Synopsis

One of the challenges MRF faces is the amount of data needed to be stored, loaded, and processed, especially when a high resolution dictionary is needed or large multi-dimensional analyses need to be taken into account. A low rank approximation to the high resolution MRF dictionary using a coarse dictionary is an effective remedy to this problem. Here we present one of many possible ways to implement low rank approximation to an arbitrary fine MRF dictionary by a coarse dictionary equipped with polynomial fitting, so as to avoid the need of pre-calculating, storing, and processing the large, finely-resolved MRF dictionary.

Purpose

Magnetic Resonance Fingerprinting (MRF) ¹ is a new quantitative MRI technique that matches the signal evolution for each voxel against a pre-calculated dictionary based on Bloch equation simulations with different combinations of parameters of interest, such as $$$T_1$$$, $$$T_2$$$, and off-resonance. One of the challenges it faces is the amount of data needed to be stored, loaded, and processed, especially when a high resolution dictionary is needed or large multi-dimensional analyses need to be taken into account ².

A low rank approximation to the high resolution MRF dictionary using a coarse dictionary is an effective remedy to this problem. Here we present one of many possible ways to implement low rank approximation to an arbitrary fine MRF dictionary by a coarse dictionary equipped with polynomial fitting, so as to avoid the need of pre-calculating, storing, and processing the large, finely-resolved MRF dictionary.

Methods

Consider an MRF-FISP ³ dictionary $$$D\in\mathbb{C}^{t\times n}$$$ with $$$T_1$$$ varying from 20ms to 1,940ms with a step size of 80ms and $$$T_2$$$ varying from 10ms to 170ms with a step size of 40ms ($$$T_2\le T_1$$$), where $$$t=3,000$$$ is the number of time frames and $$$n=119$$$ is the number of tissue type parameters. Let $$$D\approx U\Sigma V^*$$$ be the rank 3 randomized SVD ⁴ approximation of $$$D$$$, where $$$\cdot^*$$$ denotes the adjoint operator. We project the row space of $$$D$$$ into a 3-dimensional space by $$$X=U^*D$$$. We then fit the projected data with a degree 5 polynomial, resulting in a 4 times finer approximate dictionary (for a fair comparison against a 4 times finer dictionary of size $$$3,000\times1,585$$$ with step sizes of 20ms and 10ms for $$$T_1$$$ and $$$T_2$$$ respectively).

Now for a projected voxel evolution signal $$$\boldsymbol{s}$$$, it is first matched against all $$$T_1$$$, $$$T_2$$$ grids using maximal correlation to obtain the largest two $$$T_1$$$ values and two $$$T_2$$$ values (corresponding to the coarse dictionary) respectively. We then partition the fitted $$$T_1$$$ curves between the two fitted $$$T_2$$$ curves into $$$p=4$$$ even segments and find on each $$$T_1$$$ curve the point with the largest correlation with $$$\boldsymbol{s}$$$. Record the indices of these two points as $$$i$$$ and $$$j$$$ ($$$i, j=1,\ldots,p+1$$$). Then the $$$T_2$$$ value corresponding to $$$\boldsymbol{s}$$$ is estimated as $$$(T_{2,2}-T_{2,1})\times(i+j-2)/(2p)+T_{2,1}$$$. The $$$T_1$$$ value corresponding to $$$\boldsymbol{s}$$$ is estimated similarly.

To validate our model, we match an in vivo brain dataset of a healthy volunteer against a coarse dictionary, a fitted dictionary, and a dictionary that is 4 times finer than the coarse dictionary. The dataset is obtained on a Siemens Skyra 3T scanner (Siemens Healthcare, Erlangen, Germany) using the MRF-FISP sequence with a matrix size of $$$256\times 256$$$ and a field-of-view of 30cm². The brain data is also projected into the 3-dimensional random SVD space before matched against different dictionaries.

Results

Fig.1 shows a 3D visualization of the projected randomized SVD space of the coarse MRF-FISP dictionary fitted with a degree 5 polynomial with fitting statistics $$$R^2=1$$$ and adjusted $$$R^2=0.99$$$. The cyan and red curves represent different $$$T_1$$$ and $$$T_2$$$ values respectively, and the circles represent fitted values along each curve, partitioning each coarse grid into 4 segments. The figure shows that the projected coarse dictionary fits the polynomial surface perfectly.

The MRF results with different dictionaries are shown in Fig.2 and Fig.3. In Fig.2, the $$$T_1$$$, $$$T_2$$$ maps of a scanned human brain are obtained via the projected coarse dictionary, the projected coarse dictionary fitted with a degree 5 polynomial, and the projected fine dictionary. We see a clear quality degradation when the dictionary is coarse. This, however, is remediated by fitting a polynomial to the coarse dictionary. The results are more clearly displayed in the difference maps in Fig.3. The $$$T_1$$$, $$$T_2$$$ map differences between the coarse dictionary and the 4 times finer dictionary are significantly diminished by the polynomial fitted dictionary.

Discussion and Conclusion

Acknowledgements

References

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