A comparison of linear and quadratic tetrahedral finite elements for image-guided surgery applications

Timothy J Carter*, Christine Tanner and David J Hawkes

Centre for Medical Image Computing, University College London

1 Introduction

One approach to compensating for soft tissue deformation during surgery is to use a finite element model. The time constrains of surgery mean that approximations must be made, so that the model can be rapidly deformed. The use of linear shape functions is one possible approximation, which can be of particular benefit since it allows analytic integrals to be calculated, whilst higher order elements generally require numeric integration. However, due to its high water content soft tissue is typically almost incompressible and linear tetrahedral elements are particularly susceptible to locking when modelling almost incompressible material. Locking is an excessive stiffness of the mesh, which results in smaller displacements being calculated than actually occur.

Figure 1. Linear (four-noded) and quadratic (ten-noded) elements. Nodes are marked by dots.

We compare here the displacement accuracy of finite element models using linear elements with the finite element models using quadratic elements (illustrated in Figure 1) when simulating the compression of the breast.

2 Method

In this paper we employ the technique presented by Tanner et al. [1]. MR images were acquired of two volunteers. Each volunteer was positioned in a fixation device designed for performing breast biopsies. The right breast was placed between the two plates, in the sagittal plane, without any compression for the first image. The plate on the breasts medial side was then kept immobile whilst the plate on the lateral side was moved medially by 15mm, resulting in compressions of around 20%, and a second image was acquired.

The MR images of the breast were segmented into fatty and fibroglandular tissue. The segmentations were then blurred, and downsampled to reduce the number of elements created during the meshing step. A 3D triangulation of the outer surface of the breast was obtained using marching cubes and decimation techniques. The triangulated volumes were meshed into ten-noded tetrahedral elements with quadratic shape functions. The midpoint nodes were then removed from the elements to provide a geometrically similar four-noded mesh with linear shape functions.

The material properties and number of elements for the models are shown in Table 1. The Poissons ratios were selected according to the values which were found to minimise the maximum landmark error (defined in the results section) for quadratic tetrahedral elements.

Efat /kPa

Efibroglandular /kPa

Poissons ratio Number of elements

Volunteer 1 1 5 0.495 34873 Volunteer 2 1 5 0.2 40636

Table 1. Youngs Modulus (E), Poissons ratio and number of elements for each model.

Displacements were established by using a full 3D non rigid registration algorithm [2] to register the MR image of the compressed breast to the MR image of the uncompressed breast. Displacements were applied to nodes on

*email : t.carter@cs.ucl.ac.uk

the medial and lateral surfaces of the breast, and to the nodes along the chest wall. All other nodes were unconstrained.

The effect of element order on accuracy was assessed by comparing the accuracy with which the two models were able to recover the displacements of 12 landmarks identified in the pre- and post-deformation images (the landmark error), and by taking the magnitude of the difference in displacement for the vertex nodes between the linear and quadratic elements, excluding the nodes which were displaced as boundary conditions (the node differences).

3 Results

The results are shown in Table 2, and the node differences are visually illustrated for Volunteer 1 in Figure 2. Examination of the node differences indicates that regions surrounded by the nodes upon which the boundary conditions are imposed have smaller node difference, whilst larger node differences appear to occur in regions of high curvature.

Landmark error Linear element

mean (maximum) /mm

Landmark error Quadratic element mean (maximum)

/mm

Node differences

mean (95th percentile) /mm

Volunteer 1 2.02 (3.54) 2.18 (3.65) 0.56 (1.17) Volunteer 2 2.36 (3.67) 2.31 (3.45) 0.23 (0.77)

Table 2. Comparison of linear and quadratic landmark errors, and the node differences.

Figure 2 Surface rendering and section showing node differences for Volunteer 1

4 Discussion and future work

The difference between the node displacements using linear elements and the node displacements using quadratic elements is small compared to the errors which are likely to arise from other causes, such as imperfect boundary conditions being applied to the model. This difference is also relatively small compared to the accuracy necessary for our intended application, namely image-guided breast surgery, where a system accuracy of around 5mm is desirable. The landmark errors for the linear and quadratic elements are comparable. That the landmark errors for Volunteer 1 is smaller when using linear rather than quadratic elements does not indicate that linear elements are more appropriate, but instead illustrates that the material model does not perfectly reflect the behaviour of tissue. There is no indication that locking has occurred during the deformation of the model of Volunteer 1, which might have been expected due to the high Youngs modulus.

We conclude that linear quadratic elements may be appropriate for modelling soft tissue during surgery. We now intend to investigate the effect of element size on model accuracy, and to repeat these experiments for scenarios which replicate our surgical scenario more closely.

Acknowledgements This work is supported by the MIAS-IRC (EPSRC GR/N14248) through a Doctoral Training Award to TJC. References 1. C.Tanner, A. Degenhard, J.A. Schnabel et al. A method for the Comparison of Biomechanical Breast Models In IEEE

Workshop on Mathematical Models in Biomedical Image Analysis, Kauai, USA, pp. 11-18, 2001 2. D. Rueckert, L.I. Sonoda, C. Hayes et al. Non-rigid Registration using Free-Form Deformations: Application to Breast

MR images. IEEE Transactions on Medical Imaging, 7, pp. 1-10, 1999