  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.
