Prior Image based Temporally Constrained Reconstruction for Magnetic Resonance guided HIFU

Jaya Prakash^{1}, Nick Todd^{2}, and Phaneendra K Yalavarthy^{3}

*TCR*: The below
cost function is minimized iteratively for estimating the image m, from
the acquired data d^{1, 2}:

\[m=\min_{m'}\{\lambda \Arrowvert (\Delta m')\Arrowvert_{2}^2 + {\Arrowvert \mathbf{W} \mathbf{F}m'- d\Arrowvert^2_2} \} ,\]

here F is the Fourier Transform, W represents the binary function that represents which phase encoding lines have been acquired, λ is a regularization parameter, and m’ is the image estimate. Aliasing artifacts arise by performing the reconstruction with under-sampled data, these aliasing artifacts are removed by usage of penalty terms (that act like filters) in the TCR algorithm. The reconstruction of the 12x under-sampled 4-D data set of 128 x 108 x 24 voxels (22 slices with 9% oversampling) and 100 iterations takes 54.32 seconds to converge.

*Prior Image based Temporally
Constrained Reconstruction (PITCR)*: The following
cost function is minimized iteratively for estimating the image m, from
the acquired data d^{3}:

\[m=\min_{m'}\{\tilde{\lambda} \left[ \alpha \Arrowvert \triangle (m'-m'_{pr})\Arrowvert_{2}^2 + (1-\alpha) \Arrowvert (\triangle m')\Arrowvert_{2}^2 \right] + {\Arrowvert \mathbf{ F} m'- d \Arrowvert^2_2 \}},\]

here α represents the weighting factor and is set as
0.3 in this work. The prior image (m'_{pr}) is given as:

\[ m'_{pr} = \left\{ \begin{array}{ll} m'_{j+1} & \mbox{if } j > t*, \\ m'_{j-1} & \mbox{if } j < t*. \end{array}\right. ,\]

here t* represents the time point at which maximum temperature arises. The reconstruction of the 12x under-sampled 4-D data set of 128 x 108 x 24 voxels (22 slices with 9% oversampling) and 100 iterations takes 88.48 seconds to converge.

*Testing*. HIFU heating was performed on ex-vivo pork muscle sample, the imaging
parameters for the acquired under-sampled data were: 1.5 x 1.5 x 3.0 mm
resolution, 128 x 108 x 24 imaging matrix (22 slices plus 9% slice
oversampling), TR = 25 ms, TE = 10 ms, EPI Factor = 9, bandwidth = 738
Hz/pixel, flip angle = 20^{o}, 6x under sampling, and time interval of
1.2 seconds per under-sampled time frame. This acquired 6x under-sampled data
was further under-sampled by 50% to show the efficacy of PITCR method.

1. Todd et al. MRM; 71, 1394-1404, 2014.

2. Todd et al. MRM; 67, 724-30, 2012.

3. Prakash et. al. Med. Phys; 42, 6804-6814 2015.

Comparison of
temperature map reconstruction of the standard temporally constrained reconstruction
(TCR) with prior image based temporally constrained reconstruction (PITCR) method.
The sampling used is shown in the parenthesis. The difference between the truth
and TCR, and truth and PITCR is also shown. The plots show the maximum temperature
increase over time with 6X under-sampling and for acquired 6X under-sampled data
further reduced by 50%.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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