Functional Spectral Imaging by Ultrasound (FSIU): A Spectral-Theoretic Basis for Functional Ultrasound

Abstract

Functional Spectral Imaging (FSI) models image formation as the recovery of tissue surrogates such as density and stiffness from spectral perturbations of a self-adjoint elliptic operator. Rather than relying on reflectivity or relaxation kinetics, FSI tracks shifts of a truncated set of eigenmodes under controlled excitation, providing a non-ionizing and operator-theoretic route to contrast. Tissue heterogeneity is modeled as a small perturbation of L = -div(D grad) + gamma, with first-order Hadamard formulas linking local contrasts to eigenvalue shifts. Frechet derivatives and their adjoints yield gradients for variational inversion, stabilized by Tikhonov or total-variation regularization and modal truncation. Finite-element simulations show submillimetric localization (about 0.1-0.3 mm) and milligram-scale detectability (thresholds near 1 mg) under ideal noise. Retaining 10-15 modes preserves about 85 percent of anomaly contrast while suppressing noise. A spectral-entropy index separates compact from diffuse inclusions and acts as a morphology surrogate. FSI thus provides a mathematically controlled, non-ionizing framework for localized functional imaging, motivating validation in physical phantoms and in vivo studies.