Disease Is a Spectral Perturbation

TL;DR

This paper introduces a spectral perturbation framework to analyze disease transformation from healthy states using biomarker covariance matrices, enabling mechanistic insights and improved prognosis.

Contribution

It formalizes a spectral method to characterize disease-induced changes in biomarker coordination, providing a new explainability and prognostic tool.

Findings

Eigenvectors define normal modes of biomarker coordination.

Eigenvalues quantify energy in each mode.

Projection onto disease eigenmodes improves prognosis accuracy.

Abstract

We propose a novel method of understanding disease transformation from a healthy baseline with biomarker-level explainability. By modeling the biomarker covariance matrices of healthy controls and disease states, the perturbation can be individually characterized to accomplish mechanistic explanations of disease trajectories, both at a molecular level and for individual patients. Given a cohort of n patients each measured on p biomarkers, we define the biomarker "Hamiltonian" H = X^T X / n \in R^{p \times p}, where X \in R^{n \times p} is the covariant biomarker matrix. The eigenvectors of H define a set of normal modes of biomarker coordination, and the eigenvalues quantify the energy carried by each mode. In the healthy state, the reference Hamiltonian H_0 governs this structure where disease perturbs H_0 by an additive operator \Delta H, thus shifting eigenvalues and rotating eigenvectors in proportion to the severity of pathological disruption. We formalize this framework, derive the spectral change given a disease perturbation, and demonstrate that the projection of a newly diagnosed patient's cumulative biomarker covariance structure onto disease-discriminant eigenmodes constitutes an optimal prognostic statistic for greater precision in disease prognosis. This work serves as a veritable white paper with application across a panoply of disease frameworks from cancer to neurodegenerative disorders.