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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