Quantitative Magnetic Resonance Imaging (qMRI) provides a measure of physical tissue properties that are ideally independent from scanner hardware and the employed sequence. This allows a better inter- and intra-patient comparison, thus bearing the potential to be a good biomarker for pathology. However, long acquisition times are usually required for qMRI in comparison to conventional MRI, a disadvantage which is an obstacle for its use in clinical research and routine. Several iterative reconstruction methods have been developed to accelerate qMRI sequences; these algorithms usually go along with long computation times and a reduced robustness compared to a direct Fourier transform. Here, we propose to split the optimization problem of a model-based reconstruction of T2 maps into smaller sub-problems, with the purpose of increasing its robustness as well as decreasing its computational cost. This can be generalized to other, similarly posed problems.

It is common practice for model-based methods to define a cost-function that incorporates the model behavior directly within a data fidelity term^1-3. This is also done in the “Model-Based Accelerated Relaxometry by Iterative Nonlinear Inversion” (MARTINI)², a T2 mapping algorithm using a multi-echo spin-echo (MESE) sequence. MARTINI’s cost-function is defined as follows:

$$\phi(M_{0},T2)=\frac{1}{2}\sum_{c=1}^N\sum_{t\in{TE}}^{}{\parallel}PF\left\{S_{c}M_{0}\exp\left(-\frac{t}{T2}\right)\right\}-Y_{t,c}{\parallel}_2^2$$

with TE being the echo times, N the number of coil elements, P a binary mask representing the sampling pattern, F the Fourier transform operator, S the coil sensitivities, M₀ the equilibrium magnetization, T2 the transverse relaxation and Y the acquired k-space data. Minimizing this cost-function will result in an estimation of T2 and M₀. However, the minimization of this nonlinear problem is numerically challenging and may lead to image artifacts and long reconstruction times. We suggest splitting up the problem similarly to what was proposed for compressed sensing⁴, resulting in a 2-step algorithm which we term “Split-Algorithm for Fast T2 mapping” (SAFT).

Step 1: The MESE magnetization is calculated based on an initial guess of T2 and M₀ using the forward signal model:$$\hat{M}_{t}=M_{0}\exp\left(-\frac{t}{T2}\right)$$Using this first guess of the magnetization, the following problem is solved with a linear least-squares algorithm:$$\phi_{1}(M)=\frac{1}{2}\sum_{c=1}^N\sum_{t\in{TE}}^{}{\parallel}PF\left\{S_{c}M_{t}\right\}-Y_{t,c}{\parallel}_2^2+\sum_{t\in{TE}}^{}\alpha{\parallel}M_{t}-\hat{M}_{t}{\parallel}_2^2$$estimating the magnetization M that best fits the acquired data. The second l₂-norm of the cost function forces the magnetization to be similar to the previously calculated $$$\hat{M}$$$ with the similarity weighted by α.

Step 2: The MESE signal model is fitted onto the previously estimated M by solving the following problem with a nonlinear least-squares algorithm,$$\phi_{2}(T2,M_{0})=\frac{1}{2}\sum_{t\in{TE}}^{}{\parallel}M_{t}-M_{0}\exp\left(-\frac{t}{T2}\right){\parallel}_2^2$$yielding a new estimate of T2 and M₀. Subsequently, steps 1 and 2 are iteratively repeated until the algorithm converges to a minimum, providing an approximation of T2 and M0. Optionally, a spatial regularization can be added to Φ₂. Here, we performed an additional reconstruction using a wavelet sparsity constraint for both T2 and M₀.

After obtaining written consent, three whole-brain MESE datasets (TA 3:28min, acq. matrix 260x512, resolution 0.75x0.45x3mm³, slice gap 0.3mm, TR/∆TE 4000/10.9ms, Number of echoes/slices/concatenations 16/43/2) of healthy volunteers were acquired at 3T (MAGNETOM Skyra, Siemens Healthcare, Germany) using commercially available 20- and 32-channel head/neck coils. The used prototype MESE sequence was 10x undersampled according to a GRAPPATINI sampling pattern⁵. The datasets were reconstructed using MARTINI² and the proposed algorithm, with and without a spatial regularization (SAFT vs. regularized SAFT).To compare the convergence of MARTINI and SAFT, the cost value of each iteration according to Φ(T2,M0) is plotted in Figure 1. For a fair comparison, all algorithms were initialized with the same guess for T2 and M₀. The plot demonstrates that both SAFT reconstructions converge smoothly and reach a minimum after ~25 iterations. MARTINI’s cost values jump initially until the algorithm starts converging at ~15 iterations and reaches a minimum similar to SAFT after ~60 iterations. Example T2 maps, reconstructed using MARTINI, non-regularized and regularized SAFT, are illustrated in Figure 2. It can be seen that the non-regularized SAFT reconstruction resembles the results of MARTINI. The wavelet-regularized SAFT reconstruction yields similar results but with less noise in the parameter maps. It should be noted that an even faster computation performance can be achieved solving Φ₂ with a log-linear regression, because then the algorithm comprises only two linear problems. However, we used a nonlinear approach to avoid noise-induced T2 overestimation⁶.We suggest a method that splits the nonlinear problem of model-based reconstruction into smaller sub-problems which are solved alternately. The splitting results in a convex convergence, improving the robustness of the inverse reconstruction problem. Furthermore, we demonstrate that the improved robustness can be used to add non-convex regularization (e.g. sparsity constraints) to the optimization in order to further improve the estimated quantitative maps.