Fractional Calculus-Based Modeling of Electromagnetic Field Propagation in Arbitrary Biological Tissue

Abstract

The interaction of electromagnetic fields and biological tissues has become a topic of increasing interest for new research activities in bioelectrics, a new interdisciplinary field combining knowledge of electromagnetic theory, modeling, and simulations, physics, material science, cell biology, and medicine. In particular, the feasibility of pulsed electromagnetic fields in RF and mm-wave frequency range has been investigated with the objective to discover new noninvasive techniques in healthcare. The aim of this contribution is to illustrate a novel Finite-Difference Time-Domain (FDTD) scheme for simulating electromagnetic pulse propagation in arbitrary dispersive biological media. The proposed method is based on the fractional calculus theory and a general series expansion of the permittivity function. The spatial dispersion effects are taken into account, too. The resulting formulation is explicit, it has a second-order accuracy, and the need for additional storage variables is minimal. The comparison between simulation results and those evaluated by using an analytical method based on the Fourier transformation demonstrates the accuracy and effectiveness of the developed FDTD model. Five numerical examples showing the plane wave propagation in a variety of dispersive media are examined.

Fractional Calculus Based FDTD Modeling of Layered Biological Media Exposure to Wideband Electromagnetic Pulses

“…Despite its simplicity, it was proven that this simple planar model produces SAR values very close to those obtained by using more complex ones [31]. Moreover, it has been demonstrated that the developed fractional derivative-based FDTD scheme allows an accurate space-time evaluation of electromagnetic field profiles in a broad frequency range [22][23][24][25][26]. The reflection and transmission of a electromagnetic wave at the tissue interface depends on the frequency, permittivity and conductivity of the involved biological materials.…”

“…In order to evaluate the electromagnetic field propagation inside the layered structure involving HN dielectric materials, the FDTD scheme proposed in [22][23][24][25][26] has been used. In particular, it implements a more general series representation of the Riemann-Liouville fractional derivative operator, and it takes into account multiple relaxation times, as well as ohmic losses occurring in common biological media displaying different dispersion mechanisms.…”

“…Moreover, the conventional total field/scattered field approach and the Courant stability criterion are considered. Thus, applying a second order accurate finite-difference scheme and the procedure detailed in [22][23][24][25][26] for the finite-difference discretization, the FDTD update equations for the electric field, E, the magnetic field H and the auxiliary displacement current density, J i , can be calculated within the HN media and UPML regions. In particular, by considering the standard Cartesian Yee cell grid for the FDTD scheme and applying the central difference discretization, a second order O (∆z) 2 truncation error is achieved.…”

Fractional‐Calculus‐Based Electromagnetic Tool to Study Pulse Propagation in Arbitrary Dispersive Dielectrics

“…On the other hand, it suffers of a reduced numerical accuracy for , , and . To overcome this limitation, the authors have extended the FDTD scheme by implementing a more general series representation of the fractional derivative operators . This study was motivated to seek for an extended model flexibility enabling a better parametrization of the dispersive media properties as well as a better fitting, over broad frequency ranges, of the experimental dielectric response.…”

“…Applying a second‐order accurate finite‐difference scheme at the time instant and using the semi‐implicit approximation: and the numerical approximation after some mathematical manipulations, well detailed in ref. , it is possible to find the following updating equations for the magnetic and electric fields as well as for the displacement current density: …”