Tendon-driven continuum robots are compliant and capable of assuming complex curves, making them ideal for minimally invasive surgery in confined spaces. These robots can achieve an infinite family of curves, determined by both actuation and design considerations, such as the routing paths of the tendons along the robot. However, it has thus far been challenging to design arbitrary tendon paths because one is working in an infinite design space with mechanics-based models that require solving differential equations to compute robot kinematics. Toward solving this problem, this paper presents a new, expressive tendon routing parameterization for tendon-driven robots. We leverage the parameterization to optimize the tendon routings for a design objective drawn from a practical neurosurgical application: choroid plexus cauterization for hydrocephalus. Our method found a set of tendon routings that enabled the robot to reach a full set of 20 physician-specified points with less than 0.8 mm of error and a mean error across all points of 0.5 mm.