Patient-specific 3D spine models serve as a foundation for spinal treatment and surgery planning as well as analysis of loading conditions in biomechanical and biomedical research. Despite advancements in imaging technologies, the reconstruction of complete 3D spine models often faces challenges due to limitations in imaging modalities such as planar X-Ray and missing certain spinal structures, such as the spinal or transverse processes, in volumetric medical images and resulting segmentations. In this study, we present a novel accurate and time-efficient method to reconstruct complete 3D lumbar spine models from incomplete 3D vertebral bodies obtained from segmented magnetic resonance images (MRI). In our method, we use an affine transformation to align artificial vertebra models with patient-specific incomplete vertebrae. The transformation matrix is derived from vertebra landmarks, which are automatically detected on the vertebra endplates. The results of our evaluation demonstrate the high accuracy of the performed registration, achieving an average point-to-model distance of 1.95 mm. Additionally, in assessing the morphological properties of the vertebrae and intervertebral characteristics, our method demonstrated a mean absolute error (MAE) of 3.4{\deg} in the angles of functional spine units (FSUs), emphasizing its effectiveness in maintaining important spinal features throughout the transformation process of individual vertebrae. Our method achieves the registration of the entire lumbar spine, spanning segments L1 to L5, in just 0.14 seconds, showcasing its time-efficiency. Clinical relevance: the fast and accurate reconstruction of spinal models from incomplete input data such as segmentations provides a foundation for many applications in spine diagnostics, treatment planning, and the development of spinal healthcare solutions.