We present an efficient finite element method to simulate a transversely isotropic non-linear material for ligaments. The approach relies on tetrahedral elements and exploits the geometry to optimize computation of the derivatives of the strain energy. To better support incompressibilty, deviatoric and dilational responses are uncoupled and a penalty term controls volume preservation. We derive stress and elasticity tensors required for implicit solvers and verify our model against the FEBio software using a variety of load scenarios with synthetic shapes. The maximum node positioning error for ligament materials is < 5% for strains under physiological conditions. To generate subject-specific ligament models, we propose a novel technique to estimate fiber orientation from segmented ligament geometry. The approach is based on an automatic centerline extraction and generation of the corresponding diffusion field. We present results for a medial collateral ligament segmented from standard MRI data. Results show the general viability of the method, but also the limitations of current MRI acquisitions. In the future, we hope to employ the presented techniques for real-time simulation of knee surgery.