The anatomical muscle geometry and fibre orientations of the medial gastrocnemius from a female cadaver were acquired from the Living Human Digital Library (LHDL) (Viceconti, et al., 2007). The geometry acquired from the LHDL project required further processing (smoothing, cropping and fibre vector interpolation) before they could be meshed and used for a finite element study. A simplified process map of this work-flow is shown in Figure 5.1.
Figure 5.1 Process chart showing the procedures followed from original imaging data to finite element simulation.
Acquire anatomical data from LHDL (splines and geometry. Pre-process crop and smooth geometry, fit mesh and calculate centroid co- ordinates Superimpose centroids to splines and apply inverse distance interpolation algorithm to extract interpolated vector field Apply unit vectors to each element via material model definition Set boundary conditions and run finite element simulation
81
The muscle was a surface membrane that had small discontinuities and only consisted of the posterior region of the muscle. Therefore, to get this to a geometry that could be used in finite element mechanics, the general order of pre-processing occurred as follows:
(i) The discontinuities were filled by an arbitrary automated hole fitting function.
(ii) A triangulation algorithm was applied, followed by a Delaunay three- dimensional function to re-create a close surface and three-dimensional volumetric geometry.
(iii) The geometry was then cropped, at specific regions to provide a musculo- tendon complex
(iv) The geometry was then smoothed using a Lagrangian function which therefore allowed a mesh to be fitted to the muscle.
This process is covered in detail below, and Figure 5.1 outlines the full process required to get the muscle geometry ready for meshing, and the process of incorporating the anatomical fibre orientation data into the finite element workflow. The muscle geometry of the medial gastrocnemius muscle acquired from the LHDL project is shown in Figure 5.4a. The images acquired from LHDL consisted of the rough surface path of the posterior side of the muscle, as shown in Figure 5.3 a. Before the model could be used, the geometry needed to be processed and refined, as the surface of the geometry had discontinuities (small holes). These were resolved by using the ‘hole-filing’ function in the LHPBuilder2 (Kohout & Clapworthy, 2012). This then provided a continuous anterior surface of the muscle. Some of the holes in the surface were very small, however have been shown in the image below (note, there were several holes on the surface and edge of the geometry, where most of them were situated towards the proximal end of the gastrocnemius. Figure 5.2 is representative of some of the miniscule holes that were present on the geometry surface and edge). Effectively, the holes consisted of small discontinuities in the surface of the muscle, as shown schematically in the image below. The hole filing algorithm has not been
2 LHPBuilder is an application developed using the Multimod Application Framework that provide
to the LHP participants a software tool to import, fuse, and store biomedical data on the Living Human Digital Library. www.swmath.org/software/6798
82
well documented in the LHPBuilder; however, the algorithm used is based on the mathematical framework reported in (Kohout & Clapworthy, 2012).
Figure 5.2 (a) The anterior view of the muscle showing areas in which the hole-filling algorithm was used in LHPBuilder (red elements). (b) Shows the posterior-lateral view of the geometry and (c) shows a closer view of the hole- filling algorithm by LHPBuilder as per element/pixel, where green dots denote geometry vertex boundaries automatically detected by LHPBuilder.
The continuous anterior surface was then imported into Paraview 5.2.0 – RC33 and the triangulate function was used there (Ayachit, 2016). This triangulate function created a boundary by creating surfaces between adjacent vertices. Consequently, the posterior surface of the muscle was estimated using the adjacent soleus muscle as a reference (as shown in Figure 5.3 b), to ensure that the estimated posterior side of the medial gastrocnemius does not penetrate the soleus muscle, as shown in Figure 5.4.
3 Paraview is an open source multiple-platform application for interactive, scientific visualization.
83
The 3D estimation of the muscle was based on creating a triangulation of the geometry information and its neighbouring points; so that the estimation a) created a closed 3D surface and b) ensured that one closed 3D surface did not extend/penetrate the geometry boundary of the neighbouring tissues. The three-dimensional Delaunay triangulation algorithm (Delaunay, 1934) - also available in Paraview, was used to define the entire closed surface of the 3D medial gastrocnemius.
Figure 5.3 (a) The anterior surface geometry of the medial gastrocnemius obtained from LHDL data. (b) Processed surface after using the hole-filing function available in LHP Builder to fix discontinuity. (c) An enclosed surface
describing the three-dimensional geometry of the medial gastrocnemius following three-dimensional triangulation.
The muscle was also cropped at the calcaneal distal tendon, so that while up to 40 mm of tendon at the insertion remained, the rest of the Achilles tendon was omitted,
84
as illustrated in Figure 5.4. This was done by obtaining the muscle-tendon geometry co-ordinates of these geometrical regions and cross-checked visually using the data from the palpated landmarks from LHDL data. The landmarks marked where the muscle fibres on the geometry began and the tendon fibres ended, so it was visually easy to differentiate between the muscle tissue and the tendon tissue. The soleus is shown in Figure 5.4 to ensure that the 3D estimation of the gastrocnemius geometry did not penetrate the boundary of the Soleus muscle. This was also done to qualitatively determine the accuracy of the estimation of the gastrocnemius geometry. Following the determination of the three-dimensional geometry outlined above, the surface of the muscle was still very coarse and could not be meshed for finite element simulations, therefore smoothing was carried out to ensure that a good mesh would fit the surface (Figure 5.4b). This was done by exporting the surface geometry as a stereolithography ASCII (STL - ASCII) file and imported into an FE Bio module - Preview4. Once in Preview, smoothing was done using its Laplacian smoothing function. This smoothing function is widely used to smooth a polygonal mesh in various computational geometry models (Herrmann, 1976; Sorkine, et al., 2004). The general method this function follows is to smooth each vertex or node, by moving the vertex to a new location based on the average location of its neighbours. The smoothing operation per vertex can be described by the following Equation 5.61:
1 1 N i j j x x N = =
(5.61)Where N is the number of neighbouring vertices to node
i
. The position of the j-th neighbouring vertex isx
j and xiis the new node location.The number of iterations selected in this context was the minimum required to allow for a simple mesh to be fitted onto the geometry, with minimal element distortion (aspect ratio <1.1 and skewness <0.2).
The data points of the medial gastrocnemius were originally defined in an anatomical co-ordinate system, so the long axis (middle) of the muscle was not aligned
4 PreView is a Finite Element (FE) preprocessor that has been designed specifically to set up finite
element problems. Is was designed for FEBio, and is a module of the open source FEBio software. https://febio.org/preview
85
with any orthogonal axes. The data points therefore needed to be transposed which resulted in the x-axis running down the long axis of the muscle (Figure 5.4c). The transposition was done to ensure that the displacement being applied at the insertion site was applied along the muscle axis, rather than at an angle to the muscle. This process of cropping, smoothing, transposition is illustrated in Figure 5.4.
Figure 5.4 (a) Original anterior surface of the gastrocnemius muscle, showing the tendon and muscle region (blue box). (b) After the three- dimensional Delaunay triangulation function, the volumetric muscle geometry is cropped and smoothed, in its original co-ordinate system. (c) The muscle was
then transposed into the new co-ordinate system with the x-axis running down the muscle length. The geometry is now ready for meshing.
Finally, the processed geometry was exported and meshed in ANSYS ICEM CFD5. A 10-node tetrahedral volumetric mesh was fitted with a maximum element size of 1 mm based on the mesh convergence analysis carried out later in this chapter.
The resulting mesh contains 144995 elements and 497016 nodes and a mesh density of 0.96 elements/mm3, as shown in Figure 5.5.
5 A pre-processing and meshing module for use in ANSYS finite element simulations that provides
86
Figure 5.5 Mesh of the medial gastrocnemius muscle
It should be noted that the cuboid model was meshed with both hexahedral and tetrahedral elements, and the results were very similar between the two. Tetrahedral elements however, were chosen for the anatomical geometry, as the mesh fitted better over the surface of the muscle in comparison to the hexahedral elements.