Mathematical
modeling of cardiovascular flow.
"Complex nonlocal
cardiovascular system mathematical modeling"
The
study related to hemodynamics in human body as a
whole under the influence of periodically contracting heart. Attention is paid
to cross-influence of various organs (in particular the kidney) on pressure in
the cardiovascular system. Study is also envisaged of the influence of
different factors connected with deviations from normal of functional
characteristics of vessels, upon the condition of a system as a whole, as well
as study of ways of compensating for the vessels’ deficiencies, e.g. bypass. One
of the ways of putting this study into practice would be to analyze the impact
and transportation within the cardiovascular system of pharmacological
substances. Considerations of external influences, e.g.
vibration, on the functioning of cardiovascular system, is also planed.
The
cardiovascular system is formally described with a graph consisting of edges
and nodes. Edges correspond to individual large vessels or to groups of similar
small vessels. Nodes correspond to functional properties of either the zones of
bifurcation of vessels or tissues of various organs.
The
laws of conservation of mass and impulse have been chosen as a basis for
description of blood flow in the system, with such laws in the form of partial
differential equations in partial derivatives. The vessels are assumed
sufficiently long compared with their diameter, which permits using the quasi
one-dimensional approximation for their mathematical description. The length of
the arc which connects the cross-section centers of the vessel (the vessel’s
axis) has been chosen as the spatial coordinate. The area of cross-section
depends upon the coordinates, time, pressure, etc. The blood density is assumed
constant and the blood flow is directed along vessels axis.
In
order to numerically implement the above- mentioned non-linear mathematical
model in a graph, a conservative homogeneous implicit finite-difference scheme
has been constructed on a directed graph. A system of finite-difference
equations is completed with discrete analogues of correlations simulating the
performance of organs corresponding to certain nodes. Since such system of
finite-difference equations of hemodynamics
represents a system of non-linear algebraic equations for values of functions
in the dots of a discrete grid at a new time row, Newton method was used for
solving this non-linear system.
The
authors have created a software providing for
mathematical modeling of hemodynamics at any graph of
the cardiovascular system. These programs permit to modify the properties of
graph elements in an interactive mode along with visualizing the results of
calculations
Brief
description of Research
The
goal of this project, which was started in collaboration with the Faculty of
Basic Medicine of MSU, was to create mathematical model, numerical methods and
corresponding software for numerical simulations of cardiovascular flow. For
this purpose cardiovascular system is associated with the graph of vessels
(edges) and tissues (nodes). Each vessel is taken as a one-dimensional flexible
pipe, which is oriented in 3-D space and connected either with other vessels or
with tissues. Diameters of vessels are not constant and depend upon a great
number of physiological and physical parameters, such as pressure, coefficient
of flexibility, gravitation, etc. Vessel can be taken as a certain vessel or as
a group of similar vessels. Tissues are characterized with their volume, its
ability to produce or sorb a certain amount of blood,
Darcy coefficient, etc. Series of models of heart with different complexity are
considered. Pressure, velocity of blood, diameter of vessel, which
are estimated in any point of cardiovascular graph, are taken as basic function
to be computed in the result of numerical simulation.
From
mathematical point of view, the problem is stated as a system of nonlinear
partial differential equations of viscous fluid dynamics on a graph. This
system is approximated with explicit conservative finite-difference scheme with
specially designed linking conditions on a graph. This extremely complex system
of nonlinear equations is solved with the help of iteration method.
Possibilities of Applications
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