Disease Is a Spectral Perturbation

John D. Mayfield; Matthew S. Rosen

arxiv: 2605.02949 · v1 · submitted 2026-05-01 · 💻 cs.LG · stat.ML

Disease Is a Spectral Perturbation

John D. Mayfield , Matthew S. Rosen This is my paper

Pith reviewed 2026-05-09 20:07 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords biomarker covariancedisease perturbationHamiltonianeigenmodesprognostic statisticspectral analysiscovariance matrix

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The pith

The projection of a newly diagnosed patient's biomarker covariance onto disease-discriminant eigenmodes constitutes an optimal prognostic statistic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper frames disease as an additive perturbation to the healthy biomarker covariance matrix, called the Hamiltonian. This perturbation shifts eigenvalues and rotates eigenvectors that represent normal coordination patterns among biomarkers. The central tool projects an individual patient's covariance structure onto the modes that best separate disease from health. If the model holds, this yields more precise forecasts of disease course while supplying mechanistic detail at both molecular and patient-specific levels. The approach is offered for use across conditions such as cancer and neurodegeneration.

Core claim

The biomarker Hamiltonian is defined as H = X^T X / n from the data matrix X of n patients and p biomarkers. In the healthy state this matrix is H0; disease acts through an additive operator ΔH that alters eigenvalues and eigenvectors in proportion to severity. The projection of a new patient's cumulative covariance onto the resulting disease-discriminant eigenmodes is presented as the optimal statistic for greater precision in prognosis.

What carries the argument

The biomarker Hamiltonian, defined as the covariance matrix of measured biomarkers, whose eigen-decomposition supplies normal modes that disease perturbs additively via ΔH.

If this is right

Individual patients receive mechanistic explanations through their specific disease perturbation ΔH.
Eigenvalues shift and eigenvectors rotate in proportion to the severity of pathological disruption.
The same framework supplies prognostic statistics across multiple disease types including cancer and neurodegeneration.
Disease trajectories can be tracked at the level of biomarker coordination rather than isolated marker values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Monitoring how ΔH shrinks under treatment could serve as a dynamic marker of therapeutic response.
The same covariance-perturbation approach might extend to other high-dimensional biological measurements such as genomic or imaging profiles.
Disease classification could shift from single-marker thresholds toward spectral signatures derived from the full covariance structure.

Load-bearing premise

That the covariance matrix of biomarkers and its eigen-decomposition meaningfully encode disease mechanisms and that an additive perturbation ΔH accurately captures how disease alters those mechanisms in real patients.

What would settle it

A validation cohort in which the proposed projection statistic shows no gain in prognostic accuracy over conventional clinical scores or single-biomarker models, or where observed patient data deviate systematically from the additive perturbation assumption.

Figures

Figures reproduced from arXiv: 2605.02949 by John D. Mayfield, Matthew S. Rosen.

**Figure 1.** Figure 1: Spectral signatures of four biologic terrain states across 36 biomarkers. Each row represents a distinct covariance regime visualized across four panels. Left: Three-dimensional terrain of the 36×36 biomarker covariance matrix, where surface height encodes covariance magnitude. Center-left: Two-dimensional heatmap of the same covariance matrix (Magma colormap; diagonal fixed at 1.0). Center-right: Top 10 e… view at source ↗

**Figure 2.** Figure 2: From multimodal biomarkers to latent disease geometry via Hamiltonian spectral decomposition. (A) Raw biomarker space. Ten-dimensional multimodal biomarker trajectories for Group A (progressive disease, red) and Group B (stable, blue), shown projected onto three representative biomarker axes. The two groups are indistinguishable in the original measurement space. (B) Hamiltonian eigenvector frame. The biom… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a conceptual proposal framing disease as an additive spectral perturbation on biomarker covariance matrices, but the derivations and validation are missing so the optimality claim is hard to evaluate.

read the letter

The main takeaway is that the paper offers a Hamiltonian-style model where healthy biomarker covariances define normal modes and disease acts as an additive ΔH that rotates eigenvectors and shifts eigenvalues. The central assertion is that projecting a new patient's covariance onto the resulting disease-discriminant modes gives an optimal prognostic statistic. That framing is the only real novelty here; the underlying use of covariance matrices and eigen-decomposition for classification is already standard in the field, though the explicit normal-mode language is a fresh analogy. The paper does a reasonable job sketching how this could supply mechanistic, patient-level explanations across conditions like cancer or neurodegeneration. The write-up is clear enough on the high-level idea. The soft spots are more substantial. The abstract promises a derivation of the spectral change and a demonstration of optimality, yet neither appears in the supplied text. The optimality claim looks circular because the modes are built from the same disease data used to define the projection. The additive perturbation assumption is stated without justification or any check against the nonlinear, threshold, and feedback effects common in real biomarker networks. No data, synthetic tests, error bars, or comparisons to existing prognostic tools are provided, so the practical advantage remains unshown. This is the sort of paper that might interest a reading group working on physics-inspired methods in health data, but it is not yet a finished empirical or theoretical contribution. I would send it for peer review. The analogy is worth letting referees pressure the authors to supply the missing steps and some validation experiments; with that work it could become a useful modeling direction.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes modeling disease transformation via an additive perturbation ΔH to the healthy biomarker covariance 'Hamiltonian' H0 = X^T X / n. Eigenvectors of H0 represent normal modes of biomarker coordination; disease induces eigenvalue shifts and eigenvector rotations. The projection of a newly diagnosed patient's cumulative biomarker covariance onto the resulting disease-discriminant eigenmodes is claimed to constitute an optimal prognostic statistic, providing mechanistic explanations and improved precision across diseases from cancer to neurodegeneration.

Significance. If the additive perturbation model and optimality of the projection can be rigorously derived and validated, the framework could supply a mathematically grounded, biomarker-level approach to disease mechanism analysis and individualized prognosis. The absence of any empirical results, error analysis, or benchmarks in the current text limits immediate impact, but the conceptual framing has potential relevance for explainable ML in biomedicine.

major comments (3)

[Abstract] Abstract: the claim that the projection 'constitutes an optimal prognostic statistic' is presented without derivation, external benchmark, or comparison to existing prognostic methods; optimality appears to follow tautologically from the definition of the disease-discriminant modes themselves.
[Abstract] Abstract: no derivation steps, equations, or proof are supplied for the spectral change induced by the additive operator ΔH, despite the explicit statement that the spectral change is derived; this derivation is load-bearing for all subsequent claims of mechanistic insight and prognostic optimality.
[Abstract] Abstract: the modeling choice that disease perturbs the healthy Gram matrix via an additive operator ΔH is asserted without justification or synthetic test; biomarker networks routinely exhibit multiplicative, threshold, and feedback effects that would produce non-additive covariance changes, and no check is shown that the projection remains superior under such violations.

minor comments (1)

The manuscript describes itself as a 'white paper' yet supplies neither data, error analysis, nor comparison to standard prognostic statistics; this should be stated explicitly in the introduction so readers understand the scope.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major point below and have outlined the specific revisions that will be incorporated into the next version of the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the projection 'constitutes an optimal prognostic statistic' is presented without derivation, external benchmark, or comparison to existing prognostic methods; optimality appears to follow tautologically from the definition of the disease-discriminant modes themselves.

Authors: We agree that the optimality statement requires an explicit derivation rather than appearing definitional. Within the model, the disease-discriminant modes are the eigenvectors most aligned with the perturbation ΔH, and the projection is optimal in the sense that it maximizes the captured variance attributable to the disease-induced spectral shift (a direct consequence of the variational characterization of eigenvectors). However, we acknowledge the absence of external benchmarks or comparisons to existing prognostic methods. In the revision we will add the mathematical derivation of this optimality property and include a dedicated discussion section that explicitly notes the current lack of empirical comparisons and outlines plans for future validation against standard clinical prognostic models. revision: partial
Referee: [Abstract] Abstract: no derivation steps, equations, or proof are supplied for the spectral change induced by the additive operator ΔH, despite the explicit statement that the spectral change is derived; this derivation is load-bearing for all subsequent claims of mechanistic insight and prognostic optimality.

Authors: The referee is correct that the submitted manuscript does not contain the explicit derivation steps or equations for the spectral perturbation, even though the abstract asserts that the spectral change is derived. We will revise the main text to include the full first-order perturbation expansion: the eigenvalue shift Δλ_k = v_k^T ΔH v_k and the eigenvector correction Δv_k = Σ_{m≠k} (v_m^T ΔH v_k)/(λ_k - λ_m) v_m, derived from standard non-degenerate perturbation theory applied to the Gram matrix H_0. These equations will be presented with intermediate steps immediately following the definition of ΔH. revision: yes
Referee: [Abstract] Abstract: the modeling choice that disease perturbs the healthy Gram matrix via an additive operator ΔH is asserted without justification or synthetic test; biomarker networks routinely exhibit multiplicative, threshold, and feedback effects that would produce non-additive covariance changes, and no check is shown that the projection remains superior under such violations.

Authors: We accept that the additive form of ΔH is introduced without sufficient justification or robustness checks. The choice is motivated by the linear-response regime of perturbation theory, which provides a first-order approximation for small pathological changes, but we recognize that biomarker interactions can be multiplicative or threshold-based. In the revised manuscript we will add a paragraph justifying the additive model as the leading term in a Taylor expansion of the covariance operator and include a short synthetic simulation section that compares the prognostic projection under additive versus multiplicative perturbations to demonstrate its relative stability. revision: yes

Circularity Check

1 steps flagged

Optimality of patient projection onto disease-discriminant eigenmodes reduces to construction of those modes from ΔH

specific steps

self definitional [Abstract]
"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"

The disease-discriminant eigenmodes are obtained precisely by diagonalizing the difference induced by the additive operator ΔH between H_0 and the disease Hamiltonian. Projecting a patient's covariance onto these modes therefore extracts, by linear algebra construction, the component aligned with the modeled disease perturbation; labeling this projection 'optimal' for prognosis is tautological within the assumed additive spectral model rather than a separately derived or validated property.

full rationale

The paper's core derivation begins with the standard Gram matrix H = X^T X / n and assumes an additive perturbation ΔH to obtain disease-altered eigenvectors. Standard first-order perturbation theory then yields eigenvalue shifts and eigenvector rotations. The subsequent claim that the projection of a new patient's covariance onto the resulting 'disease-discriminant eigenmodes' is an 'optimal prognostic statistic' is not an independent result; it follows directly from the definition of those modes as the directions of maximal healthy-to-disease difference. No external benchmark, synthetic non-additive test, or comparison to alternative statistics is required for the claim to hold inside the model, producing partial circularity confined to the optimality assertion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the untested premise that biomarker covariance matrices behave like physical Hamiltonians and that disease is well-approximated by a linear additive operator on that matrix. No free parameters are named, but the choice of which eigenmodes count as 'disease-discriminant' is left unspecified and therefore functions as an implicit modeling choice.

axioms (2)

domain assumption Biomarker data can be represented by the matrix H = X^T X / n whose eigenvectors are normal modes of coordination
Invoked in the definition of the healthy reference Hamiltonian H_0 without further justification.
ad hoc to paper Disease perturbs the healthy Hamiltonian by an additive operator ΔH
Central modeling assumption stated directly in the abstract.

invented entities (1)

disease-discriminant eigenmodes no independent evidence
purpose: Directions in biomarker space that separate disease from health for prognosis
Introduced as the target of the patient projection; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5515 in / 1442 out tokens · 34352 ms · 2026-05-09T20:07:03.553632+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 17 canonical work pages

[1]

Hasin et al. (2017). Multi-omics approaches to disease.Genome Biology,18. https://doi.org/10. 1186/s13059-017-1215-1

2017
[2]

Multi-Omics Profiling for Health.Molecular & Cellular Proteomics,22

Babu & Snyder (2023). Multi-Omics Profiling for Health.Molecular & Cellular Proteomics,22. https: //doi.org/10.1016/j.mcpro.2023.100561

work page doi:10.1016/j.mcpro.2023.100561 2023
[3]

Nagin et al. (2018). Group-based multi-trajectory modeling.Statistical Methods in Medical Research,27, 2015–2023.https://doi.org/10.1177/0962280216673085

work page doi:10.1177/0962280216673085 2018
[4]

Hjaltelin et al. (2023). Visualising disease trajectories from population-wide data.Frontiers in Bioinfor- matics,3.https://doi.org/10.3389/fbinf.2023.1112113

work page doi:10.3389/fbinf.2023.1112113 2023
[5]

Proust-Lima et al. (2022). Describing complex disease progression using joint latent class models for multivariate longitudinal markers and clinical endpoints.Statistics in Medicine,42, 3996–4014. https://doi.org/10.1002/sim.9844

work page doi:10.1002/sim.9844 2022
[6]

Jiang et al. (2025). Abstract P1006: Trajectories of Cardiometabolic Diseases and Cancer.Circulation. https://doi.org/10.1161/cir.151.suppl_1.p1006

work page doi:10.1161/cir.151.suppl_1.p1006 2025
[7]

Metwally et al. (2021). Robust identification of temporal biomarkers in longitudinal omics studies. Bioinformatics,38, 3802–3811.https://doi.org/10.1093/bioinformatics/btac403

work page doi:10.1093/bioinformatics/btac403 2021
[8]

Maghsoudi et al. (2022). A comprehensive survey of the approaches for pathway analysis using multi-omics data integration.Briefings in Bioinformatics,23.https://doi.org/10.1093/bib/bbac435

work page doi:10.1093/bib/bbac435 2022
[9]

Nagin et al. (2024). Recent Advances in Group-Based Trajectory Modeling for Clinical Research.Annual Review of Clinical Psychology.https://doi.org/10.1146/annurev-clinpsy-081122-012416

work page doi:10.1146/annurev-clinpsy-081122-012416 2024
[10]

Li et al. (2021). A flexible joint model for multiple longitudinal biomarkers and a time-to-event outcome. Biometrical Journal,63, 1575–1586.https://doi.org/10.1002/bimj.202000085

work page doi:10.1002/bimj.202000085 2021
[11]

Singh (2025). A Multivariate Joint Modeling Framework for Disease Progression in Chronic Illnesses Using Longitudinal Data.International Journal of Scientific Research in Engineering and Management. https://doi.org/10.55041/ijsrem48268

work page doi:10.55041/ijsrem48268 2025
[12]

Morabito et al. (2025). Algorithms and tools for data-driven omics integration to achieve multilayer biolog- ical insights.Journal of Translational Medicine,23. https://doi.org/10.1186/s12967-025-06446-x

work page doi:10.1186/s12967-025-06446-x 2025
[13]

Carrasco-Zanini et al. (2024). Proteomic prediction of diverse incident diseases.The Lancet Digital Health,6, e470–e479.https://doi.org/10.1016/s2589-7500(24)00087-6

work page doi:10.1016/s2589-7500(24)00087-6 2024
[14]

Ouwerkerk et al. (2023). Multiomics Analysis Provides Novel Pathways Related to Progression of Heart Failure.Journal of the American College of Cardiology,82, 1921–1931. https://doi.org/10.1016/j. jacc.2023.08.053 14

work page doi:10.1016/j 2023
[15]

Distribution of eigenvalues for some sets of random matrices.Mathematics of the USSR-Sbornik,1, 457–483

Marˇ cenko & Pastur (1967). Distribution of eigenvalues for some sets of random matrices.Mathematics of the USSR-Sbornik,1, 457–483

1967
[16]

On the distribution of the largest eigenvalue in principal components analysis.Annals of Statistics,29, 295–327

Johnstone (2001). On the distribution of the largest eigenvalue in principal components analysis.Annals of Statistics,29, 295–327

2001
[17]

Baik et al. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.Annals of Probability,33, 1643–1697

2005
[18]

Guo et al. (2025). Multi-omics profiling reveals two distinct trajectories in the progression from mild cognitive impairment to Alzheimer’s disease.Journal of Alzheimer’s Disease,107, 1080–1096. https://doi.org/10.1177/13872877251365210

work page doi:10.1177/13872877251365210 2025
[19]

Huang et al. (2025). Predicting Alzheimer’s disease subtypes and understanding their molecular char- acteristics in living patients with transcriptomic trajectory profiling.Alzheimer’s & Dementia,21. https://doi.org/10.1002/alz.14241

work page doi:10.1002/alz.14241 2025
[20]

Argelaguet et al. (2018). MOFA: a statistical framework for comprehensive integration of multi-modal single-cell data.Molecular Systems Biology,14.https://doi.org/10.15252/msb.20178124

work page doi:10.15252/msb.20178124 2018
[21]

Lock et al. (2013). JIVE: joint and individual variation explained.Annals of Applied Statistics,7, 523–542.https://doi.org/10.1214/12-AOAS597

work page doi:10.1214/12-aoas597 2013
[22]

Watanabe et al. (2023). Multiomic signatures of body mass index identify heterogeneous health pheno- types and responses to a lifestyle intervention.Nature Medicine,29, 996–1008. https://doi.org/10. 1038/s41591-023-02248-0 15

2023

[1] [1]

Hasin et al. (2017). Multi-omics approaches to disease.Genome Biology,18. https://doi.org/10. 1186/s13059-017-1215-1

2017

[2] [2]

Multi-Omics Profiling for Health.Molecular & Cellular Proteomics,22

Babu & Snyder (2023). Multi-Omics Profiling for Health.Molecular & Cellular Proteomics,22. https: //doi.org/10.1016/j.mcpro.2023.100561

work page doi:10.1016/j.mcpro.2023.100561 2023

[3] [3]

Nagin et al. (2018). Group-based multi-trajectory modeling.Statistical Methods in Medical Research,27, 2015–2023.https://doi.org/10.1177/0962280216673085

work page doi:10.1177/0962280216673085 2018

[4] [4]

Hjaltelin et al. (2023). Visualising disease trajectories from population-wide data.Frontiers in Bioinfor- matics,3.https://doi.org/10.3389/fbinf.2023.1112113

work page doi:10.3389/fbinf.2023.1112113 2023

[5] [5]

Proust-Lima et al. (2022). Describing complex disease progression using joint latent class models for multivariate longitudinal markers and clinical endpoints.Statistics in Medicine,42, 3996–4014. https://doi.org/10.1002/sim.9844

work page doi:10.1002/sim.9844 2022

[6] [6]

Jiang et al. (2025). Abstract P1006: Trajectories of Cardiometabolic Diseases and Cancer.Circulation. https://doi.org/10.1161/cir.151.suppl_1.p1006

work page doi:10.1161/cir.151.suppl_1.p1006 2025

[7] [7]

Metwally et al. (2021). Robust identification of temporal biomarkers in longitudinal omics studies. Bioinformatics,38, 3802–3811.https://doi.org/10.1093/bioinformatics/btac403

work page doi:10.1093/bioinformatics/btac403 2021

[8] [8]

Maghsoudi et al. (2022). A comprehensive survey of the approaches for pathway analysis using multi-omics data integration.Briefings in Bioinformatics,23.https://doi.org/10.1093/bib/bbac435

work page doi:10.1093/bib/bbac435 2022

[9] [9]

Nagin et al. (2024). Recent Advances in Group-Based Trajectory Modeling for Clinical Research.Annual Review of Clinical Psychology.https://doi.org/10.1146/annurev-clinpsy-081122-012416

work page doi:10.1146/annurev-clinpsy-081122-012416 2024

[10] [10]

Li et al. (2021). A flexible joint model for multiple longitudinal biomarkers and a time-to-event outcome. Biometrical Journal,63, 1575–1586.https://doi.org/10.1002/bimj.202000085

work page doi:10.1002/bimj.202000085 2021

[11] [11]

Singh (2025). A Multivariate Joint Modeling Framework for Disease Progression in Chronic Illnesses Using Longitudinal Data.International Journal of Scientific Research in Engineering and Management. https://doi.org/10.55041/ijsrem48268

work page doi:10.55041/ijsrem48268 2025

[12] [12]

Morabito et al. (2025). Algorithms and tools for data-driven omics integration to achieve multilayer biolog- ical insights.Journal of Translational Medicine,23. https://doi.org/10.1186/s12967-025-06446-x

work page doi:10.1186/s12967-025-06446-x 2025

[13] [13]

Carrasco-Zanini et al. (2024). Proteomic prediction of diverse incident diseases.The Lancet Digital Health,6, e470–e479.https://doi.org/10.1016/s2589-7500(24)00087-6

work page doi:10.1016/s2589-7500(24)00087-6 2024

[14] [14]

Ouwerkerk et al. (2023). Multiomics Analysis Provides Novel Pathways Related to Progression of Heart Failure.Journal of the American College of Cardiology,82, 1921–1931. https://doi.org/10.1016/j. jacc.2023.08.053 14

work page doi:10.1016/j 2023

[15] [15]

Distribution of eigenvalues for some sets of random matrices.Mathematics of the USSR-Sbornik,1, 457–483

Marˇ cenko & Pastur (1967). Distribution of eigenvalues for some sets of random matrices.Mathematics of the USSR-Sbornik,1, 457–483

1967

[16] [16]

On the distribution of the largest eigenvalue in principal components analysis.Annals of Statistics,29, 295–327

Johnstone (2001). On the distribution of the largest eigenvalue in principal components analysis.Annals of Statistics,29, 295–327

2001

[17] [17]

Baik et al. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.Annals of Probability,33, 1643–1697

2005

[18] [18]

Guo et al. (2025). Multi-omics profiling reveals two distinct trajectories in the progression from mild cognitive impairment to Alzheimer’s disease.Journal of Alzheimer’s Disease,107, 1080–1096. https://doi.org/10.1177/13872877251365210

work page doi:10.1177/13872877251365210 2025

[19] [19]

Huang et al. (2025). Predicting Alzheimer’s disease subtypes and understanding their molecular char- acteristics in living patients with transcriptomic trajectory profiling.Alzheimer’s & Dementia,21. https://doi.org/10.1002/alz.14241

work page doi:10.1002/alz.14241 2025

[20] [20]

Argelaguet et al. (2018). MOFA: a statistical framework for comprehensive integration of multi-modal single-cell data.Molecular Systems Biology,14.https://doi.org/10.15252/msb.20178124

work page doi:10.15252/msb.20178124 2018

[21] [21]

Lock et al. (2013). JIVE: joint and individual variation explained.Annals of Applied Statistics,7, 523–542.https://doi.org/10.1214/12-AOAS597

work page doi:10.1214/12-aoas597 2013

[22] [22]

Watanabe et al. (2023). Multiomic signatures of body mass index identify heterogeneous health pheno- types and responses to a lifestyle intervention.Nature Medicine,29, 996–1008. https://doi.org/10. 1038/s41591-023-02248-0 15

2023