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Magn Reson Imaging. 2015 Nov;33(9):1182-1186. doi: 10.1016/j.mri.2015.07.001. Epub 2015 Jul 26.

Stejskal's formula for multiple-pulsed diffusion MRI.

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Center for Biomedical Imaging, Medical University of South Carolina, Charleston, SC, USA; Department of Radiology and Radiological Science, Medical University of South Carolina, Charleston, SC, USA. Electronic address:


A fundamental theoretical result for diffusion MRI is the formula by Stejskal showing that the diffusion NMR signal is proportional to the Fourier transform of the diffusion displacement probability density function. Here this result is extended to multiple-pulsed diffusion MRI (MP-dMRI) by using a higher dimensional q-space formalism to express the diffusion-weighted signal for a sequence with N diffusion wave vectors in terms of a Fourier transform of a diffusion displacement probability density function in a 3N-dimensional space. As an illustration of the application of this extended version of Stejskal's formula, it is used to analyze the cumulant expansion of the signal magnitude for MP-dMRI.


Cumulant expansion; Diffusion MRI; Kurtosis; Multiple wave vector; Pulsed field gradient; q-space

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