Article Contents
Article Contents

# On the parameter estimation problem of magnetic resonance advection imaging

• * Corresponding author: Simon Hubmer

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.

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

 Citation:

• Figure 1.  Example image of a clinical MRI scanner

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

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)

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

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)

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)

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)

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)

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)

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)

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

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

Figure 13.  Result of the algorithm applied to the modified problem ($\delta =1%$), where all involved velocities were multiplied by a factor of ^4$with initial signal (58), using the weak divergence-free and the wavelet embedding option. Velocity norm MIP (left) and colour direction MIP (right) Figure 14. Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 16 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right) Figure 15. Results of our proposed algorithm (upper two figures, 20 seconds of data) and the regression-based algorithm (lower two figures, 15 minutes of data), applied to subject 2 of the data set. Velocity norm MIPs (left) and colour direction MIPs (right) Table 6.1. Comparison of the results of the reconstruction algorithm applied to the test problem ($\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

Figures(15)

Tables(1)