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Fractional calculus in bioengineering, part 2.

Richard L Magin1

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

Critical Reviews in Biomedical Engineering
|September 18, 2004
PubMed
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Fractional calculus, a mathematical tool for noninteger order operations, offers new ways to model complex biological systems. Its application in bioengineering, particularly for cell membranes and biomaterials, provides a more accurate understanding of natural phenomena.

Area of Science:

  • Biomedical Engineering
  • Mathematical Modeling
  • Biophysics

Background:

  • Fractional calculus, involving noninteger order integral and differential operations, has historical roots but limited application in biological systems.
  • Recent advancements highlight its use in physics and engineering, yet bioengineering applications remain underexplored.
  • Early studies on nerve cell electrical properties by Cole and Hodgkin demonstrated the utility of fractional calculus.

Purpose of the Study:

  • Introduce fractional calculus operations and their relevance to bioengineering.
  • Demonstrate the application of fractional calculus to model biological phenomena.
  • Highlight the potential of fractional calculus for developing novel functional relationships in biological system modeling.

Main Methods:

Related Experiment Videos

  • Reviewing the historical development and mathematical foundations of fractional calculus.
  • Illustrating basic fractional calculus operations on standard engineering functions.
  • Presenting specific examples of fractional calculus applications in electrochemistry, physics, bioengineering, and biophysics.

Main Results:

  • Fractional calculus accurately models phenomena like heat transfer, electrode/electrolyte behavior, and nerve signal propagation.
  • The Mittag-Leffler function, a generalization of the exponential function, provides a better fit for cell membrane data.
  • Fractional derivatives naturally incorporate hereditary integrals and power-law relationships for biomaterial modeling.

Conclusions:

  • Fractional calculus offers a rigorous and direct approach to modeling complex biological systems.
  • Expanding mathematical operations to include fractional calculus can lead to new functional relationships for bioengineering.
  • This review emphasizes the underutilized potential of fractional calculus in biomedical research and bioengineering.