Debra McGivney^{1}, Rasim Boyacioglu^{1}, Yun Jiang^{1}, Charlie Wang^{2}, Dan Ma^{1}, and Mark Griswold^{1,2}

Magnetic resonance fingerprinting is a framework for creating quantitative tissue property maps from a single acquisition. The accuracy and precision of these maps depend upon a precomputed dictionary of simulated signal evolutions, to which acquired signals are matched using the inner product to determine the tissue property values. We propose to approximate the inner product as a quadratic function of the tissue properties in a neighborhood around the correct match in order to reduce the effect of tissue property step size in the dictionary. Results from data acquired with different MRF sequences demonstrate the value of the proposed approach.

The MRF dictionary
is a large matrix of size *m*x*n*,
where *m* is the number of time points in the sequence,
and *n* represents the number of tissue property combinations.
For a given sequence, we can represent each dictionary entry as a function of
the tissue property vector $$$\theta$$$ by $$$d = d(\theta)$$$.

Assuming that a given voxel is represented by tissue $$$\theta_0$$$ , the acquired MRF signal evolution from this voxel can be written as

$$s = d(\theta_0) + \epsilon$$

with noise term $$$\epsilon$$$. The signal is matched to the dictionary by comparing the inner product values between $$$s$$$ and each dictionary entry to find the maximum in absolute value. The inner product is written as

$$f(\theta) = s^\ast d(\theta) = \left(d(\theta_0) + \epsilon \right)^\ast d(\theta) = d(\theta_0)^\ast d(\theta) + \epsilon^\ast d(\theta), (1)$$

where * denotes conjugate transpose. Note
that in the case where $$$\theta$$$ is in a neighborhood of $$$\theta_0$$$,
then a quadratic approximation of the inner product (i.e., $$$f(\theta) \approx ||d(\theta)||^2$$$)
is appropriate. After applying a Taylor series expansion of $$$d$$$ in a neighborhood of $$$\theta_0$$$, we can write the inner product as a
quadratic function of $$$\theta = (T_1, T_2)$$$.
For example, in MRF-FISP^{2}, the inner product is approximated as the quadratic

$$f(T_1,T_2) \approx p_{00} + p_{10}T_1 + p_{01}T_2 + p_{11}T_1 T_2 + p_{20}T_1^2 + p_{02}T_2^2, (2)$$

for coefficients $$$p_{ij}, 0\leq i+j \leq 2$$$.
We
can estimate a tissue property neighborhood which contains the true values by
matching $$$s$$$ to a MRF
dictionary with larger tissue property step sizes, i.e., a “coarse dictionary.”
Using the corresponding inner product values associated with this neighborhood,
we compute the coefficients as in equation (2) for MRF-FISP, and then find its
critical point. It is straightforward to generalize this model to more than two
tissue properties, as in the case of MRF-bSSFP^{1} or MRF with quadratic
phase (qRF-MRF)^{3}.

The quadratic inner
product model was tested on a simulated brain phantom using MRF-FISP. A dictionary
of size 3000x5970 was used to create simulated MRF signal evolutions and to generate the true values. The coarse dictionary had dimensions 3000x1510,
formed by downsampling the larger dictionary tissue properties each by 2. At
each pixel, a neighborhood of points was found by matching the signal evolution
to the coarse dictionary, and then equation (2) was approximated and the
critical point found.
Three volunteers were
consented and scanned under an IRB-approved study. They were a normal subject
scanned with MRF-FISP, a brain tumor patient (adenocarcinoma metastasis) scanned
with MRF-bSSFP, and a normal volunteer scanned with qRF-MRF^{4}. The MRF-bSSFP benchmark dictionary contained 3307 T_{1}, T_{2} combinations and 77 off-resonance values and the coarse dictionary contained 854 T_{1}, T_{2} combinations and 39 off-resonance values. For qRF-MRF, only a coarse dictionary was used due to the fact that there are four tissue property dimensions. The dictionary contained 1590 T_{1}, T_{2} combinations, 51 off-resonance values, and 26 T_{2}* values. For both MRF-bSSFP and qRF-MRF, a modified version of equation (2) was used.

A one-shot spiral trajectory was used in each case and SVD compression^{5,6} was used for both the benchmark
and coarse dictionaries for compression.

- Ma, D. et al. Magnetic resonance fingerprinting. Nature 495, 187–192 (2013).
- Jiang, Y., Ma, D., Seiberlich, N., Gulani, V. & Griswold, M. MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout. Magn. Reson. Med. 74, 1621–1631 (2015).
- Wang, C. Y. et al. Magnetic resonance fingerprinting with quadratic RF phase for measurement of T2 * simultaneously with δf, T1 , and T2. Magn. Reson. Med. 1–14 (2018). doi:10.1002/mrm.27543
- Wang, C. et al. Magnetic resonance fingerprinting with pure quadratic RF phase. Submitted as an abstract, 27th annual ISMRM, Montreal (2019).
- McGivney, D. et al. SVD compression for magnetic resonance fingerprinting in the time domain. IEEE Trans. Med. Imaging 33, 2311–2322 (2014).
- Yang, M. et al. Low rank approximation methods for MR fingerprinting with large scale dictionaries. Magn. Reson. Med. 79, 2392–2400 (2018).

Figure 1. T_{1}, T_{2} maps for the numerical brain phantom. In the left column are the
true values, in the middle are maps found matching to a coarse MRF dictionary,
and on the right are the values found by applying quadratic interpolation on
the coarse MRF results. In the table, four pixels were chosen to compare the
results from the coarse match and quadratic fit to the true values.

Figure
2. T_{1} and T_{2} maps from a normal volunteer with MRF-FISP. Difference
maps are on the right, with the discrepancy between the coarse and fine
dictionary matches on the left, and discrepancy between quadratic
interpolation and the fine dictionary match on the right. In most
regions, the discrepancy between the quadratic interpolation and
fine match is smaller than that between the coarse and fine matches,
indicating that the quadratic interpolation outperforms MRF
matching with a coarse dictionary. In CSF, it may be that the inner product values are too flat, causing an overestimation of T_{1} and T_{2}.

Figure 3. T_{1}, T_{2}, and off-resonance maps from a MRF-bSSFP
acquisition of a brain tumor patient. The results from a fine MRF dictionary
are in the left column, using a coarse MRF dictionary are in the middle, and
the results from quadratic interpolation of the coarse dictionary are on the
right. In particular, the T_{2} and off-resonance maps from the coarse dictionary are quite flat, with improvement shown in the corresponding maps from quadratic interpolation.

Figure 4. Differences between T_{1}, T_{2}, and off-resonance maps
compared to the fine MRF dictionary match. The left column represents the
discrepancy between the coarse MRF dictionary match and the fine dictionary match, whereas the right column is the discrepancy between the quadratic
interpolated results and the fine dictionary match. In most regions of the brain,
the interpolated values are closer to the benchmark, except in the case of CSF,
which is likely due to the fact that the coarse and fine dictionaries have a value for T_{2} in common and the inner product is generally flat in these regions.

Figure 5. Off-resonance and T_{2}* maps using qRF-MRF. In the first
row are off-resonance maps, with the coarse dictionary fit and the quadratic
interpolated result. In the bottom row are the T_{2}* maps. Note the increase in
structural detail seen in the off-resonance map after quadratic interpolation
has been applied. The qRF-MRF method also generates T_{1} and T_{2} maps, which yield similar results seen in Figures 2 and 3 after quadratic interpolation.