Formulation of a mathematical model of bone remodeling taking accont of bone resorption by overload and its application to clinical problems

骨骼改变其结构和力学性能以适应力学环境的现象称为机械性骨重建。在本研究中，我们建立了一个机械性骨重建的数学模型来描述骨重建随时间的唯象过程，并检验了所提出的模型的有效性和准确性。文中还讨论了该模型在ABAQUS有限元程序中的应用。研究结果总结如下：1.通过n+1个宏观内状态变量为n+1的序列演化方程，描述了从力学刺激到骨密度变化的重建生理信号传递过程，建立了单轴载荷的数学模型。该模型可以定量描述骨改建的时间依赖性过程。该模型还体现了加载历史的独立性。将该模型推广到多轴加载情况，并对其进行了改进，以考虑超载引起的骨吸收。所提出的模型显示了临床上足够的骨重建的历史依赖性：该模型可能适用于康复和训练问题。该模型被纳入有限元程序ABAQUS中，以检验该模型的适用性。建立了单元件模型，并与单独使用该模型的仿真结果进行了比较。这两个结果在性质上是一致的。并将该模型应用于股骨的有限元模型。结果表明，骨小梁的结构设计是最佳的。

The phenomenon where bone changes its structure and mechanical property to adapt to mechanical circumstances is called mechanical bone remodeling. In the present study, we formulate a mathematical model of mechanical bone remodeling to describe phenomenological process of bone remodeling with time, and the validity and the accuracy of the proposed model are examined. The incorporation of the model into the finite element code ABAQUS is also discussed. The results obtained from the present study are summarized in the following :1. A mathematical model for uniaxial loading is formulated by representing physiological signal transmission processes of remodeling from the mechanical stimulus to change of bone density by n+1 sequential evolution equations with n+1 macroscopic internal state variables.2. This model can quantitatively describe a time dependent process of bone remodeling. The model also shows the independence of loading histories.3. The model was extended to the multiaxial loading case, and is improved to take into account bone resorption by overload.4. The proposed model showed clinically adequate history dependence of bone remodeling : the model may be applicable to problems of rehabilitation and training.5. The model was incorporated into the finite element code ABAQUS to examine the applicability of the model. A single element model is constructed, and the results are compared with the simulation results using only the model. The two results coincide with each other qualitatively.6. The model was also applied to a finite element model of femur. The results showed the optimum structural design of bone trabecula.