# Machine Learning Approach to Quadratic Programming-Based Microwave Imaging for Breast Cancer Detection

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{b}and permeability 𝜇

_{0}. Non-magnetic scatterers, characterized by the relative dielectric constant ε

_{r}(r), are located within the domain of interest D ϵ R

^{2}, and illuminated by the Ni line sources, located at the point ${r}_{p}^{i}$ with p = 1, 2, …, Ni. For each incidence, the scattered field is measured by Nr antennas, located at the point ${r}_{q}^{s}$ with q = 1, 2, …, Nr.

^{jωt}is assumed), describing the interaction of the wave scatterer in the D-domain. It is also known as the state equation [8], and is reported as follows:

- −
- ${E}^{t}\left(r\right)$ is the total electric field;
- −
- ${E}^{i}\left(r\right)$ is the incident electric field;
- −
- ${k}_{b}=w\sqrt{{\epsilon}_{b}{\mu}_{0}}$ is the wavenumber of the homogeneous medium background;
- −
- $J\left(r\right)$ is the contrast current density, defined as $J\left(r\right)=\chi \left(r\right){E}^{t}\left(r\right)$, where the contrast function is given by:

^{2}-dimensional form and, from the definition of the contrast current density J(r), the discretized forms of J(r) can be computed as $\overline{J}=\mathrm{diag}\left(\overline{\chi}\right)\xb7{\overline{E}}^{t}$ [12].

## 3. Method

#### 3.1. Quadratic Programming Approach

- ${E}_{m}^{s}$ is the scattered electric field at the position r
_{m}on the surface S; - g
_{mn}is the discretization of the Green function, ${a}_{n}=\sqrt{\Delta x\Delta y/\pi ,}{J}_{1}$ is the Bessel function of the first type, and r_{n}is the vector position of the n-th pixel; - ${\chi}_{n}$ is the contrast value at r
_{n}; - ${E}_{n}^{t}$ is the total electric field at r
_{n}.

#### 3.2. Proposed Machine Learning Method

## 4. Numerical Results

#### 4.1. Circular Model

^{2}, is assumed. In terms of discretization, 150 × 150 pixels were considered for the initial image, and 64 × 64 pixels were assumed for the reconstructed image. With reference to the configuration illustrated in Figure 4, 18 line sources and 18 line receivers were arranged in a circle with a radius equal to 10 cm, centered at (0.0) cm.

- − large circle diameter = 8 cm, medium with ε
_{r}= 40; - − medium circle diameter = 6 cm, with ε
_{r}= 4.5; - − small circle (tumor) diameter = 1 cm, with ε
_{r}= 57.

_{r}= 10, has been assigned.

#### 4.2. Breast Phantoms

^{2}, namely, 3λ/5. In terms of discretization, 150 × 150 pixels were considered for the initial image, and 64 × 64 pixels were considered for the reconstructed image. In total, 18 line sources and 18 line receivers were arranged in a circle with a radius equal to 10 cm, centered at (0.0) cm. Scattered field sampling was carried out using 15 different frequencies, assuming a background relative permittivity ε

_{r}= 10 [37,38,39]. A relative permittivity range of from 2.5 to 67 is assumed for the breast, with the initial solution of the contrast map using the minimum value. An image of 150 × 150 pixels was considered, from which the scattered fields were obtained. In the reconstruction, an image of 64 × 64 pixels was obtained, this being helpful for significantly reducing the execution time of the quadratic algorithm to approximately 20 min for 5 iterations. Simulations were performed for the four classes of phantoms mentioned above, by also adding three false tumors (of 3 mm, 5 mm, and 8 mm in diameter) at different locations. Specifically, the tumor was placed within the breast model, with coordinates (X cm, Y cm) in nine different positions, namely: (0 cm, 0 cm), (0 cm, 1.9 cm), (0 cm, −1.9 cm), (1.9 cm, 0 cm), (−1.9 cm, 0 cm), (1.9 cm, 1,9 cm), (−1.9 cm, 1,9 cm), (1.9 cm, −1,9 cm), and (−1.9 cm, −1,9 cm), thus obtaining a total of 108 images. These are used as input for the CNN to be trained.

_{r}retrieved by the adopted machine learning approach, so as to finally have a full characterization of the breast phantoms. The conductivity model is expressed by the following equation [24]:

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Numerical results from the quadratic programming-based microwave imaging for the circular breast model, showing the original (

**a**) and retrieved (

**b**) relative permittivity.

**Figure 6.**Permittivity (upper) and conductivity (lower) results, obtained with the CNN method applied to quadratic programming-based microwave imaging: (

**a**) Class 1, Phantom 1, Breast ID: 071904; (

**b**) Class 2, Phantom 1, Breast ID: 012204; (

**c**) Class 3, Phantom 2, Breast ID: 070604PA2; (

**d**) Class 4, Phantom 1, Breast ID: 012304.

Error Quadratic BIM | Error Quadratic BIM + CNN | Accuracy Quadratic BIM + CNN | |
---|---|---|---|

Circular model | 44% | - | |

Class 1, Phantom 1 | 47.1% | 7.6% | 92.4% |

Class 2, Phantom 1 | 74.9% | 9.8% | 90.2% |

Class 3, Phantom 2 | 78.8% | 7,9% | 92.1% |

Class 4, Phantom 1 | 88.5% | 7.05% | 92.95% |

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**MDPI and ACS Style**

Costanzo, S.; Flores, A.; Buonanno, G.
Machine Learning Approach to Quadratic Programming-Based Microwave Imaging for Breast Cancer Detection. *Sensors* **2022**, *22*, 4122.
https://doi.org/10.3390/s22114122

**AMA Style**

Costanzo S, Flores A, Buonanno G.
Machine Learning Approach to Quadratic Programming-Based Microwave Imaging for Breast Cancer Detection. *Sensors*. 2022; 22(11):4122.
https://doi.org/10.3390/s22114122

**Chicago/Turabian Style**

Costanzo, Sandra, Alexandra Flores, and Giovanni Buonanno.
2022. "Machine Learning Approach to Quadratic Programming-Based Microwave Imaging for Breast Cancer Detection" *Sensors* 22, no. 11: 4122.
https://doi.org/10.3390/s22114122