Fractional Calculus Models of Magnetic Resonance Phenomena: Relaxation and Diffusion

Area of Science:

• Applies advanced mathematical concepts of fractional calculus to the field of magnetic resonance imaging (MRI).
• Integrates computational, mathematical, and biophysical perspectives for enhanced biological tissue analysis.

Background:

• Fractional calculus provides novel methods for characterizing complex biological tissues at various scales.
• It models molecular dynamics like transport and rotation using power-law kernels in governing equations.

Purpose of the Study:

• To review the principal fractional order models applied to nuclear magnetic resonance (NMR) and MRI phenomena.
• To identify connections, limitations, and future applications of fractional calculus in MRI.
• To explore the potential of these models in identifying imaging biomarkers for diseases.

Main Methods:

• Incorporation of power-law convolution kernels into time and space derivatives of NMR/MRI equations.
• Development of fractional order generalizations of the Bloch and Bloch-Torrey equations.
• Utilizing coarse-graining, simulation, and accelerated computation for complex problem-solving.

Main Results:

• Fractional calculus models capture complex molecular dynamics and non-Gaussian behavior in biological tissues.
• Early studies explored fractal dimensions and power-law decays, suggesting stretched exponential relaxation.
• Fractional generalizations of Bloch equations enable characterization of NMR/MRI relaxation and diffusion.

Conclusions:

• Fractional calculus offers powerful tools for understanding biological complexity in MRI.
• Challenges remain in obtaining analytical solutions and predicting image contrast changes.
• Future research focuses on a multifaceted approach for disease biomarker discovery using advanced computational methods.