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

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Complex Permittivity

#### 2.2. Electromagnetic Analysis

**E**, the magnetic field

**H**and the auxiliary displacement current density, ${\mathbf{J}}_{i}$, 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 $O{\left(\Delta z\right)}^{2}$ truncation error is achieved.

#### 2.3. Bioheat Equation with Thermoregulation

**E**is the electric field at the generic z coordinate.

## 3. Numerical Results

**E**,

**H**and ${\mathbf{J}}_{i}$ fields can be Fourier transformed with respect to the space variable z to provide the spectrum of monochromatic plane-wave modes propagating along the computational lattice. In this way, it has been checked that the resulting characteristic polynomial equation has zeros inside the stability circle [20,24].

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FDTD | Finite difference time domain |

PEF | Pulsed electric field |

EM | Electromagnetic |

CSF | Cerebro-spinal fluid |

SAR | Specific absorption rate |

HN | Havriliak-Negami |

UPML | Uniaxial perfectly-matched layer |

RF | Radio frequency |

EWQPSO | Enhanced weighted quantum particle swarm optimization |

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**Figure 2.**Sketch illustrating the thermoreceptor signals and the thermoregulation mechanisms involved in Gagge’s two-node model.

**Figure 3.**Reflectance and transmittance spectrum of the six-layered dielectric slab shown in Figure 1.

**Figure 4.**(

**a**) PEF source having ${E}_{\mathrm{max}}=1600\phantom{\rule{0.277778em}{0ex}}\mathrm{V}/\mathrm{m}$ and ${t}_{\mathrm{hold}}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{n}\mathrm{s}$; (

**b**) the corresponding SAR profile inside the multilayered head structure.

**Figure 5.**(

**a**) Time-dependent temperature rise in the skin, bone, and brain layers; (

**b**) comparison between non-thermoregulated and thermoregulated models.

**Figure 6.**SAR profile obtained by considering a PEF source with (

**a**) three different pulse amplitudes; (

**b**) three different hold times.

**Figure 7.**Temperature versus the time for three different PEF amplitudes; (

**a**) at the air-skin boundary and (

**b**) at the outer surface of the brain layer.

**Figure 8.**Temperature versus time for three different PEF durations; (

**a**) at the air-skin boundary and (

**b**) at the outer surface of the brain layer.

Parameter | Skin | Fat | Bone | Dura | CSF | Brain |
---|---|---|---|---|---|---|

${\alpha}_{1}$ | 0.93 | 0.92 | 0.91 | 0.88 | 0.92 | 0.93 |

${\alpha}_{2}$ | 0.92 | 0.91 | 0.70 | 0.81 | 0.99 | 0.99 |

${\beta}_{1}$ | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

${\beta}_{2}$ | 0.87 | 0.85 | 0.99 | 0.97 | 0.85 | 0.79 |

${\tau}_{1}\phantom{\rule{0.277778em}{0ex}}\left(\mathrm{p}\mathrm{s}\right)$ | 8.02 | 8.34 | 13.79 | 8.02 | 8.03 | 8.01 |

${\tau}_{2}\phantom{\rule{0.277778em}{0ex}}\left(\mathrm{n}\mathrm{s}\right)$ | 67.68 | 23.98 | 63.96 | 105.85 | 24.71 | 117.25 |

$\Delta {\u03f5}_{{r}_{1}}$ | 36.94 | 2.35 | 8.18 | 37.89 | 63.23 | 42.93 |

$\Delta {\u03f5}_{{r}_{2}}$ | 1789.8 | 79.20 | 130 | 1087.4 | 315.09 | 2823.9 |

$\sigma \phantom{\rule{0.277778em}{0ex}}(\mathrm{S}/\mathrm{m})$ | 0 | 0.01 | 0.006 | 0.5 | 2 | 0.02 |

${\u03f5}_{{r}_{\infty}}$ | 4 | 2.5 | 2.5 | 4 | 4 | 4 |

$d\left(\mathrm{m}\mathrm{m}\right)$ | 0.7 | 1.6 | 20.5 | 0.5 | 2 | ∞ |

Medium | $\mathit{\rho}\phantom{\rule{0.277778em}{0ex}}(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3})$ | $\mathit{C}\phantom{\rule{0.277778em}{0ex}}(\mathbf{J}/\mathbf{k}\mathbf{g}\mathbf{K})$ | $\mathit{k}\phantom{\rule{0.277778em}{0ex}}(\mathbf{W}/\mathbf{m}\mathbf{K})$ | ${\mathit{W}}_{\mathit{b}}\phantom{\rule{0.277778em}{0ex}}(\mathbf{1}/\mathbf{s})$ |
---|---|---|---|---|

Skin | 1100 | 3600 | 0.42 | 0.0025 |

Fat | 920 | 3000 | 0.25 | 0.0005 |

Bone | 1850 | 3100 | 0.39 | 0.0005 |

Dura | 1050 | 3600 | 0.5 | 0.0003 |

CSF | 1060 | 4000 | 0.62 | 0 |

Brain | 1030 | 3650 | 0.535 | 0.011 |

Blood | 937 | 3889 | - | - |

**Table 3.**Thermal increase comparison inside the brain layer by considering the thermal model published in [42] and the proposed thermoregulated model.

Model | $\mathit{f}=900\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{MHz}\right)$ | $\mathit{f}=1500\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{MHz}\right)$ |
---|---|---|

Thermal model in [42] | $0.160\phantom{\rule{0.277778em}{0ex}}\mathrm{K}$ | $0.132\phantom{\rule{0.277778em}{0ex}}\mathrm{K}$ |

Proposed thermoregulated model | $0.156\phantom{\rule{0.277778em}{0ex}}\mathrm{K}$ | $0.117\phantom{\rule{0.277778em}{0ex}}\mathrm{K}$ |

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**MDPI and ACS Style**

Mescia, L.; Bia, P.; Chiapperino, M.A.; Caratelli, D. Fractional Calculus Based FDTD Modeling of Layered Biological Media Exposure to Wideband Electromagnetic Pulses. *Electronics* **2017**, *6*, 106.
https://doi.org/10.3390/electronics6040106

**AMA Style**

Mescia L, Bia P, Chiapperino MA, Caratelli D. Fractional Calculus Based FDTD Modeling of Layered Biological Media Exposure to Wideband Electromagnetic Pulses. *Electronics*. 2017; 6(4):106.
https://doi.org/10.3390/electronics6040106

**Chicago/Turabian Style**

Mescia, Luciano, Pietro Bia, Michele Alessandro Chiapperino, and Diego Caratelli. 2017. "Fractional Calculus Based FDTD Modeling of Layered Biological Media Exposure to Wideband Electromagnetic Pulses" *Electronics* 6, no. 4: 106.
https://doi.org/10.3390/electronics6040106