Simulation de la navigation intravasculaire de cathéters par la théorie des tiges de Cosserat

Ischemic stroke, caused by a blood clot blocking blood flow to the brain, is a leading cause of death, responsible for approximately three million fatalities worldwide each year. Timely intervention plays a key role in clinical outcomes, as the shorter the time between symptom onset and treatment, the higher the chances of recovery and the lower the risk of long-term disability or lasting effects. The standard treatment is endovascular thrombectomy, a procedure in which a physician manually steers a slender and flexible tool called a catheter from its proximal end, typically inserted through the femoral artery, to navigate the vascular system and retrieve the blood clot from the brain. This procedure is mechanically challenging, requiring a high level of practitioner expertise. Difficulties arise both from the physical complexity of manually steering the catheter through tortuous, narrow vessels and from the variability in vascular anatomy between patients. These challenges highlight the need for computational tools that can assist in training and preoperative planning, with the goal of reducing both planning and intervention time. This work aims to develop a robust and time-efficient numerical simulation of catheter navigation that maintains good physical fidelity. Such a simulation could support physicians by enabling them to determine in advance which vascular path to follow or which type of catheter to use. The simulation models the catheter, the steering at its proximal end, the vascular geometry, and the frictional contact between the catheter and the vessel walls. To achieve a balance between physical fidelity and modeling complexity, the catheter is modeled using the one-dimensional geometrically exact Cosserat rod, which is well suited to capture the large deformations and rotations present in catheter navigation. Though well suited for the task, simulating catheter behavior using this model presents several numerical challenges. Solving the Cosserat rod equations in a robust and time-efficient way is nontrivial, particularly when accounting for contact and friction. The shooting method, while commonly used to solve Cosserat boundary value problems due to its high time efficiency, becomes unstable and fails to converge for soft catheters and small time steps because of the singular nature of the problem in the dynamic regime. Alternatively, the strain parameterization in the Lagrangian framework is known for its robustness and its ability to significantly reduce the number of degrees of freedom, but it can lead to stiff systems that require prohibitively small time steps for integration. This work addresses these challenges in two complementary ways. First, a dynamic simulation based on an orthogonal collocation method with Chebyshev polynomials is proposed. We demonstrate through numerical experiments that this approach robustly addresses the inherent singularity in the boundary value problem of dynamic Cosserat rods. In particular, we show that in every difficult scenario where the shooting method fails, increasing the polynomial degree beyond a certain threshold ensures convergence of the collocation to a correct solution. This method is extended to include frictional contact and tested for catheter navigation inside a model of a real patient's carotid artery. While robust, this approach remains computationally intensive. To improve efficiency, a quasi-static assumption is introduced, justified by the spatial confinement of the catheter within the vascular network. In this regime, a Lagrangian strain parameterization is used. To address the stiffness of the underlying equations, we model the blood vessel walls using implicit functions, which enable the use of implicit time integration. Thanks to the smoothness of these implicit functions, the Jacobian matrix can be derived analytically. This approach led to a 100x speed improvement over explicit integration and a 10x improvement over the collocation method.