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2. Wave Tomography
Efficient iterative methods are proposed for solving the inverse problem of wave tomography.
The inverse problems are considered as coefficient inverse problems for second-order differential equation of hyperbolic type.
Mathematical models describe diffraction, refraction, multiple scattering and attenuation.
Mathematical task is inverse non-linear ill-posed problem.
Efficient methods have been developed for solving 2-D inverse tomography problems (layer-by-layer diagnostics of a 3D object).
Efficient methods have been developed for solving 3-D inverse tomography problems. The methods are based on direct computation of the gradient of the residual functional by solving the conjugate problem for the wave equation.
The results of our study can be used in such fields as ultrasonic diagnosis, non-destructive testing in industry,
civil engineering, electromagnetic sounding of the Earth's subsurface layers, and so on.
In medicine the currently developed ultrasound tomography devices can be an alternative to x-ray tomography and MRT. The primary task of ultrasound tomography is to address the problems of differential diagnosis of breast cancer — one of the most pressing issues of modern health care. Ultrasonic tomography devices can solve this problem by making it possible to perform regular examinations using safe, non-ionizing radiation.
From a medical viewpoint, diagnostic facilities for the differential cancer diagnosis should have a resolution of 3 mm or better.
Common medical devices for ultrasound examinations usually employ the reflection-based scheme.
In these cases we usually cannot sound the object from all directions, and both the detectors and the sources are located on the same side of the domain studied, or even in the same plane. Ultrasonic tomography schemes allow to use both reflected and transmitted data. The methods of solving inverse problems as coefficient inverse problems can provide information about the internal structure of the object even in the case of limited data tomography. In application to breast cancer diagnostics, we are not able to place sources and detectors on the thorax side.
Currently, the development of ultrasonic tomography devices is at the stage of prototyping. One of the main challenges faced by ultrasonic tomography is the development of mathematical methods for solving inverse problems. Attempts to use the well-established ray transmission tomography scheme appear quite natural. Unlike X-ray tomography, where rays have the form of straight-line segments, rays in ultrasonic tomography are bent by refraction. In this formulation the inverse problem is nonlinear, and iterative methods are needed to solve it.
Supercomputers are needed to address such inverse problems in terms of the wave model described by second-order
hyperbolic equations. The algorithms developed in this study are easily scalable
on supercomputers, including GPU, running up to several tens of thousands of processes in parallel.
Low frequency ultrasonic tomography method has been developed. In this case the wavelength about 5mm can be used in ultrasonic tomography devices. The frequencies corresponding to this wavelength are several times lower than those used in existing prototypes. It is important because attenuation in soft tissues is strongly dependent on frequency: the higher the frequency the greater is attenuation.
Recent results are published in the following papers:
Goncharsky AV, Romanov SY. Supercomputer technologies in inverse problems of ultrasound tomography. Inverse Probl. 2013;29:075004.
Goncharsky AV, Romanov SY, Seryozhnikov SY. Inverse problems of 3D ultrasonic tomography with complete and incomplete range data. Wave motion 2014;51:389-404.
Inverse problems of ultrasound tomography in models with attenuation. Phys Med Biol. 2014 Apr 21;59(8):1979-2004. doi: 10.1088/0031-9155/59/8/1979. Epub 2014 Apr 2.
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