An insertability constraint for shape optimization

Patient-specific implants offer a host of benefits over their generic counterparts. Nonetheless, the design and optimization of these components present several technical challenges, among them being the need to ensure their insertability into the host bone tissue. This presents a significant challenge due to the tight-fitting nature of the bone-implant interface. This paper presents a novel insertability metric designed to efficiently assess whether a rigid body can be inserted into a tight-fitting cavity, without interference. In contrast to existing solutions, the metric is fully differentiable and can be incorporated as a design constraint into shape optimization routines. By exploiting the tight-fitting condition, the problem of planning an interference-free insertion path is reformulated as the search for a single interference-free movement, starting from the inserted configuration. We prove that if there exists any outward movement for which no interference is indicated, then the body can be fully extracted from or, equivalently, inserted into the cavity. This formulation is extremely efficient and highly robust with respect to the complexity of the geometry. We demonstrate the effectiveness and efficiency of our method by applying it to the optimization of two-dimensional (2D) and three-dimensional (3D) designs for insertability, subject to various design requirements. We then incorporate the proposed metric into the optimization of an acetabular cup used in total hip replacement (THR) surgery where geometric and structural requirements are considered.