Modelling of magnetic vortex microdisc dynamics under varying magnetic field in biological viscoelastic environments

Magnetically driven microparticles provide a versatile platform for probing and manipulating biological systems, yet the physical framework governing their actuation in complex environments remains only partially explored. Within the field of cellular magneto-mechanical stimulation, vortex microdiscs have emerged as particularly promising candidates for developing novel therapeutic approaches. Here, we introduce a simplified two-dimensional model describing the magneto-mechanical response of such particles embedded in viscoelastic media under varying magnetic fields. Using a Maxwell description of the medium combined with simplified elasticity assumptions, we derive analytical expressions and support them with numerical simulations of particle motion under both oscillating and rotating magnetic fields. Our results show that rotating fields typically induce oscillatory dynamics and that the transition to asynchronous motion occurs at a critical frequency determined by viscosity and stiffness. The amplitude and phase of this motion is governed by the competition between magnetic and viscoelastic contributions, with particle motion being strongly impaired when the latter dominate. Energy-based considerations further demonstrate that, within the frequency range explored of few tens of Hertz, no heat is generated -- distinguishing this approach from magnetic hyperthermia -- while the elastic energy transferred to the surrounding medium is, in principle, sufficient to perturb major cellular processes. This work provides a simple framework to anticipate the first-order influence of rheological properties on magnetically driven microdisc dynamics, thereby enabling a better understanding of their impact in cells or extracellular materials and bridging the gap between experimental observations and theoretical modelling.