An adaptive numerical method for multi-cellular simulations of tissue development and maintenance.

Area of Science:

• Computational Biology
• Biophysics
• Mathematical Modeling

Background:

• Multi-cellular models are increasingly popular for simulating biological systems.
• Existing simulation tools utilize various numerical methods.
• Understanding the suitability of these methods for tissue dynamics is crucial.

Purpose of the Study:

• To review and assess numerical methods for multi-cellular modeling, particularly for tissue development, maintenance, and disease.
• To introduce and evaluate an adaptive time-stepping algorithm for improved simulation efficiency and accuracy.
• To focus on off-lattice, mechanics-based models using ordinary differential equations for cell movement.

Main Methods:

• Review of numerical integration techniques including Forward Euler, Runge-Kutta 4, and Adams-Bashforth 2.
• Implementation of an adaptive time-stepping algorithm.
• Analysis of convergence properties under specific conditions (event synchronization, boundary handling).
• Comparative simulations to assess error and runtime.

Main Results:

• All tested numerical methods can achieve correct order convergence with careful handling of events and boundaries.
• An adaptive time-stepping method using Runge-Kutta 4 with moderate adaptivity provides the best balance between L∞ error and computational time.
• Judicious selection of numerical methods can accelerate simulations by 10-60 times compared to basic Forward Euler methods.
• Adaptive time-stepping further enhances simulation speed by a factor of approximately 4.

Conclusions:

• The choice of numerical method significantly impacts the efficiency of multi-cellular simulations.
• Adaptive time-stepping, particularly with Runge-Kutta 4, is highly effective for optimizing simulations of tissue and organ dynamics.
• Accurate and efficient multi-cellular modeling is essential for advancing research in development, maintenance, and disease.