Harrigan T P, Hamilton J J
Department of Orthopedic Surgery, University of Missouri-Kansas City School of Medicine 64108.
J Biomech. 1992 May;25(5):477-88. doi: 10.1016/0021-9290(92)90088-i.
The origin of unstable bone remodelling simulations using strain-energy-based remodelling rules was studied mathematically in order to assess whether the unstable behavior was due to the mathematical rules proposed to characterize the processes, or to the numerical approximations used to exercise the mathematical predictions. A condition which is necessary for the stability of a strain-energy-based remodelling theory was derived analytically using the calculus of variation. The analytical result was derived using a simple elastic model which consists of a long beam loaded by an axial force and a bending moment. This loading situation mimics the coupling between local density and global density distributions seen in vivo. A condition necessary for a stable remodelling scheme is arrived at, but the conditions necessary to guarantee a stable remodelling scheme are not. In this remodelling scheme, the elastic modulus is proportional to volumetric density raised to an exponent n, and the microstructural stimulus is taken as the strain energy density divided by volumetric density raised to an exponent m. In order for a remodelling scheme to be stable in this loading situation, m must be greater than n. Finite-difference time-stepping is used to verify the predictions of the analytical study. These numerical studies appear to confirm the analytical studies. Physiologic interpretation of the behavior found with n greater than m indicates that this type of unstable behavior is unlikely to be observed in vivo. Since numerical approximations are not made in deriving this stability condition, we conclude that the mathematical rules proposed to characterize bone remodelling based on strain energy density should meet this condition to be relevant to physiologic bone remodelling.
为了评估基于应变能的重塑规则进行的不稳定骨重塑模拟的起源，从数学角度进行了研究，以确定不稳定行为是由于用于描述这些过程的数学规则，还是用于实现数学预测的数值近似方法。利用变分法解析推导了基于应变能的重塑理论稳定性的必要条件。分析结果是通过一个简单的弹性模型得出的，该模型由一根承受轴向力和弯矩的长梁组成。这种加载情况模拟了体内局部密度和整体密度分布之间的耦合。得出了稳定重塑方案的必要条件，但未得出保证稳定重塑方案的必要条件。在这个重塑方案中，弹性模量与体积密度的n次方成正比，微观结构刺激被视为应变能密度除以体积密度的m次方。为了使重塑方案在这种加载情况下稳定，m必须大于n。使用有限差分时间步长法来验证分析研究的预测。这些数值研究似乎证实了分析研究的结果。对n大于m时发现的行为进行生理学解释表明，这种类型的不稳定行为在体内不太可能被观察到。由于在推导这个稳定性条件时没有进行数值近似，我们得出结论，基于应变能密度提出的用于描述骨重塑的数学规则应满足此条件，才与生理性骨重塑相关。
