Nkisi-Mbangu: Memory Modelling of the Arterial Pulse Wave (APW) Using the Atangana-Baleanu-Caputo Fractional Windkessel Model (W-ABC)

Background: This article introduces and validates the Fractional Windkessel (W-ABC) model for accurate computational reproduction of the arterial pulse wave (APW). Accurate modelling of the APW is crucial for non-invasive assessment of cardiovascular function. Classic integer-order Windkessel models (WK2/WK3/WK4) systematically fail to reproduce the non-exponential diastolic decay of the APW, a signature of arterial viscoelasticity. Objective: To formulate, implement, and validate the W-ABC model—a fractional Windkessel model using the Atangana-Baleanu Caputo (ABC) operator—for high-fidelity computational synthesis of the APW, including systolic rise, heavy-tailed diastolic decay, and dicrotic notch. Methods: The W-ABC integro-differential equation is solved numerically via a second-order Adams-Bashforth scheme (Δt = 1 ms). The cardiac input Q_in(t) is modelled by a gamma-variate function; the dicrotic notch is a localised Gaussian perturbation I^d(t). Parameters (R, C, α, D, t^d, σ) were calibrated by Levenberg-Marquardt nonlinear least-squares minimisation against a synthetic reference APW (normotensive adult, 60 bpm, 500-point window, noise-free). Results: At α = 0.85, absolute diastolic RMSE decreased from 3.2 mmHg (α = 1, WK2 baseline) to 1.9 mmHg (W-ABC), representing a 40% reduction. Maximum systolic slope error remained <5% for both models. The dicrotic notch was accurately reproduced by the Gaussian term. Conclusion: The W-ABC model provides a mathematically rigorous, physiologically interpretable framework for APW synthesis. The fractional order α constitutes a candidate non-invasive biomarker of arterial viscoelasticity, pending prospective clinical validation.