  In heavy-ion collisions at intermediate energies, the production and
propagation of $\Delta$ particles are crucial to understanding the nuclear
equation of state and inferring the properties of nuclear matter at high
densities. Based on the self-consistent relativistic
Boltzmann-Uehling-Uhlenbeck (RBUU) transport theory, by introducing the
isovector $\rho$ meson exchange into the effective Lagrangian and adopting the
density-dependent coupling constants, the detailed expressions for
isospin-dependent in-medium $N\Delta\rightarrow N\Delta$ elastic cross sections
$\sigma_{N\Delta}^{*}$ have been calculated. The energy and density dependence
of the isospin-related $\sigma_{N\Delta}^{*}$ as well as the total
contributions of $\sigma$, $\omega$ and $\rho$ meson fields are analyzed. It is
found that the total $\sigma_{N\Delta}^{*}$ has a sensitive center-of-mass
energy dependence at lower energies while exhibiting a slight increase as the
center-of-mass energy increases. The isospin effect between different
isospin-separated channels weakens as the energy and/or density increases. The
isospin effect on the density- and energy-dependent in-medium $N\Delta$ elastic
cross sections is dominantly caused by the delicate balance of the isovector
$\rho$ meson exchange.
