### Abstract

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 diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm^{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)^{α}], where α 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.

Original language | English (US) |
---|---|

Pages (from-to) | 255-270 |

Number of pages | 16 |

Journal | Journal of Magnetic Resonance |

Volume | 190 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2008 |

### Fingerprint

### Keywords

- Anomalous diffusion
- Bloch-Torrey equation
- Diffusion-weighted MRI
- Fractional calculus
- Stretched exponential

### ASJC Scopus subject areas

- Biophysics
- Biochemistry
- Nuclear and High Energy Physics
- Condensed Matter Physics

### Cite this

*Journal of Magnetic Resonance*,

*190*(2), 255-270. https://doi.org/10.1016/j.jmr.2007.11.007

**Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation.** / Magin, Richard L.; Abdullah, Osama; Baleanu, Dumitru; Zhou, Xiaohong Joe.

Research output: Contribution to journal › Article

*Journal of Magnetic Resonance*, vol. 190, no. 2, pp. 255-270. https://doi.org/10.1016/j.jmr.2007.11.007

}

TY - JOUR

T1 - Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation

AU - Magin, Richard L.

AU - Abdullah, Osama

AU - Baleanu, Dumitru

AU - Zhou, Xiaohong Joe

PY - 2008/2/1

Y1 - 2008/2/1

N2 - 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 diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm2/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)α], where α 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.

AB - 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 diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm2/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)α], where α 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.

KW - Anomalous diffusion

KW - Bloch-Torrey equation

KW - Diffusion-weighted MRI

KW - Fractional calculus

KW - Stretched exponential

UR - http://www.scopus.com/inward/record.url?scp=38349041965&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38349041965&partnerID=8YFLogxK

U2 - 10.1016/j.jmr.2007.11.007

DO - 10.1016/j.jmr.2007.11.007

M3 - Article

C2 - 18065249

AN - SCOPUS:38349041965

VL - 190

SP - 255

EP - 270

JO - Journal of Magnetic Resonance

JF - Journal of Magnetic Resonance

SN - 1090-7807

IS - 2

ER -