Measuring how much drug has crystallized inside an amorphous solid dispersion: whole-pattern PLS against the peak-area ratio

Background and motivation

An amorphous solid dispersion places a poorly soluble drug in the amorphous state, intimately mixed with a polymer, to raise its apparent solubility and, potentially, its bioavailability; the paper calls this "one potential method" and cites the classic reviews of Chiou and Riegelman (1971), Leuner and Dressman (2000), Serajuddin (1999) and Ford (1986). C3 The advantage holds only while the drug stays amorphous. The free energy of the drug in the dispersion is typically higher than that of its crystal, which means the drug is above its "solubility limit" in the polymer (Tao et al., 2009; Marsac et al., 2006b, 2009). Crystallization therefore cannot be held off indefinitely and "may proceed to varying degrees" during storage, as the same group had shown for felodipine dispersions in Rumondor et al. (2009b). C4 A formulator who follows the physical stability of a dispersion thus needs a number: how much of the drug has become crystalline.

Ways to quantify crystallinity were well developed for simpler systems. Differential scanning calorimetry, infrared and Raman spectroscopy and powder X-ray diffractometry (PXRD) had been used to measure crystallinity in pure drugs or the ratio of polymorphs (for example Clas et al., 1995; Taylor and Zografi, 1998; Shah et al., 2006), and crystalline drug had been quantified in the presence of other components (Suryanarayanan and Herman, 1991; Moore et al., 2008, 2009). What had "not been widely reported" was the quantification of crystalline drug inside an otherwise amorphous, molecular-level dispersion, where drug and polymer are mixed molecule by molecule. C5 PXRD was the natural candidate: it is "well-suited" to detecting crystalline material, and detection limits of 5% crystallinity or lower had been reported for mixtures of amorphous and crystalline drug (Fix and Steffens, 2004; Nunes et al., 2005; Surana and Suryanarayanan, 2000, among others). C6·R1

The difficulty lies in the amorphous part of the pattern. Earlier work by the group (Rumondor et al., 2009a) and by Newman et al. (2008) had shown that the X-ray pattern of a molecular-level dispersion cannot be described as a linear combination of the patterns of the pure amorphous components; intimate drug–polymer mixing may change the shape of the amorphous halo. C8 A calibration built from physical mixtures of amorphous drug, crystalline drug and polymer would therefore not represent a real, partly crystallized dispersion. The authors argue that the calibration samples must capture this multiphase nature and that the mathematical model must accommodate "subtle and not so subtle changes" in the patterns, and they set out to test whether multivariate PLS modelling gives more accurate numbers than the traditional peak-area or peak-height methods. C2·C5

The model system was felodipine (FEL) with poly(vinylpyrrolidone) (PVP K29-32), a pair that the group had studied extensively and that is "known to" form an intimately mixed amorphous dispersion (Marsac et al., 2006a, 2006b, 2008, 2010; Konno and Taylor, 2006). C7·M1·M2

Background concepts

Bragg peaks and the amorphous halo. In a PXRD pattern, crystalline material gives sharp peaks at angles set by its lattice (for felodipine the strongest lie near 10.3, 16.5 and 23.3° 2θ, read from Figure 1), while amorphous material gives one or two broad humps, the halo (near 12 and 22° 2θ for the felodipine–PVP dispersion). R6 In a partly crystallized dispersion both are present, and the question is how to turn their relative size into a percentage.

What "% crystallinity" means here. Throughout the paper, % crystallinity is the percent of the drug, not of the sample, that is crystalline: a sample with 50% PVP and "25% crystallinity" contains 50% felodipine, of which a quarter (12.5% of the sample) is crystalline (Figure 1 caption). P2·X12 The paper converts once between the two scales, for the most dilute system:

In plain terms: an error of 4.7% of the drug in a dispersion that is 25% drug corresponds to $0.047\times 0.25\approx 0.012$, about 1.2% of the sample mass (Section 3.3). E3·R19

The area method. The traditional estimate, taken by the authors from Shah et al. (2006) and Nunes et al. (2005), divides the area of the Bragg peaks by the total area under the pattern (Equation 1):

In plain terms: $A_c$ is the area contributed by the crystalline phase (the Bragg peaks) and $A_a$ the area contributed by the amorphous phase (the halo), both between 5 and 35° 2θ; their ratio equals the crystalline fraction only if the measured crystalline and amorphous intensities are proportional to the crystalline and amorphous fractions. E1·C13 The authors used areas rather than peak or halo intensities "to minimize variations due to lattice strain and particle size". C14

Vainshtein's law. The total X-ray intensity a sample scatters is conserved when its material changes between crystalline and amorphous states at constant composition; what the peaks gain, the halo loses. The paper invokes it in words, not as an equation, to interpret the PLS loadings (Section 3.3, citing Chen et al., 2001). E2·C22

PLS regression. Partial least-squares regression is a multivariate calibration method. It treats each pattern (here one intensity per 0.04° step between 5 and 35° 2θ) as one vector, finds a few latent "components" that are linear combinations of the intensities chosen to predict the property of interest (here % crystallinity), and regresses the property on them. Each component has loading weights, one per 2θ position, that show which parts of the pattern it uses. R2Y is the fraction of the variation in the property that a component explains; the cumulative R2Y adds the components. E5·R14·R15

Calibration statistics. For both methods the paper reports the slope of predicted against actual crystallinity (ideally 1), the goodness of fit R² (ideally 1) and the root-mean-squared error of estimation, RMSEE, in units of % drug crystallinity (Table 1). R9·E4 RMSEE is an error of estimation: it measures how well the model reproduces the samples it was built on. It is not the same as a cross-validated error or an error on new samples, and it is not a detection limit; the paper reports neither of the latter. X3 The 95% prediction bands in Figures 2 and 6 follow De Gryze et al. (2007) and show the range in which single new predictions are expected. E6

Methods in detail

The study has one experimental core, a set of calibration samples, and two ways of reading their diffractograms. The flow below summarizes it.

Contribution 1: calibration samples that contain the real dispersion

Problem. A calibration set must look like the samples it will measure. A partly crystallized dispersion contains crystals plus a molecular-level amorphous mixture whose halo differs from that of pure amorphous drug, so mixtures of amorphous drug, crystals and polymer would miss the feature the authors cared about. C8

What they did. Crystalline felodipine and PVP K29-32 (kept over P₂O₅ for at least a week) were dissolved at the required ratio in a 1:1 (weight basis) mixture of ethanol and dichloromethane, rotary-evaporated, dried overnight under vacuum and gently ground in a mortar below 10% relative humidity; PXRD confirmed that these dispersions were X-ray amorphous. M2·P1·R2 Crystalline felodipine was then added, and each mixture was cryo-milled (two impacts per second, 4 min in total) and passed through a 150-µm sieve. P2 The design crossed three overall PVP contents (25, 50 and 75% w/w) with six crystallinity levels (0, 10, 25, 50, 75 and 90% of the drug added as crystals), with three independent replicate samples per point, so each calibration model rests on 18 samples. P2·A2·A3

Note by the AI writer (not discussed in the paper). Because the overall drug:polymer ratio is fixed within each model while the crystalline share rises, the amorphous part of the sample loses drug as crystallinity rises. From the stated levels, its drug content falls from 75% to about 23% between 0 and 90% crystallinity at 25% PVP, from 50% to about 9% at 50% PVP, and from 25% to about 3% at 75% PVP. X8 This mimics what happens when drug crystallizes out of a real dispersion, which is presumably the point of the design; it also means the halo shape changes along each calibration series, which is one of the "subtle" changes the authors expected a model to accommodate. C2·X8

Controls on the preparation. Cryo-milling could, in principle, dissolve some crystals into the dispersion or damage them. Three samples with 25% PVP and half the drug added as crystals were milled for 2, 4 and 6 min and examined by attenuated total reflectance infrared spectroscopy and PXRD. P4 The IR spectra overlapped; the bands that Marsac et al. (2006a) assigned to specific felodipine–PVP interactions did not grow, and the crystalline felodipine band did not change, which "indicates" that milling did not form detectable new dispersion. R3·C9 The diffractograms were almost identical, apart from a slight drop in Bragg-peak height that the authors read as milling that "may have introduced some crystalline defects". R4·C10 The calculated crystallinity of samples milled for 2 and 6 min differed by less than 2%, "well within" the errors of the calculation methods. R5·C11

Data collection. Patterns were collected on a Shimadzu XRD-6000 with Cu-Kα radiation in Bragg–Brentano geometry, from 5 to 35° 2θ at 8°/min with 0.04° steps. Before each day of measurement the Si [1 1 1] peak had to fall between 28.423 and 28.463°, and three consecutive readings of every sample were overlaid to check that no crystallization happened during the measurement. P3 Each pattern was then normalized to the photon intensity of the Si standard recorded the same day and to unit area. P5 Slower scans were rejected for a specific reason: in preliminary runs at 2 or 4°/min on mixtures of amorphous and crystalline nifedipine, the Bragg peaks grew from one reading to the next (data not shown), which the authors take to "suggest" that the X-ray exposure itself induced crystallization. R11·C19

Contribution 2: the area method as baseline

What they did. The total area under each normalized pattern ($A_c+A_a$) was integrated from 5 to 35° 2θ by the trapezoidal rule. Fourier-transform filtering separated the sharp Bragg peaks from the halo, and the Bragg-peak area $A_c$ was integrated separately; Equation 1 then gave one crystallinity value per pattern. P6·E1 The calculated values were plotted against the known values with a fitted straight line and 95% prediction bands (Figure 2). R7·R8 The authors also tried variants: correlating crystallinity with the total Bragg-peak area, and several other area normalizations. R10

Why it is a fair baseline. It is the standard univariate approach, it uses the same normalized patterns as PLS, and it rests on a clear physical assumption (Equation 1) whose failure can be diagnosed. E1·C13

Contribution 3: PLS models of the whole pattern

What they did. For each PVP level, a PLS model was built in SIMCA-P+ v.11 with the normalized pattern as the predictor and the known crystallinity as the response. Two components were "found sufficient to fit the data without over-fitting"; the criterion is not stated. P7·R12·X4 Normal probability plots of the first models showed nonlinearity, especially at 25 and 50% PVP, and log-transformed models were better; which variable was transformed is not stated, and since the response includes 0% values, a plain logarithm of the response would not be defined. R13·X4 The paper reports the slope, R² and RMSEE of the final models (Table 1), the R2Y of each component (Table 2), the loading weights (Figures 3–5) and the predicted-against-actual plots with prediction bands (Figure 6). R9·R14·R15·R16·R17·R18

How it differs from the area method. PLS does not need the halo and the peaks to be separated, and it can use any feature of the pattern that varies with crystallinity, including shifts or changes of shape, not only areas. The cost is that the model is a fitted, empirical calibration: its statistics describe how well it fits the samples it was built on unless it is validated on others. C33·X3

Experimental analysis

What the patterns look like

Figure 1 shows the normalized patterns of the 50% PVP series. The 0% sample is a smooth double halo with maxima near 12 and 22° 2θ; as the crystalline share rises, felodipine Bragg peaks (near 10.3, 16.5, 20.6, 23.3, 24.6, 25.5 and 26.5° 2θ, read from the figure) grow out of it, and the 10.3° peak is already visible at 10% crystallinity. R6 The authors state that the 25 and 75% PVP series gave similar profiles, which are not shown. R6 This is the expected picture: Bragg intensity rises relative to the halo as crystalline drug replaces amorphous drug. C12

The area method misjudges crystallinity

Figure 2 plots, for each PVP level, the actual crystallinity against the value calculated with Equation 1. The scatter is large (values read from Figure 2, approximate). At 25% PVP, the three 90% samples came out at about 108, 125 and 126%, and one 10% sample at about 47%. At 75% PVP, the 0% samples ranged from about −12 to 14% and the 75% samples from about 47 to 119%. R7 Values below 0 and above 100% are physically impossible, which shows how far the single ratio can drift. R8 The 95% prediction bands lie about 21, 39 and 46 percentage points on either side of the fitted line for 25, 50 and 75% PVP (read from Figure 2). R8

Table A. Calibration statistics of predicted against actual % felodipine crystallinity for the two methods (from Table 1; RMSEE in % drug crystallinity; 18 samples per model). R9

Table A gives the summary. As the polymer content rises from 25 to 75%, R² of the area method falls from 0.93 to 0.80 to 0.75 and RMSEE rises from 11.2 to 17.0 to 23.6% drug crystallinity. R9 The authors explain the trend by the shrinking contribution of the Bragg peaks to the total area at higher polymer content, and read the slopes of 1.26, 1.03 and 1.25 as a sign of "poor overall correlation". C15·C16 Other area-based calculations did no better: none improved the prediction, linearity was lost, and RMSEE was "unacceptably high (>20%)" at both extremes of crystallinity (data not shown). R10·C17

Note by the AI writer (not discussed in the paper). Two details matter when reading Table 1. First, Figure 2 plots actual crystallinity on the y-axis, and its drawn lines have slopes of about 0.79, 0.97 and 0.80; the reciprocals (about 1.27, 1.03 and 1.25) match the tabulated slopes, so Table 1 slopes are those of predicted on actual, as its caption says, and a slope above 1 means overestimation at high crystallinity. X2 Second, the paper does not say how RMSEE was computed for the area method. The root mean square of the raw differences between calculated and actual values, read from Figure 2a, is about 26%, whereas the scatter about the drawn line is about 9%; the tabulated 11.2% is therefore closer to an error after linear recalibration than to the raw error of Equation 1. X1

The authors discuss why the area method fails. "One possible cause" is a poor signal-to-noise ratio, which a slower scan could improve, but slower scans induced crystallization in the nifedipine tests. C18·C19 "Other factors that may have introduced errors" are a lack of sample uniformity and markedly different proportionality constants for the amorphous and crystalline phases, which would break the assumption behind Equation 1. C20·C13 None of these causes was measured. X11

PLS reproduces the known crystallinity closely

With two-component PLS models, the calculated values lie close to the actual ones at all three PVP levels (Figure 6). Read from the figure (approximate): at 25% PVP the 90% samples came out at about 91–92% and the 0% samples at about −3 to 1%; at 50% PVP the largest spread was at 90%, from about 86 to 94%; at 75% PVP the 75% samples came out at about 63, 66 and 76%. R17 The prediction bands lie about 6, 6 and 10 points on either side of the line, against about 21, 39 and 46 for the area method; the authors call them "much narrower". R18·R8·C30

RMSEE fell from 11.2 to 2.2% at 25% PVP, from 17.0 to 1.9% at 50% and from 23.6 to 4.7% at 75%, a five- to nine-fold reduction; R² rose to 0.996, 0.997 and 0.978 and the slopes to 0.998, 0.997 and 0.984 (Table A). R9 The authors call the PLS predictions "far better" in the abstract and "much better" in the results, and point out that the gain is "particularly apparent" at 75% PVP, where the Bragg signal is weakest. C27·C28 The 75% PVP model still has the widest prediction band, "most likely as a result of lower sensitivity caused by reduced intensity of the Bragg peaks". C31

For the 75% PVP dispersion the drug is a quarter of the mass, so an RMSEE of 4.7% of the drug is about 1.2% of the sample. The authors say this "indicates" that PLS can estimate crystallinity at that level, comparable with results for blends of pure crystalline and amorphous material on a laboratory diffractometer (Surana and Suryanarayanan, 2000). R19·E3·C29 As discussed under Limitations, this compares a calibration error with a detection limit. X3

What the PLS components capture

Table B. Fraction of the variation in % felodipine crystallinity explained by each PLS component (from Table 2, where the first column is headed "Loading"). R14

The first component explains most of the variation at 25 and 50% PVP (R2Y 0.872 and 0.948) but only half at 75% PVP (0.511), where the second component adds almost as much (0.448). R14 Its loading weights (Figures 3–5, trace a) follow the pattern of crystalline felodipine (trace c), with maxima at the felodipine Bragg positions. R15 The authors report that the halo contributes negatively and the Bragg peaks positively, so the first component describes intensity moving from the halo into the peaks as crystals replace amorphous drug at a fixed drug:polymer ratio; the total diffracted intensity is preserved, "in agreement with" Vainshtein's law, and the authors note that this result is "in agreement with expectations". C21·C22·E2

The second-component loading weights (trace b) are noisy, with sharp features at the main Bragg positions near 10.3, 16.5 and 23.3° 2θ. R16 The authors interpret them as shifts in both the Bragg peaks and the halo and slight broadening of the Bragg peaks, and offer three "potential explanations": experimental inaccuracies, since "experimental and sample limitations may have contributed to shifts"; strain in the crystal lattice and small crystallites, which "may have been introduced by the cryo-milling process"; and intimate drug–polymer mixing in the dispersion, which changes the halo. C23·C24·C25·C26

Note by the AI writer (not discussed in the paper). The loading traces are printed normalized and offset, with no zero line, so the signs the authors describe cannot be read from the figures, and no peak positions or widths are reported to support the shift and broadening reading. X7 The cumulative R2Y values (0.992, 0.994, 0.958) also differ from the PLS R² of Table 1 (0.996, 0.997, 0.978); the paper does not say whether the two are computed on different scales, for example before and after the log transformation. X5

Controls

Three controls support the comparison. The preparation step barely affects the result: milling for 2 or 6 min changed calculated crystallinity by less than 2% and left the IR spectra unchanged. R3·R4·R5 Three consecutive readings of every sample were overlaid to check that the 8°/min scan did not crystallize the samples, whereas slower scans on nifedipine did produce growing Bragg peaks. P3·R11 Finally, the poor performance of the area method is not an artefact of one normalization, because several variants were tried. R10

Discussion

The authors' argument can be laid out as a short chain that splits at the choice of method.

The conclusion the authors draw is that drug crystallinity in these dispersions "can be well predicted" by PLS, while the area-based calibration is worse, and that "It is thought" the improvement comes from subtle variations in the patterns (small crystallites or crystal strain, and intimate drug–polymer mixing) captured by the second-component loading weights. C27·C32 They add two practical advantages: PLS "requires less subjective and time intensive data analysis", because the halo and the Bragg peaks need not be separated, and a PLS model "enables further analysis of the data", such as the detection of outliers and nonlinearity and, through the loadings, insight into which parts of the pattern matter. C33·C34

What sets the accuracy. In both methods the error grows where the Bragg peaks are weakest. With the area method RMSEE doubles from 25 to 75% PVP; with PLS the 75% PVP model has the largest RMSEE (4.7%) and the widest prediction band, which the authors attribute ("most likely") to lower sensitivity from weaker Bragg peaks. R9·C15·C31 Expressed per sample mass the picture changes, because the 75% PVP samples hold the least drug: their 4.7% error is about 1.2% of the sample, the figure the authors quote. R19·X12

Note by the AI writer (not discussed in the paper): how much of the gain needs the second component? The data in Table B do not single out the second component as the source of the improvement. At 50% PVP, the first component alone explains R2Y 0.948, already above the area-method R² of 0.80, and the second adds only 0.0465, yet this is the level with the largest drop in RMSEE (17.0 to 1.9%). At 25% PVP, by contrast, the first component alone (0.872) is below the area-method R² (0.93), and at 75% PVP the second component carries almost half of the fit. R14·R9·X6 Whole-pattern regression may therefore help mainly by using all the intensity, with the second component mattering most when the Bragg signal is weak; the paper gives no RMSEE for one-component models that would settle this. X6

What it suggests for formulation work (the AI writer's reading). For physical-stability studies of a dispersion, the paper gives a recipe for a calibration set that resembles the samples: make the real dispersion at the target overall drug loading and spike it with crystalline drug, rather than mixing amorphous drug, crystals and polymer. C8·P2 Whole-pattern PLS then reaches errors of a few percent of the drug on the calibration set, far below what the peak-area ratio achieves. R9 Before such a model is used to accept or reject samples, it needs a validation the paper does not show: cross-validation or an independent test set, and ideally a measured detection limit. X3 Crystallinity should be reported per drug and per sample mass, since the same per-drug error means different amounts of crystals at different loadings. X12 Finally, the calibration crystals were milled with the dispersion; crystals grown slowly in storage may be larger and less strained, and the second component, which the authors link to strain and small crystallites, may treat them differently. X13·C25

Limitations

Stated by the authors. The authors list possible error sources rather than limitations of scope. For the area method they name poor signal-to-noise ("One possible cause"), which they could not fix by scanning slower because slow scans appeared to crystallize amorphous nifedipine ("suggest"), and a possible lack of sample uniformity and different proportionality constants for the two phases ("may have introduced errors"). C18·C19·C20 For the PLS loadings they note that "experimental and sample limitations may have contributed to shifts", and that cryo-milling may have introduced strain and small crystallites. C24·C25 They also acknowledge that milling may have introduced some crystalline defects, although its effect on calculated crystallinity was below 2%. C10·R5

The AI writer's own judgement.

• No validation. RMSEE describes the fit to the 18 calibration samples of each model; no cross-validation statistic, no test set and no limit of detection or quantitation is reported. The statement that PLS "can estimate" about 1.2% crystalline drug in the 75% PVP dispersion, and its comparison with detection limits in simple blends, rests on this fit error. X3·C29
• Unclear area-method statistic. How RMSEE was computed for the area method is not stated, and the raw errors of Equation 1 read from Figure 2a (about 26%) are larger than the tabulated 11.2%. X1
• Model details missing. Which variable was log-transformed, how the number of components was chosen, and why the cumulative R2Y of Table 2 differs from the R² of Table 1 are not explained; the models cannot be rebuilt from the paper. X4·X5
• Untested mechanism. The attribution of the PLS gain to second-component features is hedged ("It is thought"), the weight of that component differs widely between PVP levels, and the loadings cannot be checked for sign or shift shape on the printed figures. C32·X6·X7
• Scope. One drug, one polymer grade, one diffractometer and one milling protocol; the amorphous composition changes along each calibration series by design; the calibration crystals were milled. X8·X13
• A claim not demonstrated. Outlier detection is offered as an advantage of PLS, but no outlier is reported or examined. X9
• A small inconsistency. The heading of Section 3.2 refers to the "relative height" of the Bragg peaks, while the method uses areas. X10

How solid overall (AI assessment). Moderate. The improvement is large, consistent at all three polymer levels and read directly from the same diffractograms, which makes the comparison between the two methods fair. What is not established is how well the PLS models predict samples they were not fitted to, and why they work as well as they do. R9·X3·X6

Conclusion

The paper shows, for felodipine–PVP dispersions at 25, 50 and 75% polymer, that a calibration set built from the real dispersion plus crystalline drug, read by a two-component PLS model of the whole PXRD pattern, reproduces the known drug crystallinity with RMSEE of 2.2, 1.9 and 4.7%, while the classical Bragg-peak-to-total-area ratio gives 11.2, 17.0 and 23.6% and even impossible values below 0 and above 100%. R9·R7·C27 The first PLS component captures the exchange of intensity between the amorphous halo and the Bragg peaks; the second, which the authors read as peak and halo shifts and broadening, is "thought" to explain the remaining gain. C21·C23·C32 The approach gives a route to following crystallization in a dispersion by PXRD, with the caveat that its reported errors are calibration-fit errors that still need validation on independent samples. X3