Discrete Dipole Approximation-Based Microwave Tomography for Fast Breast Cancer Imaging

Abstract

This paper describes a fast microwave tomography reconstruction algorithm based on the two-dimensional discrete dipole approximation. Synthetic data from a finite-element based solver and experimental data from a microwave imaging system are used to reconstruct images and to validate the algorithm. The microwave measurement system consists of 16 monopole antennas immersed in a tank filled with lossy coupling liquid and a vector network analyzer. The low-profile antennas and lossy nature of system make the discrete dipole approximation an ideal forward solver in the image reconstructions. The results show that the algorithm can readily reconstruct a 2D plane of a cylindrical phantom. The proposed forward solver combined with the nodal adjoint method for computing the Jacobian matrix enables the algorithm to reconstruct an image within 6 seconds. This implementation provides a significant time savings and reduced memory requirements and is a dramatic improvement over previous implementations.

I. Introduction

BREAST CANCER is usually among the top five most frequent cancers in most countries and ranks first as the cause of death for women diagnosed with cancer [1], [2]. Regular breast screening has proven to be the most effective method to improve long term survival [3]. While frequent screening enhances chances of early diagnosis, the inherent limitations of conventional imaging methods sometimes reduces their effectiveness. Numerous screening modalities are used currently, with X-ray mammography being the most common. Generally, screening devices are expected to be user-friendly, harmless, and comfortable for patients. Additionally, they should be both sensitive and specific to various tissues and lesions [4]. While mammography is considered the gold standard, its sensitivity and specificity are weak for heterogeneously dense and extremely dense breasts, exposes patients to ionizing radiation and is often uncomfortable [3], [5]. An alternative for women with dense breasts is to use other screening methods such as ultrasound, magnetic resonance imaging (MRI) or positron emission tomography (PET) [3]. These techniques cannot be used in large scale screening programs due to cost inefficiency and increasing likelihood of false positive results [3]. Microwave imaging is a new and potentially complementary approach to mammography due to the endogenous tumor versus benign tissue dielectric property contrast [6]–[9].

Microwaves have been proposed for interrogating biological tissue for the purpose of imaging for many decades [10]. In cases such as breast cancer, there is known dielectric property contrast between benign and malignant tissue [6]–[9]. For other applications such as stroke diagnosis, there is a high property contrast between blood and brain tissue [11]. For bone health imaging [12], it has been shown that the dielectric properties correlate with bone density [13]. Interaction of the electromagnetic waves with tissue can cause substantial perturbation to the propagating fields, especially when there is high contrast between adjoining tissue types [14]. Through the use of different imaging techniques, the scattered fields can be used to generate microwave images of the interrogated tissue types.

Microwave imaging has been an active area of research among academic groups over the last four decades. The imaging techniques can mainly be classified into four categories: radar, holography, thermoacoustic imaging, and tomography. Radar imaging has been analyzed in simulation studies and has obtained some acceptance in clinical experiments [15]–[18]. Its primary implementations are confocal microwave imaging [19], [20] and near field synthetic focusing [21] and generally involves the summations of the fields at each pixel in the imaging domain where the phases of each have been augmented so that each contributor is synchronized. These can utilize both reflection and transmission data. Holography can generally utilize both transmission and reflection data. It involves using Fourier transforms of the measurements to project back to the target and then an inverse Fourier transform to convert back to the spatial domain. Holography approaches have been primarily studied in simulations and phantom experiments [22], [23]. Thermoacoustic imaging involves exciting a tissue domain with a low duty-cycle microwave pulse and receiving the induced mechanical waves with an ultrasound detection system. Thermoacoustic imaging has advanced to early clinical studies after several phantom studies [24], [25]. Microwave tomography has received more attention in simulation and phantom experiments [26]–[29], and clinical investigations have been performed [30]–[33].

Focusing on microwave tomography, from a mathematical perspective, the image reconstruction algorithms include more traditional approaches such as different modifications of Born and Rytov approximation methods [34], [35] for low contrast cases, Newton and gradient minimization methods for high contrast cases [30], [36] and more innovative studies such as evolutionary algorithms [37], [38] and Newton-based methods formulated in Lebesgue spaces [39], [40]. In the latter, contrary to common minimization problems, the optimization approach is formulated on the general l^p Banach spaces.

The microwave tomography field has been plagued by challenges with the reconstruction algorithms where they often converge to unwanted solutions and require exorbitant computational resources, the latter of which additionally drives unrealistic and expensive hardware requirements. To the first order, computational costs are driven by whether an algorithm is performed in either 2D or 3D, with the latter often being orders of magnitude slower. This is due partially by requiring more measurement data to serve the needs of more parameters being reconstructed in the larger 3D domain, but also due to the 3D forward solutions individually taking more time to compute. These reconstruction times can range from many hours to even days [41], [42]. In contrast, 2D reconstruction algorithms are much more computationally efficient. While numerous groups have concentrated on 3D approaches motivated by the presumption that better matches can be achieved between the measurement and computed data [42], [43], 2D approaches have proven to be effective and have been successfully demonstrated in numerous phantom and clinical studies [26], [30], [43]. While a primary motivation for microwave tomography is for it to either replace or complement conventional X-ray mammography and MRI, for the burgeoning markets of underserved and under-resourced countries, microwave tomography could be a viable alternative. The primary motivation in this paper is to demonstrate an accurate reconstruction technique that is both fast and requires minimal memory resources.

In this paper, we describe a two-dimensional microwave tomography algorithm and report in detail regarding its computational efficiency. The new microwave tomography algorithm is based on a discrete dipole approximation (DDA) forward solver which facilitates fast image reconstructions with high accuracy. As is the case for all tomography approaches, the forward solution computation time is the time-limiting factor. This algorithm benefits from a strong synergism between the hardware implementation and novel algorithm concepts. For instance, because of the log transform, the algorithm is able to converge to a unique solution without encountering local minima which plague many algorithms [44]. In addition, the lossy coupling bath subdues most scattering from the adjacent array antennas making the field patterns closely mimic those for purely dielectric media [45]. This enables the use of the DDA in the first place, since it requires inconveniently high levels of discretization when highly scattering objects are present - all but eliminating the proposed speed efficiency. The DDA forward solver exploits the fact that the defining forward solution matrix can be formulated into a block-Toeplitz structure. These can be individually broken down and padded into circulant matrices. Using iterative, conjugate gradient techniques, the matrix factorization can be reduced to a series of matrix vector multiplications for which advanced FFT-based algorithms can further reduce the N² operation into an N log N operation exploiting properties of the circulant matrix. This can ultimately improve the image reconstruction time by an order of magnitude or more.

The paper is organized in the following format: Section II describes the algorithm and its computational costs; Section III discusses the imaging system used in this study; Section IV shows simulated numerical results and those for phantom experiments; and Sections V and VI present the discussion and conclusions.

II. Methods

This section describes the formulation of the two- dimensional Gauss-Newton reconstruction algorithm using the 2D-DDA forward solver and a logarithmic transformation format. The expected outcome is to retain the excellent convergence behavior of our previous approach as well as dramatically reduce the computational time. This section begins with a description of the 2D-DDA forward solver. The subsequent subsection describes the 2D reconstruction algorithm in detail, which is followed by a brief discussion of the computational complexity of the 2D-DDA and the efficiency of the proposed algorithm on uniform grids when computing the Jacobian matrix by the nodal adjoint method.

A. Frequency-Domain Forward Computation and 2D-DDA

The forward solver is an essential part of an image reconstruction algorithm and impacts both precision and acceleration. The forward solver method, domain decomposition scenarios, and physical volume of the imaging domain all influence computation time during reconstruction. For high efficiency, the 2D-DDA requires a uniform grid discretization of the computational domain into dipoles [46]–[48] where each dipole simultaneously represents a pixel in the image domain. The DDA is capable of dealing with irregular geometries; however, it is most powerful when it is applied on a uniform grid that can exploit computational efficiencies. The main premise of the DDA is to model the electric field propagation utilizing an array of dipoles with their associated dipole moments representing their impact in terms of their different material properties. Fig. 1 shows a schematic diagram of the relationship between dipoles forming the 2D forward computation domain, the antenna array, and the parameter reconstruction domain (image), with the dipole spacing defining the image resolution.

Fig. 1.

Schematic representation of the computational domains, forward and imaging.

The DDA forward solver represents the electric field at any position based on the dipole moment P at each given point. In effect, when one antenna radiates a wave, the associated electric field imposes a dipole moment P_i at each dipole i. The total electric field at any dipole location r_i is the sum of the incident electric fields due to a line source plus joint responses from the other N – 1 dipoles and is calculated by [46].

$$E_{tot}\left(r_i\right)=E_{inc}\left(r_i\right)+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N{A}_{ij}{P}_j,$$ (1)

Here E_inc(r_i) represents the incident electric field distribution due to one of the radiating antennas and measured at location r_i, and is modelled in terms of an electric line source as if it were a point source in the xy-plane:

$$E_{inc}\left(r_i\right)=\frac{I_0\omega {\mu }_0}{4}{H}_0^{\left(2\right)}\left(k_b{R}_i\right)$$ (2)

where I₀, ω, and µ₀ are the current amplitude, operating angular frequency and free-space permeability, respectively. k_b is the wave number for the background medium. $H_0^{\left(2\right)}$ is the zero-order Hankel function of the second kind, and R_i is the dipole distance from the transmitting antenna to points in the grid. In (1), P_j serves as the dipole moments. The A_ij term is the Green’s function for the 2D Helmholtz equation [49] describing the interaction between two dipoles located at r_i and r_j and is written as

$$A_{ij}=\frac{-j}{4}{H}_0^{\left(2\right)}\left(k_b{r}_{ij}\right)$$ (3)

where r_ij is the distance between nodes i and j [46]. In addition to (1), the DDA formulation of the dipole moment vector P is generally approximated in terms of the microscopic electric field at the dipole, from which format the total electric field is obtained as

$${\alpha }_i{E}_{inc}\left(r_i\right)=P_i-{\alpha }_i\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N{A}_{ij}{P}_j,$$ (4)

The α term in (4) can be modelled by the 2D version of Clausius-Mossotti relationship [47] which was previously applied to 3D tomographic imaging [45]

$${\alpha }_{2D}=2\frac{{ϵ}_t-{ϵ}_b}{{ϵ}_t+{ϵ}_b}$$ (5)

where ϵ_b and ϵ_t are the complex permittivities of the background and target, respectively. Based on the study performed by Steffan and Richter [50], we suggest computing the polarizability term via

$${\alpha }_{2D,imaging}=\frac{\pi }{6}(\frac{{ϵ}_{r,t}-{ϵ}_{r,b}}{{ϵ}_{r,t}+{ϵ}_{r,b}}+j\frac{{\sigma }_t-{\sigma }_b}{{\sigma }_t+{\sigma }_b})$$ (6)

where ϵ_r,t, ϵ_r,b, σ_t and σ_b are the relative permittivities and conductivities of the target and background, respectively. We note that for the purpose of imaging where the actual properties are unknown, the properties of target are replaced with those computed iteratively on the parameter mesh. The details of the acceleration of this forward solution technique is described in detail in Hosseinzadegan et al. [51], [52].

B. Reconstruction Algorithm

The inverse problem is formulated as a non-linear optimization problem and is expressed as

$$\underset{k^2}{\min }{f}_{obj}:={\Vert E^m-E^c\left(k^2\right)\Vert }^2$$ (7)

where E^m is the measured electric field and E^c is the calculated electric field, k² represents the wavenumber squared in the form of

$$k^2={\omega }^2{\mu }_0{ϵ}_0{ϵ}_r+j\omega {\mu }_0\sigma$$ (8)

where ω is the angular frequency, µ₀ is the free space magnetic permeability, ϵ₀ is the free space permittivity, ϵ_r is the relative permittivity, σ is the electrical conductivity and j is the imaginary unit. Studies by Meaney et al. at Dartmouth College [44], [53] showed that without loss of generality, the optimization problem can be formulated in the log-transformed format as

$$\underset{k^2}{\min }{\Vert {\mathrm{\Gamma }}^m-{\mathrm{\Gamma }}^c\left(k^2\right)\Vert }^2+{\Vert {\mathrm{\Phi }}^m-{\mathrm{\Phi }}^c\left(k^2\right)\Vert }^2$$ (9)

where Γ^m, Γ^c, Φ^m, and Φ^c are the log magnitudes and phases for the measured and computed electric fields, respectively. The least-square optimization problem (7) with the Gauss-Newton method provides the update equation for the dielectric properties at each iteration in form of

$$\left(J^{T}J\right)\left\{\Delta k^2\right\}=J^T\left\{\Delta E\right\}$$ (10)

where J is the Fréchet derivative between E and k² called the Jacobian matrix, J^T is its transpose, J^T J is the Hessian matrix, ∆k represents the electrical property variation at the current iteration compared to the previous one, and ∆E is the difference between the measured signals at multiple observation points and the associated computed electric fields. The main challenge in this problem is that the Hessian matrix is not well-conditioned, and that its illposedness requires a regularization procedure. By applying Levenberg-Marquardt regularization [54], [55], (10) is transformed to

$$\left(J^{T}J+\lambda I\right)\left\{\Delta k^2\right\}=J^T\left\{\Delta E\right\}$$ (11)

where λ is an iterative decreasing coefficient, and I is the identity matrix. The reconstructed properties are updated at each iteration such that

$$k_{i+1}^2=k_i^2+S\Delta k^2$$ (12)

where S is the step size which is typically in an interval of (0, 1) and is set empirically. Convergence occurs when the change in the L2 norm of the residual field vector falls below a predetermined threshold-typically in less than 20 iterations.

C. Computational Acceleration in Reconstruction

Tomographic reconstruction algorithms are iterative processes that require computation of multiple forward solutions. The computational costs of the reconstruction algorithm are largely dependent on the 2D-DDA forward solver. For the DDA calculations, the most significant factor in terms of computational cost is matrix-vector product A × P in (1) (note that this is with respect to the iterative conjugate gradient type techniques described in Hosseinzadegan et al. [52]). Using square cells allows deployment of the fast Fourier transformation which is highly efficient. To apply the FFT and replace a matrix-vector multiplication with a convolution, the block Toeplitz matrix A′, where A′ is a single block of A, is converted to a circulant matrix after which the circulant matrix and vector are convoluted. In this way, the lattice length is doubled (its size is a power 2). We note that FFT operation is of the order of O(n log n)- where n is the size of a block of A, while the matrix-vector multiplication, where matrix A is N × N on the order of O(N²) [48], [52]. n is typically equal to $\sqrt{N}$. Additionally, the interaction matrices A can also be computed in a preprocessor program as its elements are only functions of the dipole distances and background wavenumber and can be calculated once before the reconstruction. In this way, the computation time and in-use memory are dramatically reduced in comparison to other numerical approaches such as those for the finite-element method and finite-difference time-domain approaches which we have implemented previously [56], [57].

For the conjugate gradient conjugate orthogonal (COCG) with the FFT iterative solver, it is essential to maintain the specific coefficient matrix (i.e. block-Toeplitz configuration) format along with a modest condition number. The latter is critical since the convergence rate of CG-type algorithms is proportional to its condition number (i.e higher condition numbers result in slower solutions). The condition number of the matrix is closely related to its diagonal in the form of 1/α (see (4) - note that the 1/α term on the diagonal arises after both sides of the equation are divided by alpha). The α terms are near zero during the first iteration (α defines the contrast level in the imaging domain which is always zero for the first iteration since the starting distribution estimate for the reconstruction algorithm is that of the homogeneous bath). After the first iteration, the α terms corresponding to the imaging domain are updated according the Gauss-Newton iterative solver. While the α values are not exactly zero during the reconstruction, they are quite small owing to the homogeneous bath value initiation [52]. The associated high matrix condition number (on the order of 1e14) results in a slow COCG-FFT convergence requiring thousands of forward solution iterations. To improve the convergence rate for CG type algorithms, preconditioning has been suggested [58], [59]. Diagonal scaling, i.e. Jacobi preconditioning, is efficient and cost effective for diagonally dominant matrix cases such as this. The magnitude of the 1/α term on the diagonal is larger than the sum of the other elements in each row ensuring this criterion. Using the preconditioning M = 1/α (i.e. scaling the diagonal of the coefficient matrices during imaging) dramatically improves the convergence rate for the COCG-FFT algorithm for an accuracy level of 1e-5 at a very low computational cost. Additionally, to further improve the accuracy and convergence rate, the dipole moments calculated during each previous reconstruction iteration are used as the initial guess for the current iteration of the preconditioned COCG-FFT algorithm. Combination of these two approaches considerably reduces the required maximum number of iterations necessary for the forward solutions.

A less significant, but still considerable contributor to the overall computation costs is the calculation of the Jacobian matrix. It is essential to calculate it efficiently or the gains from implementing the DDA in the forward solution will be diminished. The Jacobian matrix has previously been calculated via the adjoint method which exploits the notion of reciprocity and has been used when the forward solver uses either finite-element or finite-difference time-domain methods [60]–[62]. The nodal adjoint method is an efficient and simplified alternative to the adjoint method [57]. The imaging group at Dartmouth College combined the nodal adjoint method with the finite-element dual mesh scheme and has been able to produce 2D images of the breast within several minutes [57]. The nodal adjoint method in the finite-element type domain requires forming a sparse matrix, performing a matrix-vector multiplication followed by a vector inner product to compute each element in the Jacobian. While this is substantially more efficient than the previous sensitivity method [56], it is still sub-optimal. For the DDA forward solver the domain is uniformly discretized into squares. For this configuration, we have previously noted that whole rows of the Jacobian can be constructed by performing a single vector-vector multiplication. This implies that every element in the Jacobian matrix can be computed as a product of three numbers at a single pixel in the imaging domain - (a) a pre-computed weighting parameter, (b) the field value due to a source at the associated transmitting antenna, and (c) the field value due to a source at the associated receiving antenna. This is essentially the least possible number of computations necessary and dramatically reduces its overall computation time. This notion is very broad and applies to both 2D and 3D situations. More details on the derivation and the computational aspects of this method are reported by the authors in [63] and is beyond of the scope of this study.

III. Microwave Imaging System

Our imaging system [26], [64] is constructed based on similar engineering principles as those of the tomographic system at Dartmouth College. The Dartmouth system has been tested and validated in multiple phantom experiments and clinical studies [30], [45]. Fig. 2 shows the measurement system and its 16 transmitting/receiving monopole antennas and an immersion tank filled with a coupling liquid which is a mixture of glycerin and water. The concentric array of antennas is positioned on a 15.2 cm diameter circle with equal spacing between each. The antennas are connected to the Rohde & Schwarz ZNBT8 16 channel vector network analyzer via flexible coaxial cables. The VNA operates over the frequency range of 9 kHz to 8.5 GHz and has a dynamic range up to 140 dB. In this study we used a mixture of 80% glycerin and 20% water which reduces the dielectric contrast between the breast tissue and background and is also highly lossy which acts to suppress multipath signals and surface waves [65], [66]. Data is collected for each antenna transmitting with the remainder acting as receivers - a total of 240 measurements - 16 transmitters × 15 receivers per transmitter. The data is calibrated by first collecting a set of measurements for the homogeneous bath followed by the set with the target present. The calibration involves subtracting the homogeneous bath data from that of the target set in its log transformed format. This cancels all contributions from cable differences, receiver transfer characteristics and others. The exact nature of the calibration process is described in more detail in Epstein et al. [67]. It is assumed that the users have a means of measuring the liquid dielectric properties using an external probe such as that manufactured by Keysight Technologies (Dielectric Probe Kit - Santa Clara, CA, USA). In addition, the phases of the measurements and computed values need to be unwrapped to ensure that each is on the same Riemann sheet. The unwrapping processes for both the measured and computed values are described in Meaney et al. [44].

Fig. 2.

The imaging system at Chalmers University of Technology.

IV. Results

In this section, we report on studies performed in both simulation and phantom experiments to assess imaging capabilities. In addition, the computational costs in terms of reconstruction times and memory requirements are investigated. First, we examine the accuracy of the 2D-DDA forward solution with respect to a benchmark solver - COMSOL Multiphysics (COMSOL AB - Stockholm, Sweden) and the actual measurements at the receiving antennas. We then reconstruct the target electrical properties (permittivity and conductivity) of the 2D imaging zone via the iterative Gauss-Newton algorithm using synthetic data from a finite-element based solver (COMSOL). This is followed by reconstructions using real measurement data. Finally, the computation time and memory requirements for the newly developed reconstruction algorithm are calculated and compared to previous implementations.

A. Simulated Measured Data

In this section, we investigate the 2D-DDA solutions at the antenna positions as well as the reconstructed images recovered with the 2D tomographic algorithm from simulation data. For the simulations, we devise a case where the imaging tank encompasses the antenna array and is filled with a mixture of 80% glycerin and 20% water. The dielectric properties of the mixture are measured over the frequency range 500 MHz - 2 GHz with a dielectric probe (Agilent 85070 Performance Probe) (Fig. 3).

Fig. 3.

Measured relative permittivity (real and imaginary parts) of 80:20 glycerin-water mixture as a function of frequency.

The background relative permittivity ϵ_r,b is 25.4 with conductivity σ_b of 1.44 (S/m) at 1.3 GHz. We assume that a cylindrical object (a circle for the purpose of 2D imaging) is used as an object and placed in the imaging zone. The cylinders are 20 cm long and placed such that their centers are at the same vertical height of the centers of the antennas. The antennas are positioned 10.5 cm above the tank base and 3.0 cm below the liquid to minimize corruption from multipath signals. Two different configurations and their dielectric properties are shown in Fig. 4a and b. In case 1, the permittivity of the object has a value of ϵ_r,obj = 22 and conductivity of σ_obj = 1 (S/m); the object is positioned at x = 3.0 and y = 3.0 cm and has a diameter d = 3.0 cm. In case 2, the permittivity of Object 1 has a value of ϵ_r,obj = 18.5 and conductivity of σ_obj = 0.9 (S/m); the object is positioned at x = −3.0 and y = 3.0 cm and has a diameter d = 2.0 cm. The permittivity of Object 2 has a value of ϵ_r,obj = 30 and conductivity of σ_obj = 1.8 (S/m); the object is positioned at x = 2.0 and y = −2.0 cm and has a diameter d = 3.0 cm.

Fig. 4.

Schematic representation of simulation setups; a) one circular object at (3,3) cm; b) two circular objects at (2,−2) and (−3,3) cm.

1). Electromagnetic field distributions:

Fig. 5 shows the 1.3 GHz magnitude (dB) and phase (degrees) distributions for the first phantom configuration (Fig. 4a) due to a source which is positioned where antenna # 1 is located using both the DDA and finite-element (COMSOL) methods, respectively. As can be seen, the distributions appear almost identical except for a slight scaling factor in the magnitude distributions which is due primarily to differences in the source amplitudes. This difference is subsequently cancelled out as part of the calibration process for reconstructing the images. Likewise, the source phases for the FE and DDA solutions are 180 degrees out of phase. These differences are also canceled out as part of the calibration process. The perturbations in the field distributions due to the scatter are slight when viewed on these large magnitude and phase scales; however, when the contours within the highlighted areas are compared with those for the mirror image (i.e. near x = −0.03 m), the differences are evident. For this reason, the associated difference plots for the values at the antennas are shown in Fig. 6 to emphasize the perturbations.

Fig. 5.

Electric field distributions for the forward domain: the DDA and FE magnitudes in dB (left column) and phase in degrees (right column) are shown for case 1 when antenna # 1 is transmitting.

Fig. 6.

Case 1 magnitude and phase projections for four different transmitters. Projections are calculated via the 2D-DDA.

Fig. 6 shows the calibrated magnitude and phase measurements (i.e. with the homogeneous solutions subtracted out), referred to as projections, as a function of relative receiver number for the first phantom due to single transmitters (antennas #1, 5, 9, and 13). It is notable that the relative receiver number refers to the position of the receiving antenna with respect to the transmitting antenna. The first absolute antenna is positioned at 6:00 and the remaining ones in sequence in clockwise order. For example, in the case where absolute antenna #11 is transmitting to absolute receiving antenna #16, antenna #16 is actually relative receiver #5 (see Fig. 7a and 7b). The plots illustrate that when antenna #1 is transmitting, the projection values (both magnitudes and phase) at the receivers #8 to #11 are most impacted due to the presence of the object. The measurement projections for antennas closest to the transmitter remain close to zero while the maximum perturbations occur for receiving antennas where the object is located directly in the line of sight between it and the transmitting antenna. We also note that the overall shapes and amplitudes of the projections do not change significantly but move around depending on the spatial orientation of the transmitter and target.

Fig. 7.

Schematic representation of the antenna array.

To validate the 2D-DDA results, the 2D projections from a single transmitter are compared with those for the COMSOL finite-element based solver and the results for case 1 are presented in Fig. 8. In general the DDA and FE based magnitude and phase solutions are consistent with respect to their overall patterns. The larger amplitude and phase projection differences generally occur for the lowest absolute amplitudes (i.e. where the receive antennas are furthest from the transmitter). It should also be noted that for the COMSOL solution the regions for the target are defined in terms of the areas of the respective elements within the finite-element mesh. In contrast, the target for the DDA simulations is defined by the properties imposed at the various nodes of the grid. Defining a homogeneous object in this way implies that it is effectively larger than expected because the associated properties on the object perimeter nodes blend with those of the surrounding background in the interface zone between the two. As such, one would not expect the results of a DDA and finite-element solution to be exactly identical. Notwithstanding, the overall trends are encouraging and acceptable.

Fig. 8.

Case 1 comparison of calculated magnitude and phase projections of the signals transmitted from Antenna 1 for the 2D-DDA and FE solutions.

2). Image Reconstruction:

The DDA based reconstruction algorithm is developed and tested with the synthetic data simulated in the finite-element solver, COMSOL Multiphysics. The FE solutions are computed for the same antenna locations and used to avoid inverse crime issues. To accomplish this, FE solutions with additive noise are computed in the same configuration as would actual measured data in the imaging system. Then the data are employed in the minimization scheme according to the reconstruction algorithm (9). For all DDA reconstructions, the forward solution is on a 25.0 × 25.0 cm square grid composed of 64 ×64, 3.91 mm squares (65 × 65 nodes). The imaging zone is comprised of step-wise circle with 1012 squares and 1085 nodes (i.e. the number of reconstruction unknowns). The effective diameter of the imaging zone is 14.0 cm. Fig. 9 and 10 show the 1.3 GHz reconstructed relative permittivity and conductivity images for cases 1 and 2, respectively, where the simulated measured FE electric fields (with and without noise) are used. To produce the images, the convergence criterion consists of the algorithm stopping when the error difference between the current and previous iteration is less than 1e-3. These images illustrate that the algorithm is capable of recovering the objects in the imaging domain even in presence of noise. As the noise levels increase, especially for the −80 dBm case, the images start degrading where relative permittivty images are affected more by the noisy data compared to that for the conductivity images. In addition to the images, transects of the true property values along with the recovered relative permittivity and conductivity at different noise levels are presented in Fig. 11 for case 1. It should be noted that the recovery of both the permittivity and conductivity in the lower noise cases is good. The transects confirm that the properties in the background are uniform and close to that of the bath. The recovered objects are in the correct location and the correct size. The recovered properties of the object are quite good - almost exact for the conductivity and only a minor overshoot for the permittivity. It is clear that as the noise level increases, it has debilitating effects on the quality of the recovered images.

Fig. 9.

Case 1 image pairs of reconstructed relative permittivity (left) and conductivity (right) at the final iterations and frequency of 1.3 GHz. The FE simulated data are used as measured data. Dashed circles represent the objects’ borders.

Fig. 10.

Case 2 image pairs of reconstructed relative permittivity (left) and conductivity (right) at the final iterations and frequency of 1.3 GHz. The FE simulated data are used as measured data. Dashed circles represent the objects’ borders.

Fig. 11.

Case 1 reconstructed and actual data along the line through the center of the object located at (3,3) cm at 1.3 GHz.

The projection data computed from the reconstruction process are expected to match the original field distribution if the reconstruction is successful. The electric field pairs should match for all combinations of transmitter-receiver antennas. Fig. 12 presents magnitude and phase projections due to transmitter #1 for iterations 1, 5, 10, 15 and 16 compared to the FE data with −100 dBm noise level for case 1. The calculated values follow the general shape of the FE data projections, and approach the exact values quite closely during the iterative process. Although the iteration curves do not fit the exact plot perfectly, they demonstrate a good shape rendering of the original. The relative L2 norm errors of projections with respect to the first iteration are also calculated for each step in the reconstruction and are presented as function of iteration in Fig. 13. The low level of errors confirms the agreement between the DDA and FE solutions. We note that the calculated errors for case 1 are lower than that for case 2 as was expected since the contrast level of the imaging domain is higher for case 2 compared to that for case 1.

Fig. 12.

Case 1 with noise level of −100 dBm. Comparison of calculated magnitude and phase projections of the signals transmitted from Antenna 1 for multiple iterations as a function of receiver number. The FE solutions as base-line measurements are also shown.

Fig. 13.

Computed relative L2 norm error of projections for cases 1 (left) and 2 (right) as a function of iteration number.

B. Phantom Experiments

We test the 2D-DDA reconstruction algorithm in a similar configuration for the experimental setup with a simple cylindrical phantom filled with a mixture of glycerin and water. The measurement setup is arranged to resemble the simulation scenarios as close as possible (Fig. 14). The imaging tank is filled with an 80:20 glycerin-water mixture and a 4 cm diameter cylindrical phantom is positioned in the centre of imaging domain and is filled with an 88:12 glycerin-water mixture. The actual properties of the bath and target are ϵ_r,bath = 25.4 and conductivity of σ_bath = 1.44 (S/m) and ϵ_r,obj = 16.5 and conductivity of σ_obj = 0.90 (S/m), respectively.

Fig. 14.

Measurement setup for the case where a cylindrical inclusion is positioned at the center of imaging tank.

Fig. 15a shows the 1300 MHz reconstructed permittivity and conductivity images. The algorithm does a good job of recovering the object with respect to size, location and properties with only minimal artifacts surrounding the object. In addition, Fig. 15b and c show transects through the recovered images compared with the actual phantom property values. Fig. 16a and b show the magnitude and phase projections due to a single transmitter at selected iterations in comparison to the actual measured data. The projections at all iterations present similar shapes as the measured data and monotonically converge to distributions close to that of the same. Overall, the phase values have a closer match to the measured values than that for the magnitude. The relative L2 norm error decreased to 0.038 after 18 iterations from a normalized value of 1 at the start, as shown in Fig. 17.

Fig. 15.

a) Image pairs of reconstructed relative permittivity (left) and conductivity (right) at iteration 18 for 1.3 GHz; b,c) Reconstructed and actual data along the line through center of the imaging domain.

Fig. 16.

Comparison of calculated magnitude and phase projections of the signals transmitted from Antenna 1 for multiple iterations as a function of receiver number. The actual measurement projection is also shown.

Fig. 17.

Computed relative L2 norm error of projections for the experimental study as a function of iteration number.

C. Computational Costs

Our previous 2D finite element-based algorithm took 139.7 seconds to reconstruct a 2D image in 20 iterations (559 unknowns) using software code written in FORTRAN77 that was compiled with gfortran (v. 7.5.0) and running on a single 4-core Xeon E3-1270 3.60GHz processor with hyperthreading enabled and 32GB of DDR4 2133MHz of RAM running the Ubuntu (18.04 LTS) operating system. It required 225.7 Mbytes of memory at run time. In contrast, running on the same machine, the new DDA-based algorithm took 6.01 seconds to reconstruct a 2D image in 20 iterations (1085 unknowns) using software code written in MATLAB (v. R2019b MathWorks, Inc., Natick, MA). It required 13.7 Mbytes of memory at run time. In both cases, they utilized the same input data.

V. Discussion

The 2D results were broken into two primary parts - simulations and actual measurements. For the simulations, we first demonstrated that the forward solutions utilizing the DDA matched those of a known finite element-based standard, namely COMSOL. This is shown for both the full field distributions and for a more refined representation at the associated receiver antenna locations as dictated for the reconstruction process. There are minor differences between the DDA and FE distributions which are primarily due to how the dielectric property distributions are depicted in the associated domains. In addition, noise is artificially added to the true signal to assess the impact of noise on the field distributions. In all cases, the majority of the deviations occur for the antennas furthest from the transmitter since these absolute field values are orders of magnitude less than those for the received signals at closer antennas. It should be noted that we can routinely achieve noise floor levels of −140 dB compared to that of the transmitter when using the Rohde & Schwarz ZNBT vector network analyzer. This implies that the three noise levels used (−120, −100 and −80 dB) are dramatically greater than that achieved in actual measurements, subsequently implying that these are robust tests of the algorithm under challenging circumstances. The associated reconstructed images are performed for two different phantom cases. In both cases, the convergence is monotonic and rapid and the recovered images are good renderings of the actual target. In addition, plots are shown of the computed field values for a given transmitter at selected iterations and compared to that for the simulated measured data. Similarly to previous experiments, the phase data more closely matches the measured data at the end of convergence than for the magnitude data. These results confirm that when starting from a homogeneous bath distribution, the algorithm is capable of accurate property recovery without converging to local minima. Similarly, for the measured object case, the algorithm does a good job of recovering the object size, location and properties, albeit, the reconstruction of the permittivity images is more refined than that for the conductivity. The sequence of phase and amplitude projections as a function of iteration further emphasizes that the reconstruction process is well behaved and robust and does not exhibit unwanted convergence to local minima.

Although the paper presents the results from a simple cylindrical phantom, the imaging system utilized in this study is capable of producing 2D images in more complicated scenarios. The overall hardware configuration, especially including the microwave electronics, antenna array, liquid tank and coupling liquid, are identical to previous implementations used for realistic breast phantoms and extensively for actual patient exams. In addition, the approach here is identical in all algorithmic aspects of the previous implementations except for the forward solver. These include the Gauss-Newton iterative reconstruction method, the log transformation, and phase unwrapping processes. Given that the DDA technique has demonstrated to provide equally accurate forward solutions and image reconstructions as those using our FE-based and FDTD-based approaches, we do not anticipate any differences between this and our previous implementations when we translate the algorithm to patient exams.

Finally, these results confirm that the DDA can be used with tomographic approaches to dramatically reduce the computation times. To our knowledge, tomographic reconstructions with our FE-based algorithm were among the fastest and took roughly 140 seconds for a 20 iteration reconstruction. The results from the new DDA approach confirm that it can perform 20 iterations in roughly 6 seconds. This is a dramatic improvement. In addition, because this new approach exploits the efficiencies of circulant matrices, the memory requirements are roughly 16.5 times less than our FE algorithm. This opens the door for utilizing low-cost computing in the future.

VI. Conclusion

We have successfully implemented the discrete dipole approximation for the forward solution portion of the 2D tomographic reconstruction algorithm. The work is validated using both simulated and measurement data. The algorithm performs well compared with our previous FE-based technique, especially with regards to rapid and monotonic convergence and not being plagued by convergence to unwanted local minima, even when using a simple starting distribution estimate of the homogeneous background. Utilizing conventional preconditioning techniques in conjunction with the DDA, we are able to reduce the 20 iteration reconstruction time from 140 seconds to roughly 6 seconds. The memory requirements were also dramatically reduced. This approach is readily translatable to full 3D reconstruction algorithms. In fact, the DDA was originally developed for reducing the forward solution computation time for certain classes of computationally heavy 3D experiments. These 2D results are a significant piece of the puzzle for simplifying and reducing system size, cost and speed towards eventual deployment in under-resourced environments which are currently not able to establish conventional breast screening programs.

Biography

Samar Hosseinzadegan (StM’17) received her B.S. in Mathematics from K. N. Toosi Universit of Technology, Tehran, Iran; and the M.S. in Engineering Mathematics and Comptational Scienec from Chalmers University of Technology, Gothenburg, Sweden. She is currently working on her Ph.D. in biomedical engineering at Chalmers University of Technology. Her research interests include computational electromagnetics and numerical and experimental biomedical electromagnetics.

Andreas Fhager (M’07) received the M.Sc. degree in engineering physics and the Ph.D. degree in electrical engineering from the Chalmers University of Technology, Gothenburg, Sweden, in 2001 and 2006, respectively. He was appointed Docent in Electrical Engineering at Chalmers University of Technology in 2011. His current research interests include electromagnetic imaging methods for breast cancer detection, stroke diagnostics, and other biomedical applications of microwaves.

Mikael Persson (M’10) received the M.Sc. and Ph.D. degrees from the Chalmers University of Technology, Gothenburg, Sweden, in 1982 and 1987, respectively. He became a Professor in Electromagnetics in 2000 and a Professor in Biomedical Electromagnetics in 2006 with the Department of Signals and Systems, Chalmers University of Technology, where he has been the Head of the Division of Signal Processing and Biomedical Engineering since 2010. He has authored and co-authored more than 200 refereed journal and conference papers. His current research interests include electromagnetic diagnostics and treatment.

Shireen D. Geimer received the B.S. degree in physics from Purdue University, Fort Wayne, IN, USA, in 1991 and the M.S. degree in engineering (with a specialization in space physics) from Dartmouth College, Hanover, NH, USA, in 1995. Since 1994, she has been a Research Engineer with the Numerical Methods Laboratory, Thayer School of Engineering, Dartmouth College. Her current research interests include mesh generation in two and three dimensions for remote sensing and biomedical applications and execution of simulations and processing of modeling solutions

Paul M. Meaney (M’91-SM’11-F’17) received the AB degree in electrical engineering and computer science from Brown University, Providence, RI, USA, in 1982, the master’s degree in microwave engineering from the University of Massachusetts, Amherst, MA, USA, in 1985, and the Ph.D. degree from Dartmouth College, Hanover, NH, USA, in 1995. He was involved in the millimeter-wave industry with Millitech Corporation, South Deerfield, MA, USA, and Alpha Industries, Woburn, MA. He spent two years as a Post-Doctoral Fellow, including one year at the Royal Marsden Hospital, Sutton, U.K. He has been a Professor with Dartmouth College since 1997 and is also the President of Microwave Imaging System Technologies, Inc., Hanover, which he co-founded with Dr. K. D. Paulsen in 1995. The Dartmouth group has authored several clinical studies in various settings including breast cancer diagnosis, breast cancer chemotherapy monitoring, bone density imaging, and temperature monitoring during thermal therapy. He has also explored various commercial spin-off concepts, such as detecting explosive liquids and noninvasively testing whether a bottle of wine has gone bad. He has co-authored over 80 peer-reviewed journal papers, co-written 1 textbook and presented numerous invited papers related to microwave imaging, and holds 14 patents. His current research interests include microwave tomography, which exploits the many facets of dielectric properties in tissue and other media, and also include breast cancer imaging, where his group was the first to translate an actual system into the clinic.