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Fractional calculus in bioengineering, part 3.

Richard L Magin1

  • 1University of Illinois at Chicago, Department of Bioengineering, Chicago, Illinois 60607-7052, USA. rmagin@uic.edu

Critical Reviews in Biomedical Engineering
|January 18, 2005
PubMed
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Fractional calculus, a mathematical tool for noninteger order operations, offers powerful methods for modeling biological systems. Its application in bioengineering, particularly for nerve impulse propagation and biomaterials, reveals new functional relationships for complex biological phenomena.

Area of Science:

  • Bioengineering and Biophysics
  • Applied Mathematics
  • Biomedical Research

Background:

  • Fractional calculus, involving noninteger order integral and differential operations, has historical roots but limited application in biological systems.
  • While established in physics and engineering, its potential in bioengineering remains largely unexplored despite suitability for biomedical problems.
  • Early studies on nerve cell electrical properties by Cole and Hodgkin hinted at the utility of fractional calculus.

Purpose of the Study:

  • To introduce fractional calculus operations and demonstrate their application to biological systems.
  • To highlight the suitability of fractional calculus for modeling phenomena in electrochemistry, bioengineering, and biophysics.
  • To showcase the potential of fractional calculus in developing novel functional relationships for complex biological modeling.

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Main Methods:

  • Reviewing the historical development and fundamental operations of fractional calculus, following Heaviside's approach.
  • Demonstrating basic fractional calculus operations on standard engineering functions (step, ramp, pulse, sinusoid).
  • Analyzing specific examples from electrochemistry, physics, bioengineering, biophysics, neuroscience, bioelectricity, and tissue biomechanics.

Main Results:

  • Fractional derivatives accurately describe natural phenomena like heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation.
  • Generalization of exponential functions to Mittag-Leffler functions provides better fits for cell membrane data.
  • Fractional calculus naturally incorporates hereditary integrals and power-law stress-strain relationships for biomaterials.

Conclusions:

  • Fractional calculus offers a rigorous and direct approach to modeling complex biological systems.
  • Its application extends to distributed systems, electrochemical analysis, impedance spectroscopy, and biomechanics.
  • Expanding mathematical operations to include fractional calculus can lead to new insights and functional relationships in biomedical research.