MRI researchers with interest in model-based reconstruction, relaxometry, and parameter mapping techniques; Clinicians with interest in quantitative MRI techniques

A fast and accurate determination of T1 values can be accomplished by Look-Locker type MRI sequences^1,2. Additionally, model-based image reconstruction techniques exploit compressibility in the parameter domain^3-5 and allow for reconstruction of high-resolution parameter maps even from strongly undersampled data. Recently, we developed a model-based reconstruction algorithm for high-resolution T1 maps using a radially sampled single-shot inversion-recovery (IR) FLASH experiment⁶. Major improvements in the signal model as well as in the implementation now enable us to perform fast and precise T1 mapping in vivo with high accuracy.

We used a radial FLASH sequence (resolution 0.75 x 0.75 x 5 mm³, 4° nominal flip angle, TR = 2.99 ms, 1500 projections, T_acq = 4.5 s) in combination with a preceding inversion module (non-selective, adiabatic pulse). All experiments were performed at 3 T (MAGNETOM Prisma, Siemens Healthcare, Erlangen, Germany) using a 64 channel head coil. To keep the TR at minimum we utilized a spoiling scheme based on random RF phases⁷ rather than the more conventional RF spoiling. The radial view order was based on the small Golden Angle⁸ of about 68.75°.

The reconstruction was formulated as a nonlinear optimization problem and solved by the iteratively regularized Gauss Newton (IRGN) method similar to [9]. Since regularization strength always translates into a tradeoff-off between bias and noise, a proper regularization is essential in the context of quantitative MRI. Here, we selected new parameters for the original signal equation governing the magnetization time course of a Look-Locker sequence exploiting the relationship $$$M_0⁄M_{\text{SS}} =R_1^*⁄R_1$$$:

$$M (M_{\text{SS}},r_1,r_1^{'}) = M_{\text{SS}}\left[1-(r_1 r_1^{'} )^{(t/\Delta t)} \left(\frac{\ln⁡ r_1^{'} }{\ln⁡ r_1} +2 \right) \right]$$

where $$$M_{\text{SS}}$$$ is the steady-state magnetization, $$$M_0$$$ the equilibrium magnetization and $$$R_1^*$$$ the effective relaxation rate. We further introduced the dimensionless variables $$$r_1=e^{-R_1 \Delta t}$$$ and $$$r_1^{'}=(\cos \alpha)^{-\Delta t / \text{TR}}$$$ that describe the relative signal loss per unit time due to the two different sources of relaxation, namely inherent T1 relaxation and successive signal loss due to the train of RF pulses with flip angle $$$\alpha$$$.

For $$$r_1^{'}$$$ we expect a similar smoothness as for $$$\alpha$$$ and hence a Sobolov norm penalization can be used to ensure a proper smoothness of this map. For the remaining two transformed variables, simple Tikhonov regularization can be used without biasing the quantitative solution inappropriately. For the remaining two transformed variables, simple Tikhonov regularization can be used without biasing the quantitative solution inappropriately.

The task of T1 mapping can now be stated as a nonlinear inverse problem of the form

$$F_{j,t}(x)=y_{j,t},\,\,\,\, F_{j,t}:x\rightarrow P_k \;\mathcal{F} \;P_{\text{FoV}} \;C_j \;M(x;t)$$

where $$$y_{j,t}$$$ is the raw data from the $$$j$$$-th coil at time $$$t$$$, $$$C_j$$$ are the (predetermined) coil sensitivity profiles, $$$\mathcal{F}$$$ is the Fourier transform, and $$$P_k$$$ and $$$P_{\text{FoV}}$$$ are the orthogonal projections onto the measured k-space trajectory and the field of view, respectively. Initial values for the IRGN method are $$$M_{\text{SS}}^{\text{ini}}=0$$$, $$$r_1^{'}=1$$$ and $$$r_1=e^{-1/2}$$$. Reconstruction was performed with the Parallel Computing Toolbox of MATLAB (The MathWorks, Inc., Natick, Massachusetts) in less than 2 min on a single GPU (NVIDIA TITAN GPX).Figure 1 shows the reconstructed T1 map from a phantom containing 6 compartments with different T1 values (Diagnostic Sonar LTD., Scotland, UK). For analysis we calculated the average and standard deviation per ROI and plotted this against the number of IRGN steps. As a reference we used values obtained by a Cartesian FSE sequence with 13 different inversion times. Figure 2 shows the transverse brain T1 map (iteration #13) from a volunteer with no known illnesses using the very same parameters as in the phantom study. Also a proton density map and a flip angle map were calculated from the transformed set of parameters.We presented a model-based reconstruction algorithm for high-resolution T1 maps from a radially sampled IR-FLASH experiment. The chosen parametric formulation allows for separation of the effective relaxation rate into two parts: The inherent T1 relaxation and the contribution due to imaging. The latter one is governed by the flip angle distribution over the FoV and hence a high degree of smoothness can be exploited as prior knowledge. The presented method results in relevant T1 maps for most in vivo applications with very high accuracy and precision as validated in phantom studies.