Microwave image reconstruction utilizing log-magnitude and unwrapped phase to improve high-contrast object recovery

On the utilization of the adjoint method in microwave tomography

In microwave imaging, the adjoint method is widely used for the efficient calculation of the update direction, which is then used to update the unknown model parameter. However, the utilization and the formulation of the adjoint method differ significantly depending on the imaging scenario and the applied optimization algorithm. Because of the problem-specific nature of the adjoint formulations, the dissimilarities between the adjoint calculations may be overlooked. Here, we have classified the adjoint method formulations into two groups: the direct and indirect methods. The direct method involves calculating the derivative of the cost function, whereas, in the indirect method, the derivative of the predicted data is calculated. In this review, the direct and indirect adjoint methods are presented, compared, and discussed. The formulations are explicitly derived using the two-dimensional wave equation in frequency and time domains. Finite-difference time-domain simulations are conducted to show the different uses of the adjoint methods for both single source-multiple receiver, and multiple transceiver scenarios. This study demonstrated that an appropriate adjoint method selection is significant to achieve improved computational efficiency for the applied optimization algorithm.

Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix

This paper focuses on the construction of the Jacobian matrix required in tomographic reconstruction algorithms. In microwave tomography, computing the forward solutions during the iterative reconstruction process impacts the accuracy and computational efficiency. Towards this end, we have applied the discrete dipole approximation for the forward solutions with significant time savings. However, while we have discovered that the imaging problem configuration can dramatically impact the computation time required for the forward solver, it can be equally beneficial in constructing the Jacobian matrix calculated in iterative image reconstruction algorithms. Key to this implementation, we propose to use the same simulation grid for both the forward and imaging domain discretizations for the discrete dipole approximation solutions and report in detail the theoretical aspects for this localization. In this way, the computational cost of the nodal adjoint method decreases by several orders of magnitude. Our investigations show that this expansion is a significant enhancement compared to previous implementations and results in a rapid calculation of the Jacobian matrix with a high level of accuracy. The discrete dipole approximation and the newly efficient Jacobian matrices are effectively implemented to produce quantitative images of the simplified breast phantom from the microwave imaging system.

Review of Microwaves Techniques for Breast Cancer Detection

Conventional breast cancer detection techniques including X-ray mammography, magnetic resonance imaging, and ultrasound scanning suffer from shortcomings such as excessive cost, harmful radiation, and inconveniences to the patients. These challenges motivated researchers to investigate alternative methods including the use of microwaves. This article focuses on reviewing the background of microwave techniques for breast tumour detection. In particular, this study reviews the recent advancements in active microwave imaging, namely microwave tomography and radar-based techniques. The main objective of this paper is to provide researchers and physicians with an overview of the principles, techniques, and fundamental challenges associated with microwave imaging for breast cancer detection. Furthermore, this study aims to shed light on the fact that until today, there are very few commercially available and cost-effective microwave-based systems for breast cancer imaging or detection. This conclusion is not intended to imply the inefficacy of microwaves for breast cancer detection, but rather to encourage a healthy debate on why a commercially available system has yet to be made available despite almost 30 years of intensive research.

3-D Microwave Tomography Using the Soft Prior Regularization Technique: Evaluation in Anatomically Realistic MRI-Derived Numerical Breast Phantoms

OBJECTIVE

Fusion of magnetic resonance imaging (MRI) breast images with microwave tomography is accomplished through a soft prior technique, which incorporates spatial information (from MRI), i.e., accurate boundary location of different regions of interest, into the regularization process of the microwave image reconstruction algorithm.

METHODS

Numerical experiments were completed on a set of three-dimensional (3-D) breast geometries derived from MR breast data with different parenchymal densities, as well as a simulated tumor to evaluate the performance over a range of breast shapes, sizes, and property distributions.

RESULTS

When the soft prior regularization technique was applied, both permittivity and conductivity relative root mean square error values decreased by more than 87% across all breast densities, except in two cases where the error decrease was only 55% and 78%. In addition, the incorporation of structural priors increased contrast between tumor and fibroglandular tissue by 59% in permittivity and 192% in conductivity.

CONCLUSION

This study confirmed that the soft prior algorithm is robust in 3-D and can function successfully across a range of complex geometries and tissue property distributions.

SIGNIFICANCE

This study demonstrates that our microwave tomography is capable of recovering accurate tissue property distributions when spatial information from MRI is incorporated through soft prior regularization.

Impact of Information Loss on Reconstruction Quality in Microwave Tomography for Medical Imaging

This paper studies how limited information in data acquired by a wideband microwave tomography (MWT) system can affect the quality of reconstructed images. Limitations can arise from experimental errors, mismatch between the system and its model in the imaging algorithm, or losses in the immersion and coupling medium which are required to moderate this mismatch. We also present a strategy for improving reconstruction performance by discarding data that is dominated by experimental errors. The approach relies on recording transmitted signals in a wide frequency range, and then correlating the data in different frequencies. We apply this method to our wideband MWT prototype, which has been developed in our previous work. Using this system, we present results from simulated and experimental data which demonstrate the practical value of the frequency selection approach. We also propose a K-neighbour method to identify low quality data in a robust manner. The resulting enhancement in imaging quality suggests that this approach can be useful for various medical imaging scenarios, provided that data from multiple frequencies can be acquired and used in the reconstruction process.

Integrating prior information into microwave tomography Part 1: Impact of detail on image quality

PURPOSE

The authors investigate the impact that incremental increases in the level of detail of patient-specific prior information have on image quality and the convergence behavior of an inversion algorithm in the context of near-field microwave breast imaging. A methodology is presented that uses image quality measures to characterize the ability of the algorithm to reconstruct both internal structures and lesions embedded in fibroglandular tissue. The approach permits key aspects that impact the quality of reconstruction of these structures to be identified and quantified. This provides insight into opportunities to improve image reconstruction performance.

METHODS

Patient-specific information is acquired using radar-based methods that form a regional map of the breast. This map is then incorporated into a microwave tomography algorithm. Previous investigations have demonstrated the effectiveness of this approach to improve image quality when applied to data generated with two-dimensional (2D) numerical models. The present study extends this work by generating prior information that is customized to vary the degree of structural detail to facilitate the investigation of the role of prior information in image formation. Numerical 2D breast models constructed from magnetic resonance (MR) scans, and reconstructions formed with a three-dimensional (3D) numerical breast model are used to assess if trends observed for the 2D results can be extended to 3D scenarios.

RESULTS

For the blind reconstruction scenario (i.e., no prior information), the breast surface is not accurately identified and internal structures are not clearly resolved. A substantial improvement in image quality is achieved by incorporating the skin surface map and constraining the imaging domain to the breast. Internal features within the breast appear in the reconstructed image. However, it is challenging to discriminate between adipose and glandular regions and there are inaccuracies in both the structural properties of the glandular region and the dielectric properties reconstructed within this structure. Using a regional map with a skin layer only marginally improves this situation. Increasing the structural detail in the prior information to include internal features leads to reconstructions for which the interface that delineates the fat and gland regions can be inferred. Different features within the glandular region corresponding to tissues with varying relative permittivity values, such as a lesion embedded within glandular structure, emerge in the reconstructed images.

CONCLUSION

Including knowledge of the breast surface and skin layer leads to a substantial improvement in image quality compared to the blind case, but the images have limited diagnostic utility for applications such as tumor response tracking. The diagnostic utility of the reconstruction technique is improved considerably when patient-specific structural information is used. This qualitative observation is supported quantitatively with image metrics.

Advanced Fast 3-D Electromagnetic Solver for Microwave Tomography Imaging

This paper describes a fast-forward electromagnetic solver (FFS) for the image reconstruction algorithm of our microwave tomography system. Our apparatus is a preclinical prototype of a biomedical imaging system, designed for the purpose of early breast cancer detection. It operates in the 3-6-GHz frequency band using a circular array of probe antennas immersed in a matching liquid; it produces image reconstructions of the permittivity and conductivity profiles of the breast under examination. Our reconstruction algorithm solves the electromagnetic (EM) inverse problem and takes into account the real EM properties of the probe antenna array as well as the influence of the patient's body and that of the upper metal screen sheet. This FFS algorithm is much faster than conventional EM simulation solvers. In comparison, in the same PC, the CST solver takes ~45 min, while the FFS takes ~1 s of effective simulation time for the same EM model of a numerical breast phantom.

Two-step inversion with a logarithmic transformation for microwave breast imaging

PURPOSE

The authors have developed a new two-step microwave tomographic image reconstruction process specifically designed to incorporate logarithmic transformed microwave imaging algorithms as a means of significantly improving spatial resolution and target property recovery. Log transform eliminates the need for a priori information, but spatial filtering often integrated as part of the regularization required to stabilize image recovery, generally smooths image features and reduces object definition. The new implementation begins with this smoothed image as the first step, but then utilizes it as the starting estimate for a second step which continues the iterative process with a standard weighted Euclidean distance regularization. The penalty term of the latter restricts the new image to a multi-dimensional location close to the original but allows the algorithm to optimize the image without excessive smoothing.

METHODS

The overall approach is based on a Gauss-Newton iterative scheme which incorporates a log transformation as a way of making the reconstruction more linear. It has been shown to be robust and not require a priori information as a condition for convergence, but does produce somewhat smoothed images as a result of associated regularization. The new two-step process utilizes the previous technique to generate a smoothed initial estimate and then uses the same reconstruction process with a weighted Euclidean distance penalty term. A simple and repeatable method has been implemented to determine the weighting factor without significant computational burden. The reconstructions are assessed according to conventional parameter estimation metrics.

RESULTS

We apply the approach to phantom experiments using large, high contrast canonical shapes followed by a set of images recovered from an actual patient exam. The image improvements are substantial in regards to improved property recovery and feature delineation without inducing unwanted artifacts. Analysis of the residual vector after the reconstruction process further emphasizes that the minimization criterion is efficient with minimal biases.

CONCLUSIONS

The outcome is a novel synergism of an established stable reconstruction algorithm with a conventional regularization technique. It maintains the ability to recover high quality microwave tomographic images without the bias of a priori information while substantially improving image quality. The results are confirmed on both phantom experiments and patient exams.

Electrical Characterization of Glycerin: Water Mixtures: Implications for Use as a Coupling Medium in Microwave Tomography

We examine the broadband behavior of complex electrical properties of glycerin and water mixtures over the frequency range of 0.1 - 25.0 GHz, especially as they relate to using these liquids as coupling media for microwave tomographic imaging. Their combination is unique in that they are mutually miscible over the full range of concentrations which allows them to be tailored to dielectric property matching for biological tissues. While the resultant mixture properties are partially driven by differences in the inherent low frequency permittivity of each constituent, relaxation frequency shifts play a disproportionately larger role in increasing the permittivity dispersion while also dramatically increasing the effective conductivity over the frequency range of 1 to 3 GHz. For the full range of mixture ratios, the relaxation frequency shifts from 17.5 GHz for 0% glycerin to less than 0.1 GHz for 100% glycerin. Of particular interest is the fact that the conductivity stays above 1.0 S/m over the 1-3 GHz range for glycerin mixture ratios (70-90% glycerin) we use for microwave breast tomography. The high level of attenuation is critical for suppressing unwanted multipath signals. This paper presents a full characterization of these liquids along with a discussion of their benefits and limitations in the context of microwave tomography.

A multistage selective weighting method for improved microwave breast tomography

Microwave tomography has shown potential to successfully reconstruct the dielectric properties of the human breast, thereby providing an alternative to other imaging modalities used in breast imaging applications. Considering the costly forward solution and complex iterative algorithms, computational complexity becomes a major bottleneck in practical applications of microwave tomography. In addition, the natural tendency of microwave inversion algorithms to reward high contrast breast tissue boundaries, such as the skin-adipose interface, usually leads to a very slow reconstruction of the internal tissue structure of human breast. This paper presents a multistage selective weighting method to improve the reconstruction quality of breast dielectric properties and minimize the computational cost of microwave breast tomography. In the proposed two stage approach, the skin layer is approximated using scaled microwave measurements in the first pass of the inversion algorithm; a numerical skin model is then constructed based on the estimated skin layer and the assumed dielectric properties of the skin tissue. In the second stage of the algorithm, the skin model is used as a priori information to reconstruct the internal tissue structure of the breast using a set of temporal scaling functions. The proposed method is evaluated on anatomically accurate MRI-derived breast phantoms and a comparison with the standard single-stage technique is presented.

Microwave Imaging of the Breast

Microwave imaging for medical applications has been of interest for many years. Recently, microwave imaging for breast cancer detection has gained attention due to advances in imaging algorithms, microwave hardware and computational power. The breast is relatively translucent to microwaves, accessible for imaging, and there appears to be a significant electromagnetic property contrast between tumors and healthy tissues. Therefore, breast imaging may be the first clinically viable application of microwave imaging. This paper reviews recent developments in passive, hybrid, and active approaches to microwave breast cancer detection.

3D microwave tomography of the breast using prior anatomical information

PURPOSE

The authors have developed a new 3D breast image reconstruction technique that utilizes the soft tissue spatial resolution of magnetic resonance imaging (MRI) and integrates the dielectric property differentiation from microwave imaging to produce a dual modality approach with the goal of augmenting the specificity of MR imaging, possibly without the need for nonspecific contrast agents. The integration is performed through the application of a soft prior regularization which imports segmented geometric meshes generated from MR exams and uses it to constrain the microwave tomography algorithm to recover nearly uniform property distributions within segmented regions with sharp delineation between these internal subzones.

METHODS

Previous investigations have demonstrated that this approach is effective in 2D simulation and phantom experiments and also in clinical exams. The current study extends the algorithm to 3D and provides a thorough analysis of the sensitivity and robustness to misalignment errors in size and location between the spatial prior information and the actual data.

RESULTS

Image results in 3D were not strongly dependent on reconstruction mesh density, and the changes of less than 30% in recovered property values arose from variations of more than 125% in target region size-an outcome which was more robust than in 2D. Similarly, changes of less than 13% occurred in the 3D image results from variations in target location of nearly 90% of the inclusion size. Permittivity and conductivity errors were about 5 times and 2 times smaller, respectively, with the 3D spatial prior algorithm in actual phantom experiments than those which occurred without priors.

CONCLUSIONS

The presented study confirms that the incorporation of structural information in the form of a soft constraint can considerably improve the accuracy of the property estimates in predefined regions of interest. These findings are encouraging and establish a strong foundation for using the soft prior technique in clinical studies, where their microwave imaging system and MRI can simultaneously collect breast exam data in patients.

3D parallel-detection microwave tomography for clinical breast imaging

A biomedical microwave tomography system with 3D-imaging capabilities has been constructed and translated to the clinic. Updates to the hardware and reconfiguration of the electronic-network layouts in a more compartmentalized construct have streamlined system packaging. Upgrades to the data acquisition and microwave components have increased data-acquisition speeds and improved system performance. By incorporating analog-to-digital boards that accommodate the linear amplification and dynamic-range coverage our system requires, a complete set of data (for a fixed array position at a single frequency) is now acquired in 5.8 s. Replacement of key components (e.g., switches and power dividers) by devices with improved operational bandwidths has enhanced system response over a wider frequency range. High-integrity, low-power signals are routinely measured down to -130 dBm for frequencies ranging from 500 to 2300 MHz. Adequate inter-channel isolation has been maintained, and a dynamic range >110 dB has been achieved for the full operating frequency range (500-2900 MHz). For our primary band of interest, the associated measurement deviations are less than 0.33% and 0.5° for signal amplitude and phase values, respectively. A modified monopole antenna array (composed of two interwoven eight-element sub-arrays), in conjunction with an updated motion-control system capable of independently moving the sub-arrays to various in-plane and cross-plane positions within the illumination chamber, has been configured in the new design for full volumetric data acquisition. Signal-to-noise ratios (SNRs) are more than adequate for all transmit/receive antenna pairs over the full frequency range and for the variety of in-plane and cross-plane configurations. For proximal receivers, in-plane SNRs greater than 80 dB are observed up to 2900 MHz, while cross-plane SNRs greater than 80 dB are seen for 6 cm sub-array spacing (for frequencies up to 1500 MHz). We demonstrate accurate recovery of 3D dielectric property distributions for breast-like phantoms with tumor inclusions utilizing both the in-plane and new cross-plane data.

Integration of microwave tomography with magnetic resonance for improved breast imaging

PURPOSE

Breast magnetic resonance imaging is highly sensitive but not very specific for the detection of breast cancer. Opportunities exist to supplement the image acquisition with a more specific modality provided the technical challenges of meeting space limitations inside the bore, restricted breast access, and electromagnetic compatibility requirements can be overcome. Magnetic resonance (MR) and microwave tomography (MT) are complementary and synergistic because the high resolution of MR is used to encode spatial priors on breast geometry and internal parenchymal features that have distinct electrical properties (i.e., fat vs fibroglandular tissue) for microwave tomography.

METHODS

The authors have overcome integration challenges associated with combining MT with MR to produce a new coregistered, multimodality breast imaging platform--magnetic resonance microwave tomography, including: substantial illumination tank size reduction specific to the confined MR bore diameter, minimization of metal content and composition, reduction of metal artifacts in the MR images, and suppression of unwanted MT multipath signals.

RESULTS

MR SNR exceeding 40 dB can be obtained. Proper filtering of MR signals reduces MT data degradation allowing MT SNR of 20 dB to be obtained, which is sufficient for image reconstruction. When MR spatial priors are incorporated into the recovery of MT property estimates, the errors between the recovered versus actual dielectric properties approach 5%.

CONCLUSIONS

The phantom and human subject exams presented here are the first demonstration of combining MT with MR to improve the accuracy of the reconstructed MT images.

Surface scanning through a cylindrical tank of coupling fluid for clinical microwave breast imaging exams

PURPOSE

Microwave tomographic image quality can be improved significantly with prior knowledge of the breast surface geometry. The authors have developed a novel laser scanning system capable of accurately recovering surface renderings of breast-shaped phantoms immersed within a cylindrical tank of coupling fluid which resides completely external to the tank (and the aqueous environment) and overcomes the challenges associated with the optical distortions caused by refraction from the air, tank wall, and liquid bath interfaces.

METHODS

The scanner utilizes two laser line generators and a small CCD camera mounted concentrically on a rotating gantry about the microwave imaging tank. Various calibration methods were considered for optimizing the accuracy of the scanner in the presence of the optical distortions including traditional ray tracing and image registration approaches. In this paper, the authors describe the construction and operation of the laser scanner, compare the efficacy of several calibration methods-including analytical ray tracing and piecewise linear, polynomial, locally weighted mean, and thin-plate-spline (TPS) image registrations-and report outcomes from preliminary phantom experiments.

RESULTS

The results show that errors in calibrating camera angles and position prevented analytical ray tracing from achieving submillimeter accuracy in the surface renderings obtained from our scanner configuration. Conversely, calibration by image registration reliably attained mean surface errors of less than 0.5 mm depending on the geometric complexity of the object scanned. While each of the image registration approaches outperformed the ray tracing strategy, the authors found global polynomial methods produced the best compromise between average surface error and scanner robustness.

CONCLUSIONS

The laser scanning system provides a fast and accurate method of three dimensional surface capture in the aqueous environment commonly found in microwave breast imaging. Optical distortions imposed by the imaging tank and coupling bath diminished the effectiveness of the ray tracing approach; however, calibration through image registration techniques reliably produced scans of submillimeter accuracy. Tests of the system with breast-shaped phantoms demonstrated the successful implementation of the scanner for the intended application.

Fast 3-d tomographic microwave imaging for breast cancer detection

Microwave breast imaging (using electromagnetic waves of frequencies around 1 GHz) has mostly remained at the research level for the past decade, gaining little clinical acceptance. The major hurdles limiting patient use are both at the hardware level (challenges in collecting accurate and noncorrupted data) and software level (often plagued by unrealistic reconstruction times in the tens of hours). In this paper we report improvements that address both issues. First, the hardware is able to measure signals down to levels compatible with sub-centimeter image resolution while keeping an exam time under 2 min. Second, the software overcomes the enormous time burden and produces similarly accurate images in less than 20 min. The combination of the new hardware and software allows us to produce and report here the first clinical 3-D microwave tomographic images of the breast. Two clinical examples are selected out of 400+ exams conducted at the Dartmouth Hitchcock Medical Center (Lebanon, NH). The first example demonstrates the potential usefulness of our system for breast cancer screening while the second example focuses on therapy monitoring.

Clinical microwave tomographic imaging of the calcaneus: a first-in-human case study of two subjects

We have acquired 2-D and 3-D microwave tomographic images of the calcaneus bones of two patients to assess correlation of the microwave properties with X-ray density measures. The two volunteers were selected because each had one leg immobilized for at least six weeks during recovery from a lower leg injury. A soft-prior regularization technique was incorporated with the microwave imaging to quantitatively assess the bulk dielectric properties within the bone region. Good correlation was observed between both permittivity and conductivity and the computed tomography-derived density measures. These results represent the first clinical examples of microwave images of the calcaneus and some of the first 3-D tomographic images of any anatomical site in the living human.

Microwave imaging for breast cancer detection: advances in three--dimensional image reconstruction

Microwave imaging is based on the electrical property (permittivity and conductivity) differences in materials. Microwave imaging for biomedical applications is particularly interesting, mainly due to the fact that available range of dielectric properties for different tissues can provide important functional information about their health. Under the assumption that a 3D scattering problem can be reasonably represented as a simplified 2D model, one can take advantage of the simplicity and lower computational cost of 2D models to characterize such 3D phenomenon. Nonetheless, by eliminating excessive model simplifications, 3D microwave imaging provides potentially more valuable information over 2D techniques, and as a result, more accurate dielectric property maps may be obtained. In this paper, we present some advances we have made in three-dimensional image reconstruction, and show the results from a 3D breast phantom experiment using our clinical microwave imaging system at Dartmouth Hitchcock Medical Center (DHMC), NH.

Bone dielectric property variation as a function of mineralization at microwave frequencies

A critical need exists for new imaging tools to more accurately characterize bone quality beyond the conventional modalities of dual energy X-ray absorptiometry (DXA), ultrasound speed of sound, and broadband attenuation measurements. In this paper we investigate the microwave dielectric properties of ex vivo trabecular bone with respect to bulk density measures. We exploit a variation in our tomographic imaging system in conjunction with a new soft prior regularization scheme that allows us to accurately recover the dielectric properties of small, regularly shaped and previously spatially defined volumes. We studied six excised porcine bone samples from which we extracted cylindrically shaped trabecular specimens from the femoral heads and carefully demarrowed each preparation. The samples were subsequently treated in an acid bath to incrementally remove volumes of hydroxyapatite, and we tested them with both the microwave measurement system and a micro-CT scanner. The measurements were performed at five density levels for each sample. The results show a strong correlation between both the permittivity and conductivity and bone volume fraction and suggest that microwave imaging may be a good candidate for evaluating overall bone health.

Comparison of no-prior and soft-prior regularization in biomedical microwave imaging

Microwave imaging for medical applications is attractive because the range of dielectric properties of different soft tissues can be substantial. Breast cancer detection and monitoring of treatment response are areas where this technology could be important because of the contrast between normal and malignant tissue. Unfortunately, the technique is unable to achieve the high spatial resolution at depth in tissue which is available from other conventional modalities such as x-ray computed tomography (CT) or magnetic resonance imaging (MRI). We have incorporated a soft-prior regularization strategy within our microwave reconstruction algorithm and compared it with the images obtained with traditional no-prior (Levenberg-Marquardt) regularization. Initial simulation and phantom results show a significant improvement of the recovered electrical properties. Specifically, errors in the microwave property estimates were improved by as much as 95%. The effects of a false-inclusion region were also evaluated and the results show that a small residual property bias of 6% in permittivity and 15% in conductivity can occur that does not otherwise degrade the property recovery accuracy of inclusions that actually exist. The work sets the stage for integrating microwave imaging with MR for improved resolution and functional imaging of the breast in the future.

Microwave tomography in the context of complex breast cancer imaging

The notion of applying microwave imaging to breast cancer imaging has been studied at various levels by numerous scientists. The earliest appeal of this concept related to the presumably high property contrast between benign and malignant tissue that was unique to the breast. Subsequent published studies have shown that this assumption was overly simplistic and that the tissue property heterogeneity is considerable within the breast. As we have expanded the clinical use of our microwave tomographic system, we are now using this approach to monitor tumor progressions during neoadjuvant chemotherapy. In these cases, while we can still characterize and track the tumor progression, we have observed a new phenomenon. Very often these cancer patients exhibit skin thickening near the tumor site. Our images have reconstructed elevated dielectric properties along the breast surface associated with the accompanying edema. These observations further add to the complex nature of breast dielectric properties and the challenges for imaging them using microwave interrogation.

Importance of phase unwrapping for the reconstruction of microwave tomographic images

Microwave image reconstruction is typically based on a regularized least-square minimization of either the complex-valued field difference between recorded and modeled data or the logarithmic transformation of these field differences. Prior work has shown anecdotally that the latter outperforms the former in limited surveys of simulated and experimental phantom results. In this paper, we provide a theoretical explanation of these empirical findings by developing closed form solutions for the field and the inverted electromagnetic property parameters in one dimension which reveal the dependency of the estimated permittivity and conductivity on the absolute (unwrapped) phase of the measured signal at the receivers relative to the source transmission. The analysis predicts the poor performance of complex-valued field minimization as target size and/or frequency and electromagnetic contrast increase. Such poor performance is avoided by logarithmic transformation and preservation of absolute measured signal phase. Two-dimensional experiments based on both synthetic and clinical data are used to confirm these findings. Robustness of the logarithmic transformation to variation in the initial guess of the reconstructed target properties is also shown. The results are generalizable to three dimensions and indicate that the minimization form with logarithmic transformation offers image reconstruction performance characteristics that are much more desirable for medial microwave imaging applications relative to minimizing differences in complex-valued field quantities.

Importance of phase unwrapping for the reconstruction of microwave tomographic images

Microwave image reconstruction is typically based on a regularized least-square minimization of either the complex-valued field difference between recorded and modeled data or the logarithmic transformation of these field differences. Prior work has shown anecdotally that the latter outperforms the former in limited surveys of simulated and experimental phantom results. In this paper, we provide a theoretical explanation of these empirical findings by developing closed form solutions for the field and the inverted electromagnetic property parameters in one dimension which reveal the dependency of the estimated permittivity and conductivity on the absolute (unwrapped) phase of the measured signal at the receivers relative to the source transmission. The analysis predicts the poor performance of complex-valued field minimization as target size and/or frequency and electromagnetic contrast increase. Such poor performance is avoided by logarithmic transformation and preservation of absolute measured signal phase. Two-dimensional experiments based on both synthetic and clinical data are used to confirm these findings. Robustness of the logarithmic transformation to variation in the initial guess of the reconstructed target properties is also shown. The results are generalizable to three dimensions and indicate that the minimization form with logarithmic transformation offers image reconstruction performance characteristics that are much more desirable for medial microwave imaging applications relative to minimizing differences in complex-valued field quantities.

Enhancement of the Krylov subspace regularization for microwave biomedical imaging

Although Krylov subspace methods provide fast regularization techniques for the microwave imaging problem, they cannot preserve the edges of the object being imaged and may result in an oscillatory reconstruction. To suppress these spurious oscillations and to provide an edge-preserving regularization, we use a multiplicative regularizer which improves the reconstruction results significantly while adding little computational complexity to the inversion algorithm. We show the inversion results for a real human forearm assuming the 2-D transverse magnetic illumination and a cylindrical object assuming the 2-D transverse electric illumination.

Microwave thermal imaging of scanned focused ultrasound heating: phantom results

We are developing a microwave tomographic imaging system capable of monitoring thermal distributions based on the temperature dependence of the recovered dielectric properties. The system has been coupled to a high intensity focused ultrasound (HIFU) therapy device which can be mechanically steered under computer control to generate arbitrarily shaped heating zones. Their integration takes advantage of the focusing capability of ultrasound for the therapy delivery and the isolation of the microwave imaging signal from the power deposition source to allow simultaneous treatment monitoring. We present several sets of phantom experiments involving different types of heating patterns that demonstrate the quality of both the spatial and temporal thermal imaging performance. This combined approach is adaptable to multiple anatomical sites and may have the potential to be developed into a viable alternative to current clinical temperature monitoring devices for HIFU, such magnetic resonance (MR) imaging.

3-D-microwave breast tumor detection: study of system performance

This paper presents a study of performance for a 3-D breast tumor detection system. The system is based on processing the scattered signals when a narrow pulse is transmitted from a set of antennas placed surrounding the breast. The 3-D system performance is evaluated by placing the tumor at any breast position, even at locations near the breast muscle where detection is more difficult. Results are obtained for different sizes and positions of the abnormalities that are defined as spherical and bunch-shaped geometries, which approach the well-known breast tumors morphology. These results show that the array distribution considered will have a high impact on detection.

Combined optical imaging and mammography of the healthy breast: optical contrast derived from breast structure and compression

In this paper, we report new progress in developing the instrument and software platform of a combined X-ray mammography/diffuse optical breast imaging system. Particularly, we focus on system validation using a series of balloon phantom experiments and the optical image analysis of 49 healthy patients. Using the finite-element method for forward modeling and a regularized Gauss-Newton method for parameter reconstruction, we recovered the inclusions inside the phantom and the hemoglobin images of the human breasts. An enhanced coupling coefficient estimation scheme was also incorporated to improve the accuracy and robustness of the reconstructions. The recovered average total hemoglobin concentration (HbT) and oxygen saturation (SO2) from 68 breast measurements are 16.2 microm and 71%, respectively, where the HbT presents a linear trend with breast density. The low HbT value compared to literature is likely due to the associated mammographic compression. From the spatially co-registered optical/X-ray images, we can identify the chest-wall muscle, fatty tissue, and fibroglandular regions with an average HbT of 20.1+/-6.1 microm for fibroglandular tissue, 15.4+/-5.0 microm for adipose, and 22.2+/-7.3 microm for muscle tissue. The differences between fibroglandular tissue and the corresponding adipose tissue are significant (p < 0.0001). At the same time, we recognize that the optical images are influenced, to a certain extent, by mammographical compression. The optical images from a subset of patients show composite features from both tissue structure and pressure distribution. We present mechanical simulations which further confirm this hypothesis.

Log transformation benefits parameter estimation in microwave tomographic imaging

Microwave tomographic imaging falls under a broad category of nonlinear parameter estimation methods when a Gauss-Newton iterative reconstruction technique is used. A fundamental requirement in using these approaches is evaluating the appropriateness of the regression model. While there have been numerous investigations of regularization techniques to improve overall image quality, few, if any, studies have explored the underlying statistical properties of the model itself. The ordinary least squares (OLS) approach is used most often, but there are other options such as the weighted least squares (WLS), maximum likelihood (ML), and maximum a posteriori (MAP) that may be more appropriate. In addition, a number of variance stabilizing transformations can be applied to make the inversion intrinsically more linear. In this paper, a statistical analysis is performed of the properties of the residual errors from the reconstructed images utilizing actual measured data and it is demonstrated that the OLS algorithm with a log transformation (OLSlog) is clearly advantageous relative to the more commonly used OLS approach by itself. In addition, several high contrast imaging experiments are performed, which demonstrate that different subsets of data are emphasized in each method and may contribute to the overall image quality differences.

Initial clinical experience with microwave breast imaging in women with normal mammography

RATIONALE AND OBJECTIVES

We have developed a microwave tomography system for experimental breast imaging.

MATERIALS AND METHODS

In this article, we illustrate a strategy for optimizing the coupling liquid for the antenna array based on in vivo measurement data. We present representative phantom experiments to illustrate the imaging system's ability to recover accurate property distributions over the range of dielectric properties expected to be encountered clinically. To demonstrate clinical feasibility and assess the microwave properties of the normal breast in vivo, we summarize our initial experience with microwave breast exams of 43 women with negative mammography according to the Breast Imaging Reporting and Data System (BI-RADS 1).

RESULTS

The clinical results show a high degree of bilateral symmetry in the whole breast average microwave properties. Focal assessments of microwave properties are associated with breast tissue composition evaluated through radiographic density categorization verified through magnetic resonance image correlation in selected cases. Specifically, both whole-breast average and local microwave properties increase with increasing radiographic density, in which the latter exhibits a more substantial rise.

CONCLUSION

These findings support our hypothesis that water content variations in the breast play an influential role in dictating the overall dielectric property distributions and indicate that the microwave properties in the breast are more heterogeneous than previously believed based on ex vivo property measurements reported in the literature.

The multidimensional phase unwrapping integral and applications to microwave tomographical image reconstruction

Spatial unwrapping of the phase component of time varying electromagnetic fields has important implications in a range of disciplines including synthetic aperture radar (SAR) interferometry, MRI, optical confocal microscopy, and microwave tomography. This paper presents a fundamental framework based on the phase unwrapping integral, especially in the complex case where phase singularities are enclosed within the closed path integral. With respect to the phase unwrapping required when utilized in Gauss-Newton iterative microwave image reconstruction, the concept of dynamic phase unwrapping is introduced where the singularity location varies as a function of the iteratively modified property distributions. Strategies for dynamic phase unwrapping in the microwave problem were developed and successfully tested in simulations and clinical experiments utilizing large, high contrast targets to validate the approach.

Spherical harmonics microwave algorithm for shape and location reconstruction of breast cancer tumor

A reconstruction algorithm to simultaneously estimate the shape and location of three-dimensional breast cancer tumor is presented and its utility is analyzed. The approach is based on a spherical harmonic decomposition to capture the shape of the tumor. We combine a gradient descent optimization method with a direct electromagnetic solver to determine the coefficients in the harmonic expansion as well as the coordinates of the center of the tumor. The results demonstrate the potential advantage of collecting data using a multiple-view/tomographic-type strategy. We show how the order of the harmonic expansion must be increased to capture increasingly "irregularly" shaped tumors and explore the resulting increase in the central processing unit (CPU) time required by the algorithm. Our approach shows accurate reconstruction of the tumor image regardless of the source polarization. This work demonstrates the promise of the algorithm when used on data corrupted with Gaussian noise and when perfect knowledge of the tumor electrical properties is not available.

Reconstruction Quality and Spectral Content of an Electromagnetic Time-Domain Inversion Algorithm

A tomographic time-domain reconstruction algorithm for solving the inverse electromagnetic problem is described. The application we have in mind is dielectric breast cancer detection but the results are of general interest to the field of microwave tomography. Reconstructions have been made from experimental and numerically simulated data for objects of different sizes in order to investigate the relation between the spectral content of the illuminating pulse and the quality of the reconstructed image. We have found that the spectral content is crucial for a successful reconstruction. The work has further shown that when imaging objects with different scale lengths it is an advantage to use a multiple step procedure. Low frequency content in the pulse is used to image the large structures and the reconstruction process then proceed by using higher frequency data to resolve small scale lengths. Good agreement between the results obtained from experimental data and simulated data has been achieved.

Ultrasound-guided microwave imaging of breast cancer: Tissue phantom and pilot clinical experiments

Microwave imaging promises high contrast between tumor and normal breast tissues, but its spatial resolution is limited. Here, we present a multimodality approach for high-resolution microwave imaging, where microwave image reconstruction is structurally guided by ultrasound imaging. The combined imaging concept is demonstrated using tissue phantom measurements obtained from a 16 x 15 transmitter/receiver microwave imaging system and a modified B-mode ultrasound system. With the geometry of the target and background known a priori from ultrasound, successful dielectric property images are recovered using a finite element-based reconstruction algorithm. We show that a target as small as 1.2 mm in diameter can be imaged with the multimodality approach, whereas it is impossible to detect such a small-size object using microwave imaging alone. The pilot clinical studies on two cases suggest that breast tumors can be much more accurately detected by the multimodality method.

Microwave thermal imaging: initial in vivo experience with a single heating zone

The deployment of hyperthermia as a routine adjuvant to radiation or chemotherapy is limited largely by the inability to devise treatment plans which can be monitored through temperature distribution feedback during therapy. A non-invasive microwave tomographic thermal imaging system is currently being developed which has previously exhibited excellent correlation between the recovered electrical conductivity of a heated zone and its actual temperature change during phantom studies. To extend the validation of this approach in vivo, the imaging system has been re-configured for small animal experiments to operate within the bore of a CT scanner for anatomical and thermometry registration. A series of 5-7 day old pigs have been imaged during hyperthermia with a monopole antenna array submerged in a saline tank where a small plastic tube surgically inserted the length of the abdomen has been used to create a zone of heated saline at pre-selected temperatures. Tomographic microwave data over the frequency range of 300-1000 MHz of the pig abdomen in the plane perpendicular to the torso is collected at regular intervals after the tube saline temperatures have settled to the desired settings. Images are reconstructed over a range of operating frequencies. The tube location is clearly visible and the recovered saline conductivity varies linearly with the controlled temperature values. Difference images utilizing the baseline state prior to heating reinforces the linear relationship between temperature and imaged saline conductivity. Demonstration of in vivo temperature recovery and correlation with an independent monitoring device is an important milestone prior to clinical integration of this non-invasive imaging system with a thermal therapy device.

Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation

An efficient Gauss-Newton iterative imaging technique utilizing a three-dimensional (3-D) field solution coupled to a two-dimensional (2-D) parameter estimation scheme (3-D/2-D) is presented for microwave tomographic imaging in medical applications. While electromagnetic wave propagation is described fully by a 3-D vector field, a 3-D scalar model has been applied to improve the efficiency of the iterative reconstruction process with apparently limited reduction in accuracy. In addition, the image recovery has been restricted to 2-D but is generalizable to three dimensions. Image artifacts related primarily to 3-D effects are reduced when compared with results from an entirely two-dimensional inversion (2-D/2-D). Important advances in terms of improving algorithmic efficiency include use of a block solver for computing the field solutions and application of the dual mesh scheme and adjoint approach for Jacobian construction. Methods which enhance the image quality such as the log-magnitude/unwrapped phase minimization were also applied. Results obtained from synthetic measurement data show that the new 3-D/2-D algorithm consistently outperforms its 2-D/2-D counterpart in terms of reducing the effective imaging slice thickness in both permittivity and conductivity images over a range of inclusion sizes and background medium contrasts.

Pre-scaled two-parameter Gauss-Newton image reconstruction to reduce property recovery imbalance

Gauss-Newton image reconstruction in microwave imaging can be formulated in terms of a single complex quantity, the wave number squared (k2), with the understanding that the relative permittivity and conductivity images can be extracted afterwards through a simple constitutive relationship. However, this approach ignores the fact that the magnitude of the average real and imaginary components can be considerably out of balance depending on the operating frequency and tissue characteristics which can inadvertently imbalance the process in favour of one parameter over the other. In an effort to achieve property recovery which is balanced, we introduce a pre-scaling procedure at the property update stage of the reconstruction. Utilization of this concept in conjunction with our two-step regularization process for both simulation and phantom experiments demonstrates that the penalty term weighting parameters for the optimal mean-squared property errors for the two recovered distributions (relative permittivity and conductivity) together with that yielding the lowest least-squared electric field error coincide only when the scaling is applied. The scheme provides a means for simultaneous optimization of the two permittivity and conductivity images.