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Background  The erythrocyte sedimentation rate (ESR) is a simple and inexpensive laboratory test, which is widespread in clinical practice, for assessing the inflammatory or acute response. This work addresses the theoretical and experimental investigation of sedimentation a single and multiple particles in homogeneous and heterogeneous (multiphase) medium, as it relates to their internal structure (aggregation of solid or deformed particles).    Methods  The equation system has been solved numerically. To choose finite analogs of derivatives we used the schemes of directional differences.    Results  (1) Our model takes into account the influence of the vessel wall on group aggregation of particles in tubes as well as the effects of rotation of particles, the constraint coefficient, and viscosity of a mixture as a function of the volume fraction. (2) This model can describe ESR as a function of the velocity of adhesion of erythrocytes; (3) Determination of the ESR is best conducted at certain time intervals, i.e. in a series of periods not exceeding 5 minutes each; (4) Differential diagnosis of various diseases by means of ESR should be performed using the aforementioned timed measurement of ESR; (5) An increase in blood viscosity during trauma results from an increase in rouleaux formation and the time-course method of ESR will be useful in patients with trauma, in particular, with traumatic shock and crush syndrome.    Conclusion  The mathematical model created in this study used the most fundamental differential equations that have ever been derived to estimate ESR. It may further our understanding of its complex mechanism.

Mathematical model describing erythrocyte sedimentation rate. Implications for blood viscosity changes in traumatic shock and crush syndrome