Asymptotic limits for a non-linear integro-differential equation modelling leukocytes' rolling on arterial walls

TL;DR

This paper rigorously analyzes the long-time behavior of a non-linear integro-differential model for leukocyte rolling, establishing asymptotic limits as a small parameter approaches zero, including explicit solutions in special cases.

Contribution

It introduces a new scaling and provides a rigorous asymptotic analysis of the model, extending results to non-smooth elastic energy functions and explicitly solving special cases.

Findings

Asymptotic limits characterized for the model as epsilon approaches zero.

Explicit solution obtained for the case with absolute value potential.

Convergence results extended to Lipschitz continuous elastic energies with finite jumps.

Abstract

We consider a non-linear integro-differential model describing $z$, the position of the cell center on the real line presented in [Grec et al., J. Theo. Bio. 2018]. We introduce a new $\varepsilon$-scaling and we prove rigorously the asymptotics when $\varepsilon$ goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (the velocity $\dot{z}$ tends to a limit). The convergence results are first given when $\psi$, the elastic energy associated to linkages, is convex and regular (the second order derivative of $\psi$ is bounded). In the absence of blood flow, when $\psi$, is quadratic, we compute the final position $z_\infty$ to which we prove that $z$ tends. We then build a rigorous mathematical framework for $\psi$ being convex but only Lipschitz. We extend convergence results with respect to $\varepsilon$ to this case when $\psi'$ admits a finite number of jumps. In the last part, we show that in the constant force case (see Model 3 in [Grec et al], $\psi$ is the absolute value), we solve explicitly the problem and recover the above asymptotic results.