A mathematical phase field model predicts superparamagnetic nanoparticle accelerated fusion of HeLa spheroids for field guided biofabrication

Abstract

In vitro tissue models are crucial for regenerative medicine, drug discovery, and the reduction of animal testing. 3D bioprinting, particularly when utilizing magnetic manipulation of cell spheroids, provides precise control over tissue architecture. However, existing mathematical models lack the precision to capture the interplay between biological dynamics and magnetic forces during spheroid fusion. This study developed and validated a novel mathematical model that simulates magnetically assisted spheroid fusion, taking into account cell migration, adhesion, and the effects of external magnetic fields. The model integrates principles of cell mechanics, fluid dynamics, and magnetostatics, implemented in COMSOL Multiphysics. Experimental validation used HeLa cell spheroids bioprinted with superparamagnetic iron oxide nanoparticles (SPIONs). Spheroid fusion was monitored with and without an external magnetic field using confocal microscopy. Rigorous statistical analysis (MAE, RMSE, MAPE, R², Chi-Square, Bland-Altman, and variance-weighted metrics) was used to evaluate model performance. The model accurately predicted accelerated fusion under magnetic manipulation, reducing fusion time from approximately 7 days (without field) to 2 days. High R² values (> 0.99 for two-spheroid fusion and > 0.97 for multi-spheroid systems) and narrow confidence intervals demonstrated strong agreement between the simulation and the experiment. Increased system complexity introduced slightly higher error variability, but the model maintained robust predictive capabilities. Spheroid disassembly was observed in the four-spheroid case, highlighting the complex interplay of magnetic forces and cellular reorganization. This validated, high-precision model represents a significant advancement in tissue engineering, providing a powerful tool for optimizing bioprinting protocols, designing complex tissue constructs, and advancing in vitro model development. This breakthrough has implications for regenerative medicine and drug discovery while also highlighting the importance of addressing nanoparticle safety concerns.