MR parameter mapping has shown great potential but is still limited in clinical applications due to the lengthy acquisition time. Several reconstruction methods have been proposed to improve the acquisition speed^1-11. The purpose of this work is to develop and evaluate a novel image reconstruction method to accelerate parameter mapping with reduced multi-channel acquisition using alternating projections on the single-exponential parametric manifold, the subspace with data consistency, and the convex set of regularized coil sensitivities. The method is a multi-channel extension of our previous work in¹². The method is applicable to other quantitative imaging applications beyond parameter mapping.

In MR parameter mapping, the mth reconstructed image $$$I_{m}$$$ is directly related to the acquisition at the mth echo time $$$d_{m}$$$ as: $$${{\bf{d}}_m}={{\bf{F}}_m}{{\bf{S}}_m}{{\bf{I}}_m}+{{\bf{n}}_m}$$$, where $$$F_{m}$$$ the Fourier operator with a specific undersampling pattern at m, $$$S_{m}$$$ is a diagonal matrix representing the sensitivity map, and $$$n_{m}$$$ denotes k-space noise. In parameter mapping, the image series $$$I_{m}$$$ can be modeled with parameters ρ and θ as: $$${{\bf{I}}_m}={{\rm{P}}_m}({\bf{\theta}}){\bf{\rho}}$$$, where $$$\bf P_{m}(\theta)$$$ is a parametric function of θ, and also specifies scanning setting at the mth acquisition, and ρ is the parameter linearly related to the image. For example, in the case of T2 mapping, $$${{\bf{I}}_m}={\bf{\rho }}{e^{-TE(m)\cdot{\bf{\theta}}}}$$$, where θ is a vectorized 1/T₂ values at all pixels and ρ the proton density. Thereby the image series acquired at different echo times lies in a low dimensional manifold. To recover the PARametric MAnifold (named PARMA), we solve the unknown parameter maps θ and ρ and coil sensitivities $$$S_{m}$$$ by: $$\left\{{{\bf{\hat\theta}},{\bf{\hat\rho}},{{{\bf{\hat S}}}_m}}\right\}=\mathop{\arg\min}\limits_{{\bf{\rho,\theta}}}\sum\limits_{m = 1}^M {[\left|\left|{{\bf{d}}_m}-{{\bf{F}}_m}{{\bf{S}}_m}{{\bf{P}}_m}({\bf{\theta}}){\bf{\rho}}\right|\right|_2^2+\beta{\rm{TV}}({{\bf{S}}_m})]}\quad(1)$$.

Initialization: Image reconstruction (convex): We first reconstruct each individual image at the mth echo time using Sparse BLIP¹³: $$\left\{{{{{\bf{\hat I}}}_m},{{{\bf{\hat S}}}_m}}\right\}=\mathop{\arg\min}\limits_{{{\bf{I}}_m},{{\bf{S}}_m}} \left\|{{{\bf{d}}_m}-{{\bf{F}}_m}{{\bf{S}}_m}{{\bf{I}}_m}}\right\|_2^2+\lambda{\rm{R}}({{\bf{I}}_m})+\beta {\rm{TV}}({{\bf{S}}_m})\quad(2)$$where $$$\bf\widehat{I}_{m}$$$ and $$$\bf\widehat{S}_{m}$$$ are the reconstructed image and coil sensitivities, λ and β control the weights of regularization. Total variation is used to regularize both image and coil sensitivities.

Step 1: Projection onto manifold: The initialized images are projected onto a closest manifold using $${{\rm{P}}_\Omega}\left({\left|{{\bf{\hat I}}}\right|}\right):=\mathop{\arg\min}\limits_{\bf{I}}\left\{{\left|\left||{{\bf{\hat I}}}|-{\bf{I}}\right|\right|_2^2:{\bf{I}} \in\Omega}\right\}\quad(3)$$ where Ω is a manifold with parameters θ and ρ. $$$\bf I$$$ includes a series of images $$$I_{m}$$$ at all m, $$$\bf\left|\widehat{I}\right|$$$ denotes the operation of taking the magnitude of $$$\bf\widehat{I}$$$. The projection is equivalent to finding the parameters: $$\left\{{{\bf{\hat\theta}},{\bf{\hat\rho}}}\right\}=\mathop{\arg\min}\limits_{{\bf{\theta,\rho}}}\mathop\sum\limits_{m=1}^{\rm{M}} \left|\left||{{\bf{\hat I}}_{\bf{m}}}|-|{{\rm{P}}_m}\left({\bf{\theta}}\right){\bf{\rho}}|\right|\right|_2^2\quad(4)$$ Levenberg-Marquardt algorithm is used to fit the parametric model.

Step 2: Projection onto the subspace with data consistency: The updated images series on the manifold are further transformed into k-space to enforce the data consistency constraint. Specifically, the image series are projected onto a subspace by maintaining the updated k-space data at the acquired locations and replacing the k-space data at the unacquired locations using a weighted combination of the updated value and measurement.

Step 3: Projection onto a convex set: We then update the coil sensitivities by solving a convex optimization problem $${{\bf{\hat S}}_m}=\mathop {\arg\min}\limits_{{{\bf{S}}_m}}\left\|{{{\bf{d}}_m}-{{\bf{F}}_m}{{\bf{S}}_m}{{\bf{I}}_m}}\right\|_2^2+\beta{\rm{TV}}({{\bf{S}}_m})\quad(5)$$ using the images from Step 2. The above three steps are then repeated iteratively. Such an alternating projection converges if the initial value is close to the true intersection of the above three sets¹⁴.

The proposed method was evaluated using a set of 6-channel T2 brain dataset from a 3T scanner(MAGNETOM Trio, SIEMENS, Germany) with a turbo spin echo sequence (matrix size = 192 x 192, FOV = 192 x 192 mm, slice thickness = 3 mm, ETL = 16, △TE = 8.8 ms, TR = 4000ms, bandwidth = 362Hz/pixel). The full k-space data was retrospectively undersampled with a reduction factor of 5 using different 1D random undersampling patterns for different TEs. The proposed PARMA method was compared to two sparsity-constrained compressed sensing methods, one sparsified with principal component analysis (CS-PCA)² and the other with dictionaries learned using the relaxation model (MBDL)⁷. Figure 1 shows the T2 maps and the corresponding errors obtained using CS-PCA, MBDL, and the proposed PARMA. Compared to the sparsity-constrained methods, the proposed method improves the T2 accuracy.In this paper, a novel parametric manifold recovery method PARMA is proposed for accelerated MR quantitative imaging. Compared to the existing methods, the proposed method enforces the parametric model by projecting the reconstruction to its closest manifold and enforces data consistency from multiple channels at the same time. The experimental results show the potential of highly accelerated quantitative imaging by the proposed method.