Department of Mathematics/Program, in Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, IA 52242-1419, United States.
Math Biosci Eng. 2012 Apr;9(2):281-95. doi: 10.3934/mbe.2012.9.281.
Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.
骨重塑的异常与许多骨骼疾病有关，如骨质疏松症和多发性骨髓瘤。计算和数学建模可以帮助治疗和治疗，以及理解基本的生物学。不同的建模方法可以深入了解现象的不同方面，因此拥有各种计算和数学模型的武器库是很有用的。在这里，我们开发了一种可以有效描述观察到的复杂几何形状和空间行为的许多方面的骨重塑数学表示。在骨和骨髓区域之间存在一个锐利的界面。此外，由于重塑，骨的表面会进出，即在正常方向上移动。基于这些观察，我们使用水平集函数来表示重塑的空间行为。我们详细阐述了一个用于破骨细胞和成骨细胞群体动态的时间模型，以确定影响骨与骨髓之间界面变化的骨量变化。我们展示了基于我们的计算模型的模拟，这些模拟显示了骨与骨髓之间界面的运动是骨重塑的结果。模拟表明，有可能在复杂的几何形状中捕捉到骨重塑的空间行为，就像在体外和体内发生的那样。通过采用水平集方法，可以开发骨重塑空间行为的计算和数学表示。通过在这个形式主义中包含更多细节，如更复杂的细胞因子相互作用和准确的参数值，可以模拟与骨重塑有关的具有空间行为的现象，就像在体外和体内一样。这使得在硅中进行更接近实验观察的实验成为可能。
