A full vectorial contrast source inversion scheme for three-dimensional acoustic imaging of both compressibility and density profiles

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.

Study on 3-D Acoustic Imaging for Human Thorax Based on Contrast Source Inversion

In this article, we study a 3-D acoustic imaging algorithm that can reconstruct compressibility, attenuation, and density simultaneously based on the contrast source inversion (CSI) method. This is a nonlinear and ill-posed inverse problem. To deal with the nonlinearity, we introduce two asymmetrical contrast sources that are functions of the contrasts and the total field. In this case, the scattered field and the total field are linear with the two contrast sources, and the two contrast sources are also linear with the two contrasts; thus, the nonlinearity is partially alleviated. To mitigate the ill-posedness of this inverse problem, we apply a multifrequency, multitransmitter, and multireceiver setting. Besides, to ensure the robustness of the algorithm, two multiplicative regularization terms are introduced as additional constraints. The reconstruction of those acoustic parameters can be achieved by alternately updating the contrast sources and the contrasts from the knowledge of the pressure field. Numerical studies show good reconstruction of compressibility, attenuation, and density of the synthetic thorax model, which validates the feasibility of imaging human thorax using low-frequency ultrasound.

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.

Using Filter Factors for Regularization in Ultrasound Tomography

The motivation for ultrasound tomography is the location and identification of malignant human breast tissues for the purpose of detecting breast cancer. Although mammography is widely used for breast cancer detection, it has a high false positive rate and does not always accurately separate malignant tissue from benign tissue. Ultrasound tomography is used to compensate for these shortcomings. The computational model for ultrasound tomography is based upon solving an inverse scattering problem by finding the approximate total field and unknown scattering function using an iterative method. The principal computational problem involved is the solution of an ill-conditioned linear system, $Xy\approx b$, arising from an ill-posed problem written as an integral equation. In this paper, we explore the DSVD and the DGSVD regularization methods to solve the inverse scattering problem. The DGSVD algorithm gives better results than the DSVD algorithm when we introduce noise in either one or both sides of the linear system $Xy\approx b$.

Feasibility study of acoustic imaging for human thorax using an acoustic contrast source inversion algorithm

In this work, an acoustic imaging method based on contrast source inversion and its feasibility in quantitatively reconstructing compressibility, attenuation, and density of human thorax is studied. In the acoustic wave equation, the inhomogeneity in density makes the relationship between the contrasts and the total pressure highly nonlinear. To reduce this nonlinearity, two contrast sources are introduced to ensure symmetry in the equation, such that the inverse problem can be solved efficiently by alternately updating two contrast sources and two contrasts. Moreover, to improve the stability of the algorithm, the multiplicative regularization scheme with two additive regularization factors is applied. Using this algorithm, acoustic parameters of human thorax from low frequency ultrasound measurement are reconstructed. Numerical results show that the acoustic parameters of human thorax can be properly reconstructed at frequency of tens of kHz using this algorithm.

First-arrival traveltime sound speed inversion with a priori information

PURPOSE

A first-arrival travel-time sound speed algorithm presented by Tarantola [Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, Philadelphia, PA, 2005)] is adapted to the medical ultrasonics setting. Through specification of a covariance matrix for the object model, the algorithm allows for natural inclusion of physical a priori information of the object. The algorithm's ability to accurately and robustly reconstruct a complex sound speed distribution is demonstrated on simulation and experimental data using a limited aperture.

METHODS

The algorithm is first demonstrated generally in simulation with a numerical breast phantom imaged in different geometries. As this work is motivated by the authors' limited aperture dual sided ultrasound breast imaging system, experimental data are acquired with a Verasonics system with dual, 128 element, linear L7-4 arrays. The transducers are automatically calibrated for usage in the eikonal forward model.A priori information such as knowledge of correlated regions within the object is obtained via segmentation of B-mode images generated from synthetic aperture imaging.

RESULTS

As one illustration of the algorithm's facility for inclusion ofa priori information, physically grounded regularization is demonstrated in simulation. The algorithm's practicality is then demonstrated through experimental realization in limited aperture cases. Reconstructions of sound speed distributions of various complexity are improved through inclusion of a priori information. The sound speed maps are generally reconstructed with accuracy within a few m/s.

CONCLUSIONS

This paper demonstrates the ability to form sound speed images using two opposed commercial linear arrays to mimic ultrasound image acquisition in the compressed mammographic geometry. The ability to create reasonably good speed of sound images in the compressed mammographic geometry allows images to be readily coregistered to tomosynthesis image volumes for breast cancer detection and characterization studies.

Comparing different ultrasound imaging methods for breast cancer detection

Ultrasound is frequently used to evaluate suspicious masses in breasts. These evaluations could be improved by taking advantage of advanced imaging algorithms, which become feasible for low frequencies if accurate knowledge about the phase and amplitude of the wave field illuminating the volume of interest is available. In this study, we compare five imaging and inversion methods: time-of-flight tomography, synthetic aperture focusing technique, backpropagation, Born inversion, and contrast source inversion. All methods are tested on the same full-wave synthetic data representing a 2-D scan using a circular array enclosing a cancerous breast submerged in water. Of the tested methods, only contrast source inversion yielded an accurate reconstruction of the speed-ofsound profile of the tumor and its surroundings, because only this method takes effects such as multiple scattering, refraction, and diffraction into account.

Iterative reconstruction of the transducer surface velocity

Ultrasound arrays used for medical imaging consist of many elements placed closely together. Ideally, each element vibrates independently. However, because of mechanical coupling, crosstalk between neighboring elements may occur. To quantify the amount of crosstalk, the transducer velocity distribution should be measured. In this work, a method is presented to reconstruct the velocity distribution from far-field pressure field measurements acquired over an arbitrary surface. The distribution is retrieved from the measurements by solving an integral equation, derived from the Rayleigh integral of the first kind, using a conjugate gradient inversion scheme. This approach has the advantages that it allows for arbitrary transducer and pressure field measurement geometries, as well as the application of regularization techniques. Numerical experiments show that measuring the pressure field along a hemisphere enclosing the transducer yields significantly more accurate reconstructions than measuring along a parallel plane. In addition, it is shown that an increase in accuracy is achieved when the assumption is made that all points on the transducer surface vibrate in phase. Finally, the method has been tested on an actual transducer with an active element of 700 × 200 μm which operates at a center frequency of 12.2 MHz. For this transducer, the velocity distribution has been reconstructed accurately to within 50 μm precision from pressure measurements at a distance of 1.98 mm (=16λ0) using a 200-μm-diameter needle hydrophone.

An acoustic inverse scattering problem for spheres with radially inhomogeneous compressibility

An inverse acoustic scattering problem the main aim of which is to reconstruct the one-dimensional variation of the acoustical parameters of a spherical object is investigated. The problem is first formulated conventionally through a coupled system of integral equations, and then this system is reduced to one-dimensional form by using the orthogonality properties of spherical harmonics. The inverse problem is solved in an iterative fashion via classical Newton algorithm. Some numerical simulations are carried out to test the feasibility of the method as well as to see the effects of some parameters on the solution. It is shown that the method is very effective for the profiles having smooth variations provided that an appropriate initial guess is chosen. However, some of the classical disadvantages of the Newton type algorithms are also observed in numerical experiments which may limit the applicability of the method to a certain extent.

Multi-parameter acoustic imaging of uniform objects in inhomogeneous soft tissue

The problem studied in this paper is ultrasound image reconstruction from frequency-domain measurements of the scattered field from an object with contrast in attenuation and sound speed. The case in which the object has uniform but unknown contrast in these properties relative to the background is considered. Background clutter is taken into account in a physically realistic manner by considering an exact scattering model for randomly located small scatterers that vary in sound speed. The resulting statistical characteristics of the interference are incorporated into the imaging solution, which includes application of a total-variation minimization-based approach in which the relative effect of perturbation in sound speed to attenuation is included as a parameter. Convex optimization methods provide the basis for the reconstruction algorithm. Numerical data for inversion examples are generated by solving the discretized Lippman-Schwinger equation for the object and speckle-forming scatterers in the background. A statistical model based on the Born approximation is used for reconstruction of the object profile. Results are presented for a two-dimensional problem in terms of classification performance and compared with minimum-l2-norm reconstruction. Classification using the proposed method is shown to be robust down to a signal-to-clutter ratio of less than 1 dB.

Density imaging using a multiple-frequency DBIM approach

Current inverse scattering methods for quantitative density imaging have limitations that keep them from practical experimental implementations. In this work, an improved approach, termed the multiple-frequency distorted Born iterative method (MF-DBIM) algorithm, was developed for imaging density variations. The MF-DBIM approach consists of inverting the wave equation by solving for a single function that depends on both sound speed and density variations at multiple frequencies. Density information was isolated by using a linear combination of the reconstructed single-frequency profiles. Reconstructions of targets using MF-DBIM from simulated data were compared with reconstructions using methods currently available in the literature, i.e., the dual-frequency DBIM (DF-DBIM) and T-matrix approaches. Useful density reconstructions, i.e., root mean square errors (RMSEs) less than 30%, were obtained with MF-DBIM even with 2% Gaussian noise in the simulated data and using frequency ranges spanning less than an order of magnitude. Therefore, the MFDBIM approach outperformed both the DF-DBIM method (which has problems converging with noise even an order of magnitude smaller) and the T-matrix method (which requires a ka factor close to unity to achieve convergence). However, the convergence of all the density imaging algorithms was compromised when imaging targets with object functions exhibiting high spatial frequency content.

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

Although real imaging problems involve objects that have variations in three dimensions, a majority of work examining inverse scattering methods for ultrasonic tomography considers 2-D imaging problems. Therefore, the study of 3-D inverse scattering methods is necessary for future applications of ultrasonic tomography. In this work, 3-D reconstructions using different arrays of rectangular elements focused on elevation were studied when reconstructing spherical imaging targets by producing a series of 2-D image slices using the 2-D distorted Born iterative method (DBIM). The effects of focal number f/#, speed of sound contrast c, and scatterer size were considered. For comparison, the 3-D wave equation was also inverted using point-like transducers to produce fully 3-D DBIM image reconstructions. In 2-D slicing, blurring in the vertical direction was highly correlated with the transmit/receive elevation point-spread function of the transducers for low c. The eventual appearance of overshoot artifacts in the vertical direction were observed with increasing c. These diffraction-related artifacts were less severe for smaller focal number values and larger spherical target sizes. When using 3-D DBIM, the overshoot artifacts were not observed and spatial resolution was improved. However, results indicate that array configuration in 3-D reconstructions is important for good image reconstruction. Practical arrays were designed and assessed for image reconstruction using 3-D DBIM.

Fast computation of the acoustic field for ultrasound elements

A fast method for computing the acoustic field of ultrasound transducers is presented with application to rectangular elements that are cylindrically focused. No closed-form solutions exist for this case but several numerical techniques have been described in the ultrasound imaging literature. Our motivation is the rapid calculation of imaging kernels for physics-based diagnostic imaging for which current methods are too computationally intensive. Here, the surface integral defining the acoustic field from a baffled piston is converted to a 3-D spatial convolution of the element surface and the Green's function. A 3-D version of the overlap-save method from digital signal processing is employed to obtain a fast computational algorithm based on spatial Fourier transforms. Further efficiency is gained by using a separable approximation to the Green's function through singular value decomposition and increasing the effective sampling rate by polyphase filtering. The tradeoff between accuracy and spatial sampling rate is explored to determine appropriate parameters for a specific transducer. Comparisons with standard tools such as Field II are presented, where nearly 2 orders of magnitude improvement in computation speed is observed for similar accuracy.

Density imaging using inverse scattering

Inverse scattering is considered one of the most robust and accurate ultrasonic tomography methods. Most inverse scattering formulations neglect density changes in order to reconstruct sound speed and acoustic attenuation. Some studies available in literature suggest that density distributions can also be recovered using inverse scattering formulations. Two classes of algorithms have been identified. (1) The separation of sound speed and density contributions from reconstructions using constant density inverse scattering algorithms at multiple frequencies. (2) The inversion of the full wave equation including density changes. In this work, the performance of a representative algorithm for each class has been studied for the reconstruction of circular cylinders: the dual frequency distorted Born iterative method (DF-DBIM) and the T-matrix formulation. Root mean square error values lower than 30% were obtained with both algorithms when reconstructing cylinders up to eight wavelengths in diameter with moderate density changes. However, in order to provide accurate reconstructions the DF-DBIM and T-matrix method required very high signal-to-noise ratios and significantly large bandwidths, respectively. These limitations are discussed in the context of practical experimental implementations.

Shape-based ultrasound tomography using a Born model with application to high intensity focused ultrasound therapy

A shaped-based ultrasound tomography method is proposed to reconstruct ellipsoidal objects using a linearized scattering model. The method is motivated by the desire to detect the presence of lesions created by high intensity focused ultrasound (HIFU) in applications of cancer therapy. The computational size and limited view nature of the relevant three-dimensional inverse problem renders impractical the use of traditional pixel-based reconstruction methods. However, by employing a shape-based parametrization it is only necessary to estimate a small number of unknowns describing the geometry of the lesion, in this paper assumed to be ellipsoidal. The details of the shape-based nonlinear inversion method are provided. Results obtained from a commercial ultrasound scanner and a tissue phantom containing a HIFU-like lesion demonstrate the feasibility of the approach where a 20 mm x 5 mm x 6 mm ellipsoidal inclusion was detected with an accuracy of around 5%.