How to implement constrained mixture growth and remodeling algorithms for soft biological tissues.

Area of Science:

• Biomechanics
• Computational Biology
• Materials Science

Background:

• Biological soft tissues exhibit adaptive remodeling in response to mechanical stimuli to maintain homeostasis.
• The constrained mixture theory provides a framework for modeling tissue growth and remodeling, involving material degradation and replacement.

Purpose of the Study:

• To provide a comprehensive mathematical overview of the constrained mixture theory and its homogenized equivalent.
• To offer practical guidelines for the numerical implementation of these theories in modeling soft tissue adaptation.
• To evaluate the performance and efficiency of different implementation approaches.

Main Methods:

• Detailed mathematical formulation of the constrained mixture theory and its homogenized counterpart.
• Development and testing of numerical implementation strategies for growth and remodeling models.
• Comparative analysis of the classical and homogenized theories using multiple simulation examples.
• Conducting a time step convergence study to assess numerical stability and accuracy.

Main Results:

• Demonstrated good agreement between the classical constrained mixture theory and its homogenized version.
• The homogenized theory showed superior time efficiency compared to the classical approach.
• A time step convergence study highlighted the critical need for small time steps, particularly for the classical theory, to ensure accurate results.

Conclusions:

• The constrained mixture theory and its homogenized equivalent offer effective tools for modeling biological soft tissue growth and remodeling.
• The homogenized theory presents a more time-efficient computational approach.
• Careful selection of numerical parameters, such as time step size, is crucial for reliable simulations of tissue adaptation.