Fractional calculus in bioengineering: A tool to model complex dynamics

Abstract

The premise of this work is that fractional (non-integer order) calculus can provide the basis for a greater understanding of the dynamic processes that occur in biological tissues. Such an understanding is fundamental in bioengineering where engineers seek a simpler description of the underlying multi-scale processes that occur, for example, when tissues are mechanically stressed or strained. Fractional order models work well in physics, electrochemistry and rheology, particularly in describing dielectric, magnetic and viscoelastic materials over extended ranges of time and frequency. In heat transfer and electrochemistry, for example, the half-order fractional integral is the natural integral operator connecting applied gradients (thermal or material) with the resultant diffusion of ions or heat. Can fractional calculus be applied in bioengineering to uncover similar relatively simple links between stress and strain in load-bearing tissues, such as cartilage, the electrical impedance of implanted cardiac pacemaker electrodes, or in predicting changes in the shear modulus of tumors developing in breast tissue? Since the constitutive properties of tissue depend on the micro-scale architecture of the cellular and extracellular networks, the challenge for the bioengineer is to develop new modeling, visualization and assessment tools that better predict the macro-scale mechanical performance from measurements observations at the micro- and nano-scale. In this paper I describe some of the characteristics of fractional calculus that I believe make it well suited for this application, and outline three areas of bioengineering research where fractional calculus is being applied.