Fast computation of soft tissue deformations in real-time simulation of surgery with HEML: a comparison of computation time with conventional FEM

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1 Noname manuscript No. (will be inserted by the editor) Fast computation of soft tissue deformations in real-time simulation of surgery with HEML: a comparison of computation time with conventional FEM Zhuowei Chen François Goulette Received: date / Accepted: date Abstract Virtual surgery simulators evoke a lot of interests in the world of surgery training, where they allow to improve the quality of formalization of surgeons gesture. One of the current major technical difficulties for the real-time simulation of surgery is the possibility to realize fast and realistic deformation algorithms of organs hyperelastic behavior. However today, few models are available, they are still time costly and limited in number of tetrahedrons by algorithm complexity. We present in this paper a new method that we call HEML (HyperElastic Mass Links), including Saint Venant-Kirchhoff, Neo-Hookean and Mooney- Rivlin hyperelastic constitutive laws, which is particularly fast, derived from the finite element method, is valid for large displacements and large deformations. The algorithm complexity is linear in the number of tetrahedra; the comparisons in computation time per iteration with a commercial finite element engineering software on the same cases show a gain of more than 100 times faster. Keywords Real-time simulation Hyperelasticity Soft tissue HEML 1 Introduction Fast and realistic modeling of soft biological tissue is a significant challenge due to its complex nature. In this paper, we focus on the simulation of soft tissue (like uterus) deformation in the context of surgery training. The overall objective is to build a simulator for learning of the obstetric gestures related to childbirth. In such case, it is required that soft tissue deformation be simulated in real-time which refers to a computation time of the discrete differential equations small enough to allow a reasonable refreshment rate, at least 25 frames per second for visual display. Z.-W. Chen, F. Goulette Centre de Robotique, Mines ParisTech, 60 bd Saint Michel, Cedex 06, France Tel.: Fax: zhuowei.chen@mines-paristech.fr francois.goulette@mines-paristech.fr

2 2 Zhuowei Chen, François Goulette Several computational methods and models have been developed to simulate real-time soft tissue deformations. Many of these models have addressed the issue of computing the internal elastic force of the material, related to its intrinsic material. Some researchers have been interested in mass-spring models (10; 18), due to their simplicity of implementation and their low computation complexity properties. However, they suffer from a lack of realism, which lead to further research on extensions of the model. Cotin et al (3) proposed a so-called masse-tensor model which is as simple to implement and as efficient as mass-spring models, but it is based on continuum mechanics and linear elasticity theory; this model has been developed further to handle large deformations and large displacements with the Saint Venant-Kirchhoff constitutive law (17; 5) which exhibit a linear stress-strain relationship. However, this model is limited to this specific material. Marchesseau et al (9) proposed a Multiplicative Jacobian Energy Decomposition (MJED) method for discretizing hyperelastic materials on linear tetrahedral meshes which leads to faster matrix assembly than the standard Finite Element Method. This approche is not limited to one specific hyperelastic material but can not reach the ideal 25 FPS needed for the real-time simulation. Other researchers preferred to derive discrete computational algorithms from the equations of Continuum Mechanics, in order to obtain real-time computations: based on the Boundary Element Method (BEM) (11), the Finite Difference Method (FDM) (4) and the Finite Element Method (FEM) (1; 13). However, the heavy complexity of these methods makes computation time a real challenge. For these reasons, we try to find a compromise between biomechanical accuracy and computational efficiency to realize a real-time simulator. A framework has been proposed to design fast algorithms to compute the elastic force field for any hyperelastic model, handling large deformations and large displacements (6). The algorithms are designed under the P1-finite element approximation in homogeneous isotropic cases. Hyperelastic models include the Saint Venant-Kirchhoff constitutive law (used in mass-tensor), and other important hyperelastic constitutive laws such as Neo-Hookean and Mooney-Rivlin. We chose to call this approach HyperElastic Mass Link (HEML), for the following reasons: Link, because forces at a given node are given as a sum of forces proportional to the links (vectors) to all connected neighbors; the formulation of the modulus of these incident forces depend only on the square lengths of the links of adjacent tetrahedrons. Mass : as in mass-spring or mass-tensor, masses are affected to the mesh nodes, used in the discrete differential equations. HyperElastic, because the framework presented may be used to design algorithms for computation of any hyperelastic material. In the present work, the intention is to present and validate the HEML method for hyperelastic materials on linear tetrahedral meshes. Two numerical examples are performed in this study to show the validity of the developed algorithm, the same examples are also performed in an engineering software using finite element method to show the efficiency of this algorithme.

3 Title Suppressed Due to Excessive Length 3 2 HyperElastic Mass Link method 2.1 Hyperelastic bodies undergoing large deformations Rubber and some biological soft tissue materials are said to be hyperelastic (7; 20; 16). Usually, these kind of materials undergo large deformations. In order to describe the geometrical transformation problems, the deformation gradient tensor is introduced by : F = I + u (1) where I is the unity tensor. u is the displacement vector. Because of large displacements and rotations, Green-Lagrangian strain is adopted for the non-linear relationships between strains and displacements. We note C the stretch tensor or the right Cauchy-Green deformation tensor (C = F T F). The Green-Lagrangian strain tensor E is defined by : E = 1 (C I) (2) 2 In the case of hyperelastic law, there exists an elastic potential function W (or strain energy density function) which is a scale function of one of the strain tensors, whose derivative with respect to a strain component determines the corresponding stress component. This can be expressed by : S = 2 W C where S is the second Piola-Kirchoff stress tensor. To construct the tangent stiffness matrix for the analysis of nonlinear structures by the finite element method, one has to determine the stress-strain tangent operator D, which is a fourth-order tensor resulting from the derivation of S with respect to E. In fact, the assumption of isotropy of the material allows to write the potential W depends only on the three invariants of C: (3) W = W (I 1, I 2, I 3 ) where I 1 = tr(c), I 2 = 1 ( I 2 1 tr(c 2 ) ), 2 I 3 = det(c) (4) 2.2 Notations Our model is based on a tetrahedral mesh of the studied material (Fig. 1). The tetrahedron is the simplest primitive to mesh any shape without having to introduce topological anisotropy. We consider a linear interpolation of finite elements (P1 approximation finite elements). Without loss of generality, a numbering is chosen for the four vertices of a given tetrahedron, which are denoted M i and m i (0 i 3) for initial and deformed configurations respectively. The six edge vectors are denoted as V i and v i (1 i 6). Any three edge vectors out of six are enough to express the others, so we denote V and v the matrices: V = (V 1, V 2, V 3 ),

4 4 Zhuowei Chen, François Goulette v = (v 1, v 2, v 3 ). Considering non-degenerate tetrahedra in initial state, V is invertible. We denote Li the lengths of edges V i, l i the lengths of edges v i, L and l the vectors of the six square lengths, and l = l L the vector of differences. Fig. 1 Initial and deformed configurations 2.3 Energy and forces As we discussed earlier, hyperelasticity means that there exists a volumetric energy function W from which derives the stress tensor. In this work, We consider a homogeneous, isotropic and hyperelastic material. The assumption of isotropy of the material allows to write the potential W depends only on the three invariants of C: W = W (I 1, I 2, I 3 ). In the P1 approximation of finite elements, mesh elements are tetrahedral; the approximation states that the deformation gradient tensor F is constant over a given tetrahedron T k, it is also the same case for tensor C. We can decompose the energy W over each tetrahedron T k, the total energy of the material being the sum of W k over all the tetrahedrons. It can be demonstrated that, under the P1 approximation, the value of C depends linearly on the vector l (edge square lengths) and on the initial state (for more informations, please see (6)), in other word, it depends only on the vertices positions and on the initial state: W k = W k (l k ) = W k (m 0, m 1, m 2, m 3 ) (5) Instead of computing the force and stiffness matrix using the first and second derivatives of the energy with respect to C (leading respectively to S and D), we compute them directly by deriving the energy with respect to the nodal positon: F (X) = W X For a given tetrahedron T k, the force at a node m i can be expressed by: (6) F i,k = W k m i (7) F i,k is the contribution of the force of tetrahedron T k at node m i. It s a vector of 3 dimension.

5 Title Suppressed Due to Excessive Length 5 As we mentioned before, W k is a function of the vector l, then equation (7) can be expressed by: W k (l k ) = W k l k (8) m i l k m i From the relation between the edges square length li 2 and the edge vectors v i, one can express the edges square length by the four nodes. This leads to a formulation of the derivative of l k over each vertex m i of the tetrahedron, linear in the matrix v, with 4 (6 3) constant matrices DLM i that depend on the initial state: 0 i 3, l k m i = DLM i vk T. Then, the force of each vertex m i of tetrahedron T k can be expressed by a linear formulation: 0 i 3, F i,k = W k l k l k m i = W k l k DLM i v T k (9) 2.4 Isotropic hyperelastic model We have used this approach (Equations 5, 9) to derive elastic forces for various specific isotropic hyperelastic materials: Saint Venant-Kirchhoff, Neo-Hookean and Mooney-Rivlin. Saint Venant-Kirchhoff model is a hyperelastic material model which is an extension of the linear elastic material model to the nonlinear regime. The volumetric energy function is usually formulated with the Green-Lagrange tensor E, and the Lame coefficients λ and µ. Neo-Hookean (15) and Mooney- Rivlin (12; 19) are popular hyperelastic material models that can be used for predicting the nonlinear stress-strain behavior of rubber or biomechanical materials undergoing large deformations (21; 14; 8). For each type of hyperelastic model, the energy density is a function of the three invariants I 1, I 2, I 3, and we express it by the edges square lengths l or l. Then we derive a formulation of forces with respect to l or l. We present in table 1 the energy density and forces derived at the nodes of a given tetrahedron, for each material models. Table 1 Energy and forces for different hyperelastic material models Material model Energy density W Force at node i General type W (l) = W (I 1, I 2, I 3 ) Saint Venant -Kirchhoff W l l m i λ 2 (tr(e))2 + µtr(e 2 ) l T (MW ST V k DLM i ) v T = l T MW ST V k l Neo-Hookean C 1 (I 1 3) = C 1 (Vtr l 3) C 1 (Vtr DLM i ) v T Mooney-Rivlin C 1 (I 1 3) + C 2 (I 2 3) C 1 (Vtr DLM j ) v T + = C 1 (Vtr l 3)+ C 2 l T (MW M R DLM i ) v T C 2 [ 1 2 lt (Vtr T Vtr Mtr)l 3] For a tetrahedral element, we can express the right Cauchy-Green strain tensors in a formulation of the six square lengths: C = 6 i=1 l i C i. We denote Vtr and Mtr the 6-vector of traces of C i and the 6 6-matrix of traces of C i C j respectively.

6 6 Zhuowei Chen, François Goulette The three invariants of tensor C can be expressed by the vector of edge square lengths l. In the Saint Venant-Kirchhoff model, MW ST V K is a 6 6 matrix: MW ST V K = λ 8 (Vtr VtrT ) + µ 4 Mtr. For the Neo-Hookean and Mooney- Rivlin models, C 1 and C 2 are material parameters, MW M R is a 6 6 matrix: MW M R = 1 2 (VtrT Vtr Mtr). Generally, the soft biological tissues are assumed to be incompressible. In Saint Venant-Kirchhoff case, the material is compressible: incompressibility is a special, limit case, when the Poisson coefficient tends towards to 0.5, which is in practice not possible for Lamé coefficients to become infinite. For the Neo-Hookean and Mooney-Rivlin models, a term can be added to the energy function to take into account the incompressible behavior of material: K 2 (J 1)2. In which the K is the initial bulk modulus and J is the determinant of F, from the physical point of view, J is always positive, otherwise it means that the body could be interpenetrated which is no physical at all. From a numerical point of view, K can be considered as a penalty factor, the higher it is, the more J tends to 1, and therefore we have a more tendency towards to an incompressible behavior. 2.5 Implementation for hyperelastic materials In order to compute an elastic force field on a material from the equations presented above, one has to perform two kinds of operations: an initialization regarding the initial state of the material (geometry and material parameters); a computation of forces in deformed state, at each step of the numerical integration process. Concerning initialization, one has to compute, for each tetrahedron, four characteristic matrices from its initial state and the material parameters. Concerning the force field in deformed state, for each tetrahedron one has to compute the six edge vectors and square lengths, and then, using the four characteristic matrices, the four forces at each vertex using the corresponding equation are given in table 1. In the end one has to sum up, for each node of the element, the forces coming from incident tetrahedra. This algorithm has been implemented into a laptop, with the Visual C++ developing environment under Windows XP. The laptop was a Intel Core 2 Duo at 2.40 GHz, 3.45 Go RAM. 3 Comparison of computation time Before we consider the complex biological models, we simulate first a simple cube in compression in order to validate the accuracy of the HEML method, in comparing with the classical finite element method software ANSYS The Neo-Hookean constitutive law is chosen, with the material parameters C 1 = 70kP a and K = 10MP a. The results of the reaction force of a node in the direction of compression during it s deformation in two methods are given in Figure 2. It can be seen that the computed results of two methods are very close.

7 Title Suppressed Due to Excessive Length 7 Fig. 2 Comparison of the computed force by FEM and HEML 3.1 Fetus In order to test our algorithms on solving of hyperelastic problems, we considered in our present work to simulate a fetus. The three-dimensional solid model was obtained from static images of MRI using free software segmentation ITK-SNAP based on the evolution of an active contour. The mesh model of a fetus is provided by the SAARA Team of LIRIS of University of Lyon 1 as described in (2). It includes 4430 tetrahedral elements and 1128 nodes (fig. 3). For the simplification, we didn t decompose the fetus in three parts as presented in the article, it was considered as a whole part. For the mechanical behavior, it was modeled as Neo- Hookean material with a density of 950 kg/m 3 and C 1 = 70kP a as in the article, and K = 10MP a for the bulk modulus. For a complete computation of material deformations, we add gravity to simulate the fall of fetus on a plan. The same model with same material parameters has been simulated by using the finite element software ANSYS. The average computation times per iteration for the elastic force field are given in the first line of table 2. It is noticeable that, for the fetus model, the computation can reach an average time of 4.02 ms. Compared to the conventional finite element method, our algorithm is much faster which can reach a real-time simulation. 3.2 Abdomen The second example is an abdomen of the parturient woman model which is adopted from the same article (2). This model includes tetrahedral elements and 5591 nodes (fig. 4). Same as in this article, the Neo-Hookean constitutive law is chosen with a density of 2500 kg/m 3 and C 1 = 5kP a, K = 10MP a for the bulk modulus. Similar as the fetus model, we add gravity to simulate the fall of abdomen on a plan. The same model with same material parameters has been performed by using

8 8 Zhuowei Chen, François Goulette ANSYS. The average computation times per iteration for the elastic force field are given in the second line of table 2. It is noticeable that, for the abdomen model, containing as many as tetrahedrons, the computation is feasible and can reach an average time of ms. Compared to the conventional finite element method, our algorithm shows a gain of more than 100 times faster. However, our programme can t show the Von Mises stress field result as ANSYS dose, we can t compare quantitatively the stress level or the energy of this two methods, which will be our future work. Fig. 3 Mesh of the fetus Fig. 4 Mesh of the abdomen of the parturient woman

9 Title Suppressed Due to Excessive Length 9 Table 2 Computation time per iteration for the fetus model and the abdomen model in second ANSYS (s) HEML (s) Fetus (4430 elements) e-3 Abdomen (21436 elements) e-3 4 Conclusion We have presented the HyperElastic Mass Link framework, a methodology for fast computation of deformable bodies that handles the hyperelastic materials, which is a compromise between biomechanical accuracy and computational efficiency. Equations of energy density and forces derived at each node of a given tetrahedron have been presented for three different material models (Saint Venant-Kirchhoff, Neo-Hookean and Mooney-Rivlin). The complexity of the algorithms is linear in the number of tetrahedra. This algorithm has been validated by comparing the computed elastic force with the classical finite element method. Two biomechanical models (a fetus and a parturient woman s abdomen), with the Neo-Hookean hyperelastic constitutive law, have been performed in our algorithm. The algorithm can handle the computation of a mesh of more than tetrahedrons in an average operating time of ms, it presents an improvement over the finite element method: the computation time per iteration shows improvement of more than 100 times. Further works can be focused on these issues: the method can be extended to other hyperelastic materials like Ogden material, or viscoelastic materials, and to add plastic deformations; it could also be extended to non homogeneity, and to anisotropy like transversally isotropic materials; to improve computation time with specific coding for fast computation, or parallelization; to have a more abundant performance of the results (visualize and manipulate mesh data structures via the mouse, interactive animations, cutting planes or texture transparency), develop our algorithm on an open software platform. Acknowledgements This work was partly supported by the SAGA project of French Agence Nationale de la Recherche. The authors would also like to thank the SAARA Team of LIRIS of University of Lyon 1 for providing the abdomen mesh model. Conflict of interest The authors declare that they have no conflict of interest. References 1. Berkley J, Weghorst S, Gladstone H, Raugi G, Berg D, Ganter M (1999) Banded matrix approach to finite element modelling for soft tissue simulation. Virtual Real 4(3): Buttin R, Zara F, Shariat B, Redarce T, Grangé G (2013) Biomechanical simulation of the fetal descent without imposed theoretical trajectory. Comput Meth Programs Biomed 111: Cotin S, Delingette H, Ayache N (2000) A hybrid elastic model allowing realtime cutting, deformations and force-feedback for surgery training and simulation. Vis Comput 16(8):

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