David T. Borup

Use of the Finite-Difference Time-Domain Method in Calculating EM Absorption in Human Tissues

Although there are acceptable methods for calculating whole body electromagnetic absorption, no completely acceptable method for calculating the local specific absorption rate (SAR) at points within the body has been developed. Frequency domain methods, such as the method of moments (MoM) have achieved some success; however, MoM requires computer storage on the order of (3N) 2 and computation time on the order of (3N) 3 where N is the number of cells. The finite-difference time-domain (FDTD) method has been employed extensively in calculating the scattering of metallic objects, and recently is seeing some use in calculating the interaction of EM fields with complex, lossy dielectric bodies. Since the FDTD method has storage and time requirements proportional to N, it presents an attractive alternative to calculating SAR distribution in large bodies. This paper describes the FDTD method and evaluates it by comparing its results to analytic solutions in two and three dimensions. The utility of the FDTD method is demonstrated by a 3D scan of the human torso. The results obtained demonstrate that the FDTD method is capable of calculating internal SAR distribution with acceptable accuracy. With the availability of supercomputers, such as the CRAY II, the calculation of SAR distribution in a man model of 50 000 cells (1.27 cm per cell) appears to be feasible.

Fast-Fourier-Transform Method for Calculation of SAR Distributions in Finely Discretized Inhomogeneous Models of Biological Bodies

The paper describes a novel iterative approach for calculations of specific absorption rate (SAR) distributions in arbitrary, lossy, dielectric bodies. To date, the method has been used for 2-D problems where its accuracy has been confirmed by comparison with the analytic solutions for homogeneous and layered, circular, cylindrical bodies. With computation times that are proportional to N log/sub 2/N rather than N/sup 2/ to N/sup 3/ for the method of moments, the present approach should be extendable to 3-D bodies with N= 10/sup 4/to 10/sup 5/ cells allowing, thereby, details of SAR distributions that are needed for EM hyperthermia, as well as for assessing biological effects.

Comparison of the FFT Conjugate Gradient Method and the Finite-Difference Time-Domain Method for the 2-D Absorption Problem

The need for high-resolution distributive dosimetry demands a numerical method capable of handling finely discretized, arbtrarily inhomogeneous models of biological bodies. At present, two of the most promising methods in terms of numerical efficiency are the fast-Fourier-transform conjugate gradient method (FFT-CGM) and the finite-difference time-domain (FD-TD) method. In this paper, these two methods are compared with respect to their ability to solve the 2-D Iossy dielectric cylinder problem for both the TM and TE incident polarizations. Substantial errors are found in the FFT-CGM solutions for the TE case. The source of these errors is explained and a modified method is developed which, although inefficient, alleviates the problem and illuminates the difficulties encountered in applying the pulse-basis method of moments to biological problems. In contrast, the FD-TD method is found to yield excellent solutions for both polarizations. This, coupled with the numerical efficiency of the FD-TD method, suggests that it is superior to the FFT-CGM for biological problems.

Inverse electromagnetic scattering models for sea ice

Inverse scattering algorithms for reconstructing the physical properties of sea ice from scattered electromagnetic field data are presented. The development of these algorithms has advanced the theory of remote sensing, particularly in the microwave region, and has the potential to form the basis for a new generation of techniques for recovering sea ice properties, such as ice thickness, a parameter of geophysical and climatological importance. Moreover, the analysis underlying the algorithms has led to significant advances in the mathematical theory of inverse problems. In particular, the principal results include the following. (1) Inverse algorithms for reconstructing the complex permittivity in the Helmholtz equation in one and higher dimensions, based on layer stripping and nonlinear optimization, have been obtained and successfully applied to a (lossless) laboratory system. In one dimension, causality has been imposed to obtain stability of the solution and layer thicknesses can be obtained from the recovered dielectric profile, or directly from the reflection data through a nonlinear generalization of the Paley-Wiener theorem in Fourier analysis. (2) When the wavelength is much larger than the microstructural scale, the above algorithms reconstruct a profile of the effective complex permittivity of the sea ice, a composite of pure ice with random brine and air inclusions. A theory of inverse homogenization has been developed, which in this quasistatic regime, further inverts the reconstructed permittivities for microstructural information beyond the resolution of the wave. Rigorous bounds on brine volume and inclusion separation for a given value of the effective complex permittivity have been obtained as well as an accurate algorithm for reconstructing the brine volume from a set of values. (3) Inverse algorithms designed to recover sea ice thickness have been developed. A coupled radiative transfer-thermodynamic sea ice inverse model has accurately reconstructed the growth of a thin, artificial sea ice sheet from time-series electromagnetic scattering data.

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.

Reconstruction of Elasticity and Attenuation Maps in Shear Wave Imaging: An Inverse Approach

Acoustic shear waves of low frequency can be detected and measured using a phase contrast based magnetic resonance imaging technique called MR Elastography or phase measurement based ultrasound techniques. Spatio-temporal variations of displacements caused by the propagating waves can be used to estimate local values of the elasticity of the object being imaged. The currently employed technique for estimating the elasticity from the wave displacement maps, the local frequency estimator (LFE), has fundamental resolution limits and also has problems with shadowing and other refraction-related artifacts. These problems can be overcome with an inverse approach using Green’s function integrals which directly solve the wave equation problem for the propagating wave. The complete measurements of wave displacements as a function of space and time over the object of interest obtained by the above techniques permit an iterative approach to inversion of the wave equation to obtain elasticity and attenuation maps.

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.