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Anomalous NMR Relaxation in Cartilage Matrix Components and Native Cartilage: Fractional-Order Models

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journal contribution
posted on 27.05.2011 by Richard L. Magin, Weiguo Li, M. Pilar Velasco, Juan Trujillo, David A. Reiter, Ashley Morgenstern, Richard G. Spencer
We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T(1) and T(2)). The model has solutions in the form of Mittag-Leffler and stretched exponential functions that generalize conventional exponential relaxation. Such functions have been shown by others to be useful for describing dielectric and viscoelastic relaxation in complex, heterogeneous materials. Here, we apply these fractional-order T(1) and T(2) relaxation models to experiments performed at 9.4 and 11.7 Tesla on type I collagen gels, chondroitin sulfate mixtures, and to bovine nasal cartilage (BNC), a largely isotropic and homogeneous form of cartilage. The results show that the fractional-order analysis captures important features of NMR relaxation that are typically described by multi-exponential decay models. We find that the T(2) relaxation of BNC can be described in a unique way by a single fractional-order parameter (α), in contrast to the lack of uniqueness of multi-exponential fits in the realistic setting of a finite signal-to-noise ratio. No anomalous behavior of T(1) was observed in BNC. In the single-component gels, for T(2) measurements, increasing the concentration of the largest components of cartilage matrix, collagen and chondroitin sulfate, results in a decrease in α, reflecting a more restricted aqueous environment. The quality of the curve fits obtained using Mittag-Leffler and stretched exponential functions are in some cases superior to those obtained using mono- and bi-exponential models. In both gels and BNC, α appears to account for micro-structural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T(2) NMR relaxation processes in biological tissues.


National Institute of Bioengineering and Biomedical Imaging, National Institutes of Health (NIH), Grant Number R01 EB 007537, and the Intramural Research Program of the NIH, Institute on Aging.


Publisher Statement

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Magnetic Resonance. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Magnetic Resonance, [Vol 210, Issue 2, (June 2011)] DOI: 10.1016/j.jmr.2011.03.006. The original publication is available at







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