Microwave Breast Imaging Using a Dry Setup

This article demonstrates for the first time, both numerically and experimentally, the feasibility of radar-based microwave imaging of anthropomorphic heterogeneously dense breasts in prone position, requiring no immersion liquid. The dry, contactless approach greatly simplifies the setup, favors patient comfort, and further avoids lengthy sanitation procedures after each exam. We use a radar-type technique with the antennas distributed in cylindrical configuration around the breast phantom. The reflectivity map is reconstructed using a wave-migration algorithm in the frequency domain. This article presents new developed strategies to cope with the challenges of a dry setup, namely increased skin artifact due to the concomitant absence of matching liquid and nonuniform breast shape. We propose an iterative and adaptive algorithm based on singular value decomposition that effectively removes the skin backscattering under the abovementioned conditions. It is compatible with automatic processing, and computationally fast. One of its inputs is the breast three-dimensional surface information, and its distance to the antennas, all obtained automatically from a proposed low-cost procedure based on a webcam. The imaging method is reasonably resilient to the presence of fibroglandular tissues, and to uncertainties of tissue permittivity. Another tackled challenge is the miniaturization of the antenna in air, which is achieved with an optimized balanced antipodal Vivaldi of the same size as counterparts used in dense immersion liquids. Finally, all the building blocks are combined to demonstrate experimentally the overall dry system performance, with very good detection of the tumor at three different positions in the breast, even in low-contrast scenarios.


I. INTRODUCTION
O VER the last two decades, Microwave Imaging (MWI)   has been investigated as an alternative imaging modality for breast cancer screening.De facto techniques, such as mammography (X-rays) and Magnetic Resonance Imaging (MRI), have well known limitations [1], which contributed to the emergence of new technologies.
Active MWI systems illuminate the breast with electromagnetic (EM) energy usually comprised in the frequency spectrum between 1 GHz and 10 GHz.The microwave energy is radiated by antennas distributed around the breast.Given the permittivity contrast of different tissues, the EM waves are scattered and picked up by the same or separate antennas [1].In the specific case of breast cancer detection, MWI benefits from high contrast between healthy and malignant tissues [1].
It is common to categorize MWI methods as quantitative or qualitative.The first type aims at spatially mapping the dielectric properties of media in the frequency-domain, for instance by means of the distorted Born's or Rytov's linear approximation [2], [3].Quantitative methods involve timeconsuming iterative calculations to find the solutions of illposed inverse problem, which depends greatly on the accuracy of the forward model and on good initializations [2], [4].In contrast, qualitative imaging aims at reconstructing the reflectivity of a given volume [5] and is compatible with realtime examinations.In the present work we focus on qualitative techniques based on radar-approach.
Due to the restricted bandwidth and relatively large wavelengths, the attainable resolution of microwaves is of a few centimeters in free-space, which contrasts with the fine resolution of X-rays.Nevertheless, unlike mammography, microwaves do not pose any health risk and are compatible with contactless imaging setup.Moreover, medical MWI is intended as a primary screening examination method and not as the ultimate and single-validation examination.In case of such detection, patients would be forwarded to other exams, such as MRI or ultrasound.We highlight that this is already standard procedure with mammography.The attainable resolution using microwaves is addressed in more detail ahead in this paper.
Given the high permittivity contrast between skin and air, the received signals are dominated by this early-time reflection [6].This artifact is easily hundreds of times larger in magnitude than the tumor response, which can potentially mask the tumor, thus precluding its detection.Consequently, before addressing the image reconstruction, it is necessary to eliminate (or at least significantly reduce) the early-time response from the skin.This processing step is known as artifact removal.
A typical strategy to mitigate the skin backscattering is to immerse the breast in a coupling medium [7]- [9].Nevertheless, some researchers have recently explored dry breast imaging setups using computationally heavy tomographic techniques for image inversion [10], [11], or faster holographic techniques [12], [13].The latter make no reference to the artifact removal algorithm and enable only two-dimensional images of the breast, whereas [10], [11] use a simplified anatomic structure to model the breast, and benefit from a more favorable contrast between fibroglandular and tumor tissues than what is generally reported in the literature [14].Also, the papers do not address issues related to the real measurement scenario and experimental setup.In fact, no one has yet fully analyzed the impact of a contactless dry approach [15].The contact liquid makes the setup much more complex and bulky than a dry one.Moreover, immersing the breast poses sanitation issues, complicates the maintenance of the examination setup, and makes the exam longstanding and somewhat uncomfortable to the patient.Therefore, we investigate whether it is imperative to use a coupling medium.To this end, we assume a transmission line model (TLM) representing each layer of the body plus the external medium, and calculate the corresponding scattering matrix.The dry setup shows some increase of skin backscatter, but, as demonstrated in this paper, not enough to rule it out.
Without the coupling medium, larger efforts are required by the artifact removal algorithm, which relies on a precise retrieval of the breast 3D shape and the antenna distances to the breast skin.This issue has been addressed in [16] and [17].However, the solutions proposed in those works are either rather expensive [16] or require complex measurement schemes [17].We propose a technique based on a commercial webcam, which is not only low-cost, but also involves less measurement points.The information is fed into our proposed artifact removal technique, which is based on the Singular Value Decomposition (SVD) factorization [18].This method decomposes the measured data into a set of reflections, allowing removing the stronger echoes originated by the breast skin.We introduce an automation-compatible iterative procedure that adapts the echo removal algorithm to the reflection information in the measured data, without introducing distortion in the tumor position.The algorithm is fast, and compatible with real-time or near-real-time processing.As discussed in [19], precise breast shape and antenna distance information not only benefit the effectiveness of the artifact removal algorithm but also improves the accuracy of the final tumor image reconstruction.The methods described in this paragraph are key to make dry setups viable.SVD is used in other works as well [8], [20], but using simplistic assumptions such as that of revolution-symmetric phantoms.
The absence of coupling medium no longer benefits the antenna size and the antenna-breast interaction.Therefore, it requires the design of miniaturized broadband antennas compatible with the available volume around the breast.This subject is also one of the goals of the present study.
As an ultimate objective, leveraging all the aforementioned elements, we test experimentally a complete dry MWI system for breast cancer screening in prone position.The advantages of this examination posture over supine posture are discussed in [21].Some authors also suggested compressing the breast to make it planar and more easily scanned [15], but this is what we want to avoid, since it may be painful and contrary to the contactless exam that we investigate in this paper.
The breast in prone position is non-symmetric.Nevertheless, few works have replicated the breast shape in their experimental tests.Indeed, most authors use a symmetric-shaped phantom to represent the breast [8], [22]- [24].Symmetricshaped breast allows using much simpler artifact removal and imaging algorithms that are not, however, applicable to anthropomorphic shapes.Yet, the symmetry assumption is not realistic and leads to poor results in real applications.The use of symmetric antenna distribution illuminating the breast 1) simplifies the setup, 2) is easily adjustable to any breast size, and 3) helps decreasing costs by avoiding the use of complex positioning mechanisms.Nevertheless, it increases the difficulty of separating the tumor response from the measured signals, given that the antennas will be at different distances from the skin, as a consequence of the non-symmetric shape of the breast.
In summary, we experimentally demonstrate the correct operation of a MWI system using an anthropomorphic breast model in prone posture and a cylindrical antenna distribution without resorting to immersion medium.The breast includes fibroglandular tissues and, therefore, the dielectric constrast between healthy tissues and tumor is very much reduced.The image reconstruction algorithm relies on wave-migration [5], which was adopted due to its achievable accuracy, compatibility with the assumption of tissue dispersive properties and potential for real-time processing.To the authors' best knowledge, the present paper is the first experimental work successfully operating under all of these conditions.This paper is organized as follows: in Sections II and III we address the problem formulation and signal processing, including the artifact removal and imaging algorithms.Section IV studies the impact of using an immersion liquid and concludes about the viability of imaging the body using a dry system.The experimental setup is described in Section V, including the antennas and phantom utilized in the measurements.The breast-shape estimation system and associated algorithms are presented in Section VI.The experimental results are discussed in Section VII and, finally, the main conclusions are drawn in Section VIII.

II. PROBLEM FORMULATION
In microwave breast imaging, the tissues are illuminated by N a probing antennas distributed around the breast, which transmit broadband signals and retrieve the echoes originated by the contrast between malignant and healthy tissues.In monostatic systems, the same antenna transmits and receives the signals; alternatively, one can use a single antenna as transmitter, while multiple antennas pick up the echoes at different coordinates -multistatic system.An example of the imaging scenario under study here is sketched in Fig. 1.
In this work we focus on MWI techniques for reflectivity map reconstruction based on monostatic observations.Such inverse problems may be described by the following matrix linear model [25] (1) g = Ar + w, where r is the sought reflectivity vector defined in suitable set of spatial coordinates, in which the operator (.) T denotes the transpose, and N x , N y , and N z are the number of test points along each axis.In turn, g contains the measurements indexed by frequency and antenna position, w is an unknown perturbation vector, which we designate noise vector, that, in addition to additive noise, accounts for model uncertainties and degradations (such as the interference signal due to the air-skin interface), and matrix A is a linear operator accounting for the phase of the travelling wave.In more detail, we have where s a = s a (f 1 ) . . .s a f N f T is the measured input reflection coefficient of the a-th antenna at coordinates (x a , y a , z a ) in the frequency band of interest ∆f where N f designates the number of useful frequencies.The dimensions of g and r are N a N f × 1 and N x N y N z × 1, respectively.For convenience, we may also denote the measured input reflection coefficient at a-th antenna by s a (f ) = s (x a , y a , z a , f ).
Assuming the antenna to be a perfect radiator (i.e., with no internal reflections), only target point scatterers [26], and neglecting multiple reflections between the scatterers due to high losses of tissues, A is given by where A has dimensions N a N f × N x N y N z .In turn, each sub-matrix A a is given by in which the terms E (x, y, z, f ) a account for the phase delay between the a-th antenna and the test point coordinate, (x, y, z), at frequency f , i.e., in which k 0 (f ) is the free-space wavenumber of the propagating wave at frequency f , n b (f ) is the dispersive refraction index of breast tissues, and d a,air and d a,b are the distances travelled by the radiated wave from the a-th antenna to the test point, (x, y, z), through air and tissues, respectively: The coordinates (x s , y s , z s ) define the air-breast interface entry point (see Fig. 1).The dimensions of sub-matrix Note that in this formulation we considered the breast medium to be homogeneous, only captured by the term n b (f ), since its internal structure is hardly known.Yet, this term can be adjusted according to some criteria, as to obtain the best imaging results [27].Also, we assumed only scalar model.Both assumptions are common in medical MWI applications based on radar approaches.Lastly, we note that operator A does not account for the amplitude of the travelling wave, which is a multiplicative exponential term.The main reason for this is that the attenuation suffered by the waves in the breast tissue is difficult to estimate.As a result, if incorrectly estimated, even if just slightly, it may have a negative impact on the estimation of r.

III. SIGNAL PROCESSING
This section discusses the signal processing algorithms used for the artifact removal and for the reflectivity map reconstruction of the breast tissues.We build up on the formulation described in the previous section, as it will be useful for the implementation of the algorithms.

A. Artifact removal algorithm
The following signal processing assumes that all signals retrieved in the presence of the breast have the free-space antenna input reflection coefficient subtracted.This correction step ensures that the unwanted antenna internal reflections are removed from the scattered signals, thus improving the measurable dynamic range [28].
To reduce the dominating short-distance skin backscattering that precludes the tumor detection, we extend the technique used in [8] based on Singular Value Decomposition (SVD).In that work, the analysis is restricted to uniform breast shapes, immersed in a coupling medium with the antenna at a constant distance from the skin.That greatly simplifies the application of the artifact removal algorithm.In the present work, we extend the method to more realistic setups by increasing its adaptability according to the retrieved signal reflection content.
Let us consider, for each antenna, the propagation scenario sketched in Fig. 2. There, d init s designates the shortest distance between the antenna and the skin, d back s , the physical distance between opposite walls of the breast, and d t is the distance from the skin to the tumor.The reflection coefficients of the front skin interface, opposite skin interface, and tumor are denoted by Γ init s , Γ back s , and Γ t , respectively.The transmission coefficient of the skin (considered equal for inward and outward waves, apart from 180º phase shift) is denoted by T s .Note that d init s is a particular case of d a,air .For now, we consider an homogeneous breast.The method will be generalized to heterogeneous cases ahead.Building up on the assumptions discussed in Section II, the measured signals in the scenario sketched in Fig. 2, and in the absence of noise, are well approximated by a sum of complex exponentials [26] described as (8) are coefficients related to the reflection, transmission, and dissipation losses, respectively, all dependent on body material properties, as well as the radial spreading of the propagating spherical wave, χ.Each term in (8) corresponds to a single scatterer.Moreover, since c init |c t | and c init c back , then the first term on the right hand side of (8) dominates s a (f ), which hinders the estimation of the second term, corresponding to the tumors.It is important to take into account the reflection on the back skin interface, so that the algorithm is robust to fatty, less dense breasts.We will show that this coefficient does not have any relevant impact when the breast is denser.
The terms on the right hand side of (8) have the structure 2, 3 are different, and that β 1 ∆f 2π, then the terms γ i (f ) for i = 1, 2, 3 are orthogonal and may be filtered out in the frequency domain.However, the last condition is not satisfied at least for signal γ 1 (f ); as an example, take ∆f = 3 GHz and d init s = 10 mm thus yielding β 1 ∆f = 0.4 rad.Therefore, the signal γ 1 (f ) cannot be filtered out in the frequency domain.
We attack the filtering problem just described with the SVD [31], which factorizes any matrix M of size m × l as of size l × l that hold the vectors u i , for i = 1, . . ., m and v i , for i = 1, . . ., l, are the left-and right-singular vectors respectively, forming two orthonormal basis of the space spanned by the columns and rows of M, respectively.The matrix Σ is diagonal and contains the singular values, σ ≥ 0, in non-increasing magnitude.Among the various applications of SVD, the best q-rank approximation of matrix M, which minimizes the Frobenius norm, M q − M F , is calculated as where (σ i , u i , v i ), for i = 0, . . ., q, are the first q singular values, left singular vectors, and right singular vectors, ordered by non-increasing value of σ i .Below, we exploit the lowrank approximation property of the SVD in the direction (or directions) of maximum power corresponding to the skin signal.
Contrarily to [8], we drop the assumption of uniform breast shapes.Yet, we still assume locally smooth shapes, which imply that the skin backscattering is locally identical; i.e., within a region of 2 × N n neighbor antennas relative to the ath central position, the antennas "see" similar skin responses [30].To take advantage of this assumption, we arrange the S a matrix containing the (2 × N n + 1) input reflection coefficients versus frequency, centered at the a-th antenna, as (11) S a = s a−Nn . . .s a . . .s a+Nn .
Based on expression (8), each of the columns of S a , i.e., s p , for p = a − N n , . . ., a + N n , may be written as the sum of three terms: where γ p,1 , γ p,2 and γ p,3 correspond to the signals originated by the skin (closest to the antenna), tumor and skin (farthest to the antenna), respectively.Given the above referred similarity, vector γ p,1 is almost constant for p = a−N n , • • • , a+N n since in this interval their elements present only slight variations of the distance between the antenna and the skin.Therefore the elements of γ p,1 have common β i across the region around 2N n neighboring antennas.The same happens for γ p,3 .However, since β i are different for γ p,1 , γ p,2 and γ p,3 , this makes them orthogonal.Moreover, since c init |c t |, we assume that γ p,1 has norm much greater than the norm of γ p,2 .At some antenna positions the reflection on the back skin wall is also greater than the tumor response, i.e. c back > |c t |.
Fig. 3 schematizes the relative location of vectors s p .There, s p,1 designates the projection of s p onto the subspace orthogonal to span(u 1 , . . ., u q ).Vector s p,2 , which is the projection of s p onto the subspace orthogonal to span(u q+1 , . . ., u 2Nn+1 ), is close to γ p,2 and has the bulk of the skin response removed.
In order to explore the properties of s p , matrix S a is factorized using the SVD.According to the above rationale, and since c init |c t | and c init c back , then the initial reflection on the skin is well-approximated by the first singular value, which has the greatest magnitude.We then project S a onto the orthogonal subspace spanned by the first q singular vectors U q = [u 1 . . .u q ], where q is to be determined according to some criteria.This way, we get the calibrated matrix (i.e.without the skin artifact), S cal a , by subtracting the contribution from the unwanted q scatterers: Therefore, the calibrated matrix contains the frequency response of the inner tissues only, for the a-th antenna position.From S cal a we extract the frequency response corresponding to the central antenna a, s cal a .Note that if q = 0 we remain with the original signal, which contains the initial skin reflection.In the opposite scenario, if we remove all the q singular values min(2 × N n + 1, N f ), where the number of frequencies is generally much greater than the number of antenna positions, the filtered signal becomes zero.
In [8] not only the skin reflection was assumed to be the same across all the antenna positions, but also the value of q was taken constant.Nevertheless, in a realistic non-uniform breast scenario, q cannot be considered equal for all antennas.In fact, q in (13) needs to be adapted for each antenna position, since the skin reflection varies significantly along the spatial scan.We propose an iterative method to automatically set q for each antenna position.We start by considering q = 0.The signals are then converted to the spatial domain by means of the Inverse Discrete Fourier Transform (IDFT), from which we obtain s a,0 (d), where d is the roundtrip distance.This allows estimating the initial skin reflection distance, d init s .Then, we increment q, and apply the SVD followed by the IDFT to calculate s a,1 (d).The process is repeated, and q is incremented, until the maximum of s a,q (d) falls out of the breast walls.These are defined by the electric length intervals d init This procedure is repeated for all N a antenna positions.In the end, we merge the s cal a in the according order into a single matrix, ( 15) in which s cal a is defined along frequency, This method falls into the data adaptive class of artifact removal algorithms, since the orthogonal subspace is learned from the measured input reflection coefficients.
We highlight that the proposed procedure to define q can be easily automated.Furthermore, it is computationally fast, making it compatible with real-time systems, and it does not introduce any distortion in the tumor position.Lastly, it is capable of detecting tumors at distances from the skin below the theoretic image resolution (equation ( 14)), as will be shown ahead.The latter is an important feature of this artifact removal, since many algorithms struggle under these conditions, in particular those based on time/spatial gating techniques.
From this point onwards, we modify the observation model in (1) to accommodate the impact of the artifact removal, as follows ( 16) where w cal corresponds to the noise after the air-skin interference has been removed, and g cal is the corresponding vector of observations.
As an example of the proposed iterative artifact removal algorithm, let us consider the two-dimensional numerical setup depicted in Fig. 4, which was simulated using Computer Simulation Technology (CST) Microwave Transient Solver [33].It consists of a simplified cylindrical breast phantom with elliptical cross-section illuminated by a broadband bowtie antenna [34].The breast model includes a 0.7 mm thick skin layer and homogeneous fatty tissues with complex permittivity of s = 36 − j3.4 and f at = 7.34 − j0.24 at 3.5 GHz, respectively.The tumor, also cylindrical, is located at coordinates (x, y) = (15, 10) mm and has a complex permittivity of tmr = 40.7 − j4 at 3.5 GHz.Later on, we perform an additional simulation with the same breast shape and tumor, but considering heterogeneous breast filling with xy cross-section depicted in the inset in Fig. 4.These tissues intend to represent the higher-density fibroglandular tissues of the breast, which we assume to have a permittivity of f g = 30.5 − j3 [14].The used bowtie antenna operates in the 2-5 GHz range (|S 11 | ≤ −10 dB).A total of N a = 24 antenna positions are considered around the breast, distributed over a circle of radius 75 mm centered at (0, 0, 0).As a consequence of the breast elliptical shape, the antenna-to-breast distance, d init s , varies between 35 mm and 40 mm.We applied the proposed artifact removal to the above homogeneous model considering N n = 2, ∆f = 3 GHz and N f = 34 frequencies.The corresponding theoretical resolution in the breast tissue, given by ( 14), is 9.2 mm.
An example of the spatial signals, s a,q (d), from which the value of q was determined, is presented in Fig. 5.At the illustrated position, d init s = 39 mm, d back s = 72 mm and d t = 36 mm approximately.The abscissa in this figure represents the electrical distance.The vertical dashed lines identify the front and back breast walls, whereas the red circle identifies the electrical tumor distance.Note that the distance of the skin reflections is assumed to be known -this subject is handled in section VI.
First, we note that q = 1 is enough to successfully remove the initial skin reflection.However, the tumor is still masked by the reflection on the opposite wall.By filtering another singular vector, the second skin reflection is mitigated and the tumor is observable at the correct distance (red circle).A maximum is still observable near the back wall, but the same does not happen for the other antenna positions.Therefore, the imaging reconstruction algorithm, presented in the next sub-section, inherently filters it out.

B. Image reconstruction algorithm
The imaging problem that we face aims at solving the inverse problem under the linear observation model (16).Its final objective is to estimate the reflectivity of the breast tissues, r.In this sub-section, we formulate an imaging algorithm based on wave-migration approach to get a useful estimation of r, r, thus allowing detecting the tumor.
The present imaging inverse problem is quite challenging due to its ill-posedness.The reasons that account to this are: 1) The presence of noise and model uncertainties captured by the term w cal .We have identified at least three sources of uncertainties that contribute to the term w cal .Firstly, the microwave energy refracts at the breast boundary.However, we are considering only direct ray propagation for the sake of simplicity and computational effort.We note that this is common practice in MWI based on radar algorithms [26].Secondly, we assume static antenna phase center.Nevertheless, the phase center moves with frequency and depends on angle of arrival, leading to errors in the calculation of distances [28].Lastly, the variability of the permittivity of the internal tissues that is not possible to predict which also accounts to the errors.2) Matrix A is fat, corresponding to an under-determined system of equations.
3) The number of singular values of A with significant magnitude, N σ , is considerably smaller than the number of rows in A, N rows .As an example, Fig. 6 shows the singular values of A calculated based on the twodimensional scenario depicted in Fig. 4. In this case, we have considered N x = 61, N y = 61 and N z = 1 (two-dimensional example), while keeping N a and N f unchanged.The number of rows of A in this example is N rows = N a N f = 24 × 34 = 816, but N σ is around 100, thus proving this is a challenging inverse problem.
Such an ill-posed inverse problem calls for regularization.However, the quality of the inference obtained by solving an inverse problem relies strongly on the quality of the direct model.This is not the case in the problem in hands, owing to the uncertainties captured by the term w cal .As a result, we solve this problem through a matched-filter approach.Although matched-filter does not provide the best resolution, it is robust to model errors and yields the best signal-tonoise ratio [35].This way, we are able to retrieve reliable information about the presence of tumors inside the breast.As such, the reflectivity map is estimated as Hence, each element i of r can be calculated as ( 18) in which d a,b and d a,air are defined according to eq. ( 7), and g cal a,f is the calibrated input reflection coefficient at frequency f measured at antenna a after the skin reflection has been removed.Since we do not have prior knowledge of the internal structure of the breast, the reconstruction algorithm assumes propagation in homogeneous medium inside the breast, accounted by term n b that may be dispersive.Lastly, given that r is defined over a known volume, the intensity of each pixel forming the image, I (x, y, z), is proportional to the power of the backscattered signals at each coordinate (x, y, z), i.e.,

C. Performance metrics
We use quantitative figures of merit to assess the quality of the image reconstruction algorithm and to perform comparisons between available methods.
1) Imaging quality metrics: The tumor-to-clutter ratio (TCR) and tumor-to-mean ratio (TMR) assess the imaging detection quality [36].TCR compares the maximum intensity corresponding to the tumor response (T ) and the larger unwanted artifact intensity (clutter -C) occurring anywhere in the whole 3D reconstructed image: 2) Tumor detection metric: Lastly, we compute two metrics to assess the ability of the overall signal processing to detect the tumor in the correct position.To this end, we use the positioning error (PE), which quantifies the deviation of the detected tumor, P detection , position with respect to the actual position, P tumor , known from the measurement setup:

IV. QUANTIFICATION OF IMMERSION LIQUID IMPACT ON SKIN REFLECTION
The use of a dense coupling medium between antennas and breast in Medical MWI systems is common practice among research works, as explained in the Introduction [7]- [9].In order to conclude about the real need of a coupling medium, we analyze the skin reflection using a cascaded transmission line model (TLM), see Fig. 8.The TLM provides useful information about the energy reflected by the skin and the energy coupled to the breast.Each transmission line section represents a propagation medium: background, skin, fat and fibroglandular.The cascade of tissues should approximate reasonably the internal structure of the breast, given that fat is immediately below the skin, whereas fibroglandular tissues are internal.A similar study was performed in [37], but considering homogeneous breast with high density.We emphasize that the results presented in this section were validated by full wave simulations using CST (not shown here for conciseness).The impedance of each medium is defined as where µ 0 and 0 are the permeability and permittivity of vacuum, respectively, and medium (f ) = medium (f ) − j medium (f ) is the complex relative permittivity of each medium.For simplicity, the equations will not show onwards the dependence on frequency.The propagation constant is written as κ medium = jω √ µ 0 0 medium given that µ r = 1 for all involved materials.Parameter ω designates the angular frequency, ω = 2πf .We considered constant permittivity of tissues across frequency, with values of s = 37 − j10.36, f at = 8 − j0.8, f g = 48 − j9.6, for the skin, fat and fibroglandular tissues, respectively.These dielectric properties correspond to the approximate properties of each tissue at 3.5 GHz, according to references [40] and [14], respectively.For the purpose of this study, the permittivity of the background medium, backgr , is swept between 1 and 100 with loss tangent of tan (δ) = backgr backgr = 0.01.According to transmission line theory, the magnitude of input reflection coefficient, |S 11 |, of the TLM shown in Fig. 8 is given by ( 24) s + Z backgr where input impedance at the background/skin interface is given by Z in s = Z s Z in f at +Zs tanh(κsls) Zs+Z in f at tanh(κsls) , and, lastly, the input impedance at the fat/fibroglandular interface is given by Z f at +Z f g tanh(κ f at l f at ) .These parameters take into account the losses in each dielectric.
We assume that the distance between the excitation port and skin is l backgr = 20 mm and breast path length is l f at = 35 mm and l f g = 40 mm.Regarding the skin thickness, l s , reference [42] reports that the minimum thickness for a healthy breast is around 0.7 mm.On the contrary, a diseased breast can have 2.5 mm skin thickness.For the purpose of the present study we present results for both thicknesses.As for the frequency range, we analyzed the bandwidth from 1 GHz up to 6 GHz, which is the most commonly used bandwidth for breast imaging.Fig. 9 presents the numerical results based on TLM of the backscattered power at the background medium/skin interface, |S 11 | 2 .Although the results in Fig. 9 provide useful information about the impact of the coupling medium for a wide range of values of backgr , for the purpose of this paper let us concentrate our analysis on three cases: backgr = 1, backgr = 3, and backgr = 30.The first represents the dry setup we seek, the second represents the immersion in canola oil, which is common practice in medical MWI [38], [39], and the last represents the case which was identified by the authors of reference [37] as the best case scenario to minimize the reflection on the skin.These results are summarized in Table I.We quantified the skin reflection ratio (SRR) as the ratio between the average of the reflected power over the entire frequency band [1,6] GHz for given values of l s , and backgr , and the average for backgr = 1: This indicator gives a measure of the skin reflection reduction by the coupling media compared to the dry case.The maximum SRR for a breast immersed in oil is about −2.75 dB obtained for l s = 0.7.In turn, for backgr = 30 the skin suppression improves at most −7.92 dB obtained for l s = 2.5.
These values should be analyzed in light of the noise floor of the measurement setup.To this end, we have measured the noise floor of our dry experimental setup (to be described in section V), and concluded that an increase of skin reflection up to 8 dB does not preclude the use of dry imaging setups, since the tumor response after applying our artifact removal should be well above the noise floor.As a result, the use of immersion liquid only partially tackles the skin artifact challenge, therefore making its use not imperative.Based on this conclusion, we have developed an imaging system without immersion liquid, which we will address in the remaining of this paper.
However, for practical reasons related to the 3D-printing fabrication method, the skin of our experimental breast phantom does not present the actual skin electrical properties.Instead, its refractive index is n s = √ 2.75 [43] corresponding to PLA plastic.Of course, this raises the doubt of representativeness of such model in the present context.In order to clear that, we have repeated the S 11 study for the "air-PLAfat-fibroglandular" scenario, considering the skin thickness parameter fixed at l s = 1.2 mm.The results are plotted in Fig. 10.For better comparison, we superimpose the same figure the |S 11 | 2 curves for l s = 0.7 mm and l s = 2.5 mm, already plotted in Fig. 9, calculated for backgr = 1 (dry setup).In order to assess the representativeness of the plastic breast relative to the real case, we calculated the SRR of curves represented in Fig. 10, where the numerator is the S 11 l s , backgr = 1 2 for l s = 0.7 mm or l s = 2.5 mm, and the denominator is the S 11 l s = 1.2 mm, backgr = 1 2 considering n s = √ 2.75.For l s = 0.7 mm the SRR is about 0.9 dB, which means that the PLA breast reflects more energy than the real breast.As for thicker skin value, l s = 2.5 mm, the SRR is approximately −1.04 dB.These values show that the PLA breast is a reasonable approximation to a realistic breast and, therefore, the algorithms and measurements described in this paper should be representative of the real examination scenario.

V. EXPERIMENTAL SETUP
In this section we describe our experimental dry setup.Moreover, we demonstrate the feasibility of a functional minia-turized antenna, despite not benefiting from a dense immersion medium.

A. Breast phantoms
For the experimental tests we use a MRI-derived breast phantom taken from the University Wisconsin-Madison repository (ID: 062204) [44], corresponding to American College of Radiology (ACR) 3 classification -heterogeneously dense breast.We 3D printed it using polylactic acid (PLA, r = 2.75 [43] on a Ultimaker 2+ Extended [45]).The model includes the breast shell, with 1.2 mm wall thickness, the fibroglandular container -Fig.11 (a) and the tumor -an ellipsoidal container with the internal dimensions depicted in Fig. 11 (b).The 3D printer settings used for the tumor container were the same as for the remaining parts.We highlight that the breast repository of the University of Wisconsin-Madison is accepted in the literature as fair representation of actual breasts for the different ACR classification in terms of density.The containers were filled with different mixtures of TX-100, distilled water and sodium chloride (NaCl) that approximate the dielectric properties of breast and tumor tissues.The recipes were taken from [46].We measured the complex permittivity of the mixtures using the cavity method described in [47].Fig. 12 presents the results of the dielectric constant and equivalent conductivity.The latter is calculated as σ e = ω 0 .The agreement between phantom liquids and the real tissues is reasonably good.Fig. 12. Permittivity and equivalent conductivity of healthy fatty tissues, fibroglandular tissues and tumor taken from [40] and [14] (dashed lines), respectively, and of liquid phantoms used in measurements (solid lines).

B. Antenna design and antenna distribution
We opted for a Balanced Antipodal Vivaldi Antenna (BAVA), printed on a RT/duroid 6010 substrate ( r = 10.2, tan(δ) = 2.3 ×10 −3 ) of thickness 25 mils (0.635 mm).The antenna topology falls within the class of travelling-wave, end-fire antennas that cover a wide frequency spectrum [48].Furthermore, its small cross-section minimizes the undesired interactions between the breast and antenna.
The geometry of our BAVA is detailed in Fig. 13.It consists of two antipodal exponential "fins" capped by truncated ellipses to achieve a compact antenna design [49].Two exponential curves of the form ( 26) describe the "fins" and the feeding microstrip line [39].Thirteen other parameters, marked in Fig. 13, define the remainder geometry.The fine-tuning of all parameters was achieved through full-wave simulation using CST Microwave Transient solver [33].One of the optimization goals was to achieve an input reflection coefficient below -15 dB (|S 11 | ≤ -15 dB) between 2.5 GHz and 5 GHz.This frequency interval offers the best trade-off between penetration depth, image resolution, and antenna size.The second goal (results not shown for the sake of conciseness), was to ensure that the radiation maximum is directed towards the u-axis.
The optimized values of coefficients A a , C a , P a , A t , C t , and P t are presented in Table II, along with the remaining geometry parameters.The 90º mitre bend in the microstrip line ensures that the feeding connector has minimal influence on the antenna performance, which also contributes to improve the S 11 .The bandwidth is reasonably achieved, as illustrated in Fig. 14.Differences between the simulated and measured results are attributed to manufacturing inaccuracy.We emphasize that, compared to the antenna presented in [39], which has a dimension of 44 mm but uses a dense embedding medium, our BAVA design is quite compact considering that it radiates in air.This proves that it is possible to have miniaturized antennas even in dry systems.
The antenna elements are intended to be positioned in a cylindrical configuration around the breast, as sketched in Fig. 1 by rotating the antenna around the breast.We lay the antennas horizontally to allow for more measurement planes along the z-direction.

C. Experimental setup
For the measurements, we fabricated an elevated styrofoam structure with the breast model placed at the center.The styrofoam material ( r ≈ 1) ensures that the structure does not influence the measurements.In addition, the height at which the breast model and the antenna are placed minimizes any influence from the floor.The measurement setup is shown in Fig. 15.It approximates the examination scenario, with the patient laying in the prone position and the breast pending from a cavity in the examination bed.Measurements were performed in the frequency-domain between 2 GHz and 5 GHz using a Vector Network Analyzer (VNA).A single BAVA element was connected to the VNA.The cylindrical antenna configuration is emulated by rotating the breast phantom while keeping the antenna fixed.The diameter of the cylindrical antenna distribution is approximately 120 mm, corresponding approximately to average 17 mm distance to the breast wall.
The setup allows for continuous z-axis positioning of the antenna.We used 4 antenna heights, each one with 24 azimuth positions over the circumference.The upper height (i.e.closer to the chest) is at z = -13 mm and the subsequent ones are spaced with 10 mm steps.Note that according to our axes system, the breast is at negative z-values (Fig. 1).

VI. BREAST SURFACE RECONSTRUCTION AND DISTANCE ESTIMATION SYSTEM
The exact antenna distance to the non-symmetric breast is needed for effective skin artifact removal and improved image retrieval from the inversion algorithm.Therefore, it is imperative to estimate the breast shape in a way that is simultaneously practical for a real exam.
The proposed system is based on a Logitech C930e webcam [50].The camera, located in one side of the setup, takes 12 snapshots for different azimuth angles of the breast (between 0º and 165º), against a contrasting background.An example is shown in Fig. 16 (a).The distance scale is estimated by placing two Red/Green (R/G) markers at a known distance.We then apply a Template Matching Correlation Technique (TMCT) [51], to identify the markers in each snapshot.This technique is widely used in digital image processing for pattern finding, due to its simplicity, rapid execution, robustness [52], and resilience to brightness variations.
The process allows calculating 12 breast meridian profiles from 12 snapshots.This is a relatively straightforward process, based on breast edge detection against a background contrast and pixel counting.In our setup, the distance between breast and antenna varies from around 9 mm up to 30 mm.This variation is due solely to the non-symmetric shape of the breast, since the cylindrical antenna distribution has constant radius.
Two steps are used to obtain a full three-dimensional reconstruction of the breast shape from the determined 12 vertical profiles.First, we obtain an interpolation function for each meridian profile.This step is followed by another interpolation of the points from all profiles contained in each z-plane, in order to achieve a closed curve (the breast is approximately elliptical for constant z).By repeating this procedure for all zplanes with a desired resolution along this coordinate (0.5 mm in our case) we get the tridimensional surface of the breast.This process is simple, compatible with real-time processing and sufficiently accurate for MWI.Comparison between the reconstructed breast dimensions and the actual breast phantom size revealed a maximum error of about 2 mm.Moreover, it can be implemented using any commercial webcam.

VII. EXPERIMENTAL RESULTS
Prior to imaging the breast we quantified the noise floor of our fully-featured setup to be around −110dB (relative to an input power of 10 dBm), whereas the tumor response magnitude lies between -45 dB and -80 dB.The latter was calculated as S cal 11 − S tmr 11 , where S tmr 11 is the measured S 11 in the presence of the breast with tumor and S cal 11 is the measured S 11 in the presence of the healthy breast (i.e.without tumor).Thus, an increase in skin backscattering by as much as 8 dB, as concluded in section IV, is manageable and, therefore, does not preclude a dry imaging setup.
In order to fully assess the capability of the dry MWI system, we performed experimental tests with the tumor located at different positions in the breast volume, both for homogeneous and heterogeneous breasts.The artifact removal was applied considering N n = 2 antennas, contained in the same height as the central antenna (the different antenna heights are considered independently).For each antenna position, we considered N f = 34 frequency points in the 2-5 GHz interval.
In the first detection scenario, we considered the tumor at (x t , y t , z t ) = (15, -20, -32) mm immersed in homogeneous breast.This location is intentionally far enough from the breast walls, where the imaging resolution is expected to easily distinguish between the skin and the tumor.The imaging results from our system are shown in Fig. 17 in the main cuts.The white contours identify the limits of the breast and tumor.We successfully detect the tumor in the correct position, with TCR = 1.5 dB and TMR = 14.4 dB.However, the tumor shape is elongated along the z-direction.Two reasons contribute to this result.Firstly, as mentioned, our cylindrical antenna distribution uses only four antenna heights, limited by the breast z-extent with reasonably vertical wall.As a result, the achievable resolution along the z-direction is lower.This could be improved by adding more antenna rings, but at the expense of a more complex setup.Secondly, less important, the empty plastic tube used to fill the tumor cavity also presents some dielectric contrast relative to the background liquid, contributing to the backscattering.
In order to highlight the advantage of the proposed artifact removal algorithm, we compare it to state-of-the-art methods.We tested the artifact removal algorithms described in [30] and [36].The first algorithm calculates the average of the S 11 measured for the neighboring antenna positions and subtracts it from the S 11 measured for the position of interest.As for [36], its algorithm weighs and combines the signals based on finite impulse response (FIR) filters, in an attempt to minimize the early portion of the time domain signal.The reconstructed images using these two algorithms are shown in Fig. 18.The tumor is clearly not detected in the first case, mainly because the algorithm is affected by the non-symmetric distance between the antennas and the breast surface.The FIR-based algorithm provides better results, but the clutter level is too high to distinguish the tumor from unwanted artifacts (TCR = 0.04 dB, TMR = 7.1 dB).This algorithm fails mostly because the skin reflection varies too much to be able to combine the signals and successfully remove the artifacts.
Due to space restrictions, the comparison of these methods is omitted for the next scenarios.Yet, we note that in all cases the alternative methods exhibit the same poor performance as in Fig. 18.
The second scenario is intended to demonstrate the capability of the artifact removal to preserve the response of the tumor when the distance from the latter to the skin is less than the theoretical resolution of the imaging system, ∆d = 12.5 mm in the present case.This is a significant challenge for artifact removal algorithms, especially for those implementing time or spatial gating, which struggle to distinguish between two close scatterers.To this end, we have positioned the same tumor at (x t , y t , z t ) = (-20, -22, -32) mm.At these coordinates the tumor is estimated to be about 5 mm from the skin.The reconstructed reflectivity of the breast is presented in Fig. 19.
Again, the tumor is detected at the correct position, despite being closer to the skin than the theoretical image resolution.In this case we computed TCR = 4.1 dB showing very Lastly, in the third scenario we evaluate the capability of the imaging system to detect tumors in heterogeneously dense breasts.To this end, we included the fibroglandular container (Fig. 11) filled with the corresponding liquid.The tumor was placed at (x t , y t , z t ) = (15, -25, -32) mm.Fig. 20 illustrates the imaging results obtained under these conditions.Even in low contrast scenarios, the tumor is correctly detected with TCR = 2.33 dB.However, the clutter level has increased due to the backscattering of the fibroglandular tissues, which resulted in TMR = 13.4 dB.Yet, the positioning error is just 10.8 mm, which we consider good for the intended primary screening technique.
The metrics calculated in the three discussed scenarios are summarized in Table III.Although tumor shape is not correctly reproduced, we stress that a sharp isolated detection clearly above the clutter is a much valuable result, as it gives a reliable indication of a diseased tissue, since it results from the known high contrast of unhealthy tissues.
We have performed additional experimental tests for other tumor positions (not shown here), and the obtained detection metrics were similar to the above.

VIII. CONCLUSION
We assessed the feasibility of a dry MWI setup, with the advantage of being contactless, more hygienic and potentially     more practical.To this end, we quantified the skin backscattering reduction when the breast is immersed in a matching media.In the best case scenario, we concluded that the reflection is mitigated by 8 dB for dense embedding medium of permittivity r = 30.We showed that this reduction is insufficient to rule out a dry imaging setup, given that the noise floor of imaging setups is sufficiently low, as to accommodate for the lower power reaching the tumor.However, more robust artifact removal algorithm had to be developed not only to tackle the stronger skin artifacts, but also the additional challenge posed by the non-uniform shape of the pending breast.We proposed an adaptive artifact removal algorithm based on SVD.The method is robust and automation-compatible, allowing isolating the tumor response even when its distance to the skin is less than the theoretical spatial resolution of the system.This feature is enabled by the surface estimation method based on a webcam, which feeds the breast surface data to the imaging algorithms contributing to improve its performance.
Moreover, we designed a very compact broadband antenna with adequate performance despite not benefiting from a dense contact medium, in order to fit more antennas around the breast.All these elements were integrated in a demonstrator that included an anthropomorphic heterogeneous breast phantom.We obtained very good detection of the tumor in different positions, according to established quantitative metrics, despite the low contrast in the presence of fibroglandular tissue.
In summary, this work proved that a dry setup with proper signal processing is feasible for operation in real exams.It also showed that the setup can be compact and cost effective.In the future, we plan to improve imaging results using regularization techniques.We will explore the possibility of characterizing the antenna, in order to minimize the level of uncertainty in the linear observation model.

Fig. 1 .
Fig. 1.Imaging domain representation with the antennas distributed around the breast in a cylindrical configuration.

Fig. 2 .
Fig. 2. Simplified propagation scenario for one antenna, used in the artifact removal formulation.
frequency dependent refractive index of the breast tissues and ∆d is the theoretical range resolution of the imaging system in the breast tissues[32] (14) ∆d = v 4n avg b ∆f .

Fig. 4 .
Fig. 4. Two-dimensional numerical setup used to exemplify the application of the proposed artifact removal algorithm (dimensions in millimeters).The top right close-up represents the xy-cross-section of the heterogeneous breast used in the numerical tests.

Fig. 5 .
Fig.5.Example of sa,q (d) for different q values: (a) q = 0; (b) q = 1; (c) q = 2.These results correspond to a single antenna position.The vertical dashed lines identify the electrical distance of the front and back breast walls, whereas the red circle identifies the actual tumor distance.

Fig. 6 .
Fig. 6.Singular values of A, σ i , calculated from the numerical example shown in Fig. 4.

( 20 )
TCR [dB] = 10 log 10 max (T ) max (C) Additionally, the TMR evaluates the ratio between the intensity of the tumor response and the mean intensity of the background medium response: (21) TMR [dB] = 10 log 10 max (T ) mean (C)

( 22 )
PE = P tumor − P detected Additionally, we calculate the normalized PE, NPE, relative to the theoretical resolution of the imaging system, ∆d (expression (14)): (23) NPE = PE ∆d We applied the described imaging algorithm to the numerical results obtained from the setup shown in Fig. 4, after using the proposed artifact removal algorithm.The reconstructed images for the homogeneous and heterogeneous breasts are represented in Fig. 7 (a) and (b), respectively.The small white circle marks the actual position of the tumor.The results show very good tumor detection in both scenarios.For the homogeneous breast we obtained TCR = 3.44 dB, TMR = 10.2 dB, PE = 3.6 mm, and NPE = 0.39; whereas for the heterogeneous breast, TCR = 1.54 dB, TMR = 9.7 dB, PE = 7.3 mm, and NPE = 0.8.In both cases, the error in the tumor position is better than the theoretical resolution of the system.Moreover, the imaging results slightly deteriorate with the presence of the fibroglandular tissues, which decrease the contrast between tumor and background medium, and also increase the signals backscattered inside the breast.

Fig. 7 .
Fig. 7. Two-dimensional image obtained from the numerical results: (a) homogeneous breast; (b) ACR-3 heterogeneously dense breast.The outer white contour identifies the breast shape and the smaller white circle identifies the actual tumor position.

Fig. 8 .
Fig. 8. Schematic of the transmission line model used to study the impact of the immersion liquid in mitigating the energy reflected from the skin and energy coupled to the body.

Fig. 11 .
Fig. 11.3D-printed phantom used in the experimental setup: (a) breast phantom; (b) internal dimensions of the tumor container.

Fig. 14 .
Fig. 14.Simulated and measured input reflection coefficient, S 11 , of the BAVA in free-space.

Fig. 16 .
Fig. 16.(a) Example of snapshot taken by the webcam; (b) Reconstructed breast surface using the proposed webcam system.

Fig. 16 (
b) shows the final breast surface estimation of the anthropomorphic model used in the setup.

Fig. 17 .
Fig. 17.Imaging results of the homogeneous breast at (a) z = -30 mm and (b) x = 12 mm.The tumor is located at (15,-20,-32) mm.The white contours identify the limits of the breast and tumor.

Fig. 18 .
Fig.18.Imaging results of the homogeneous breast (z = -30 mm) using skin artifact removal methods from the literature (a) neighborhood-based[30] and (b) FIR filter-based[36].The tumor is located at (15,-20,-32) mm.The white contours identify the limits of the breast and tumor.

Fig. 19 .
Fig. 19.Imaging results of the homogeneous breast at (a) z = -26 mm and (b) x = -14 mm.The tumor is located at (-20,-22,-32) mm, at 5 mm distance from the skin.The white contours identify the limits of the breast and tumor.

Fig. 20 .
Fig. 20.Imaging results of the heterogeneous breast at (a) z = -24 mm and (b) x = 8 mm.The tumor is located at (10,35,-32) mm.The white contours identify the limits of the breast and tumor.

TABLE III CALCULATED
IMAGING METRICS.