To analyze contrast in fast steady-state sequences, a given set of acquisition parameters can be considered: prescribed flip angle (alpha), Repetition Time (TR), phase cycling increment (phase) and spoiling gradient (characterized by a, the distance along which a dephasing of 2π occurs between 2 TRs). The measured signal then depends on: proton density (M₀), background phase (Phi₀), effective flip angle alpha_eff, longitudinal and transversal relaxation time (T1, T2) and the apparent diffusion (ADC). Our objective is to define the optimal set of acquisition parameters that will give the smallest estimation error over a predefined range of physical parameters.

The Fisher information matrix (FIM) and the Cramer-Rao lower bounds (CRLB) are standard tool for experiment-design optimization [2,3] or for fitting error evaluation [4]. Here, it is shown that the acquisition parameters can be optimized by minimizing the CRLB of each physical parameters, thus reducing scan time/enhancing precision for multi-parametric mapping.

For any unbiased joint estimation method of the above-mentioned N_par=6 physical parameters, a lower bound of the error σ_θk in the parameter θ_k, is given by the CRLB: $$\sigma_{\theta_{k}}\geq CRLB = \sqrt{FIM^{-1}_{kk}}$$

When N_meas observed data Y_i are normally distributed about S_i=S(x_i,θ_k) with variance σ the FIM is :

$$FIM_{jk}=\sum_{i=1}^{N_{meas}}\frac{1}{\sigma^{2}}\frac{\partial S_{i}}{\partial \theta_{j}} \frac{\partial S_{i}}{\partial \theta_{k}}$$

Note that the MR signal is kept as a complex value, thus the assumption of Gaussian noise is reasonable. For all simulations, a SNR of 500 (σ=1/500) was used. The FIM was calculated numerically with a fast algorithm [1]. In order to optimize the acquisition protocol, the objective function J was defined as the sum of the normalized CRLB: $$J=\sum_{k=1}^{N_{par}}\frac{\sigma_{\theta_{k}}^{2}}{\theta_k^2}$$

To normalize the CRLB for $$$alpha_{eff}$$$ and Phi₀, nominal references of 90° and π were used, respectively. To obtain optimized acquisition parameters, the minimization of J was perform with a stochastic optimization algorithm, SOMA [5].

Four sets of acquisition parameters were considered and compared. The first one was defined empirically and the others were obtained by considering 12 acquisitions and the following constraints: a constant TR between 6 and 60 ms. alpha between 2° and 180°, the phase increment between 0 and 359° by 1° steps and a between 125µm and 1250µm.

Protocol1: empirically defined set with 28 acquisitions: TR=12.5ms, alpha=24°, a=312µm, and 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°]

Protocol2: optimization performed considering T1=0.9s, T2=0.08s and D=1.10-⁹m²s-¹

Protocol3: optimization performed considering T1=5s, T2=0.05s and D=1.10-⁹m²s-¹

Protocol4: optimization performed by summing the normalized CRLB over 4 values : T1=[0.5/5/0.5/5]s, T2=[0.05/0.05/2/2]s, and D=1.10-⁹m²s-¹

optimized parameters:

Protocol2: TR=12.5ms, alpha=[2/8/12/20/20/21/21/27/32/38/48/49], phase increments =[356/4/1/178/283/3/3/357/240/356/357/182] a=[280/742/1222/125/126/125/125/147/131/1076/943/1022].

Protocol3: TR=12ms, alpha=[4/4/5/9/11/12/13/13/14/16/16/18], phase increments =[359/147/140/360/1/359/359/1/359/359/358/359] a=[845/547/928/965/168/157/146/154/143/816/1234/1060].

Protocol4: TR=10.3ms, alpha=[4/8/12/14/14/15/16/19/19/23/23/61], phase increments =[2/0/359/1/359/178/2/182/357/0/2/180] a=[915/1006/146/142/138/1204/131/163/1117/872/940/1219].

It covers very different flip angles, phase increments and spoiling gradients.

Table1 provides the expected error for each protocol estimated for typical brain tissue relaxation and diffusion parameters. As protocol2 was specially designed for these expected relaxation and diffusion parameters, it provides the minimal error. It can be seen that it provides more precise estimations than protocol1 with much less N_meas acquisitions demonstrating the gain in total scan time when using optimization as compared to empirically defined protocols.

To estimate if the protocols provide adequate precision over a wide range of relaxation rates, the relative error was computed for a large range of T1 and T2 (from 0.5 to 5s and 50ms to 2000ms respectively) (figure 1-4). Results are shown for D=1.10^-9mm²s^-1, but were similar for values between 0.5 to 3 demonstrating a limited influence of diffusion. For all parameters, excepted for T2, the relative error was increasing with increasing T1 and decreasing T2. Protocol3, which was optimized for targeted T1/T2=5s/50ms provides the smallest maximum relative error over the range of T1 and T2 values (reduced by a factor 2 as compare to protocol2 for M0, T1 and ADC). When using an objective function designed for extreme T1 and T2 values, such as in protocol4, more precise estimates with more uniform errors are obtained in a wider range.

Optimization of acquisition parameters through the CRLB may provide a valuable tool for objective protocol definition in MRI, especially with SSFP sequence in which the contrast mechanisms are leading to complex signal models. In the context of multi-parametric mapping, it was shown that protocols can be optimized for proton density, background phase, effective flip angle, relaxation times and apparent diffusion coefficient mapping with targeted relative errors on the fitted physical parameters. Based on an adequate choice of the objective function, it was shown that error on fitted parameters can be minimized for a given range of T1 and T2.