Calibration-Free Second Harmonic Generation (SHG) Image Analysis for Quantification of Trace Crystallinity Within Final Dosage Forms of Amorphous Solid Dispersions

Abstract

A statistical model enables auto-calibration of second harmonic generation (SHG) images for quantifying trace crystallinity within amorphous solid dispersions (ASDs) over a wide dynamic range of crystallinity. In this paper, we demonstrate particle-counting approaches for quantifying trace crystallinity, combined with analytical expressions correcting for particle overlap bias in higher crystallinity regimes to extend the continuous dynamic range of standard particle-counting algorithms through to the signal averaging regime. The reliability of the values recovered by these expressions was demonstrated with simulated data as well as experimental data obtained for an amorphous solid dispersion formulation containing evacetrapib, an Eli Lilly and Company compound. Since particle counting independently recovers the crystalline volume and the SHG intensity, the average SHG intensity per unit volume can be used as an internal calibrant for quantifying crystallinity at higher volume fractions, for which particle counting is no longer applicable.

Keywords

Second harmonic generation SHG image analysis amorphous solid dispersion calibration-free nonlinear optics autocalibration pharmaceuticals

Introduction

Modern drug discovery efforts fueled by the growth of high throughput screening and combinatorial chemistry as tools in the pharmaceutical industry have led to more drug candidates with greater chemical complexity and correspondingly lower aqueous solubilities.^1,2 Poorly water-soluble active pharmaceutical ingredients (APIs) lead to reduced bioavailability, which in turn results in an inadequately effective drug formulation.² Consequently, the preparation of amorphous forms of an API has become increasingly favorable due to their higher apparent solubilities and faster dissolution rates.^3,4 However, amorphous APIs are typically metastable forms with a thermodynamic driving force to crystallize over a range of timescales dependent on the composition of the formulation and the external environment. A common strategy used to prevent crystallinity is the addition of crystallization-inhibiting polymers that contributes to the stabilization of amorphous formulations, resulting in an amorphous solid dispersion (ASD).^4–6 However, even trace residual crystallinity can provide nuclei for crystal formation during storage and subsequently following introduction to the body, with corresponding reduction in bioavailability.^7,8 Therefore, the ability to detect and characterize crystallinity within drug products is critical for the assessment of the safety, stability, and efficacy of ASD drug products.

Several methods are currently available for analysis of residual crystallinity within nominally amorphous formulations, including infrared (IR) and Raman spectroscopy,^9–13 powder X-ray diffraction (PXRD),^14–15 solid-state nuclear magnetic resonance (ssNMR),^16–19 and differential scanning calorimetry (DSC).^20,21 All of the methods mentioned are often limited to detecting total crystallinity no lower than about 1% (w/w%) under ideal conditions (i.e., high drug load and limited excipient interference) utilizing benchtop instrumentation, which may not be sufficient as even trace amounts of crystals can impact efficacy. Thus, it is important that the analytical tool used for the characterization of crystallinity in ASDs is selective and as sensitive as possible.

Nonlinear optical (NLO) imaging has proven to be a successful method in the pharmaceutical field for the sensitive detection of trace crystallization in amorphous systems.^22,23 Second harmonic generation (SHG) is a coherent process, in which the incoming light is converted to light of twice the frequency. The particular symmetry properties of SHG only permit this second-order process in ordered non-centrosymmetric systems; amorphous or disordered systems do not generate coherent SHG. Crystals of homochiral molecules generally fall into space groups that are symmetry-allowed for SHG, providing highly selective and sensitive detection of homochiral crystalline APIs.^22–28 Limits of detection for chiral crystallinity using SHG have been reported in the parts per trillion (ppt) regime in melt-quenched films and in the parts per million (ppm) regime for powders.^23,29 More recently, SHG has been used to monitor changes in crystallinity in amorphous solid dispersions in spray-dried dispersions.³⁰ In that work, the integrated SHG intensity was found to scale with the fraction of crystallinity arising during storage at high temperature and humidity, with limits of detection for crystallinity down to ∼0.01%. This capability greatly reduced the timescale required for early identification of conditions leading to crystal formation. Complementary NLO methods including coherent anti-Stokes Raman (CARS) imaging involving interactions from multiple beams have also been demonstrated for compositional analysis of pharmaceutical materials.^31,32 In general, CARS lacks the symmetry-dictated selectivity of even-ordered NLO processes, although multimodal imaging integrating both SHG and CARS has the potential to recovers such information.³²

Despite successes in detection, precise quantification of residual crystallinity by SHG continuously spanning a wide dynamic range of crystalline content remains challenging. To date, quantification of crystallinity has relied almost exclusively on the integrated SHG intensity produced within a field of view (FOV). This approach has significant limitations for developing predictive models for crystallization kinetics. Quantification of SHG by the integrated SHG intensity also typically requires the need for calibration standards,²³ in which measurements of the pure substance provide a reference point for connecting the measured SHG intensity back to a percent crystallinity. However, access to reliable standards is not always trivial. For many APIs, several different crystal forms are potentially kinetically accessible during crystal formation. Given the high sensitivity of SHG to crystal form,²⁷ standards prepared based on the assumption of one polymorphic form would be inapplicable for measurements obtained with a different crystal form. In addition, the integrated SHG can be dependent on the crystal size distribution, such that sets of measurements acquired with identical percentage crystallinities can produce quite different integrated SHG activities. Algorithms have been developed to address bias from the particle size distribution,²³ but represent yet another potential complication for routine quantification of crystallinity.

In one recent departure from the signal integration approach for quantification, Schmitt et al. directly recovered the particle size distribution via image analysis for poorly soluble particulates present in commercially available formulations of Abraxane_.²² Second harmonic generation-active particles as large as 120 µm in diameter were observed within the commercial lyophilized powders. In comparison to signal integration, this particle counting approach has the potential to further lower the limits of detection by SHG (by analogy with photon counting). Differences in particle size and orientation greatly influence the SHG intensity, but not the integrated area in the image. Despite the numerous advantages of particle counting in the limit of low crystallinity (<1%), the dynamic range is quite limited; as the overall crystallinity increases, so too does the probability of observing spatially overlapping crystals. This effect is analogous to paralysis in photon counting, in which overlap in the temporal transients ultimately sets the upper limit on the accessible dynamic range. Unlike photon counting, in which transients of closely separated photons in time are counted as a single event, particle counting can be biased instead from overlap in space between adjacent particles. Unfortunately, the bias introduces a measurement gap, in which neither technique (particle counting or signal integration) yields an optimal, unbiased signal-to-noise (S/N) ratio.

In this work, the two methods (signal integration and particle counting) are bridged through the derivation of a statistical model based on analytical expressions for the optimization of signal to noise. Specifically, a particle-counting algorithm capable of unbiased extension of the counting regime to overlap with the regime optimized for signal integration was developed. Based on the relative variance of the recovered volume, an analytical expression was derived for the percent crystallinity for which both bias-corrected particle counting and signal integration produce an identical average S/N ratio. Crystalline fractions higher than this point are optimally determined by signal integration and fractions below by bias-corrected particle counting. The average SHG-activity per unit volume measured in the particle-counting regime served as an internal calibration standard for quantifying crystallinity in the signal integration regime, removing the need for external standards. Monte Carlo simulations were performed to assess the merits of the approach. Measurements were performed on evacetrapib, a cholesteryl ester transfer protein inhibitor, formulated tablet³³ to experimentally demonstrate the bridging of the two regimes and autocalibration.

Theoretical Framework

In the particle-counting regime, in which particles are well-separated and non-overlapping, volume determination from image analysis provides a S/N advantage relative to integration of the SHG intensity. This advantage in particle counting comes from the suppression of dark counts and the neglect of variance in the SHG intensity due to particle orientation and size. However, at higher percentage crystallinity, particle-counting approaches begin to introduce bias from “pulse-pile-up,” in which overlap between adjacent particles within the depth of field results in counting as a single particle rather than as a pair.^34,35 Analogous to single-channel measurements with pulse pileup, methods to correct for this bias can extend the dynamic range of particle counting, but at the expense of signal to noise.³⁵ Segmentation algorithms can separate overlapping particles that are relatively large (i.e., span many pixels), but are not reliable for particles only a few pixels in dimension as studied herein.³⁶ The primary objective of the theoretical framework outlined below is to extend the dynamic range of particle counting by correcting for bias from particle–particle overlap, and identifying the point at which integration of the SHG intensity offers a S/N advantage over bias-corrected particle counting. Furthermore, the SHG intensities measured in the particle-counting regime can directly serve to calibrate the intensity-based volume assessment without the need for an external reference sample.

In the absence of overlap between spherical particles and in the limit of a large depth of field, the volume of an individual particle can be determined from the cross-sectional area, A, by $V_{i} = \frac{4}{3 \sqrt{π}} A_{i}^{3 / 2}$ . The total volume is given by the summation of the volumes of all n particles present in a given FOV. The mean of the total volume, $μ_{V_{Tot}}$ , can be estimated by the mean particle area, μ _Α , and the mean number of particles, μ _n .

μ_{V_{tot}} = μ_{n} μ_{V} ≅ μ_{n} \frac{4}{3 \sqrt{π}} μ_{A}^{\frac{3}{2}}

(1)

The approximation in Eq. 1 neglects computational bias arising from the nonlinear relationship between the variances in particle size and the recovered volume, but should be reasonably reliable for narrow distributions in particle sizes. The relative variance in the total volume in the limit of low volumes can be related back to the Poisson distributed number of particles and the distribution in particle sizes. Summing the contributions from both the variance in the particle radius and the Poisson number of particles yields the following expression for the relative variance, $σ_{V_{meas}}^{2}$ , in the measured volume in the limit of no overlapping particles, normalized by the mean measured volume, $μ_{V_{meas}}$ .

(\frac{σ_{V_{meas}}}{μ_{V_{meas}}})^{2} = \frac{1}{μ_{n}} [1 + 9 (\frac{σ_{r}}{μ_{r}})^{2}]

(2)

As the density of particles increases, the probability of particle–particle overlap becomes non-negligible and the mean ground truth area μ _Α will no longer equal the mean measured area by particle counting, $μ_{A_{meas}}$ . Considering only the probability of two-particle overlap, the measured volume $V_{meas} = \sum_{i} \frac{4}{3 \sqrt{π}} A_{i}^{\frac{3}{2}}$ overestimates the ground truth total volume V_Tot. This bias in the volume can be removed by incorporating a correction term C, together with knowledge of the difference between the observed number of particles, N, in the presence of some overlap from naïve particle counting and the underlying ground truth number of particles, n. (see the Supplemental Material for the derivation of Eq. 3 and the analytical form for the constant C ).

μ_{V_{recov}} = \frac{4}{3 \sqrt{π}} μ_{A_{meas}}^{\frac{3}{2}} (2 μ_{N} - μ_{n}) (1 - 2 C)

(3)

Recovery of a reasonable estimate for the mean number of underlying particles μ _n is more challenging. As described in the Supplemental Material, an approximate value for μ _n can be calculated from foreknowledge of the average particle area μ _A (e.g., measured at low concentration) and the FOV through the dimensionless parameter $A_{O} = (μ_{A} / FOV)$ .

μ_{n} ≅ \frac{1 - \sqrt{1 - 8 {\bar{A}}_{O}} μ_{N}}{4 {\bar{A}}_{O}}

(4)

From Eq. 4, it should be clear that the estimation of μ _n is only applicable in the regime in which ${\bar{A}}_{O} μ_{N} < 1 / 8$ . Allowing for a reasonable buffer, the particle-counting analysis algorithm presented herein should be applicable provided that ${\bar{A}}_{O} μ_{N} < 0.1$ . Experimentally, ${\bar{A}}_{O} μ_{N} ≅ 0.1$ corresponds to approximately 10% of the pixels within the FOV occupied by particles. In this regime, the probability of three or more particle overlap events becomes significant and are not accounted for in the two-particle corrections considered herein. Computationally, the factor of $\sqrt{1 - 8 {\bar{A}}_{O} μ_{N}}$ cannot be evaluated for ${\bar{A}}_{O} μ_{N} > 1 / 8$ , for which the square root becomes negative. Fortunately, integration of the SHG intensity provides an alternative approach for quantification of crystallinity in the regime ${\bar{A}}_{O} μ_{N} > 0.1$ , enabling further extension of the dynamic range by bridging the counting and averaging regimes.

The relative variance in the recovered volume when incorporating the two-particle correction is given by the following expression, the derivation of which is provided in the Supplemental Material.

(\frac{σ_{V_{recov}}}{μ_{V_{recov}}}) ≅ [\frac{μ_{n} - 2 μ_{p} + 4 C^{2} μ_{p}}{(μ_{n} - 2 μ_{p} + 2 C μ_{p})^{2}}] [1 + 9 (\frac{σ_{r}}{μ_{r}})^{2}]

(5)

In Eq. 5, μ _p is the mean number of dimer particles, in which two actual particles are treated as a single particle by naïve shape-independent particle-counting algorithms, and is given by μ _p = μ _n – μ _N . As expected, the relative variance from Eq. 5 simplifies to that in Eq. 2 in the limit of μ_p approaching zero, corresponding to negligible particle–particle overlap.

For practical purposes of quantification, the recovered volume is converted into the volume fraction (i.e., the crystalline fraction relative to the total interrogated volume). Conversion from the volume fraction to the more commonly used mass fraction requires knowledge of the packing density of the sample, which can be independently determined.

VolumeFraction = V_{recov} \sqrt{\frac{π}{\bar{A} \times {FOV}^{3}}}

(6)

The volumes recovered from particle counting can be integrated with the corresponding intensity information for autocalibration of intensity-based volume assessments. In brief, the average SHG intensity per unit volume can be estimated by integrating the SHG intensity of all the counted particles and dividing by the recovered volume of crystalline material. Once this calibration has been performed using images that are amenable to particle-counting analysis, the integrated intensity can be converted to a crystalline volume in regions where particle counting is inapplicable. Equation 6 rescales the recovered crystalline volume from the SHG intensity measurements to be in a volume fraction of the total FOV. In this manner, the total crystallinity can be determined without the need for preparation of calibration standards of known crystallinity.

Experimental and Computational Methods

Monte Carlo simulations were performed to assess the accuracy of the equations outlined in the theory section. A script written in Matlab, version R2014a, modeled the amorphous solid dispersion by generating a set number of particles with random positions in a 512 × 512 pixel image. Particles were simulated as spheres to the best approximation with pixels. Particle radii were randomly selected from a lognormal distribution, using μ = 0.693 and σ = 0.336 in units of pixels. These numbers were selected to produce images similar to the particle size distribution produced in the experimental data. The intensity of each particle was dictated by two factors: (1) an orientational term accounting for the difference in SHG intensity for different crystal orientations; and (2) a term related to crystal size. For the orientational term, the intensity was rescaled by a random number from an exponential distribution, consistent with previous observations of the modeled and measured distribution in intensities for BaTiO₃ nanoparticles.²³ Building on previous SHG microscopy analyses,²³ the size dependence of the intensity scales differently depending on the size of the particle relative to the depth of field and beam waist. In Regime I, corresponding to particles smaller than the beam waist, the SHG intensity scales with the squared volume of the particle. In Regime II, in which particles are larger than the beam waist but smaller than the depth of field, the individual pixels were scaled quadratically with particle radius. For particles larger than the depth of field, consistent with Regime III, the SHG intensity is independent of particle size and was not rescaled.

The average single particle area relative to the FOV, ${\bar{A}}_{0}$ , is a key factor in the analysis. For the simulations, the value of ${\bar{A}}_{0}$ was determined to be $5.9 \times 10^{- 5}$ (pix²/pix²). To determine ${\bar{A}}_{0}$ for the experimental data, particle areas were averaged from images that showed very little particle overlap namely, 13 FOVs for the blank sample, 13 FOVs for the 0.05% crystallinity by mass sample, and three FOVs for the 0.10% crystallinity by mass sample. See the Materials section below for details on the preparation of these samples. ${\bar{A}}_{0}$ for the experimental data was determined to be $4.2 \times 10^{- 5}$ .

Materials

Evacetrapib was obtained from Eli Lilly and Company. Evacetrapib tablets (unit formula shown in Table I) prepared identically to the final dosage form. The actual tablets, which contain 50% drug in the sodium salt form, were used to make the standards. No crystal forms are known for the sodium salt. The free-form (protonated) of the crystalline drug is then spiked into the ground tablet powder by light mixing. Tablets were confirmed to contain amorphous sodium salt after grinding and before mixing using PXRD and Fourier transform infrared (FT-IR) analysis. Spiked standards of crystalline drug substance in the drug product were prepared by adding known amounts of crystalline drug substance to the ground evacetrapib tablets powder followed by additional mixing in the mortar and pestle. The crystalline evacetrapib used for spiking was roughly 98–100% crystalline. With the exception of the API, none of the intragranular or extragranular excipients produces substantial SHG. Although microcrystalline cellulose produces some weak background SHG from locally ordered domains,³⁷ all the bulk crystalline materials adopt centrosymmetric and SHG-inactive space groups.

Table I.

Unit formula for Evacetrapib tablets.

	Amount (mg/tablet)	Function
Crystalline evacetrapib	0	API
Intragranular ingredients
Spray dried dispersion	260.0	Active dispersion
Evacetrapib free acid	–	Crystalline active
Hypromellose	–	Dispersion polymer
Croscarmellose sodium	13.00	Disintegrant
Microcrystalline cellulose	97.50	Filler
Sodium chloride	84.50	Disintegrant
Sodium bicarbonate	84.50	Disintegrant
Colloidal silicon dioxide	2.600	Glidant
Magnesium stearate	0.8125	Lubricant
Extragranular ingredients
Croscarmellose sodium	13.00	Disintegrant
Microcrystalline cellulose	93.275	Filler
Magnesium stearate	0.8125	Lubricant
Tablet core weight	650.00
Film coating ingredients
Color mixture	26.00	Film coat
Coated tablet weight	676.00

Infrared Spectroscopy

Infrared spectra of the powders (∼20 mg) were obtained in absorbance mode using a Nicolet 6700 FT-IR (ThermoFisher Scientific, USA) equipped with a globar IR beam source, KBr beam splitter, DTGS detector, and a Smart OMNI sampler attenuated total reflectance (ATR) module with a germanium crystal and low-pressure sample tower. The scan range was set at 500–4000 cm⁻¹ with 2 cm⁻¹ resolution, and 64 scans were co-added. All spectra were acquired and altered using OMNIC 8.3 software package. An automatic baseline correction and ATR correction were applied to the spectra when plotted and all spectra were normalized to the peak at approximately 1560–1570 cm⁻¹.

Powder X-ray Diffraction

The powder samples were packed into a 25 mm diameter cavity steel containment cell ensuring a flat level powder surface was obtained. The PXRD data were obtained using CuKα radiation with a Bruker D4 Endeavor powder diffractometer (Bruker AXS, USA) operating at 40 kV and 40 mA. The X-ray measurements were conducted in Bragg–Brentano geometry using a scan range of 4–40° 2θ and 6–14° 2θ at scan time per step (step size 0.028°) of 0.5 and 10 s, respectively. The diffraction data were baseline corrected utilizing the Bruker software.

Second Harmonic Generation

Second harmonic generation images were acquired using a modified SONICC (second order nonlinear imaging of chiral crystals) instrument from Formulatrix (Formulatrix, USA), following custom modifications performed in-house to enable epi-detection of powders. In brief, a polarizing dichroic mirror was added to the beam path of the Formulatrix system in the backward-collected direction, which was focused onto the entrance of a multimodal liquid light guide (Newport, USA) terminating in an additional narrow bandpass 530 nm dichroic mirror (Semrock, USA) designed to separate the SHG from two-photon excited ultraviolet fluorescence (TPE-UVF). Use of a polarizing dichroic mirror retained direct compatibility with two photon excited ultraviolet fluorescence (TPE-UVF), which requires 532 nm incident light to pass through the same dichroic mirror used to reflect the epi-detected SHG of the same wavelength. The 1064 nm fundamental beam was generated by a Fianium (Fianium, UK) fiber laser with a pulse repetition rate of 50 MHz, and pulse width of ∼150 fs. An image was created by scanning the incident beam across the sample to generate a 512 × 512 pixel image. The laser was focused onto the sample using a 10× objective with a depth of field of 200 µm. For each sample investigated, approximately 10 mg of powder was placed in a DSC bottom pan set into wells of a 96-well plate. The particle-counting analysis was performed using the Analyze Particles built-in algorithm in ImageJ³⁸ to generate lists of particle numbers, areas, and average intensities. Reduction of background was accomplished by applying size and intensity thresholds for counting. The size threshold used excluded anything less than two pixels and the intensity threshold used excluded any pixel with intensity less than three counts.

Results

Simulations and Statistical Analysis

Analysis of the simulation data shown in Fig. 1 demonstrated quantitative agreement between the ground truth number of particles (black line) present in an image and the number recovered by the statistical model (blue diamonds). This high degree of correlation was maintained while spanning the regimes of both low (<1%) and high crystallinity (>1%), for which representative images are provided within the figure. The expression in Eq. 4 significantly extends the range over which the true underlying mean number of particles can be recovered before introducing bias. For reference, particle counting performed without bias correction indicated clear deviation between the naively measured versus bias-corrected crystallinity, introducing significant systematic errors for ∼1000 particles (corresponding to ∼6% of the area occupied by particles with ${\bar{A}}_{0} = 5.9 \times 10^{- 5}$ ).

The most compelling outcome is the reliability in the volume recovery. Both the volumes measured before bias correction (red Xs) and the volumes recovered (blue diamonds) are plotted alongside the ground truth particle volumes in Fig. 2. Notably, the volumes calculated using naïve analysis techniques resulted in significant bias in the crystallinity that was largely removed by the correction described in the “Theory” section. For particle densities higher than ∼2100 particles per image, the probability of ${\bar{A}}_{O} μ_{N} > 0.1$ becomes significant, such that Eq. 4 is no longer applicable. In this case, the probability of three or more particles overlapping is too great for the corrected counting algorithm to reliably operate. The bias introduced by the bias-corrected counting algorithm is small throughout this entire operational regime. However, the increase in relative variance at high crystalline fractions is indicative of the anticipated reduction in the S/N ratio associated with larger correction factors.

A theoretical comparison of the predicted S/N ratio from the analytical modeling for the counting and integrating approaches is provided in Fig. 3. In the hypothetical scenario of particles of identical size and brightness, the S/N ratio is dictated solely by the Poisson statistics describing the number of particles in a given FOV, in which the variance is equal to the mean. All results in Fig. 3 are rescaled by this theoretical limiting result in order to better highlight the differences. The upper limit (dash–dot) corresponds to the S/N ratio produced considering only the Poisson distributed number of particles and the inherent variance in their cross-sectional areas. As the number of particles increases, rescaling by the bias correction extends the dynamic range, but reduces the S/N ratio achievable by particle counting. By comparison, the relative S/N ratio from integration in both Regimes II and III is independent of the mean number of particles. In Regime III, the reduction in S/N ratio arises from the additional variability in intensity due to crystal orientation that does not directly impact particle-counting approaches. In Regime II, additional variance in intensity from differences in particle thickness further reduces the S/N ratio. In the present case, the microscope has an effective FOV of 1.4 × 1.4 mm and the particles probed had an average diameter of 10.4 µm. In the commonly encountered Regime II, the S/N ratio of particle counting is increased by > 3.5-fold compared to intensity integration in the low crystallinity limit, corresponding to a > 10-fold reduction in measurement time for a comparable S/N ratio. Over the entire accessible range of volumes considered in the simulations in all particle size regimes, particle counting resulted in higher S/N ratio for the recovered volume relative to intensity integration, although the advantage is reduced as the particle density increases.

In addition to the improvement in the S/N in the low crystallinity regime, the particle-counting approach described herein has the distinct advantage of minimizing potential complications associated with bias from crystal polymorphism. All crystals of homochiral molecules by necessity must adopt non-centrosymmetric space groups. With the rare exception of octahedral symmetry, all other space groups of homochiral molecules are symmetry-allowed for SHG. However, the relative brightness of the SHG activity can vary significantly depending on the crystal form. If crystals initially adopt one crystal form, then convert to a different form (e.g., as per the Ostwald rule of stages), measurements based on the SHG activity would result in a bias from the change in crystal form. In contrast, the particle counting approach described herein is based exclusively on the cross-sectional area irrespective of the integrated SHG intensity, removing bias from variance in crystal form.

Experimental Implementation

Physical mixtures of crystalline evacetrapib drug substance in an amorphous solid dispersion (ASD) formulated tablet were made to assess the merits of the proposed auto-calibration algorithms, the SHG images of which are shown in Fig. 4. Thirteen FOVs were collected for the lowest two concentrations of the physical mixture (the 0% and 0.05%) and three FOVs were collected for the rest of the concentrations (0.1%, 0.25%, 0.5%, 1%, 2.5%, 5%, and 10% crystallinity by weight). The higher number of measurements performed at lower concentrations was chosen to reduce the Poisson uncertainties from the finite number of particles present at those concentrations. The corrections for the particle count, the total volume, and the volume fraction were performed using the particle-counting correction algorithm. The limits of detection were calculated for both the integrated intensity and the bias-corrected, particle-counting method, they were found to be 0.4% and 0.17% (1700 ppm), respectively.

In the present study, the size distribution of crystals was consistent with Regime II based on measurements of both the beam waist and depth of field. In brief, the beam waist was smaller than a single pixel, such that all particles counted spanning ≥ 3 pixels were large relative to the beam waist. The depth of field was measured to be 200 µm, which was significantly greater than the upper limit of the size distribution observed by particle counting. This regime is the most likely to be encountered in analyses of pharmaceutical materials; particles much smaller than the focal volume produce relatively weak SHG and appear as puncta in images, while particles larger than the depth of field are of a size compatible with numerous alternative conventional analyses. In Regime II, the effective interaction length is dictated by the crystal size, such that changes in the crystal size distribution can potentially influence the autocalibration. Fortunately, knowledge of the crystal size distribution from microscopy measurements provides a route to analytical model and correct for such effects.²³

The limits of detection by SHG were significantly higher than those observed in previous studies of model systems, dictated largely by weak but non-zero background contributions from the excipients. The most likely source of the background SHG is the microcrystalline cellulose, from which weak SHG activity has been reported previously.²⁸ Native cellulose as generated biologically has been shown to be SHG-active.³⁹ While the cellulosic materials commonly used as excipients in pharmaceutical formulations generally undergo substantial treatments before use, residual weak SHG from a sparse population of locally ordered domains may potentially contribute to the observed background. Analysis of SHG active areas by complementary methods such as Raman spectroscopy or XRD, previously successfully demonstrated by Schmitt et al.²² and Newman et al.,²⁹ respectively, may provide routes for more definitively determining the chemical composition of the SHG-active domains arising within the excipients.

Despite the advantages of both noise reduction and the absence of independent calibration standards, several application spaces are not yet well suited for the particle-counting autocalibration approach described herein because of the assumptions made in the model. One assumption of the model is that the particle size distribution does not change substantially over time, with increase in crystallinity arising primarily from interconversion on a per particle basis. In cases in which the sizes of individual crystals evolve during the analysis, (e.g., through molecular diffusion, crystal growth, and/or Ostwald ripening), additional parameters related to the anticipated evolution of the size distribution could be integrated into the approach but are not described herein. Furthermore, the model is currently limited exclusively to spheroidal particles. In practice, many APIs crystalize in needle-like high aspect ratio crystal, for which the relationship between cross-sectional area and volume is not as concise as described herein. Extension of the model to crystals with a greater diversity of morphologies is still in progress.

It should be noted that the method described recovers the volume fraction by image analysis, but not the mass fraction. Independent knowledge of the void volume in the probed powder would enable conversion of the volume fraction to the mass fraction. In the present case, the correction for void volume was performed using the known mass fraction as a calibrant. In practice, gravimetric or volumetric methods could be used to determine the void volume and recover the mass fraction.

The observed ∼2.5-fold improvement in the limit of detection is in excellent agreement with the 3.5-fold enhancement predicted by the statistical modeling. The particle sizes encountered in this study were consistent with Regime II, in which the particles were large relative to the beam diameter, but small relative to the long depth of field of the 10× objective. The agreement between theory and experiment supports the reliability of both the statistical modeling and the simulations.

In addition to removing bias and increasing S/N ratio in the low crystallinity limit, the particle-counting algorithm has the distinct advantage of enabling autocalibration. In the low crystallinity regime, the volume fraction is determined independently of the SHG activity. As such, the SHG activity per unit volume determined in the low crystallinity limit can serve to calibrate extension to higher crystallinity beyond the point at which particle counting remains reliable. Results from the counting algorithm allowed for the determination of both ${\bar{A}}_{0}$ and the SHG per unit volume, which were in turn used for autocalibration to extend the dynamic range beyond the particle-counting regime. For the experimental data of this study ${\bar{A}}_{0}$ was determined to equal $4.2 \times 10^{- 5}$ and the SHG intensity per unit volume was found to equal 3 counts per cubic pixel. In the high crystallinity limit, the total SHG intensity is divided by the determined SHG per unit volume to give the equivalent volume, and that volume is rescaled to the volume fraction using Eq. 6. In Fig. 5 the high crystallinity points in red (5% and 10%) agree quite well with the extrapolated calibration curve determined by the low crystallinity blue points. Two important caveats are noteworthy: (1) uncertainties in the slope will generally compound upon extrapolation; and (2) the average SHG/V calculated in this manner is ensemble-averaged over the entire FOV and may not be representative of all positions within the image for non-uniform illumination (e.g., in the presence of vignetting).

The same samples were analyzed by a suite of complementary methods for comparison with the SHG microscopy approach. The data from FT-IR absorption spectroscopy are shown in Fig. 6. The peak at 1736.6 cm⁻¹ (highlighted with a black box in Fig. 6) was chosen for quantification of the crystalline fraction. A calibration curve gives the limit of detection at 0.8% using the peak height. The same samples were also analyzed using PXRD, the baseline corrected data is shown in Fig. 7. The strongest peak at 12.05° was used to make a calibration curve based on peak height. The inset in Fig. 7 shows the calibration curve that was used to determine the limit of detection. A comparable limit of detection of 0.9% was obtained using the peak height for quantification of crystallinity. In all cases considered, the proposed particle-counting approach provides a clear advantage for lowering the limits of detection.

Conclusion

A statistical model for removing bias due to particle overlap and for improving the S/N ratio of particle counting in the regime of low crystallinity was presented and validated both theoretically and experimentally. The analytical expressions derived herein provide a correction term for volume determinations from particle counting, in addition to allowing for the calculation of total crystallinity based on SHG intensities from the particle-counting regime without needing a known calibration sample. It is further possible now to define a transition point based on the relative variance in volume such that the crystalline volume is optimally evaluated by particle counting below this point and by the integrated SHG intensity above it. Data calculated by applying this model to simulations of crystalline samples were shown to reliably recover the true particle count and crystallinity volume, notably extending the dynamic range of particle-counting methods by approximately an order of magnitude. The model was further validated using experimental data and was shown to successfully recover the particle count, total volume, and the volume fraction from SHG images of an amorphous solid dispersion of evacetrapib. The approach described here further offers the advantage of autocalibration by enabling the use of the SHG intensity per unit volume measured by particle counting in the low crystallinity regime for the evaluation of the volume in the high crystallinity regime. This statistical model significantly improves the accessibility of quantitative crystallinity information from NLO imaging, laying the foundation for broader adoption in pharmaceutical analyses.

Supplemental Material

Supplemental material1 - Supplemental material for Calibration-Free Second Harmonic Generation (SHG) Image Analysis for Quantification of Trace Crystallinity Within Final Dosage Forms of Amorphous Solid Dispersions

Supplemental material, Supplemental material1 for Calibration-Free Second Harmonic Generation (SHG) Image Analysis for Quantification of Trace Crystallinity Within Final Dosage Forms of Amorphous Solid Dispersions by Casey J. Smith, Janny Dinh, Paul D. Schmitt, Paul A. Stroud, Jeremy Hinds, Michael J. Johnson and Garth J. Simpson in Applied Spectroscopy

Supplemental Material

Supplemental material2 - Supplemental material for Calibration-Free Second Harmonic Generation (SHG) Image Analysis for Quantification of Trace Crystallinity Within Final Dosage Forms of Amorphous Solid Dispersions

Supplemental material, Supplemental material2 for Calibration-Free Second Harmonic Generation (SHG) Image Analysis for Quantification of Trace Crystallinity Within Final Dosage Forms of Amorphous Solid Dispersions by Casey J. Smith, Janny Dinh, Paul D. Schmitt, Paul A. Stroud, Jeremy Hinds, Michael J. Johnson and Garth J. Simpson in Applied Spectroscopy

Footnotes

Conflict of Interest

The authors report there are no conflicts of interest.

Funding

The authors gratefully acknowledge funding through the Lilly Research Awards Program (grant no. 107309) and from the NSF Chemical Measurement and Imaging (CMI) Grant Opportunity for Academic Liaison with Industry (GOALI) award (grant no. 209682). They also collectively acknowledge Formulatrix for extended loan of a SONICC instrument and support of its modification for powders analysis.

Supplemental Material

All supplemental material mentioned in the text is available in the online version of the journal.

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Supplementary Material

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