"Fractional Calculus: A Tool to Model Complex Dynamics in Multi-scale Structures"
-- Richard L. Magin
Abstract
Fractional-order models provide a heuristic approach to the description of complex circuits and systems.
Instead of simply extending the structure, composition or number of components in a linear system, the "fractional" approach is to generalize the order of the integer derivatives that describe key dynamic processes (e.g., battery charging, viscoelastic creep or electric/magnetic polarization). As noted by the mathematician M. Kac, ("Some Mathematical Models in Science," Nov. 7, 1969, Vol. 166 pp. 695-699, Science), “success is characterized by the fidelity with which such models fit the observed phenomena, and by the sharpness of the questions they pose about the underlying physics.”
Fractional-order models are sometimes criticized for what they are not – conceptual models of fundamental laws or theories. Therefore, it is important to identify the properties of a complex system that suggest it is appropriate for fractional-order generalization. Just as Brownian motion in a stochastic system follows from assumptions that the component particles are identical and independent of each other, fractional-order models appear to be most useful for systems that relax these assumptions, and exhibit some degree of memory or nonlocal behavior.
The theme of this presentation is that fractional (non-integer order) calculus can provide a basis for a greater understanding of the molecular events that occur in biological tissues.
Such an understanding is fundamental in bioengineering when engineers seek to describe the underlying multi-scale processes that occur, for example, when tissues are electrically stimulated or mechanically stressed. Fractional-order models work well in physics, chemistry and rheology, particularly in describing dielectrics and viscoelastic materials over extended ranges of time and frequency. Further, in heat transfer and electrochemistry, the half-order fractional integral is a natural convolution operator connecting the applied gradients (thermal or material) with the diffusion of ions or heat.
Can fractional calculus uncover similar relatively simple links between stress and strain in load-bearing tissues, the electrical impedance of implanted cardiac pacemaker electrodes, or in predicting changes in the shear modulus of tumors developing in breast tissue? Because the constitutive properties of tissue depend on the composition and micro-scale architecture of the cellular and extracellular networks, the challenge is to develop non-invasive modeling, visualization and assessment tools that predict macro-scale mechanical performance from micro-scale observations or measurements.
In this seminar I will describe some of the characteristics of fractional calculus that make it well suited for modeling engineered systems, and outline three areas of bioengineering research where fractional calculus is being applied.
Biography
Magin completed undergraduate and graduate studies in physics at Georgia Tech (bachelor's, 1969, master's, 1972) followed by additional graduate work in biophysics at the University of Rochester (Ph.D., 1976). He worked as a postdoctoral student for three years in the Laboratory of Chemical Pharmacology at the National Cancer Institute. In 1979, he joined the faculty in the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign. He worked in Urbana for 18 years as an assistant, associate and full professor before joining the Department of Bioengineering at the University of Illinois at Chicago in 1998. He served as Bioengineering Department head at UIC from 1998 to 2010. He is a professor of bioengineering at UIC and directs the Diagnostic NMR Systems Laboratory.
Magin is a Fellow of the IEEE and AIMBE, and former editor of the Critical Reviews in Biomedical Engineering. In 1989 and 1990, Magin completed a sabbatical year in the Department of Radiology of Shands Medical Center at the University of Florida in Gainesville. In 2006, he Magin received a Fulbright grant to lecture and conduct research at the Technical University of Kosice in Kosice, Slovakia. In 2012, he was designated a distinguished professor of bioengineering at UIC. His research interests focus on the applications of magnetic resonance imaging (MRI) in science and engineering. He can be reached at rmagin@uic.edu rmagin@uic.edu.
