Sen Ma^{1,2}, Anthony G Christodoulou^{2}, Christopher T Nguyen^{2,3}, Fei Han^{4}, Nan Wang^{1,2}, Yibin Xie^{2}, and Debiao Li^{1,2}

Three-dimensional, multi-parametric quantitative mapping of relaxation and diffusion parameters is desirable for many clinical imaging application, including the diagnosis and follow-up assessment of tumors. Traditionally, this is performed by separate scans which are time-consuming. We propose a novel Multitasking framework to achieve 3D whole-brain simultaneous T1/T2/ADC mapping in ~9min. The underlying multidimensional image is modeled as a low-rank tensor, with a time-resolved phase correction technique to compensate for the phase inconsistency induced by pulsatile motion during diffusion preparation. T1/T2/ADC measurements in healthy volunteers agree with reference methods, yielding intraclass correlation coefficients > 0.80 for both gray and white matter.

**Sequence Design:** T1-T2-diffusion
weighting was generated using hybrid T2- and diffusion-preparation (Fig. 1a). A
crusher gradient scheme was implemented to prevent the inconsistent phase from
being tipped into magnitude at the cost of half signal loss^{8,9},
yielding a T1-decay signal evolution which was sampled using FLASH readouts
(flip angle=5, TR/TE=5.6/2.5ms). An 1s gap was
placed at the end of each shot to allow long-T1 tissues to recover towards
thermal equilibrium. Imaging data ($$$\mathbf{d}_\text{img}$$$) were sampled with Gaussian variable
density along *ky* and *kz*, while training data ($$$\mathbf{d}_\text{tr}$$$) were sampled every 8 readouts at
k-space center (Fig. 1b). The resulting signal equation is:$$S_{n}=\frac{1}{2}\cdot A \cdot (e^{-\frac{TR}{T1}}\cos(\alpha ))^{n}\cdot e^{-\frac{TE_{prep}}{T2}}\cdot e^{-bD}\cdot \sin(\alpha ),$$where $$$A$$$ absorbs proton density and T2* weighting, $$$n$$$ is the readout index, and $$$\alpha$$$ denotes FLASH flip angle.

**Reconstruction:** The multidimensional image can be represented as a 5-way tensor $$$\mathcal{A}$$$ with one dimension indexing voxel location $$$\mathbf{r}=(x,y,z)$$$ and four indexing different time/parameter dimensions: T1 decay $$$t_{\text{T1}}$$$, T2prep duration $$$t_{\text{T2}}$$$, b-value $$$t_{\text{b}}$$$, and diffusion direction $$$t_{\text{dir}}$$$. A time-resolved phase map $$$\mathbf{P}$$$ is applied to compensate for shot-to-shot phase inconsistency, restoring spatiotemporal image correlation thus lowering the effective tensor rank. We express this phase-corrected LRT model as $$$\mathbf{A}_{(1)}=\mathbf{P}\circ (\mathbf{U\Phi})$$$ where $$$\mathbf{A}_{(1)}$$$ is the unfolded matrix form of $$$\mathcal{A}$$$, $$$\mathbf{U}$$$ contains spatial coefficient maps, and $$$\mathbf{\Phi}$$$ contains multidimensional temporal basis functions characterizing the four temporal dimensions^{10}. $$$\mathbf{P}$$$ is estimated from a real-time reconstruction:$$\mathbf{P}=\angle S(\mathbf{I}_{0}),\: \text{with}\: \mathbf{I}_{0}=\mathbf{U}_{0}\mathbf{\Phi}_{0,\text{rt}},\: \text{where}\: \mathbf{U}_{0}=\underset{\mathbf{U}}{\text{argmin}}\left \| \mathbf{d}_{\text{img}}-E(\mathbf{U}\mathbf{\Phi}_{0,\text{rt}}) \right \|^{2},$$where the real-time temporal subspace $$$\mathbf{\Phi}_{0,\text{rt}}$$$ is estimated from SVD of $$$\mathbf{d}_{\text{tr}}$$$, $$$E(\cdot )$$$ combines spatial encoding and sampling, and $$$S(\cdot )$$$ performs smoothing. $$$\mathbf{\Phi}$$$ is determined via Bloch-constrained LRT completion and high-order SVD^{7} of the multidimensional tensor subspace. $$$\mathbf{U}$$$ is recovered by mapping the spatial coefficient maps from the phase-varying real-time subspace to the phase-corrected multidimensional subspace via applying the time-resolved $$$\mathbf{P}$$$ and projection operator.

**Experiment Design:** Data were collected on a 3T Siemens Vida scanner on an ISMRM/NIST system phantom and $$$n$$$=4 healthy volunteers. The imaging protocol included: 1) IR-TSE for T1 reference; 2) Multiecho-SE for T2 reference; 3) Single-shot EPI for ADC reference; 4) T1/T2/ADC Multitasking sequence with 4 T2-prep durations of 13.62, 31.4, 80, and 110ms, and 6 diffusion-preps of b=400 and 800s/mm^{2}, 3 noncolinear directions. Reference scans took ~18min while Multitasking scan took ~9min. All protocols were scanned with resolution=1.5x1.5x5mm^{3}, FOV=240x240mm^{2}, 20 slices/partitions with matched positions.

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