On the parameter estimation problem of magnetic resonance advection imaging

Simon Hubmer; Andreas Neubauer; Ronny Ramlau; Henning U. Voss

doi:10.3934/ipi.2018007

Article Contents

2018, Volume 12, Issue 1: 175-204. Doi: 10.3934/ipi.2018007

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On the parameter estimation problem of magnetic resonance advection imaging

1.
Doctoral Program Computational Mathematics, Johannes Kepler University Linz, Altenberger Strasse 69, A-4040 Linz, Austria
2.
Industrial Mathematics Institute, Johannes Kepler University Linz, Altenberger Strasse 69, A-4040 Linz, Austria
3.
Johann Radon Institute, Altenberger Strasse 69, A-4040 Linz, Austria
4.
Department of Radiology, Weill Cornell Medical College, 516 E 72nd Street New York, NY 10021, USA

* Corresponding author: Simon Hubmer
* Corresponding author: Simon Hubmer

Received: September 30, 2016

Revised: August 31, 2017

Published: January 2018

The first author is funded by the Austrian Science Fund (FWF): W1214-N15, project DK8. The fourth author acknowledges support by the Nancy M. and Samuel C. Fleming Research Scholar Award in Intercampus Collaborations, Cornell University

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Abstract

We present a reconstruction method for estimating the pulse-wave velocity in the brain from dynamic MRI data. The method is based on solving an inverse problem involving an advection equation. A space-time discretization is used and the resulting largescale inverse problem is solved using an accelerated Landweber type gradient method incorporating sparsity constraints and utilizing a wavelet embedding. Numerical example problems and a real-world data test show a significant improvement over the results obtained by the previously used method.

Keywords:

Mathematics Subject Classification: 65M32, 68U10, 92C50.

Citation:

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Figure 1. Example image of a clinical MRI scanner

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Figure 2. Simulation phantom: Magnitude of the norm of the velocity vector field (left figure) and colour direction MIP of the velocity (right figure)

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Figure 12. Result of the algorithm applied to the modified problem ($\delta =1%$), where all involved velocities were multiplied by a factor of $10^4$, using the weak divergence-free, the wavelet embedding and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 3. Magnitudes of the velocity vector field components. Left: First component. Middle: Second component. Right: Third component

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Figure 4. Result of the algorithm applied to the test problem ($\delta =1%$), using no additional options. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 5. Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free option. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 6. Result of the algorithm applied to the test problem ($\delta =1%$), using the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 7. Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free and the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 8. Result of the algorithm applied to the test problem ($\delta =1%$), using the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 9. Result of the algorithm applied to the test problem ($\delta =1%$), using the wavelet embedding and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 10. Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP and colour direction MIP

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Figure 11. Result of the algorithm applied to the test problem ($\delta =1%$), using the weak divergence-free, the wavelet embedding and the sparsity option with $\alpha =10^{-3}$. Velocity norm MIP and colour direction MIP

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Figure 13. Result of the algorithm applied to the modified problem ($\delta =1%$), where all involved velocities were multiplied by a factor of ^4$ with initial signal (58), using the weak divergence-free and the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right)

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Figure 14. Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 16 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right)

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Figure 15. Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 2 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right)

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Table 6.1. Comparison of the results of the reconstruction algorithm applied to the test problem ($\delta =1%$), achieved using combinations of the different computation options

	div-free	wavelets	sparsity	$k_*$	${\left\\| {\left( {\vec v_{{k_}}^\delta ,\vec \rho _{0,{k_}}^\delta } \right) - \left( {{{\vec v}^\dagger },\vec \rho _0^\dagger } \right)} \right\\|_{{\ell _2}}}$
Figure 4	no	no	no	90	16.9658
Figure 5	yes	no	no	126	9.2904
Figure 6	no	yes	no	121	16.4324
Figure 7	yes	yes	no	162	8.8878
Figure 8	no	no	yes	71	17.179
Figure 9	no	yes	yes	100	16.7834
Figure 10	yes	no	yes	99	5.6577
Figure 11	yes	yes	yes	138	5.5245

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