Accelerated MR parameter mapping using compressed sensing with model-based sparsifying transform

Introduction: The estimation of MR parameters, such as the relaxation times T1, T2 and diffusion coefficients D, requires the acquisition of multiple images at different sequence parameters, which is often associated with long acquisition times. These data show a high temporal correlation, which can be described by a model facilitating accelerated image acquisition by data undersampling as shown in [1]. Recently, Compressed Sensing (CS) [2-4] was demonstrated for image reconstruction from incomplete k-space data. In this work we show that prior knowledge about the data can be used to define a model-based sparsity transform for improved CS reconstruction for MR parameter estimation. Theory: CS relies on two premises: data compressibility and incoherent sampling. In Cartesian sampling, incoherence can be achieved by random undersampling in the phase encoding direction. An important factor for the quality of CS reconstruction is the appropriate choice of a sparsifying transform. Transforms like wavelets or finite differences can be applied to sparsify a large class of signals. In parameter estimation, there is a strong correlation of the data in the temporal dimension described by the underlying model. We use that knowledge to define a sparsifying transform by means of Principal Component Analysis (PCA). This is demonstrated here for T2 mapping, in which the signal is described by an exponential decay. However, the method is not restricted to exponentials and could be generalized for other models. The model-based sparsity transform is obtained as follows. Training data S are generated, based on a uniformly distributed set of T2 times covering a broad range of values. The matrix S contains a set of exponentials corresponding to these T2 in its columns. The matrix U, taken from the Singular Value Decomposition (SVD) of the correlation matrix R = SS = UΣU achieves a compact representation of the training set and also of any other exponentially decaying signal with T2 in the given range. The reconstruction was performed by solving the unconstrained optimization problem: