ISSN Print: 0278-940X
Volumes:Volume 43, 2015 Volume 42, 2014 Volume 41, 2013 Volume 40, 2012 Volume 39, 2011 Volume 38, 2010 Volume 37, 2009 Volume 36, 2008 Volume 35, 2007 Volume 34, 2006 Volume 33, 2005 Volume 32, 2004 Volume 31, 2003 Volume 30, 2002 Volume 29, 2001 Volume 28, 2000 Volume 27, 1999 Volume 26, 1998 Volume 25, 1997 Volume 24, 1996 Volume 23, 1995
Critical Reviews™ in Biomedical Engineering
Fractional Calculus in Bioengineering, Part 1
Richard L. Magin
Department of Bioengineering, University of Illinois at Chicago, Chicago, IL, USA 60607
Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stressstrain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner.
|Home||Begell Digital Library||eBooks||Journals||References & Proceedings||Research Collections|