Mingrui Yang^{1}, Dan Ma^{1}, Yun Jiang^{2}, Jesse Hamilton^{2}, Nicole Seiberlich^{2}, Mark Griswold^{1}, and Debra McGivney^{1}

Direct calculation of MRF dictionaries can be prohibitive when high resolution or multi-component chemical exchange effects are taken into account. To address this problem, we propose a new approach based on the randomized SVD (rSVD) to generate a low rank approximation of the large sized dictionary without the need of pre-calculating, storing, or loading the dictionary. This in return saves significant amounts of memory, and speeds up the template matching process of MRF. In addition, when combined with polynomial fitting, one can generate MRF maps with arbitrary high resolution dictionaries.

MRF^{1}
matches each voxel signal evolution
against a pre-calculated dictionary based on the simulations of signal
evolutions with different combinations of parameters of interest, such as $$$T_1,T_2$$$,
and off-resonance. The direct calculation of a dictionary, however, could be prohibitive
due to memory constraint even for modern computers, especially when a
high resolution dictionary is needed or multi-component chemical exchange
effects are taken into account^{2}.

We propose a new approach based on the randomized SVD (rSVD)^{3}
to generate a low rank approximation to large sized dictionaries without the
need of pre-calculating, storing, or loading the dictionary. This can
save significant amounts of memory, and speed up the template matching process
of MRF. In addition, when combined with polynomial fitting, one can generate
MRF maps with arbitrary high resolution dictionaries.

Previous studies have shown that
MRF dictionaries can be significantly compressed using e.g. the SVD^{4}.
Consider a dictionary $$$D\in\mathbb{C}^{t\times{n}}$$$,
with
$$$t$$$ the number of time frames and $$$n$$$ the number of tissue parameters. Let $$$\Omega\in\mathbb{C}^{n\times{k}}$$$ be a Gaussian matrix drawn from
normal distribution with mean 0 and variance 1, where $$$k$$$ is the low rank desired. We embed the tissue
property dimension $$$n$$$ into a much lower dimension $$$k$$$ by $$$X=(DD^*)^pD\Omega$$$ with the power iteration index $$$p=0,1,2,\ldots$$$, and
perform QR factorization $$$X=QR$$$, where
$$$p$$$ depends on the MRF sequence generating the
dictionary. Form the matrix $$$Y=Q^*D$$$ and compute its SVD $$$Y=\tilde{U}\Sigma{V}^*$$$. Form
the orthonormal matrix $$$U=Q\tilde{U}$$$. We have
$$$D\approx U\Sigma{V}^*$$$, where
the dimensions of $$$U,\Sigma$$$, and $$$V$$$ are $$$t\times{k},k\times{k}$$$, and $$$n\times{k}$$$ respectively. The
template matching for given test signal $$$\boldsymbol{s}$$$ is then computed via $$$\max{D^*\boldsymbol{s}\approx(V\Sigma^*)(U^*\boldsymbol{s})}$$$. Note
that in real implementation, $$$D$$$ does not need to be pre-computed and stored.
Only one tissue property entry at a time is computed on-the-fly to
update the calculation of $$$U,\Sigma$$$ and $$$V$$$, which are much smaller than $$$D$$$, resulting in a significant reduction in the required memory.

We use three types of dictionaries
to test the performance of our method: an MRF-FISP^{5} dictionary, an
MRF-bSSFP^{1} dictionary, and an MRF-X^{2} dictionary. The MRF-FISP
dictionary contains 3,000 time frames and 5,970 different tissue property combinations with $$$T_1$$$ values ranging from 10ms to 2,950ms and $$$T_2$$$ values ranging from 2ms to 500ms ($$$T_2\le{T_1}$$$). The MRF-bSSFP
dictionary contains 500 time frames, 500,112 tissue property and off-resonance combinations with $$$T_1$$$ values ranging from 20ms to 5,900ms, $$$T_2$$$ values ranging from 5ms to 2,900ms ($$$T_2\le{T_1}$$$), and
off-resonance frequencies ranging from -300Hz to 300Hz.
The MRF-X dictionary contains 3,000 time frames and 1,409,232 different combinations of
two-component $$$T_1,T_2$$$,
volume fraction parameters , and a chemical exchange rate of 0.1s^{-1}.

The brain datasets used for evaluation include
three in vivo scans of healthy volunteers. One was obtained on a Siemens Espree 1.5T
(Siemens Healthcare, Erlangen, Germany) using the MRF-bSSFP sequence with a
matrix size of $$$128\times128$$$ and a
field-of-view of 25cm^{2}. The other two were collected on a Siemens Skyra 3T (Siemens Healthcare, Erlangen, Germany)
using the MRF-FISP and MRF-X sequences respectively with matrix sizes of $$$256\times256$$$ and $$$192\times192$$$, and field-of-view of 30cm^{2}.

Fig.1 shows the results of applying rSVD to MRF with rank $$$k=30$$$ to the brain dataset obtained from the MRF-FISP acquisition. Fig.2 and Fig.3 demonstrate the results of applying the method with $$$k=100$$$ to the brain data obtained from the MRF-bSSFP acquisition, with power $$$p=0,2$$$ respectively. Fig.4 shows the potential of this approach applied to the very large MRF-X dictionary, where pre-computing, storing, and loading the dictionary is prohibitive.

The memory consumption details are shown in Table.1. The memory savings for calculating the MRF-FISP dictionary is almost 1,000 times using our rSVD method against the direct computation. For the MRF-bSSFP calculation using rSVD, the savings is still around 15 times. The direct calculation of the MRF-X dictionary is infeasible on a 64Gb-memory computer, while the memory consumption using rSVD is 3.2Gb.

1. D. Ma, V. Gulani, N. Seiberlich, K. Liu, J. L. Sunshine, J. L. Duerk, and M. A. Griswold, Magnetic resonance fingerprinting. Nature 2013, 495:187–192.

2. J. Hamilton, A. Deshmane, M. A. Griswold, and N. Seiberlich, MR Fingerprinting with Chemical Exchange (MRF-X) for In Vivo Multi-Compartment Relaxation and Exchange Rate Mapping, Proc. Intl. Soc. Mag. Reson. Med. 24 (2016), 0431.

3. N. Halko, P. G. Martinsson, and J. A. Tropp, Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions, SIAM Review 2011 53:2, 217-288.

4. D. F. McGivney, E. Pierre, D. Ma, Y. Jiang, H. Saybasili, V. Gulani, and M. A. Griswold, SVD Compression for Magnetic Resonance Fingerprinting in the Time Domain. IEEE Transactions on Medical Imaging, vol. 33, no. 12, pp. 2311-2322, Dec. 2014.

5. Y. Jiang, D. Ma, N. Seiberlich, V. Gulani, and M. A. Griswold, MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout. Magn. Reson. Med. 2015, 74: 1621–1631.

Fig.1 Performance of randomized SVD on MRF-FISP
sequence with rank $$$k=30$$$ and power iteration index $$$p=0$$$. Top:
$$$T_1$$$ maps. Bottom: $$$T_2$$$ maps. From left to right: MRF maps without
compression, MRF maps with randomized SVD with $$$k=30$$$,
difference maps between the two cases. The
MRF-FISP dictionary contains 3,000 time frames and 5,970 different tissue property combinations
with $$$T_1$$$ values ranging from 10ms to 2,950ms and $$$T_2$$$ values ranging from 2ms to 500ms ($$$T_2\le T_1$$$).

Fig.2 Performance of randomized SVD on
MRF-bSSFP sequence with rank $$$k=100$$$ and power iteration index $$$p=0$$$.
Top to bottom: $$$T_1$$$ maps, $$$T_2$$$ maps, and off-resonance maps. Left to right:
MRF maps without compression, MRF maps with randomized SVD with $$$k=100$$$ and $$$p=0$$$,
and difference maps between the two cases. The MRF-bSSFP dictionary contains
500 time frames, 500,112 different tissue property and off-resonance combinations with $$$T_1$$$ values ranging from 20ms to 5,900ms, $$$T_2$$$ values ranging from 5ms to 2,900ms ($$$T_2\le T_1$$$),
and off-resonance frequencies ranging from -300Hz to
300Hz (displaying -100Hz to 100Hz).

Fig.3 Performance of randomized SVD on MRF-bSSFP
sequence with rank $$$k=100$$$ and power iteration index $$$p=2$$$.
Top to bottom: $$$T_1$$$ maps, $$$T_2$$$ maps, and off-resonance maps. Left to right:
MRF maps without compression, MRF maps with randomized SVD with $$$k=100$$$ and $$$p=2$$$,
and difference maps between the two cases.

Fig.4 Performance of randomized SVD on the
MRF-X sequence for fixed chemical exchange rate 0.1s^{-1} with
rank $$$k=200$$$.
Top to bottom: species A, species B. Left to right: $$$T_1$$$ maps, $$$T_2$$$ maps, and volume fraction maps of species A. The
two component (AB) MRF-X dictionary has $$$T_1^A$$$ ranging from 60ms to 2,500ms, $$$T_2^A$$$ ranging from 10ms to 300ms, $$$T_1^B$$$ ranging from 60ms to 5,000ms, $$$T_2^B$$$ ranging from 10ms to 1,000ms ($$$T_2^A\le T_1^A,T_2^B\le T_1^B,T_2^A\le T_2^B,T_1^A\le T_1^B$$$),
exchange rate fixed to be 0.1s^{-1},
and volume fraction of species A ranging from 0 to 99%.

Table.1 Memory consumption for calculating dictionaries with rSVD and direct approach