The complex permittivity of materials underlies their interaction with electromagnetic (EM) fields and provides information on how EM energy is absorbed and dissipated. This information is particularly important for many fields of research and development, and in recent years, considerable efforts were made to address gaps in knowledge identified by the medical technology industry. Accurate knowledge of the complex permittivity of human tissues is crucial in the design and development of diagnostic and therapeutic EM medical devices which are gaining increasing attention. This is because RF and MW radiation is non-ionising and has potential application in developing non- or minimally invasive theranostic devices. Examples of such applications are MW hyperthermia, ablation and imaging being investigated for diagnosis and treatment of a number of clinical conditions, such as early stroke detection and cancer [

1,

2,

3,

4,

5,

6]. Accurate knowledge of the dielectric properties of human tissue is not only required in the device design phase but is also essential for the creation and use of pre-treatment planning protocols, involving patient-specific 3-D EM field simulations. This is important in the case of microwave ablation, where an accurate estimate of the specific absorption rate (SAR) at various points around the MW applicator allows for the adjustment of input power, frequency, duration and insertion point in order to optimise the treatment outcome.

#### Measurement Methods

Numerous methods exist for measuring the dielectric properties of different materials. Some of the most common include cavity perturbation, transmission line, tetrapolar impedance and open-ended coaxial probes [

7,

8].

The open-ended coaxial probe method is widely used for measuring the dielectric properties of liquids and semi-solids, such as biological tissues [

7,

9,

10,

11,

12,

13]. A typical setup used to perform dielectric measurements is shown in

Figure 1b and includes an open-ended coaxial probe, connected to a VNA. The open tip of the coaxial transmission line is either embedded in or placed in contact with the material under test (MUT) such that the fringing fields emanating from the open tip reside within the MUT, as shown in

Figure 1b. The VNA test port emits a MW signal swept over a pre-selected frequency range and receives the reflection from the probe tip. It then computes the reflection coefficient (S11) at the pre-selected frequency points and the corresponding complex permittivity is computed [

10,

11,

12,

13,

14,

15].

Several publications have reported different methods to calculate the complex permittivity from the measured S11 [

15,

16,

17,

18,

19]. Reference [

20] compares two methods for converting S11 to complex permittivity; one is based on a lumped-element equivalent circuit while the other implements a full-wave analysis, requiring solution of the theoretical model for the fringing EM fields, which is referred to as the forward problem. Once the forward problem is formulated, the inverse problem is solved iteratively to determine the complex permittivity of the MUT from the measured S11 [

21]. This is computationally intensive and the inverse problem can be ill-posed, meaning that the solution does not always converge. On the other hand, the lumped-element equivalent circuit model provides a simpler approach by modelling the fringing fields at the tip of the coaxial probe through lumped equivalent circuits [

22,

23,

24,

25,

26,

27,

28,

29,

30,

31], and has become more popular over the years. The parameters for the equivalent circuits can be calculated from measurements on samples whose complex permittivity is accurately known, such as deionised water and NaCl solutions of precise concentration. Once these parameters are obtained for the standard liquids, they can be used to determine unknown dielectric parameters for other materials.

Whilst these methods are not as computationally intensive as the full-wave analysis, they depend on the initial measurements of three standard liquids [

32]. This means that the computed complex permittivity is highly dependent on the initial three reflection measurements and minor errors could propagate significantly to the computed complex permittivity of the MUT.

Recently, commercially available open-ended coaxial lines have gained popularity for measurements conducted and reported by various research groups. Some manufacturers provide dielectric measurement kits, along with software to compute the complex permittivity from reflection coefficients measured by a VNA. The commercial software is unfortunately a black-box, providing little or no control on the computational algorithms used and present compatibility issues when using hardware and software from different manufacturers. Moreover, these kits require a three-step calibration at reference plane A in

Figure 1; open-circuit, short circuit and a standard liquid which is usually deionised water [

10,

11,

12,

13]. This procedure requires skill and experience to guarantee acceptable repeatability, especially when terminating the coaxial open end with a short circuit so as to ensure good and uniform contact with the conducting surface.

In this paper, we propose an alternative method which overcomes such limitations by using an Artificial Neural Network (ANN) to convert measured reflection coefficients to complex permittivity. With this method, S11 measurements can be performed with any arbitrary VNA-probe configuration and then converted to the complex permittivity of a MUT. In our proposed method, any in-house designed probe can be used since the determination of the complex permittivity depends solely on the measurement of the reflection coefficient, without requiring VNA calibration extension to the measurement plane at the probe-material interface. This provides complete flexibility in the choice of hardware and experimental setup.

The reflection coefficient

Γ (S11) at the boundary separating two semi-infinite regions with complex permittivities

${\epsilon}_{1}$ and

${\epsilon}_{2}$, assuming non-magnetic materials of finite conductivity, is obtained by considering the tangential electric and magnetic intensity components which are continuous across the boundary, leading to

where

${Z}_{1}$ and

${Z}_{2}$ are respectively the wave impedances in medium 1, from where the wave is incident on the boundary, and medium 2, such that

where

${\mu}_{0}$ is the free space permeability since the material media are assumed non-magnetic. Thus, substituting for

${Z}_{1}$ and

${Z}_{2}$,

Equation (3) is non-linear and implies that

Γ can be transformed to

${\epsilon}_{2}$ at all frequencies, providing

ε_{1} is known. Clearly, the fringing fields at the antenna/MUT interface have non-tangential components that are not considered in the above simple analysis. However, the one-to-one relationship between

Γ and

${\epsilon}_{2}$ still exists.

Figure 2 describes conceptually how the conventional open-ended coaxial technique obtains

${\epsilon}_{2}$ for a MUT, and compares this process with the proposed ANN conversion approach.

Training of the ANN was based on a large number of measurements (>50), thus reducing reliance on precise calibration and the subsequent validation required by equivalent circuit methods [

10,

11,

12,

13]. The proposed method can be reused, without the necessity of repeated measurements on all samples, as long as no changes occur in the measurement setup. In this work, we propose a shift of the calibration plane to the VNA test port (reference plane B in

Figure 1), using only standard and repeatable coaxial open and short circuits as well as a matched load, which greatly simplifies the procedure and facilitates measurements in challenging scenarios such as sterile environments where in-vivo measurements are sought. Our alternative conversion technique is suitable for wideband and, even more so. for single frequency measurements.