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Critical Reviews™ in Biomedical Engineering
SJR: 0.26 SNIP: 0.375 CiteScore™: 1.4

ISSN Druckformat: 0278-940X
ISSN Online: 1943-619X

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Critical Reviews™ in Biomedical Engineering

DOI: 10.1615/CritRevBiomedEng.2020033925
Forthcoming Article

Fractional Calculus Models of Magnetic Resonance Phenomena: Relaxation and Diffusion

Richard Magin
University of Illinois-Chicago, Department of Bioengineering, Chicago, Illinois, USA
Matt Hall
UCL Great Ormond Street Institute of Child Health, University College, London, London, UK and National Physical Laboratory, Teddington, UK,
Muge Karaman
University of Illinois at Chicago, Department of Bioengineering, Department of Radiology, Center for Magnetic Resonance Research Chicago, Illinois, USA
Viktor Vegh
Centre for Advanced Imaging, ARC Centre for Innovation in Biomedical Imaging, The University of Queensland, Brisbane, Queensland, Australia


Fractional calculus applications in magnetic resonance imaging (MRI) have grown over the last twenty years. From the mathematical, computational, and biophysical perspectives, fractional calculus provides new tools for describing the dynamic complexity of proteins, membranes and cells. Specifically, fractional order models concisely capture molecular transport, rotation, and vibration by incorporating power law convolution kernels into the time and space derivatives appearing in the equations that govern nuclear magnetic resonance (NMR) phenomena. Hence, it is natural to expect fractional calculus models of relaxation and diffusion to be applied to problems in NMR and MRI. Early studies incorporated the fractal dimensions of multi-scale materials in the non-linear growth of the mean squared displacement, assumed power-law decays of the spectral density, and suggested stretched exponential signal relaxation to describe non-Gaussian behavior. Subsequently, fractional order generalization of the Bloch, and Bloch-Torrey equations were developed to characterize NMR relaxation, and diffusion-weighted MRI. However, even for simple geometries, analytical solutions of fractional order equations in time and space are difficult to obtain, and predictions of the corresponding changes in image contrast are not always possible. Currently a multifaceted approach using coarse graining, simulation, and accelerated computation is being developed to glean ‘imaging’ biomarkers of disease. This review surveys the principal fractional order models used to describe NMR and MRI phenomena, identifies limitations and shortcomings, and finally points to future applications of the approach.