# A stochastic model for cell adhesion to the vascular wall

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## Abstract

Cell dynamics in the vicinity of the vascular wall involves several factors of mechanical or biochemical origins. It is driven by the competition between the drag force of the blood flow and the resistive force generated by the bonds created between the circulating cell and the endothelial wall. Here, we propose a minimal mathematical model for the adhesive interaction between a circulating cell and the blood vessel wall in shear flow when the cell shape is neglected. The bond dynamics in cell adhesion is modeled as a nonlinear Markovian Jump process that takes into account the growth of adhesion complexes. Performing scaling limits in the spirit of Joffe and Metivier (Adv Appl Probab 18(1):20, 1986), Ethier and Kurtz (Markov processes: characterization and convergence, Wiley, New York, 2009), we obtain deterministic and stochastic continuous models, whose analysis allow to identify a threshold shear velocity associated with the transition from cell rolling and firm adhesion. We also give an estimation of the mean stopping time of the cell resulting from this dynamics. We believe these results can have strong implications for the understanding of major biological phenomena such as cell immunity and metastatic development.

## Keywords

Cell adhesion Metastatic development Immune response Atherosclerosis Stochastic process## Mathematics Subject Classification

60J70 92C17## Notes

### Acknowledgements

The authors are very grateful to V.C. Tran and R. Voituriez for very helpful discussions and suggestions.

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