The Penn State Computational Biomechanics Group, Department of Mechanical and Nuclear Engineering, Pennsylvania State University , University Park, PA , USA.
Department of Anthropology, Pennsylvania State University , University Park, PA , USA.
Front Bioeng Biotechnol. 2015 Mar 19;3:24. doi: 10.3389/fbioe.2015.00024. eCollection 2015.
Bones of the cranial vault are formed by the differentiation of mesenchymal cells into osteoblasts on a surface that surrounds the brain, eventually forming mineralized bone. Signaling pathways causative for cell differentiation include the actions of extracellular proteins driven by information from genes. We assume that the interaction of cells and extracellular molecules, which are associated with cell differentiation, can be modeled using Turing's reaction-diffusion model, a mathematical model for pattern formation controlled by two interacting molecules (activator and inhibitor). In this study, we hypothesize that regions of high concentration of an activator develop into primary centers of ossification, the earliest sites of cranial vault bone. In addition to the Turing model, we use another diffusion equation to model a morphogen (potentially the same as the morphogen associated with formation of ossification centers) associated with bone growth. These mathematical models were solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data showing skull bone formation in mouse at different embryonic days in mice carrying disease causing mutations and their unaffected littermates. The results show that the relative locations of the five ossification centers that form in our model occur at the same position as those identified in experimental data. As bone grows from these ossification centers, sutures form between the bones.
颅骨的骨骼是由间充质细胞在围绕大脑的表面分化为成骨细胞形成的，最终形成矿化骨骼。导致细胞分化的信号通路包括由基因信息驱动的细胞外蛋白的作用。我们假设与细胞分化相关的细胞和细胞外分子的相互作用可以使用图灵的反应扩散模型来建模，该模型是由两种相互作用的分子（激活剂和抑制剂）控制的图案形成的数学模型。在这项研究中，我们假设激活剂的高浓度区域会发展成初级成骨中心，这是颅骨骨骼的最早部位。除了图灵模型之外，我们还使用另一个扩散方程来模拟与骨生长相关的形态发生素（可能与形成成骨中心相关的形态发生素相同）。这些数学模型使用有限体积法求解。使用包含大量实验数据的数据集来确定计算域和模型参数，这些数据显示了在携带致病突变及其未受影响的同窝仔鼠的不同胚胎日龄的小鼠中头骨骨形成。结果表明，我们模型中形成的五个成骨中心的相对位置与实验数据中确定的位置相同。随着骨骼从这些成骨中心生长，骨缝在骨骼之间形成。
