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The authors have declared that no competing interests exist.

Conceived and designed the experiments: SL MS JTCL. Performed the experiments: SL MS. Analyzed the data: SL MS JTCL. Wrote the paper: SL MS JTCL.

Surface-enhanced Raman scattering (SERS) nanoparticles have been engineered to generate unique fingerprint spectra and are potentially useful as bright contrast agents for molecular diagnostics. One promising strategy for biomedical diagnostics and imaging is to functionalize various particle types (“flavors”), each emitting a unique spectral signature, to target a large multiplexed panel of molecular biomarkers. While SERS particles emit narrow spectral features that allow them to be easily separable under ideal conditions, the presence of competing noise sources and background signals such as detector noise, laser background, and autofluorescence confounds the reliability of demultiplexing algorithms. Results obtained during time-constrained

The field of biomedical optics has traditionally approached disease detection by deducing tissue status through the measurement of optical signals generated either intrinsically by tissue constituents

Recently, a number of groups

(a) Multiple flavors of nanoparticles exist where each nanoparticle contains a gold core coated with a Raman-active layer, encased in a silica shell. (b) Raman spectra of five nanoparticle flavors. (c) Example result from a least-squares routine showing the ability to demultiplex two different nanoparticles from a mixture under noisy conditions.

Therefore, in this report, we propose a general method for quantifying the reliability of particle concentrations and ratios that are computed from a least-squares demultiplexing algorithm. First, this method is developed through numerical simulations of various mixtures of SERS nanoparticles in the context of noise and background signals. We define a metric, the spectral reliability index (SRI), which serves as a predictor of error in single- and multi-flavor applications. We further provide results from well-controlled experiments to assess the feasibility and accuracy of our approach. While initial experiments are intentionally simplified to verify the accuracy and reproducibility of these methods, our strategy could potentially be of value for a range of technologies that utilize targeted SERS-based nanoparticles to provide multiplexed measurements of molecular biomarkers both

Since our ultimate goal is to provide experimentalists with a reliable measure of nanoparticle concentrations or concentration ratios, we set out to simulate spectral measurements of SERS nanoparticles under the varying noise and background conditions that may be encountered experimentally. Simulated spectra were generated first for single-flavor applications, then for two-particle applications where the relative concentrations between particle flavors ranged from 1∶1 to 5∶1, and finally for two-flavor ratios within three-flavor mixtures. The rationale for a 5∶1 maximum range of relative nanoparticle concentrations is based on the observation that signal contrast between tumor and normal tissues rarely extends beyond a factor of five for

It bears mentioning that actual measurements of SERS nanoparticles in cells and tissues can include endogenous Raman background signals from tissues as well as variable autofluorescence and laser background components. With this in mind, our study makes a few assumptions that are often relevant for biomedical spectroscopy with SERS nanoparticles: 1) background signals are spectrally broadband and do not contain sharp narrowband spectral features that are morphologically similar to the spectral peaks generated by SERS nanoparticles (i.e., we assume that endogenous Raman signals are orders of magnitude weaker than the signals from SERS nanoparticles); 2) since the Stokes shift of Raman signals is higher than fluorescence signals, autofluorescence background signals are due to the slowly varying (broadband) tail at the long-wavelength side of the autofluorescence spectra; 3) a significant amount of stray laser light contributes to a broadband background at the CCD detector despite efforts taken to filter out the illuminating laser radiation (see section 2.5).

In order to simulate a realistic spectral measurement, pure spectra from single- or multi-flavor particle mixtures were mixed with varying magnitudes of broadband background signals and zero-mean Gaussian-distributed white noise (Eq. (1)). Note that Eq. (1) incorporates all broadband background signals (e.g. laser background and autofluorescence background) into a single spectral component, _{n}_{n}

Other than sources of noise, it is assumed that each measured spectrum consists of a weighted sum of fixed nanoparticle spectra (_{n}_{n}_{n}_{n}_{m}^{th}-order polynomial term_{m}^{th}-order polynomial term (for baseline correction)

Below we describe metrics called ‘relative fitting error’ (RFE) and ‘spectral reliability index’ (SRI). RFE and SRI are “goodness-of-fit” metrics that quantify how well the simulated spectra can be decomposed into the reference spectra defined

A common approach to quantify the reliability of a spectral fitting routine is to compare the norm of the fit to the norm of the input signal, a metric that has previously been termed the relative fitting error, RFE, given in Eq. (3)

Although both RFE and SRI are informative consolidated measures of goodness of fit, the biomedical application of these fitting algorithms requires us to examine errors in the actual nanoparticle weights computed. For single-particle cases, we use the standard percent error definition. In cases with more than one particle, we compute each individual particle’s error and generate a single value that we call the composite error for the entire mixture (Eq. (5)). This will be discussed further in section 2.4.

An example of the difference between RFE and SRI is shown in

Simulations of composite error as a function of two different spectral quality metrics, RFE (Eq. (3)) and SRI (Eq. (4)). Note that the RFE metric is highly sensitive to the broadband background level (

In an ideal situation, an experimentalist would obtain a single measurement and assume that the nanoparticle weights computed by a DCLS routine are accurate and reproducible in subsequent measurements. In reality, the randomness of the noise and broadband background signals results in a distribution of measurement errors. Thus, the goal of our simulations is to provide researchers with a minimum necessary SRI value to guarantee a maximum bound on concentration error (e.g., 10% error for any single measurement). To achieve this, a measure of confidence must be assigned to the maximum bound on concentration error that is desired for a single measurement. We first note that the concentration errors generated by the DCLS routine are normally distributed (Gaussian) for any single nanoparticle flavor. Second, we define a composite error for multi-flavor mixtures, which is the root-mean-squared error of all nanoparticle flavors (Eq. (5)). Defining the composite error in this way implies that the composite error for a mixture of nanoparticle flavors (

In order to assign a level of confidence to the error of a single measurement that an experimentalist may perform, we simulate 50,000 spectral measurements corresponding to a particular SRI, process them through a DCLS demultiplexing routine, and then construct a histogram of concentrations errors (results shown later in section 3.2). After fitting a Gaussian (_{80}, corresponding to a certain percentile (in our case, 80%) of the cumulative distribution function is found. In other words, since 80% of the possible error values corresponding to a particular SRI are less than this error value, _{80}, an experimentalist who has taken a single measurement and obtained an SRI value from the DCLS routine can be 80% confident that the error in their measurement is at most _{80} (results shown later in section 3.2).

Various flavors of SERS nanoparticles were obtained from Cabot Corporation, formerly Oxonica Materials (Mountain View, CA). Droplets (5 µL) of varying nanoparticle concentrations and mixtures between 0.8 and 800 pM were placed on a glass slide and point measurements were obtained using a custom fiberoptic probe (Fiberguide Industries, Stirling, NJ) that is angled at ∼45 degrees to the glass slide to minimize specular reflections. On the illumination side of the device, light from a 785-nm diode laser (30 mW) is first filtered with a narrow-bandpass filter (Semrock, LD01-785) prior to being coupled into the illumination fiber at the center of the fiber bundle. This removes off-resonant laser noise (including amplified spontaneous emission) that may be collected by the multimode fibers and contribute background signal and shot noise to the Raman spectra. Raman signals are collected via a bundle of 36 close-packed multimode fibers (200-µm core) surrounding one singlemode illumination fiber (

(a) A spectrometer with CCD detector is used to capture Raman signals from a nanoparticle sample illuminated with a 785-nm laser source. See text for details. (b) Example of a strong signal with a high SRI and (c) a weak signal with a lower SRI, in which noise and broadband background signals increasingly dominate over the SERS signals. Representative SERS peaks are numbered 1–4 and the peak of the broadband background is labeled with a star.

Experiments are performed with a custom spectrometer from Bayspec (RamSpec NIR) outfitted with a cooled deep-depletion spectroscopic CCD (Andor DU920P-BR-DD). The Bayspec spectrograph utilizes low-f/# optics (f/1.8 or NA = 0.28) to effectively image Raman signals from the multimode collection fibers (NA = 0.21) onto a proprietary volume phase grating and then onto the Andor CCD. The Andor DU920P CCD contains 1024 x 256 pixels, with a 6.2-mm height (256-pixel dimension) that is exactly matched to the height of the image of our linear fiber bundle array with 4∶3 de-magnified imaging between the entrance slit and the CCD detector (the NA increases from 0.21 to 0.28). Since a large amount of the laser light (785-nm illumination wavelength) is collected by our multimode collection fibers, a relay extension is built into the front of the spectrograph to allow for the placement of a longpass interference filter (Semrock LP02-830RU-25) to reject illumination light at 785-nm, as well as any autofluorescence background at shorter wavelengths than the Raman peaks (<830-nm). This 4∶3 demagnification relay extension contains a 150-µm slit on the far end where the image of the linear array of fibers is refocused. This slit spatially filters out the diffuse stray light in the relay chamber from photons that are rejected by the longpass filter (

The reference spectra of the nanoparticles (_{n}

Experiments were first conducted with single nanoparticle flavors at various concentrations. _{80}). Finally, _{80}) varies as a function of SRI. A comparison between simulations and experiments (

(a) Weights returned by the least-squares routine are linear over the range of measured concentrations. (b) Errors from multiple measurements of the same single-flavor sample are normally distributed. Shaded region indicates where 80% of the errors lie. (c) A plot of error vs. SRI, in which the reported error is the 80% confidence bound (_{80}). The simulations agree well with the results found experimentally.

Next, a dual-flavor mixture is analyzed through simulations and experiments (^{th} percentile (80% confidence level) for our analyses (_{80}). By constructing composite-error histograms (e.g.

(a) Composite errors for multi-particle mixtures are gamma-distributed. Vertical lines indicate bounds for the 80^{th} percentile of error values (_{80}) that may occur for a particular SRI. (b) A plot of error (_{80}) vs. SRI for a 1∶1 mixture of particle flavors and a 5∶1 mixture of flavors. Simulations (solid lines) closely predict experimental results. (c) A plot of the minimum SRI required to ensure a composite error ≤10% with 80% confidence. This dual-flavor example shows how the minimum SRI value depends upon the mixture ratio.

Finally, a triple-flavor mixture is investigated via simulations and experiments, and the error in the concentration ratios is presented (

The plot shows percent error (_{80}) in the ratio between particle flavors as a function of SRI.

In summary, we have presented an analysis of the accuracy of least-squares demultiplexing for spectral measurements of SERS particle mixtures. In particular, we have proposed and demonstrated the feasibility of a spectral reliability index (SRI) that correlates with measurement accuracy. The flowchart shown in _{n}_{n}

A flowchart illustrating how SRI may be used in practice to ensure that a SERS-based multiplexed molecular diagnostic is reliable. See text for details.

In practice, a set of look-up tables could be generated through simulations in order to establish the relationships between error vs. SRI and particle ratios. This would allow one to rapidly assess the reliability of spectral demultiplexing measurements in real time. These simulations and look-up tables would be different for every device, experimental application (e.g., number of particle flavors), and user-defined error threshold since the relationship between error and SRI depends upon all of these instrument- and application-dependent factors. However, the general algorithm depicted in the flow chart in

As SERS-coded nanoparticles gain popularity in animal studies and eventually for clinical diagnostics, the reliability of demultiplexing algorithms will need to be constantly assessed. Although spectroscopy is powerful in its ability to convey large amounts of information, performing accurate spectroscopic measurements is far from trivial. It has been our intent to develop a general metric and algorithm for rapidly assessing the experimental accuracy of spectral measurements involving multiplexed SERS nanoparticles. While we have performed simulations and experiments to demonstrate the basic feasibility of this approach using well-controlled experimental conditions under the simplifying assumptions mentioned in section 2, future studies are needed to demonstrate the accuracy and utility of these methods in a variety of biomedical applications such as

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The authors thank Anushree Srivastava and Jack Zhou for technical support.