James Wiskin

Mapped continuation methods for computing all solutions to general systems of nonlinear equations

Abstract In many fields of engineering and science, use of mathematical models leads to systems of linear algebraic and nonlinear algebraic and transcendental equations. When all equations are linear, software packages such as LINPACK and YSMP can be applied to obtain the single solution. When the system contains nonlinear equations, more than one solution may exist, but until recently software packages were designed to obtain at best just one solution from a specified starting guess. If the nonlinear equations are all of polynomial form, recent software packages such as HOMPACK and CONSOL can systematically locate all solutions. The study reported here addresses the general case where the system may contain nonlinear equations with transcendental terms. By forming a fixed-point global homotopy and applying differential arclength continuation in finite mapped space, the two methods described (toroidal mapping and boomerang mapping) have located all solutions from a single starting guess for all cases studied. The two methods are illustrated for the case of an adiabatic continuous stirred-tank reactor operating in a steady-state mode with two consecutive reactions taking place, one of which is catalytic and irreversible with the other noncatalytic and reversible. All five steady-state solutions are found by each method.

Inverse scattering and refraction corrected reflection for breast cancer imaging

Reflection ultrasound (US) has been utilized as an adjunct imaging modality for over 30 years. TechniScan, Inc. has developed unique, transmission and concomitant reflection algorithms which are used to reconstruct images from data gathered during a tomographic breast scanning process called Warm Bath Ultrasound (WBU™). The transmission algorithm yields high resolution, 3D, attenuation and speed of sound (SOS) images. The reflection algorithm is based on canonical ray tracing utilizing refraction correction via the SOS and attenuation reconstructions. The refraction correction reflection algorithm allows 360 degree compounding resulting in the reflection image. The requisite data are collected when scanning the entire breast in a 33° C water bath, on average in 8 minutes. This presentation explains how the data are collected and processed by the 3D transmission and reflection imaging mode algorithms. The processing is carried out using two NVIDIA® Tesla™ GPU processors, accessing data on a 4-TeraByte RAID. The WBU™ images are displayed in a DICOM viewer that allows registration of all three modalities. Several representative cases are presented to demonstrate potential diagnostic capability including: a cyst, fibroadenoma, and a carcinoma. WBU™ images (SOS, attenuation, and reflection modalities) are shown along with their respective mammograms and standard ultrasound images. In addition, anatomical studies are shown comparing WBU™ images and MRI images of a cadaver breast. This innovative technology is designed to provide additional tools in the armamentarium for diagnosis of breast disease.

Three-dimensional nonlinear inverse scattering: Quantitative transmission algorithms, refraction corrected reflection, scanner design and clinical results

Research in quantitative whole breast ultrasound imaging has been developing rapidly. Recently we published results from 2D transmission inverse scattering algorithms, based on optimization, incorporating diffraction, refraction, and limited multiple scattering effects, using data collected from an early prototype, which showed the feasibility of high resolution quantitative imaging of the breast tissue speed and attenuation, and concomitant refraction corrected reflection imaging. However, artifact problems in speed and attenuation result from the 2D algorithms, and the data characteristics. The reflection algorithm uses the speed map to model refractive effects of rays, so these artifacts are unacceptable. The 3D inverse scattering algorithm presented here, using data from a new prototype, overcomes most of these artifacts. We then use a 3D refraction corrected 360 degree compounded reflection algorithm for high resolution speckle free reflection images. We discuss the transmission and reflection algorithms and the advanced scanner used to collect the data, as well as initial clinical results from the Mayo Clinic, Breast Cancer Imaging Center, Orange County, and the University California, San Diego. We show examples of fibroadenomas, calcifications, cancers (IDC), in dense, fatty and average breast tissue, and compare these with hand-held ultrasound, MRI and mammography, where available.

Inverse scattering from arbitrary two-dimensional objects in stratified environments via a Green’s operator

An important problem in geophysics, medical imaging, and nondestructive imaging today is the construction of a practical, accurate, and efficient means of imaging geophysical anomalies, tumours, or material defects in layered media. This paper discusses such a method. The use of a “stratified Green’s function” for the solution of the forward problem is detailed. This forward problem is then incorporated into an efficient and accurate inversion algorithm based on optimization. The method is nonperturbative, unlike diffraction tomography, which relies on linearization to make the problem tractable. In the inversion, a pair of Lippmann–Schwinger-like integral equations are solved simultaneously via the Galerkin procedure for the unknown total internal fields and speed distribution. The computational burden is high, but made manageable by utilizing BiConjugate gradients, fast fourier transforms, and “sinc” basis functions to speed up the solution of the forward problem. The size and contrasts for which the me...

Quantitative volumetric breast imaging with 3D inverse scatter computed tomography

A method was developed to map tissue properties of the entire breast including sound speed and attenuation using fully 3D nonlinear inverse-scattering tomography. Clinical measurements suggest that in breast tissue benign and cancerous lesions may be identified in part by these inherent acoustic parameters. Sound speed accuracy and linearity are very high over a wide range (1325-1700 m/sec) with ~1.5 mm resolution at 2 MHz in transmission mode. Attenuation tomograms provide image contrast over a wide range (0-4 dB/cm/MHz) and assist classification of masses. High resolution 0.6 mm volumetric reflection tomograms are acquired with bandwidth 2-8 MHz, are refraction-corrected with the transmission tissue data and are precisely registered in 3D with the transmission volumes. USCT promises an automated whole-breast scan providing a global view of the entire breast in 3D, facilitating comparison to prior exams in a reproducible geometry. Scanner design, automated operation and results of our trial with over 125 subjects with confirmed breast masses will be presented with detailed comparison to conventional sonography and MRI.

Imaging performance of Quantitative Transmission Ultrasound

Quantitative Transmission Ultrasound (QTUS) is a tomographic transmission ultrasound modality that is capable of generating 3D speed-of-sound maps of objects in the field of view. It performs this measurement by propagating a plane wave through the medium from a transmitter on one side of a water tank to a high resolution receiver on the opposite side. This information is then used via inverse scattering to compute a speed map. In addition, the presence of reflection transducers allows the creation of a high resolution, spatially compounded reflection map that is natively coregistered to the speed map. A prototype QTUS system was evaluated for measurement and geometric accuracy as well as for the ability to correctly determine speed of sound.

Imaging method utilizing attenuation and speed parameters in inverse scattering techniques

Methods for imaging the internal structures of an object using an acoustic wave field are provided. In one aspect, for example, a method of imaging internals of a physical object using acoustic waves may include transmitting an acoustic wave field toward the object, receiving a resultant acoustic wave field with a receiver, where the resultant acoustic wave field is in response to the transmitted acoustic wave field reflected from or transmitted through the object, and determining a predicted resultant acoustic wave field derived from a model of the object. The method may also include determining a residual between the predicted resultant acoustic wave field and the resultant acoustic wave field, and back propagating the residual to determine corrections to the model of the object. In another aspect, the above recited steps may be further iterated to successively refine the model of the object over a number of iterations until a predefined condition is reached.

Clinical Results with Ultrasound Computed Tomography of the Breast

Although the science and engineering of ultrasound computed tomography (USCT) has been explored for over four decades, there have been relatively few instances of a system being developed and applied to patients. Nonetheless, there have been notable results from the clinical setting, especially recently, that illustrate how a successful USCT scanner may provide significant advances to women’s health. For practical anatomical reasons, this work has almost exclusively addressed imaging of the female breast. Other quantitative ultrasound techniques have been applied to characterizing the female breast, including quantitative backscatter analysis, shear wave speed, computer-aided diagnosis, etc., but USCT is the focus of this chapter. We highlight the evolution of scanner design and image reconstruction by presenting key results from patient measurements by the major researchers in the field. There has been steady progress in electronics, parallel processors, reconstruction algorithms, understanding of the physical properties of breast tissue and a resurgence of interest in the medical community for dedicated breast ultrasound systems. It is understood today that USCT may be able to contribute in many aspects of the medical management of breast disease including detection, diagnosis and treatment of breast cancer.

Inverse Scattering Theory

This paper discuss a fully 3D nonlinear algorithm that results in a 3D quantitative estimate of breast tissue characteristics and a refraction corrected reflection algorithm (RFCR) that utilizes these estimates. The data are obtained from a specially designed clinical ultrasound breast scanner and processed on the device. We discuss the data collection process, a fast solution to the forward problem and a concomitant fast inverse scattering solution for the imaging problem. We show how the resulting 3D tissue map is used in a refraction corrected reflection algorithm.

Full inverse scattering vs. Born-like approximation for imaging in a stratified ocean

This Oceans 93 paper describes full nonlinear inversion that gives quantitative reconstructions of submerged and/or buried objects with the aid of a layered Greens function. The computational speed is preserved by use of Fast Fourier Transforms and stabilized biconjugate gradients, for both the homogeneous background and layered background case.<<ETX>>