We present a fractional-order extension of the Bloch equations to describe anomalous NMR relaxation phenomena (T1 and T2). 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 T1 and T2 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 T2 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 T1 was observed in BNC. In the single-component gels, for T2 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 microstructural complexity in the setting of an altered distribution of relaxation times. This work suggests the utility of fractional-order models to describe T2 NMR relaxation processes in biological tissues.

Abstract:The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences the applicability of the fractional Sturm-Liouville theory. PACS

Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra ℓ 1 (N0) and the algebraic structure of Cesàro sums of a linear operator T ∈ B(X) is established. In particular, we show that every (C, α)-bounded operator T induces -and is in fact characterized -by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the (C, α) boundedness of the resolvent operator for temperated α-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.2010 Mathematics Subject Classification. 47C05, 47A35; 44A55, 65Q20.

Purpose To compare the ordinary monoexponential model with three anomalous relaxation models—the stretched Mittag-Leffler, stretched exponential, and biexponential functions—using both simulated and experimental cartilage relaxation data. Methods Monte Carlo simulations were used to examine both the ability of identifying a given model under high signal-to-noise ratio (SNR) conditions and the accuracy and precision of parameter estimates under more modest SNR as would be encountered clinically. Experimental transverse relaxation data were analyzed from normal and enzymatically degraded cartilage samples under high SNR and rapid echo sampling to compare each model. Results Both simulation and experimental results showed improvement in signal representation with the anomalous relaxation models. The stretched exponential model consistently showed the lowest mean squared error in experimental data and closely represents the signal decay over multiple decades of the decay time (e.g. 1-10ms, 10-100ms, and >100ms). The stretched exponential parameter αse showed an inverse correlation with biochemically-derived cartilage proteoglycan content. Conclusion Experimental results obtained at high field suggest potential application of αse as a measure of matrix integrity. Simulation reflecting more clinical imaging conditions, indicate the ability to robustly estimate αse and distinguish between normal and degraded tissue, highlighting its potential as a biomarker for human studies.