two-step model. They assume ("Here we assume") that polymers "most likely" affect nucleation by hindering the reorganization of a cluster of solute molecules into an ordered structure, the step proposed as rate-limiting C5. Following the simulations of Anwar et al., an effective inhibitor "should thus interact favorably with the solute": a strong affinity for the solute, relative to the solvent and to other polymer molecules, both brings the polymer to the solute aggregates and disrupts their reorganization C35. Considering only nonspecific interactions, the polymer's hydrophobicity relative to the drug sets this affinity: a very hydrophilic polymer would be expected to prefer water, and a too-hydrophobic polymer "is likely to interact more favorably with other polymer molecules" C4. A polymer should therefore have a hydrophobicity similar to the drug's, or form highly favourable specific interactions with it, to inhibit nucleation C36. The chain below sets out the argument with its two outcomes.

Structure, not only hydrophobicity. For efavirenz and ritonavir the synthetic polymers were ineffective even when their solubility parameter matched the drug's, as for Pn-IPAAmd and PAlAmn C38. The authors suggest that the structure of the monomer units "may also influence" the ability to disrupt solute packing, and that the ineffectiveness of the synthetic polymers "perhaps can be attributed to their flexible structures"; the bulky substituents and fewer degrees of freedom of the cellulose esters would cause more disruption through steric effects (refs 17, 46, 47) C39. In the Conclusion this becomes the second of two key properties: cellulose derivatives with bulky side groups were more effective than synthetic polymers of similar hydrophobicity C38.

Ionization. The synthesized esters are amphiphilic, and their amphiphilic nature "appears to further enhance" inhibition: at pH 6.8 they were more effective than the neutral cellulose derivatives C41. The authors argue that ionized carboxylate groups repel each other between monomer units, preventing self-association and so maximizing interaction with the solute, which "apparently increases their impact on solute nucleation" C42. At pH 3.8, with less than a quarter of the groups ionized, the loss of effect "may be attributed to increased self-association of additive molecules" C43.

Supersaturation level. Since nucleation slows as supersaturation falls, polymers "are likely" to inhibit more strongly at lower supersaturation, which the celecoxib experiment confirmed in the authors' view: a moderate supersaturation was held far longer than a high one, giving a larger area under the curve and a higher supersaturation factor C45·R15·R16. For the rapid crystallizers efavirenz and celecoxib, the polymers' effect at the high supersaturations generated here was of limited practical value, with crystallization within minutes even with the best inhibitor C44. The authors call the pursuit of very high supersaturation "a double-edged sword": it raises the nucleation tendency, and a supersaturating dosage form "may fail" through rapid crystallization; "it may be appropriate" to aim for a lower supersaturation sustained for longer, perhaps controlled in vivo with controlled-release dosage forms or less soluble high-energy forms C46.

What it suggests for formulation

The authors draw two practical messages. Polymers should not be chosen arbitrarily, because their effect "seems to be linked" to the properties of the drug C34. And the identified properties, hydrophobicity similar to the drug and a bulky cellulose structure, matter for developing new excipients with better crystallization-inhibiting properties C3·C38·C47.

Note by the AI writer (not discussed in the paper): for hydrophobic drugs such as ritonavir and efavirenz, the polymers most often used as amorphous solid dispersion carriers, PVP, PVPVA and HPC, barely delayed nucleation from solution in this test, while carboxylated cellulose esters of similar hydrophobicity did R9. Because growth inhibitors and nucleation inhibitors ranked differently here, a screen based only on seeded growth could select the wrong polymer for a formulation that relies on preventing nucleation R19·C40. The loss of effect of the acid esters at pH 3.8 suggests that their protection may be weaker in the stomach than in the intestine R14.

Limitations

Limitations the authors state

The authors note that the measured induction time is not a fundamental property and depends on the detection method C13. They describe the gains at the amorphous-solubility supersaturation as limited: for celecoxib the increase from 2 to 13 min is "rather modest" against a 3–4 h absorption window, and for efavirenz and celecoxib the solutions crystallized within minutes even with the best inhibitor C32·C44. Their mechanistic statements carry explicit hedges ("we assume", "most likely", "perhaps", "apparently", "may be attributed") C5·C39·C42·C43.

The AI writer's own judgement

What holds. The induction times were measured in triplicate under one protocol for three drugs and 21 polymers, with less than 15% scatter R9·R17. The central pattern is large and visible directly in the bar charts: the best polymers for ritonavir and efavirenz are nearly inactive for celecoxib, and the reverse R9. The 31-fold gain for celecoxib with HPMC at lower supersaturation has tight replicates R15. The drugs' own behaviour is cross-checked by three methods (amorphous solubility, PLM and induction time) R2·R4·R8.

What weakens it.

• The solubility-parameter rule is a correlation over three drugs, and the Table 6 averages include polymers coded ineffective R13·X3.
• The Table 5 colour classes and the stated count of effective ritonavir inhibitors do not follow the bar heights or the footnote thresholds X1·X2.
• The structure rule is stated generally in the Conclusion, but for celecoxib the synthetic Pn-IPAAmd was among the four best polymers and PAA beat most cellulose esters X11.
• The claim that the ionizable esters beat neutral cellulose derivatives at pH 6.8 holds for ritonavir and efavirenz but not for celecoxib, where HPMC was best; which derivatives count as neutral is not stated, although HPMCAS, CAPh and CMCAB carry acidic groups X10.
• The pH test changed the buffer as well as the pH, and the "about 50%" applies to the raw times rather than to the effectiveness ratio as defined X9·X8.
• The induction time contains the growth to detectable size; the assumption that this is small was not tested, and with the strongest inhibitor the extinction rose gradually before the fitted onset (about 180 against 260 min for ritonavir), so the longest induction times depend on where the regression lines are drawn X14·C14.
• The authors call the polymers' effect on crystalline solubility "negligible", yet for celecoxib two polymers raised it by 40–53%, which would lower the true starting supersaturation with those polymers X7.
• The cluster mechanism, polymer self-association and chain rigidity were argued from simulations and structure, not measured C5·C39·C43.
• Only one polymer concentration (5 µg/mL) was used, and the lower-supersaturation benefit was shown for one drug and one polymer A2·P12.

Conclusion

At 5 µg/mL, the 21 polymers lengthened the induction time of three drugs started at their theoretical amorphous solubility by at most about 2-fold (ritonavir, about 120 to 244 min) and about 5- to 6-fold (efavirenz about 5.1 to 28 min, celecoxib about 2.4 to 12.6 min) R9·R12. Which polymers worked depended on the drug: hydrophobic cellulose adipate esters for ritonavir and efavirenz, moderately hydrophilic polymers such as HPMC for celecoxib, and the effective polymers averaged close to the drug's own solubility parameter R9·R13. The authors conclude that effective inhibitors have "an optimal level of hydrophobicity" similar to the drug and a bulky cellulose structure, factors that "are likely" to act by promoting polymer-solute interactions that disrupt cluster reorganization C3·C38·C5. The celecoxib experiment adds a formulation lesson the authors state directly: halving the supersaturation let HPMC delay nucleation 31-fold and delivered more supersaturation over 4 h, so a lower, sustained supersaturation "may be appropriate" R15·R16·C46. The mechanism remains untested, and the class boundaries in Table 5 should be read from the bar heights rather than from the colours X1·C5.