Sampling Error: Impact on the Quantitative Analysis of Nanoparticle

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Sampling Error: Impact on the Quantitative Analysis of Nanoparticlebased Surface-enhanced Raman Scattering Immunoassays Alexis Cathrine Crawford, Aleksander Skuratovsky, and Marc D. Porter Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.6b01263 • Publication Date (Web): 24 May 2016 Downloaded from http://pubs.acs.org on May 26, 2016

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Analytical Chemistry

1

Sampling Error: Impact on the Quantitative Analysis of Nanoparticle-based Surface-enhanced Raman Scattering Immunoassays Alexis C. Crawford,a,b Aleksander Skuratovsky,b,c and Marc D. Portera,b,c* a

Department of Chemistry, University of Utah, Salt Lake City, UT 84112 The Nano Institute of Utah c Department of Chemical Engineering, University of Utah, Salt Lake City, UT 84112 * Corresponding author ([email protected]) b

ACS Paragon Plus Environment


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2 Abstract This paper examines the impact of the sampling error caused by the small size of the focused laser spot when using surface-enhanced Raman scattering (SERS) as a quantitative readout tool to analyze a sandwich immunoassay. The assay consists of a thin film gold substrate that is modified with a layer of capture monoclonal antibodies (mAbs), and extrinsic Raman labels (ERLs) that consist of gold nanoparticle cores (60 nm diameter) coated with a monolayer of a Raman reporter molecule and a layer of human IgG mAbs to tag the captured antigen. The contribution of sampling error to the measurement is delineated first by constructing and analyzing an antigenic random accumulation model; this is followed by an experimental study of the analysis of an assay substrate using two different laser spot sizes. Both sets of findings indicate that the analysis with a small laser spot can lead to a sampling error (i.e., undersampling) much like that found when the size of a measured soil sample fails to accurately match that of a larger, more representative sample. That is, the smaller the laser spot size, the larger probable deviation in the accuracy of the measurement and the greater the imprecision of the measurement. Possible implications of these results with respect to the general application of SERS for quantitative measurements are also briefly discussed.


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3 Introduction Surface-enhanced Raman scattering (SERS) is a powerful tool for analyzing a number of materials.1-5 However, the reliability of this surface-sensitive technique for quantitative measurements remains an ongoing question.6-11 Part of the issue rests with the difficulties in the reproducible preparation and/or structural stability for some of nanostructured architectures which have exceedingly large enhancements.12-14 As a result, several laboratories,6-10,

15-22

including our own,4,

23-31

have pursued tactics aimed at

overcoming these reproducibility problems. Our approach uses particle geometries (e.g., spherically shaped particles) that can be prepared with a more reproducible and stable structure but do not have as large of an enhancement32-34 as some types of nanostructured materials (e.g., nanocavity arrays,7,

10, 35

nanostructured metal surfaces,36-39 and porous

metal films40). This form of reproducibility management32, 41-43 has begun to demonstrate the merits of quantitative SERS in health care diagnostics testing4,

9, 26, 44-46

and other

areas.3, 47-50 This paper examines another obstacle to exploiting the strengths of SERS as a quantitative measurement tool: sampling error. We are referring here to the error introduced in a measurement when the size of the sample analyzed is below that needed to reliably represent the composition of the sample. To this end, this paper shows how sampling error occurs from the small size of the focused laser spot typically used to measure biolytes captured and labeled in the SERS-based sandwich immunoassay shown in Figure 1. Indeed, this type of sampling error, often referred to as undersampling, has strong parallels with the classic sampling problem that occurs when the size of the sample analyzed fails to accurately match that of a larger, more representative sample.



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4 The statistical basis of undersampling can be found in the central limit theorem, which states that data collected from a sufficiently large sampling of a population will be normally distributed about the true mean of the population.51-52 Furthermore, this theorem specifies that the variance of the measured distribution will decrease as the size of the sampled population increases. Such an improvement can also be accomplished by increasing the number of samplings of a population at a fixed analysis area, , which improves the variance by . We hypothesized that a significant and unrecognized portion of the error associated with SERS measurements, especially when working near the limit of detection (LoD), is due to undersampling. A test like an enzyme-linked immunosorbent assay (ELISA)53 measures the absorbance of the homogenous solution produced by the assay, which inherently averages the response across the reactive surface of a microplate well. However, Raman spectrometers are typically designed with high light collection efficiencies by means of large numerical aperture optics; this results in small laser spot sizes. Indeed, the area of the focused laser spot can be tens of microns in diameter, which is often a very small fraction of the total area of the sample surface (e.g. a 10-µm laser spot interrogates ~10−6 of the geometric surface area of a 3-mm diameter address). It follows that an analysis carried out for such a small relative surface area can inherently result in sampling error. This level of undersampling can may lead to a large measurement variance and biased mean value, both being more problematic as the amount of analyte decreases. There are several statistical approaches for assessing the impact of measurement variance. The overall variance54-56 of the analytical result, , is the sum of the variance


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5 due to sampling, , and that of the actual measurement, . This relationship can be written as:

= +

(1)

If all of the other assay components are under statistical control,57 can be found by repeatedly measuring a sample that has a consistent signal in order to quantify contributions from, for example, laser power fluctuations and detector noise. then equals the difference between and . Often, however, attempts to improve precision neglect the importance of by focusing on lowering . Youden’s Rule of Thumb55 states that if is nine times larger than , reducing will only marginally affect . Larger gains in will be achieved by lowering through increases in nreplicate of a fixed size or increases in the size of sample analyzed in one measurement. Another approach to the analysis of variance determines the amount of sample needed to produce results at a predefined accuracy and precision. Classically, this is done by means of the sampling constant, :

= (%)

(2)

where is the mass of the sample and % is the percent relative standard deviation from sampling error. While the use of is frequently associated with the analysis of solids and particulates,58-59 it can be applied to other types of analyses potentially plagued by undersampling. This approach is particularly useful when the accuracy and precision



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6 of a measurement is sample size dependent. can then be applied to determine the mass required to realize, for example, a 1% RSD for a given sample type. Herein, we examine the impact of sampling error when using SERS for the quantitative readout of the sandwich immunoassay in Figure 1. As a starting point, we constructed a simple random antigen accumulation model to serve as a statistical framework to test for the possible impact of the laser spot size on the accuracy and precision of the measurement. We then show that the findings from the SERS analysis of a sandwich (heterogeneous) immunoassay for human immunoglobulin G (H-IgG) antigen is subject to the same sampling problem found from the model. The paper concludes with a brief discussion of the potential implications of our results on the overall applicability of SERS as a quantitative measurement tool.

Experimental The preparative process for the SERS immunoassay has been reported previously and is reviewed in the Supplemental Information, along with other experimental details. Instrumentation. SERS maps of the samples were acquired using a DXR Raman microscope

(Thermo

Scientific)

equipped

with

a

HeNe

laser

(632.8

nm),

thermoelectrically cooled CCD detector, and 50-µm entrance slit. The spectral resolution of the instrument changes from 5.2 to 8.8 cm−1 from 50 to 3500 cm−1, respectively. Spectra are the average of two 1-s integrations at each sample location. The microscope was fitted with a 10× objective, which yields a 5-µm diameter laser spot, or a 50× objective (MPlan N objectives, Olympus), which yields a spot diameter of 0.5 µm. Maps were collected in 20-µm steps over the entire assay address (2.0 mm) using


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7 a sample translation stage (1-µm resolution). The maps consist of ~120 rows and columns (~14,400 data points). The intensity at each location was determined from the peak height of υs(NO2) at 1336 cm−1 (Figure 1) for the DSNB coating on the ERLs after baseline correction (see Results and Discussion for the mapping and analysis procedures). Computational modeling. To delineate the impact of the area interrogated by the focused laser spot, Monte Carlo simulations were designed to mimic the SERS analysis. The first simulation step produced a two-dimensional (2D) surface (3.0 mm diameter) composed of a random distribution of point-sized adsorbates (PSAs). The next step simulated the analysis of the surface by a focused laser spot by randomly placing a disk of predetermined area on the simulated assay surface and counting the number of PSAs within the disk. Predetermined numbers of replicate measurements ( ) were made with each analysis area, with the average and standard deviation of PSAs per unit area used to assess the accuracy and precision of the results, respectively. To produce a 2D surface with a random PSA distribution, a 3×1015 by 3×1015 x-y network was populated using a pseudo-random number generator with 15-digit accuracy. A location was rejected if a portion of it was outside the 3.0-mm diameter address. This network was filled with 1.000×107 PSAs (1.415 PSA/µm2), emulating a density of PSAs of an assay for a sample with a ~1 ng/mL H-IgG (see Supplemental Information Section 2 for details). We did not account for the possibility of PSA co-occupancy, but the probability of two or more PSAs occupying the same location was extremely low. To simulate the analysis by a laser spot, the pseudo-random number generator was used to select an x- and y-coordinate as the center of a predefined circular analysis area. If the entire analysis area resided within the 3.0-mm address boundary, the number



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8 of PSAs within that area was counted and recorded; a location was rejected if any portion of the analysis area resided beyond the boundary, and a new location was selected. The results of the simulations are represented as the area analysis ratio (AAR) or the spot diameter. The AAR equals the area contained within a single circular analysis area divided by the total area of the simulated substrate. The simulations spanned AARs from 1×10−7 to 0.99. Using the 3.0-mm diameter of the emulated address (7.07×106 µm2) these AARs range from 1×10−7 and 0.99, which equate to laser spots of 0.71 µm2 (2.9mm diameter) and 7.0×106 µm2 (0.95-µm diameter), respectively.

Results and Discussion Development of the Monte Carlo simulation and analysis. To set the stage for an unbiased statistical analysis of the potential sampling problem associated with SERS analysis, we devised a random accumulation model to document the impact of the surface area of the sample analyzed on the accuracy and precision of the measurement. To do so, we first defined an expectation value (1.415 PSA/µm) for the random accumulation model from SEM images of a SERS substrate after the completion of an assay, which is detailed in Supplemental Information Section 2. Figure 2a presents an example of the random accumulation of PSAs on a 3.0-mm diameter substrate. For visualization, only 5,000 of 1.000×107 randomly distributed PSAs are marked by crosses that are ~1031 times larger than their actual size, which would not be visible at this length scale. The example in Figure 2b shows five (nreplicate=5) pseudorandom generated locations of 300-µm diameter analysis areas (AAR of 1.0×10−2). The analysis counts the number of PSAs within each disk. The average and standard deviation


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9 of the results for different sizes and numbers of sampling areas are then compared to the true value for the simulated accumulation to determine accuracy and precision. Benefits of replicate measurements on accuracy and precision. If all other components of the assay are under statistical control, the reliability of a measurement can be improved by increasing the sample size (AAR) or by increasing the number of samplings, nreplicate, of the same sized sampling. Supplemental Information Table S1 lists the AARs examined and their corresponding diameters. Figure 3 shows data from 10 separate simulations for assessing the impact of nreplicate. The data are numerically listed in Supplemental Information Table S2. The results are presented as plots of the PSA density (Figure 3a), average absolute deviation from the mean (DAvg) (Figure 3b), and standard deviation (s) from the mean (Figure 3c) as a function of nreplicate for three different AARs. The values of nreplicate spanned from 1 to 100 (1, 2, 3, 4, 5, 10, 25, 50, 75, and 100). The three AARS [1.0×10−7 (0.95 µm diameter), 1.0×10−6 (3.0 µm diameter), and 1.0×10−5 (9.5 µm diameter)] were chosen to have sizes similar to those typically used in our laboratory. As evident in Figure 3b, increases in both nreplicate and AAR improve the accuracy of the measurement in that DAvg approaches zero. The improvement with nreplicate is the notable for the smallest AAR, 1.0×10−7. That is, the rate of convergence in accuracy depends on the total area sampled: the largest of AAR (1.0×10−5) has the highest rate of convergence and the smallest AAR (1.0×10−7) has the lowest rate of convergence. Moreover, the deviation in the accuracy at an AAR of 1.0×10−7 for nreplicate=1 is ~80% larger than that for the same AAR at nreplicate ≥25. Similar improvements are evident for



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10 the larger AARs (1.0×10−6 and 1.0×10−5), though not as dramatic. These results indicate that increases in both nreplicate and AAR improve measurement accuracy. Figure 3c plots how the precision of the simulation measurements is influenced by nreplicate for the same three AARs. The results show that s for each AAR fluctuates at low values of nreplicate and begins to converge to different limiting levels as nreplicate increases. The smallest AAR, 1.0×10−7, approaches a limit in s at an nreplicate of ~50. This value of s corresponds to a measurement imprecision that is larger than the true value of the sample and is symptomatic of a significant level of undersampling. The two larger AARs, 1.0×10−6 and 1.0×10−5, approach smaller limiting values in s after only 10 measurements, demonstrating improvements in precision associated with replicate sampling. Note that the changes in s qualitatively follow the expected decrease with respect to

!"#$

(Supplemental Information Table S2). These results also predict that the analysis of a sample with lower levels of accumulation will be subject to larger deviations in accuracy and greater levels of imprecision. We will show later that undersampling can be a large source of the error found in our SERS-based immunoassay. By way of projection, these results can be used to estimate the minimum value of nreplicate needed for a given AAR that will produce a measurement within an acceptable level of tolerance. The minimum value of nreplicate, calculated for a chosen confidence level and target %RSD, is expressed as:

!"#$

=

%


(3)

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11 where % is from the Student’s t-table. As an example, we used a confidence interval of 95%, a 1% RSD (RSD of 0.014 from a true value of 1.415 PSAs µm⁻2), and the values of s from the simulations for nreplicate >50 (Supplemental Information Table S3). Using these parameter values, the simulation results indicate that an nreplicate of 1 is sufficient to measure the true PSA density of 1.415 PSAs/µm2 for AARs larger than 1.0×10−2 (spot size diameters ≥300 µm). For the smaller AARs of 1.0×10−3 (95-µm diameter), 1.0×10−4 (30-µm diameter), 1.0×10−5 (9.5-µm diameter), 1.0×10−6 (3.0-µm diameter), and 1.0×10−7 (0.95-µm diameter), the required values of nreplicate are 1, 8, 857, 8,768, and 81,359, respectively. The latter numbers represent clearly onerous analysis times. If we increase the %RSD to a level the is similar to the measurement variance found for the instrument used herein (5%, see below), the values of nreplicate decrease to 3, 34, 355, and 3,254 for AARs of 1.0×10−4, 1.0×10−5, 1.0×10−6, and 1.0×10−7; larger AARs require only a single measurement to meet the specified performance. Impact of larger analysis areas. Figure 4 shows the effect of increases in AAR. The inserts plot the x- and y-axes on a log scale in order to accentuate the differences at smaller AARs. The progression in DAvg and s indicates that nreplicate has a larger impact at lower AARs. The most notable improvements are observed for AARs from 1.0×10−7 to ~1.0×10−2. At an AAR of ~1.0×10−2 (~300-µm diameter spot), the simulation measurement has an accuracy >99% and an RSD of 1% or better for nreplicate >2. These results can also be used to predict a lower limit for the AAR required to meet a given level of performance by using the sampling constant ( ) formulation in Equation 2. Using a 1% RSD as an example, 10 separate simulations were carried out for



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12 the above level of accumulation at evenly spaced increments in AAR (1.0×10−3 to 0.5). This analysis is detailed in Supplemental Information Section 3 and the results are plotted in Figure S2. The results indicate that a at a AAR of 3.4×10−2 (spot diameter of 560 µm) is required to achieve a 1% RSD measurement for a single sampling. If we lower the analysis criteria to match instrument noise at 5% RSD (see below), the required AAR drops to 1.4×10−3 (i.e., a 110-µm spot diameter). It is important to note that the measurement improvements derived from an increase in AAR fundamentally differ from those gained by an increase in nreplicate. Larger AARs improve both the accuracy and precision of the measurement. However, while increasing nreplicate can increase accuracy, the improvement in precision at large values of nreplicate is subject to the law of diminishing returns (i.e., s improves with

). That said, the advantages of a larger laser spot comes with a tradeoff: the smaller numerical apertures required to enlarge the laser spot also have lower light collection efficiencies. This dictates use of a higher laser power in order to maintain signal strength. Measurement distribution as a function of analysis area. Another way to gage the impact of sample size is to analyze the results in terms of a normal distribution. Plots to this end are shown in Figure 5 for 100 simulations and an nreplicate=1 (the numerical data are listed in Supplemental Information Table S4). For these plots, the center of the distribution is the mean of the results, and the precision is directly linked to the width of the distribution. The results for an AAR of 1.0×10−7 are not shown because the extremely small sample size produced discretized results that failed to fit a normal distribution.


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13 As expected, the distributions indicate that larger AARs result in the highest probabilities of obtaining an accurate and precise measurement. Both accuracy and precision degrade with decreases in AAR. The loss of accuracy is evident by shifts in the maxima of the distributions from the true value. The increases in the widths of the distributions are diagnostic of an erosion of precision. In other words, a decrease in AAR lowers the accuracy of the measurement due to an increase in the distribution of the measurements; i.e., a smaller AAR imparts a greater heterogeneity to the measurement. Analysis of SERS immunoassay substrates. This section examines the spot size dependence of the signal strength and its distribution in the form of a high density SERS map for an actual sample. The sample was prepared at 10 ng/mL H-IgG according to the procedure described in the Supplemental Information Section 2. Figure 6a presents a ~2×2 mm Raman map of the sample that extends slightly past the border of the 2.0-mm disk-shaped address. The map was constructed by mounting the sample on an x-y translation stage (20-µm steps) in a Raman microscope and measuring the signal at each location using a 5.0-µm laser spot (632.8 nm at 5 mW). Only the locations in the red circular area, which has a ~700-µm diameter and surrounds ~4,000 data points, were analyzed in order to omit inclusion of the obvious defects in the sample. For analysis simplicity, intensities were normalized to the average intensity of the data within the red circular area. By way of contrast, the image in Figure 6a has a color-scaled, normalized intensity of 1.000±0.085 (raw signal of 1223±104 cts/s). The green color represents the normalized signal of 1.0. The orange and red (warm) colors are normalized signals of 1.5 and 2.0, respectively. The blue and violet (cool) colors are the normalized signals of 0.5



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14 and no observable signal, respectively. The normalized value for s translates to a normalized overall variance, , of 0.007. To examine the signal distribution across the substrate, the map was converted from Cartesian to polar coordinates and plotted in Figure 6b as the normalized intensity versus radial distance from the address center. The average normalized intensity and the corresponding 95% confidence interval surrounding the mean (±1.96×s) are represented by the solid and dashed black lines, respectively. These data were then binned in 0.01 increments to generate the histogram in Figure 6c in order to qualitatively assess the fit of the data to a normal distribution by a least squares analysis. Since the residuals shown in Figure 6d of the least squares fit to the histogram in Figure 6c do not have an identifiable pattern, we conclude that the data is reasonably fit by a normal distribution. To quantify the contribution from sampling error, , to the overall signal variance, , the instrument signal variance, , was determined by collecting 100 spectra from a glassy carbon plate for the D band intensity60 at 1332 cm−1. This determination gave a raw signal of 175±6 cts/s, a normalized signal of 1.000±0.034, and a normalized of 0.001. Based on this result and that of the normalized variance for (0.007) from Figure 6a, Equation 1 yields a normalized of 0.006, which is 6 times greater than . This difference indicates that sampling error, not instrument error, limits the performance of SERS in our assays with respect to quantitation, and that greater improvements in accuracy and precision will be achieved by increasing the laser spot size and not nreplicate. The impact of sampling error was further assessed by analyzing the sample in Figure 6 at 100 random locations in the red-circled portion of sample using both 5.0- and 0.5-µm diameter laser spots, AARs of 6.25×10−6 and 6.25×10−8, respectively. Note that


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15 the higher light collection efficiency of the 50× microscope objective required a reduction of laser power to 0.5 mW in order to avoid saturating the detector response. The two sets of results were therefore normalized independently of each other to facilitate comparisons. The analysis of the data gave signals of 1235±108 cts/s (1.00±0.09 when normalized) for the 5.0-µm diameter spot and 4782±862 cts/s (1.00±0.18 when normalized) for the 0.5-µm diameter laser spot. The analysis of the two experiments in Figure 7 shows that the accuracy and precision of the measurement improve with a larger laser spot size. While only a qualitative comparison can be made based on the assumptions invoked in the simulations, the sensitivity of the experimental measurements to the change in spot size found in the experimental study is less than that of the simulations because of the lower level of antigen accumulation in the simulations. Nonetheless, the experimental data clearly indicate that measurements with the smaller laser spot introduce a larger fundamental error in the sample analysis, i.e., a larger negative impact on accuracy and precision. By way of a more informative comparative, if we set the %RSD to 5% and the confidence level to 95%, Equation 3 indicates that a minimum of 4 measurements are required to meet this tolerance for the 10× objective, but that 51 or more measurements are necessary with the 50× objective. Alternatively, we can project the size of the laser spot required for 1 read by means of the sampling constant, , in Equation 2. For a 1% RSD, this analysis yields a equal to 5.5×10−2 AAR or a 470-µm laser spot. If the tolerance is reduced to 5% RSD, the requisite AAR is 5.5×10−4. This AAR translates to a 47-µm laser spot which is readily achievable with minor optical system modifications of most Raman spectrometers or



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16 potentially by using approaches that raster the laser spot across the sample.61-63

Conclusions There is no question as to the strength of SERS as a technique for low level detection. However, challenges in the “reproducibility of enhancement” continue to slow the widespread application of SERS to quantitative analysis. This work has shown, both through model simulations and experiments, how the sampling error that is related to the small size of the laser spot can have a strong, negative impact on the accuracy and precision of a SERS measurement. The smaller the size of the laser spot, the greater the probability for a larger deviation in the accuracy of the measurement and the larger the imprecision of the measurement. In other words, the smaller the laser spot size, the greater the bias from undersampling and, therefore, the difficulty in realizing an accurate and precise SERS measurement. To take full advantage of the predicted benefits of an increase in the laser spot size, however, it is important to keep in mind that the most straightforward approach to do so involves a change in the optical magnification of the objective. This change, nonetheless, dictates an increase in the absolute power of the laser in order to maintain the same power density used at the small spot size, which may not be part of the standard design or options available in some of today’s commercially available Raman spectrometers. Adjusting the absolute laser power to maintain the same power density at the sample may, nonetheless, requires a laser with a sufficient output power, which may dictate the use of a higher powered laser that is part of the standard design or options available in some but not all of today’s commercially available Raman spectrometers.


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17 The change in magnification must also, for example, take into account the accompanying differences in the collection efficiency of the scattered signal and the angular dependence of the scatter SERS signal. Herein, we have demonstrated that the classic tools for dissecting the impact of sampling error (e.g., number of replicate samples, nreplicate and the sampling constant, ). These methods can be applied for the same diagnostic purposes when using SERS for quantitative analysis to increase the accuracy and precision of results. We are currently devising experiments to more fully assess the implications of these results.

Acknowledgements The authors acknowledge many valuable discussions with Ronald Wampler and Nicholas Schlotter. This work was supported by the Critical Paths Initiative of the U.S. Food and Drug Administration (U18FD004034) and the Innovative Molecular Analysis program of the National Cancer Institute (R33CA155586).

Supporting Information The Supporting Information is available free of charge on the ACS Publications website. This information includes Preparation of assay components, SEM image of completed SERS immunoassay substrate, the determination of the sampling constant (K ' ) for the simulated assay, and a series of figures and tables that summarize and/or support pertinent findings in the paper.

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22 Figure Captions Figure 1. The three primary steps of the SERS-based immunoassay: (a) preparation of extrinsic Raman labels (ERLs); (b) preparation of the capture substrate; and (c) procedure for the assay. The first two steps are completed prior to running the assay. The assay is performed by incubating a 20-µL droplet on the capture substrate. The sample is then rinsed, exposed to a 20-µL suspension of ERLs, rinsed again, dried under ambient conditions, and analyzed. (d) Analysis is performed via excitation of the SERS substrate with a focused laser spot. The SEM image shows ERLs on the capture surface and the SERS spectrum collected from a focused laser spot with a 5-µm diameter. Figure 2. Visual representation of pseudo-random distribution of PSAs on a 3.0-mm diameter substrate. (a) 5,000 oversized PSAs (1×1031 times larger than actual size) on the simulated address. (b) Same simulated substrate with five pseudo-random number generated locations for the analysis areas, each with a 300-µm diameter or an AAR of 1.0×10−2. Figure 3. Results from 10 simulations of the random accumulation of PSAs with a true value of 1.415 PSAs µm−2 show the impact of nreplicate >1 on: (a) PSA density; (b) accuracy expressed as DAvg; and (c) precision expressed as . A listing of the numerical values in this figure is given in the Supplemental Information Table S2. Figure 4. Monte Carlo simulation results for the impact of sample size in terms of the AAR on: (a) PSA density; (b) accuracy expressed as the DAvg; and (c) precision expressed as s. Insets show the same data with the x- and y-axes on a log scale to highlight small changes in the data. A listing of the numerical values in this figure is given in the Supplemental Information Table S2 Figure 5. Normal distribution curves for data collected from 100 simulations: (a) normal distribution curves for AARs between 99 and 0.1; (b) normal distribution curves for AARs between 0.10 and 1.0×10−3; and (c) AARs between 1.0×10−3 and 1.0×10−6. Each yaxis has a different scale to show differences in the frequencies of the distributions. For aid visualization, the x-axis tick marks in (a) equate to the same spacing as the smaller tick marks in (b) and (c). A listing of the numerical values in this figure is given in the Supplemental information Table S4. Figure 6. SERS immunoassay substrate analyzed with (a) high density Raman color contrast mapping of the normalized signal intensity. Green represents the mean signal from the central 4,000 data points (within the red circle). Warmer and cooler colors indicate higher and lower signal intensities, respectively. (b) Normalized SERS signal is shown in polar coordinates to highlight differences in the signal distribution from the center to the outer edge of the address. (c) The central 4,000 data points represented in a histogram with the data are binned into 0.01 normalized increments with a linear least squares fit to a normal distribution. (d) Residuals for the least squares fit to a normal distribution. The solid black lines represent a normalized signal of one, the dashed black lines indicate the 95% CI, and red lines indicate the 4,000 data point cutoff.


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23 Figure 7. Raman analysis of ERL-based SERS-based immunoassay substrate for the detection of H-IgG at 10.0 ng mL−1. The impact of nreplicate on the (a) accuracy expressed as DAvg, and (b) precision expressed as s for the two different spot size. (c) Normalized distributions of the signal from 100 nreplicate measurements on the same substrate using 5 and 0.5 µm laser spots. A listing of the numerical values for (a) and (b) in this figure is given in the Supplemental Information Table S5.



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24 for TOC only


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Figure 1. The three primary steps of the SERS-based immunoassay: (a) preparation of extrinsic Raman labels (ERLs); (b) preparation of the capture substrate; and (c) procedure for the assay. The first two steps are completed prior to running the assay. The assay is performed by incubating a 20-µL droplet on the capture substrate. The sample is then rinsed, exposed to a 20-µL suspension of ERLs, rinsed again, dried under ambient conditions, and analyzed. (d) Analysis is performed via excitation of the SERS substrate with a focused laser spot. The SEM image shows ERLs on the capture surface and the SERS spectrum collected from a focused laser spot with a 5-µm diameter. 126x139mm (300 x 300 DPI)



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Figure 2. Visual representation of pseudo-random distribution of PSAs on a 3.0-mm diameter substrate. (a) 5,000 oversized PSAs (1×1031 times larger than actual size) on the simulated address. (b) Same simulated substrate with five pseudo-random number generated locations for the analysis areas, each with a 300-µm diameter or an AAR of 1.0×10−2. 109x185mm (300 x 300 DPI)


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Figure 3. Results from 10 simulations of the random accumulation of PSAs with a true value of 1.415 PSAs µm−2 show the impact of nreplicate >1 on: (a) PSA density; (b) accuracy expressed as DAvg; and (c) precision expressed as s. A listing of the numerical values in this figure is given in the Supplemental Information Table S2. 92x179mm (300 x 300 DPI)



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Figure 4. Monte Carlo simulation results for the impact of sample size in terms of the AAR on: (a) PSA density; (b) accuracy expressed as the DAvg; and (c) precision expressed as s. Insets show the same data with the x- and y-axes on a log scale to highlight small changes in the data. A listing of the numerical values in this figure is given in the Supplemental Information Table S2. 92x177mm (300 x 300 DPI)


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Figure 5. Normal distribution curves for data collected from 100 simulations: (a) normal distribution curves for AARs between 99 and 0.1; (b) normal distribution curves for AARs between 0.10 and 1.0×10−3; and (c) AARs between 1.0×10−3 and 1.0×10−6. Each y-axis has a different scale to show differences in the frequencies of the distributions. For aid visualization, the x-axis tick marks in (a) equate to the same spacing as the smaller tick marks in (b) and (c). A listing of the numerical values in this figure is given in the Supplemental information Table S4. 92x171mm (300 x 300 DPI)



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Figure 6. SERS immunoassay substrate analyzed with (a) high density Raman color contrast mapping of the normalized signal intensity. Green represents the mean signal from the central 4,000 data points (within the red circle). Warmer and cooler colors indicate higher and lower signal intensities, respectively. (b) Normalized SERS signal is shown in polar coordinates to highlight differences in the signal distribution from the center to the outer edge of the address. (c) The central 4,000 data points represented in a histogram with the data are binned into 0.01 normalized increments with a linear least squares fit to a normal distribution. (d) Residuals for the least squares fit to a normal distribution. The solid black lines represent a normalized signal of one, the dashed black lines indicate the 95% CI, and red lines indicate the 4,000 data point cutoff. 150x161mm (300 x 300 DPI)


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Figure 7. Raman analysis of ERL-based SERS-based immunoassay substrate for the detection of H-IgG at 10.0 ng mL−1. The impact of nreplicate on the (a) accuracy expressed as DAvg, and (b) precision expressed as s for the two different spot size. (c) Normalized distributions of the signal from 100 nreplicate measurements on the same substrate using 5- and 0.5-µm laser spots. A listing of the numerical values for (a) and (b) in this figure is given in the Supplemental Information Table S5. 100x168mm (300 x 300 DPI)


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