Medical concentric tubes have recently been designed using auxetic structures with negative Poisson's ratios to reduce undesirable snapping instability and improve mechanical performance. Despite the vast range of stiffness levels, prior auxetic designs were restricted to relatively simple geometries like square or triangular holes. Here, we present a novel free-form optimization for designing the bending and tor-sional stiffness of tubular structures using the multi-objective Bayesian optimization (MBO) method. The initial dataset of hole shapes is generated using a non-parametric Bezier curve, and the corresponding mechanical properties are estimated via numerical analysis. The acquisition function suggests a new hole shape with higher performance metrics based on the dataset, and updates to the regression model are made with new data points to improve prediction accuracy. Compared to a conventional tube with rect-angular holes, the optimized design improves key factor, a ratio of bending stiffness to torsional rigidity, and torsional stiffness by 32% and 25%, respectively, by expanding the existing design space. The perfor-mance of the optimally designed tube was validated by resonant vibration tests.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).