Hypothesis-Based Particle Detection for Accurate Nanoparticle Counting and Digital Diagnostics

TL;DR

This work addresses the challenge of accurately counting nanoparticle reporters in imaging-based digital assays without relying on training data or empirical thresholds. It introduces a physics-grounded, multiple-hypothesis framework that jointly fits particle parameters under a Poisson image model and selects the best count via an information-criterion-like penalty, yielding interpretable outputs tied to imaging physics. Extensive simulations demonstrate robustness to weak signals, variable backgrounds, magnification changes, and moderate PSF mismatch, while experiments on dark-field nanoparticle images and a SARS-CoV-2–like DNA assay reveal statistically significant differences between control and target samples and reveal over-dispersion in count statistics. The method provides a reliable, calibration-free tool for digital molecular diagnostics and offers a principled basis for optimizing acquisition and assay design, with avenues for extension to multiplexed imaging and other modalities.

Abstract

Digital assays represent a shift from traditional diagnostics and enable the precise detection of low-abundance analytes, critical for early disease diagnosis and personalized medicine, through discrete counting of biomolecular reporters. Within this paradigm, we present a particle counting algorithm for nanoparticle based imaging assays, formulated as a multiple-hypothesis statistical test under an explicit image-formation model and evaluated using a penalized likelihood rule. In contrast to thresholding or machine learning methods, this approach requires no training data or empirical parameter tuning, and its outputs remain interpretable through direct links to imaging physics and statistical decision theory. Through numerical simulations we demonstrate robust count accuracy across weak signals, variable backgrounds, magnification changes and moderate PSF mismatch. Particle resolvability tests further reveal characteristic error modes, including under-counting at very small separations and localized over-counting near the resolution limit. Practically, we also confirm the algorithm's utility, through application to experimental dark-field images comprising a nanoparticle-based assay for detection of DNA biomarkers derived from SARS-CoV-2. Statistically significant differences in particle count distributions are observed between control and positive samples. Full count statistics obtained further exhibit consistent over-dispersion, and provide insight into non-specific and target-induced particle aggregation. These results establish our method as a reliable framework for nanoparticle-based detection assays in digital molecular diagnostics.

Figures (5)

• Figure 1: Workflow of the particle detection algorithm. The input image is tested against multiple hypotheses ($H_0, H_1, \ldots, H_{n_{\text{max}}}$), corresponding to 0 to $n_{\text{max}}$ particles, each modeled as a sum of Gaussian PSFs on a uniform background. For each hypothesis $H_n$, the algorithm determines the model parameters by maximum likelihood estimation, computes a penalized log-likelihood ($\ell_p$) and Fisher information matrix ($\mathbb{M}$) under the assumption of Poisson noise. The corresponding scores $\xi_n = \ell_p - \tfrac{1}{2} \log\det(\mathbb{M})$ are then calculated. The hypothesis with the highest $\xi_n$ is selected, yielding the estimated particle count $\hat{N}$, background level, particle positions and scattering intensities.
• Figure 2: Baseline confusion matrix. Confusion matrix for the particle count estimation algorithm under baseline simulation conditions (Table \ref{['tab:baseline']}), illustrating the probability of the estimated count $\hat{N}$ (columns) given the true count $N$ (rows). Strong diagonal dominance is evident reflecting minimal over- or under-counting.
• Figure 3: Weighted accuracy under simulated conditions. Weighted accuracy (solid lines), over-count rate (dashed) and under-count rate (dotted) versus (a) particle scatterong strength, (b) background intensity, (c) relative SNR at fixed SBR, (d) magnification (type I), where both signal and background scale as $1/\text{magnification}^2$ (optical stray-light–dominated), (e) magnification (type II), where background remains constant (sensor/electronic-noise–dominated), and (f) the ratio of the assumed PSF width used in MLE to the true PSF width. Each panel shows results for average particle densities per image ($\bar{N}$) of 0.25 (blue), 0.5 (orange), and 1.0 (purple). Each point aggregates outcomes from 10,000 test images per true count (0--4). Baseline conditions are given in Table \ref{['tab:baseline']} and depicted by vertical dashed gray lines. Hypotheses up to five particles were evaluated in all cases.
• Figure 4: Two-particle resolution. Stacked plot for the fraction of estimation outcomes $\hat{N}$ for a true two-particle image ($N=2$), as a function of normalized separation for PSF widths of $\sigma = 1$ (top panel), $\sqrt{2}, 2, 2\sqrt{2}, 4$ and $4\sqrt{2}$ pixels (bottom), each aggregated over 10,000 simulations. At large separations, correct two-particle estimates dominate. At small separations, under-counting is the primary error mode. Over-counting arises only near the transition region. Example particle images at the resolution limit (vertical dashed line corresponding to the fraction of $\hat{N}=2$ estimates falling to 50%), for each PSF width are also shown. White arrows on inset denote corresponding threshold separation.
• Figure 5: Nanoparticle imaging based SARS-CoV-2 assay. (a) Schematic of nanoparticle functionalisation (see Supplementary Information) and target induced nanoparticle clustering. (b) Example dark-field images of deposited nanoparticles for different nanoparticle concentrations. For each concentration, images shown correspond to a single $50\times 50$ pixel sub-region input into the particle counting algorithm. High nanoparticle concentrations (13.2 pM), however can frequently produce high numbers of particles ($>n_{\text{max}}$) in a given sub-image (orange dashed bounding box), such that smaller $35\times35$ pixel regions were used (blue dashed bounding box). (c)-(e) Distribution of estimated particle counts across all sub-images, for both control (blue markers) and SARS-CoV-2 ('covid') derived biomarkers positive (orange markers) samples, overlaid with GPD fits (solid curves). Error bands correspond to inter-image standard deviations. (f) Fitted density parameters (per $50\times50$ pixel region) for different nanoparticle concentrations as found from GPD fitting (note, the 13.2 pM density was rescaled) shown with corresponding linear fits (dashed curves).