The foundation for an accurate and unifying Fourier-based theory of diffusion weighted magnetic resonance imaging (DW–MRI) is constructed by carefully re-examining the first principles of DW–MRI signal formation and deriving its mathematical model from scratch. The derivations are specifically obtained for DW–MRI signal by including all of its elements (e.g., imaging gradients) using complex values. Particle methods are utilized in contrast to conventional partial differential equations approach. The signal is shown to be the Fourier transform of the joint distribution of number of the magnetic moments (at a given location at the initial time) and magnetic moment displacement integrals. In effect, the k-space is augmented by three more dimensions, corresponding to the frequency variables dual to displacement integral vectors. The joint distribution function is recovered by applying the Fourier transform to the complete high-dimensional data set. In the process, to obtain a physically meaningful real valued distribution function, phase corrections are applied for the re-establishment of Hermitian symmetry in the signal. Consequently, the method is fully unconstrained and directly presents the distribution of displacement integrals without any assumptions such as symmetry or Markovian property. The joint distribution function is visualized with isosurfaces, which describe the displacement integrals, overlaid on the distribution map of the number of magnetic moments with low mobility. The model provides an accurate description of the molecular motion measurements via DW–MRI. The improvement of the characterization of tissue microstructure leads to a better localization, detection and assessment of biological properties such as white matter integrity. The results are demonstrated on the experimental data obtained from an ex vivo baboon brain.
Keywords: magnetic resonance imaging, diffusion weighted imaging, fourier transform
Citation: Özcan A (2013) Complete fourier direct magnetic resonance imaging (CFD-MRI) for diffusion MRI. Front. Integr. Neurosci. 7:18. doi: 10.3389/fnint.2013.00018
Received: 01 November 2012; Paper pending published: 29 January 2013;
Accepted: 08 March 2013; Published online: 02 April 2013.
Edited by:Elizabeth B. Torres, Rutgers University, USA
Reviewed by:Bennett A. Landman, Vanderbilt University, USA
Copyright © 2013 Özcan. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and subject to any copyright notices concerning any third-party graphics etc.
*Correspondence: Alpay Özcan, Health Research, Arlington Innovation Center, Virginia Polytechnic Institute and State University, 900 N. Glebe Road, Arlington, VA 22203, USA. e-mail: firstname.lastname@example.org