Original Article

A mathematical framework to study the mechanical and cellular activity in regulating the fracture healing process

¹ College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou, China
² College of Optical, Mechanical and Electrical Engineering, Zhejiang A and F University, Hangzhou, China
³ Key Laboratory of Special Purpose Equipment and Advanced Processing Technology, Ministry of Education and Zhejiang Province, Zhejiang University of Technology, Hangzhou, China
⁴ Department of Mechanical and Electrical Engineering, Quzhou College of Technology, Quzhou, China
⁵ Department of Orthopedics, People's Hospital of Quzhou, The Affiliated Hospital of Wenzhou Medical University, Quzhou, China

^* e-mails: xunyi@zjut.edu.cn; yqh@zafu.edu.cn

Received: 19 May 2025
Accepted: 1 August 2025

Abstract

The coupling mechanism between mechanical stimulation and biological response during fracture healing has not been fully elucidated. This study establishes an innovative mathematical model to analyze this complex process. Based on the phenomenon of early weight-bearing in clinical application of Taylor external fixation frames, we constructed a bidirectional coupling mechanic-biological computing framework. The model contains two interactive modules: the mechanical module uses porous media theory to simulate the fluid-solid coupling behavior of tissues, and the biological module uses reaction-diffusion equations to characterize processes such as cell proliferation, migration, differentiation and matrix metabolism. The two modules realize real-time data interaction through customized developed interface programs. The core of the model is to establish a quantitative relationship between mechanical stimulation (through physical quantities such as fluid shear stress) and the final morphology of mesenchymal stem cells (differentiated into chondrocytes or osteoblasts). As the simulation progresses, dynamic changes in the extracellular matrix will feedback and adjust the mechanical parameters of the tissue to form a closed-loop system. The axisymmetric finite element method was used to numerical verification of long bone fracture cases, and the results showed that the model could accurately reproduce the spatiotemporal characteristics of callus evolution. Comparison with clinical observational data confirms the reliability of the model. Parameter analysis shows that this model can quantify the impact of multiple factors such as stem cell microenvironment, initial callus characteristics and fixation device stiffness on healing mode, providing a theoretical basis for personalized treatment.