Synopsis

Keywords: MR Fingerprinting/Synthetic MR, MR Fingerprinting

We introduce an optimization framework using the longitudinal steady-state equilibrium for FLASH MRF sequences, and taking into account signal noise by using a Monte Carlo method. We use this framework to optimize echo times, flip angles, timings and recovery time for T1_H2O and FF mapping while shortening our original MRF T1-FF sequence. The resulting Fast MRF T1-FF sequence yields comparable estimation results to the original sequence with an almost twice reduced acquisition time.

Introduction

In the field of neuromuscular disorders (NMD), fat fraction (FF) is an established biomarker of disease severity and water T1 (T1_H2O) has been shown to be a potential biomarker of disease activity [1]. The use of MR Fingerprinting (MRF) enables the simultaneous quantification of various parameters, such as FF and T1_H2O [2]. However FLASH T1 MRF sequences (such as MRF T1-FF proposed in [3]) generally require long recovery times between repetitions for allowing the longitudinal magnetization to grow back to equilibrium. Shortening the sequence would highly benefit 3D imaging, where multiple acquisitions of the MRF scheme are required to encode along the partition encoding direction. However a shorter sequence is likely to degrade the parameter estimation quality. Hence we need to optimize the acquisition parameters of our shortened sequence for maintaining the quality of FF and T1_H2O quantification.

In this work, we introduced an optimization framework that takes into account the longitudinal steady-state equilibrium of the MRF sequence and handles acquisition noise using Monte Carlo simulations [4]. Through this framework, we optimized the echo times (TE), flip angles (FA) and recovery time of the MRF T1-FF sequence and compared the mapping accuracy and precision to the original implementation.

Methods

The optimization framework consisted of 4 blocks : i) a simulation block for steady-state MRF FLASH sequences with intial inversion pulse and variable TE,TR and FA, ii) an MRF pattern matching algorithm, iii) a cost function taking into account noise and iv) an optimizer (see Fig [1]).

Regarding the MRF FLASH simulation block, for N impulsions with flip angles (θ_i)_i=1..N of phase (φ_i)_i=1..N, echo times (TE_i)_i=1..N and repetitions times (TR_i)_i=1..N, the magnetization has the following dynamics:
$$
M_x(t+TE_i)=M_z(t)sin(\theta_i)sin(\phi_i)e^{⁻\frac{TE_i}{T_2}}\\M_y(t+TE_i)=-M_z(t)sin(\theta_i)cos(\phi_i)e^{⁻\frac{TE_i}{T_2}}\\M_z(t+TR_i)=M_0+(M_z(t)cos(\theta_i)-M_0)e^{⁻\frac{TR_i}{T_1}}
$$
$$$m_p=M_z(p*T)$$$ represents the initial magnetization after p repetitions of the MRF scheme, where T is the total duration of the sequence.
We have :
$$m_p=Am_{p-1}+B$$
with $$$A=\prod_{i=1}^N{e^{⁻\frac{TR_i}{T_1}}cos(\theta_i)}$$$ and $$$B=M_0\sum_{i=0}^{N-1}{(1-e^{⁻\frac{TR_{N-i}}{T_1}})\prod_{j=N-i}^N{e^{⁻\frac{TR_j}{T_1}}cos(\theta_j)}}$$$.
$$$m_p$$$ is an arithmetic-geometric sequence with A<<1. Hence it reaches a steady-state after very few repetitions of the sequence, and the steady-state value is equal to B/(1-A).
MRF is mainly impacted by noise due to undersampling that we approximated with a Gaussian distribution representing an average SNR of 20. For pattern matching, we used exhaustive search into a bicomponent dictionary of fingerprints [5].

The following cost function was calculated to assess parameters mapping accuracy ([4]) :
$$C((\theta_i)_{i=1..N},(TE_i)_{i=1..N},TR_N)=\lambda_{T1}\frac{1}{N_s}\sum_{s=1}^{N_s}{\frac{|\widehat{T1_{H_2O}^s}-T1_{H_2O}^s|}{T1_{H_2O}^s}}+\lambda_{FF}\frac{1}{N_s}\sum_{s=1}^{N_s}|\widehat{FF^s}-FF^s|+\lambda_{T}\sum_{i=1}^{N}{TR_i}$$
where $$$T1_{H_2O}^s,FF^s $$$(resp.$$$\widehat{T1_{H_2O}^s},\widehat{FF^s}$$$ ) are the ground truth (resp. estimated) parameters for each simulated signal and N_s is the total number of simulated signals. We fixed N=760 spokes, the smallest number of spokes for which estimation did not worsen drastically. As using 760 spokes instead of 1400 spokes already reduced the sequence duration, we used $$$ \lambda_{T}=0$$$. $$$\lambda_{T1}$$$ and $$$\lambda_{FF}$$$ were set empirically to 1 and 2 for those terms to have similar order of magnitudes.
To reduce the dimension of the problem, we assumed that TE and FA had piecewise constant evolutions with respectively 3 and 5 pieces, and that (TR_i)_i=1..(N-1) were kept at their minimum possible value allowed by the gradient system and bandwidth, which left us with a total of 15 parameters to optimize for : Flip angles values (θ_i)_i=1..N , echo time values (TE_i)_i=1..N, breaks timings and recovery time TR_N at the end of each repetition.
The signals were simulated for the following grid of tissue parameters : T1_H2O =500:100:1900 (ms), FF = 0.:0.1:0.5, df = [-30.,0,30] (Hz), B1=[0.7,1.0].

The algorithm used for minimizing the cost function was differential evolution [6], which is an efficient global optimization method when the gradient of the cost function is not easily accessible. The maximum functions evaluations was set at 150000 but the algorithm stopped earlier.

We compared the reference (MRF T1-FF) and optimal (Fast MRF T1-FF) sequences on 3D numerical phantoms (16 slices – 64 square subregions of size 4x4 per slice with randomly varying T1_H2O, FF, Df and B1) and 3D in vivo data acquired at 3T (Prisma^Fit, Siemens Healthineers) on the thighs of one healthy control (20 partitions, resolution 1x1x5 mm³, FOV 10x40x40cm³,10 regions of interest (ROIs) drawn on the left thigh muscles).

Results

Each repetition of Fast MRF T1-FF lasts 6.5s against 10.9s for MRF T1-FF (Fig [2]). The recovery of 3s at the end of each repetition did not allow for the longitudinal magnetization to grow back to intial equilibrium, however the steady-state was reached after a maximum of two repetitions of the Fast MRF T1-FF sequence (Fig [3]).
On numerical phantoms, T1_H2O and FF estimation accuracy and precision were similar to the reference sequence (Figure 4).
On in vivo data, accuracy was comparable to MRF T1-FF. Precision was degraded for T1_H2O, with a 40 ms increase in standard deviation on average (Figure 5).

Discussion & Conclusion

Using a new optimization framework taking into account steady-state magnetization and noise, we designed a new sequence (Fast T1-FF MRF) for fast quantification of T1_H2O and FF, with reduced time and comparable results to those of the reference sequence. In vivo, the new sequence was less precise, however this could be alleviated by MRF denoising techniques ([7]).
The explicit calculation of the steady-state value removed the need to simulate several repetitions of the MRF sequence and allowed to reduce the computation time for sequence optimization.

Acknowledgements

This study was funded by ANR-20-CE19-0004.