Abstract
Understanding biological complexity requires new techniques, such as high-resolution magnetic resonance imaging (MRI), for visualizing heterogeneous materials. Fundamental to the application of techniques based on nuclear magnetic resonance (NMR) is the use of new conceptual tools from mathematics and the physical sciences (e.g., fractional calculus, fractals, and non-Gaussian stable distributions). In complex synthetic or biological systems, such as Sephadex gels, articular cartilage, and human brain tissue, multi-exponential decay models are often used to describe the measured NMR relaxation and diffusion processes. Recently, however, a stretched-exponential decay model was derived using fractional calculus from a fractional-order generalization of the Bloch–Torrey equation. In this case the stretched-exponential function was shown to provide an excellent fit to the signal attenuation in diffusion-weighted MRI, and, therefore, provides a direct link between fractional calculus and NMR. In this chapter we show that the fractional order of the stretched-exponential model of diffusion can be related to the complexity of biological tissues through its elaboration as a distribution of rate constants.
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Acknowledgment
The authors R.L. Magin and Y.Z. Rawash would like to acknowledge the support of NIH grant R01 EB007537.
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Magin, R.L., Rawash, Y.Z., Berberan-Santos, M.N. (2012). Analyzing Anomalous Diffusion in NMR Using a Distribution of Rate Constants. In: Baleanu, D., Machado, J., Luo, A. (eds) Fractional Dynamics and Control. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0457-6_22
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DOI: https://doi.org/10.1007/978-1-4614-0457-6_22
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