To facilitate the relativistic heavy-ion calculations based on transport equations, the binary collisions involving a ∆ resonance in either the entrance channel or the exit channel are investigated within a Hamiltonian formulation of πN N interactions. An averaging procedure is developed to define a quasi-particle ∆ * and to express the experimentally measured N N → πN N cross section in terms of an effective N N → N ∆ * cross section. In contrast to previous works, the main feature of the present approach is that the mass The ∆ potential in nuclear matter has been calculated by using a BrucknerHartree-Fock approximation. By including the mean-field effects on the ∆ propagation, the effective cross sections of the N N → N ∆ * , N ∆ * → N N and N ∆ * → N ∆ * reactions in nuclear matter are predicted. It is demonstrated that the density dependence is most dramatic in the energy region close to the pion production threshold.