Breast tumors detection using multistatic microwave imaging with antipodal Vivaldi antennas utilizing DMAS and it-DMAS techniques

Abstract

This work presents a study where a sinusoidal corrugated antipodal Vivaldi antenna (SC-AVA) operating in the ultra-wideband (UWB) region is employed as a transducer for microwave imaging (MWI) of a cancerous breast. The functionality of the antenna within the UWB range is confirmed based on performance parameters like return loss, gain, radiation pattern, fidelity factor, and group delay. E-field distribution, H-field distribution, and near field directivity simulations in the presence of the breast phantom have also been carried out and reported. The practical application of the developed antenna for biomedical imaging is evaluated by measuring the specific absorption rate (SAR) readings at multiple frequencies within its operating range. The SAR readings are obtained from an electromagnetic simulator by modelling a realistic heterogeneous breast phantom with multiple embedded tumors, and placing them in close proximity to the transducer. The modelled SC-AVA is further utilized for imaging multiple tumors hidden inside the gland layer of the heterogeneous breast phantom developed in-house. The fabricated breast phantom is scanned using the in-house developed multistatic MWI setup. Based on the data obtained from the scanning setup the images are reconstructed using both the delay multiply and sum (DMAS) and iterative DMAS imaging algorithms. Furthermore, a comparison of the reconstructed images is done to check in which case the obtained images are closer to the fabricated breast phantom.

Introduction

According to available data, it has been determined that more than 1.8 million new cases of breast tumors are diagnosed annually. Attaining a 97% survival rate in breast cancer treatment relies on early detection [1]. Therefore, there is a critical need for an on-site, rapid, and portable diagnostic device to ensure the prompt detection of breast cancer [2]. Several recognized methods exist for the detection of breast cancer. The primary method involves clinical breast examination, followed by techniques such as X-ray mammography, ultrasonography, magnetic resonance imaging (MRI), and positron emission tomography (PET). X-ray mammography is the most commonly used method for detecting cancerous growths in various parts of the human body. However, this method utilizes low-dose X-rays to image internal body parts, which can be harmful, and its repeated exposure may even induce cancer in individuals undergoing mammography. Additionally, the breast compression involved in the procedure can cause discomfort to the patient. On the other hand, MRI utilizes radio waves to generate images of the target without causing harm to the human body. MRI scans also provide excellent visualization of hidden tumors within the body. Nevertheless, MRI scans are expensive, and obtaining test results takes longer compared to other procedures. As a result, MRI devices cannot be repeatedly used for imaging the human breast. PET scans involve the injection of a radioactive tracer directly into the body to identify regions with tumors by tracking the absorption of the tracer by the body tissues. However, apart from the time required to complete the tumor identification process, the primary concern is the injection of an external substance into the human body. These limitations in existing technologies have served as motivation to develop a novel imaging technique that overcomes the constraints imposed by the current methods.

Microwave imaging (MWI), which employs electromagnetic (EM) radiation, is a nondestructive technique used to detect the presence, size, and location of various hidden objects. MWI has extensive applications, including the diagnosis of breast tumors [3], detection of brain stroke [4], identification of corrosion in metal beam used for construction [5], detection of oil well leakages [6], etc. MWI uses nonionizing radiation to scan the target without causing any harm. This technology is also cost-effective as compared to the globally used imaging methods. Moreover, once the entire system is operational, qualified personnel are not required to operate an MWI system. These advantages highlight the benefits of MWI over the widely popular biomedical imaging techniques used at present.

The design of antenna plays a vital role in MWI as it significantly impacts the sharpness of the reconstructed image. Various antenna designs, including pyramidal horn antennas [7], Vivaldi antennas [8–10], coplanar waveguide-fed antennas [11], and electromagnetic bandgap antennas [12], have been recommended for breast tumor detection. For an efficient MWI system, the transducer must possess broadband capabilities, be compact, directional, and have high radiation efficiency. Therefore, in this work, an antipodal Vivaldi antenna (AVA) that meets all these requirements is proposed to image various breast phantoms to detect the tumors.

In imaging applications, ultra-wideband (UWB) antennas are essential as the low-frequency section controls the depth of penetration, while the high-frequency section ensures high resolution. Different types of Vivaldi antennas and their antipodal counterparts are discussed in papers [13–16]. This manuscript presents a sinusoidal corrugated AVA (SC-AVA) operating in the UWB range. The manuscript comprehensively analyzes the performance of the antenna in both the time and frequency domains. Furthermore, the imaging capability of the antenna is examined with widely recognized image reconstruction algorithms such as delay multiply and sum (DMAS) algorithm and iterative DMAS (it-DMAS) algorithm [17, 18].

The entire manuscript is organized as follows: The “Antenna design and analysis” section provides an in-depth explanation of the modeling and analysis of the UWB SC-AVA, discussing both the time domain and frequency domain analyses. The “Breast phantom design and performance assesment in the simulator” section details the design of the breast phantom and validates it through specific absorption rate (SAR) measurement. Also, the E-field, H-field, and near field directivity (NFD) analysis are done in the same section. The “Breast phantom fabrication and imaging” section elaborates on the fabrication procedure of the breast phantom, dielectric characteristics of the breast phantom, and finally the image reconstruction using DMAS and it-DMAS imaging techniques. Lastly, the conclusion, summarizing the complete work, is discussed in the “Conclusion” section.

Antenna design and analysis

Antennas used for medical applications usually operate in the near field. It is desirable to use an antenna with wide bandwidth and directional radiation pattern throughout the operating band for MWI of breast tumors. All the design equations used in the modelling of the SC-AVA is elaborated in equations (1–8).

In a conventional AVA (CAVA), length (L) should be greater than half of the wavelength (λ),

(1)\begin{equation}L \gt \frac{{{\lambda }}}{2}\end{equation}

where λ is the maximum operating wavelength.

Width (W) of the Vivaldi antenna should be greater than one-fourth of the wavelength (λ),

(2)\begin{equation}W \gt \frac{{{\lambda }}}{4}.\end{equation}

Equation of outer tapering edge is given by

(3)\begin{equation}{x_1} = {A_1}\left( {{e^{y{r_1}^{n1}}} - 1} \right) - \frac{1}{2}TW.\end{equation}

Equation of inner tapering edge is given by

(4)\begin{equation}{x_2} = {A_2}\left( {{e^{y{r_2}^{n2}}} - 1} \right) - \frac{1}{2}TW.\end{equation}

The taper rates are given by

(5)\begin{equation}{r_1} = {\left\{ {\frac{{{\text{log}}\left( {\frac{{{w_1}/2 + \,{A_{1\,}} + TW/2}}{{{A_1}}}} \right)}}{H}} \right\}^{\frac{1}{{n1}}}}\end{equation}

and

(6)\begin{equation}{r_2} = {\left\{ {\frac{{{\text{log}}\left( {\frac{{w/2\, + \,{A_2} + \,TW/2}}{{{A_2}}}} \right)}}{{{H_1}}}} \right\}^{\frac{1}{{n2}}}}.\end{equation}

Figure 1(a) displays the modelled CAVA in the simulator and Fig. 1(b) displays the modelled SC-AVA in the simulator. The device (SC-AVA) has overall dimensions (L × W) of 161.56 × 136.68 mm² and is made on an FR-4 substrate with a thickness of 1.6 mm and loss tangent 0.025. A CAVA is designed first. This is followed by introducing the sinusoidal corrugations in the antenna to make it the SC-AVA. The exponential taper profile amplitudes (A ₁ = 1.6 and A ₂ = 1.2), the opening of the flares (W ₁ = 82.1243 mm), the taper length (TL or H = 148.54 mm), and the inner profile height (H ₁ = 41.3 mm) are also listed here. The exponent terms n ₁ and n ₂ are taken as 2 and 5 respectively. The feed line width (TW = 2.9 mm) is chosen such that the input impedance of the antenna is 50 Ω. The mouth opening or the taper width (W ₁ = 82.1243 mm) was chosen to be greater than $\frac{{{\lambda _{max}}}}{2}$, with ${\lambda _{max}}$ being the free space wavelength corresponding to the lowest frequency of operation. To improve the return loss and improve the impedance matching, the feed design on the ground plane was modified to vary exponentially. The design equation for the tapered ground elaborated based on equation (7) is

(7)\begin{equation}{x_3} = S\exp ( - a(t - \frac{{TW}}{2}))\end{equation}

Figure 1. The modelled AVA in the simulator. (a) CAVA and (b) SC-AVA.

where ${x_3}$ is the taper profile of the feed, S (13 mm) is the feed length and a (0.1) is the decay constant of the feed. In an endeavor to enhance the low-frequency performance of AVA and achieve a nearly uniform gain across the UWB frequency range, sinusoidal corrugations are introduced on the outer edges of the flares of the AVA design. The design of the sinusoidal corrugations on the flares is done based on equation (8):

(8)\begin{equation}{x_4} = A\left( {Sin\,\left( {f \times pi \times t} \right) + 1} \right)\end{equation}

where ${x_4}$ is the sinusoidal curve, A = 4, f = 0.122046, and pi = 3.14. The antenna design as well as all the full-wave simulations in this study were performed in the EM solver CST Microwave Studio 2016, which is developed and distributed by Computer Simulation Technology, which is a subsidiary of Dassault Systems, with its headquarters in Germany. Please note that all the dimensions of CAVA and SC-AVA are same. The fabricated SC-AVA is displayed in Fig. 2, with the front section depicted in Fig. 2(a) and the back section depicted by Fig. 2(b). Figure 3 illustrates the equivalent circuit model of the antenna. In this model, the microstrip to double slot line transition within the SC-AVA is represented as a series combination of multiple parallel RLC circuits. To ensure a closer match between the circuit simulation response and the response obtained from the simulator, appropriate optimizations of R, L, and C values are performed. For simplicity, only the equivalent circuit model corresponding to four resonances is presented in Fig. 3. The R, L, and C values are separately noted down from the Advanced Design System software developed by Keysight USA, and are tabulated and displayed in Table 1. This work also extensively examines the antenna through various frequency and time domain analyses.

Figure 2. The fabricated SC-AVA. (a) Front side and (b) back side.

Figure 3. The equivalent circuit model of the SC-AVA.

Table 1. R, L, and C values of SC-AVA circuit model

Frequency domain analysis

The simulated reflection coefficient of the designed SC-AVA is compared with the return loss of the CAVA and is displayed in Fig. 4(a). The −10 dB frequency band of the CAVA is from 2 to 10 GHz. With the addition of the sinusoidal corrugations and ground plane tapering, the lower cutoff frequency changes to 1.1 GHz. The final operating range of the SC-AVA is from 1.1 to 10 GHz which is an improvement over its conventional counterpart. Also, the gain performance with and without the sinusoidal corrugations are displayed in Fig. 4(b). It is observed that the designed SC-AVA obtains a higher maximum gain of 10.3 dBi as opposed to 9.3 dBi with the CAVA. The antenna measurement setup used to measure the antenna performance parameters are displayed clearly in Fig. 5(a) and the antenna mounting stand is clearly displayed in Fig. 5(b). The developed SC-AVA is measured in the anechoic chamber environment and the return loss performance and gain performance is compared with its simulated counterpart. It is displayed in Fig. 6. Figure 6(a) depicts the comparison between the simulated and measured return loss performance. It is found that the measured and simulated responses are close to each other. Also, the circuit simulation response is displayed in the same figure. It is also observed that the circuit simulation plot is also in agreement with the other two plots. The gain of the designed SC-AVA has been measured and compared with the simulated response. It is observed that the simulated and measured responses are in agreement with minor mismatches. It is displayed in Fig. 6(b). Slight mismatch between the measured and simulated curves are due to fabrication inaccuracies as well as due to soldering the SubMiniature version A (SMA)connector with the antenna. The simulated near field distribution of the antenna is shown in Fig. 7. Figure 7(a) depicts the near field distribution at 3.5 GHz and Fig. 7(b) depict the near field distribution at 4.5 GHz. The plane with respect to the antenna in which the field distribution is observed is depicted in Fig. 7(c). As it can be clearly observed from the figure, the fields are observed in the XZ plane with respect to the antenna. Hence, the proposed antenna design, with its significant capabilities, proves to be a valuable asset for microwave-based breast imaging applications.

Figure 4. The simulated responses of the antenna. (a) Return loss and (b) gain response.

Figure 5. The antenna measurement setup. (a) Complete setup and (b) antenna mounting stand.

Figure 6. The comparison of the simulated antenna response with its measured counterpart. (a) Return loss and (b) gain response.

Figure 7. The simulated near field distribution of the antenna at (a) 3.5 GHz and (b) 4.5 GHz. The plane with respect to the antenna in which the field distribution is observed is depicted in (c).

Time domain analysis

In order to assess the suitability of the proposed antenna for employment in MWI, a comprehensive analysis of its time domain characteristics is essential. This analysis encompasses the examination of various factors, such as the transmit-received signal, fidelity factor (FF), and group delay. To evaluate the signal shaping properties and potential signal distortion of the antenna, three distinct configurations were investigated. These configurations involved two front-to-front arrangement and one side-by-side arrangement, with a 200 mm gap maintained between the transmitter and receiver antennas. The normalized magnitude with respect to time is displayed in Fig. 8(a–c). The same figure also presents the various antenna arrangements for assessing the reliability of the antenna for imaging applications. Various combinations like front to front with antipodal sections opposite to each other (Fig. 8(a)), front to front with antipodal sections on the same side (Fig. 8(b)), and side-by-side (Fig. 8(c)) configurations are considered. In the case of face-to-face configurations (Fig. 8(a) and (b)), it is evident that the received signals closely resemble the emitted signals as compared to the side-by-side configuration, indicating strong guided radiation. Therefore, for breast imaging applications using microwave technology, the face-to-face setup is recommended. The calculation of the cross-correlation between the transmitted and received signals is performed using equation (9), commonly referred to as the FF [19]. This factor serves as a measure of the similarity and accuracy between the transmitted and received signals:

(9)\begin{equation}{\text{FF}} = \;ma{x_{\tau \,}}\left| {\frac{{\smallint_{ - \infty }^{ + \infty } S\left( t \right)r\left( {t\, - \,\tau } \right)dt}}{{\smallint_{ - \infty }^{ + \infty } S{{\left( t \right)}^2}.\smallint_{ - \infty }^{ + \infty } r{{\left( t \right)}^2}dt\,}}} \right|\end{equation}

Figure 8. Normalized magnitude of transmitted signal and received signal for (a) front to front with antipodal sections opposite to each other, (b) front to front with antipodal sections on the same side, and (c) side-by-side configuration.

where s(t) and r(t) represent transmitted and received signals, respectively, and τ represents the group delay. The FF values for different configurations of the antenna are as follows: for the front-to-front configuration with antipodal sections on the opposite side, the FF value is 98.4%; for the front-to-front configuration with antipodal sections on the same side, the FF value is 90.6%; and for the side-by-side configuration, the FF value is 70.7%. The high FF value observed in the front-to-front configuration indicates that the transmitted signal experiences less distortion in this arrangement.

Another crucial aspect in evaluating the performance of the antenna in time domain is the group delay, which characterizes the phase distortion of the signal. The group delay is defined as the negative derivative of the transfer function phase, φ(ω), with respect to frequency [19]. It provides an estimate of the time required for a signal to pass through the antenna. The group delay can be computed using equation (10) as

(10)\begin{equation}{{\tau }}\left( {{\omega }} \right){\text{ }} = {\text{ }} - \frac{{{{d\varphi }}\left( {{\omega }} \right)}}{{{{d\omega }}}}\, = {\text{ }} - \frac{{{{d\varphi }}\left( {{\omega }} \right)}}{{2{{\pi df}}}}.\end{equation}

A flat group delay response is critical for UWB applications. Figure 9 illustrates the estimated group delay for the proposed work using the three different combinations. The variations in group-delay remains within an acceptable range of up to 3.8 ns for both the front-to-front configurations and is distorted in the case of side-by-side configuration. Considering both the FF and group delay, the front-to-front setup is recommended for breast imaging applications using microwave technology.

Figure 9. The simulated group delay for (a) front to front with antipodal sections opposite to each other, (b) front to front with antipodal sections on the same side, and (c) side-by-side configuration.

Breast phantom design and performance assessment in the simulator

Breast phantom design

In order to evaluate the SAR value, a heterogeneous breast phantom is modelled within the CST Microwave Studio simulator. The designed breast phantom consists of three distinct layers: the skin layer with a radius of R ₁ = 60 mm, the fat layer with a radius of R ₂ = 58 mm, and the gland layer with a radius of R ₃ = 50 mm. Within the gland layer, four tumors are inserted for testing purposes. Two of these tumors have a radius of R ₄ = 5 mm each, while the other two tumors have a radius of R ₅ = 4 mm each. The tumors are carefully positioned within the gland layer of the modelled heterogeneous breast phantom. The modelled heterogeneous breast phantom with four tumors is displayed in Fig. 10(a). Various orientations of the designed phantom are also displayed in the same figure (Fig. 10(b–c)).

Figure 10. The modelled heterogeneous breast phantom in the simulator with four embedded tumors. (a) Top view, (b) front view, and (c) perspective view.

Multistatic SAR analysis in the simulator

The simulator is also utilized to conduct the SAR analysis of the proposed antenna. SAR quantifies the level of radio frequency (RF) power absorbed by the human body. This analysis provides valuable insights into the potential impact of the antenna on human tissue, contributing to the overall understanding of its performance in medical applications. The SAR is examined for testing the utility of the microwave transducer for medical imaging. It is expressed according to equation (11) as

(11)\begin{equation}{\text{SAR = }}\;\sigma \frac{{{{\left| E \right|}^2}}}{{2\rho }}\end{equation}

where σ is the conductivity (S/m) of the tissue, $\left| E \right|$ is the maximum value of the electric field induced in the human body (V/m), and ρ is the mass density (kg/m³). In this work, a multistatic antenna configuration is utilized to check the SAR readings as a multistatic arrangement is considered for imaging. The SAR analysis setup top view is displayed in Fig. 11(a) and the perspective view is displayed in Fig. 11(b). The SAR values are determined at frequencies of 4 GHz, 6 GHz, 8 GHz, and 10 GHz by averaging over 1 g of tissue. The power radiated by the antenna is 0.5 W. The SAR evaluation is performed in the simulator to ensure compliance with the safety standards set by the Federal Communication Commission (FCC). All the obtained SAR results are tabulated in Table 2 for clarity. Notably, it is evident that the SAR values remain below 1.6 W/kg, thereby confirming the compliance of the antenna with the guidelines outlined by FCC.

Figure 11. The SAR analysis setup in the simulator. (a) Top view and (b) perspective view.

Table 2. Multistatic heterogeneous breast phantom SAR values

E-field and H-field analysis with a single antenna

In order to evaluate the performance of the antenna for MWI, a comprehensive analysis was conducted to assess the penetration of both the electric field (E-field) and magnetic field (H-field) from various perspectives. Figure 12(a–d) depict the infiltration of the E-field in the YZ-plane within the heterogeneous breast model at frequencies of 4 GHz, 6 GHz, 8 GHz, and 10 GHz respectively. The illustrations clearly indicate that the E-field effectively penetrates the breast tissue, reaching a significant depth. Similarly, the penetration of the H-field in the YZ-plane within the same breast model is showcased in Fig. 13(a–d). These figures demonstrate the propagation of the H-field and its penetration within the breast tissue at the aforementioned frequencies. Due to the lossy nature of the breast tissues, the intensity of the H-field decreases as it spreads. However, the antenna exhibits nearly directional propagation characteristics, leading to enhanced infiltration of the EM fields within the breast tissues. By analyzing the E-field and H-field penetration from different perspectives, it becomes evident that the antenna can effectively penetrate the breast tissue, allowing for the capture of valuable information for imaging the breast tumors.

Figure 12. The simulated E-field distribution of the antenna with heterogeneous breast phantom at (a) 4 GHz, (b) 6 GHz, (c) 8 GHz, and (d) 10 GHz. The scaling is displayed in inset.

Figure 13. The simulated H-field distribution of the antenna with heterogeneous breast phantom at (a) 4 GHz, (b) 6 GHz, (c) 8 GHz, and (d) 10 GHz. The scaling is displayed in inset.

NFD analysis

The NFD is defined as the ratio of the power emitted inside the phantom (P_F) to the emitted power over the surface of the phantom (P_T). It is depicted by equation [19]. This term quantifies the directivity of the antenna in the near field region. The calculation of NFD is determined using equation (12) as follows:

(12)\begin{equation}NFD = \,\frac{{{P_F}}}{{{P_T}}}.\end{equation}

Figure 14 illustrates the percentage of NFD in the work. From the figure, it can be observed that the NFD is around 80%.

Figure 14. The NFD plot in the proposed work.

Breast phantom fabrication and imaging

Heterogeneous breast phantom fabrication

The preparation of the heterogeneous breast phantom involves several steps. First, propylene glycol and distilled water are mixed together in a beaker. This mixture is then placed in a heater and heated until the temperature reaches 50°C. Agar–agar and gelatin powder is then added and mixed with the hot solution until it dissolves completely. After this step, surfactant, formalin, and safflower oil are added and mixed with the heated solution. The obtained mixture is removed from the heater and then placed in an ice bath for cooling. The mixture is later poured into a hemispherical container for molding. This is the how the skin layer was made. After the skin layer is made, the fat layer prepared using the same process discussed above, but with different concentration of materials. The fabricated fat layer is placed adjacent to the already made skin layer. After this, the gland layer is made. In order to incorporate the gland layer into the breast phantom, a section of the solidified fat layer is carefully removed. The gland layer is then poured into the space created by removing the fat layer. Subsequently, holes of varying sizes are created within the gland layer using a glass rod. These holes serve as the designated locations for inserting the tumor mixture. Each hole is filled with the tumor mixture. To distinguish and visually differentiate the different layers within the phantom, food colors of distinct hues are utilized. Once the fabrication process is complete, the breast phantom is placed inside a refrigerator for preservation and solidification. This step ensures that the phantom maintains its structure and stability. Figure 15(a) displays the heterogeneous phantom with four tumors. In a realistic case, the tumors are buried inside the gland layer. Thus, a gland is layer placed above the embedded tumors to make it concealed as displayed in Fig. 15(b). The embedded tumors are at a depth of 3–5 mm from the top surface of the gland layer. The chemicals used for fabrication of the heterogeneous breast phantom are displayed in Table 3.

Figure 15. (a) The fabricated heterogeneous breast phantoms with four tumors and (b) the heterogeneous breast phantom with concealed tumors.

Table 3. Composition of different materials for breast phantom fabrication

Dielectric characteristics of the breast phantom

The dielectric properties of different layers of the designed breast phantom are measured using Vector Network Analyzer (VNA)with the waveguide measurement technique (depicted in Fig. 16(a)). The measured relative permittivity values are displayed in Fig. 16(b). In the case of the fabricated heterogeneous breast phantom, the dielectric constants of the various layers are as follows: The relative permittivity of the skin layer is approximately 28 throughout the band. The relative permittivity of the fat layer varies from 3 to 4 throughout the band. The relative permittivity of the gland layer varies from 20 to 10. Finally, the relative permittivity of the tumor layer varies from 47 to 37, in the measured band.

Figure 16. The (a) measurement setup and (b) measured relative permittivity values of the modelled breast phantom.

Breast phantom imaging process

Figure 17 showcases the complete imaging setup employed in this work. Please note, the multistatic approach is used for image reconstruction. In the starting stage, the breast phantom is placed on the turntable of the imaging setup. The turn table is surrounded by nine similar antennas used for scanning the breast phantom. Then, the placed phantom is rotated in 40° step size to obtain a total of nine sets of readings per rotation from the VNA, which are serially stored in the computer connected to it. Here only one antenna acts as the transmitter and the remaining eight antennas act as the receivers. The frequency domain data (S ₁₁ and S ₂₁ data) corresponding each antenna rotation is extracted from the VNA. The frequency domain data is then converted to time domain data before further processing. A total of two different image reconstruction techniques are used to obtain the target image and a comparison is made between them.

Figure 17. The multistatic imaging setup used in the proposed work.

The DMAS is the first technique used to image the heterogeneous breast phantom. The DMAS [20] image reconstruction process is a derivative of the conventional delay and sum (DAS) [21] process. The DMAS process involves pairing multiplication of the delay compensated signals to obtain a higher contrast ratio with the expense of higher computational complexity as compared to the widely popular DAS.

The DMAS algorithm conducts a correlation process for each pixel position in the imaging grid to acquire highly similar data, resulting in improved image accuracy, attributed to a significant increase in the sample size. In a DMAS imaging process, the delay is initially calculated using equation (13):

(13)\begin{equation}\tau \left( {{P_i}} \right) = \,\frac{{\left| {{T_x} - \,{P_i}} \right| + \,\left| {{R_{x1}} - \,{P_i}} \right|}}{{\frac{C}{{\sqrt {{\varepsilon _r}} }}}}.\end{equation}

Here, P_i is the pixel under consideration, T_x and R_x1 represents the transmitting and first receiver antenna, $\left| {{T_x} - \,{P_i}} \right|$ represents the distance from the pixel under observation to the transmitting antenna, $\left| {{R_{x1}} - \,{P_i}} \right|$ represents the distance from the pixel under consideration to the receiver antenna, $\tau \left( {{P_i}} \right)$ is the calculated time delay corresponding to the pixel ${P_i}$, C is the speed of light in vacuum and ${\varepsilon _r}$ is the dielectric constant of the layer through which the radiation is passing. Consider an imaging grid of 100 × 100 pixels cutting the breast phantom and let Pi be a random pixel on this grid. Based on the distance from the transmitting and receiving antenna to the pixel under consideration ${P_i}$, the delay is calculated. The amplitude corresponding to the particular time delay is noted from two time domain signals. The first time domain signal is the S ₁₁ signal converted to time domain and the second time domain signal is the S ₂₁ signal converted to time domain. These amplitudes are then multiplied together. In similar manner, for the same pixel, the amplitudes are found out using the S ₂₁ and S ₃₁ readings. The obtained amplitudes are then multiplied together. In a similar manner the amplitudes from S ₄₁–S ₅₁, S ₅₁–S ₆₁, S ₆₁–S ₇₁, S ₇₁–S ₈₁, S ₈₁–S ₉₁, and finally S ₉₁–S ₁₁ were paired up and multiplied together. All these separately multiplied amplitudes, corresponding to the pixel under consideration P_i is then added together to complete the DMAS process corresponding to that pixel. This process is then repeated for all the pixels in the imaging grid to complete the image reconstruction procedure. Please note that the delay value changes depending upon the considered pixel as well as the location of the transmitter and receiver antennas. The DMAS image reconstruction algorithm is mathematically represented based on equation (14) as

(14)\begin{equation}{Y_{DMAS}}\, = \,\mathop \sum \limits_{i = 1}^M \mathop \sum \limits_{j = i + 1}^{M - 1} {S_i}\left[ k \right]{S_j}\left[ k \right]\end{equation}

where Y_DMAS is the DMAS signal, M is the total number of received signals, ${S_i}\left[ k \right]$ and ${S_j}\left[ k \right]$ are delayed signal intensities obtained from the time domain signals based on the position of the transmitter and receiver antennas.

The second image reconstruction method used in this work is an iterative version of the DMAS technique. The it-DMAS process adopts the iterative form of the maximum likelihood expectation maximization (MLEM) algorithm utilized in PET [22]. The approach on how the MLEM algorithm is included with the DMAS process to obtain the it-DMAS process is discussed in steps (1)–(4). They are as follows:

1. (1) Initialization: Start with an initial image estimate (${I^0}$), often initialized as a uniform or low-resolution image. This is the starting stage of the imaging process.

2. (2) Iteration (for each nth iteration):

   1. (a) Forward projection of the unity matrix: The forward projection of a unity matrix U involves simulating the expected measurements that would be obtained if the imaging system were to scan an object with uniform properties. The notation $\hat F$[U] represents the forward projection operator applied to the unity matrix U.

   2. (b) Combining operation (D.$\,\widehat {\boldsymbol{F}}$[U]): The combining operation involves multiplying the experimentally measured data D by the simulated measurements from the forward projection of the unity matrix $\hat F$[U]. This operation is an element wise multiplication, where each element in D is multiplied by the corresponding element in $\hat F$[U]. Thus, D.$\,\hat F$[U] represents a combination of the real-world measured data D and the simulated measurements $\hat F$[U] that would be expected if the imaging system interacted with an object having uniform properties.

   3. (c) Forward projection of the current image estimate: Simulate the expected radar measurements based on the current image estimate to obtain $\hat F$[${I^n}$]. The forward projection operation transforms the initial image estimate image into a simulated dataset representing the expected measurements that an imaging system would obtain if it were to interact with an object resembling that image.

   4. (d) Division of the combined data by the forward projected data: Mathematically, the division involves dividing each element of the combined data matrix D. $\,\hat F$[U] by the corresponding element of the forward projection of the previous image estimate matrix $\hat F$[${I^n}$]. The output $\frac{{D.\hat F\left[ U \right]}}{{\hat F\left[ {{I^n}} \right]}}$ is a matrix representing the normalized combined data. This matrix provides a refined set of measurements that have been adjusted based on the relationship between the experimentally measured data, simulated measurements from the unity matrix, and the initial image estimate.

   5. (e) Back projection operation: The back projection operator is a mathematical operation that transforms data from the time or frequency domain back into the spatial domain. It is a fundamental step in the reconstruction process, where spatial information is reintroduced. Mathematically, the back projection operation $\hat B\left[ {\frac{{D.\hat F\left[ U \right]}}{{\hat F\left[ {{I^n}} \right]}}} \right]$ is applied to each element of the normalized data matrix.

   6. (f) Normalization and multiplication: The normalization by the back projection of the unity matrix $\hat B$[U] is an important step in the iterative reconstruction process. The normalization step is applied to ensure that the back projected data is scaled appropriately relative to the back projection of the unity matrix. It corrects for any scaling effects introduced during the back projection process. This data is then multiplied with the initial image estimate ${I^n}$ to get the new images estimate ${I^{n + 1}}$. The final iterative expression that encompasses all the steps is as given by equation (15):

(15)\begin{equation}{I^{n + 1}}\, = \frac{{{I^n}}}{{\hat B\left[ U \right]}}\hat B\left[ {\frac{{D.\hat F\left[ U \right]}}{{\hat F\left[ {{I^n}} \right]}}} \right].\end{equation}

1. (g) Repeat: Repeat steps (a)–(f) for a predefined number of iterations or until convergence criteria are met. Here we are running the simulation up to six iterations.

1. (3) Convergence check: Monitor convergence by assessing the change in the image estimate between consecutive iterations or by comparing the simulated and actual measurements.

2. (4) Final reconstructed image: The iterative process results in a final reconstructed image that best matches the experimentally measured data, representing the microwave properties of the imaged breast tissue.

The rotation subtraction method elaborated in paper [23] is utilized as the clutter removal technique in this work. A generalized flow chart of the image reconstruction process is depicted in Fig. 18.

Figure 18. The imaging process flow chart.

The reconstructed images of the fabricated breast phantom with four tumors using DMAS and it-DMAS algorithms are displayed in Fig. 19(a–b). The embedded tumors are appropriately identified up to a reasonable extent. Also it can be observed that the it-DMAS algorithm obtains the image with reduced false positive intensities. Minor deviations from the exact tumor locations are also observed in all the cases. This can be attributed to the measurement errors while taking the readings with the phantoms, irregularity while placing the phantom in the turntable, losses from the RF switch, cable losses, and the reflections from the nearby surfaces.

Figure 19. The reconstructed images of the breast phantom with four tumors using (a) DMAS and (b) it-DMAS image reconstruction methods.

The work done is compared to some of the recent works in literature and is listed in Table 4 [11, 24–39]. Four tumors of varying sizes embedded inside the gland layer of the complex breast phantom are detected clearly in this work. Also, the antenna attains the highest gain as compared to some of the other works in literature. The proposed work also reconstructed tumors that are compact as compared to some of the recent works in literature.

Table 4. Comparison the imaging results with similar breast imaging works

NG = not given; BW = bandwidth; FBW = fractional bandwidth; IA = imaging algorithms; TR = tumor radius; UWB = ultra-wideband; AMC = artificial magnetic conductor; CPW = coplanar waveguide; DRA = dielectric resonator antenna.

Conclusion

An SC-AVA operating in the range of 1.1–10 GHz, having a peak gain of 10.3 dBi is introduced in the proposed work for MWI applications. All the frequency and time domain analysis of the antenna are depicted clearly. A heterogeneous breast phantom with multiple tumors of varying sizes is modelled in this work followed by fabricating the same in the lab. SAR analysis of the SC-AVA with the modelled breast phantom is done first. This is followed by using the antenna to reconstruct the tumors embedded inside the modelled breast phantoms. Four tumors embedded inside the gland layer of the heterogeneous breast phantom are reconstructed clearly in this work using the DMAS and it-DMAS image reconstruction techniques. The proposed antenna along with multiple algorithms will be used to image several complex heterogeneous breast phantoms in the future. Also several clutter removal algorithms will be utilized in conjunction with this work to further reduce the false positive intensities. The developed SC-AVA will be further utilized for nondestructive testing of various concealed metallic and nonmetallic targets.