Geris Liesbet, Gerisch Alf, Sloten Jos Vander, Weiner Rüdiger, Oosterwyck Hans Van
Division of Biomechanics and Engineering Design, Katholieke Universiteit Leuven, Celestijnenlaan 300C (PB 2419), 3001 Leuven, Belgium.
J Theor Biol. 2008 Mar 7;251(1):137-58. doi: 10.1016/j.jtbi.2007.11.008. Epub 2007 Nov 19.
The process of fracture healing involves the action and interaction of many cells, regulated by biochemical and mechanical signals. Vital to a successful healing process is the restoration of a good vascular network. In this paper, a continuous mathematical model is presented that describes the different fracture healing stages and their response to biochemical stimuli only (a bioregulatory model); mechanoregulatory effects are excluded here. The model consists of a system of nonlinear partial differential equations describing the spatiotemporal evolution of concentrations and densities of the cell types, extracellular matrix types and growth factors indispensable to the healing process. The model starts after the inflammation phase, when the fracture callus has already been formed. Cell migration is described using not only haptokinetic, but also chemotactic and haptotactic influences. Cell differentiation is controlled by the presence of growth factors and sufficient vascularisation. Matrix synthesis and growth factor production are controlled by the local cell and matrix densities and by the local growth factor concentrations. Numerical simulations of the system, using parameter values based on experimental data obtained from literature, are presented. The simulation results are corroborated by comparison with experimental data from a standardised rodent fracture model. The results of sensitivity analyses on the parameter values as well as on the boundary and initial conditions are discussed. Numerical simulations of compromised healing situations showed that the establishment of a vascular network in response to angiogenic growth factors is a key factor in the healing process. Furthermore, a correct description of cell migration is also shown to be essential to the prediction of realistic spatiotemporal tissue distribution patterns in the fracture callus. The mathematical framework presented in this paper can be an important tool in furthering the understanding of the mechanisms causing compromised healing and can be applied in the design of future fracture healing experiments.
骨折愈合过程涉及多种细胞的作用与相互作用，受生化和机械信号调控。成功愈合过程的关键是恢复良好的血管网络。本文提出了一个连续数学模型，该模型描述了不同的骨折愈合阶段及其仅对生化刺激的反应（一种生物调节模型）；此处排除了机械调节作用。该模型由一个非线性偏微分方程组组成，描述了愈合过程中不可或缺的细胞类型、细胞外基质类型和生长因子的浓度及密度的时空演变。该模型在炎症期之后开始，此时骨折骨痂已经形成。细胞迁移不仅通过接触运动来描述，还包括趋化和趋触影响。细胞分化由生长因子的存在和充足的血管化控制。基质合成和生长因子产生由局部细胞和基质密度以及局部生长因子浓度控制。给出了该系统的数值模拟，其参数值基于从文献中获得的实验数据。通过与标准化啮齿动物骨折模型的实验数据进行比较，证实了模拟结果。讨论了对参数值以及边界和初始条件的敏感性分析结果。愈合情况受损的数值模拟表明，响应血管生成生长因子建立血管网络是愈合过程中的关键因素。此外，正确描述细胞迁移对于预测骨折骨痂中现实的时空组织分布模式也至关重要。本文提出的数学框架可以成为深入理解导致愈合受损机制的重要工具，并可应用于未来骨折愈合实验的设计。
