Blood system functions are very diverse and important for most processes in human organism. One of its primary functions is matter transport among different parts of the organism including tissue supplying with oxygen, carbon dioxide excretion, drug propagation etc. Forecasting of these processes under normal conditions and in the presence of different pathologies like atherosclerosis, loss of blood, anatomical abnormalities, pathological changing in chemical transformations and others is significant issue for many physiologists. In this connection should be pointed out that global processes are of special interest as they include feedbacks and interdependences among different regions of the organism. Thus the main goal of this work is to develop the model allowing to describe effectively blood flow in the whole organism. As we interested in global processes the models of the four vascular trees (arterial and venous parts of systemic and pulmonary circulation) must be closed with heart and peripheral circulation models. As one of the model applications the processes of the blood loss is considered in the end of the paper.
Investigations of the blood flow mechanics were started long time ago. Most likely Euler in 1775 was the first who wrote equations for the blood flow in arteries. In addition, Young was the first who calculated the pulse wave velocity [1] in the artery of a human. Womersley summarized the physics of pulsative flow in the channels having flexible walls [2,3] and others. However, great numbers of nowadays publications affirm urgency, importance and incompleteness of the problem of cardiovascular system modeling.
Often in biomechanics, the blood is considered as a fluid. In simplest cases it supposed to be single-component, non-viscous, non-compressible while the most complicated models include chemical reactions between the components dissolved in blood. In any case, it should be mentioned that blood has very complicated rheological properties. It may be considered in terms of continuum media due to the certain conditions taken place in most parts of the circulatory system of the organism under the normal conditions [4].
Wide range of the proposed models may be classified by their dimensionality. Twodimensional [5,6] and three-dimensional [7,8] models are widely used for the blood flow analysis in specific part of the circulatory system. If applied to the tasks of global circulation such approaches require huge computational resources. One-dimensional models [4,[9][10][11][12][13][14][15][16] are often exploited for investigations of the pulse wave propagation and reflection from the vessel bifurcation. Such approach is more effective for the tasks of global circulation but it requires identification of the great number of ill-conditioned and strongly variable parameters describing the model. It also exist lumped models [7,[17][18][19] that in one of the most known cases are based on electro-mechanical analogies [20]. The latest approaches are based on the multiscale technique that combines the models of different dimensionality in order to include possibly greater parts of circulatory system e.g. [7].
In fact, circulatory system is a single whole structure that results in non-linear internal effects. Thus, ideal model must include possibly greater parts of circulatory system. In addition, such model allows to simulate different global processes such as feedbacks, interdependences among different regions of the circulatory system, matter transfer and others.
As a basis for such model, we propose to use the model of non-linear pulsative flow of viscous incompressible fluid streaming through the collapsible tube [14][15][16]. The flow is supposed to be pseudo one-dimensional as all values are supposed to be averaged over crosssectional area of the vessel. By means of specific boundary conditions it may be generalized for the case of the pulsative flow through the graph of the collapsible tubes. Every edge of such graph corresponds to the vessel in cardiovascular system. Every node corresponds to the vessels bifurcation. The heart in these terms corresponds to the one specific node and performs pumping function. The other group of specific nodes must be considered in the graph that corresponds to the junction points with microcirculation. In fact peripheral circulation is hard to describe in terms of such model. It requires to determine dramatically huge number of the parameters and computational resources. Thus some averaging procedures required to produce virtual vessels having properties corresponding to some macro area of the capillary channel. Another way to deal with this problem is to include the model of peripheral circulation based on the principle of liquid filtration through the porous media. The multiscale approach can be realized using this model by substituting some part of the vascular network with appropriate local 3D model.
To the knowledge of the authors quite a few reconstructions of the whole circulatory system using at least 1D approach are exist at present [13,21]. In addition linear theory was used in some approaches that simplified analytical and computational analysis and allows obtaining experimentally proven results but it still carried some inaccuracy in the model.
Physically blood flow is as pulsatile flow of incompressible fluid streaming through the network of the vessels. As the base model for such flow, we propose to use pseudo onedimensional model of non-stationary incompressible fluid streaming through the collapsible tube. All such tubes must be connected each other by the appropriate boundary conditions.
For every vessel of k th generation the mass and momentum conservation are [4,[9][10][11][12][13][14][15][16]:
( )
where t -time; x -distance counted from the junction point with vessel of younger generation; ρ -blood density; k -index of the vessel; ( )
-linear velocity of the flow averaged over cross-section;
( ) , k p t x -pressure in the vessel counted off from atmospheric; k ψ -the impact from the external forces (gravitation, friction and others); ki χ -param
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