# American Institute of Mathematical Sciences

January  2018, 12(1): 175-204. doi: 10.3934/ipi.2018007

## 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

Received  October 2016 Revised  September 2017 Published  December 2017

Fund Project: 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

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.

Citation: Simon Hubmer, Andreas Neubauer, Ronny Ramlau, Henning U. Voss. On the parameter estimation problem of magnetic resonance advection imaging. Inverse Problems & Imaging, 2018, 12 (1) : 175-204. doi: 10.3934/ipi.2018007
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##### References:
Example image of a clinical MRI scanner
Simulation phantom: Magnitude of the norm of the velocity vector field (left figure) and colour direction MIP of the velocity (right figure)
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)
Magnitudes of the velocity vector field components. Left: First component. Middle: Second component. Right: Third component
Result of the algorithm applied to the test problem ($\delta =1%$), using no additional options. Velocity norm MIP (left) and colour direction MIP (right)
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)
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)
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)
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)
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)
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
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
Result of the algorithm applied to the modified problem ($\delta =1%$), where all involved velocities were multiplied by a factor of ^4$with initial signal (58), using the weak divergence-free and the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right) Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 16 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right) Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 2 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right) Comparison of the results of the reconstruction algorithm applied to the test problem ($\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}}}$Figure 4 no no no 90 16.9658 Figure 5 yes no no 126 9.2904 Figure 6 no yes no 121 16.4324 Figure 7 yes yes no 162 8.8878 Figure 8 no no yes 71 17.179 Figure 9 no yes yes 100 16.7834 Figure 10 yes no yes 99 5.6577 Figure 11 yes yes yes 138 5.5245  div-free wavelets sparsity$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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