# Point cloud method for image based biomechanical stress analysis

This paper introduces a point-cloud method for the stress analysis in biological systems. The method takes a point cloud as the geometric input. Each point represents a small material volume which is assumed to undergo a uniform deformation during motion. The strain at each point is computed from the relative displacements of a set of neighboring points. Delaunay tessellation is utilized to provide the neighboring relation and the point volume. An eﬃcient method is developed to extract point-cloud model from medical images. An aorta inﬂation problem and a skull impact problem are presented to demonstrate the utility of the method.

Medical images provide a pixel-resolution point-cloud depiction of the anatomy of biological organs. In the traditional ﬁnite element analysis, a point cloud needs to be ﬁrst converted into a CAD (Computer Aided Design) solid and then meshed into a ﬁnite element model. The conversion from is by no means a trivial task. Biological organs often contain intricate geometry features that are diﬃcult to describe in CAD geometry. Even if a CAD model is derived, meshing the solid body may still present a signiﬁcant challenge. With the advance of mesh generators, it is now possible to generate kinematically admissible tetrahedron meshes for an arbitrary 3D geometry. But it is yet impossible to automatically generate analysis-quality mesh for complicate geometries. Presently mesh generation is semi-automatic at the best, and often user’s interference is needed. For biomedical systems, mesh generation remains the most tedious and time-consuming step in the procedure of analysis. Recently, the present authors developed a family of discrete gradient methods (DGM) [1, 2] for stress analysis. The method is motivated by imaged-based applications; an underlying goal is to perform stress analysis directly on point clouds without converting them into a ﬁnite element model. The discrete gradient method computes the strain at a point from the relative displacements in a set of neighboring points. This pointwise gradient is utilized to establish the governing equation from a discrete weak form. The method consists of two essential ingredients, one is the identiﬁcation of the neighboring set, and the other is the computation of discrete gradient. In [1, 2], we described two approaches of deﬁning neighbors and derived the respective discrete gradient formula. These two formulations pass the constant strain patch test by construction. It has been demonstrated that DGM retains the accuracy and convergence rate of bilinear quadrilateral elements, exhibits a locking-free behavior, and is more tolerant to mesh distortion. The method is easy to implement since it does not construct the continuum assumed solutions and thus bypasses the computation of basis functions, either element-based or meshfree. 2More importantly, the method has the potential of lending itself to a fully automated procedure for image-based analysis. The objective of this paper is to describe such a procedure. The proposed paradigm is markedly diﬀerent from the voxel ﬁnite element method (VFEM) whereby voxels are directly converted into eight-node hexahedral elements [3, 4, 5, 6, 7, 8]. A limitation of the voxel method, as pointed put by [9], lies in the jagged stair-step surfaces. The nonsmooth surface can cause numerical problems and sacriﬁces the algorithmic robustness. To overcome this shortcoming, smoothing techniques were adapted [10, 11]. Algorithms were developed to modify the hexahedrons near the surface while keeping the interior mesh intact [12, 13, 14]. These strategies amend the hexahedron mesh with layers of tetrahedron elements near the boundary. In contrast, the current method is not element-based; the continuum assumed solution is never constructed. The paper starts with a review of the discrete gradient method. An eﬃcient method for segmenting point clouds from medical images is introduced in Section 3, where the entire procedure pipelining from point-cloud registration to stress analysis is also described. Numerical examples are presented in Section 4 to demonstrate the method.

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