LANS Informal Seminar
"Fractional Order Modeling of the Bloch-Torrey Equation"
DATE: November 28, 2007
TIME: - Description:
SPEAKER: Richard Magin, Department of Bioengineering, UIC
LOCATION: Building 221, A-216, Argonne National Laboratory
The Bloch equation is the starting point for modeling the
dynamic behavior of nuclear magnetic resonance (NMR) phenomena. Although originally derived as a semi-classical description of the precession of nuclear magnetic moments (spins), the Bloch equation is valid over a wide range of experimental conditions ranging from the molecular spectroscopy of liquids to the diffusion-weighted imaging of the human brain. Extension of the Bloch equation to describe chemical shifts in macromolecules or contrast in brain MRI requires only integer order differential operators. However, recent studies involving diffusion-weighted MRI and diffusion measurements in porous materials, suggest non-exponential behavior. Such behavior can be manifest through the fractional order generalization of the time and the space derivatives in the Bloch-Torrey equation. In this analysis modifications to the Bloch-Torrey equation are developed that provide a basis for the observed non-exponential behavior in the time decay of the transverse magnetization for a spin echo pulse sequence. 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 the anomalous diffusion in tissues with developing pathology.
Please send questions or suggestions to Debojyoti Ghosh: ghosh at mcs dot anl dot gov.