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

Abstract

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

Kopyltsov A.V.

Incoming article date: 01.10.2018

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