Theoretical derivation of blood velocity from TOF-MRA based artery centerline

Time-of-flight magnetic resonance angiography (TOF-MRA) is widely used for noninvasive visualization of arterial anatomy, but extracting hemodynamics like blood velocity typically requires supplementary phase-contrast scans, tagging or multi-TE images. This study proposes a novel, physics-informed computational framework to extract variable fluid velocity directly from standard TOF-MRA signal profiles. We analytically expand the approach-to-steady-state Bloch equations to include convective flow, establishing a mathematical relationship between the spatial decay of longitudinal magnetization and fluid velocity. The velocity derivation was further extended to pointwise estimation over a 1-D centerline, overcoming the limitations of constant-velocity assumptions. To validate and solve this problem, a MATLAB (R2025b) simulation framework was developed to model fluid flow in two variable-geometry flowing tube cases, i.e., continuous narrowing and focal stenosis, under synthetic scanner noise. A global inverse optimization approach utilizing Dual-Tikhonov regularization was applied to stably invert the ill-posed transit time integral, actively penalizing high-frequency numerical ringing while preserving structural curves. The computational simulations successfully recovered ground-truth point-wise velocities, tracking gradual hemodynamic accelerations and sharp stenotic jets. This theoretical framework and the example centerline TOF-MRA signal intensity provide a robust mathematical proof-of-concept that quantitative, localized functional hemodynamic metrics can be extracted from standard structural MRA imaging, establishing a foundation for advanced flow quantification without requiring additional scan time.