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  • Mathematical model of electromagnetic interaction of red blood cells in a narrow capillary

    Red blood cells (RBC) roll like tractor caterpillars along narrow capillaries. On the erythrocyte surface there are charges that, when moving together with the erythrocyte membrane, create a magnetic field in the vicinity of the RBC. Discrete charges are distributed uniformly on the surface of the RBC, their number can reach several million and the charges move together with the RBC membrane. The surface of the RBC is approximated by a truncated cylinder. Discrete charges are located evenly over the surface of the RBC and move along closed curves that are rectangular trapezoids. A mathematical model has been constructed that allows calculating the intensity of the magnetic field produced by mobile charges located on the RBC membrane. According to the Bio-Savart law, the magnetic field strength can be calculated at some point in space if the coordinates and velocity of the charge are known, the distance from the charge to the point and the angle between the velocity vector and the radius vector connecting the charge and the point. If we assume in the first approximation that the medium is isotropic and magnetic currents are absent, then Maxwell's equations can be written out. These equations can be rewritten in the form of equations in finite differences, solving by numerical methods one can obtain distributions of electric and magnetic field strengths in the vicinity of the RBC. Calculations were carried out for different values of input parameters. In the case when the RBCs move through the capillary network, in which the narrow capillaries are located close to each other, the magnetic fields of the RBCs in different capillaries interact, and, as a result, we obtain a new distribution of the magnetic field strength in the vicinity of the capillary network, which varies with time.

    Keywords: mathematical model, algorithm, magnetic field strength, electromagnetic interaction, erythrocyte, narrow capillary

  • Mathematical modeling of the magnetic field distribution in the vicinity of the erythrocyte

    Erythrocyte, when moving through a narrow capillary, stretches out and rolls like a tractor's caterpillar. Charges located on the surface of the erythrocyte, move together with the membrane and generate a magnetic field in the vicinity of the erythrocyte membrane, which can affect both elements of blood flow outside the erythrocyte, and its contents, in particular, the iron atoms that make up the hemoglobin. A three-dimensional model of the erythrocyte is constructed. The shape of the erythrocyte is approximated by a truncated cylinder of radius r with generators L1 and L2. It is assumed that all charges on the surface of the erythrocyte are the same and evenly distributed over the surface of the erythrocyte. Charges move along with the membrane along closed curves (trapezoids). Moving charge creates a magnetic field, the strength of which depends on the magnitude and speed of the charge. On the erythrocyte membrane there are several charges and each of them at some selected point creates a magnetic field. The total strength of the magnetic field is defined as the vector sum of the strains created by each of the charges. The following parameter values ​​were used in the calculations. The charge of the erythrocyte is 20 million elementary charges. The number of charges on the erythrocyte membrane is 38594. The rotation frequency of the erythrocyte membrane is 20 revolutions per second. The erythrocyte radius is 2 μm. The erythrocyte volume is 94 μm3. The erythrocyte surface area is 135 μm2, the lengths of the truncated cylinder forming are 3.4 μm and 11.5 μm. The erythrocyte speed is 100 μm / sec. The step along the space is 0.1 μm. The performed calculations of the magnetic field strength H have shown that the rotation of the erythrocyte membrane with the charges placed on it leads to a significant redistribution of the magnetic field in the vicinity of the erythrocyte. And with an increase in the frequency of rotation of the erythrocyte membrane, the heterogeneity of H increases significantly and can lead to a change in hemodynamics in the microcirculation system.

    Keywords: mathematical model, erythrocyte, narrow capillary, magnetic field, microcirculation

  • Two-dimensional model of the distribution of the magnetic field between erythrocytes in a narrow capillary

    The article deals with the movement of erythrocytes along the narrow capillaries with a diameter smaller than the erythrocyte diameter. Red blood cell in narrow capillary has tank-treading motion. The erythrocyte rotation frequency reaches several dozen revolutions per second. Electric charges located on the surface of the erythrocyte, move together with the erythrocyte membrane and create a magnetic field in the surrounding space. A two-dimensional model of erythrocyte movement along narrow capillaries was constructed. If the erythrocyte surface area and erythrocyte charge are known, then the density of charges on the erythrocyte membrane can be determined. The magnetic field strength of a moving charged particle can be determined if the particle charge, the particle velocity, the distance from the particle to the point at which the magnetic field strength is determined, the angle between the direction of the particle velocity and the direct connecting particle, and the point at which the tension is determined are known. The total strength of the magnetic field of several moving charged particles is defined as the vector sum of the magnetic field strengths of the individual moving charged particles. In the two-dimensional model it is assumed that the red blood cells are rectangles that move along the capillary, and the erythrocyte membrane is the sides of the rectangle. Discrete charges are located on the sides of the rectangle and move either clockwise or counterclockwise. It is possible two variants. Their membranes either rotate in the same direction or in opposite directions. Calculations were carried out for both variants and at different rates of rotation of erythrocyte membranes (from 0 to 50 revolutions per second) and distances between red blood cells. It is shown that at distances between erythrocytes smaller than the two capillary diameters, the influence of neighboring red blood cells can be neglected (the difference is less than 3%).

    Keywords: "mathematical model, magnetic field, erythrocytes, narrow capillaries, magnetic field strength "