Robot-assisted percutaneous needle insertion is expected to significantly increase targeting accuracy in minimally invasive operations. For this, it is necessary to provide mathematical models that can accurately capture the underlying dynamics of medical needles. Here, we present a novel nonlinear mathematical model of flexible medical needles based on the Absolute Nodal Coordinate Formulation. The model allows the description of large needle deflections and arbitrarily large rigid body motions. Tailored to the requirements of transperineal prostate biopsy and brachytherapy, it can correlate both the translational and rotational coordinates of the needle's base with its deflection, provide force feedback and accept arbitrary loading conditions. The model is optimised in terms of computational efficiency in order to allow real-time simulation and control. Experiments show that the proposed model allows for submillimeter precision in both static and dynamic needle deflection settings. Due to its accuracy and computational efficiency, it is expected to constitute a valuable tool for both real-time visual/haptic simulation and control of percutaneous needle insertion.