Purpose: To present a novel generalized MR image reconstruction based on pseudoinversion of the encoding matrix (Pinv-Recon) as a simple yet powerful method, and demonstrate its computational feasibility for diverse MR imaging applications. Methods: MR image encoding constitutes a linear mapping of the unknown image to the measured k-space data mediated via an encoding matrix ($ data = Encode \times image$). Pinv-Recon addresses MR image reconstruction as a linear inverse problem ($image = Encode^{-1} \times data$), explicitly calculating the Moore-Penrose pseudoinverse of the encoding matrix using truncated singular value decomposition (tSVD). Using a discretized, algebraic notation, we demonstrate constructing a generalized encoding matrix by stacking relevant encoding mechanisms (e.g., gradient encoding, coil sensitivity encoding, chemical shift inversion) and encoding distortions (e.g., off-center positioning, B$_0$ inhomogeneity, spatiotemporal gradient imperfections, transient relaxation effects). Iterative reconstructions using the explicit generalized encoding matrix, and the computation of the spatial-response-function (SRF) and noise amplification, were demonstrated. Results: We evaluated the computation times and memory requirements (time ~ (size of the encoding matrix)$^{1.4}$). Using the Shepp-Logan phantom, we demonstrated the versatility of the method for various intertwined MR image encoding and distortion mechanisms, achieving better MSE, PSNR and SSIM metrics than conventional methods. A diversity of datasets, including the ISMRM CG-SENSE challenge, were used to validate Pinv-Recon. Conclusion: Although pseudo-inversion of large encoding matrices was once deemed computationally intractable, recent advances make Pinv-Recon feasible. It has great promise for both research and clinical applications, and for educational use.
Motivation: There remains an absence of imaging modalities capable of probing the neuroinflammatory processes that precede the well-defined brain structural changes in Primary Progressive Multiple Sclerosis (PPMS). Goal(s): We investigated whether hyperpolarized [1-13C]pyruvate MRI can delineate alterations in cerebral glycolytic and oxidative metabolism between treatment naïve PPMS and healthy volunteers. Approach: Two treatment naïve PPMS patients and two sex matched healthy volunteers underwent [1-13C]pyruvate MRI to characterise cerebral glycolytic and oxidative metabolism. Results: A global increase in [1-13C]lactate: [1-13C]pyruvate was found in both PPMS patients relative to sex-matched healthy controls (0.23 ± 0.12 vs 0.16 ± 0.08). The 13C bicarbonate:[1-13C]pyruvate ratio was no different. Impact: These preliminary findings demonstrate a global increase in cerebral glycolytic metabolism in treatment naïve PPMS relative to age and gender matched healthy controls. This may reflect diffuse neuroinflammatory processes and suggests [1-13C]pyruvate MRI could be used to monitor disease activity.
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