Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Analysis of Stationary Solutions of One-dimensional Models of Hemodynamics

Author(s):

Gerasim Vladimirovich Krivovichev

Saint-Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
Assoc. Professor, Candidate of sc.
199034, Russia, Saint-Petersburg, Universitetskaya nab. 7/9;

g.krivovichev@spbu.ru

Nikolay Vasil'evich Egorov

Saint-Petersburg State University,
Faculty of Applied Mathematics and Control Processes
Full Professor,Doctor of Sc.
199034, Russia, Saint-Petersburg, Universitetskaya nab. 7/9;

n.v.egorov@spbu.ru

Abstract:

In last years, one-dimensional (1D) models of hemodynamics are widely used for the diagnostics of cardiovascular diseases, surgical operations, and for the analysis of the vascular pathologies effects. Models of this type are constructed by the averaging of the equations of the hydrodynamics of a viscous incompressible fluid on the vessel cross-section, taking some simplifications into account. The paper presents 1D models of blood flow, where the non-Newtonian properties of blood are considered. For the construction of models, the rheological relations for generalized Newtonian fluids are used. The following models, applied in 2D and 3D simulations are considered: the power law model, the Carreau, Carreau --- Yasuda and Cross models, the simplified Cross model, the Yeleswarapu model, and Quemada and the modified Yeleswarapu models, which are dependent on hematocrit. For the closure of models, a model power-law representation of the dimensionless velocity profile is used. The parameter of this dependence is varied during the calculations. The steady flow regime leads to the consideration of the nonlinear ordinary differential equation on the cross-sectional area. For the power law model, the simplified Cross model, and the Quemada model, integrals of this equation are obtained. Conditions for the existence and uniqueness of the solution of the initial problem are obtained. During the calculations, the parameters for the iliac artery are considered. The influence of the velocity profile and hematocrit on the obtained solutions is investigated. It is shown, that the flattening of the velocity profile leads to a decrease in the length of the interval, where the stationary solutions exist. A similar situation occurs with an increase of hematocrit. The case of a vessel with stenosis, with the shape described by a model function, is considered. It is shown that a change in the geometric parameters affects the length of the interval of existence of the solution. The solutions obtained can be useful for the comparison of different 1D models of blood as a viscous fluid and for testing programs that implement algorithms of numerical methods.

Keywords

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