Synopsis

Incorporating TGV-regularization to accelerated model-based parameter-quantification can lead to improved image quality, however, adds non-differentiability to the problem which poses a problem for commonly used first order optimization methods like non-linear Conjugate-Gradient. The proposed method overcomes this limitation by handling the problem within a Gauss-Newton-framework and applying a Primal-Dual-algorithm to solve the inner TGV-regularized problem. Numerical simulations exhibit high agreement to references for four different parameter mapping problems up to 18-fold acceleration. In-vivo results for T1-VFA and T2-MESE models strengthen these findings. The proposed method offers huge acceleration potential for model-based parameter-quantification with similar quantification quality as fully sampled data.

Purpose

Methods

MBPQ problems can be described by a data fidelity term, reflecting the mismatch between the physical model and measured data, and a regularization term $$$R(x)$$$ that accounts for numerical stabilization and enforces a-priori information on the unknown parameters $$$x$$$. $$\underset{x}{\min}\frac{\lambda}{2}\|A(x)-d\|^2_2+R(x)$$ The model $$$A(x)$$$ consists of receiver coil sensitivites, Fourier Transform, k-space sampling pattern and a model-specific connection between sequence parameter variations and MR signal and is typically non-linear. By linearization of $$$A$$$, termed $$$DA$$$ (Gauss-Newton method) the non-convex problem above rewrites as $$\underset{x}{\min}\frac{\lambda}{2}\|DAx-d_k\|^2_2+R(x)+\frac{1}{2\gamma}\|x-x_k\|_2^2$$ where $$$d_k$$$ contains the data and constant terms of the linearization. This problem is the inner problem of the Gauss-Newton method and has to be solved in every linearization step. In case of TGV-regularization a Primal-Dual (PD) algorithm⁶ with line-search⁷ is used for optimization. $$$\|x-x_k\|_2^2$$$ serves as step size restriction on the inner optimization problem ($$$x_k$$$ linearization-point).

The proposed method was validated on different MBPQ problems and compared to GN with Tikhonov regularization, i.e. $$$R(x)=\|x\|_2^2$$$ (GN-Tikh). T1 was quantified using a model for an Inversion-Recovery-Look-Locker(IR-LL) sequence^8,9 and with the Variable-Flip-Angle model (VFA)^10,11,12. For T2 quantification the standard mono-exponential model of T2 decay (MESE) was used as well as a VFA approach^11,12. First, numerical Shepp-Logan phantom data was generated and artificially subsampled to validate the algorithm. In a second step this subsampling was applied to VFA-T1 and MESE-T2 in-vivo datasets.

The numerical T1 phantom consisted of areas with 3000/1200/800ms, the T2 of 1200/120/80ms. Artificial coil sensitivities were generated with Biot-Sarvats law. Data for VFA- and IR-LL models was undersampled by $$$AF\in\{1,6,12,18\}$$$ using a variable-density pseudo-random pattern¹³. For MESE data, a pseudo-radial pattern was used since data was acquired slice by slice. All phantoms were 128x128x1 in size and a reasonable amount of Gaussian noise was added to every coil data. A constant value for T1 and T2 was used as an initial guess except for the VFA approach where a low-resolution DESPOT1^11,12 fit was used. Coil sensitivities were estimated with the method of Uecker et al.¹⁴. B1 correction for VFA was done using the Bloch-Siegert approach¹⁵. VFA-T2 was initialized with the results of the VFA-T1 fit and a constant value for T2.

In-vivo data for the VFA model was acquired in 3D. For reconstruction, a FFT in z-direction was applied and undersampling was performed retrospectively in 2D on the central slice. For evaluation, segmentation of gray and white matter was carried out using atlas-based registration with the SPM12-Toolbox. Fully sampled references were generated with DESPOT-T1-method and pixel-wise MESE-exponential fit.

Results

T1 quantification of simulated data with the proposed method is shown in Figure 1a on the left, the GN-Tikh references are shown on the right. Parameter maps are displayed in ms. Results for T2 quantification are given in Figure 1b. Quantitative evaluation of three different ROIs, corresponding to the three phantom regions, are show as boxplot in Figure 2. In-vivo results for VFA-T1 and MESE-T2 quantification are shown in Figure 3 a) and b), respectively. Boxplots for segmented gray and white matter and increasing acceleration are given in Figure 4.

Discussion

Conclusion

Acknowledgements

References

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