Performance Estimation and Ex Vivo Validation of Untethered Magnetic Robots in Soft Tissue

The navigation and control of untethered magnetic robots (UMRs) within soft brain tissue critically depend on their ability to maintain synchronized rotation under applied rotational magnetic fields. The fundamental upper limit of this magnetic synchronization, known as the step-out frequency, directly affects propulsion efficiency and control stability. This study presents an empirical model for predicting the step-out frequency of UMRs operating in ex vivo brain tissue with viscoelastic properties representative of the in vivo environment. Using Buckingham Pi dimensional analysis, we derive a compact set of dimensionless groups that capture the effects of robot geometry and tissue viscoelasticity, with magnetic torque used as the scaling basis. Experimental validation with spiral-type UMRs in gelatin-based phantoms shows that the step-out frequency decreases exponentially, from approximately 30 Hz in low-stiffness media (complex shear modulus $\sim$ 400 Pa) to below 1 Hz in high-stiffness media (complex shear modulus $\sim$ 1600 Pa). The found relationship allows for rapid estimation of robot performance limits using a single calibration point, enabling efficient evaluation of robot design and actuation parameters. The resulting framework supports the clinical translation of magnetic microrobots for precise therapeutic and surgical interventions in brain tissue.