Comprehensive normalization and binary classification methods for enhanced sensitivity and reproducibility in Luminex assay quantitation

Abstract

The Luminex assay is a powerful tool for large-scale quantitation of antibody levels and cytokines, but its utility can be limited by issues of specificity, sensitivity, and reproducibility. The corrections for background fluorescence and machine drift are essential steps in the normalization process. However, traditional methods often oversimplify these steps, failing to account for the complexity of the data, leading to the introduction of error and decreasing the sensitivity and reproducibility of the analysis. Furthermore, conventional methods to determine cut-points in binary measures do not consider the true distribution of the data, leading to arbitrary cut-points that compromise the integrity of the analysis. Here, we present a novel approach to normalize Luminex data and split the normalized bimodal data. Our method uses orthogonal regression of the measured fluorescence of a negative control bead and a blank bead to correct for background fluorescence, enhancing accuracy by preventing overcorrection due to cross-reactivity. To account for machine drift, we use a generalized additive model (GAM) on the standard curves to calculate a plate correction, thus reducing error and improving reproducibility. To distinguish between positive and negative results in bimodal measures, we use a clustering analysis to accurately split the data based on distribution. Finally, we developed a web application to easily carry out the developed method. These methods collectively increase sensitivity, specificity, and reproducibility of Luminex assay data analysis by effectively addressing the limitations of current normalization techniques, correcting for background fluorescence and machine drift, and improving the specificity and accuracy in splitting bimodal data.

1. Introduction

The Luminex assay is a versatile and widely used platform for large-scale analyte quantification, such as the quantification of cytokines (Dupont et al., 2005), antibodies (Lachmann et al., 2013), disease biomarkers (Lucas et al., 2017), specific DNA sequences (Dunbar, 2006), and protein-protein interactions (Blazer et al., 2010). This technique increases the scale of the more commonly used enzyme-linked immunosorbent assay (ELISA) by concurrently measuring multiple target analytes (Fulton et al., 1997). Rather than capturing the target antigen on a coated plate and using a colorimetric substrate to quantify its presence, the Luminex assay captures the target on coated beads in suspension and uses fluorescence as the quantifiable reporter (Kellar and Iannone, 2002; Fulton et al., 1997; Dupont et al., 2005). This allows for simultaneous detection of multiple analytes, substantially reducing required sample volume and hands-on experimental time and permits large-scale quantification of multiple targets (Kellar and Iannone, 2002; Lachmann et al., 2013). Despite its strengths, the assay can be limited by specificity and sensitivity introduced by various factors including antibody cross-reactivity, manufacturing variability, sample autofluorescence, and instrument drift (Rountree et al., 2024; Elshal and McCoy, 2006; Kellar and Iannone, 2002). Both experimental and analytical techniques can be employed to minimize error (Curion and Theis, 2024; Eckels et al., 2013). Here, we propose a comprehensive normalization method to analytically reduce the error introduced by background fluorescence and machine drift.

Antibody cross-reactivity, sample autofluorescence, bead fluorescence, and machine noise all contribute to the measured background fluorescence of each sample, thereby compromising the specificity of analyses (Rountree et al., 2024; Kellar and Iannone, 2002). Several experimental controls can be used to analytically correct for the background fluorescence. The most used control is the blocking protein-coated bead, often BSA, included in bead mixture, which is referred to as the blank bead. Additionally, there could be a blank well containing the bead mixture without sample serum and a negative well containing sample serum without the bead mixture. Typically, the background fluorescence for each experimental sample is corrected by subtracting the measured fluorescence of the blank bead from the fluorescence measured of the target-coated bead (Rausch et al., 2016; Dunbar, 2006; Kellar and Iannone, 2002). However, the blank beads exhibit a high propensity for non-specific binding, leading to overestimation of background fluorescence (Shadbahr et al., 2023; Cox et al., 2012). This overcorrection can subsequently decrease the sensitivity of the assay as it artificially reduces the true signal of the target analytes. Therefore, we propose using a negative control bead, coated with a target protein for which all patients are known to be seronegative, in addition to the blank bead to correct for background fluorescence using an orthogonal regression, a regression method that accounts for error in both variables to prevent overcorrection.

Another major consideration with the Luminex assay is the technological variation that occurs between runs. Specifically, gradual and often subtle changes in the machine’s performance can result from environmental changes, mechanical wear, and calibration inconsistencies (Rountree et al., 2024; Dunbar, 2006). Correcting for machine drift is essential to reliably execute longitudinal studies and multi-plate analyses and increase accuracy and reproducibility of the data. To address the error introduced by machine drift, a standard curve is run on each plate. Commonly, the linear range of the standard curve is used to fit a linear regression model to calculate a correction to combine data from multiple plates (Rausch et al., 2016; Ellington et al., 2010). However, these linear models can oversimplify the data as a true calibration curve is often non-linear (Cox et al., 2012; Kingsmore, 2006). Therefore, we propose using a generalized additive model (GAM), a flexible regression technique that captures non-linear relationships, rather than a linear model to fit a smooth curve to correct for machine drift.

While some target antigens measured in a Luminex assay will resemble a normal distribution, some targets, such as the detection of antibodies, are binary and exhibit a bimodal distribution. In binary measures, there is often a need to determine the fluorescence levels which separate the positive node from the negative node (Japp et al., 2021). The data are often split in half to determine positive or negative, but this type of split fails to incorporate the complexity of human sampling, where it is unlikely that a specific antibody target is present or absent in exactly half of the population (Sharma and Jain, 2014; Japp et al., 2021). This method of splitting may result in poor accuracy, leading to decreased sensitivity and reproducibility (Shadbahr et al., 2023). Therefore, we propose using a computational clustering method, which identifies natural groupings within the data to determine a cut point based on the true distribution of the data.

Given the prevalence of such issues when using the Luminex assay, it is essential to devise methods to take these limitations into consideration in order to fully maximize the strengths of this otherwise powerful assay. We propose a novel three-step solution to address the current shortcomings in normalization: (1) using an orthogonal regression on the measured fluorescence of the blank and negative control beads to correct for background fluorescence, (2) using a GAM to calculate a plate normalization factor (PNF) to correct for machine drift, and (3) using a clustering algorithm to determine the division between the two populations in a binary assay.

2. Materials and methods

2.1. Serum sample collection

Patients were recruited and enrolled on a protocol reviewed and approved by the Institutional Review Board at Baylor College of Medicine during standard outpatient clinic appointments at Texas Children’s Cancer and Hematology Center, and whole blood samples were collected in serum separator tubes and processed as previously reported (Tubman, 2024). All samples were collected prior to the emergence of SARS-CoV-2 in 2019, thereby allowing SARS-CoV-2 antigens to be utilized as a negative control protein.

2.2. Luminex assay

The Luminex assay was initially conducted with the blank bead, four negative controls beads, and ten different target beads. A subset of these data (measured fluorescence of the blank bead, three negative control beads, and four antigen-target beads) were selected to demonstrate the computational method described. Luminex magnetic beads (MagPlex Microspheres) were coupled to the antigens that included BSA (blocking protein, Sigma Cat# A7030-500 g, 3.32 μg/million beads), SARS-CoV-2 (negatives 1–3: RBD, S1, and N, kind gift from Pr. Lisa Cavacini at UMass Biologics, 3.2 μg/million beads, 5.11 μg/million beads, 5.8 μg/million beads), flu (hemagglutinin 1, BEI Cat# NR-34587, 5.1 μg/million beads), human coronavirus (HCOV) (NL63 spike, Sino Biology Cat#40604-V08B, 1.424 μg/million beads), Epstein-Barr virus (EBV) (EBNA1, Cyto-Barr, kind gift from Pr. Jaap Middeldorp at Cyto-Barr, 3 μg/million beads), and cytomegalovirus (CMV) (pp65, Prospec Cat# CMV-215, 3.91 μg/million beads). One bead region per antigen was coupled following the protocol from the Luminex xMAP Cookbook 4th version. Briefly, five million magnetic beads from each region were at first washed with distilled water and activated for 20 min with rotation in the dark using EDC and Sulfo-NHS (50 mg/ml each) in activation buffer (0.1 M Monosodium Phosphate, pH = 6.2). After being washed 3-times with MES buffer (0.05 M MES, pH = 5.0) the appropriate amount of each protein was added to the bead and rotated for 2 h in the dark. After 3 washes with storage buffer (PBS, 0.1 % BSA, 0.02 % Tween-20 and 0.05 % Sodium azide, pH = 7.4) the beads were counted using a hemocytometer and kept at 4 degrees in the dark until use. The bead mix to run the assay was prepared the day before and contained the amount of beads necessary to distribute 50 μl per well of 500 beads per region per well (768 wells total) with 15 % overage.

The serological assay used p384 plates (manufacturer) and ABE buffer (PBS, 0.1 % BSA, 0.05 % Tween-20 and 0.05 % sodium azide, pH = 7.4) for the standards and sample dilutions as well as the washes. Briefly, 50 μl of the bead mix was added to each well. After 3 min on the magnet, the supernatant was discarded, and 50 μl of the standard (2-fold serial dilution S1 to S13) and 50 μl prepared samples (dilution 1/500) were added in the appropriate wells. The plate was placed on a plate shaker at 300 rpm for 2 h in the dark. After 3 washes with ABE, 50 μl of biotinylated anti-IgG (BD Cat#555785, 1/1000 dilution) was added per well and the plate was placed back on the plate shaker at 300 rpm for 1 h in the dark. After 3 washes with ABE, 50 μl of PE-streptavidin (BD Cat#554061, 1/1000 dilution) was added per well and the plate was placed back on the plate shaker at 300 rpm for 30 min in the dark. Finally, after 3 more washes with ABE, 75 μl of ABE buffer was added per well and after 30 s on the plate shaker at 400 rpm, the plate was read by the FlexMap3D (Luminex). Using 50 μl per well, the FlexMap3D acquired a minimum of 50 beads per bead regions per well to establish the median intensity fluorescence (MFI). The standard curve for analysis was generated using a pooled mixture of all collected samples.

2.3. Computation

All data processing and analyses were conducted using R version 3.6.1. This study is representative of a single Luminex assay in which n = 316 patient and control samples were analyzed. The antigens were biologically independent and represented distinct disease models, except for the negative controls, which measured three antigens from the same virus. The full dataset (n = 316) was used to compute the orthogonal regression, GAM, and clustering, but only the negative node of the control data set was used to compare methods (Flu: n = 26, HCOV: n = 44, EBV: n = 41, CMV: n = 54). For statistical analysis, comparisons were conducted at the antigen level (n = 4), where each antigen’s statistic was derived from the aggregate of samples in the negative node. The orthogonal regression was performed on log-scaled data but was untransformed prior to subtracting from the measured intensity. The following R packages were used for key analytical steps: the generalized additive model (GAM) was computed using mgcv; the orthogonal regression was computed using pracma; the clustering was computed using mclust; figures were generated using ggplot2. The code detailing the computational method described in this study is available at Zenodo (DOI:https://doi.org/10.5281/zenodo.14026727).

3. Theory and calculations

3.1. Correcting for background fluorescence

In Luminex assays, background fluorescence can be attributed to any measured fluorescence not specific to the fluorescently bound target, including sample autofluorescence, reagent fluorescence, and non-specific binding (Rountree et al., 2024). Background fluorescence is inherent in all fluorescence-based assays and must be corrected to reduce error in the data. Based on this concept, the measured intensity (MI) of a sample (x) is equal to the true intensity (TI) of the target plus the background fluorescence of that sample (Eq. 1).

To derive a value for the background of each sample, we utilized an orthogonal regression of the measured fluorescence of the negative control bead against the blank bead. The blank beads tend to have a higher level on non-specific binding compared to a negative control bead (Cox et al., 2012; Shadbahr et al., 2023). This points towards the use of the negative control for the background correction. However, the use of the negative control can result in overcorrection if there is cross-reactivity with the negative control within a sample. Therefore, we fit the negative control to the blank using an orthogonal regression to reduce the risk of overcorrection.

Unlike traditional linear regression which assumes that the independent variable is measured without error, an orthogonal regression model accounts for error in both variables (Pallavi, 2022; Linnet, 1993). Using this method, the orthogonal regression line provides a fitted value for the background fluorescence, which is then subtracted from the measured intensity of each sample. This correction could improve the specificity and sensitivity of the assay by accurately accounting for background fluorescence without overcorrecting, thereby preserving the true fluorescence of the target analyte.

$$M{I}_x=T{I}_x+{background}_x+error$$ Equation 1 Correction for background fluorescence

3.2. Correcting for machine drift

Machine drift refers to the subtle variations in measurements obtained from an instrument over time (Pallavi et al., 2022). Correcting for machine drift is essential to ensure the reliability and reproducibility of data collected on different days (Rausch et al., 2016; Dunbar, 2006). In this experiment, data were collected from two separate plates measured on different days. To normalize the data across these plates, we use a PNF derived from a standard curve. The relationship between MI of a sample ($\mathrm{x}$) of different plates can be expressed with Eq. 2.

The PNF is derived from the standard curve, which is constructed by plotting known concentrations of target analyte or known dilutions of pooled samples against their corresponding fluorescence intensities. The curve can be interpolated to predict the expected fluorescence intensity for a given analyte concentration or known dilution factor (Rausch et al., 2016; Engvall, 1971; Ziegel, 2004). In this experiment, the standard curve was run with known dilution factors of pooled samples since the concentration of the targets were unknown within a given sample. Therefore, the PNF refers to the fluorescence intensity measured for the specific target at the experimental dilution factor.

While using a linear regression within the linear range of the standard curve has been commonly used to derive a PNF, we employ a GAM. A GAM can fit a smooth curve to data by using spline functions, which can adapt to the complexity and non-linearity of experimental data (Lucas et al., 2017; Hastie and Tibshirani, 1990). Unlike polynomial regression, which can be overly sensitive to outliers (Draper and Smith, 2014; Hastie et al., 2009), or local regression (LOESS), which is computationally intensive and lacks generalizability (Cleveland and Devlin, 1988; James et al., 2013), a GAM provides a balance between flexibility and interpretability by using smooth functions to model trends without overfitting. This approach leads to more precise normalization factors, which are critical to reduce residual error and improve the accuracy of interpolated values.

$$\frac{M{I}_{xp1}}{PN{F}_{p1}}=\frac{M{I}_{xp2}}{PN{F}_{p2}}$$ Equation 2. Correction for machine drift

3.3. Combination of normalized sample plates

After deriving the corrections for background fluorescence and machine drift, we combined these corrections to calculate the normalized intensity (NI) of each sample. The NI of a sample ($\mathrm{x}$) run on plate 1 is derived by subtracting the background fluorescence from the measured intensity of the sample and then dividing by the PNF (Eq. 3). By integrating these correction steps, we achieve a comprehensive normalization process that enhances the accuracy, sensitivity, and reproducibility of Luminex assay data.

$$N{I}_x=\frac{M{I}_{xp1}-{\mathit{background}}_x}{PN{F}_{p1}}$$ Equation 3. Full normalization

3.4. Clustering of binary data

In Luminex assays, the distribution of measured fluorescence intensities can vary depending on the target antigen. While some exhibit a normal distribution, other measures are binary and will demonstrate a bimodal distribution. This bimodality indicates the presence of distinct positive and negative populations within the data set. Binary measures require a precise determination of the fluorescence threshold that separates the positive and negative populations (Sharma and Jain, 2014). Conventional methods that arbitrarily split the data into halves or tertiles lead to oversimplification resulting in significant misclassification, reducing the accuracy and reliability of the results. To address these limitations, we use the Mclust clustering method, which leverages the true distribution of the data to determine an optimal cut point. Mclust is a model-based clustering algorithm that uses Gaussian finite mixture models fitted by the Expectation-Maximization (EM) algorithm to identify clusters within the data (Eckels et al., 2013). Once the Mclust algorithm has been applied, each data point is assigned to a specific cluster. For binary data, each data point will be assigned to a positive or negative node. By using a clustering approach, a more accurate and data-driven determination of the cut point may be achieved, improving the sensitivity and reproducibility of the assay.

4. Results

4.1. Deriving background fluorescence by fitting negative control to the blank

To accurately correct for background fluorescence in Luminex assays, we applied an orthogonal regression model to fit the measured fluorescence of the negative control bead to the blank bead. This approach allows us to account for the variability and non-specific binding inherent in the blank, thereby improving the precision of the background correction. To estimate the intensity of the background fluorescence, we plotted the relationship between the fluorescence intensity of the negative control and the blank and fit an orthogonal regression line (Fig. 1A). The regression computes subtle shifts in the intensities of both the negative and the blank, resulting in a fitted value on the derived regression line. Since we utilized three negative controls, the median fitted value of the three was used as the calculated background, and the consistency across these measurements confirms the reliability of the derived background fluorescence (Fig. 1B). Further, we demonstrated the practical benefit of this approach by showing the improvement in the signal-to-noise ratio (SNR) of a target antigen after subtracting the derived background fluorescence from the measure intensity (Fig. 1C). The significant increase in SNR indicated that removing the background fluorescence allows for better isolation of the true fluorescent signal of the target antigen, leading to more precise quantification.

Fig. 1.

Derivation of background fluorescence by orthogonal regression. (A) Plot of initial negative bead intensity vs blank bead intensity with the orthogonal regression line and the fitted negative values. (B) Orthogonal regression line of each negative control with the derived background. (C) Comparison of the SNR in the raw data compared to the background corrected data. (D) Comparison of the CV and the SNR of blank subtracted data compared to derived background corrected data. p-values were calculated using a paired two-tailed t-test in R at the antigen level (n = 4), with each value derived from an aggregate of samples (Flu: n = 26, HCOV: n = 44, EBV: n = 41, CMV: n = 54).

We also compared two different subtraction methods: the subtraction of the measured fluorescence of the blank bead to the orthogonal regression-derived background fluorescence. The result showed a decrease in the coefficient of variance (CV) and a corresponding increase in the SNR when using the orthogonal regression-derived background (Fig. 1D). This demonstrates that our method not only reduces variability but also enhances the overall quality of the fluorescence data when compared with current methods. These results demonstrate that applying an orthogonal regression to fit the negative control to the blank effectively reduces noise and variability in the data, thereby improving the sensitivity and specificity of Luminex assay measurements.

4.2. Machine drift correction by deriving a PNF

Correcting for machine drift is critical to ensuring the reliability and reproducibility of the data collected from different plates in Luminex assays. Here, we corrected for machine drift by deriving a PNF by fitting a GAM to the standard curve of each plate, allowing for accurate normalization of sample intensities. The standard curves of each plate revealed notable difference at the experimental sample dilution, highlighting the variability introduced by machine drift (Fig. 2A). By comparing the mean difference in fluorescence intensity between plates, we demonstrate the discrepancies introduced (Fig. 2B). To address this issue, we employed a GAM to fit a smooth curve to the standard curve. The GAM produced a continuous and flexible fit that precisely captured the non-linear relationship between analyte concentration and fluorescence intensity, while the linear model often failed to fit the true data (Fig. 2C). The effectiveness of the GAM-based normalization was further demonstrated by comparing the mean difference in fluorescence intensity between the two plates after correcting for machine drift using the linear or GAM models. The GAM-corrected samples had a significantly reduced mean difference between the plates compared to the linear model, indicating a more effective correction for machine drift (Fig. 2D). This improvement in consistency across plates suggests that deriving a PNF using a GAM provides a more accurate and reliable method for correcting machine drift in Luminex assays.

Fig. 2.

Derivation of PNF to correct for machine drift. (A) Comparison of standard curve and GAM curve between each plate. (B) Comparison of mean difference between plates without a machine drift correction and correction with GAM-derived PNF. (C) Comparison of GAM curve and linear model on standard curve. (D) Comparison of mean difference between plates with GAM-derived PNF and linear model-derived PNF. p-values were calculated using a paired two-tailed t-test in R at the antigen level (n = 4), with each value derived from an aggregate of samples (Flu: n = 26, HCOV: n = 44, EBV: n = 41, CMV: n = 54).

4.3. Splitting binary data with a clustering algorithm

In the context of Luminex assays, accurately distinguishing between positive and negative populations within binary measures is crucial for reliable data interpretation. Traditional methods of splitting data, such as dividing the data at the midpoint, often fail to reflect the true distribution, leading to substantially decreased accuracy and sensitivity. To address this problem, we employed a clustering algorithm to optimally split the binary data based on the actual distribution. The data measured the presence of antibodies in human samples and is therefore binary in that certain individuals will have produced reactive antibody and others will have never been exposed to the target antigen and are therefore antibody negative. The fully normalized data set demonstrates the bimodal distribution of binary data (Fig. 3A). By applying a clustering algorithm, we were able to split the data into positive and negative nodes that correspond with the initial data distribution (Fig. 3B). In contrast, splitting the data in half resulted in arbitrary cut point placing the division in the middle of a distribution, leading to significant overlap and misclassification between positive and negative nodes (Fig. 3C).

Fig. 3.

Dividing data into positive and negative nodes with clustering or traditionally used half-split. Distribution of fully normalized data (A) with overlaid distribution of positive and negative nodes split by clustering (B) or halving the data (C). (D) Comparison of KL divergence in half-split or cluster-split data. p-values were calculated using a paired two-tailed t-test in R at the antigen level (n = 4), with each value derived from an aggregate of samples (Flu: n = 26, HCOV: n = 44, EBV: n = 41, CMV: n = 54).

To quantitatively demonstrate the improved fit of the clustering method, we compared the Kullback-Leibler (KL) divergence between the cluster-split and half-split data. The KL divergence measures the difference between two probability distributions, with lower values indicating a closer fit to the true distribution. Our results showed a significant decrease in divergence for the cluster-split data compared to the half-split data, underscoring that the clustering algorithm better fits the data and accurately reflects the true population distribution (Fig. 3D). The use of a clustering algorithm to split binary data in Luminex assays significantly improves the accuracy and sensitivity of the classification, leading to reduced misclassification and enhanced reliability and reproducibility.

5. Discussion

The primary aim of this study was to enhance the accuracy, sensitivity, and reproducibility of Luminex assay data through improved methods for background fluorescence correction, machine drift normalization, and binary data splitting. Our application of orthogonal regression to fit the measured fluorescence of the negative control bead to the blank bead significantly improves the precision of background fluorescence correction by reducing noise and enhancing signal clarity compared to the more commonly used blank subtraction. Deriving a PNF from the standard curves with a GAM rather than a linear model increases accuracy and reliability for machine drift corrections. The use of a clustering algorithm to split binary data significantly improved the accuracy and sensitivity of the classification.

Our findings build upon previous research by providing more precise and reliable methods for data normalization and classification in Luminex assays by addressing critical limitations in the current methodologies. These results have broad implications for the field of multiplex immunoassays and other fluorescence-based assays and will benefit a wide range of applications, from biomarker discovery to clinical diagnostics (Rountree et al., 2024). These advancements will lead to more reliable data and accelerate scientific progress.

We have also designed an easy-to-use web application to allow anybody that performs these assays to implement this normalization strategy. The app was developed using the R Shiny framework and is accessible at http://brianaamicarellaburns.shinyapps.io/luminex_normcut allowing users to interact with and replicate analyses described in this study.

Future research should explore the application of this method to other types of multiplex and fluorescence-based assays. Additionally, integrating these techniques with automated data analysis pipelines could streamline their adoption in high-throughput settings.

While our method improves normalization and classification in Luminex assays, potential limitations should be considered. The method’s performance may be affected by small sample sizes, as clustering and regression-based normalization typically require sufficient data points to achieve robust estimations. Additionally, the approach may be sensitive to high levels of noise in the data, particularly if background fluorescence exhibits unpredictable variability. Future work should assess these limitations by benchmarking performance across datasets of varying sizes and noise levels and developing adaptive strategies, such as robust smoothing approaches, noise filtering methods, or outlier detection, to enhance the method’s robustness.

In summary, this study presents advanced methods for improving the accuracy, sensitivity, and reproducibility of Luminex assay data. By addressing critical limitations in background fluorescence corrections, machine drift normalization, and binary data classification, our approach offers significant advancement for the field. These improvements will enhance the reliability of multiplex immunoassays and other fluorescence-based measures, ultimately contributing to better scientific outcomes in basic and clinical applications.

Abbreviations:

GAM

generalized additive model

PNF

plate normalization factor

ELISA

enzyme-linked immunosorbent assay

MI

measured intensity

TI

true intensity

NI

normalized intensity

EM

expectation maximization

SNR

signal-to-noise ratio

CV

coefficient of variance

KL

Kullback-Leibler

Data availability

The Shiny app can be accessed at http://brianaamicarellaburns.shinyapps.io/luminex_normcut. The app’s source code is archived on Zenodo (DOI:10.5281/zenodo.1414026727)