Numerical simulation of nanopulse penetration of biological matter using the z -transform

Short duration, fast rise time ultra-wide-band (UWB) electromagnetic pulses (“nanopulses”) are generated by numerous electronic devices in use today. Moreover, many novel technologies involving nanopulses are under development and expected to become widely used soon. Study of nanopulse bioeffects is needed to probe their useful range in possible biomedical and biotechnological applications, and to ensure human safety. Based on the well-known dispersive properties of biological matter and their expression as a summation of terms corresponding to the main polarization mechanisms, the Cole-Cole expression is commonly employed to describe the frequency dependence of the dielectric properties of a tissue. Solving the Maxwell\u27s equations coupled with the Cole-Cole expression, however, is difficult because it is not easy to convert the equations from the frequency domain to the time domain. In this work we develop a computational approach to investigating electromagnetic fields in biological matter exposed to nanopulses, where the relative dielectric constant is given by the Cole-Cole expression for the frequency dependence of the dielectric properties of tissues. The Cole-Cole expression is first transformed to the z-domain using the z-transform method and then approximated by a second-order Taylor series of variable z. After converting the result from the frequency domain to the time domain, the finite-difference time-domain method (FDTD) is used to solve Maxwell\u27s equations coupled with the Cole-Cole expression, and a perfectly matched layer is applied to eliminate reflections from the boundary. The method is then applied to investigating the penetration of a short electromagnetic pulse into biological matter, where the relative dielectric constant is given by the Cole-Cole expression. Transmission, reflection, and absorption are calculated as a function of pulse width. It is found that these properties depend substantially on pulse characteristics. Future work in this direction could be examining the relevance of pulse rise time and pulse shape to tissue penetration. Such study could help to elucidate non-thermal mechanisms of nanopulse bioeffects

Numerical simulation of nanopulse penetration of biological matter using the ADI-FDTD method

Nanopulses are ultra-wide-band (UWB) electromagnetic pulses with pulse duration of only a few nanoseconds and electric field amplitudes greater than 105 V/m. They have been widely used in the development of new technologies in the field of medicine. Therefore, the study of the nanopulse bioeffects is important to ensure the appropriate application with nanopulses in biomedical and biotechnological settings. The conventional finite-difference time-domain (FDTD) method for solving Maxwell\u27s equations has been proven to be an effective method to solve the problems related to electromagnetism. However, its application is restricted by the Courant, Friedrichs, and Lewy (CFL) stability condition that confines the time increment and mesh size in the computation in order to prevent the solution from being divergent. This dissertation develops a new finite difference scheme coupled with the Cole-Cole expression for dielectric coefficients of biological tissues to simulate the electromagnetic fields inside biological tissues when exposed to nanopulses. The scheme is formulated based on the Yee\u27s cell and alternating direction implicit (ADI) technique. The basic idea behind the ADI technique is to break up every time step into two half-time steps. At the first half-step, the finite difference operator on the right-hand side of the Maxwell\u27s equation is implicit only along one coordinate axis direction. At the second half-step, the finite difference operator on the right-hand side of the Maxwell\u27s equation is implicit only along the other coordinate axis direction. As such, only tridiagonal linear systems are solved. In this numerical method, the Cole-Cole expression is approximated by a second-order Taylor series based on the z-transform method. In addition, the perfectly matched layer is employed for the boundary condition, and the total/scattered field technique is employed to generate the plane wave in order to prevent the wave reflection. The scheme is tested by numerical examples with two different biological tissues. For the purpose of comparison, both the proposed ADI-FDTD scheme and the conventional FDTD scheme are employed to the numerical examples. The results show that the proposed ADI-FDTD scheme breaks through the CFL stability condition and provides a stable solution with a larger time step, where the conventional FDTD scheme fails. Results also indicate that the computational time can be reduced with a larger time step