Fast sequences involving repeated RF pulses sensitize signal to a variety of MR parameters [1]. Various parametric mapping approaches based on fitting signals acquired with different RF cycling, spoiling gradients or flip angles have been proposed for e.g. flip angle, T1 and/or T2 mapping [1-5]. Here, a refined modelling of the Bloch-Torrey equation with free diffusion was used to enable 3D co-localized mapping of proton density, background phase, flip angle, relaxation rates and apparent diffusion coefficients based on fast low angle sequences. Optimization approaches based on the Fisher information matrix were applied to define acquisition parameters minimizing the error on the expected parameters in a realistic scan time feasible in vivo.

Forward model and signal fitting: To model the Bloch-Torrey equation in such RF and gradient spoiled sequences, the configuration state formalism [6] was used. This formalism enables to handle free diffusion as a repetitive Gaussian filtering [7] and a fast calculation of the steady-state can be derived [8]. Based on several acquisitions with different acquisition parameters, the measured complex signal was fitted using least-squares and the Fisher information matrix provided the expected variance on the 6 fitted parameters: proton density, background phase at TE, flip angle, relaxation rates and apparent diffusion coefficient. Monte-Carlo simulations were performed to demonstrate the ability to fit unbiased parameters and fully predict noise propagation.

Experiments: Imaging was performed at 1.5 T using an 8-channel head coil for reception on a healthy volunteer. A 3D gradient echo sequence was modified to allow sequential acquisition of volumes with different flip angles, RF phase increments and spoiling gradients. The spoiling gradient was restricted to the readout direction and was characterized by the effective dephasing distance applied during each TR, a, which was varied from volume to volume. Fixed scan parameters were: TE=3 ms, acquisition matrix 160x120x20 and voxel size 1.25 mm x1.25 mm x5 mm, reconstructed voxel size 0.7 mm x0.7 mm x2.5mm, bandwidth 293 Hz/pix, Tacq =30 s per volume, acquisition of the k=0 state. To demonstrate the ability to fit all parameters with fixed TR (12 ms), prescribed flip angle amplitude (25°) and spoiling gradients (a=312 µm), series 1 was acquired with 28 phase increments (0°/1°/-1°/2°/-2°/4°/-4°/8°/-8°/16°/-16°/180°/90°/-90°/45°/-45°/0°/117°/-117°/150°/-150°/32°/-32°/64°/-64°/128°/-128°/0°) leading to a total scan time of 14 min. To demonstrate the ability to control and enhance the precision on the fitted parameters, series 2 was acquired with 13 volumes in 6.5 min with TR=12.5 ms, flip angles (26°/18°/26°/33°/14°/9°/6°/50°/22°/16°/17°/2°/26°), a (312, 125, 312, 312, 312, 179, 312, 312, 125, 140, 125, 312, 312 µm) and phase increments (0°/-2°/3°/180°/-1°/-1°/180°/0°/-3°/2°/2°/177°/0°). The acquisition parameters were obtained by minimizing the sum of the relative errors provided by the inverse of the Fisher information matrix for a range of expected relaxation and diffusion parameters. The minimization was performed with a stochastic optimization algorithm [9], and with predefined bounds on maximum flip angle and a, and a fixed TR.

Table 1 and 2 provides the results of a Monte-Carlo simulation considering realistic physical parameters. The average fitted values are unbiased, the simulated standard deviation on the parameters corresponded to the theoretical one (square root of the diagonal elements of the inverse Fisher information matrix) indicating that the precision can be predicted theoretically, and that optimization approaches can be used to define acquisition parameters providing preselected precision on the fitted parameters.

Both protocols were able to recover parametric maps (reconstruction time ~min/slice). Exemplary in vivo images obtained using series 2 are shown in Fig.1. As can be seen, the signal is modulated in a complex way mixing all contrasts. The reconstructed parametric maps are displayed in Fig. 2 demonstrating the feasibility to map in vivo all 6 parameters with tractable precision.

Co-localized multi-parametric mapping using fast low flip angle sequences with various flip angle amplitudes, phase cycling and gradient spoiling is feasible in vivo. Proton density, background phase, effective flip angle, apparent diffusion coefficient and relaxation parameters can be measured accurately and precisely. Error propagation can be analyzed to estimate the fitted parameter precision, and to optimize the acquisition protocol for a realistic scan time in vivo with targeted precision on preselected parameters. This work generalizes previous parametric mapping approaches, notably including free diffusion and the effective flip angle, and could be further extended to take into account partial voxel, exchange or magnetization transfer, and restricted diffusion to further provide the ability to map the associated parameters using fast low angle sequences.