MOTION OF ERYTHROCYTES ALONG THE CAPILLARIES Alexander V. Kopyltsov

Erythrocyte in the capillary

Equations of Reynolds for the gap between the erythrocyte and the capillary wall

The boundary conditions u = W1 w = 0 при r = R u = U1 w = 0 при r = R + h The kinematics conditions

Integrating equations of Reynolds we obtain:

Parameters Diameter of the capillary (D) Viscosity of the plasma (μ) Volume of the erythrocyte (V) Area of the surface of the erythrocyte (S) Form of the erythrocyte (a, b, V, S) Size of the erythrocyte in the capillary (l, β) Frequency of rotation of the erythrocyte membrane (f)

Conditions Membrane: Summa of forces = 0. Frequency of rotation of the erythrocyte membrane f = 0. Erythrocyte: Summa of forces = 0. Summa of moments = 0. Erythrocyte is an elastic body, V and S are constants.

The magnitudes of parameters Viscosity of the plasma 1-2 cP Velocity of the erythrocyte 0,3-1,0 mm/с Diameter of the capillary 3-10 mkm Long of the capillary mkm Hematocrite in the capillary 0-20 % Volume of the erythrocyte mkm3 Area of the surface mkm2

The pressures difference at the ends of the erythrocyte

Dependence of the pressure difference (N/m2) at the ends of the erythrocyte on the speed of the erythrocyte U (mkm/с). 1 – Lighthill, 2 – Secomb, 3 – Kopyltsov

Constancy of V and S is cause of: Asymmetry of the form and the size of the erythrocyte in the capillary. Tank-treading motion of the erythrocyte membrane. Down of the resistance of the erythrocyte moving on 10-20%.

Increase of the frequency f is cause of: Down of the resistance of the erythrocyte movement on 10-20%. Down of the probability of the destruction of the erythrocyte membrane in the narrow capillaries. Increase of the velocity of oxygen from erythrocyte to tissue. Change of the velocity of the plasma in the erythrocyte neighborhood.

3 dimension model of the erythrocyte is developed. The model take into account: the volume, the area of surface, the elastic properties of the erythrocyte, viscosity of plasma and the erythrocyte interior and allow describe: the form and the size of the erythrocyte in the capillary, the movement of the erythrocyte along of the capillary, the tank-treading movement of the membrane of the erythrocyte.

Erythrocytes in the capillary

Bifurcation

Capillary network

Difference of the pressures at the ends of the capillary

Viscosity of the blood plasma Normalized viscosity of the blood plasma

Analysis of the influence of ∆P, ς, D, L, V and S on the blood flow: At the increase ς, D, S and the decrease V the velocity of the blood flow is increase, and the amplitude of its oscillations is decrease. At the increase ∆P or decrease L the velocity and the amplitude of its oscillations is increase.

Analysis of the influence of ∆P, ς, D, L, V and S on the blood flow: Influence on the blood flow on basic parameters (in order of the decrease) is following: ∆P, D, S, V, L, ς.

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