Martijn Nagtegaal^{1}, Thomas Amthor^{2}, Peter Koken^{2}, and Mariya Doneva^{2}

A comparative analysis between IR-bSSFP and MR Fingerprinting was performed in numerical simulations for single and multi-component parameter mapping. The single component matching works for both methods, although the accuracy for T2 is better for MR Fingerprinting. The multi component matching for a constant flip angle IR-bSSFP sequence can only match to the T1* values and cannot distinguish between the underlying T1/T2 values. Using the MR Fingerprinting sequence with a varying flip angle it is possible to match to the T1/T2 components.

In magnetic resonance fingerprinting (MRF) [1]
an acquisition scheme with varying pulse sequence parameters is used to enable simultaneous multi-parameter mapping. The original MRF sequence is closely related to the IR-bSSFP
approach for simultaneous T_{1}, T_{2}, and proton density mapping [2], in which constant flip angle α
and TR are applied after an initial inversion and a preparation pulse with α/2
at TR/2, resulting in an exponential signal recovery with an effective
relaxation time $$$T_1^*$$$.

Both methods can be applied for the simultaneous estimation
of T_{1}, T_{2} and proton density maps. Multi-component
analysis for IR-bSSFP has been done based on the effective relaxation time $$$T_1^*$$$ [3] and
for MRF based on the T_{1} and T_{2} values [4,5]. Combined with radial or spiral
acquisition with high undersampling and dictionary matching both scans can be
performed in a very short scan time. The main difference that remains is the flip
angle variation. So what are the advantages of the flip angle variation?

In this work we try to answer this question and investigate the differences between the two sequence types in single component matching and multi-component analysis experiments.

Four sequences with different FA variations ((1) constant
45°,
(2) constant 25°, (3) smooth and (4) high frequency variation) all consisting
of 500 measurements with TR=3.6ms were considered (Fig.1).
Dictionaries containing the signal evolutions for 6240 T_{1}/T_{2} combinations (T_{1} from 5ms to 2s, T_{2} from 4ms tot 1s both with an adaptive
step size) were computed for each sequence. The exponential signal for a
constant FA α is described through the effective relaxation
time$$T_1^*=\left(\frac{1}{T_1}\cos^2\alpha+\frac{1}{T_2}\sin^2\alpha\right)^{-1}$$and the signal value in steady-state by the ratio between T_{1} and T_{2}.

The accuracy and precision of the single component parameter
mapping of the four sequences were compared in numerical simulations. 65 test signals with different T_{1}/T_{2} combinations were
considered, for each combination one signal with 100 noise realizations was
simulated with an SNR of 10. These noisy signals were matched to the dictionary
using an inner product matching as proposed in [1].
The quality of the matching was determined by the mean relative error and
standard deviation in the mapped relaxation times.

The sequences were further compared in a multi-component matching task. Different
mixtures of two T_{1} and T_{2} relaxation times were tested for different levels of
additive Gaussian noise. For each simulated signal the multi-component matching
is performed through the non-negative least squares (NNLS) algorithm [6].
This process was repeated 400 times with different noise realizations leading
to a distribution of matched components.

The NNLS algorithm finds sparse results [7], which makes it an effective method to obtain a decomposition with a small number of components.

Results from the single component parameter mapping are shown
in Fig.2,3
. Sequences 1 and 2 show a reduced accuracy and precision compared to the
varying FA sequences for T_{2}. For most considered T_{1}/T_{2} combinations the error
is low (<1%), but it increases for small T_{1} and T_{2} for every sequence. For
the constant FA, the FA influences the precision of the matched relaxation
times. Using a smaller FA improves the precision of the T_{1} matching and reduces
the precision in T_{2}, which is expected from the relation through $$$T_1^*$$$.

Fig.4
shows an example of the distribution of the matched components in the multi-component
matching for the sequences, omitting sequence 2. The multi-component matching is
exact for noiseless signals (Fig.4a,b,c).
With SNR=10000 sequences 3 and 4 (Fig.4e,f)
find some extra components, all with
weights below 1%. For SNR=100 (Fig.4g,h)
the matched components are spread around the true values but still give an
accurate decomposition. For SNR = 10000 the
IRbSSFP sequence returns more than 2 large (>10%) components per signal,
spread over the corresponding $$$T_1^*$$$-contour lines, which are shown in Fig.5. The matched components can still
separate tissues with different $$$T_1^*$$$, but cannot return the
corresponding T_{1} and T_{2} values for the individual components.

The variation in the flip angle breaks
the relationship in Eq.1, making it easier to distinguish different T_{1} and T_{2}
combinations. The results in Fig.4
are reproducible for mixtures of different components and varying weights.

[1] D. Ma et al., “Magnetic resonance fingerprinting,” Nature, vol. 495, no. 7440, pp. 187–192, Mar. 2013.

[2] P. Schmitt et al., “Inversion recovery TrueFISP: Quantification of T1,T2, and spin density,” Magn. Reson. Med., vol. 51, no. 4, pp. 661–667, Apr. 2004.

[3] J. Pfister, M. Blaimer, P. M. Jakob, and F. A. Breuer, “Simultaneous T1/T2 measurements in combination with PCA-SENSE reconstruction (T1* shuffling) and multicomponent analysis,” in Proc. Intl. Soc. Mag. Reson. Med. 25, 2017, p. 0452.

[4] D. McGivney et al., “Bayesian estimation of multicomponent relaxation parameters in magnetic resonance fingerprinting: Bayesian MRF,” Magn. Reson. Med., Nov. 2017.

[5] S. Tang et al., “Multicompartment magnetic resonance fingerprinting,” Inverse Probl., vol. 34, no. 9, p. 094005, Sep. 2018.

[6] C. L. Lawson and R. J. Hanson, Solving least squares problems. SIAM, 1974.

[7] A. M. Bruckstein, M. Elad, and M. Zibulevsky, “Sparse non-negative solution of a linear system of equations is unique,” 2008, pp. 762–767.

Figure 1. The
different flip angle patterns used in this comparison. The upper two plots show
the IRbSSFP sequences with the preparation pulse of α/2. The lower left plot shows the smoothly varying
flip angle sequence, the lower right the high frequency variating flip angle
sequence.

Figure 2. The
results of the single component matching is visualized for different T1 and T2
combinations for an SNR of 10 for the four different sequences. 100 signals per T1, T2 combination are
considered. The crossings of the grid lines represent the true T1, T2
combinations, the location of the center of the dots visualize the matched
relaxation times. The width and length of the dot indicate the variation in T1
and T2 respectively. The color of the dots is based on the sum of the relative
standard deviation of the found T1 and T2 values.

Figure 3. The
relative error for T1 and T2 in the single component matching. Each column of plots represents
a different sequence. The upper plots shows the relative error for T1 and the
lower plots the relative error for T2. Each data point shows the relative error for
the corresponding combination of relaxation times. Note that the scale for e)
and f) is 10 times higher than the scale for g) and h).

Figure 4. The
distribution of the matched components in the multicomponent matching for the
three sequence types. Each column represents a sequence. The rows show the results without noise,
SNR=10000 and SNR=100. The analyzed signals have underlying relaxation times of
T1 = 250,1000ms and T2 = 45,310ms. These combinations correspond to a T1* of 150 ms and 758 ms and the ratios are 5.6
and 3.2. The black crosses indicate the ground truth.

Figure 5. The
effective relaxation time T1* for different T1 and T2 combinations
for a FA of 45 degrees.