Introduction

Echo-planar imaging is an efficient acquisition scheme, capable of providing multi-contrast and multi-parametric images.¹ Moreover, multi-echo readout and radial sampling strategies emerge as a promising alternative to conventional breath-holding Cartesian sampling, e.g. for free-breathing quantitative $$$R_2^*$$$ mapping of the liver.^2-4 However, these techniques require moderate undersampling factors and hence physiological gating methods to overcome motion problems. In contrast, this work presents a model-based reconstruction technique for dynamic $$$R_2^*$$$ and $$$B_0$$$ field inhomogeneity quantification in combination with simultaneous water/fat separation. Acquisitions are based on the multi-echo multi-spoke radial FLASH sequence, while first validating applications study the effects of brain activation and abdominal breathing on quantitative parameter maps.

Methods

The multi-echo multi-spoke radial FLASH sequence⁵ has been applied to dynamic water/fat separation⁶ based on triple-echo acquisition with asymmetric readouts⁷. Here, the echo train length (ETL) is elongated to allow for the inclusion of transversal magnetization relaxation ($$$R_2^*$$$) in the forward model,

$$F_{j,m}(x) = P_m \mathcal{F} \big\{ (\text{W} + \text{F} \cdot z_m) \cdot e^{- R_2^* \text{TE}_m} \cdot e^{i2\pi f_{B_0} \text{TE}_m} \cdot c_j \big\} \;\; \text{with} \; x=(\text{W}, \text{F}, R_2^*, f_{B_0}, c_1, \cdots, c_N)^T \; . \;\;\; (1)$$

$$$P_m$$$ and $$$\mathcal{F}$$$ is the sampling pattern of the $$$m$$$th echo and 2D FFT, respectively. $$$z_m$$$ is the six-peak fat spectrum at the $$$m$$$th echo time ($$$\text{TE}_m$$$). The unknown $$$x$$$ contains the water (W), fat (F), $$$R_2^*$$$, and $$$B_0$$$ field inhomogeneity ($$$f_{B_0}$$$) maps as well as one set of coil sensitivity maps ($$$c_j$$$). The joint estimation of all unknowns in Equation (1) poses a non-linear non-convex inverse problem,

$$\Phi(\hat{x}) = \arg\!\min_{\hat{x}} \underbrace{\sum_{j=1}^{N} \sum_{m=1}^{\text{ETL}}\left\lVert y_{j,m} - F_{j,m}(T\hat{x}) \right\rVert_2^2 + \alpha_n \left\lVert \hat{x} - \hat{x}_0 \right\rVert_2^2}_{\text{smooth}} + \beta_n \left\lVert W ( \hat{x} - \hat{x}_0 ) \right\rVert_1 \;\;\; (2)$$

where a Sobolev-norm⁸ weight matrix ($$$T$$$) is applied onto $$$f_{B_0}$$$ and $$$c_j$$$ to enforce spatial smoothness. In addition, $$$\ell_1$$$ wavelet joint sparsity constraint is applied onto the W, F and $$$R_2^*$$$ maps. Given the non-smoothness of the $$$\ell_1$$$ regularization term, IRGNM-FISTA⁹ is adapted to solve this problem. The derivative of the smooth part of Equation (2) supplies the normal equation for FISTA. This reconstruction technique has been implemented on BART¹⁰ and all reconstructions were run on TITAN Xp GPU (Nvidia, Santa Clara, CA, USA).

The initialization of the first frame is given by the estimated W, F and $$$f_{B_0}$$$ using only the first three echoes, while both $$$R_2^*$$$ and $$$c_j$$$ are initialized with 0. The rational is that this initialization strategy can better counteract potential phase wrapping from later echoes and foster convergence. The later frames are initialized by the estimate from their preceding frame, while $$$\hat{x}_0$$$ is set as the initialization damped by a factor of 0.9. The regularization strength is defined as $$$\alpha_n = \beta_n = (1/3)^{n-1}$$$ with the Newton iteration $$$n \in [1,~10]$$$.

Data acquisition were performed on a 3 T scanner (Magnetom Prisma, Siemens Healthineers, Erlangen, Germany) with a 64-channel head coil and an 18-channel body matrix coil for brain and liver studies, respectively. Details of the acquisition parameters are listed in Table 1. The brain functional MRI (fMRI) measurement lasted 258 seconds, and used a repetitive visual stimulation pattern (18-second black-and-white checkerboards) and control pattern (12-second gray field). In this preliminary study, a relatively low bandwidth and large echo spacing were used, so higher spatiotemporal resolution might be achieved with further optimization. The liver studies were conducted during free breathing without the use of gating methods.

Results

With radial sampling, as shown in Figure 1, the echo images with parallel imaging as nonlinear inverse reconstruction⁸ are resistant to spatial distortion artifacts. Moreover, the combined RSS image shows no visible signal loss caused by $$$B_0$$$ field inhomogeneity. Figure 2 shows the water, fat, $$$R_2^*$$$ and $$$f_{B_0}$$$ maps via the proposed model-based reconstruction and the time course of $$$T_2^*$$$ values (i.e. $$$1000 / R_2^*$$$) from three selected brain pixels. The $$$T_2^*$$$ time course from the purple pixel within visual cortex shows significant change of $$$T_2^*$$$ values, i.e. 27 ms increase from its baseline value 40 ms, corresponding to a relative change of 68% , while the other two pixels show only small noise-like $$$T_2^*$$$ variations.

Figures 3 and 4 depict model-based reconstructions from multi-echo radial FLASH acquisitions with 33 and 15 shots per frame, respectively. No visible image degradation can be found for these degrees of undersampling. However, respiration motions perturb both $$$B_0$$$ field inhomogeneities (see the $$$f_{B_0}$$$ maps in Figure 3) and $$$R_2^*$$$ relaxation (time course of mean $$$R_2^*$$$ in Figure 4), especially in the lower segment of liver and spleen. These changes most likely reflect the strong spin dephasing during organ movements when an air-tissue-like interface is created between lung and liver.

Discussion and Conclusion

Multi-echo radial sampling is well-suited for dynamic and quantitative imaging given its immunity to spatial distortion and resistance to motion. Moreover, dynamic $$$R_2^*$$$ mapping via the proposed model-based reconstruction offers a direct measure of time-resolved $$$R_2^*$$$ alterations in response to physiological changes.

Acknowledgements

No acknowledgement found.