Mathematical modeling of the magnetic field generated by a network of parallel narrow capillaries

A three-dimensional mathematical model was constructed that simulates the movement of red blood cells (RBCs) through narrow capillaries. The charges located on RBC membrane generate a magnetic field (MF) in the surrounding space. The movement of erythrocytes along a capillary network built from parallel narrow capillaries is considered. The distribution of magnetic field strength (MFS) in planes parallel to the plane of the capillary network has been constructed. In the plane of a parallel capillary network, the dependence of the maximum and minimum values of MFS on the distance between the centers of the capillaries has been determined. It is shown that as the distance between the capillaries decreases, the magnitude of MFS in the plane parallel to the plane in which the capillaries lie increases, and the difference between the maximum and minimum values of MFS decreases.


Introduction
When a red blood cell (RBC) moves along a narrow capillary, it rolls at a speed that can reach up to 50 revolutions per second and depends on various factors, in particular geometric (capillary diameter Dc, surface area Se, volume Ve of RBC), physical (erythrocyte speed VELe in the capillary, plasma viscosity), etc. [1,2].On the erythrocyte membrane there are clusters of anionic and cationic groups, which create a total negative charge Qe of the erythrocyte, the value of which can range from 22 thousand to 2.2 million elementary charges, equal to the electron charge of 1.6 × 10 -19 С [3].It is assumed that negative charges are located on the erythrocyte membrane discretely and evenly distributed over the surface of the erythrocyte.
Previous publications [4,5] examined MFs generated by a single RBC and a sequence of RBCs in a narrow capillary.At present, it is not possible to create a device for measuring MFS in the vicinity of one capillary.Therefore, it is important to carry out calculations of MFS in the vicinity of the capillary network and, based on these calculations, to develop principles for constructing instruments that will allow measuring MFS in the microcirculation system.If such devices are developed, it will be possible to diagnose micro vascular blockage at an early stage, which is important for diagnosing cardiovascular diseases, primarily heart attack, stroke, etc.

Mathematical modeling
In the model, an erythrocyte is approximated by a truncated cylinder with a maximum (Lmax) and minimum (Lmin) generators.It is assumed that the volume (Ve) and surface area (Se) of the red blood cell are constant quantities.
It is assumed that the negative charges on the erythrocyte membrane are discrete, distributed evenly over the surface of the erythrocyte and move along with the membrane in planes parallel to the XOZ plane (Figure 1).
where R is the distance from a single charge Q on the erythrocyte membrane to a point in threedimensional space at which MFS (H) is determined, VEL is the speed of charge movement, α is the angle between VEL and R [6,7].
The mathematical model has input and output parameters.The input parameters of the model include: the charge of the erythrocyte, the number of discrete charges on the erythrocyte membrane, the trajectories along which the charges move, the coordinates of the points at which H is determined, the speed of the erythrocyte in the capillary, the rotation frequency of the erythrocyte membrane, hematocrit (Ht), the distance between the capillaries.The output parameters include: the magnitude of H at selected points in three-dimensional space.

Results and Discussion
Calculations were carried out using the input parameters presented in Table 1.When erythrocytes move in a capillary network, which is a set of parallel narrow capillaries, in the surrounding space, due to moving charges on the erythrocyte membranes, a magnetic field is generated, the intensity distribution of which is presented in Figures 2-4.
Figure 5 shows graphs of the dependence of the maximum Hmax and minimum Hmin values of MFS (A/m) on the distance between the centers of the capillaries D (μm).The graphs were constructed using the least squares method.The dependence of Hmax (A/m) and Hmin (A/m) in the plane z=34 µm on D (µm) has the form Hmax=-0.087×D+8.82,Hmin=-0.114×D+9.48.
It can be seen that as the distance D decreases, the values of Hmax and Hmin increase and the difference between them decreases.Thus, when the distance D changes from 41 µm to 51 µm (by 24.39%), Hmax decreases from 5.16×10 -5 A/m to 4.43×10 -5 A/m (by 14.15%), and Hmin decreases from 4.57× 10 -5 A/m to 3.74×10 -5 A/m (by 18.16%).The difference Hmax-Hmin decreases from 0.79×10 -5 A/m to 0.69×10 -5 A/m (by 12.99%) when the distance D changes from 41 µm to 51 µm (by 24.39%).This is explained by the fact that since MFS decreases in inverse proportion to the square of the distance, then at large distances between capillaries the influence of neighboring capillaries can be neglected.
Table 1.Model input parameters.If the distance D decreases, then the contribution of neighboring capillaries to the total MFS increases and must be taken into account.When calculating MFS, the contribution of five neighboring capillaries was taken into account, which turned out to be sufficient for distances between capillaries of 30-80 μm.

Conclusion
Calculations of MFS near a flat capillary network consisting of parallel narrow vessels showed that the maximum and minimum values of the intensity in the plane parallel to the plane of the capillary network increase with increasing hematocrit in the capillaries and decreasing the distance between the capillaries.As the density of the capillary network increases (at the same hematocrit), the difference between the maximum and minimum values of MFS decreases, i.e. the field becomes more homogeneous.This indicator can be taken into account when developing devices that measure MFS near capillary networks, which can be used for local diagnosis of vascular capacity at the early stage of cardiovascular diseases (heart attacks, strokes, etc.).

Figure 1 .
Figure 1.XOZ plane cross-section of a red blood cell model.Moving charges generate MF in the space surrounding RBC, the intensity of which is equal to