David Leitão^{1}, Joseph V. Hajnal^{1,2}, Rui P. A. G. Teixeira^{1}, and Shaihan Malik^{1}

This work proposes a neutral measure of encoding efficiency per square-root of time excluding any effects of image reconstruction from the analysis in order to compare spoiled gradient-echo based Magnetic Resonance Fingerprinting and steady-state methods for $$$T_1$$$ and $$$T_2$$$ estimation. The results obtained indicate that gradient spoiled Fingerprinting is up to $$$50\%$$$ more efficient per square-root-time than steady-state methods. The optimal sequences of pulses found have striking features with different duration fingerprint strategies having highest efficiency under different boundary conditions.

The Cramér-Rao Lower Bound (CRLB) provides a direct means to determine the lower bound on variance for an estimated parameter $$$\theta$$$ from a set of measured data, hence an upper bound on the parameter-to-noise ratio $$$\theta NR^{(3-6)}$$$. This bound is a function of the signal-to-noise ratio (SNR) of the input data. To compare pulse sequences with different durations it is instructive to normalise to the square-root acquisition time $$$(T_{acq})$$$:

$$\theta NR_{u.t.}=\frac{\theta{}NR}{\sqrt{T_{acq}}}\equiv\eta(\theta)\cdot{}SNR_0$$

where $$$\eta (\theta)$$$ is the efficiency per square-root-time in
parameter $$$\theta$$$ and $$$SNR_0$$$ is the single measurement SNR (*i.e.,* one
readout), defined with reference to maximum possible signal, $$$M_0$$$.

Sequence properties, such as spoiling, impact on encoding power, so to achieve an initial direct comparison we consider only spoiled gradient-echo (SPGR) based MRF$$$^{(7)}$$$. We investigated behaviour for fingerprints using different numbers of excitations pulses $$$(N)$$$, both starting from thermal equilibrium, and also in a driven-equilibrium (DE) case where the sequence is repeated, which may be more relevant to 3D imaging. For each case the acquisition settings $$$(\alpha_n, TR_n)$$$ were optimized by maximising the minimum $$$\eta(\theta_i=\{T_1,T_2\})$$$ subject to inequality constraints $$$g_n(TR\geq5ms,0^\circ\leq\alpha_n\leq180^\circ)$$$:

$$\mathrm{max}_{\alpha_n,TR_n}\{\mathrm{min}_i\quad\eta(\theta_i)\} $$

$$\mathrm{s.t.}\quad\quad\quad{}g_n\leq0$$

Additional optimizations used temporal smoothness (small flip angle steps) and minimum flip angle constraints proposed by Zhao$$$^{(3)}$$$. $$$\eta$$$, which has units of $$$s^{-1/2}$$$, was computed via CRLB calculations based on extended-phase-graph simulations, validated by Monte-Carlo simulations (Figure 1). All optimizations were performed for $$$\{T_1,T_2\}=\{781ms,65ms\}$$$ which approximates white matter. SS methods for mapping $$$T_1$$$ and $$$T_2$$$ require both SPGR and steady state free precession (SSFP) sequences, so that is the point of reference used.

This work is funded by the King's College London & Imperial College London EPSCR Centre for Doctoral Training in Medical Imaging (EP/L015226/1).

This work was supported by the Wellcome EPSRC Centre for Medical Engineering at Kings College London (WT 203148/Z/16/Z) and by the National Institute for Health Research (NIHR) Biomedical Research Centre based at Guy’s and St Thomas’ NHS Foundation Trust and King’s College London. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR or the Department of Health.

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$$$(2)$$$ Deoni, S. C. L., Peters, T. M. & Rutt, B. K. High-resolution T1 and T2 mapping of the brain in a clinically acceptable time with DESPOT1 and DESPOT2. *Magn. Reson. Med.* **53**, 237–241 (2005).

$$$(3)$$$ Zhao, B., Haldar, J. P., Setsompop, K., & Wald, L. L. Optimal experiment design for magnetic resonance fingerprinting. *In Engineering in Medicine and Biology Society (EMBC), 2016 IEEE 38th Annual International Conference of the* (pp. 453-456). IEEE.

$$$(4)$$$ Assländer, J., Lattanzi, R., Sodickson, D. K., & Cloos, M. A. Relaxation in Spherical Coordinates: Analysis and Optimization of pseudo-SSFP based MR-Fingerprinting. *arXiv preprint *arXiv:1703.00481 (2017).

$$$(5)$$$ Assländer, J.,Novikov, D. S., Lattanzi, R., Sodickson, D. K., & Cloos, M. A. Hybrid-State Free Precession in Nuclear Magnetic Resonance. arXiv preprint arXiv:1807.0342 (2018).

$$$(6)$$$ Teixeira, R. P. A., Malik, S. J., & Hajnal, J. V. Joint system relaxometry (JSR) and Crámer‐Rao lower bound optimization of sequence parameters: A framework for enhanced precision of DESPOT T1 and T2 estimation. *Magnetic resonance in medicine*, **79**(1), 234-245.(2018).

$$$(7)$$$ Jiang, Y., Ma, D., Seiberlich, N., Gulani, V., & Griswold, M. A. MR fingerprinting using fast imaging with steady state precession (FISP) with spiral readout. *Magnetic resonance in medicine*, **74**(6), 1621-1631.(2015).

Figure 1 - Comparison
of the theoretical efficiency computed via Cramér-Rao Lower Bound calculation, and Monte Carlo
experiment using least squares fitting. There is a very good agreement between
the two measures.

Figure 2 - Efficiency
of spoiled gradient-echo MRF for $$$T_1$$$ (a,c) and $$$T_2$$$ (b,d)
encoding for different duration fingerprints. The ‘original’ published sequence$$$^{(7)}$$$ is shown in purple. From the top to the bottom row, a relaxation period
of $$$5\times{}T_1$$$ is added to ‘opt. IR-FISP MRF’ such the magnetization returns
to thermal equilibrium, allowing to repeat the sequence in a 3D experiment.
Note that the driven-equilibrium (DE) version does not need such a relaxation
period because it runs in a loop as many times as necessary.

Figure 3 - Optimal acquisition settings
(left column) and respective signal (right column) for some example optimized
fingerprint sequences. (a,b) IR-FISP with $$$N=50$$$ pulses/readouts; (c,d) Driven-Equilibrium (DE) with $$$N=50$$$ pulses/readouts; (e,f)
IR-FISP with $$$N=400$$$; (g,h) Constrained IR-FISP with $$$N=400$$$.

Figure 4 - Example
of an optimized fingerprint for $$$N=10$$$, optimized only for efficiency of $$$T_1$$$ estimation. This turns out to be a classic inversion recovery sequence. The
optimization determines that it is more effective to allow magnetization to
recover than to acquire more signals in the period between first and last
pulse.

Figure 5 - Average
efficiency for different techniques. The pale bar represents the gain in
encoding efficiency of a single run compared to adding a relaxation period to
allow repeating the sequence for a 3D experiment. For the case of ‘opt. IR-FISP
MRF’ only the average efficiency for a prolonged single run is shown as it was
the best-case scenario. The ‘opt. cons. IR-FISP MRF’ is the efficiency of the
prolonged MRF when constraining the acquisition settings like Bo Zhao *et al.*$$$^{(3)}$$$.