Advances in Model-Based Reconstruction of High Resolution T1 maps
Volkert Roeloffs1, Xiaoqing Wang1, and Jens Frahm1

1Biomedizinische NMR Forschungs GmbH, Max Planck Institute for Biophysical Chemistry, Goettingen, Germany

Synopsis

Fast and accurate determination of T1 values can be accomplished by Look-Locker type MRI sequences. Here, we formulate T1 mapping as a nonlinear optimization problem which we subsequently solve by the iteratively regularized Gauss Newton (IRGN) method. Our choice of the model parameterization allows to exploit smoothness of the spatial flip angle distribution as additional prior knownledge. This model-based reconstruction allows accurate and precise reconstruction of high resolution T1 maps from radial, highly undersampled data as validated in phantom studies and demonstrated in the human brain.

Target Audience

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

Purpose

A fast and accurate determination of T1 values can be accomplished by Look-Locker type MRI sequences1,2. Additionally, model-based image reconstruction techniques exploit compressibility in the parameter domain3-5 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.

Methods

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, Tacq = 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).

Results

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.

Discussion and Conclusion

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.

Acknowledgements

No acknowledgement found.

References

[1] Look D., Locker D. Time saving in measurement of NMR and EPR relaxation times. Rev. Sci. Instrum. 1970;41:250-251

[2] Deichmann R., Haase A. Quantification of T1 values by SNAPSHOT-FLASH NMR imaging. JMR 1992;96(3):608-612

[3] Sumpf T., Model-based nonlinear inverse reconstruction for T2 mapping using highly undersampled spin-echo MRI. JMRI 2011;34(2):420-428

[4] Tran-Gia J. Model-based Acceleration of Parameter mapping (MAP) for saturation prepared radially acquired data. MRM 2013;70(6):1524-1534

[5] Block KT. Model-based iterative reconstruction for radial fast spin-echo MRI. IEEE Trans. Med. Imag. 2009;28(11):1759-1769

[6] Roeloffs V. High resolution T1 mapping within seconds: Model-Based reconstruction without regularization. In: Proc. Intl. Soc. Mag. Reson. Med 2015; 23:3713.

[7] Roeloffs V. Spoiling without additional gradients: Radial FLASH MRI with randomized radiofrequency phases. MRM 2015:1522-2594

[8] Winkelmann S. An optimal radial profile order based on the Golden Ratio for time-resolved MRI. IEEE Trans Med Imaging. 2007;26(1):68-76

[9] Uecker M. Nonlinear inverse reconstruction for real-time MRI of the human heart using undersampled radial FLASH. MRM 2010;63(6): 1456--1462

Figures

Figure 1: (left) Reconstructed T1 map of a phantom containing 6 tubes with known T1 values. (right) ROI-wise mean and standard deviation as a function of IRGN iteration. Solid lines are reference values.

Figure 2: Transversal section of a human brain. Reconstructed maps obtained after the 13th IRGN iteration.



Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
4236