Debye Model. ∆t βb = . 2ϵ 0∆ϵ∆t 2τ + ∆t The main features of the dielectric spectrum of tissues are well-known and have been reviewed and reported by [96]. The dielectric spectrum of tissues is characterised by three main relaxation regions at low, medium and high frequencies. In its simplest form, each of these relaxation regions is the manifestation of a polarisation mechanism characterised by a single time constant τ. The Debye model and single-pole Xxxx–Xxxx model are commonly used to describe human tissues. This is because a lossy dispersive wave equation is conducted completely by the frequency-dependent conductivity. The Debye relation is used to model the dispersive behaviour of the complex relative permittivity of breast tissue [64]. The single-pole Xxxx–Xxxx model is defined as ϵ r (ω) = ϵ ′ (ω) − j ϵ ′′(ω) = ϵ + ϵ s − ϵ∞ + σs ∞ 1 + ( jωτ)1−α jωϵ 0 where ω is the angular frequency, ϵ ′(ω) is the frequency-dependent dielectric constant, and ϵ ′′(ω) is the frequency-dependent dielectric loss which can be converted into the effective conductivity, σ(ω) = ωϵ 0ϵ ′′(ω). Note that α is a fitting parameter. In many cases such as biological tissue within certain frequency ranges, the Xxxx-Xxxx model in (2.20) can be simplified to a single-pole Debye model by setting α = 1 in (2.20) [34] ϵ(ω) = ϵ∞ + ϵ s − ϵ∞ + σs ϵ 0 1 + jωτ jωϵ 0 ∆ϵ σs ≡ ϵ∞ + 1 + jωτ + jωϵ 0 where ϵ s is the zero-frequency relative permittivity and ϵ∞ is the relative permittivity at infinite frequency. τ is the pole relaxation time and σs denotes the static conductivity. Finally, ω denotes angular frequency. In this thesis, the frequency range of interest for the MWI is from 0.5 to 3.5 GHz. For this study, the approximations of the Debye model of different breast tissues were obtained from the University of Wisconsin Cross-Disciplinary Electromagnetics Laboratory (UWCEM), which provides a number of anatomically-realistic MRI-derived numerical breast phantoms for breast cancer detection and treatment applications. The breast tissues in these phantoms have the realistic ultra wideband dielectric properties reported in [97].
