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Fractional calculus in bioengineering.

Richard L Magin1

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

Critical Reviews in Biomedical Engineering
|July 14, 2004
PubMed
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Fractional calculus offers a powerful mathematical approach for modeling complex biological systems. This method accurately describes phenomena in bioengineering, physics, and electrochemistry, expanding modeling capabilities.

Area of Science:

  • Bioengineering
  • Biophysics
  • Applied Mathematics

Background:

  • Fractional calculus, involving noninteger order integral and differential operations, has historical roots but limited application in biological system modeling.
  • While established in physics and engineering, its potential in bioengineering remains underexplored despite suitability for biomedical research problems.

Purpose of the Study:

  • Introduce fractional calculus operations and their relevance to biological systems.
  • Demonstrate the application of fractional calculus in various scientific and engineering fields.
  • Highlight the utility of fractional derivatives in modeling complex biological phenomena.

Main Methods:

  • Reviewing the historical development and mathematical foundations of fractional calculus.

Related Experiment Videos

  • Demonstrating basic fractional calculus operations on standard engineering functions (step, ramp, pulse, sinusoid).
  • Presenting specific examples of fractional calculus applications in electrochemistry, physics, bioengineering, and biophysics.

    • Main Results:

    • Fractional calculus provides a more accurate description of phenomena like electrical properties of nerve cell membranes and signal propagation compared to traditional methods.
    • The Mittag-Leffler function, a generalization of the exponential function, better fits observed cell membrane data.
    • Fractional derivatives naturally incorporate hereditary integrals and power-law relationships, essential for modeling biomaterials and phenomena like heat transfer and electrode/electrolyte behavior.

    Conclusions:

    • Fractional calculus offers a rigorous and direct approach to modeling complex biological systems.
    • Its application can lead to new functional relationships and improved understanding of natural phenomena in bioengineering and related fields.
    • The methods of fractional calculus are well-suited for solving diverse problems in biomedical research, including sub-threshold nerve propagation.