Dynamic-Contrast-Enhanced MRI(DCE-MRI) is a powerful MR technique that provides both anatomy and perfusion information. Low-Rank(LR)/Partially-Separable^[1,2,3] models are widely applied in DCE-MRI for acceleration and improved sparse-reconstruction by exploiting spatial-temporal correlation. Existing LR methods model the spatial-temporal signal as a sparse and linear combination of basis-signals. However, these linear models cannot fully capture the complex contrast enhancement and may lead to inaccurate contrast dynamics in reconstruction. Dictionary-Learning based methods^[4-5] tackle this issue using over-complete dictionaries, which needs expensive computation and training data. Here we propose a generalized LR model with Adaptive-Nonlinear-Kernels. This Kernelized-Low-Rank(KLR) model assumes the LR property in a nonlinear-transform domain instead of the original spatial-temporal domain. The proposed method captures the spatial-temporal dynamics with sparser nonlinear representations, and achieves more accurate reconstruction results.

1. Kernelized-Low-Rank(KLR) Model

Due to non-uniform contrast changes and motion, dynamic MRI signals may not be completely rank-deficient as desired with linear Singular-Value-Decomposition(SVD). However, the spatial-temporal signal $$$X$$$ can be better represented (more rank-deficient) after certain nonlinear transforms $$$T$$$ with nonlinear kernel $$$K$$$.

Based on the relationship between SVD and Principle-Component-Analysis (PCA), we proposed to use Kernel-PCA^[6] to compute singular-values and corresponding LR representation of nonlinear-transformed data $$$T_K(X)$$$. In Kernel-PCA the decomposition is achieved more efficiently by conducting Eigen-Decomposition of the kernel-transformed Covariance Matrix of $$$T_K(X)$$$ which is $$$cov(T_K(X))=K(cov(X))=K(X^T X)$$$.

For the nonlinear transform, we apply Radial-Basic-Function (rbf) kernels as proof-of-concept: $$$K(X^TX)_{i,j}=\exp{\left(-\gamma\Vert x_i-x_j\Vert^2\right)}$$$. The parameter $$$\gamma$$$ (default 0.5) is adaptively tuned to different scale to make the representation more rank-deficient.

2. KLR Projection

The original Linear-LR Projection $$$P_{LR}^r$$$ is formulated by thresholding singular-values (SVT):

$$P_{LR}^r(X)=\arg\min_x\left[\frac{1}{2}\Vert T_K(X)-x\Vert_F^2\ s.t.\ Rank(T_K(x))\leq r \right]=\sum_{i=1}^r\sigma_i (X)u_iv_i^T$$

The fundamental idea of the proposed KLR Projection $$$P_{KLR}^r$$$ is to conduct Linear-LR Projection with SVT on the nonlinear transformed data, which is achieved efficiently by thresholding Eigenvalues (EVT) after Kernel-PCA. Then the KLR projection is achieved by finding the best match in the original spatial-temporal domain based on the EVT results in the nonlinear-transform domain. Similar idea (Pre-image method) has been used for image denoising applications^[6]:

$$P_{KLR}^r(X)=\arg\min_x\frac{1}{2}\Vert T_K(x)-P_{LR}^r(T_K(X))\Vert_F^2$$

3. KLR based DCE Reconstruction

Next, we re-formulated the DCE reconstruction by regularizing the rank of nonlinear-transformed data, with $$$y$$$ as acquired k-space samples, $$$F_u$$$ as undersampling-Fourier operator including coil-sensitivities.

$$min_x\Vert y-F_ux\Vert_2^2\ s.t.Rank(T_K(x))\leq R$$

Or equivalently,

$$min_x\Vert y-F_ux\Vert_2^2+\lambda\Vert T_K(X)\Vert_*$$

The SVT in nonlinear-transform domain is achieved efficiently using EVT with Kernel-PCA as described previously.

Linear-LR is a special case of the formulation above when $$$K$$$ is a linear-kernel. Nonlinear representations are shown to provide sparser and more accurate representations(Fig-1.3).

1. Algorithm

The optimization was solved using Iterative Projections(Fig-1):

1) Data Consistency is enforced using weighted-ESPIRiT (wESPIRiT)^[7] method to “soft-gate” the acquired k-space data based on estimated motions from navigators. The data is then inverse-Fourier transformed into image domain.

2) Sensitivity for Parallel-Imaging: Sensitivity-maps are estimated with ESPIRiT^[8] and used to combine multi-channel sensitivity-weighted images(POCS-SENSE).

3) KLR Projection is developed to improve conventional Linear-LR SVT projection. As described previously, here the SVT is conducted on the nonlinear-transformed signals, achieved using the KLR approximation formula with Kernel-PCA, EVT and Pre-image method^[6]. The computational complexity of KLR-EVT with Kernel-PCA is the same as original SVT in Linear-LR.

The KLR Projection can be applied to replace the original linear SVT in Global-Low-Rank(GLR), Locally-Low-Rank(LLR)^[3] as well as Low-Rank+Sparse(LR+S)^[2] model and Multi-scale-Low-Rank(Multi-scale-LR)^[9].

2. Experiment settings

To evaluate the performance, a free breathing DCE dataset of a 3-year-old female was acquired on a 3T GE-MR750 scanner with 32-channel cardiac array using a RF-spoiled-GRE-sequence (VDRad^[10] undersampling acquisition, Matrix Size:320x180x100, S/I 280mm FOV, R/L 224 mm FOV, 60% partial-echo, 178.51 sec for 18 contrast phases, R-factor 8.5). We integrated KLR with GLR and LLR to demonstrate the improvements(Fig-2). All the methods were performed with the same settings(Coil-Compression[11], ESPIRiT sensitivity-maps and wESPIRiT "soft-gating" and the same thresholding parameter previously tuned for Linear-LR).

In addition, a simulation dataset (DCE-phantom, retrospectively under-sampled with VDRad) was used to validate the spatial-temporal recovery performance for these methods(Fig-3).

In-vivo experiments(Fig-2) demonstrated sharper structural delineation by the proposed KLR method. As shown in Fig-3, simulation with DCE-phantom validated that KLR out-performs other low-rank methods in reconstruction. In addition, KLR showed more accurate recovery of temporal dynamics(shown in Fig-3c and Fig-3d).

Here we developed Kernelized-Low-Rank(KLR) model to further improve LR reconstruction for DCE-MRI. Generalized from Linear-LR, the proposed method better captures the nonlinearity of spatial-temporal signals by modeling them into sparser nonlinear representations. Simulation and in-vivo experiments demonstrated that KLR outperforms linear LR methods.

KLR can be easily integrated into LR methods including GLR, LLR, LR+S and Multi-scale-LR, to significantly improve reconstruction performance.