The Mechanism of Movement and Calculation of Blood Balance in the Flow Channel From the Left Atrium to the End of the Aorta

It was previously shown that the spatial dynamic geometry of the flow channel from the left atrium (LA) to the end of the aorta (Ao) corresponds to the geometric configuration of the potential swirling jet, described by exact solutions.

The aim of the work was to restore the mechanics of blood flow in the flow channel considering the balance of the medium and the geometric configuration of the streamlined surfaces of the heart and the Ao.

A swirling jet of this type arises and is maintained when a viscous medium flows over a concave surface under the following conditions: a) sufficient influent flow rate; b) the curvature corresponds to the axisymmetric surface of revolution of the involute, plotted to the helical flow streamline; c) creating conditions for maintaining the convergence of the jet after its occurrence. The evolution of the jet requires a vortex boundary layer, which structure is different from the shear boundary layer, and shear stresses are replaced by rolling stresses, which significantly reduces boundary friction.

The initial formation of a swirling blood stream occurs in the area of the dome of the LA. This becomes possible due to the fact that: a) the dynamic geometry of the concave surface of the LA in the area of its dome corresponds to the streamlines of the swirling flow; b) the directions of blood flow from the PV and the contraction of the left atrial appendage ensure the appearance of centripetal accelerations on the  concave surface necessary for swirling the jet occurrence; c) the convergence of the LA cavity in the direction towards the mitral valve (MV) provides conservation of a swirling stream structure as it is ejected into the cavity of the LV.

The above consideration allows us to obtain an equation for the balance of blood entering the generating zone of the swirling jet in the LA and injected into the LV cavity.

After the MV opens, a swirling blood stream is injected from the LA cavity into the LV. During the inflow of this blood, the LV cavity increases in volume, and a combination of forces stabilizing the geometric structure of the flow acts on the blood flow due to the intraventricular trabeculae. An increase in the volume of the LV cavity is accompanied by a change in its geometry, and a concave surface is formed on the free wall of the cavity.

The mass of blood entering the concave forming surface has an angular velocity sufficient to form a stable swirling jet. The curvature of this surface determines the structure of the jet ejected into the Ao.

A growing swirling jet increases the dynamic pressure inside its current tube, which serves as a factor in opening the aortic valve and injecting the jet into the aortic cavity. The described mechanism also allows us to get the equation of balance of the mass of blood passing through the cavity of the LV.

As a result of injection and an increase in the volume of the swirling blood stream in the Ao, a field of forces is generated, directed from the axis to the walls of the aortic channel. This force field creates the conditions for a stable proportional distribution of blood along the branches of the Ao. It becomes possible to obtain the equation of balance of the medium in the Ao.

Conclusion. Thus, it becomes possible to describe the mechanics of swirling blood flow in the LA, LV, and Ao using quantitative relationships expressed in terms of the swirling flow structure and the dynamic geometric configuration of the flow channel. The formulated equations for blood balance allow us to associate these relationships with experimental data. The results of this work will be used to develop a mathematical model of blood circulation taking into account the swirling structure of the blood flow.

Acknowledgement. The work is financially supported by the Russian Scientific Foundation (grant No16-15-00109).