An Inverse Problems Approach to Pulse Wave Analysis in the Human Brain

TL;DR

This paper addresses estimating the cerebral pulse-wave velocity (PWV) and separating forward and backward pulse components from MRI data by formulating pulse-wave splitting as an inverse problem. It introduces a frequency-domain operator framework that links unknown waves $(oot{hat}{p}_{1f},oot{hat}{p}_{Nb},u)$ to observed data $(oot{hat}{p}_k)_{k=1}^N$, and develops two regularized reconstruction strategies: linear Tikhonov with a known PWV $u$ and a joint estimation via a finite PWV grid, enhanced by an Alternate Direction Method to iteratively update waveform pieces and $u$. Numerical experiments on simulated data demonstrate robust PWV recovery, especially with $N\ge 3$ and appropriate regularization, while ADM shows sensitivity to initialization; a real MRI dataset yields a plausible PWV estimate ($u_{rec}\approx7.3$ m/s) and physically consistent split waves. The approach offers a mathematically grounded, noninvasive framework for PWV imaging in the brain, with implications for studying cerebrovascular pulsatility and its links to aging and neurodegenerative processes, and it highlights the need for higher data resolution or extended vessel segments to improve robustness at higher PWVs.

Abstract

Cardiac pulsations in the human brain have received recent interest due to their possible role in the pathogenesis of neurodegenerative diseases. Further interest stems from their possible application as an endogenous signal source that can be utilized for brain imaging in general. The (pulse-)wave describing the blood flow velocity along an intracranial artery consists of a forward (anterograde) and a backward (retrograde, reflected) part, but measurements of this wave usually consist of a superposition of these components. In this paper, we provide a mathematical framework for the inverse problem of estimating the pulse wave velocity, as well as the forward and backward component of the pulse wave separately from MRI measurements on intracranial arteries. After a mathematical analysis of this problem, we consider possible reconstruction approaches, and derive an alternate direction approach for its solution. The resulting methods provide estimates for anterograde/retrograde wave forms and the pulse wave velocity under specified assumptions on a cerebrovascular model system. Numerical experiments on simulated and experimental data demonstrate the applicability and preliminary in vivo feasibility of the proposed methods.

Figures (11)

• Figure 1.1: High resolution angiogram (left), tracked artery in 4D Flow MRI images (middle), and averaged pulse waves over color-coded voxels (right).
• Figure 2.1: Illustration of summation (left) and delay (right) equations \ref{['eq_summation']} and \ref{['eq_delay']}.
• Figure 4.1: Simulation data for $N=3$. Split waves $p_{1f},p_{2f},p_{3f},p_{1b},p_{2b},p_{3b}$ (left) and total waves $p_1,p_2,p_3$ at each measurement point with added noise level 5% (right).
• Figure 4.2: Results of the linTikh method \ref{['linTikh']} for varying regularization parameter $\alpha$ and smoothing parameter $r$. The relative noise level $\delta=5\%$ and the number of measurement points $N=2$ are fixed in all graphs.
• Figure 4.3: Quality measure $e_{res}$ for the minTikh method \ref{['minTikh']} with varying number of measurement points, compared to the direct regularization approach with hard thresholding \ref{['eq_direct']},\ref{['eq_directreg']} (top, left). Influence of the regularization parameter on reconstruction quality and reconstructed PWV for the minTikh method \ref{['minTikh']} (top, right) and ADM method \ref{['adm']} with initial value $u^0=10 \, \text{m/s}$ (bottom, left) or $u^0=4\, \text{m/s}$ (bottom, right). All experiments have a relative noise level of $\delta=5\%$, and all but the top left have a fixed number of $N=3$ measurement points.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Definition 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Proposition 2.5