Automating surgical systems enhances precision and safety while reducing human involvement in high-risk environments. A major challenge in automating surgical procedures like suturing is accurately modeling the suture thread, a highly flexible and compliant component. Existing models either lack the accuracy needed for safety-critical procedures or are too computationally intensive for real-time execution. In this work, we introduce a novel approach for modeling suture thread dynamics using control barrier functions (CBFs), achieving both realism and computational efficiency. Thread-like behavior, collision avoidance, stiffness, and damping are all modeled within a unified CBF and control Lyapunov function (CLFs) framework. Our approach eliminates the need to calculate complex forces or solve differential equations, significantly reducing computational overhead while maintaining a realistic model suitable for both automation and virtual reality surgical training systems. The framework also allows visual cues to be provided based on the thread's interaction with the environment, enhancing user experience when performing suture or ligation tasks. The proposed model is tested on the MagnetoSuture system, a minimally invasive robotic surgical platform that uses magnetic fields to manipulate suture needles, offering a less invasive solution for surgical procedures.