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Metrological approaches to organic chemical purity: primary reference materials for vitamin D metabolites

Michael A. Nelson*1, Mary Bedner

1, Brian E. Lang

2, Blaza Toman

3, Katrice A. Lippa

1

National Institute of Standards and Technology

Material Measurement Laboratory, Chemical Sciences Division1 and Biosystems and Biomaterials Division

2

Information Technology Laboratory, Statistical Engineering Division3

Gaithersburg, MD 20899-8392

* Corresponding author contact information: email michael.nelson@nist.gov;

telephone 301-975-4100; FAX 301-975-0685

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Abstract

Given the critical role of pure, organic compound primary reference standards used to characterize and certify

chemical Certified Reference Materials (CRMs), it is essential that associated mass purity assessments be fit-for-

purpose, represented by an appropriate uncertainty interval, and metrologically sound. The mass fraction purities (%

g/g) of 25-hydroxyvitamin D (25(OH)D) reference standards used to produce and certify values for clinical Vitamin

D metabolite CRMs were investigated by multiple orthogonal quantitative measurement techniques. Quantitative 1H-nuclear magnetic resonance spectroscopy (qNMR) was performed to establish traceability of these materials to

the International System of Units (SI) and to directly assess the principal analyte species. The 25(OH)D standards

contained volatile and water impurities, as well as structurally-related impurities that are difficult to observe by

chromatographic methods or to distinguish from the principal 25(OH)D species by one-dimensional NMR. These

impurities have the potential to introduce significant biases to purity investigations in which a limited number of

measurands are quantified. Combining complementary information from multiple analytical methods, using both

direct and indirect measurement techniques, enabled mitigation of these biases. Purities of 25(OH)D reference

standards and associated uncertainties were determined using frequentist and Bayesian statistical models to combine

data acquired via qNMR, liquid chromatography with UV absorbance and atmospheric pressure-chemical ionization

mass spectrometric detection (LC-UV, LC-ACPI-MS), thermogravimetric analysis (TGA), and Karl Fischer (KF)

titration.

Key Words: purity, qNMR, data combination, Vitamin D, primary standard, metrology

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Introduction

Adequate vitamin D levels are essential to proper bodily development, support several vital regulatory systems, and

serve as critical markers for multiple aspects of human health in the field of clinical medicine [1]. There may be

widespread vitamin D deficiency in the general population and the development of reference materials that support

clinical assays of vitamin D and related diagnostics is highly valuable. In particular, total levels of 25-

hydroxyvitamin D (25(OH)D) in serum are the most common indicators for assessing vitamin D status [2]. With

increased testing of these species in clinical laboratories during the past decade, standardization of 25(OH)D

measurement methods has become a prominent focus within the clinical community. Necessitated by the high-level

impact of reference materials supporting these standardization efforts, a great deal of confidence must be placed in

the accuracy of certified value assignments and the metrological soundness of the measurement methods used to

determine them. Accurate purity assessments of gravimetrically-implemented reference standards are crucial to

corresponding measurement calibrations and proper value assignments for vitamin D metabolite Certified Reference

Materials (CRM). The analytical rigor of these purity investigations is also vital to establishing the metrological

traceability of these values to the International System of Units (SI).

Mass purity characterization is critical to evaluating the suitability of chemical materials for use as primary

reference standards. Techniques which have supported this characterization include high performance liquid

chromatography with UV absorbance-detection (HPLC-UV) and gas chromatography with flame ionization

detection (GC-FID) for quantification of organic impurities, Karl Fischer titration for determination of water mass

fraction, thermogravimetric analysis (TGA) for mass assessment of volatile constituents, and differential scanning

calorimetry (DSC), when applicable, as a primary method for determination of molar purity. The combined estimate

of relative impurities by these techniques (∑𝑋% 𝐼𝑚𝑝𝑢𝑟𝑖𝑡𝑦) support a mass balance approach to purity determination

(100 % − ∑𝑋% 𝐼𝑚𝑝𝑢𝑟𝑖𝑡𝑦) [3, 4, 5, 6] that is commonly employed by producers of chemical standards and analytical

laboratories. This method assumes that all impurities have been detected and quantified and that results of the

individual techniques have been appropriately combined. However, situations occasionally arise where the

complexity of a material or chemical system prevents comprehensive identification and quantification of all

impurities. In such cases, the mass-balance method does not solely provide an accurate assessment of purity.

Often for chromatographic purity (Purityc) determinations, the relative amount of the principal component species is

assumed to be equal to the ratio of the corresponding chromatograph peak area (Aa) with respect to the aggregate

area of all quantified or estimated peaks (Ac).

𝑃𝑢𝑟𝑖𝑡𝑦𝐶 =𝐴𝑎

𝐴𝑐 (1)

Though widely available and easily implemented for mass balance purity investigations, this approach uses the

assumption (often erroneously) that all detected species have congruent g/g detection response factors.

Although crucial, TGA and Karl Fischer measurements only serve as methods to quantify a limited range of volatile

mass impurities and water. DSC cannot be implemented for some materials owing to the structural instability of

principal component species at or near the melting temperature, unsuitable melt properties, or insufficiently high

molar purities for accurate assessment (< 98 %) [7].

Quantitative 1H-NMR (qNMR) using an internal standard has been implemented to provide precise purity

assessments that do not rely on compound-specific detector calibration and provide SI traceability [8, 9]. Validated

qNMR procedures [10, 11, 12, 13] have become increasingly valued as methods for directly determining mass

purities of organic species in high-purity and composite materials. This approach provides a degree of confidence

not always afforded by more commonly-employed chromatographic techniques, largely due to a highly linear

proportionality of 1H-NMR signal intensity to amount of

1H (nearly 100 % natural H abundance) [9, 14]. These

assessments are based on accurate gravimetric sample preparation and relative signal intensity ratios of analyte 1H

moieties to those of a well-characterized internal standard material with a known, SI-traceable mass purity. NMR

spectroscopy is particularly advantageous given that DSC or other primary methods are not always viable options,

and that 1H NMR spectra provide confirmatory structural identity information.

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Often, qNMR purity investigations yield lower values than a mass balance approach, indicating that this direct

method may be favorable when the mass balance is not comprehensive or otherwise underestimates total impurity

mass. This is again attributable to a direct linearity of signal intensity and chemical selectivity for both analyte and

internal standard, and the principle that qNMR with internal standard is ideally performed irrespective of impurities.

The integrity, certified purity, and compatibility of the internal standards used for these assessments are keys to the

accuracy and traceability of qNMR purity value assignments.

Recently, qNMR has been implemented in conjunction with traditional mass balance approaches to purity

assessments as critical components of timely, cost-effective production efforts for neat organic reference materials4.

Supported by information from mass balance analytical methods, direct measurements via qNMR provide confident

evaluations of the principal component species mass fraction. This combined methodology is considered fit-for-

purpose and is commonly implemented for value assignments of organic calibrant materials and solution CRMs.

The purities (g/g) of four primary vitamin D metabolite reference standards were determined during this

investigation: 25-hydroxyvitamin D2 (25(OH)D2), two units of 25-hydroxyvitamin D3 (25(OH)D3) monohydrate, and

3-epi-25-hydroxyvitamin D3 (3-epi-25(OH)D3). These materials were used as reference standards for isotope

dilution (ID) - liquid chromatography with mass spectrometric detection (LC-MS) and HPLC-UV calibrations used

for the certification of SRM 2972a 25-Hydroxyvitamin D Calibration Solutions. Note: "Standard Reference

Material" and "SRM" are trademarks for CRMs certified by NIST.

The techniques used for 25(OH)D mass purity determinations included qNMR with an internal standard as a

primary ratio method for direct measurement of the 25(OH)D species, as well as HPLC-UV, LC-MS, TGA, and

Karl Fischer titration techniques as components of either a confirmatory or informative mass balance approach.

Information from these complementary analytical techniques facilitated characterization of the biases of each

method. Given the significant influence of these purity assessments, it is similarly vital that accurate value

assignments and associated confidence intervals be determined with appropriate statistical methods that reflect the

analysts’ expertise and state of knowledge of the measurement system. To achieve this, novel, inference-based data

combination methods were developed to evaluate purities using data from both qNMR and mass balance

approaches. Primary standards are at the top of the metrological traceability chain and influence all measurements

that reference these materials as part of unbroken, documented chains of calibration. The accuracy of purity

assignments for supporting calibration standards must be fit for purpose and is a crucial component of certifications

that influence the trueness and metrological traceability of clinical assays.

Materials and Methods

Materials

Individual lots of 25-hydroxyvitamin D metabolite reference standards were obtained from multiple vendors: one lot

of 25-hydroxyvitamin D2, two lots of 25-hydroxyvitamin D3 monohydrate (acquired 2007 and 2013), and one lot of

3-epi-25-hydroxyvitamin D3. These materials were kept in desiccators stored in the dark within freezer units at -20

˚C and were allowed to equilibrate to room temperature under low yellow incandescent light prior to removal and

weighing of sample aliquots. Measurements of 25-hydroxyvitamin D are complicated by inherent light-sensitivity

and solution lability of these species, as evidenced by the evolution of pre-Vitamin D from 25(OH)D in alcoholic

solutions [15]. Handling and storage of 25(OH)D reference standards must be performed in controlled conditions

and they are suitable for total mass fraction purity measurements for only a few hours after dilution.

Quantitative NMR measurements were made in methanol-d4 (D, 99.95 % and 99.98%) obtained from Cambridge

Isotope Laboratories. Dimethyl terephthalate (DMT, 99.99 % ± 0.16 % [Uk=2]) from Sigma Aldrich’s set of Fluka

TraceCERT® Certified Reference Materials for qNMR (refer to Disclaimer) served as an internal standard for NMR

analyses and was stored in a desiccator at room temperature.

LC Methods

Chromatographic analyses used multiple column types and multiple UV wavelengths for detection to evaluate

potential organic impurity components. Solutions of 25(OH)D were gravimetrically prepared in methanol or ethanol

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for LC analysis using an analytical balance. An Agilent Technologies (Palo Alto, CA, USA) 1100 series liquid

chromatography system with an atmospheric pressure chemical ionization (APCI) source, SL series mass

spectrometric detector (MSD), and an ultraviolet absorbance detector was used to acquire LC purity measurement

data. UV absorbance detection was evaluated at both 220 nm and 265 nm.

LC Method 1: A 150 mm x 4.6 mm ID Supelco (Bellefonte, PA, USA) Ascentis Express F5 column (SN

USBK001370), containing 2.7 μm particles, was used to resolve diastereomers 25(OH)D3 and 3-epi-25(OH)D3.

Analyses were performed at 15 °C after injection of 5 µL of sample solution. The mobile phase composed of LC-

MS grade water and methanol adhered to the following water/methanol gradient conditions: 13%/87% at injection

time to 30 min; 22%/78% at 30 min to 50 min; 13%/87% at 50 min to 56 min.

LC Method 2: A 250 mm x 4.6 mm ID Agilent Technologies (Palo Alto, CA, USA) Zorbax SB-CN (SN

USSF019040 or USSF023201) column with 5 μm particles was used. Analyses were performed at 45 °C after

injection of 4 μL of concentrated sample solution. The mobile phase composed of LC-MS grade water and methanol

adhered to the following water/methanol gradient conditions: 34%/66% at injection time to 50 min; 20%/80% at 50

min to 56 min.

Similar separation methods were used in conjunction with APCI/MS detection and selected ion monitoring (SIM) to

determine purities of the 25(OH) D2 (F5 and CN columns) and Lot 2 of 25(OH)D3 (F5 column). Ions of m/z 395

were selectively analyzed for 25(OH)D2 and those of m/z 383 for 25(OH)D3. For APCI-MS detection the optimized

parameters included 5.0 L/min drying gas flow at a temperature of 350 ˚C, 50 psi nebulizer pressure, 350 °C

vaporizer temperature, +3600 V capillary voltage, 4 μA corona current, and 150 V fragmentor potential. Multiple

aliquots of 25(OH)D standard solutions were analyzed via LC. Purity from each analysis was determined as the peak

area of the primary 25(OH)D analyte relative to the total area of all detected species’ peaks, with the assumption that

all species have similar absorbance response factors at a given wavelength. Multiple injections were performed for

subsamples of each 25(OH)D solution.

TGA

Thermogravimetric analyses were performed using an Instrument Specialists Incorporated (Twin Lakes, WI, USA)

TGAi 1000 that was calibrated with a 10 mg (nominal) weight that had a determined mass measured via a balance

with SI-traceable calibration. Subsamples of 25(OH)D reference standard units were weighed into a TGA pan using

an analytical microbalance. The TGA oven was purged with a 20 mL/min flow of dry nitrogen gas. After allowing

for an oven temperature equilibration at 25 °C, the temperature was ramped to 400 °C at a rate of 20 °C/min for each

analysis. Prior to measuring each 25(OH)D sample, this program was implemented with an empty pan to establish a

thermograph baseline. This allowed for corrections to mass determinations associated with ample buoyancy

variability resulting from variations in oven and purge gas temperatures.

The volatiles and moisture component of the vitamin D metabolite reference standard was determined as the fraction

of initial mass that was lost during the analysis. The uncertainty of TGA results for each sample was calculated by a

Monte Carlo Method (MCM) using an Excel Monte Carlo algorithm [16]. A 100,000-iteration simulation of the

volatile/moisture component calculation was implemented using variable inputs randomly assigned from rectangular

distributions that were centered on balance- and TGA-measured values and had ranges of ± 0.0015 mg from these

median values. The means of these simulation results were in agreement with those experimentally-determined. The

expanded uncertainty (Uk=2) of the volatile/moisture component determinations were assessed as twice the standard

deviation of corresponding MCM results.

Karl Fischer Titration

One day prior to sample titration, methanol was added to the titration vessel to facilitate system stabilization. The

following day, baseline drift was determined by titration prior to sample introduction. Three calibration

measurements were performed using 80 mg (nominal) of SRM 2890 Water-saturated 1-octanol (WSO) as a

standard, injected using a gas-tight syringe pierced through a silica septum injection port of the Karl Fischer titration

vessel.

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Amounts of 25(OH)D subsamples, determined using a microbalance, were added to the Karl Fischer cell in a

platinum DSC crucible for analysis. Additionally, an empty crucible was titrated as a blank analysis to estimate the

amount of water introduced to the system after piercing of the titration vessel septum. Titrations were analyzed for a

fixed length of time (40 minutes). Instrumental drift was assessed after each run via the baseline consistency during

two successive 10 minute intervals, for which appropriate adjustments were made to the value of added Karl Fischer

reagent volume to the sample cell.

Water content via Karl Fischer titration was determined using two separate calculations. With the first, the Karl

Fischer (KF) titrant was calibrated to WSO using data from the WSO titrations. The calibration factor (titer) was

determined by dividing the mass fraction of water in the WSO, certified using the same instrumental unit and

assigned the value 48.3 mg/g ± 0.8 mg/g, by the ratio of the drift-corrected KF titrant volume relative to the total

added WSO mass. The second calculation consisted of dividing the volume of delivered titrant, adjusted for drift and

atmospheric water contamination, by the added sample mass and multiplying by the calibration factor to determine

the mass fraction of water in each 25(OH)D sample.

Karl Fischer titration expanded uncertainties (U95) reported in Table 1 for 25(OH)D3 were estimated using a

parametric MCM bootstrap simulation procedure based on a Gaussian random effects model [17, 18].

qNMR

For each unit of 25(OH)D material, three subsamples were prepared that contained < 2 mg amounts of vitamin D

metabolite reference standard and DMT internal standard. Masses were determined in custom-made glass weigh

boats using a microbalance and transferred to small glass weigh bottles. Prior to analysis, glass transfer pipettes were

used to deliver ≈ 0.7 mL aliquots of methanol-d4 to each bottle under light from a 60 W incandescent bulb. These

solutions were sonicated for approximately ten min and vortexed before transfer to Bruker 5-mm internal diameter,

7 inch 600 MHz NMR tubes capped with polytetrafluoroethylene (PTFE) caps (Bruker BioSpin Corp., Billerica,

MA). To minimize 25(OH)D degradation before NMR analysis, sample dissolutions were individually performed

just before loading into the spectrometer. Experimental NMR data was acquired by a Bruker Avance II 600 MHz

spectrometer equipped with a 5-mm broadband inverse (BBI) detection probe and operating with Topspin Version

3.2 software. All measurements were performed as one-dimensional (1D) 1H{

13C}experiments using a 90 degree

pulse sequence and inverse gated GARP composite pulse decoupling (decoupler offsets ranging from 60 ppm to 120

ppm) to remove 13

C satellite peaks from the spectra. A relaxation delay of 60 s was used between each scan for all

measurements to ensure adequate total sample T1 relaxation. Data for each sample were obtained with 64 scans

(preceded by 16 dummy scans). The free induction decay data acquisition time per scan was 5.45 s. All samples

were analyzed without spinning and at a temperature of 300 K.

Spectra were processed from 131,072 acquired data points using an exponential window function with a line-

broadening factor of 0.3 Hz and Fourier Transformation for conversion from the time to the frequency domain

(spectral resolution of 0.119 Hz/point). The processed spectral width was 12,019 Hz, centered at 3,706 Hz. Phase

correction was manually performed and regions containing quantifiable peaks were selectively and manually

baseline-corrected. For peaks of interest that were discernibly overlapped with small, interfering peaks, adjustments

were made at these regions to account for their extraneous contribution to the integrals of the principal component

proton signal.

Purities (g/g) determined by qNMR were calculated according to Equation 2.

𝑃𝑢𝑟𝑖𝑡𝑦𝑃𝐶 = (𝑁𝐼𝑆

𝑁𝑃𝐶) × (

𝑀𝐶𝑀

𝑀𝐼𝑆) × (

𝐴𝑟𝑒𝑎𝑃𝐶

𝐴𝑟𝑒𝑎𝐼𝑆) × (

𝑚𝐼𝑆

𝑚𝐶𝑀) × 𝑝𝑢𝑟𝑖𝑡𝑦𝐼𝑆 (2)

Where:

𝑁𝑃𝐶 = multiplicity (# H/peak) of principal component 25(OH)D analyte peak

𝑁𝐼𝑆 = multiplicity (# H/peak) of internal standard (DMT) peak

𝑀𝐶𝑀 = relative molar mass (molecular weight, g/mol) of the composite 25(OH)D reference standard

material

𝑀𝐼𝑆 = relative molar mass (molecular weight, g/mol) of the internal standard (DMT)

𝐴𝑟𝑒𝑎𝑃𝐶 = integrated area of 25(OH)D analyte peak

𝐴𝑟𝑒𝑎𝐼𝑆 = integrated area of the internal standard peaks

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𝑚𝐶𝑀 = mass (g) of the composite 25(OH)D reference standard material weighed for sample solution

𝑚𝐼𝑆 = mass (g) of the internal standard weighed for sample solution

𝑝𝑢𝑟𝑖𝑡𝑦𝐼𝑆 = purity (mg/g) of the internal standard

25(OH)D purity values determined via qNMR are traceable to the amount-of-substance of internal standard DMT

(Sigma Aldrich), which is in turn traceable to NIST SRM 350b Benzoic Acid (Acidimetric). The stated purity of

DMT (99.99 % ± 0.16 %) was verified during this investigation by qNMR using SRM 350b and dimethyl sulfone

from the Australian Government National Measurement Institute (NMIA Collection # QNMR002, Batch 06-Q-002)

as internal standards. Purities were determined as the mass percent of principal chemical component 25(OH)D2,

25(OH)D3, or 3-epi-25(OH)D3 in the composite material.

qNMR results were calculated for each sample replicate using three unique sets of integrals extracted from three

different processed regions of the 25(OH)D 1H{

13C} spectra. For each purity calculation, measurement uncertainties

were assessed through the propagation of respective variables’ absolute uncertainties via a MCM simulation (i

=100,000) of the model defined by Equation 2 [19]. This MCM analysis calculated 𝑃𝑢𝑟𝑖𝑡𝑦𝑃𝐶 for each replicate

using a bespoke Matlab (The MathWorks, Inc., Natick, MA) program, whereby inputs for each Equation 2 variable

were iteratively defined by randomly-generated, normally-distributed values that had a mean equal to the

experimentally-measured value, and a standard deviation equal to the respective uncertainty assignment. These

uncertainties included those of the principal component peak area and composite material mass (𝐴𝑟𝑒𝑎𝑃𝐶 ,𝑚𝐶𝑀), as

well as those of the internal standard and its respective purity (𝐴𝑟𝑒𝑎𝐼𝑆, 𝑚𝐼𝑆, 𝑃𝑢𝑟𝑖𝑡𝑦𝐼𝑆). The uncertainties associated

with 𝐴𝑟𝑒𝑎𝑃𝐶 and 𝐴𝑟𝑒𝑎𝐼𝑆 of each sample were estimated as the standard deviations of the respective chemical

species’ integrated set of 1H peak areas normalized with respect to corresponding proton multiplicities (𝐴𝑟𝑒𝑎/𝑁).

The proton multiplicity of the principal component and internal standard peaks (𝑁𝑃𝐶, 𝑁𝐼𝑆) and the relative molar

mass of the internal standard were assigned zero uncertainty. The mass purity of a material and its expanded

uncertainty were estimated as the mean and twice the standard deviation of a Gaussian curve fit to the aggregate of

MCM simulation results for all sample replicates (100,000 x 3 samples).

Results and Discussion

The vitamin D metabolite reference standards contained volatile and water impurities not measureable by classical

chromatographic techniques. They also contained isomeric and structurally-related impurities that were either not

resolved via LC-UV and GC-FID or were indistinguishable in one-dimensional proton qNMR (600 MHz) spectra.

DSC was not a viable option owing to the instability of 25(OH)D at temperatures near the melting points. This

investigation demonstrates the complexity and challenges associated with making reliable organic purity

determinations using limited, viable data.

A unit of 25(OH)D2 material was initially acquired as a candidate calibration material for SRM 2972a certification

measurements. Preliminary purity screening measurements were performed to assess the material suitability for use

as a primary reference standard. HPLC-UV (λ = 220 nm and 265 nm) analyses were performed using F5 and CN

column methods similar to LC Method 1 and LC Method 2, respectively. Several impurity peaks were observed,

including one that was detected only at 220 nm, and comprised ≈ 1.2 % of the total chromatogram peak area. The

mean purity determined by detection at both wavelengths and using both LC methods was 98.2 %; however, the

results ranged from 96.7 % to 99.4 % and had a standard deviation of 1.2 %. This variability suggested inconsistent

UV absorbance responses and/or resolution amongst the organic species Therefore, qNMR was implemented to

determine the purity of this material. Results of this screening assessment indicated that the mass purity was only

approximately 65 %. The 1H{

13C} NMR spectra contained several significant peaks from impurity components that

were not observed in the LC chromatograms and not identified during this screening. In light of this, a new, more

pure standard was obtained from the manufacturer and drastic overestimation of the amount of 25(OH)D2 in

subsequent CRM value assignments was prevented. Assessments via qNMR were performed for the other vitamin D

metabolite reference standards and resulting purities were held with a higher degree of confidence than those by the

mass balance. Data from the mass balance approach provided either confirmatory results or information used to

assess observable biases of the qNMR determinations.

The compilation of all analytical results and the purities of each of the four 25(OH)D reference standard materials

are shown in Table 1. The structures of the 25(OH)D analyte species, with pertinent carbon nomenclature to the

qNMRIS analysis, are presented in Figure 1.

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Table 1 Purity assessments of 25-hydroxyvitamin D metabolite reference standards via mass balance method

(including LC-UV, LC-MS/MS, TGA, and Karl Fischer Titration), qNMR analysis, and a combination purity

estimate.

[Table 1]

Fig. 1 Structures of 25-hydroxyvitamin D2, 25-hydroxyvitamin D3, and 3-epi-25-hydroxyvitamin D3 with carbon

nomenclature relevant to identifying quantified protons of qNMR assessment.

[Figure 1]

The mass balance approach was performed with a level of completeness determined to be fit-for-purpose as a

complementary method to qNMR for characterization of reference standards of SRM 2972a 25-hydroxyvitamin D

Calibration Solutions. The “Organic Analysis” results in Table 1 are the mean of results determined using the LC

methods, with corresponding expanded uncertainties (Uk=2) that reflect Type A variability associated with the

different LC methods, combined with an estimated 0.5 % Type B uncertainty from peak collation and varying

detector responses. Twelve impurities of 25(OH)D2 were detected via LC-UV, at least nine for 25(OH)D3 Lot 1,

nineteen for 25(OH)D3 Lot 2, and nine for 3-epi-25(OH)D3. Most of the impurities were detected using SIM

detection of m/z = 395 and m/z = 383 for 25(OH)D2 and 25(OH)D3 isomers, respectively, evidencing that the

aggregates of trace-level organic impurities consist primarily of structurally-related species. An analysis of variance

performed for the LC-UV purity results of each material identified the variability to be almost entirely attributable to

that between the two chromatographic methods. Analyses using λ= 265 nm detected more impurities than λ=220

nm, though both, in addition to MS detection results, determined purity results that were in agreement with respect

to the expanded uncertainty intervals. Individual LC results presented for each method and absorbance wavelength

are estimated as the means of purity values determined via replicate analyses (injections) of sample solutions. The

means from multiple LC separation and detection methods represent reasonable estimates of “Organic Analysis”

purity for which the resolution and UV response factors of structurally similar species at a given wavelength are

likely variable. The mean “Organic Analysis Methods” values also included results from LC-MS using SIM mode

detection. Though detecting only structurally-similar species, the LC-MS results were in agreement with

corresponding consensus values and represented significant components of the expanded uncertainty intervals

associated with these approximations.

The combined TGA and KF titration results for the 25(OH)D3 monohydrate reference standards agreed with the

theoretical hydrate mass fraction of 4.3 %, confirming that the hydrate component in these materials is the primary

source of water and volatiles. Though it is recognized that these two methods provide results for measurands of a

different “kind”, they were equally treated as components of the mass balance approach. This was justified for the

25(OH)D3 Lot 1 material given that all results were well in agreement, suggesting water was the only significant

measured impurity. Although there is disparity between the two results for 25(OH)D3 Lot 2, they were similarly

treated because there was no confirmation of impurity identity or evidence that one or both results were not

influenced by unknown bias. However, the TGA method by principle provides a more comprehensive “Volatiles and

Water” assessment. For this investigation, the disparity between the mean result and the TGA result is not

statistically significant. TGA analysis also revealed that the 25(OH)D2 reference standard has a significant volatile

component of 3.74 % ± 0.11 % (Uk=2) and that the 3-epi-25(OH)D3 contains a detectable volatile component

comprising 0.4 % ± 0.1 % (Uk=2) of its total mass.

The mass balance purity estimate (PurityMB, g/g) was calculated according to:

𝑃𝑢𝑟𝑖𝑡𝑦𝑀𝐵 = (1 𝑔

𝑔− 𝐼𝑉𝑊 ) × 𝑃𝑂𝐴 (3)

whereby IVW is the mass fraction (g/g) of impurities determined using “Water and Volatiles Methods” and POA is the

purity of the principle component species relative to the estimated amount of total organics determined by the

“Organic Analysis Methods”. Uncertainties of these mass balance determinations were estimated from the relative

standard uncertainties of the “Organic Analysis” and “Volatiles and Water Methods” results combined in

quadrature. This approach was considered fit-for-purpose as part of the purity evaluations of these reference

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standards and may be implemented as part of efforts to exclusively determine the relative mass fraction estimate of a

principal component species. Other more detailed and comprehensive approaches [3] that quantify or estimate

individual impurity components, including inorganic species, may have greater fitness for purpose of specific

applications. Such approaches yield valuable information when comprehensive characterization of all components of

the standard material is desired.

The chemical shifts and coupling constants of 1H spectra were consistent with those of 25(OH)D reported in the

literature [20]. qNMR results were initially quantified with the symmetric, isolated, and well-resolved spectral

signals of 25(OH)D C6 and C7 (Figure 2) olefinic protons. The internal standard was quantified as the average of

the DMT aromatic (singlet, δH = 8.2 ppm, 4 x 1H) and methyl (singlet, δH = 3.9 ppm, 6 x

1H) signals, normalized

torespective proton multiplicities (4 and 6). The certified purity of this material (99.99 % ± 0.16 % [Uk=2]) was

confirmed using qNMR. The proton content of the internal standard was referenced to the aromatic proton content

of SRM 350b Benzoic Acid (Acidimetric) (99.9978 % ± 0.0044 % [Uk=1.96]) via cross-reference with dimethyl

sulfone (NMIA, 100.0 % ± 0.3 % [U95]).

Fig. 2 Overlay of 25-hydroxyvitamin D2, 25-hydroxyvitamin D3, and 3-epi-25-hydroxyvitamin D3 1H{

13C} 600

MHz spectra using 90 degree excitation pulses.

[Figure 2]

Comparison of qNMR results to mass balance estimates indicated that the qNMR assessment via the C6 and C7

olefinic proton peaks were positively biased, as demonstrated by the mean values for 25(OH)D3 Lot 1 and the

25(OH)D2 reference standards. Though these results are not significantly different according to 95 % confidence

intervals, the slightly higher qNMR values were not expected. Given that the DMT internal standard certified purity

was verified, this disparity suggested that the C6 and C7 peaks were positively biased by other species.

Based on LC-UV-MS and NMR assessment, 25(OH)D species isomerize to pre-25-hydroxyvitamin D in solution

[15]. Therefore, qNMR samples were measured immediately after preparation. The spectra indicated no significant

evolution of these species during the 90 min 1H{

13C} NMR analysis. For LC-UV analyses, the area of any detected

thermal isomer peaks (verified by LC-MS) were aggregated with that of the principal 25(OH)D peak. Because of the

transient nature of 25(OH)D species in solution and the relative levels of structurally-related impurities, 2D NMR

experiments only provided confirmation of species identity.

For LC-APCI-MS assessments, the MSD operating in SIM mode analyzed m/z 395 for 25(OH)D2 and m/z 383 for

25(OH)D3 to identify structurally-similar impurities. Seven of the impurities present in the 25(OH)D2 reference

standard were detected, as well as seventeen impurities in the 25(OH)D3 reference standard. This strongly indicated

that several of the organic impurities are structurally related to 25(OH)D species. A major impurity component of

the 3-epi-25(OH)D3 reference standard (≈ 1.5 %) was determined to be 25(OH)D3 via LC-UVwith reference the

25(OH)D3 standard. Though 1H-NMR provides structural selectivity, some of the low-level structurally-related and

isomeric impurities have several protons signals that are indiscernible from those of the primary species. This

limited selectivity for certain isomer moieties resulted in some peak overlap of closely structurally-related impurities

and principal component C6 and C7 peaks. Figure 2 presents 1H{

13C} spectra of 25(OH)D2, 25(OH)D3, and 3-epi-

25(OH)D3 samples. For scenarios during which these species are concomitantly present as impurities, the scaled

spectra demonstrate the overlap potential of several quantifiable peaks. A more inclusive spectral analysis using

peaks quantified from multiple chemical shift regions was performed to assess whether the suspected positive bias

was inherent in all peak integrals, or if a more accurate result could be obtained with an alternative 1H signal.

Table 1 shows three separate results for the qNMR analyses, each representing a purity determined using either C6

and C7 olefinic protons (δH = 6.0 ppm to 6.3 ppm); C9, C4, and C1 non-equivalent methylene protons (δH = 2.9

ppm, 2.6 ppm, and 2.4 ppm, respectively); or C18 and/or C21 methyl protons (δH = 0.6 ppm and 1.0 ppm). The

purities are not completely consistent with one another, indicating that the 1H signals were variably influenced by

region-specific signal overlap or difficulties associated with determining peak areas in convoluted regions of the

spectrum. The C18 and C21 methyl proton signals are singlets that have slightly varying chemical shifts amongst the

three vitamin D metabolite species (Figure 3a). This difference causes associated overlapping peaks to be more

discernible as adjustable irregularities of the = 1H peak symmetry, as was observed in most spectra. Also, the upfield

region of 25(OH)D2 1H spectra have additional methyl signals and splitting patterns that are clearly identifiable and

10

unique amongst these three metabolites. The integral adjustments had magnitudes of 0.3 % to 0.9 % of the total C18

methyl peak area, which concurs with the average approximate 0.4 % organic impurity component determined via

LC-UV for 25(OH)D2 and both 25(OH)D3 reference standards.

Fig. 3 a) Overlay of 1H{

13C} spectra of 25(OH)D regions (0.7 ppm to 0.5 ppm) containing C18 methyl proton peaks;

25(OH)D3 spectrum scaled to ≈1 % intensity of that of 3-epi-25(OH)D3 b) C18 methyl proton peak, c) C6 and C7 proton

peaks and d) C9, C4, and C1 proton peaks.

[Fig. 3]

As shown in Figure 3b, superimposition of the 25(OH)D3 C18 methyl region onto the corresponding 3-epi-

25(OH)D3 (scaled to 1 % relative intensity) spectral region illustrates the discernibility of signal interference when

these species are present as low-level impurities. This was supported by identification of 25(OH)D3 in the 3-epi-

25(OH)D3 reference standard via LC-UV (approximately 1.5 %). Figures 3c and 3d illustrate the ambiguity of these

small isomeric interferences in the two other quantified regions of the spectra, and thus the difficulty in adjusting

associated biases. For these reasons the qNMR purity was assessed exclusively from the C18 (and C21 from

25(OH)D2), methyl 1H peaks. For 25(OH)D3 Lot 1 and 25(OH)D2, the adjusted C18 and C21 methyl integral results

had higher values than those determined via the mass balance approach. Although the respective means of both

methods lie well within the 95 % confidence limits of the other, this suggests that an amount of the underlying peak

bias was not discernible and thus not mitigated. Such a case strengthens the validity of performing a diverse, multi-

method assessment for investigating total mass purity of organic chemical standards [5]. Although qNMR is a direct

and traceable measurement technique, some composite organic materials may contain structurally-similar low-level

impurities that are difficult to distinguish by one-dimensional NMR experiments, but are observable and quantifiable

with LC methods.

Primary reference materials occupy the highest position of a calibration hierarchy, and the uncertainty intervals

associated with respective purity value assignments are propagated through each step of the measurement procedure

These propagate uncertainties are significant components of the total measurement uncertainty budgets.

Cumbersomely large symmetric uncertainty intervals, often derived by combination of uncertainties from multiple

equally-weighted methods that contribute to a combined average [21], may not be fit-for-purpose for primary or

high-level calibrant characterization. Such a case exists when propagated uncertainties are large enough to render

reference materials unsuitable for widespread use within the clinical community. While assessing purity analysis

data, it is useful to develop a heuristic for determining how information from direct methods and mass balance

approaches will contribute to a consensus value. A technically-informed decision process should be implemented

that combines information according to associated degrees of confidence, identifies any need for further

investigation or invalidates inaccurate results from an erroneous measurement system .

When mass balance and direct-method results are nearly equal or have highly-overlapping probability distribution

functions (PDF), complementary results might be combined in an equally-weighted fashion. Such consensus values

are minimally affected by known potential measurement biases. This approach was taken for the qNMR and mass

balance results of most 25(OH)D purity estimates. Different approaches should be taken when the purity result PDFs

have little or no overlap, as was the case for the 3-epi-25(OH)D3 reference standard( mass balance: 98.03 % ± 1.01

% [Uk=2]; qNMR: 94.94 % ± 0.74 % [Uk=2]). Though the source of the disparity was not explicitly characterized, it

is expected that the results of the mass balance method are positively biased from incomplete quantification of all

impurities and non-equivalent UV and MS response factors. The unknown biases [17] associated with these results

give rise to a large assigned uncertainty (3 %) when the significantly different PDFs are combined in an equally-

weighted fashion.

Alternative statistical methods may be employed to infer uncertainties that draw upon expert knowledge of a

measurement system. Scenarios where these approaches might be advantageous are evaluations using pre-defined

constraints of the measurement model (informative priors) that employ analyst expertise or known limits of the

measurement system. Such situations might include observations where direct measurement results, not affected by

a known bias, are significantly different from those determined via a mass balance approach, or when the assessment

of high-purity material yields uncertainty interval bounds that exceed the logistical limit of 100 % mass fraction

purity. The purity evaluation of the 3-epi-25(OH)D3 reference standard is an instance of the former.

11

Given the direct mode of measurement via qNMR, a lower chemical mass fraction purity determined by this method

may be justified by the analyst to be closer to the true value than that from the mass balance techniques. It may

therefore be suitable to evaluate the qNMR result and its associated uncertainty as the sole component of the purity

value. Conversely, if the qNMR result is indicative of a higher purity than that of the mass balance then it becomes

ambiguous which methods are significantly biased. For such a situation, value assignment may be based upon the

approach for which the analyst has the greatest confidence.

In addition to these decision-making processes, there are statistical methods implementable for determining

consensus purity values that reflect varying degrees of confidence in the qNMR result. Since the qNMR and mass

balance purity results for 3-epi-25(OH)D3 have no overlap of Gaussian 95 % uncertainty intervals, the

corresponding purity value may be evaluated by a Two-piece normal [22]. The qNMR primary ratio method is

believed to yield results of greater “trueness” and this statistical approach may be used to estimate an asymmetric

consensus probability distribution that provides greater weighting to the qNMR result. By this approach, the

consensus distribution has a mode (μ) that is equal to the qNMR result and 2.5th

and 97.5th

percentiles that are

derived from the lower half of the qNMR uncertainty interval and the mean mass balance result, respectively. A

description of this statistical approach and its implementation is described in Online Resource 1. From the binormal

distribution, a mean consensus purity of 95.80 % (u= 0.98 %) and an asymmetric 95 % confidence interval (U95) of

[94.37 %, 98.03 %] was evaluated. Either the two-piece Normal approach or the qNMR result alone is suitable for

purity assignment of 3-epi-25(OH)D3, given the lack of agreement by these methods and the suspected unknown

bias in the mass balance value. The more conservative uncertainty interval of the two-piece Normal is the preferred

evaluation since peak integral adjustments were made during the qNMR analysis to account for observable isomeric

interferences.

Fig. 4 Probability Distribution Functions (PDFs) for 3-epi-25(OH)D3 consensus purity values evaluated using a)

two-piece Normal distribution, b) Markov Chain Monte Carlo Method (MCMC) with a triangular, qNMR-favored

prior distribution and c) MCMC that is constrained only by the mean values of the qNMR and mass balance

methods.

[Fig. 4]

The Bayesian approach to probability assessment provides powerful methodology for evaluating measurement

uncertainty that considers both empirical data and pre-existing knowledge. When there is only partial overlap of the

qNMR and mass balance PDFs, they should be combined according to a prior, expertly-informed characterization of

qNMR “trueness”. This predilection should ultimately be based on confidence that qNMR is not influenced by a

significant unknown bias. Figure 4b depicts the PDF of the combined qNMR and mass balance results for 3-epi-

25(OH)D3 evaluated by a Markov Chain Monte Carlo (MCMC) simulation, as described by Possolo and Toman

[23], using OpenBUGS [24]. The consensus purity PDF was evaluated by iterative sampling of a triangular posterior

distribution, constrained to lie between the mean of the two method PDFs, with a mode that has an abscissa equal

1/10 of this interval added to the qNMR mean. The data (results of the qNMR and mass balance methods) were first

transformed into log space by ln (100 − 𝑥), and the respective u were transformed according to 𝑢 (100 − 𝑥⁄ ). The

resulting MCMC samples of the consensus value were summarized to obtain a mean 3-epi-25(OH)D3 purity value of

95.97 % with 0.75 % standard uncertainty and an asymmetric U95 of [94.73 %, 97.59 %]. The evaluation of a

consensus PDF in this manner, using an analyst-defined prior, is applicable to this scenario given the expertly-

informed confidence in the “truth” of the qNMR measurement.

In a scenario where a direct measurement technique yields purity results that are irreconcilably higher than those

assessed by indirect methods, it is not as clear to the analyst how either or both approaches are biased. In this

instance, the PDF of a consensus result may be obtained via MCMC using a prior distribution that is constrained by

the two method means, but is non-informative in the sense that no further information is provided. The PDF of this

combined value is presented in Figure 4c. With a Bayesian MCMC, the simulated evaluations are in essence

weighted according to the respective qNMR and mass balance input uncertainty intervals. The mean combined

purity value from this evaluation is 96.39 % with a u value of 0.97 % and a U95 of [94.70 %, 98.19 %]. More

detailed description of these Bayesian approaches and their implementation are described in the Electronic

Supplementary Information for this manuscript.

Conclusion

12

The aggregate of data acquired by fundamentally-diverse methods was used to assess mass fractions of 25(OH)D

species in vitamin D metabolite reference standards. These investigations were performed to determine the amount

of vitamin D metabolite species present in gravimetrically-prepared calibrant solutions that support NIST's clinical

SRM certifications. The degree of analytical rigor put forth during this purity study was to ensure that biases

inherent in each of the implemented measurement methods were mitigated prior to assigning mass fraction purity

values. Though less-intensive purity investigations are fit for certain analytical purposes, greater confidence is

required for assessments of in-house calibrants and primary reference standards used to provide mass fraction

traceability.

Reliable purity assessments of these materials benefit from both direct and indirect measurement techniques. As

demonstrated in this particular investigation, no single authoritative technique satisfactorily and universally met all

of the measurement challenges. Though the purity via direct methods is often regarded with higher confidence, as

has largely been emphasized here, information from the indirect techniques is critical for assessing potential biases

of the direct methods. This yields valuable insight for performing thorough and proper statistical analyses that

support selective mass fraction purity assessments of reference materials. Extensive, diversified, validated, and well-

coordinated methods must be used to ensure that accurate mass fraction determinations are achieved. Additionally,

analyst-informed approaches to statistical evaluation may be advantageous and appropriate when expert knowledge

of a system is communicated and the results of a particular method are held with a relatively high degree of

confidence. The complexities associated with precisely analyzing and comprehensively evaluating the impurities of

gravimetrically-implemented higher-order 25(OH)D reference standards, developed to support accurate and

traceable Vitamin D metabolite value assignments, are fundamentally essential and a metrological necessity.

Acknowledgements

Partial funding for this work was provided by the National Institutes of Health Office of Dietary Supplements (NIH-

ODS). Some of the statistical analyses of results presented in this work were performed by James Yen of the NIST

Statistical Engineering Division, Gaithersburg, MD.

Disclaimer Certain commercial equipment, instruments, or materials are identified in this paper to specify adequately the

experimental procedure. Such identification does not imply recommendation or endorsement by the National

Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily

the best available for the purpose.

References

1. DeLuca H (1988) The vitamin D story: a collaborative effort of basic science and clinical medicine. FASEB

J. 2(3) 224-236.

2. Phinney K (2008) Development of a standard reference material for vitamin D in serum. Am J Clin. Nutr.

88 511s:512S.

3. Duewer D.L., Parris R.M., White E.W., May W.E., Elbaum H. (2004) An Approach to the Metrologically

Sound Traceable Assessment of the Chemical Purity of Organic Reference Materials. NIST Special

Publication 1012.

4. Westwood S., Choteau T., Daireaux A., Jospehs R.D., Wielgosz R.I. (2013) Mass Balance Method for the

SI Value Assignment of the Purity of Organic Compounds. Anal. Chem. 85 3118-3126.

5. Davies D.R., Jones K., Goldys A., Alamgir M., Chan B.K.H., Elgindy C., Mitchell P.S.R., Tarrant G.J.,

Maya K.R., Luo Y., Moawad M., Lawes D., Hook J.M. (2015) Purity assessment of organic calibration

standards using a combination of quantitative NMR and mass balance. Anal. Bioanal. Chem. 407 3103-

3113.

13

6. Kim S-H, Lee J, Ahn S, Song Y-S, Kim D-K, Kim B. (2013) Purity Assessment of Organic Reference

Materials with a Mass Balance Method: A Case Study of Endosulfan-II. Bull. Korean Chem. Soc. 34(2)

531-538.

7. Plato C and Glasgow A.R. Jr. (1969) Differential Scanning Calorimetry as a General Method for

Determining the Purity and Heat of Fusion of High-Purity Organic Chemicals. Application to 95

Compounds. Anal. Chem. 41(2) 330-336.

8. Saito T., Ihara T., Koike M., Kinugasa S., Fujimine Y., Nose K., Hirai T. (2009) A new traceability scheme

for the development of international system-traceable persistent organic pollutant reference materials by

quantitative nuclear magnetic resonance. Accred. Qual. Assur. 14 79-86.

9. Weber M, Hellriegel C, Rueck A, Wüthrich J, Jenks P. (2014) Using high-performance 1H NMR (HP-

qNMR®) for the certification of organic reference materials under accreditation guidelines-Describing the

overall process with focus on homogeneity and stability assessment J. Pharm Biomed. Anal. 93 102-110.

10. SaitoT., Nakaie S., Kinoshita M., Ihara T., Kinugasa S., Nomura A., Maeda T. (2004) Practical guide for

accurate quantitative solution state NMR analysis. Metrologia 41 213-218.

11. Le Gresley A., Fardus F., Warren J. (2015) Bias and Uncertainty in Non-Ideal qNMR Analysis. Crit. Rev.

Anal. Chem. 45 300-310.

12. Holzgrabe U., Deubner R., Schollmayer C., Waibel B. (2005) Quantitative NMR spectroscopy –

Applications in drug analysis. J. Pharm Biomed. Anal. 38 806-812.

13. Malz F., Jancke H. (2005) Validation of quantitative NMR. J. Pharm Biomed. Anal. 38 813-823.

14. Wells R.J., Hook J.M, Al-Deen, T.S., Hibbert D.B. (2002) Quantitative Nuclear Magnetic Resonance

(QNMR) Spectroscopy for Assessing the Purity of Technical Grade Agrochemicals: 2,4-

Dcholrophenoxyacetic Acid (2,4-D) and Sodium 2,2-Dichloropropionate (Dalapon Sodium). J. Agric.

Food. Chem. 50 3366-3374.

15. Bedner M, Lippa KA (2014) 25-Hydroxyvitamin D Isomerizes to Pre-25-hydroxyvitamin D in Solution:

Implications for Calibration in Clinical Measurements. Manuscript submitted for publication to Anal

Bioanal Chem.

16. MonteCarlito version 1.10 – Free Excel Tool for Monte Carlo Simulation. www.montecarlito.com

(accessed May 2011)

17. Ellison S.L.R., Rosslein M., Williams A. (2000) Quantifying Uncertainty in Analytical Measurements, 2nd

edition, Eds. Eurachem: ISBN 0 948926 15 5.

18. Kragten J. (1994) Calculating Standard Deviations and Confidence Intervals with a Universally Applicable

Spreadsheet Technique. Analyst 1994 2161-2165.

19. Joint Committee for Guides in Metrology (JCGM), JCGM 101:2008 Evaluation of measurement data –

Supplement 1 to the “Guide to the expression of uncertainty in measurement” – propagation of

distributions using a Monte Carlo method (2008) JCGM 101:2008 International Bureau of Weights and

Measures (BIPM), Sevres, France.

20. Kamao M, Tatematsu S., Hatakeyama S, Sakaki T, Sawada N, Inouye K., Ozono K., Kubodera N, Reddy

GS, Ocano T (2004) C-3 Epimerization of Vitamin D3 Metabolites and Further Metabolism of C-3

Epimers. J. Biol. Chem. 279(16) 15897-15907.

14

21. Joint Committee for Guides in Metrology (JCGM), JCGM 100:2008 Evaluation of measurement data –

“Guide to the expression of uncertainty in measurement” (2008) JCGM 100:2008, International Bureau of

Weights and Measures (BIPM), Sevres, France.

22. Wallis, K. (2014) The Two-Piece Normal, Binormal, or Double Gaussian Distribution: Its Origin and

Rediscoveries. Stat. Sci. 29 106 – 112.

23. Possolo A., Toman B (2007) Assessment of Measurement Uncertainty via Observation Equations.

Metrologia 44 464-475.

24. Lunn D, Spegelhalter D, Thomas A., Best N. (2009) The BUGS project: Evolution, critique and future

directions (with discussion). Stat. Med. 28 3049-3082.

15

Table 1 Purity assessments of 25-hydroxyvitamin D metabolite reference standards via mass balance method (including LC-UV, LC-MS/MS, TGA, and Karl -

Fischer Titration), qNMR analysis, and a combination purity estimate.

Organic Analysis Methods1,2 Volatiles and Water

Methods2 qNMR Purity Result3

Combined Purity

Estimate

LC-UV

(Columnλ UV)

LC-UV

(%)

LC-MS-

MS (%)

TGA

(% g/g)

Karl Fischer Titration

(% g/g)

Mass Balance

Purity Result

(% g/g)2

C6, C7 olefinic, δ=6.2-6.0 ppm

(% g/g)

C9, C4, C1

methylene δ=2.9-2.4 ppm

(% g/g)

C18 and/or

C21 methyl

δ=1.0-0.6

ppm (%

g/g)

Mass Balance and

qNMR4 (% g/g)

25(OH) D2

CN220 99.77

CN (m/z

395) 99.72 3.74 (0.11)

96.35 (0.75) 97.43 (1.27) 96.154 (0.85) 96.03 (0.66)5

CN265 99.25

F5220 99.76

F5 (m/z

395) 99.78

F5265 99.53

Mean 99.63 (1.01) 3.74 (0.11) 95.90 (1.00)

25(OH) D3

Lot 1

CN220 99.8 4.41 (0.22) 4.44 (0.16)

95.98 (0.82) 95.20 (0.62) 95.33 (0.75) 95.27 (0.63)5

CN265 99.6, 99.6 4.54 (0.20)

C18265 99.5

F5220 99.8

F5265 99.6

Mean 99.66 (1.01) 4.46 (0.11) 95.21 (1.02)

25(OH)D3

Lot 2

CN220 99.78 4.44 (0.11) 3.90 (0.22)

95.29 (0.44) 96.45 (0.64) 94.41 (1.47) 94.95 (1.07)5

CN265 99.54

F5220 99.74

F5(m/z 383)

99.75

F5265 99.49

Mean 99.66 (1.01) 4.18 (0.55) 95.49 (1.16)

3-epi-

25(OH)D3

CN220 98.4 0.4 (0.04)

96.00 (1.10) 95.94 (1.00) 94.946 (0.74)

96.48 (3.07)5 (95.80 [94.37, 98.03]7)

(95.97 [94.73, 97.59]8)

(96.39 [94.70, 98.19]9)

CN265 98.3

F5220 98.6

F5265 98.4

Mean 98.43 (1.01) 0.4 (0.04) 98.03 (1.03) 1 Results reported as % amount of the principal component species relative to the total estimated amount of organics. 2Standard uncertainties expanded by a coverage factor (k) of 2 (Uk=2) 3 qNMR result expanded uncertainties (Uk=2) are assessed as the 1-σ interval of a Gaussian curve fit to the distribution of results from an n x 100,000-iteration Monte Carlo simulation of Equation 1, expanded by a factor of 2 (n=3; no. of sample replicates).

sample data sets randomized during Monte Carlo analysis), multiplied by the appropriate coverage factor, k. 4 Purity estimate calculated with qNMR result quantified with C18 and C21 methyl 1H signal integrals. 5Combined purity estimate uncertainty (Uk=2) determined by Gaussian random effects-modeled bootstrap simulation of all contributing results and corresponding standard uncertainties.

16

6Quantification based only the integral of C18 methyl 1H signal. 7Combined purity evaluated using Two-Piece Normal PDF; useful approach when the result of direct method is held in higher confidence and the PDFs of contributing data do not demonstrate

overlap of 95 % coverage intervals. 8Combined purity evaluated using Bayes MCMC with a triangular, qNMR-favored triangular prior distribution. 9Combined purity evaluated using Bayes MCMC that is constrained by qNMR and mass balance means, but is otherwise non-informative

17

Figure 1. Structures of 25-hydroxyvitamin D2, 25-hydroxyvitamin D3, and 3-epi-25-

hydroxyvitamin D3 with carbon nomenclature relevant to identifying quantified protons of

qNMR assessment.

18

Figure 2 Overlay of 25-hydroxyvitamin D2, 25-hydroxyvitamin D3, and 3-epi-25-hydroxyvitamin D3 1H{

13C} 600 MHz spectra using

90 degree excitation pulses.

S-19

Figure 3 a.) Overlay of 1H{

13C} spectra of 25(OH)D regions (0.7 ppm to 0.5 ppm) containing C18

methyl proton peaks; 25(OH)D3 spectrum scaled to ≈1 % intensity of that of 3-epi-25(OH)D3 b.) C18

methyl proton peak, c.) C6 and C7 proton peaks and d.) C9, C4, and C1 proton peaks.

S-20

Figure 4 Probability Distribution Functions (PDFs) for 3-epi-25(OH)D3 consensus purity values

evaluated using a.) two-piece Normal distribution, b.) Markov Chain Monte Carlo Method

(MCMC) with a triangular, qNMR-favored prior distribution and c.) MCMC that is constrained

only by the mean values of the qNMR and mass balance methods.

S-21

Electronic Supplementary Information

Higher-order metrological approaches to organic chemical purity: primary reference materials

for vitamin D metabolites

Michael A. Nelson*1, Mary Bedner

1, Brian E. Lang

2, Blaza Toman

3, Katrice A. Lippa

1

National Institute of Standards and Technology

Material Measurement Laboratory, Chemical Sciences Division1 and Biosystems and

Biomaterials Division2

Information Technology Laboratory, Statistical Engineering Division3

Gaithersburg, MD 20899-8392

* Corresponding author contact information: email michael.nelson@nist.gov;

telephone 301-975-4100; FAX 301-975-0685

S-22

This supplementary information section contains additional details regarding the statistical

models used to evaluate probability distributions for mass purity of 3-epi-hydroxyvitamin D3, as

presented in Table 1 and Figure 4 of the main text. These include the corresponding formula

bases, computational platforms and respective code scripts implemented to evaluate the results

and uncertainty intervals.

Two-piece normal distribution

Presented here is a description of the two-piece normal distribution used to express the consensus

purity of 3-epi-25-hydroxyvitamin D3 (3-epi-25(OH)D3) within the interval bounded by results

determined via qNMR and Mass Balance analytical methods. These two results exhibited no

overlap of 95 % probability intervals.

The two-piece normal is derived from two incongruent half-Normal distributions, one to the left

and one to the right of a shared “best estimate” mode value (μ), that have different standard

deviations, expressed as σ1 and σ2, respectively. The purity probability density function (PDF),

P, of 3-epi-25(OH)D3 as a two-piece normal distribution with properties μ, σ1 and σ2, is defined

by

𝑓(𝑃) =

{

𝐴𝑒𝑥𝑝 ⌊−

(𝑃 − 𝜇)2

2𝜎12 ⌋ 𝑓𝑜𝑟 𝑃 ≤ 𝜇

𝐴𝑒𝑥𝑝 ⌊−(𝑃 − 𝜇)2

2𝜎22 ⌋ 𝑓𝑜𝑟 𝑃 ≥ 𝜇

Whereby,

μ = mean of qNMR mass fraction purity result (Gaussian PDF), 0.9494 g/g

σ1 = standard deviation of the qNMR mass fraction purity result, 0.0037 g/g

σ2 = standard deviation of a Normal distribution (μ =0.9494 g/g) such that the resulting 97.5th

percentile is equal to the mean of the mass balance result PDF; 0.9803 g/g

A = (√2𝜋(𝜎1+ 𝜎2)

2)−1

The left piece has a standard deviation equal to that of the qNMR result PDF (σ1= 0.0037 g/g),

while the distribution of the right piece (σ2) is the standard deviation of a half-Normal used to

derive a two-piece PDF that has a value of 2σ along the interval between the mode of the qNMR

and Mass Balance results. The two half-Normals are scaled such that 𝑓(𝜇) = 𝐴. More about the

history and usage of this distribution is described by Wallis [S-1].

This distribution was evaluated with a kernel using Wolfram programming language,

implemented with Mathematica 10.0 computational software. The code impleneted for this

evaluation is in figure S1.

S-23

Uncertainty evaluation using an Observation Equation approach

Bayesian statistical models were used to provide two alternative probability assessments of 3-

epi-25(OH)D3 using the qNMR and mass balance results (μd) and their associated uncertainties

(u). The evaluation applied an observation equation approach, as described by Possolo and

Toman [S-2]. The results μd and associated uncertainties u were first logarithmically transformed

according to ln(100- μd) and u/(100- μd), respectively. Supporting computations for Bayesian

approach evaluations were performed using a Markov Chain Monte Carlo simulation in

OpenBUGS [S3].

One assessment of 3-epi-25(OH)D3 consensus mass purity (P) according to this approach

utilized pre-existing information that characterized the confidence for greater “trueness” of

results via qNMR primary ratio direct measurement, expressed as a prior distribution. The use of

a prior distribution requires that Bayes Theorem be applied, that is, it requires the use of an

Observation equation rather than a Measurement equation.

In this approach, observable quantities D related to the measurand P, are expressed as m

simultaneous observation equations, whereby Di are given as functions i of P and k non-

observable parameters E:

𝐷𝑖 = i (𝑃, 𝐸1, … . 𝐸𝑘) [1]

Bayes theorem allows expression of the probability distribution of purity P, conditional on D in

terms of an estimated distribution of P, known as a prior probability distribution (p). The

resulting expression, which updates the prior distribution using the information provided by the

measurements, is known as a posterior distribution (q).

Supposing that the measurement data (d = [d1 , d2] sampled from Gaussian PDFs with μ and σ

equal to the logarithmically transformed mean and standard uncertainty, respectively, of the

qNMR and mass balance results) are an outcome of random vector D with probability density

𝑓(𝑑|𝑃, 𝐸1, … . 𝐸𝑘), the posterior density q of P is given by Bayes’ formula as:

𝑞(𝑃|𝑑) =𝑓(𝑑|𝑃)𝑝(𝑃)

∫ 𝑓(𝑑|𝑠)𝑝(𝑠) d𝑠𝑋

[2]

For realistic model-based applications the denominator of the Bayes’ formula is not computable

and the conditional probability of P may be expressed using the numerator by the relation:

(𝑃|𝑑) ∝ 𝑝(𝑑|𝑃)𝑝(𝑃) [3]

A Markov Chain Monte Carlo (MCMC) simulation implemented in OpenBUGS [S-3] (Figure

S2) was used to generate samples from the posterior distribution q using only the Bayes’

numerator of Equation 2. The procedure creates a Markov Chain that has an equilibrium

S-24

distribution congruent to q. After a period of sufficient iteration a sample drawn in this way

becomes like a sample drawn from q.

For probability assessment of 3-epi-25(OH)D3 purity depicted in Figure 4b., the prior

distribution p is defined to be a triangular distribution that lies between the qNMR and mass

balance sample means, with a mode that has an abscissa equal 1/10 of this interval added to the

qNMR mean (Model 1). For the probability assessment depicted in Figure 4c., the prior

distribution p specifies that P lies between the mean of the qNMR and the mean of the mass

balance results, without any additional constraint (Model 2). In essence, the outputs of this model

are weighted with respect to the qNMR and mass balance result uncertainties.

The annotated OpenBUGS scripts used to perform these probability assessments are in Figure S2

and S3.

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Figure S1 Mathematica Code for evaluation of 3-epi-25(OH)D3 purity using a Two-Piece Normal Distribution

S-26

Figure S1 (continued)

S-27

Figure S1 (continued)

S-28

Figure S2 OpenBUGS code for 3-epi-25(OH)D3 purity via Model 1, with triangular prior

S-29

Figure S3 OpenBUGS code for 3-epi-25(OH)D3 purity via Model 2

References

[S-1]Wallis, K. (2014) The Two-Piece Normal, Binormal, or Double Gaussian Distribution: Its

Origin and Rediscoveries. Stat. Sci. 29 106 – 112.

[S-2] Possolo A., Toman B. (2007) Assessment of Measurement Uncertainty via Observation

Equations. Metrologia 44 464-475.

[S-3] Lunn D., Spegelhalter D., Thomas A., Best N. (2009) The BUGS project: Evolution,

critique and future directions (with discussion). Stat. Med. 28 3049-3082.