In this paper, we introduce an efficient method for computing curves minimizing a variant of the Euler-Mumford elastica energy, with fixed endpoints and tangents at these endpoints, where the bending energy is enhanced with a user defined and data-driven scalar-valued term referred to as the curvature prior. In order to guarantee that the globally optimal curve is extracted, the proposed method involves the numerical computation of the viscosity solution to a specific static Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). For that purpose, we derive the explicit Hamiltonian associated to this variant model equipped with a curvature prior, discretize the resulting HJB PDE using an adaptive finite difference scheme, and solve it in a single pass using a generalized Fast-Marching method. In addition, we also present a practical method for estimating the curvature prior values from image data, designed for the task of accurately tracking curvilinear structure centerlines. Numerical experiments on synthetic and real image data illustrate the advantages of the considered variant of the elastica model with a prior curvature enhancement in complex scenarios where challenging geometric structures appear.