Impact of sampling error on quantitative analysis of immunoassays
Surface-enhanced Raman scattering (SERS) is a powerful technique used in healthcare diagnostics and many other areas.1 SERS can be used to detect markers of infectious disease and cancer, as well as biological warfare agents, at levels far below those of other techniques.2 Although SERS has unquestionable advantages when used for characterization, problems with measurement reproducibility have limited its reliability in quantitative analysis. For example, there are difficulties in the reproducible preparation and structural stability of some of the nanostructured materials used in SERS. This is because the enhancement in the signal that gives rise to SERS is intimately linked to the size and shape of these materials. Changes on the nanoscale can therefore affect the strength of the SERS signal and thus its reliability for measurement. As a result, research has focused on stabilizing nanostructured materials by using techniques such as the formation of metal adlayers or the adsorption of counterions.3, 4
Another important factor affecting the reproducibility of SERS measurements (in particular in immunoassays) results from the size of the laser spot used to collect the spectra. If a large laser spot is used, a small numerical aperture (which has a low light-collection efficiency) is required. In addition, a higher-powered laser is needed to maintain the strength of the signal. However, if a small laser spot is used, the sampling error (i.e., the error introduced when the composition of the sample analyzed fails to match the composition of a larger, more representative sample) can result in lower accuracy and precision of the measurements.5 Furthermore, undersampling (caused by the analysis of only a small fraction of the total surface area of a sample) can lead to large variances in measurements and to biased values of the mean. Both of these problems become more severe as the level of the analyte decreases. The theoretical basis of undersampling lies in the central limit theorem, which indicates that the variance of data will decrease as the size of the sampled population increases. This can also be achieved by keeping the sampled population at a fixed size and increasing the number of samples (which reduces the variance in proportion to the square root of the number of samples).
We have found that if the other parameters of an immunoassay are statistically controlled, the variance due to the measurement process can be determined by measuring fluctuations in the laser intensity and other contributions to measurement error. However, the overall variance of a measurement is the sum of the variance due to measurement and the variance due to sampling. Attempts to improve precision often focus on reducing the former and neglect the role of the latter. More notable gains can be achieved, however, by reducing the variance due to sampling (as a result of increases in either the number of samples or the size of the sample analyzed in a single measurement).
To investigate this sampling error problem, we used a random accumulation model to examine how the size of the laser spot affected the accuracy and precision of SERS measurements of an antigen surface density. With this model, the first step produces a 2D distribution of point-sized antigens across a simulated 2.0mm-wide surface. The next step mimics the analysis of this surface with a laser spot of fixed size, by randomly placing a predetermined number of disks (of the same size as the laser spot) on the surface. With this approach, we thus aimed to replicate an analytical procedure that involves changing the number of samples. We also repeated the procedure with different sizes of disks, to reflect differences in laser spot size.
We ran our model keeping the area analysis ratio (AAR) constant and increasing the number of samples. We also ran it by keeping the number of samples fixed and increasing the AAR. As we expected, increases in both the number of samples and the AAR improved the measurement accuracy (see Figure 1). However, the benefit derived from increasing the AAR fundamentally differed from that gained by increasing the number of samples. Increasing the AAR improved both accuracy and precision, whereas increasing the number of samples increased the accuracy and the improvement in precision was proportional to the square root of the number of samples (as predicted by the central limit theorem).
Next, we designed an experiment to test the effect of the laser spot size on accuracy and precision. We thus measured the SERS response in a sandwich immunoassay. We prepared a sample by spiking human immunoglobulin G antigen into phosphate-buffered saline (pH 7.4) at a concentration of 10ng/mL. We then determined the average and standard deviation of the surface density as a function of the number of samples, using laser spot sizes of 5.0 and 0.5μm. The data from the immunoassay supported the conclusions reached using our simulation, namely, that measurements with smaller laser spots introduce larger fundamental errors into the data. We used a relative standard deviation of 5% at a 95% confidence level as a measure of precision. To achieve this level of precision with a spot size of 0.5μm, more than 50 measurements were needed. However, with a spot size of 5.0μm only four measurements were necessary.
Our work has shown how the sampling error related to a small laser spot can have a strong, negative impact on both accuracy and precision in SERS measurements. We believe that the impact of sampling error will need to be taken into account to advance SERS as a technique for quantitative measurement. We are currently designing experiments to carefully quantify the contributions made by sampling error to measurements in diagnostic immunoassays. To do so, we will analyze serum samples from patients infected with tuberculosis and correlate the results with particle densities measured using gold nanoparticles as labels in SERS.
This work was supported by the US Food and Drug Administration's Critical Paths Initiative (U18FD004034) and the National Cancer Institute's Innovative Molecular Analysis Program (R33CA155586).
University of Utah
Marc Porter is director of the Nano Institute of Utah and a faculty member of the Departments of Chemistry and Chemical Engineering. He has expertise in analytical chemistry, interfacial science, and nanotechnology. His laboratory focuses on the development of sandwich immunoassays and validation using giant magnetoresistance and SERS.
Alexis Crawford recently completed her PhD in chemistry. The focus of her work is in immunoassays, with the detection of infectious diseases by SERS.
Aleksander Skuratovsky is a graduate student in chemical engineering. The focus of his work is in modeling many aspects of immunoassays and surface chemistry.