A comparison of numerical methods used for finite element modelling of soft tissue deformation

Soft tissue deformation is often modelled using incompressible non-linear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making up the mesh. The effect of these choices on the accuracy of the computed solution is investigated, using a selection of model problems motivated by typical deformations seen in soft tissue modelling. Model problems are set up with discontinuous material properties (as is the case for the breast), steeply changing gradients in the body force (as found in contracting cardiac tissue), and discontinuous first derivatives in the solution at the boundary, caused by a discontinuous applied force (as in the breast during mammography). It was found that the choice of pressure basis functions is vital in the presence of a material interface, higher-order schemes do not perform as well as may be expected when there are sharp gradients, and in general it is important to take the expected regularity of the solution into account when choosing a numerical scheme.

[1]  J. Weiss,et al.  Subject‐specific finite element analysis of the human medial collateral ligament during valgus knee loading , 2003, Journal of orthopaedic research : official publication of the Orthopaedic Research Society.

[2]  Adrian J. Lew,et al.  Discontinuous Galerkin methods for non‐linear elasticity , 2006 .

[3]  Jay D. Humphrey,et al.  Review Paper: Continuum biomechanics of soft biological tissues , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  David Gavaghan,et al.  Predicting Tumor Location by Modeling the Deformation of the Breast , 2008, IEEE Transactions on Biomedical Engineering.

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  Steven E. Benzley,et al.  A Comparison of All Hexagonal and All Tetrahedral Finite Element Meshes for Elastic and Elasto-plastic Analysis , 2011 .

[7]  Andrew D. McCulloch,et al.  Computational Methods for Soft Tissue Biomechanics , 2003 .

[8]  Hervé Delingette,et al.  Nonlinear and anisotropic elastic soft tissue models for medical simulation , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[9]  Martyn P. Nash,et al.  Mechanics and material properties of the heart using an anatomically accurate mathematical model. , 1998 .

[10]  Jonathan P. Whiteley,et al.  The solution of inverse non-linear elasticity problems that arise when locating breast tumours , 2005 .

[11]  Thomas D Brown,et al.  A finite element exploration of cartilage stress near an articular incongruity during unstable motion. , 2007, Journal of biomechanics.

[12]  Y C Fung,et al.  The degree of nonlinearity and anisotropy of blood vessel elasticity. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[13]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[14]  R S Reneman,et al.  Regional wall mechanics in the ischemic left ventricle: numerical modeling and dog experiments. , 1996, The American journal of physiology.

[15]  Robert L. Spilker,et al.  A contact finite element formulation for biological soft hydrated tissues , 1998 .

[16]  M R Drost,et al.  Finite element modelling of contracting skeletal muscle. , 2003, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[17]  Y. Fung.  Elasticity of soft tissues in simple elongation. , 1967, The American journal of physiology.

[18]  James S. Duncan,et al.  Estimation of 3D left ventricular deformation from echocardiography , 2001, Medical Image Anal..

[19]  Hamid Dehghani,et al.  Breast deformation modelling for image reconstruction in near infrared optical tomography. , 2004, Physics in medicine and biology.

[20]  Nathan Ida,et al.  Introduction to the Finite Element Method , 1997 .

[21]  K. Bathe,et al.  The inf-sup test , 1993 .

[22]  T. Belytschko,et al.  Extended finite element method for three-dimensional crack modelling , 2000 .

[23]  Gertjan Kloosterman,et al.  Contact methods in finite element simulations , 2002 .

[24]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[25]  D. Schötzau,et al.  An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity , 2006 .

[26]  M. Nash,et al.  Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias. , 2004, Progress in biophysics and molecular biology.

[27]  M. Mooney.  A Theory of Large Elastic Deformation , 1940 .

[28]  Dimitris N. Metaxas,et al.  A deformable finite element model of the breast for predicting mechanical deformations under external perturbations. , 2001, Academic radiology.

[29]  Roy C. P. Kerckhoffs,et al.  Computational Methods for Cardiac Electromechanics , 2006, Proceedings of the IEEE.

[30]  D Ambrosi,et al.  The role of stress in the growth of a multicell spheroid , 2004, Journal of mathematical biology.

[31]  Martin J. Yaffe,et al.  Biomechanical 3-D finite element modeling of the human breast using MRI data , 2001, IEEE Transactions on Medical Imaging.

[32]  I W Hunter,et al.  An anatomical heart model with applications to myocardial activation and ventricular mechanics. , 1992, Critical reviews in biomedical engineering.

[33]  Martin J Bishop,et al.  Soft Tissue Modelling of Cardiac Fibres for Use in Coupled Mechano-Electric Simulations , 2007, Bulletin of mathematical biology.

[34]  David J Gavaghan,et al.  Non-linear modelling of breast tissue. , 2007, Mathematical medicine and biology : a journal of the IMA.

[35]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[36]  A Ramos,et al.  Tetrahedral versus hexahedral finite elements in numerical modelling of the proximal femur. , 2006, Medical engineering & physics.

[37]  Frank P T Baaijens,et al.  Modeling the mechanics of tissue-engineered human heart valve leaflets. , 2007, Journal of biomechanics.

[38]  P. Hunter,et al.  New developments in a strongly coupled cardiac electromechanical model. , 2005, Europace : European pacing, arrhythmias, and cardiac electrophysiology : journal of the working groups on cardiac pacing, arrhythmias, and cardiac cellular electrophysiology of the European Society of Cardiology.

[39]  Michael I. Miga,et al.  Modality independent elastography (MIE): a new approach to elasticity imaging , 2004, IEEE Transactions on Medical Imaging.