A Systematic Framework for Harmonic Set Selection in Multiplexed Magnetic Particle Spectroscopy

Magnetic particle spectroscopy (MPS) has emerged as a powerful biosensing modality due to its high sensitivity, matrix-free detection, and compatibility with point-of-care diagnostics. However, the lack of robust multiplexing capability remains a major bottleneck that limits its broader application in complex bioassays. To address this challenge, spatial and spectral separation strategies have been explored in recent years. While spatial separation approaches enable multiplexing without sacrificing sensitivity, they require physically distinct assay regions and substantially increase assay complexity, cost, and reagent consumption. In contrast, spectral separation offers a more resource-efficient solution by decoding multiple analyte signals from a single MPS measurement, but its performance critically depends on the choice of harmonic components used for decoding. In practical multiplexed MPS bioassays, numerical instability in solving harmonic equations directly translates into analyte misquantification and degrades assay performance. Despite significant progress, there is currently no systematic framework to guide the selection of harmonic sets that maximize spectral separation accuracy across different mixture complexities. Herein, we systematically investigate truncated spectral separation across binary, ternary, and quaternary mixtures of distinct magnetic nanoparticle labels by evaluating a wide range of harmonic component combinations under identical experimental conditions. We demonstrate that the absolute determinant of the truncated reference matrix serves as a robust and effective criterion for identifying harmonic sets that yield the lowest decoding error under the evaluated experimental conditions. The proposed harmonic selection framework is demonstrated to be effective across multiple decoding strategies and provides practical design guidance for robust and scalable multiplexed MPS bioassays.