Internship: Pressure gradient drive, flow, shear, and wall motion from Doppler holography of retinal vessels using moving-wall Womersley model inversion at cardiac harmonics

We introduce a point–spread–function (PSF)–aware moving–wall Womersley
(WMW) inversion that turns retinal Doppler holography (DH) into compact,
phase-referenced readouts of flow, shear, wall motion, and impedance-like behaviour,
\emph{without} explicit wall segmentation or pulse–wave–velocity measurements.
For each cardiac harmonic \(n\), the complex velocity profile \(V_n(r)\) is modeled
as the PSF-convolved superposition of the rigid-wall Womersley basis \(B_n\) and
its log-radius sensitivity \(\Psi_n = \partial B_n/\partial\ln R\). A weighted, complex
least-squares (LS) step estimates the \emph{drive} and \emph{moving-wall} gains
\((C_n,D_n)\) at fixed \((R_0,\nu)\), while a low-dimensional outer
Levenberg–Marquardt (LM) loop refines \((R_0,\nu)\) (and optionally PSF
meta-parameters) under explicit conditioning and adequacy criteria. Within a
fixed numerical aperture, DH measures RMS velocities from angle-averaged Doppler
power; consequently, the reconstructed amplitudes \(C_n\), and hence \(Q_n\) and
\(\tau_n\), do not require per-vessel cosine correction. From \((R_0,\nu,C_n,D_n)\) we derive volumetric flow \(Q_n\), wall shear
\(\tau_n\), fractional wall kinematics \(R_n/R_0 \simeq D_n/C_n\), a per-length
impedance-like ratio \(\tilde H_{Z,n} = G_n/Q_n\) with \(G_n = i\omega_n\rho C_n\),
and harmonic AC energetics. The pipeline is PSF-aware, conditioning-controlled,
and rigorously QC-gated, with uncertainty quantified by cycle block bootstrap and
axial median pooling. Angle-robust phases and flow-referenced quantities
\(\tilde H_{Z,n}\), \(H_{R|Q}(\omega_n)\), \(H_{\tau|Q}(\omega_n)\) are strong candidates
for trial-ready, multi-site endpoints in retinal microvascular cohorts and
provocation protocols.