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Fractional Calculus in Magnetic Resonance
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Fractional Calculus in Magnetic Resonance

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 15515

Special Issue Editors

Prof. Dr. Richard L. Magin

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Guest Editor
Department of Bioengineering, University of Illinois at Chicago, Chicago, IL, USA
Interests: magnetic resonance imaging; diffusion; relaxation; fractional calculus
Dr. David Reiter

E-Mail Website
Guest Editor
Department of Radiology and Imaging Sciences and the Department of Orthopedics, Emory University, Atlanta, GA, USA
Interests: quantitative magnetic resonance imaging and spectroscopy

Special Issue Information

Dear Colleagues,

The applications of fractional calculus in the field of magnetic resonance are widespread and growing. In particular, we can extend the capabilities of nuclear magnetic resonance (NMR), electron spin resonance (ESR), and magnetic resonance imaging (MRI) by the generalization of the integer-order derivatives found in the governing equations (Bloch, and Bloch–Torrey equations). Solutions obtained using fractional calculus illuminate the structure and dynamics of materials at the molecular, cellular, and tissue length scales. In these situations, the space and time-fractional derivatives encode features that are not completely resolved using standard methods. As a consequence, molecular couplings, cell membrane permeability, and imaging biomarkers, for example, can be computed and displayed. These new techniques combine the specificity of fractional calculus with the non-perturbing sensitivity of magnetic resonance. The development of these methods and models requires cooperation between experts in magnetic resonance and applied mathematics; cooperation exhibited by the technical, review, and tutorial papers in this Special Issue.

The purpose of this Special Issue is to gather articles reflecting the latest developments of fractional calculus in the fields of nuclear magnetic resonance (NMR), electron spin resonance (ESR), and magnetic resonance imaging (MRI). Applications employing fractional calculus in the sub-disciplines of NMR/ESR spectroscopy, relaxation, diffusion, and MRI are encouraged.

Prof. Richard L. Magin
Dr. David Reiter
Guest Editors

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Keywords

  • Fractional calculus
  • Magnetic resonance
  • Magnetic resonance imaging
  • Nuclear magnetic resonance
  • Electron spin resonance
  • Spectroscopy
  • Relaxation
  • Diffusion

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Published Papers (6 papers)

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Research

16 pages, 4958 KiB  
Article
Analytical and Numerical Connections between Fractional Fickian and Intravoxel Incoherent Motion Models of Diffusion MRI
by Jingting Yao, Muhammad Ali Raza Anjum, Anshuman Swain and David A. Reiter
Mathematics 2021, 9(16), 1963; https://doi.org/10.3390/math9161963 - 17 Aug 2021
Cited by 1 | Viewed by 1773
Abstract
Impaired tissue perfusion underlies many chronic disease states and aging. Diffusion-weighted imaging (DWI) is a noninvasive MRI technique that has been widely used to characterize tissue perfusion. Parametric models based on DWI measurements can characterize microvascular perfusion modulated by functional and microstructural alterations [...] Read more.
Impaired tissue perfusion underlies many chronic disease states and aging. Diffusion-weighted imaging (DWI) is a noninvasive MRI technique that has been widely used to characterize tissue perfusion. Parametric models based on DWI measurements can characterize microvascular perfusion modulated by functional and microstructural alterations in the skeletal muscle. The intravoxel incoherent motion (IVIM) model uses a biexponential form to quantify the incoherent motion of water molecules in the microvasculature at low b-values of DWI measurements. The fractional Fickian diffusion (FFD) model is a parsimonious representation of anomalous superdiffusion that uses the stretched exponential form and can be used to quantify the microvascular volume of skeletal muscle. Both models are established measures of perfusion based on DWI, and the prognostic value of model parameters for identifying pathophysiological processes has been studied. Although the mathematical properties of individual models have been previously reported, quantitative connections between IVIM and FFD models have not been examined. This work provides a mathematical framework for obtaining a direct, one-way transformation of the parameters of the stretched exponential model to those of the biexponential model. Numerical simulations are implemented, and the results corroborate analytical results. Additionally, analysis of in vivo DWI measurements in skeletal muscle using both biexponential and stretched exponential models is shown and compared with analytical and numerical models. These results demonstrate the difficulty of model selection based on goodness of fit to experimental data. This analysis provides a framework for better interpreting and harmonizing perfusion parameters from experimental results using these two different models. Full article
(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)
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12 pages, 296 KiB  
Article
Half Way There: Theoretical Considerations for Power Laws and Sticks in Diffusion MRI for Tissue Microstructure
by Matt G. Hall and Carson Ingo
Mathematics 2021, 9(16), 1871; https://doi.org/10.3390/math9161871 - 6 Aug 2021
Viewed by 1896
Abstract
In this article, we consider how differing approaches that characterize biological microstructure with diffusion weighted magnetic resonance imaging intersect. Without geometrical boundary assumptions, there are techniques that make use of power law behavior which can be derived from a generalized diffusion equation or [...] Read more.
In this article, we consider how differing approaches that characterize biological microstructure with diffusion weighted magnetic resonance imaging intersect. Without geometrical boundary assumptions, there are techniques that make use of power law behavior which can be derived from a generalized diffusion equation or intuited heuristically as a time dependent diffusion process. Alternatively, by treating biological microstructure (e.g., myelinated axons) as an amalgam of stick-like geometrical entities, there are approaches that can be derived utilizing convolution-based methods, such as the spherical means technique. Since data acquisition requires that multiple diffusion weighting sensitization conditions or b-values are sampled, this suggests that implicit mutual information may be contained within each technique. The information intersection becomes most apparent when the power law exponent approaches a value of 12, whereby the functional form of the power law converges with the explicit stick-like geometric structure by way of confluent hypergeometric functions. While a value of 12 is useful for the case of solely impermeable fibers, values that diverge from 12 may also reveal deep connections between approaches, and potentially provide insight into the presence of compartmentation, exchange, and permeability within heterogeneous biological microstructures. All together, these disparate approaches provide a unique opportunity to more completely characterize the biological origins of observed changes to the diffusion attenuated signal. Full article
(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)
23 pages, 2115 KiB  
Article
The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging
by Thomas R. Barrick, Catherine A. Spilling, Matt G. Hall and Franklyn A. Howe
Mathematics 2021, 9(15), 1763; https://doi.org/10.3390/math9151763 - 26 Jul 2021
Cited by 5 | Viewed by 3396
Abstract
Quasi-diffusion imaging (QDI) is a novel quantitative diffusion magnetic resonance imaging (dMRI) technique that enables high quality tissue microstructural imaging in a clinically feasible acquisition time. QDI is derived from a special case of the continuous time random walk (CTRW) model of diffusion [...] Read more.
Quasi-diffusion imaging (QDI) is a novel quantitative diffusion magnetic resonance imaging (dMRI) technique that enables high quality tissue microstructural imaging in a clinically feasible acquisition time. QDI is derived from a special case of the continuous time random walk (CTRW) model of diffusion dynamics and assumes water diffusion is locally Gaussian within tissue microstructure. By assuming a Gaussian scaling relationship between temporal (α) and spatial (β) fractional exponents, the dMRI signal attenuation is expressed according to a diffusion coefficient, D (in mm2 s−1), and a fractional exponent, α. Here we investigate the mathematical properties of the QDI signal and its interpretation within the quasi-diffusion model. Firstly, the QDI equation is derived and its power law behaviour described. Secondly, we derive a probability distribution of underlying Fickian diffusion coefficients via the inverse Laplace transform. We then describe the functional form of the quasi-diffusion propagator, and apply this to dMRI of the human brain to perform mean apparent propagator imaging. QDI is currently unique in tissue microstructural imaging as it provides a simple form for the inverse Laplace transform and diffusion propagator directly from its representation of the dMRI signal. This study shows the potential of QDI as a promising new model-based dMRI technique with significant scope for further development. Full article
(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)
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13 pages, 3967 KiB  
Article
Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range
by Guangyu Dan, Weiguo Li, Zheng Zhong, Kaibao Sun, Qingfei Luo, Richard L. Magin, Xiaohong Joe Zhou and M. Muge Karaman
Mathematics 2021, 9(14), 1688; https://doi.org/10.3390/math9141688 - 19 Jul 2021
Cited by 6 | Viewed by 2527
Abstract
It has been increasingly reported that in biological tissues diffusion-weighted MRI signal attenuation deviates from mono-exponential decay, especially at high b-values. A number of diffusion models have been proposed to characterize this non-Gaussian diffusion behavior. One of these models is the continuous-time [...] Read more.
It has been increasingly reported that in biological tissues diffusion-weighted MRI signal attenuation deviates from mono-exponential decay, especially at high b-values. A number of diffusion models have been proposed to characterize this non-Gaussian diffusion behavior. One of these models is the continuous-time random-walk (CTRW) model, which introduces two new parameters: a fractional order time derivative α and a fractional order spatial derivative β. These new parameters have been linked to intravoxel diffusion heterogeneities in time and space, respectively, and are believed to depend on diffusion times. Studies on this time dependency are limited, largely because the diffusion time cannot vary over a board range in a conventional spin-echo echo-planar imaging sequence due to the accompanying T2 decays. In this study, we investigated the time-dependency of the CTRW model in Sephadex gel phantoms across a broad diffusion time range by employing oscillating-gradient spin-echo, pulsed-gradient spin-echo, and pulsed-gradient stimulated echo sequences. We also performed Monte Carlo simulations to help understand our experimental results. It was observed that the diffusion process fell into the Gaussian regime at extremely short diffusion times whereas it exhibited a strong time dependency in the CTRW parameters at longer diffusion times. Full article
(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)
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21 pages, 10258 KiB  
Article
Fractional Order Magnetic Resonance Fingerprinting in the Human Cerebral Cortex
by Viktor Vegh, Shahrzad Moinian, Qianqian Yang and David C. Reutens
Mathematics 2021, 9(13), 1549; https://doi.org/10.3390/math9131549 - 1 Jul 2021
Cited by 1 | Viewed by 2229
Abstract
Mathematical models are becoming increasingly important in magnetic resonance imaging (MRI), as they provide a mechanistic approach for making a link between tissue microstructure and signals acquired using the medical imaging instrument. The Bloch equations, which describes spin and relaxation in a magnetic [...] Read more.
Mathematical models are becoming increasingly important in magnetic resonance imaging (MRI), as they provide a mechanistic approach for making a link between tissue microstructure and signals acquired using the medical imaging instrument. The Bloch equations, which describes spin and relaxation in a magnetic field, are a set of integer order differential equations with a solution exhibiting mono-exponential behaviour in time. Parameters of the model may be estimated using a non-linear solver or by creating a dictionary of model parameters from which MRI signals are simulated and then matched with experiment. We have previously shown the potential efficacy of a magnetic resonance fingerprinting (MRF) approach, i.e., dictionary matching based on the classical Bloch equations for parcellating the human cerebral cortex. However, this classical model is unable to describe in full the mm-scale MRI signal generated based on an heterogenous and complex tissue micro-environment. The time-fractional order Bloch equations have been shown to provide, as a function of time, a good fit of brain MRI signals. The time-fractional model has solutions in the form of Mittag–Leffler functions that generalise conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex heterogeneous materials. Hence, we replaced the integer order Bloch equations with the previously reported time-fractional counterpart within the MRF framework and performed experiments to parcellate human gray matter, which consists of cortical brain tissue with different cyto-architecture at different spatial locations. Our findings suggest that the time-fractional order parameters, α and β, potentially associate with the effect of interareal architectonic variability, which hypothetically results in more accurate cortical parcellation. Full article
(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)
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29 pages, 2358 KiB  
Article
Slices of the Anomalous Phase Cube Depict Regions of Sub- and Super-Diffusion in the Fractional Diffusion Equation
by Richard L. Magin and Ervin K. Lenzi
Mathematics 2021, 9(13), 1481; https://doi.org/10.3390/math9131481 - 24 Jun 2021
Cited by 6 | Viewed by 1947
Abstract
Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct [...] Read more.
Fractional-order time and space derivatives are one way to augment the classical diffusion equation so that it accounts for the non-Gaussian processes often observed in heterogeneous materials. Two-dimensional phase diagrams—plots whose axes represent the fractional derivative order—typically display: (i) points corresponding to distinct diffusion propagators (Gaussian, Cauchy), (ii) lines along which specific stochastic models apply (Lévy process, subordinated Brownian motion), and (iii) regions of super- and sub-diffusion where the mean squared displacement grows faster or slower than a linear function of diffusion time (i.e., anomalous diffusion). Three-dimensional phase cubes are a convenient way to classify models of anomalous diffusion (continuous time random walk, fractional motion, fractal derivative). Specifically, each type of fractional derivative when combined with an assumed power law behavior in the diffusion coefficient renders a characteristic picture of the underlying particle motion. The corresponding phase diagrams, like pages in a sketch book, provide a portfolio of representations of anomalous diffusion. The anomalous diffusion phase cube employs lines of super-diffusion (Lévy process), sub-diffusion (subordinated Brownian motion), and quasi-Gaussian behavior to stitch together equivalent regions. Full article
(This article belongs to the Special Issue Fractional Calculus in Magnetic Resonance)
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