7. Proof-of-concept experiment and simulation

To show the capabilities of the NRI system as an independent system, a simple measurement was designed to firstly acquire the data from a scattering object inside a known medium, and secondly to retrieve the dialectic map of the medium and the object. This first measurement was simplified using the following requirements: (1) The medium is required to be homogenous with known dielectric properties close to those of breast fat; (2) the object needs to be a strong scatterer of simple and known shape and big size so that it can be easily detected; (3) the data acquisition is to be done in a straight line, instead of the maze-like path, passing through the centroid of the object; (4) the air gap between the NRI system (bottom container) and the under-test medium is to be eliminated. The conditions were realized by using a stainless steel bearing ball of 1-inch diameter inside a container filled with sunflower oil. It is conspicuous that the conditions described differ largely from the reality in which the breast tissue is highly heterogeneous, the contrast between malignant and healthy tissues is comparatively small, and the size of tumors in their early stages is in order of millimeters; however, this first test was intentionally designed to avoid complexities for the purpose of proof-of-concept. Besides this experiment, a simulation was carried out to illustrate how the bimodal imaging using the DBT and the near-field radar system works. The details of this simulation are given in subsection 7.2 and 7.4.

#### 7.1. The bearing ball imaging experiment setup

Required by the imaging algorithm, both the background data, when there was no scatterer in the medium, and the total field data, when the ball was placed in the oil container, had to be acquired. Accordingly, the NRI system was placed on top of the container, partially immersed in the oil, and 25 sets of data at 25 equally spaced positions on the motion path were obtained from the medium with and without the bearing ball inside it. In the case of total field measurements, the ball, seated at the bottom of the container, was positioned at various distances from the centre of the bottom acrylic sheet. The antennas' center motion path was defined to be a straight line passing through the center of the ball, when the system was viewed from the top, as shown with a hashed arrow in Figure 8. The layout of the experiment is illustrated in Figure 9.

To obtain a more accurate model of the coupling liquid that was used for both the SAR analysis and imaging algorithm, the complex permittivity of absolute ethanol was measured by the PNA-X material measurement software (Keysight Material Measurement Suite 2015), and it was compared against the one reported in the literature [95]. The result is in good agreement with the one reported by Sato et al., as shown in Figure 10.

#### 7.2. Imaging algorithm

The total electric field E(r, ω), assumed to be a function of the vector position r and frequency ω, due to electromagnetic propagation into a three-dimensional medium can be expressed by Helmholtz equation as [111]

Near-Field Radar Microwave Imaging as an Add-on Modality to Mammography http://dx.doi.org/10.5772/intechopen.69726 31

7. Proof-of-concept experiment and simulation

30 New Perspectives in Breast Imaging

simulation are given in subsection 7.2 and 7.4.

7.1. The bearing ball imaging experiment setup

with the one reported by Sato et al., as shown in Figure 10.

Figure 9.

7.2. Imaging algorithm

Helmholtz equation as [111]

To show the capabilities of the NRI system as an independent system, a simple measurement was designed to firstly acquire the data from a scattering object inside a known medium, and secondly to retrieve the dialectic map of the medium and the object. This first measurement was simplified using the following requirements: (1) The medium is required to be homogenous with known dielectric properties close to those of breast fat; (2) the object needs to be a strong scatterer of simple and known shape and big size so that it can be easily detected; (3) the data acquisition is to be done in a straight line, instead of the maze-like path, passing through the centroid of the object; (4) the air gap between the NRI system (bottom container) and the under-test medium is to be eliminated. The conditions were realized by using a stainless steel bearing ball of 1-inch diameter inside a container filled with sunflower oil. It is conspicuous that the conditions described differ largely from the reality in which the breast tissue is highly heterogeneous, the contrast between malignant and healthy tissues is comparatively small, and the size of tumors in their early stages is in order of millimeters; however, this first test was intentionally designed to avoid complexities for the purpose of proof-of-concept. Besides this experiment, a simulation was carried out to illustrate how the bimodal imaging using the DBT and the near-field radar system works. The details of this

Required by the imaging algorithm, both the background data, when there was no scatterer in the medium, and the total field data, when the ball was placed in the oil container, had to be acquired. Accordingly, the NRI system was placed on top of the container, partially immersed in the oil, and 25 sets of data at 25 equally spaced positions on the motion path were obtained from the medium with and without the bearing ball inside it. In the case of total field measurements, the ball, seated at the bottom of the container, was positioned at various distances from the centre of the bottom acrylic sheet. The antennas' center motion path was defined to be a straight line passing through the center of the ball, when the system was viewed from the top, as shown with a hashed arrow in Figure 8. The layout of the experiment is illustrated in

To obtain a more accurate model of the coupling liquid that was used for both the SAR analysis and imaging algorithm, the complex permittivity of absolute ethanol was measured by the PNA-X material measurement software (Keysight Material Measurement Suite 2015), and it was compared against the one reported in the literature [95]. The result is in good agreement

The total electric field E(r, ω), assumed to be a function of the vector position r and frequency ω, due to electromagnetic propagation into a three-dimensional medium can be expressed by

Figure 9. The bearing ball imaging experiment: (a) the entire setup with the NRI system on top of the oil container which is seated on a low-density wooden table. (b) The front and (c) perspective views of the experiment showing how the ethanol container is partially submerged in oil with the aim of air gap elimination. As shown, the ball rests on a plastic base whose diameter is smaller than the ball itself and thus shadowed by the ball when the microwaves illuminate the medium.

Figure 10. The dielectric relaxation result of ethanol; measurement versus what is reported in the literature [97]. E<sup>0</sup> and E<sup>00</sup> respectively denote the real and imaginary part of complex permittivity.

$$\nabla \times \nabla \, E(\mathbf{r}, \omega) - k^2(\mathbf{r}, \omega) E(\mathbf{r}, \omega) = j\omega\mu\_0 I(\mathbf{r}, \omega) \tag{5}$$

where <sup>k</sup>ðr, <sup>ω</sup>Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ε</sup>0εrμ0ðr, <sup>ω</sup><sup>Þ</sup> <sup>p</sup> is the wavenumber in the medium with <sup>ε</sup><sup>0</sup> being the vacuum permittivity, εr(r, ω) being the relative complex permittivity, and μ<sup>0</sup> being the vacuum permeability; and I(r, ω) is the microwave excitation source in the case of near-field radar radiation. The relative complex permittivity of a material is defined as ε<sup>r</sup> ¼ ε<sup>0</sup> <sup>r</sup> � jσ=ωε0, where ε<sup>0</sup> <sup>r</sup> is the real part, and ε<sup>00</sup> <sup>r</sup> ¼ σ=ωε<sup>0</sup> is the imaginary part that depends on the conductivity of the material σ. The total electric field E(r, ω) is composed of the background Eb(r, ω) and scattered field Es(r, ω), or mathematically:

$$E(\mathbf{r},\omega) = E\_{\mathbb{b}}(\mathbf{r},\omega) + E\_{\mathbb{s}}(\mathbf{r},\omega) \tag{6}$$

The background field is usually modeled with the help of simulations or analytical methods, and the scattered field is obtained by subtracting the background field from the measured total field. Since Eb(r, ω) also satisfies the Helmholtz equation, with kb(r, ω) instead of k(r, ω) and the solution

$$E\_b(\mathbf{r}, \omega) = j\omega \int \mathbf{G}\_b(\mathbf{r}, \mathbf{r}', \omega) I(\mathbf{r}', \omega) d\mathbf{r}' \tag{7}$$

where Gb(r,r<sup>0</sup> , ω) is dyadic Green's function and a solution of

$$\nabla \times \nabla \mathbf{G}\_b(\mathbf{r}, \omega) - k\_b^2(\mathbf{r}, \omega) \mathbf{G}\_b(\mathbf{r}, \omega) = \mathbf{\tilde{I}} \delta(\mathbf{r} - \mathbf{r}') \tag{8}$$

with ~I being the unit dyad, the scattered field Es(r, ω) can be solved as

$$E\_s(\mathbf{r},\omega) = \int \mathbf{G}\_b(\mathbf{r},\mathbf{r}',\omega) k\_b^2(\mathbf{r}',\omega) E(\mathbf{r},\omega) \chi(\mathbf{r}',\omega) d\mathbf{r}' \tag{9}$$

in which is χ = (εr(r, ω) � εr,b(r, ω))/εr,b(r, ω), the contrast parameter, and εr,b(r, ω) is the relative permittivity of the background medium. In cases where the contrast parameter is comparatively small, BORN approximation can be applied by replacing the total field with the background field in Eq. (9), which results in

$$E\_s(\mathbf{r}, \omega) \approx \int \mathbf{G}\_b(\mathbf{r}, \mathbf{r}', \omega) k\_b^2(\mathbf{r}', \omega) \mathbf{E}\_b(\mathbf{r}, \omega) \chi(\mathbf{r}', \omega) d\mathbf{r}' \tag{10}$$

that can be solved now. Making use of the FDFD method and discretization, Eq. (10) can be linearized in terms of the unknown–the contrast parameter– as follows [112]:

$$\mathbf{y} = A\boldsymbol{\chi} + \mathbf{e} \tag{11}$$

where y∈ C<sup>M</sup> is the measurement vector obtained by the receiving antenna, A ∈ C<sup>M</sup>�<sup>N</sup> is the sensing matrix, χ ∈ C<sup>N</sup> is the unknown contrast vector, and e∈ C<sup>M</sup> is the modeled noise. The sensing matrix A is generated by the Green's functions of the background medium. Note that for the DBT-NRI system, the background Green's functions are computed using a full wave model (like HFSS ANSYS) in which the dielectric properties of a healthy heterogeneous breast is derived from the fat content of the DBT image. For the results of the bearing ball experiment, presented in this chapter, the background Green's functions are computed from modeling the mechatronic system without the metallic scatterer (ball), also using HFSS ANSYS. As the number of measurements, M, is much less than the number of unknowns N, a regularization method is to be used to reduce the ill-posedness of the solution. In this case, Tikhonov regularization scheme can be applied, which seeks the solution to the succeeding optimization problem [112]:

$$\min \left| \left| \mathcal{A} \mathbf{x} - \mathbf{y} \right| \right|\_{\ell\_2}^2 + \gamma \left| \left| \mathbf{x} \right| \right|\_{\ell\_p}^p \tag{12}$$

$$\text{Subject to} \begin{cases} \text{Re}\left(\text{diag}(\varepsilon\_b)\mathbf{x} + \varepsilon\_b\right) \ge 1\\ \text{Im}\left(\text{diag}(\varepsilon\_b)\mathbf{x} + \varepsilon\_b\right) \ge 0 \end{cases} \tag{13}$$

in which ℓ<sup>p</sup> denotes norm-p and γ is the regularization parameter. When this problem is solved using p = 2, indicating that the ℓ<sup>2</sup> is used as a regularizer function, a simple closed-form solution is given by the following equation:

$$\mathbf{x} = (\mathbf{A}^H \mathbf{A} + \mathbf{y} \mathbf{I})^{-1} \mathbf{A}^H \mathbf{y} \tag{14}$$

where A<sup>H</sup> is the Hermitian of matrix A and γ is is selected by trial and error with the aim of achieving an efficient and smooth solution. Other convex optimization approaches can also be used. Specifically, when the problem is solved using p = 1, indicating the use of the ℓ<sup>1</sup> as a regularizer function, sparsity can be imposed to the solution—see for example [111]. This regularizer function is of special interest when the DBT and the NRI are operating together.

#### 7.3. Results

field. Since Eb(r, ω) also satisfies the Helmholtz equation, with kb(r, ω) instead of k(r, ω) and the

Gbðr,r 0 , ωÞIðr 0 , ωÞdr

in which is χ = (εr(r, ω) � εr,b(r, ω))/εr,b(r, ω), the contrast parameter, and εr,b(r, ω) is the relative permittivity of the background medium. In cases where the contrast parameter is comparatively small, BORN approximation can be applied by replacing the total field with the back-

that can be solved now. Making use of the FDFD method and discretization, Eq. (10) can be

where y∈ C<sup>M</sup> is the measurement vector obtained by the receiving antenna, A ∈ C<sup>M</sup>�<sup>N</sup> is the sensing matrix, χ ∈ C<sup>N</sup> is the unknown contrast vector, and e∈ C<sup>M</sup> is the modeled noise. The sensing matrix A is generated by the Green's functions of the background medium. Note that for the DBT-NRI system, the background Green's functions are computed using a full wave model (like HFSS ANSYS) in which the dielectric properties of a healthy heterogeneous breast is derived from the fat content of the DBT image. For the results of the bearing ball experiment, presented in this chapter, the background Green's functions are computed from modeling the mechatronic system without the metallic scatterer (ball), also using HFSS ANSYS. As the number of measurements, M, is much less than the number of unknowns N, a regularization method is to be used to reduce the ill-posedness of the solution. In this case, Tikhonov regularization scheme can be applied, which seeks the solution to the succeeding optimization

<sup>b</sup> <sup>ð</sup>r, <sup>ω</sup>ÞGbðr, <sup>ω</sup>Þ ¼ <sup>~</sup>Iδð<sup>r</sup> � <sup>r</sup>

, ωÞEðr, ωÞχðr

, ωÞEbðr, ωÞχðr

0

0

0

y ¼ Aχ þ e (11)

<sup>0</sup> (7)

Þ (8)

, ωÞdr<sup>0</sup> (9)

, ωÞdr<sup>0</sup> (10)

<sup>ℓ</sup><sup>p</sup> (12)

ð

2

Ebðr, ωÞ ¼ jω

, ω) is dyadic Green's function and a solution of

∇ � ∇Gbðr, ωÞ � k

with ~I being the unit dyad, the scattered field Es(r, ω) can be solved as

ð

Gbðr,r 0 , <sup>ω</sup>Þk<sup>2</sup> <sup>b</sup> ðr<sup>0</sup>

Esðr, ωÞ ¼

Esðr, ωÞ ≈

ð

Gbðr,r 0 , <sup>ω</sup>Þk<sup>2</sup> b ðr 0

linearized in terms of the unknown–the contrast parameter– as follows [112]:

min�

�jAχ � yj

� � 2 <sup>ℓ</sup><sup>2</sup> þ γ � �jχj � � p

ground field in Eq. (9), which results in

solution

32 New Perspectives in Breast Imaging

where Gb(r,r<sup>0</sup>

problem [112]:

#### 7.3.1. Images reconstructed from computational simulations

To demonstrate the efficacy of the hybrid bimodal DBT/NRI system, the fat percentage of the DBT image of a breast was used to model the electromagnetic scattering of a breast with and without a cancer lesion. Figure 11(a) shows the ground truth model. A total of six antennas at three different frequencies, 500, 600, and 700 MHz, were used on the periphery of the breast model, inside a bolus liquid, that enhanced the electromagnetic coupling into the tissue [110]. The method described in [106], which used the ℓ<sup>1</sup> norm as a regularizer function, was employed to estimate the real and imaginary parts of the contrast variable χ. As shown in Figure 11(b), the contrast source variable successfully localized the tumor, albeit of being surrounded by fibroglandular tissue.

Figure 11. (a) The ground truth complex permittivity, the real (left) and the imaginary (right) part. (b) Successfully reconstructed contrast source variable χ, as defined in Eq. (9), after using the imaging technique described in [106]; (left) real and (right) imaginary parts.

Figure 12. Reconstructed images of a bearing ball embedded in sunflower oil. Images obtained from the data acquired by a regular SMA cable (a), a phase-stable cable (b), and simulation (c), respectively, when the ball was located 1 cm off center. Parts (d)–(f) show similar images for when the ball was 5 cm off center.

#### 7.3.2. Images reconstructed from the experiment

Implementing the imaging algorithm described in the previous subsection, the images of the oil-bearing ball medium were reconstructed for different cases when the ℓ<sup>2</sup> norm is used as the regularizer. As shown in Figure 12, the measurements were carried out for various ball locations (two of which are shown here) and for two type of cables, regular SMA cable and phase-stable cable. Despite the presence of some artifacts, the normalized dielectric maps from the measurements agree well with those obtained from the simulations. In overall, though phase-stable cables improved the phase response of the system significantly, they were not as effective in the final imaging results. These images demonstrate that the NRI system is capable of collecting meaningful data and generating images for a simply configured medium.

#### 8. Conclusions

In this chapter, the basics of a bimodal imaging system aimed at early detection of breast cancer were reviewed and some preliminary computational and experimental results were presented. It was noted that the conventional mammography has raised some concerns due to the reported rates of false-positive and false-negative results, as well as overdiagnosis. Digital breast tomosynthesis has been able to compensate for some of these problems, only up to a certain degree, by enabling multi-layer imaging of the breast; and it has resulted in improved specificity and sensitivity. However, the low contrast between malignant and fibroglandular tissues in the X-ray frequencies still remains as a drawback of X-ray-based breast screening. Near-field radar imaging (NRI), as a cheap and safe modality, has the potential to alleviate the problem of low contrast inasmuch as the aforementioned tissues show more contrast at microwave frequencies. Founded on this observation, an NRI mechatronic system, compatible with the DBT, was developed to be used as an auxiliary diagnosis tool, in a co-registered manner. The performance of the NRI component of this system was experimentally evaluated in a near-ideal case, and the achieved results showed its capability in imaging a strong scatterer in a homogenous medium. Computational results were also carried out, showing the efficacy of the bimodal system to detect tumors surrounded by fibroglandular tissue. It was also shown, through specific absorption rate analysis, that the radiation of the implemented antennas is safe for humans, according to the standards. Our next step in this exciting work is to pilot the hybrid DBT/NRI system using more realistic phantoms, as well as human patients.
