MR thermometry requires estimation of accurate temperature maps with high spatio-temporal resolution with reduced data collection time. Thus, recent studies have focused on using under-sampled data for estimating the temperature map with constrained reconstruction techniques. The temporally constrained reconstruction (TCR) based scheme applies a temporal constraint in the dynamic imaging setting, but is not suitable when the needed data optimization is of the order of 12x. In this work, we present an approach which applies a prior image constraint based on the previous gradient iteration step for performing the image reconstruction and the same was validated with ex-vivo pork muscle sample.

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 d³:

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

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.

Figure 1 shows the temperature maps from the original TCR and the PITCR algorithm with 6x under-sampled data and 12x under-sampled data and the corresponding difference maps (where the truth is obtained by inverse Fourier transform). The temperature versus time profile is also shown in Fig. 1, which clearly indicates the superior performance of PITCR over the TCR approach with lesser data. Note that the reconstruction time using PITCR algorithm is much higher compared to TCR algorithm, but this is overcome by requirement of fewer measurements. This result indicates a promising step towards realizing availability of temperature maps with very fewer measurements.