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Concentric tube robots (CTRs) show particular promise for minimally invasive surgery due to their inherent compliance and ability to navigate in constrained environments. Due to variations in anatomy among patients and variations in task requirements among procedures, it is necessary to customize the design of these robots on a patient- or population-specific basis. However, the complex kinematics and large design space make the design problem challenging. Here we propose a computational framework that can efficiently optimize a robot design and a motion plan to enable safe navigation through the patient’s anatomy. The current framework is the first fully gradient-based method for CTR design optimization and motion planning, enabling an efficient and scalable solution for simultaneously optimizing continuous variables, even across multiple anatomies. The framework is demonstrated using two clinical examples, laryngoscopy and heart biopsy, where the optimization problems are solved for a single patient and across multiple patients, respectively.