Fast algorithms for computing the effective properties of cancellous bone from micro-CT scans

•  Cancellous bone is a two-phase composite with solid trabeculae and bone marrow. Effective properties of cancellous bones are important in assessing bone health. Computation of the effective properties is carried out by solving partial differential equations in the domain constructed from micro-CT scans. Due to the very complex microstructure of the trabeculae, traditional meshing softwares fail to handle the meshing task. A state-of-the-art 3D segmentation algorithm was applied to a stack of micro-CT scans of cancellous bone, followed by the construction of the signed distance function ɸ of the 3D structure (Tsai and Kim). We developed a meshing algorithm modified from Matlab-based DISTMESH, which can efficiently generate high-quality boundary mesh for the cancellous bone. Finally, the partial differential equations were solved by the Boundary Element Method (BEM) accelerated by the Fast Multipole Method (FMM), which is an O(N) fast algorithm. This poster presents key ideas of the algorithms, the implementation and the numerical results.

Abstract

•  For best performance of the FMM-BEM on the three-dimensional bone Ω1, we apply distmesh, introduced by P.-O. Persson, to generate a high-quality mesh on it. Distmesh first generates an initial mesh on ���"Ω1 by the function “isosurface” in Matlab. It then regards the output as a force-based system of truss (points) and bars (edges of triangles) with force-displacement function f(x)=k(ℓ0-ℓ) and mainly use signed distance function ɸ to project the points back to boundary after movement.

•  Distmesh2d, a branch of distmesh, works surprisingly well for a planar surface since it applies Delaunay triangulation. However, due to the bad triangulation generated by the Matlab function isosurface, distmesh always fails to generate a good mesh on the boundary of geometrically complicated non-planar surface such as that of trabecular bone.

•  Let Ω1 be the trabeculae and Ω2 the closure of R\Ω1, where R⊂ℝ3 is a bounded box containing Ω1, and define ! =Ω1⋂Ω2. The modified distmesh first generates a mesh on ! . This mesh is then used to generate meshes on ���Ω1⋂"R and Ω2⋂"R respectively by distmesh2d in a conformal fashion. With this high-quality triangular mesh, we are able to compute the effective properties of the bone by solving a partial differential equation with FMM-BEM.

Introduction

•  Distmesh sometimes fails (generating overlapping triangles) even on a surface with a smooth distance function.

•  Distmesh always fails near the points where the distance function ɸ is of bad condition. The signed distance function is of “bad condition” at a point if a small change of the point location in some direction can result in a substantial change of the direction of &ɸ.

•  Distmesh does not support for generating a mesh on a surface with non-empty boundary.

WHY modify?

Resulting mesh on Ω2 Triangle quality of mesh on Ω2

Result by solving equation Result by solving equation

The figure shows overlapping triangles even on a unit sphere,

whose distance function is clearly

smooth everywhere

We detect overlapping triangles by

considering the orientation of triangles and control it by fixing

their vertices.

The figure shows the failure on a cylinder,

whose distance function on the

circumference of its two bases is of bad

condition

We tackle this problem by fixing all initial mesh points on "R. At the end,

we collapse the fixed points on "R which are too close into one point

To generate the mesh on each plane of "R ���(i) use mesh points of ! on��� "R as fixed points ���(ii) recompute the signed ��� distance functions ɸi w.r.t. ��� this planar curves ���(iii)use adaptation 1+λ|ɸi(x)|

This figure shows the uniform mesh. The

algorithm for uniform mesh always runs fast and has less number of overlapping triangles

But for better use of the fixed number of triangles

to approximate the surface, we use curvature of ! for adaptation. The result is showed in this

figure

n 1 1.59742233 2 1.56044835 … … 12 1.57108043 13 1.57119655 14 1.57111973

•  The histogram on the left shows that the resulting mesh is of high quality. •  The two colored figures indicate the good performance of BEM-FMM using our

mesh as input. •  The convergence in each direction also provides a strong evidence of the success

of the mesh. •  The most surprising thing is that we can generate a mesh with discretized signed

distance function of size 145x145x37, (resulting in 30000 points and more than 60000 triangles on "Ω1 and "Ω2 each) in 0.5 hours and complete the task for effective properties in 2 hours with a PC Laptop of 4GB memories.

Summary

•  Since distmesh is well-known for its speed and mesh quality, we need to assure that the modifications do not slow it down significantly. This is the case according to our numerical experiments.

How fast is it?

Results with constants:

ϵ1=0.3 ϵ2=1.96 tol=1e-4

The iteration process is based on Dirichlet-to-Neumann Map

.

Formulation of effective properties

Student:Wai-Yip Chan, the Chinese University of Hong Kong Advisor: Prof. Miao-Jung Yvonne Ou, University of Delaware