A. K. Khe

Propagation of nonlinear perturbations in a quasineutral collisionless plasma

For the nonlinear kinetic equation describing the one-dimensional motion of a quasineutral collisionless plasma, perturbation velocities are determined and conditions of generalized hyperbolicity are formulated. Exact (in particular, periodical) solutions of the model are constructed and interpreted physically for the class of traveling waves. Differential conservation laws approximating the basic integrodifferential equation are proposed. These laws are used to perform numerical calculations of wave propagation, which show the possibility of turnover of the kinetic distribution function.

Hydrodynamic calculations for embolization strategy of cerebral arteriovenous malformations with fistulous component

На локальной математической модели церебральной артериовенозной мальформации (АВМ), состоящей из набора трубок различного диаметра, рассматриваются различные сценарии ее эмболизации. Показано, что наименьший перепад давления на входе в АВМ достигается в случае эмболизации на первом этапе наиболее широкой фистулы, затем крупного и мелкого компартментов мальформации. Проводится сравнение модельных расчетов с результатами эмболизации АВМ у 15 пациентов.

Measurement of viscous flow velocity and flow visualization using two magnetic resonance imagers

The accuracies of measuring the velocity field using clinical and research magnetic resonance imagers are compared. The flow velocity of a fluid simulating blood in a carotid artery model connected to a programmable pump was measured. Using phase-contrast magnetic resonance tomography, the velocity distributions in the carotid artery model were obtained and compared with the analytical solution for viscous liquid flow in a cylindrical tube (Poiseuille flow). It is found that the accuracy of the velocity measurement does not depend on the field induction and spatial resolution of the imagers.

The analysis of solutions behaviour of Van der Pol Duffing equation describing local brain hemodynamics

This article proposes the generalized model of Van der Pol — Duffing equation for describing the relaxation oscillations in local brain hemodynamics. This equation connects the velocity and pressure of blood flow in cerebral vessels. The equation is individual for each patient, since the coefficients are unique. Each set of coefficients is built based on clinical data obtained during neurosurgical operation in Siberian Federal Biomedical Research Center named after Academician E. N. Meshalkin. The equation has solutions of different structure defined by the coefficients and right side. We investigate the equations for different patients considering peculiarities of their vessel systems. The properties of approximate analytical solutions are studied. Amplitude-frequency and phase-frequency characteristics are built for the small-dimensional solution approximations.

Monitoring of hemodynamics of brain vessels

This paper describes the results of intraoperational monitoring of hemodynamic parameters (velocity and pressure) in brain vessels, carried out within the framework of 50 neurosurgical operations using a Volcano ComboMap instrumental measuring system. It is established that the introduced parameter of specific load used during the neurosurgical operations is a significant parameter for the success of the operation.

Using non-linear analogue of Nyquist diagrams for analysis of the equation describing the hemodynamics in blood vessels near pathologies

This article considers method of describing the behaviour of hemodynamic parameters near vascular pathologies. We study the influence of arterial aneurysms and arteriovenous malformations on the vascular system. The proposed method involves using generalized model of Van der Pol-Duffing to find out the characteristic behaviour of blood flow parameters. These parameters are blood velocity and pressure in the vessel. The velocity and pressure are obtained during the neurosurgery measurements. It is noted that substituting velocity on the right side of the equation gives good pressure approximation. Thus, the model reproduces clinical data well enough. In regard to the right side of the equation, it means external impact on the system. The harmonic functions with various frequencies and amplitudes are substituted on the right side of the equation to investigate its properties. Besides, variation of the right side parameters provides additional information about pressure. Non-linear analogue of Nyquist diagrams is used to find out how the properties of solution depend on the parameter values. We have analysed 60 cases with aneurysms and 14 cases with arteriovenous malformations. It is shown that the diagrams are divided into classes. Also, the classes are replaced by another one in the definite order with increasing of the right side amplitude.

Differential properties of Van der Pol — Duffing mathematical model of cerebrovascular haemodynamics based on clinical measurements

The present paper discusses the method of identification (diseased/healthy) human cerebral vessels by using of mathematical model. Human cerebral circulation as a single tuned circuit, which consists of blood flow, elastic vessels and elastic brain gel tissue is under consideration. Non linear Van der Pol-Duffing equation is assumed as mathematical model of cerebrovascular circulation. Hypothesis of vascular pathology existence in some position of blood vessel, based on mathematical model properties for this position is formulated. Good reliability of hypothesis is proved statistically for 7 patients with arterial aneurysms.

Construction and investigation of 3D vessels net of the brain according to MRI data using the method of variation of scanning plane

The blood realizes the transport of substances, which are necessary for livelihoods, throughout the body. The assumption about the relationship genotype and structure of vasculature (in particular of brain) is natural. In the paper we consider models of vessel net for two genetic lines of laboratory mice. Vascular net obtained as a result of preprocessing MRI data. MRI scanning is realized using the method of variation of slope of scanning plane, i.e. by several sets of parallel planes specified by different normal vectors. The following special processing allowed to construct models of vessel nets without fragmentation. The purpose of the work is to compare the vascular network models of two different genetic lines of laboratory mice.

Reconstruction of unbroken vasculature of mouse by varying the slope of the scan plane in MRI

Reconstruction of vascular net of small laboratory animals from MRI data is associated with some problems. This paper proposes a method of MRI data processing which allows to eliminate the fragmentation of reconstructed vascular net. Problem of vessels fragmentation occurs in the case when vessels are parallel to the scanning plane. Our approach is based on multiple scanning, object under consideration is probed by several sets of parallel planes. The algorithm is applied to real MRI data of small laboratory animals and shows good results.

Personalized mathematical modeling of cerebral arterial aneurysms

In this work we describe a mathematical and computational framework for personalized modeling of the cerebral hemodynamics in the presence of arterial aneurysms. As an example we study nonlocal hydrodynamic properties of cerebral arterial aneurysms. Using numerical simulations we determine the area of influence of the aneurysm on the surrounding blood flow, the changes induced by increase and decrease of the pressure, and we study the flow features in the case of multiple aneurysms. For our computations we use personalized clinical data obtained during intraoperative endovascular measurements.