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

Area of Science:

• Medical Imaging
• Biophysics
• Applied Mathematics

Background:

• Diffusion-weighted MRI (DW-MRI) is crucial for diagnosing various neurological conditions.
• Current DW-MRI analysis often uses the apparent diffusion coefficient (D) derived from the monoexponential model.
• Anomalous diffusion, described by the stretched exponential model, better reflects complex tissue microstructures.

Purpose of the Study:

• To propose an alternative derivation of the stretched exponential model for anomalous diffusion using fractional calculus.
• To investigate the application of fractional order space and time derivatives in the Bloch-Torrey equation.

Main Methods:

• Generalizing the spatial Laplacian in the Bloch-Torrey equation with a fractional Brownian motion model.
• Replacing the time derivative with a fractional Riemann-Liouville derivative in Caputo form.
• Validating the fractional order dynamics against experimental data from Sephadex gels, cartilage, and brain.

Main Results:

• The proposed fractional order dynamics successfully model signal attenuation in diffusion-weighted images.
• Both spatial and temporal fractional derivative approaches were explored, with the spatial case showing promising fits.
• The derived fractional order dynamics revert to classical results for integer-order operations.

Conclusions:

• Fractional calculus provides a robust framework for modeling anomalous diffusion in biological tissues.
• This approach offers a potential advancement for classifying complex diffusion patterns in pathologies.
• Future work may enhance the diagnostic capabilities of diffusion-weighted MRI.