Browsing by Author "Magin, Richard"

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Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation
(Academic Press INC Elsevier Science, 2008) Baleanu, Dumitru; Abdullah, Osama; Baleanu, Dumitru; Zhou, Xiaohong Joe
Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on d iffusi on -weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (turn (2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology. (c) 2007 Elsevier Inc. All rights reserved
Chaos in the fractional order nonlinear Bloch equation with delay
(Elsevier Science BV, 2015) Baleanu, Dumitru; Magin, Richard; Bhalekar, Sachin; Daftardar-Gejji, Varsha
The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (alpha) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, tau = 0, we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at alpha = 0.8548 with subsequent period doubling that leads to chaos at alpha = 0.9436. A periodic window is observed for the range 0.962 < alpha < 0.9858, with chaos arising again as a nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value alpha = 0.8532, and the transition from two to four cycles at alpha = 0.9259. With further increases in the fractional order, period doubling continues until at alpha = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at alpha = 0.8441, and alpha = 0.8635, respectively. However, the system exhibits chaos at much lower values of a (alpha - 0.8635). A periodic window is observed in the interval 0.897 < alpha < 0.9341, with chaos again appearing for larger values of a. In general, as the value of a decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a delay-dependent model that predicts instability in this non-linear fractional order system consistent with the experimental observations of spin turbulence.
Fractional Bloch equation with delay
(Pergamon-Elsevier Science Ltd, 2011) Baleanu, Dumitru; Daftardar-Gejji, Varsha; Baleanu, Dumitru; Magin, Richard
In this paper we investigate a fractional generalization of the Bloch equation that includes both fractional derivatives and time delays. The appearance of the fractional derivative on the left side of the Bloch equation encodes a degree of system memory in the dynamic model for magnetization. The introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation. The analysis of this system shows different stability behavior for the T(1) and the T(2) relaxation processes. The T(1) decay is stable for the range of delays tested (1-100 mu s), while the T(2) relaxation in this model exhibited a critical delay (typically 6 mu s) above which the system was unstable. Delays are expected to appear in NMR systems, in both the system model and in the signal excitation and detection processes. Therefore, by including both the fractional derivative and finite time delays in the Bloch equation, we believe that we have established a more complete and more realistic model for NMR resonance and relaxation
Generalized fractional order bloch equation with extended delay
(World Scientific, 2012) Baleanu, Dumitru; Daftardar-Gejji, Varsha; Baleanu, Dumitru; Magin, Richard
The fundamental description of relaxation (T-1 and T-2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time-and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T-1 and T-2 relaxation. The T-1 decay is stable for the range of delays tested (1 mu sec to 200 mu sec), while the T-2 relaxation in this extended model exhibits a critical delay (typically 100 mu sec to 200 mu sec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.
Solving the Fractional Order Bloch Equation
(John Wiley&Sons Inc, 2009) Baleanu, Dumitru; Feng, Xu; Baleanu, Dumitru
Nuclear magnetic resonance (NMR) is a physical phenomenon widely used in chemistry, medicine, and engineering to study complex materials. NMR is governed by the Bloch equation, which relates a macroscopic model of magnetization to applied radjofrequency, gradient and static magnetic fields. Simple models of materials are well described by the classical first order dynamics of precession and relaxation inherent in the vector form of the Bloch equation. Fractional order generalization of the Bloch equation presents an opportunity to extend its use to describe a wider range of experimental situations involving heterogeneous, porous, or composite materials. Here we describe the generalization of the Bloch equation in terms of Caputo fractional derivatives of order alpha (0 < alpha < 1) for a single spin system in a static magnetic field at resonance. The results are expressed in terms of the Mittag-Leffler function-a generalized exponential function that converges to the classical case when alpha = 1
Transient Chaos in Fractional Bloch Equations
(Pergamon-Elsevier Science LTD, 2012) Baleanu, Dumitru; Daftardar-Gejji, Varsha; Baleanu, Dumitru; Magin, Richard; 56389
The Bloch equation provides the fundamental description of nuclear magnetic resonance (NMR) and relaxation (T-1 and T-2). This equation is the basis for both NMR spectroscopy and magnetic resonance imaging (MRI). The fractional-order Bloch equation is a generalization of the integer-order equation that interrelates the precession of the x, y and z components of magnetization with time- and space-dependent relaxation. In this paper we examine transient chaos in a non-linear version of the Bloch equation that includes both fractional derivatives and a model of radiation damping. Recent studies of spin turbulence in the integer-order Bloch equation suggest that perturbations of the magnetization may involve a fading power law form of system memory, which is concisely embedded in the order of the fractional derivative. Numerical analysis of this system shows different patterns in the stability behavior for alpha near 1.00. In general, when alpha is near 1.00, the system is chaotic, while for 0.98 >= alpha >= 0.94, the system shows transient chaos. As the value of alpha decreases further, the duration of the transient chaos diminishes and periodic sinusoidal oscillations emerge. These results are consistent with studies of the stability of both the integer and the fractional-order Bloch equation. They provide a more complete model of the dynamic behavior of the NMR system when non-linear feedback of magnetization via radiation damping is present. (C) 2012 Elsevier Ltd. All rights reserved.