离散梯度法在基于图像的计算生物力学中的应用

DISCRETE GRADIENT METHOD FOR IMAGE-BASED BIOMECHANICAL STRESS ANALYSIS

介绍一种可用于计算生物力学的离散梯度方法,此方法可利用离散的点云模型直接进行数值模拟分析而不需要传统的几何模型。将离散梯度法应用于点云模型需要首先确定模型中点之间的相邻关系和每个点所分配的材料体积,然后通过用广义的有限差分的形式定义了梯度插值向量,并以此向量来近似函数在每个离散点上的梯度。从弱形式出发,推导建立了适用于弹性固体大变形问题的求解器,并具有和有限元法中双线性四边形单元一致的准确性和收敛性。着重描述了一种可以从医学图像中快速提取材料点并建立点云模型的方法,以及利用三角划分和重心划分确定材料点之间的相邻关系和每个材料点体积的具体过程。通过腹主动脉瘤膨胀的静力学模拟分析,展示了离散梯度法的实用性和准确性。该算法实现了基于医学图像进行生物力学分析的过程自动化,为病体特异性的研究和治疗提供便利和实用的工具。

This paper proposes a discrete gradient method for the stress analysis for biological systems.A point-cloud is taken as the geometric input instead of the conventional CAD model.Before applying the discrete gradient method on the point-cloud model,the neighboring relationship is deˉned among points as well as the volume occupied by each point.The gradient interpolation vectors,which can approximate the gradient of a function,are deˉned for each point in the form of the generalizedˉnite difference.The pointwise strain is calculated by using the discrete differentials involving the nodal displacements of a set of neighboring points.A mechanical solver using the discrete gradient method forˉnite strain elasticity is developed in weak form.It can be shown that this solver retains the similar accuracy and convergence rate of bilinear quadrilateralˉnite elements,with a locking-free behavior,and is more tolerant to the mesh distortion.An e±cient method is developed to extract the point-cloud model from medical images.Since a material constituent comprises pixels within a certain range of gray-scale values,the pixels within given thresholds are isolated and an initial pointcloud is formed.The physical coordinates are inferred from the image resolution.The Delaunay tessellation and the barycentric subdivision are utilized to provide the neighboring relation and the point volume.A static analysis of the abdominal aortic aneurysm in flation is carried out to demonstrate the usefulness of the method.Despite the use of the tessellation,the method is not element-based because the element-wise assumed solution is never constructed.A distinct feature of this method is that the entire process is completed with a minimal user interference or even fully automatically.This is signiˉcant for applications where a timely analysis is desired.