Cole-Cole constants for tissues of muscle and fat.

## Abstract

An auxiliary differential equation (ADE) transmission line method (TLM) is proposed for broadband modeling of electromagnetic (EM) wave propagation in biological tissues with the Cole-Cole dispersion Model. The fractional derivative problem is surmounted by assuming a linear behavior of the polarization current when the time discretization is short enough. The polarization current density is approached using Lagrange extrapolation polynomial and the fractional derivation is obtained according to Riemann definition of a fractional α-order derivative. Reflection coefficients at an air/muscle and air/fat tissues interfaces simulated in a 1-D domain are found to be in good agreement with those obtained from the analytic model over a broad frequency range, demonstrating the validity of the proposed approach.

### Keywords

- computational electromagnetics
- numerical methods
- transmission line matrix
- biological system modeling
- Cole-Cole medium
- fractional derivative

## 1. Introduction

In the last two decades there has been a growing interest in the interaction between biological tissues and electromagnetic field at microwave frequencies. New promising applications of this technology in biomedical engineering like microwave imaging [1, 2], minimal invasive cancer therapies as thermal ablations [3], ultra-wide band temperature dependent dielectric spectroscopy [4] and EM dosimetry [5], rely heavily on an accurate mathematical model of the response of these tissues to an external electromagnetic field. Numerical resolution of the propagation problem within these tissues requires a robust mathematical model and the previous incorporation of the dielectric data that at all the working frequencies.

A first model of the time response in a time varying electric field of biological tissues was formulated by Debye [6] through a time decaying polarization current

This model even if it fits the experimental results in liquids it loses its accuracy when applied over a large band of frequency or in the presence of more than one type of polar molecule. This non-Debye relaxation is attributed to the existence of different relaxation processes [7] each with its own relaxation time

where

The difficulty in the numerical implementation of such a model arises from the

The TLM method, even if it is flexible and wide band and being a time domain method, cannot deal with the dispersive aspect of the Cole-Cole medium directly. In [12] an approach based on a convolution product between the susceptibility and the electric field and the temporal behavior is deduced by a DFT and a nonrecursive summation leading to a considerable computational cost and a nonnegligible error at high frequencies. The causality principle is used to justify the minor dependence of the recent susceptibility values on previous ones, but the problem of the fractional Differentegration wasn’t addressed.

In this work an ADE-TLM algorithm to model the Cole-Cole dispersion. The polarization current density is approached using an extrapolation with Lagrange polynomial method and the fractional derivation is obtained using the Riemann definition of the

## 2. Formulation of the method

In the Cole-Cole model the relationship between the polarization current related to the

where

One could consider an analytic solution for this equation, but due to the fractional derivative this can only be done for particular values of

In order to obtain the discretized expression of

the interpolated expression of

and by applying (5) in the time interval

By substituting (7) in (6) the polynomial extrapolation of the polarization current density between time steps

which after simplification can be rewritten as:

Furthermore, by substituting (8) into (3) we derive the auxiliary differential equation with a fractional order derivative given by:

The generalization of the derivation operator to arbitrary non-integer orders, has been subject to intensive research from mathematicians [13, 14] and in electromagnetism [15]. One of the definitions in [16] for the fractional derivative, symbolized by the operator

where in our case

Hence by applying the fractional derivation operator given in (11) to (10) and for

where the update parameters

To get the update equation of the electric field components we start from the Maxwell-Ampere equation, and by including the conductivity term:

where

Finally, the updated electric field is given by:

## 3. The TLM Formalisme

The TLM method is a numerical technique based on the discretization of the computational domain according to Huygens principle as an alternative to the Maxwell equations used in the FDTD method [17]. In this method the simulation domain is discretized in cells where a series of uniform transmission lines, parallel and series stubs and additional sources are used to take account of the real characteristics of the propagation through the medium as given by Maxwell equations. Therefore, instead of electric and magnetic field components the electromagnetic field is represented by voltage and current waves, propagating through the unit cell circuit referred to as symmetrical condensed node (SCN). The relationship between the electromagnetic field and the voltage and current waves at the time step

As can be seen in Figure 3(b), the SCN consists of interconnected transmission lines. This structure models a unit cell of the propagation medium as in Figure 3(a). Each face of the cell corresponds to two ports orthogonal to each other and labeled from 1 to 12, these stubs model the propagation through free space and have therefore its characteristic impedance

The relationship between the electromagnetic field and the voltage and current waves at the time step

In this algorithm as in [11, 19] the dispersive properties of the medium are accounted for by adding voltage sources

where

When expressed in terms of incident voltages on the corresponding stubs and accounting for the voltage sources injected in the stubs (16,17,18) to complete the model of the Cole-Cole medium:

where the subscript

In the TLM formalism the update equation issued from the ADE in the Eq. (12) becomes:

and the electric field update equation is also formulated in the TLM formalism as:

The analogy between Eqs. (22) and (24) the expression of the normalized admittance of the stub added to the SCN node is obtained straightforward:

and the update equation for voltage sources at the center of the node:

that can be simplified to

with update constants:

A final step to establish the TLM formulation is to express the scattering process on the capacitive stubs:

## 4. Simulation and results

In order to verify the new proposed algorithm, and for the sake of simplicity a one dimensional problem is simulated, where a Cole-Cole medium (fat or muscle) occupies the region

The incident wave is a derivative Gaussian pulse given by

The fourth order Cole-Cole parameters for fat tissue as well as for muscle tissue are listed in Table 1 [9]. The values of the electric field are recorded at a point P located at the center of the SCN node 10 cells before the air-medium interface. Figure 7(a) shows the incident impulse at iteration 399 propagating in the air and Figure 7(b) at iteration 927 depicts the reflected and transmitted impulse on the air/muscle interface.

Muscle | 2.5 | 0.035 | 9 | 7.96 | 0.8 | 35 | 15.92 | 0.9 | 3.3E4 | 159.15 | 0.95 | 1.0E7 | 15.915 | 0.99 |

Fat | 4 | 0.2 | 50 | 7.23 | 0.9 | 7000 | 353.68 | 0.9 | 1.2E6 | 318.31 | 0.9 | 2.5E7 | 2.274 | 1 |

In Figures 8 and 9, the simulation results for both cases (fat and muscle) are compared to the theoretical reflection at the air/Cole-Cole medium interface which can be obtained by using the following equation [22]:

A good agreement over the whole frequency band is observed. The slight discrepancy between the numerical and theoretical results can be ascribed to the approximation made to the polarization current value when performing the Lagrange extrapolation on each iteration. The propagation and reflection through a 2-D TLM grid modeling the incidence on an air/muscle interface is presented in Figures 5 and 10 at different time steps, the interface is located at

## 5. Conclusion

Numerical methods are an essential part of the modeling process, new propagation media with anomalous relaxation properties impose innovative modeling methods. An effective ADE-TLM formulation way to model electromagnetic wave propagation in biological tissues using Cole-Cole dispersion model. The fractional-order derivative in the Cole-Cole model is tackled by using a polynomial extrapolation of the polarization current. This approximation reduces considerably both the numerical cost of the simulation and its complexity since only two previous values of the

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