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In microwave tomography, it is necessary to increase the amount of diverse observation data for accurate image reconstruction of the dielectric properties of the imaging area. The multi-polarization method has been proposed as a suitable technique for the acquisition of a variety of observation data. While the effectiveness of employing multi-polarization to reconstruct images has been confirmed, the physical considerations related to image reconstruction have not been investigated. In this paper, a compact-sized imaging sensor using multi-polarization for breast cancer detection is presented. An analysis of the correlation coefficient of the received data of adjacent antennas was performed to interpret the imaging results. Numerical simulation results demonstrated that multi-polarization can reconstruct images better compared to single polarizations owing to its low correlation coefficient and condition number.

In Japan, the breast cancer incidence and associated mortality rates of women are lower compared to Western countries. However, it is one of the major causes of death among women, and the incidence rate has increased since 1975, regardless of age group [

Recently, studies on the early detection of breast cancer by microwave imaging (MWI) have attracted considerable interest among researchers [

In microwave tomography, the electromagnetic waves are efficiently incident on the object. The influence of surrounding structures is avoided by immersing the antenna and imaged object into a lossy matching fluid [

Furthermore, the multiple-polarization method has been examined as a means to obtain a variety of observations data. The changes in the polarization of the scattered waves are influenced by the composition and shape of the biological tissue. For example, a dual-polarized MWI system that can simultaneously collect both TE and TM polarizations using scattering probes has been built to obtain 2-D tomographic images [

In this paper, we present a compact-sized imaging sensor using the multi-polarization method for breast cancer detection. Based on our previous paper [

This paper is organized as follows. The imaging algorithm approach employed for image reconstruction is described in Section 2. Section 3 presents our proposed compact-sized imaging sensor and breast model used for numerical simulation. Further in Section 3, the imaging results are presented and discussed. Finally, Section 4 discusses the conclusion of our paper.

In tomography, the breast model is expressed as a cube and discretized as voxels, and the dielectric properties are estimated in each voxel. The total number of voxels is K, and the dielectric properties, which consist of the relative permittivity and conductivity distribution, are represented by contrast. The calculated data group,

The governing equation for the scattering field at an observation point r for a given frequency is

Here,

Equation (1) is a nonlinear integral equation; therefore, we employ the DBIM to solve the nonlinear problem. DBIM is equivalent to the Gauss-Newton method that is used in nonlinear least-squares optimization problems to obtain an approximate solution for the contrast. In the iterative DBIM, we start with an initial guess of the contrast in the imaging region. At each iteration, we determine the contrast perturbations based on the approximate solution of the inverse problem, and then update the contrast. The updated contrastis referred to as the new contrast of the background; the total field and the scattering field at the observation points are recalculated based on the new contrast. Then, the calculated scattering field is compared with the measured scattering field to minimize the residue error until convergence is reached.

In this paper, the contrast function,

As we mentioned earlier, Equation (1) is a nonlinear equation. In order to linearize Equation (1) in the Born approximation, the incident field is adopted instead of total field. In this paper, we use the recalculated total field

In the linear model of Equation (3), an approximation of the contrast perturbations,

Here, k is the iteration number, and + denotes the conjugate transpose of a matrix. Because the inversion in Equation (4) is ill-posed, conjugate gradient (CG) regularization is applied at each iteration to estimate the contrast perturbations and to update and generate a new contrast. Then, the imaging region is reconstructed using the new contrast, and this process is repeated until the residual norm is smaller.

In the inverse scattering problem, a large amount of diverse observation data is required to achieve accurate image reconstruction with high resolution. In addition, the measurement error increases if the SNR decreases; thus, it cannot reconstruct the image accurately. Therefore, it is necessary to minimize the analysis region in order to reduce the computational cost. For this reason, the implementation of a compact sensor that involves a small distance between the antenna and the breast is preferred. In this case, large number of antennas must be arranged in a small and limited space. Various observation data can be obtained even in a small space by changing the plane of polarization.

We assumed that the imaging sensor is constructed from resin, which has properties similar to those of adipose (fatty) tissue. A hemispheric volume is providedinside the sensor to accommodate the breast model,

similar to the imaging sensor with fixed suction proposed in [

The total field within the scattering object was calculated according to the method of moment (MOM), as described in [

In Model 1, the analysis region consists of the chest wall, adipose tissue, and a tumor. The distributions of the two unknown parameters, i.e., the relative permittivity and conductivity, are shown in

Parameter | Background | Chest wall | Adipose tissue | Fibro-glandular tissue | Tumor |
---|---|---|---|---|---|

Relative permittivity, | 8.2 | 57 | 7 | 25 - 40 | 52 |

Conductivity, | 0.15 | 2 | 0.4 | 1.0 - 2.2 | 4 |

show the results of the 3-D image reconstruction after 100 iterations, using vertical, horizontal, and multi-pola- rization, respectively, for transmitting and receiving data. From the results, we cannot estimate both the relative permittivity and the conductivity of the tumor using single polarizations, i.e., vertical and horizontal polarization. In contrast,

Figures 4(a)-(b) show the setting and reconstruction value of the relative permittivity and conductivity through the tumor voxel. The x-axis indicates the x-coordinate of the voxel, and the y-axis indicates either the relative permittivity or conductivity. The setting values of the relative permittivity and conductivity of the voxel corresponding to the tumor are 52 and 4 [S/m], respectively. The reconstructed parameters of the tumor for the different polarizations are summarized in

For Model 2, the analysis region consists of the chest wall, adipose tissue, fibro-glandular tissue, and a tumor. Here, 10% of the volume ratio of the breast is occupied by fibro-glandular tissues that are distributed randomly.

Parameter | Setting value | Model 1 | Model 2 | ||||
---|---|---|---|---|---|---|---|

VP^{a} | HP^{b} | MP^{c} | VP | HP | MP | ||

Relative permittivity, | 52 | 16.24 | 19.93 | 54.48 | 24.26 | 73.38 | 48.98 |

Conductivity, | 4 | 0.91 | 1.36 | 3.50 | 2.25 | 0 | 3.86 |

^{a}Vertical polarization. ^{b}Horizontal polarization. ^{c}Multi-polarization.

show the results of the 3-D reconstructed images after 230 iterations, using vertical, horizontal, and multi- polarization, respectively. As shown in

Figures 6(a)-(b) show the setting and reconstruction values of the dielectric properties of the tumor voxel in the x-axis direction.

In Model 2, the tumor is adjacent to fibro-glandular tissues and the contrast difference between them is small. The above findings confirm that we can detect the tumor accurately with the presence of fibro-glandular tissues by employing multi-polarization. Overall, the results from both breast models indicate that vertical and horizontal polarization provide different information of the image reconstruction and dielectric properties. The multi- polarization array configuration consistently performed better than single polarizations.

In this section, we investigated the impact on image reconstruction by the analysis of the correlation coefficient of the received data between adjacent antennas in our imaging sensor. First, we examined the positions of the antennas shown in

Overall, the correlation coefficient of the receiver pair was significantly reduced when using multi-polariza- tion compared to single polarizations. At Position 1, adequate image reconstruction was obtained using multi- polarization owing to the low correlation coefficient, as shown in Figures 3-6. Furthermore, the correlation coefficient of the multi-polarization results at Position 3 is higher than that at Position 1 for both models. Figures 7(a)-(c) show the x-axis projection of the setting and reconstruction values of the dielectric properties of the tumor voxel for Model 1 obtained using multi-polarization. These results demonstrate that positions with low correlation coefficients reconstruct the dielectric properties sufficiently. Therefore, we can conclude that a low correlation coefficient is a viable condition for successful image reconstruction.

The condition number of a matrix characterizes the solution sensitivity of a linear equation system to errors in the data and it indicates the accuracy of the results from of the matrix inversion. We examined the condition number of the inverse matrix for the two breast models using different polarizations; these are summarized in ^{3} at the 230^{th} iteration. This suggests that the

reconstruction of the dielectric property distributions may not be performed correctly owing to the ill-posed problem of the inverse matrix. In contrast, when vertical polarization is used, the condition number decreases to 11.78 at the second iteration. Subsequently, it increases with increasing iteration number. Although an increase occurred in the condition number, it was small compared to that observed for the horizontal polarizations.

Lastly, when multi-polarization is applied to the imaging sensor, the condition number slightly increases to 12.16 at the second iteration. Subsequently, the condition number varies moderately with the iteration number and reaches 13.80 at the 230^{th} iteration. Model 1 showed a similar tendency. From these results, we conclude that the ill-posed problem does not occur when multi-polarization is applied. Thus, the images and dielectric properties of the breast can be reconstructed.

We have confirmed the effectiveness of applying multi-polarization to transmit and receive antennas to determine the dielectric property distributions of a simple breast model. For the imaging algorithm, this is accomplished using the MOM and DBIM in the inverse scattering problem. The numerical simulation results demonstrated that the ill-posed problem can be avoided because of the improvement of the condition number by multi-polarization. Furthermore, the correlation coefficient of multi-polarization is relatively low compared to those

Position | Position of antenna (upper, lower) mm | Model 1 | Model 2 | ||||
---|---|---|---|---|---|---|---|

VP | HP | MP | VP | HP | MP | ||

1 | 24, 8 | 0.9536 | 0.9253 | 0.1555 | 0.9469 | 0.9036 | 0.1598 |

2 | 32, 8 | 0.8422 | 0.7605 | 0.1655 | 0.8068 | 0.6748 | 0.1618 |

3 | 32, 16 | 0.9311 | 0.8475 | 0.2044 | 0.9020 | 0.7435 | 0.2092 |

Iteration number | Model 1 | Model 2 | ||||
---|---|---|---|---|---|---|

VP | HP | MP | VP | HP | MP | |

1 | 12.07 | 12.07 | 12.07 | 12.07 | 12.07 | 12.07 |

2 | 11.63 | 11.67 | 11.87 | 11.78 | 11.76 | 12.16 |

50 | 23.29 | 726.87 | 14.91 | 22.04 | 637.02 | 16.03 |

100 | 32.25 | 1.36 × 10^{3} | 13.85 | 69.37 | 1.66 × 10^{3} | 15.33 |

150 | - | - | - | 100.88 | 2.64 × 10^{3} | 14.35 |

230 | - | - | - | 98.19 | 3.08 × 10^{3} | 13.80 |

corresponding to single polarizations. For this reason, the correlation coefficient may represent a viable parameter for image reconstruction in microwave tomography aimed at breast cancer detection.

L.Mohamed,N.Ozawa,Y.Ono,T.Kamiya,Y.Kuwahara, (2015) Study of Correlation Coefficient for Breast Tumor Detection in Microwave Tomography. Open Journal of Antennas and Propagation,03,27-36. doi: 10.4236/ojapr.2015.34004