Background

Systemic lupus erythematosus (SLE) disease trajectories are governed by nonlinear immune dynamics that resist classical linear forecasting. Serial autoantibody panels (anti-dsDNA, anti-Smith, anti-RNP, anti-ribosomal P, anti-C1q) evolve through complex, interdependent trajectories with apparent chaotic behavior — sensitivity to initial conditions, strange attractors in cytokine–autoantibody phase space, and abrupt regime transitions preceding clinical flares.

The Koopman operator framework offers a powerful resolution: while the underlying dynamics are nonlinear, the Koopman operator acts on observables (functions of the state) and is linear, albeit infinite-dimensional. Its spectral decomposition — eigenvalues, eigenfunctions, and modes — captures the full nonlinear dynamics in a linear embedding without linearization approximations.

Hypothesis

We hypothesize that Extended Dynamic Mode Decomposition (EDMD) applied to serial multi-autoantibody panels (≥6 time points per patient, monthly intervals) will identify a finite-rank Koopman operator approximation whose spectral properties:

1. Eigenvalue magnitudes >1 identify patients in unstable immune trajectories progressing toward flare (SLEDAI increase ≥4) within 6 months, with AUROC >0.82
2. Koopman eigenfunctions define a low-dimensional linear embedding (≤5 dimensions) that separates flare-bound from stable trajectories with greater discriminative power than PCA or UMAP on the same data
3. Koopman mode amplitudes quantify the contribution of each autoantibody species to dominant dynamic modes, revealing which serological markers drive trajectory instability in individual patients

Methodology

• Observables: Lift raw autoantibody titers into a nonlinear dictionary (radial basis functions + time-delay embeddings, dictionary size ~50–100)
• EDMD: Compute finite-dimensional Koopman approximation from sequential snapshots using regularized least-squares
• Spectral analysis: Extract eigenvalues (stability), eigenfunctions (coordinates), and modes (autoantibody contributions)
• Validation: 10-fold cross-validation on retrospective lupus cohort (n≥300), with prospective validation on held-out cohort (n≥100)
• Comparisons: Benchmark against LSTM, Gaussian process, and standard linear models on identical input data

Testable Predictions

1. Koopman eigenvalues with |λ| > 1.0 will be present in ≥75% of patients who flare within 6 months and ≤15% of stable patients
2. The linear Koopman embedding will achieve higher silhouette scores for flare/stable clustering than nonlinear dimensionality reduction (UMAP, t-SNE)
3. Anti-C1q and anti-dsDNA will dominate the leading Koopman modes (highest mode amplitudes) in renal flare trajectories, while anti-ribosomal P will dominate neuropsychiatric flare modes
4. Prediction performance will plateau at dictionary sizes ~60–80, indicating a low intrinsic dimensionality of lupus immune dynamics

Limitations

• EDMD requires regularly sampled time series; missing visits necessitate imputation, introducing bias
• Dictionary choice (RBF kernels, polynomial, time-delay) affects the quality of the finite-dimensional approximation
• The infinite-dimensional Koopman operator is approximated by finite rank — residual spectrum effects may degrade predictions in highly chaotic regimes
• Cohort size requirements (≥300 with ≥6 serial panels each) limit feasibility to large lupus registries
• Autoantibody assay variability across laboratories may affect eigenvalue stability; harmonization protocols are essential

Clinical Significance

If validated, this framework would provide a principled mathematical bridge between nonlinear immune dynamics and clinically actionable linear predictions. Unlike black-box deep learning, Koopman spectral decomposition is interpretable — eigenvalues quantify stability, eigenfunctions define coordinates, and modes identify which biomarkers drive instability. This enables personalized flare risk stratification with mechanistic transparency, supporting treat-to-target strategies and preemptive immunosuppression titration.

The approach also establishes a template for applying operator-theoretic methods to other autoimmune diseases with complex, nonlinear trajectories — systemic sclerosis, vasculitis, and inflammatory myopathies.

RheumaAI Research • rheumai.xyz • DeSci Rheumatology