Tomographic reconstruction of three-dimensional volumes using the distorted born iterative method

Low frequency 3D transmission ultrasound tomography: technical details and clinical implications

A novel 3D ultrasound tomographic (3D UT) method (called volography) that creates a speed of sound (SOS) map and a reflection modality that is co-registered are reviewed and shown to be artifact free even in the presence of high contrast and thus shown to be applicable for breast, orthopedic and pediatric clinical use cases. The 3D UT images are almost isotropic with mm resolution and the reflection image is compounded over 360 degrees to create sub-mm resolution in plane.

METHODS

The physics of ultrasound scattering requires 3D modeling and the concomitant high computational cost is ameliorated with a bespoke algorithm (paraxial approximation - discussed here) and Nvidia GPUs. The resulting reconstruction times are tabulated for clinical relevance. The resulting SOS map is used to create a refraction corrected reflection image at ∼3.6 MHz center frequency. The transmission data are highly redundant, collected over 360 degrees and at 2 mm levels by true matrix receiver arrays yielding 3D data. The high resolution SOS and attenuation maps and reflection images are used in a segmentation algorithm that optimally utilizes this information to segment out glandular, ductal, connective tissue, fat and skin. These volumes are used to estimate breast density, an important correlate to cancer.

RESULTS

Multiple SOS images of breast, knee and segmentations of breast glandular and ductal tissue are shown. Spearman rho is calculated between our volumetric breast density estimates and Volpara™ from mammograms, as 0.9332. Multiple timing results are shown and indicate the variability of the reconstruction times with breast size and type but are ∼30 minutes for average size breast. The timing results with the 3D algorithm indicate ∼60 minute reconstruction times for pediatrics with two Nvidia GPUs. Characteristic variations of the glandular and ductal volumes over time are shown. The SOS from QT images are compared with literature values. The results of a multi-reader multi-case (MRMC) study are shown that compares the 3D UT with full field digital mammography and resulted in an average increase in ROC AUC of 10%. Orthopedic (knee) 3D UT images compared with MRI indicate regions of zero signal in the MRI are clearly displayed in the QT image. Explicit representation of the acoustic field is shown, indicating its 3D nature. An image of in vivo breast with the chest muscle is shown and speed of sound agreement with literature values are tabulated. Reference is made to a recently published paper validating pediatric imaging.

CONCLUSIONS

The high Spearman rho indicates a monotonic (not necessarily linear) relation between our method and industry gold standard Volpara™ density. The acoustic field verifies the need for 3D modeling. The MRMC study, the orthopedic images, breast density study, and references, all indicate the clinical utility of the SOS and reflection images. The QT image of the knee shows its ability to monitor tissue the MRI cannot. The included references and images herein indicate the proof of concept for 3D UT as a viable and valuable clinical adjunct in pediatric and orthopedic situations in addition to the breast imaging.

Ultrasound Tomography

Ultrasound tomography (USCT) is a promising imaging modality, mainly aiming at early diagnosis of breast cancer. It provides three-dimensional, reproducible images of higher quality than conventional ultrasound methods and additionally offers quantitative information on tissue properties. This chapter provides an introduction to the background and history of USCT, followed by an overview of image reconstruction algorithms and system design. It concludes with a discussion of current and future applications as well as limitations and their potential solutions.

Full Wave Inversion and Inverse Scattering in Ultrasound Tomography/Volography

Ultrasound breast tomography has been around for more than 40 years. Early approaches to reconstruction focused on simple algebraic reconstructions and bent ray techniques. These approaches were not able to provide high-quality and high spatial-resolution images. The advent of inverse scattering approaches resulted in a shift in image reconstruction approaches for breast tomography and a subsequent improvement in image quality. Full wave inverse solvers were developed to improve the reconstruction times without sacrificing image quality. The development of GPUs has markedly decreased the time for reconstruction using inverse scatting approaches. The development of fully 3D image solvers and hardware capable of capturing out of plane scattering have resulted in further improvement in breast tomography. This chapter discusses the state-of-the-art in ultrasound breast tomography, its history, the theory behind inverse scattering, approximations that are included to improve convergence, 3D image reconstruction, and hardware implementation of the constructions.

Inverse scattering algorithm for profile reconstruction of a buried defect beneath a dielectric rough surface based on the domain-boundary integral hybrid method

A reconstruction algorithm for the defect profile beneath a dielectric rough surface using scattered far fields is proposed. This is a fundamental technique for optical measurements, such as diffraction tomography, for defect inspection of microfabricated devices. In general, the profile reconstruction process is considerably time consuming. We propose a domain-boundary integral hybrid method to reduce the number of operations while maintaining the rigor of scattering; the polarization properties, scattering, and multiscattering in the sample are considered. We present reconstruction simulations to validate the proposed algorithm. The reconstruction limit as well as its dependency on sample illumination and rough surface shape are also discussed.

Fast Computation of Born-Approximated Multistatic Scattering

An algorithm for the fast computation of multistatic scattering patterns, produced by low-contrast inhomogeneous volumetric objects, which satisfy the Born approximation assumption, is presented. The algorithm relies on hierarchical interpolation and aggregation of optimally sampled phase-compensated partial contributions to the scattered field integral. The Born approximation enables the efficient simultaneous computation for ranges of incident and observation angles, as well as excitation frequencies. This leads to a reduction in the asymptotic computational complexity by several polynomial orders of the problem's acoustical size. The algorithm's error controllability and favorable computation time are demonstrated via representative examples.

Multi-parameter inversion with the aid of particle velocity field reconstruction

Multi-parameter inversion for medical ultrasound leads to an improved tissue classification. In general, simultaneous reconstruction of volume density of mass and compressibility would require knowledge of the particle velocity field along with the pressure field. However, in practice the particle velocity field is not measured. Here, the authors propose a method for multi-parameter inversion where the particle velocity field is reconstructed from the measured pressure field. To this end, the measured pressure field is described using outward propagating Hankel functions. For a synthetic setup, it is shown that the reconstructed particle velocity field matches the forward modelled particle velocity field. Next, the reconstructed particle velocity field is used together with the synthetically measured pressure field to reconstruct density and compressibility profiles with the aid of contrast source inversion. Finally, comparing the reconstructed speed of sound profiles obtained via single-parameter versus multi-parameter inversion shows that multi-parameter outperforms single-parameter inversion with respect to accuracy and stability.

The convolutional interpretation of registration-based plane wave steered pulse-echo local sound speed estimators

Pulse-echo reconstruction of sound speed has long been considered a difficult problem within the domain of quantitative biomedical ultrasound. However, recent results (Jaeger 2015 Ultrasound Med. Biol. 41 235-50; Jaeger and Frenz 2015 Ultrasonics 62 299-304; Jaeger et al 2015 Phys. Med. Biol. 60 4497-515) have demonstrated that pulse-echo reconstructions of sound speed are achievable by exploiting correlations in post-beamformed data from steered, plane-wave excitations in the presence of diffuse scatterers. Despite these recent advances, a coherent theoretical imaging framework for describing the approach and results is lacking in the literature. In this work, the problem of sound speed reconstruction using steered, plane-wave excitations is reformulated as a truncated convolutional problem, and the theoretical implications of this reformulation are explored. Additionally, a matrix-free algorithm is proposed that leverages the computational and storage advantages of the fast Fourier transform (FFT) while simultaneously avoiding FFT wraparound artifacts. In particular, the storage constraints of the approach are reduced down from [Formula: see text] to [Formula: see text] over full matrix reconstruction, making this approach a better candidate for large reconstructions on clinical machines. This algorithm was then tested in the open source simulation package k-Wave to assess its robustness to modeling error and resolution reduction was demonstrated under full-wave propagation conditions relative to ideal straight-ray simulations. The method was also validated in a phantom experiment.

Tomographic density imaging using modified DF-DBIM approach

Ultrasonic computed tomography based on back scattering theory is the most powerful and accurate tool in ultrasound based imaging approaches because it is capable of providing quantitative information about the imaged target and detects very small targets. The duple-frequency distorted Born iterative method (DF-DBIM), which uses density information along with sound contrast for imaging, is a promising approach for imaging targets at the level of biological tissues. With two frequencies f₁ (low) and f₂ (high) through N f 1 and N f 2 iterations respectively, this method is used to estimate target density along with sound contrast. The implications of duple-frequency fusion for the image reconstruction quality of density information along with sound contrast based ultrasound tomography have been analyzed in this paper. In this paper, we concentrate on the selection of parameters that is supposed to be the best to improve the reconstruction quality of ultrasound tomography. When there are restraints imposed on simulated scenarios to have control of the computational cost, the iteration number N f 1 is determined resulting in giving the best performance. The DF-DBIM is only effective if there are a moderate number of iterations, transmitters and receivers. In case that the number of transducers is either too large or too small, a result of reconstruction which is better than that of the single frequency approach is not produced by the implementation of DF-DBIM. A fixed sum N iter of N f 1 and N f 2 was given, the investigation of simulation results shows that the best value of N f 1 is N iter 2 - 1 . The error, when applying this way of choosing the parameters, will be normalized with the reduction of 56.11%, compared to use single frequency as used in the conventional DBIM method. The target density along with sound contrast is used to image targets in this paper. It is a fact that low-frequency offers fine convergence, and high-frequency offers fine spatial resolution. Wherefore, this technique can effectively expand DBIM's applicability to the problem of biological tissue reconstruction. Thanks to the usage of empirical data, this work will be further developed prior to its application in reality.

Quantitative assessment of breast density using transmission ultrasound tomography

Purpose

Breast density is important in the evaluation of breast cancer risk. At present, breast density is evaluated using two‐dimensional projections from mammography with or without tomosynthesis using either (a) subjective assessment or (b) a computer‐aided approach. The purpose of this work is twofold: (a) to establish an algorithm for quantitative assessment of breast density using quantitative three‐dimensional transmission ultrasound imaging; and (b) to determine how these quantitative assessments compare with both subjective and objective mammographic assessments of breast density.

Methods

We described and verified a threshold‐based segmentation algorithm to give a quantitative breast density (QBD) on ultrasound tomography images of phantoms of known geometric forms. We also used the algorithm and transmission ultrasound tomography to quantitatively determine breast density by separating fibroglandular tissue from fat and skin, based on imaged, quantitative tissue characteristics, and compared the quantitative tomography segmentation results with subjective and objective mammographic assessments.

Results

Quantitative breast density (QBD) measured in phantoms demonstrates high quantitative accuracy with respect to geometric volumes with average difference of less than 0.1% of the total phantom volumes. There is a strong correlation between QBD and both subjective mammographic assessments of Breast Imaging ‐ Reporting and Data System (BI‐RADS) breast composition categories and Volpara density scores — the Spearman correlation coefficients for the two comparisons were calculated to be 0.90 (95% CI: 0.71–0.96) and 0.96 (95% CI: 0.92–0.98), respectively.

Conclusions

The calculation of breast density using ultrasound tomography and the described segmentation algorithm is quantitatively accurate in phantoms and highly correlated with both subjective and Food and Drug Administration (FDA)‐cleared objective assessments of breast density.

Electromagnetic Imaging of Dielectric Objects Using a Multi-Directional Search-Based Simulated Annealing

In this paper, we introduce a global optimization method that is a novel combination of the simulated annealing method and the multi-directional search algorithm. We demonstrate the use of the algorithm for a microwave-imaging system to obtain the electrical properties of objects. The proposed global optimizer significantly improves the performance and speed of the simulated annealing method by utilizing a nonlinear simplex search, starting from an initial guess, and taking effective steps in obtaining the global solution of the minimization problem. Due to the efficient performance of the proposed global optimization method, we are able to obtain the shape, location, and material properties of the target without considering any a priori information about them. The accuracy and applicability of the proposed imaging method is demonstrated with some numerical results in which two-dimensional images of multiple objects are successfully reconstructed.

Quantitative transmission ultrasound tomography: Imaging and performance characteristics

PURPOSE

Quantitative Transmission (QT) ultrasound has shown promise as a breast imaging modality. This study characterizes the performance of the latest generation of QT ultrasound scanners: QT Scanner 2000.

METHODS

The scanner consists of a 2048-element ultrasound receiver array for transmission imaging and three transceivers for reflection imaging. Custom fabricated phantoms were used to quantify the imaging performance parameters. The specific performance parameters that have been characterized are spatial resolution (as point spread function), linear measurement accuracy, contrast to noise ratio, and image uniformity, in both transmission and reflection imaging modalities.

RESULTS

The intrinsic in-plane resolution was measured to be better than 1.5 mm and 1.0 mm for transmission and reflection modalities respectively. The linear measurement accuracy was measured to be, on average, approximately 1% for both the modalities. Speed of sound image uniformity and measurement accuracy were calculated to be 99.5% and <0.2% respectively. Contrast to noise ratio (CNR) measurements vary as a function of object size.

CONCLUSIONS

The results show an improvement in the imaging performance of the system in comparison to earlier ultrasound tomography systems, which are applicable to clinical applications of the system, such as breast imaging.

3-D Nonlinear Acoustic Inverse Scattering: Algorithm and Quantitative Results

We describe a novel 3-D ultrasound technology, the quantitative transmission ultrasound system and algorithm to image a pendent breast in a water bath. Quantitative accuracy is verified using phantoms. Morphological accuracy is verified using cadaveric breast and in vivo images, and spatial resolution is estimated. This paper generalizes an earlier 2-D algorithm to a full 3-D inversion algorithm and shows the importance of such a 3-D algorithm for artifact suppression as compared with the 2-D algorithm. The resultant high-resolution ultrasound images, along with quantitative information regarding tissue speed of sound/stiffness, provide a more accurate depiction of the breast anatomy and lesions, contributing to improved breast care.

Deterministic compressive sampling for high-quality image reconstruction of ultrasound tomography

Background

A well-known diagnostic imaging modality, termed ultrasound tomography, was quickly developed for the detection of very small tumors whose sizes are smaller than the wavelength of the incident pressure wave without ionizing radiation, compared to the current gold-standard X-ray mammography. Based on inverse scattering technique, ultrasound tomography uses some material properties such as sound contrast or attenuation to detect small targets. The Distorted Born Iterative Method (DBIM) based on first-order Born approximation is an efficient diffraction tomography approach. One of the challenges for a high quality reconstruction is to obtain many measurements from the number of transmitters and receivers. Given the fact that biomedical images are often sparse, the compressed sensing (CS) technique could be therefore effectively applied to ultrasound tomography by reducing the number of transmitters and receivers, while maintaining a high quality of image reconstruction.

Methods

There are currently several work on CS that dispose randomly distributed locations for the measurement system. However, this random configuration is relatively difficult to implement in practice. Instead of it, we should adopt a methodology that helps determine the locations of measurement devices in a deterministic way. For this, we develop the novel DCS-DBIM algorithm that is highly applicable in practice. Inspired of the exploitation of the deterministic compressed sensing technique (DCS) introduced by the authors few years ago with the image reconstruction process implemented using l₁ regularization.

Results

Simulation results of the proposed approach have demonstrated its high performance, with the normalized error approximately 90% reduced, compared to the conventional approach, this new approach can save half of number of measurements and only uses two iterations. Universal image quality index is also evaluated in order to prove the efficiency of the proposed approach.

Conclusions

Numerical simulation results indicate that CS and DCS techniques offer equivalent image reconstruction quality with simpler practical implementation. It would be a very promising approach in practical applications of modern biomedical imaging technology.

A computer simulation study of soft tissue characterization using low-frequency ultrasonic tomography

We investigate the potential of using ultrasonic diffraction tomography technique for characterization of biological tissues. Unlike most of other studies where ultrasonic tomography operates at frequencies higher than 1 MHz, low-frequency tomography uses lower frequencies on the order of 0.3-0.5 MHz. Such a choice is due to low attenuation at these frequencies, resulting in higher precision of input data. In this paper we explore transmission and reflection schemes for both 2D (layer-by-layer) and 3D tomography. We treat inverse tomography problems as coefficient inverse problems for the wave equation. The time-domain algorithms employed for solving the inverse problem of low-frequency tomography focus on the use of GPU clusters. The results obtained show that a spatial resolution of about 2-3mm can be achieved when operating at the wavelength of about 5mm even using a stationary 3D scheme with a few fixed sources and no rotating elements. The study primarily focuses on determining the performance limits of ultrasonic tomography devices currently designed for breast cancer diagnosis.

Fast inverse scattering solutions using the distorted Born iterative method and the multilevel fast multipole algorithm

The distorted Born iterative method (DBIM) computes iterative solutions to nonlinear inverse scattering problems through successive linear approximations. By decomposing the scattered field into a superposition of scattering by an inhomogeneous background and by a material perturbation, large or high-contrast variations in medium properties can be imaged through iterations that are each subject to the distorted Born approximation. However, the need to repeatedly compute forward solutions still imposes a very heavy computational burden. To ameliorate this problem, the multilevel fast multipole algorithm (MLFMA) has been applied as a forward solver within the DBIM. The MLFMA computes forward solutions in linear time for volumetric scatterers. The typically regular distribution and shape of scattering elements in the inverse scattering problem allow the method to take advantage of data redundancy and reduce the computational demands of the normally expensive MLFMA setup. Additional benefits are gained by employing Kaczmarz-like iterations, where partial measurements are used to accelerate convergence. Numerical results demonstrate both the efficiency of the forward solver and the successful application of the inverse method to imaging problems with dimensions in the neighborhood of ten wavelengths.

Acoustic scattering by arbitrary distributions of disjoint, homogeneous cylinders or spheres

A T-matrix formulation is presented to compute acoustic scattering from arbitrary, disjoint distributions of cylinders or spheres, each with arbitrary, uniform acoustic properties. The generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers. By employing field expansions based on orthogonal harmonic functions, continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal, analytic expressions to eliminate the need for integral equations. The effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner-object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant. Numerical results establish the validity and efficiency of the outer iteration procedure for nested objects. Two- and three-dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two-dimensional approximations to three-dimensional scattering of elevation-focused beams.