Time-dependent topology optimization of bone plates considering bone remodeling

Abstract Bone plates have been widely used for the treatment of bone defects and trauma. These fixation plates can stabilize or replace bone tissue to restore appropriate load-bearing functionality. Nevertheless, the use of bone plates may lead to the stress shielding, thereby weakening prosthetic bone substitutes (e.g. bone graft or scaffolds) due to significant change in the biomechanical environment after implantation. To address this issue, we propose a time-dependent topology optimization procedure for the design of bone plates by taking into account bone remodeling. A solid isotropic material penalization (SIMP) model is used to interpolate design variables. The objective is to maximize total bone density within a reconstruction area at the final stage of bone remodeling, subject to a volume constraint of the bone plate and maximum allowable compliance of the prosthetic system. The sensitivity of bone density at the final stage is derived with respect to the topological variables of the plate in a step-wise manner. To facilitate sensitivity analysis, a bone remodeling rule is formulated in two different ways to accommodate a C 1 continuity. A jaw reconstruction problem is exemplified in this study to demonstrate the effectiveness of the proposed approach. Through this specific case, the non-differentiability issue due to the lazy zone of a remodeling rule is smoothed; and the proposed approach is also compared with that of a time-independent design. The effects of volume fraction and compliance constraints are also investigated to gain further insights into the design of prosthetic substitutes. Together with additive manufacturing technology, the proposed time-dependent topology optimization procedure is expected to form a useful tool for the design of implantable devices ensuring favorable long-term treatment outcomes.