Racila M, Crolet J M
Department of Mathematics, University of Franche-Comté, Besançon, France.
Technol Health Care. 2006;14(4-5):379-92.
It is well known that long term behavior of implants depends on bone remodeling. In the absence of a model of this phenomenon, few numerical simulations take into account bone remodeling. Some laws have been proposed but they cannot be used in the essential area surrounding the implant. We propose a multi-scale approach: cortical bone is structured in a hierarchical way consisting of five levels. The cortical part of a given bone is made up of various areas having different physical properties adapted to locally existing conditions. A Bony Elementary Volume denotes the elementary part of such a zone which constitutes our first level. The other levels are in conformity with our previous studies: osteon, lamella, fibre and fibril. This latter is composed by collagen and hydroxyapatite (Hap) occurring in a viscous liquid containing mineral ions. Mathematical homogenisation theory is used to determine equivalent macroscopic properties of a BEV, knowing the physical properties of collagen and Hap and the architectural description of this bony structure. For improving the performance of our simulation software, a new behavior law has been introduced with no continuity between the various levels. The effect of the fluid at the nanoscopic scale is modeled by a constant pressure. Recent developments allow us to determine the magnitude of various entities at nanoscopic scale from information at the macroscopic level. Realized simulations show that the assumption of constant pressure is not sufficient to characterize the nanoscopic mechanical behaviour. This point needs a more complex model with the introduction of a coupling between structure and fluid. This aspect is in development.
众所周知，植入物的长期行为取决于骨重塑。在缺乏这种现象的模型的情况下，很少有数值模拟考虑骨重塑。已经提出了一些定律，但它们不能用于植入物周围的关键区域。我们提出了一种多尺度方法：皮质骨以由五个层次组成的分层方式构建。给定骨骼的皮质部分由具有适应局部现有条件的不同物理特性的各个区域组成。一个骨基本体积表示构成我们第一层次的这样一个区域的基本部分。其他层次与我们之前的研究一致：骨单位、骨板、纤维和原纤维。后者由存在于含有矿物质离子的粘性液体中的胶原蛋白和羟基磷灰石（Hap）组成。利用数学均匀化理论，在已知胶原蛋白和Hap的物理特性以及这种骨结构的结构描述的情况下，确定骨基本体积的等效宏观特性。为了提高我们模拟软件的性能，引入了一种新的行为定律，各层次之间不存在连续性。纳米尺度下流体的影响通过恒定压力来建模。最近的进展使我们能够根据宏观层面的信息确定纳米尺度下各种实体的大小。已实现的模拟表明，恒定压力的假设不足以表征纳米尺度的力学行为。这一点需要一个更复杂的模型，引入结构与流体之间的耦合。这方面正在开发中。
