This paper intends to address the problem of multi-vehicle tracking (MVT) by considering road constraints and interactions between vehicles. Most existing tracking algorithms assume that vehicles move independently in an open-field environment. However, due to traffic volume, road networks, and traffic rules, the movements of vehicles have to be affected by the surrounding environment (e.g., neighboring vehicles, physical road, speed limit) for the purposes of ensuring a safe distance and avoiding collisions. To address these problems, this paper proposes a novel MVT algorithm that incorporates the prior information from the surrounding environment into the estimation process by modeling it as the noisy control input of a dynamic model. In this paper, the kinematic state of vehicles is decomposed into longitudinal and lateral components, and the corresponding interacting multiple model (IMM) estimator is utilized to individually estimate longitudinal and lateral motions. Furthermore, by employing a two-stage IMM-based estimation scheme with different transition probability matrices (TPM) for lateral motions of the vehicle, the proposed algorithm significantly enhances tracking quality and continuity, especially in lane changing scenes. The effectiveness of the proposed algorithm is validated through numerical simulations and real-measured data.