ИСТИНА

Проект способствует совершенствованию методов медицинского применения ультразвука в медицине, а именно неинвазивной ультразвуковой хирургии в части теоретического обеспечения для выполнения моделирования распространения сфокусированных ультразвуковых пучков в неоднородных биологических тканях при проектировании безопасных и эффективных протоколов воздействия мощным ультразвуком на организм человека.

Currently, the field of medical applications of ultrasound is actively developing, one of the areas of which is non-invasive surgery using high intensity focused ultrasound. The range of practical applications of non-invasive surgery is very wide, ranging from the destruction of tumors in soft tissues to targeted drug delivery. For the successful development of methods of ultrasonic treatment of biological tissues, it is necessary to be able to quantitatively accurately describe the spatial distributions of acoustic pressure and intensity in ultrasonic beams during the propagation of ultrasound inside the human body. In modern research for these purposes, numerical simulation methods based on various acoustic models and equations are most often used. The model should describe the physical effects associated with beam diffraction, absorption, and in many cases the effects of acoustic nonlinearity are important. Also, since the acoustic properties of different biological tissues and organs differ and are unevenly distributed in space, the acoustic model must take into account diffraction by sound velocity and density inhomogeneities. The presence of such inhomogeneities leads to a deterioration in beam focusing, which is expressed in a decrease in acoustic pressure and intensity at the focal maximum, blurring of the focal region, as well as its shift, which reduces the safety and efficiency of tratment and makes it less predictable. The most accurate theoretical models are based on the full-wave equations of acoustics, in which time is the evolutionary coordinate. For their numerical solution, finite-difference and pseudospectral approaches, as well as methods related to boundary and volume elements, have been developed. Such models make it possible to describe the propagation of acoustic waves for arbitrarily distributed inhomogeneities and automatically take into account scattering and the appearance of reflected waves. The disadvantage of such models is their extreme computational resources demand in the case of a three-dimensional geometry of the problem, which is expressed in lengthy calculations and in the requirement of a large amount of RAM. Therefore, even the solution of linear problems requires high-end multi-core servers or workstations. In the case of wave propagation modeling taking into account nonlinear effects, cluster computing is required. In view of the foregoing, one-way propagation models are of interest, in which the evolutionary coordinate is the coordinate along the axis of the acoustic beam. Such models, as a rule, are constructed either on the basis of the Westervelt equation or on the basis of the parabolic Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which are solved in retarded time coordinate system along the beam axis. Methods for solving the parabolic equation are well known, including those in smoothly inhomogeneous media. However, the paraxial approximation limits the application of this equation to collimated beams and excludes cases of strong focusing, which is often encountered in the practice of using high-power focused ultrasound. Methods for solving the diffraction part of the Westervelt equation are more complex than in the KZK equation. For example, in homogeneous or flat-layered media, the angular spectrum approach is used. A method in the space of wave vectors has also been proposed, which has recently been generalized to the case of an inhomogeneous medium. Attempts were made to generalize the angular spectrum method to take into account the inhomogeneities of the medium, which, in essence, by the construction principle of the method, were taken into account phenomenologically in the form of an amplitude-phase screen. At the same time, in the theory of wave diffraction in inhomogeneous media, there is a mathematically substantiated method based on wide-angle parabolic equations. The main idea of this method is to approximate the dispersion relation of the one-way wave equation by some functions, among which rational functions in the form of Padé approximation are most often used. This method in various versions has earned high popularity in aero- and hydroacoustics, as well as in optics. The method looks very promising, but in the field of medical ultrasound it is used much less frequently than in other areas. Here it is worth noting the important fact that in the case of solving two-dimensional or effectively two-dimensional problems, for example, for radially symmetric beams, there are no fundamental theoretical and computational difficulties and many basic problems have already been solved, because direct numerical calculation ultimately reduces to one-dimensional second-order equations, which are effectively solved by finite-difference and finite-element methods. However, when considering three-dimensional problems, in order to propagate the pressure field from one plane perpendicular to the beam axis to another, it is necessary to solve a series of two-dimensional Helmholtz-type equations, for which there is no such effective scheme of alternating directions that is used in solving the usual parabolic equation. At the moment, the theory of solving these equations is being actively developed and, for example, iterative methods of solving have been proposed. In this project, we propose to study the possibility of applying and improving methods for solving a three-dimensional wide-angle parabolic equation for the case of focusing ultrasonic beams in inhomogeneous media such as biological tissues. As one of the options, it is proposed to rely on iterative methods using the approximation of spatial derivatives with respect to transverse coordinates in Fourier space. The solution of this problem will be an important step in the development of theoretical methods for calculating ultrasonic fields applicable to a wide class of problems in the field of medical ultrasound. Firstly, this will speed up calculations in problems of linear propagation of beams in inhomogeneous media, and secondly, with sufficient performance of the algorithms, this will also allow solving nonlinear problems, since numerical algorithms for solving the nonlinear Westervelt equation are basically reduced to partitioning the pressure field into a set of Fourier harmonics in time, whose diffraction at each small step along the beam axis can be calculated independently. The novelty of the proposed approaches is related to the combination of iterative approaches and spectral and finite-difference schemes for solving the equations of the wide-angle parabolic approximation for three-dimensional beams in an inhomogeneous medium.

В результате выполнения проекта будет разработан теоретический метод на основе подхода использования широкоугольных параболических уравнений, пригодный для моделирования распространения сфокусированных ультразвуковых пучков в неоднородной среде по типуа биологической ткани. Для теоретического метода будет также построен численный алгоритм и будет исследована возможность его параллелизации для увеличения производительности расчетов и произведена оценка этой производительности на современных многоядерных процессорах. Также будет произведена оценка точности метода путем сравнения с результатами расчетов, полученных из полноволновых моделей. Полученные в проекте результаты важны для дальнейшего развития и обновления теоретических методов описания ультразвуковых пучков, т.к. в современных исследованиях все больше и больше внимания уделяется учету и анализу влиянию влияния неоднородностей среды (скорости звука и плотности) на пространственные распределения основных параметров акустического поля, а именно давления, интенсивности и тепловых источников. Поскольку современные методы медицинских исследований позволяют получать структурную информацию о каждом индивидуальном пациенте, (например, с помощью данных компьютерной томографии, на основе которых можно строить акустическую модель тела человека,) и большое внимание уделяется безопасности и эффективности воздействий на пациента, то разработка методов моделирования ультразвуковых пучков в неоднородных средах способствует общему продвижению к персонализированной медицине.