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Corotational finite element formulation for virtual-reality based surgery simulators
D. Marinkovic, M. Zehn
Department of Structural Analysis, Technische Universität Berlin, Berlin, 10623, Germany
Surgical simulation provides a means for trainees to develop surgical competence that encompasses requisite knowledge, technical and cognitive skills and decision-making ability. Considering virtual-reality based surgery simulators, the key requirement is sufficiently accurate and numerically efficient computation of deformation behavior of soft tissues, which is highly nonlinear. The paper offers a simplified geometrically nonlinear corotational finite element formulation to meet the imposed requirements. The approach is used in combination with a rather simple type of finite element and an appropriate solver is chosen for fast computation of dynamical behavior. The finite element formulation is enriched with a coupled-mesh technique to enable modelling of complex geometries by relatively simple computational models. A few examples of models of internal organs are provided to discuss the aspects of the developed tools.
Keywords: real-time simulation, corotational finite element method, surgery, virtual reality, implicit time integration
Коротационная формулировка метода конечных элементов для виртуальных хирургических симуляторов
D. Marinkovic, M. Zehn
Берлинский технический университет, Берлин, 10623, Германия
Хирургическое моделирование служит инструментом для повышения компетенций хирургов, включая получение необходимых знаний, технических и когнитивных навыков, а также развитие способности принятия решений. Ключевым требованием к виртуальным хирургическим симуляторам является достаточно точный и эффективный численный расчет сильно нелинейного деформационного поведения мягких тканей. В статье предложена упрощенная геометрически нелинейная коротационная формулировка метода конечных элементов, которая удовлетворяет перечисленным требованиям. В рамках данного подхода используется простой тип конечного элемента, для быстрого расчета динамического поведения выбирается соответствующий алгоритм решения. Предлагаемая формулировка метода конечных элементов дополнена методом связанных ячеек, позволяющим моделировать объекты сложной геометрии с помощью относительно простых вычислительных моделей. На примере нескольких моделей внутренних органов рассмотрены особенности разработанных инструментов.
Ключевые слова: моделирование в реальном времени, коротационная формулировка метода конечных элементов, хирургия, виртуальная реальность, неявная схема интегрирования по времени
1. Introduction
Surgical simulation can be described as an instructional strategy used to teach technical skills, procedures and operations, by presenting learners with situations that resemble reality [1]. Simulation based on the virtual-reality (VR) concept is a highly multidisciplinary field that requires integration of solutions in areas such as biomechanical modelling, visualization, contact, material tearing, haptic feedback, user-model interaction, etc. A systematic study by Dawe et al. [2] led to the conclusion that the simulation-based surgery training do transfer to the operative settings, thus providing a safe, effective and ethical way for trainees
© Marinkovic D., Zehn M., 2017
to acquire surgical skills before entering the operating room. This is particularly important for complex procedures, such as laparoscopic surgery, but also for patient-specific surgery planning. This field of research has gained considerable attention in the previous two decades and already over a decade ago the emerging body of development was sufficiently large to pose the need for a comparative overview of existing simulators with the emphasis on the simulation validity [3].
Since the field of VR-based surgical simulation is characterized by a diversity of problems, it would be a cumbersome task to resolve all the aspects. The researchers
confine their work to advance specific areas of the field. This paper is focused on the development of formalisms for real-time or nearly real-time simulation of soft tissue deformation behavior, which is highly nonlinear. The finite element method (FEM) is addressed as a preferred method of choice to achieve the objective.
Sufficiently accurate and highly efficient simulation of soft tissue behavior is a critical concern in surgical simulation. Due to their simplicity, mass-spring systems, as a very promising approach to meet the requirement of numerical efficiency, were in the researcher focus over the past decade [4-8]. The crucial advantage of the mass-spring system is that it is computationally very efficient, easy to implement and handles large deformations with ease. The background physics of a large number of surgery simulators is based on this approach. However, the mass-spring systems have a significant accuracy problem and the ambiguity of mass and structural stiffness distribution is also a serious drawback. The survey by Nealen et al. [9] gives a good overview of possible problems and solutions offered by different authors for the mass-spring systems.
The rapid hardware development in the last decade allowed computationally more demanding continuum-based finite element method approaches. The first developments in the field were based on the linear finite element method with a possibility of using various techniques to improve the efficiency of computation, such as condensation [10]. The approach based on the linear finite element method yields very good results in modelling stiff structures like bones [11]. On the other hand, the linear finite element method is incapable of providing plausible results for deformations involving already moderately large rotations. Such deformations are quite common in surgical simulations involving soft tissues of internal organs, thus requiring at least a geometrically nonlinear approach for acceptable, i.e. plausible results. The rigorous geometrically nonlinear finite element method is computationally very demanding and, with the available hardware components, a real-time simulation based on it would require relatively small models. At the present state of development of hardware components, carefully chosen simplifications of the rigorous geometrically nonlinear finite element method are required in order to meet the aforementioned objectives and several approaches are mentioned below to give the trend of development. In order to account for geometrical nonlinearities, Zhuang [12] included the quadratic strain terms in the finite element method and extended the approach with a graded mesh technique to reduce the complexity of 3D models. Capell et al. [13] proposed division of objects into subdomains, while their local rigid-body rotations were considered by means of local coordinate frames. A similar but extended idea was proposed by Mueller et al. [14] who implemented local coordinate frames at nodes. The idea was modified by Etzmuss et al. [15] by using local coor-
dinate frames attached to triangular elements in order to model cloth behavior. Comas et al. [16] used the rigorous nonlinear finite element method approach with viscoelas-ticity and anisotropy included and, to bring the computational complexity of such a formulation closer to realtime applications, they have used parallel execution threads by means of GPU based computations. Niroomandi [17] used asymptotic numerical methods to account for non-linearities whereby the numerical efficiency was improved by means of model reduction and extended finite element method techniques. Meshless finite element method in combination with total Lagrangian adaptive relaxation algorithm was applied by Jin et al. [18] with the aim of resolving the cutting-induced discontinuity.
In the present work, a corotational finite element method formulation is used to account for large geometrically nonlinear deformations of 3D objects. It is a simplified geometrically nonlinear formulation, which aims at results with acceptable deviation, regarding the application, from those of the rigorous geometrically nonlinear formulation, and that enables rather efficient computation.
2. The simplified corotational finite element method formulation
Although the below presented corotational finite element method formulation can be used in combination with any type of finite element, for this specific application the authors use it with a rather simple type of solid—the linear tetrahedron. The reasons for the choice are high numerical efficiency of the such elements combined with high meshing ability. The authors are aware of the fact that those advantages come together with the disadvantage of relatively stiff behavior of the element, but since 'plausible deformation behavior' is sufficient from the point of accuracy in the considered application, the choice appears to be a reasonable compromise. The formulation of the linear tetrahedral element is widely available in the finite element method literature. An interested reader may address Bathe [19] for more information on the element formulation. Thus, in the further text, it will be assumed that the element linear stiffness Ke and mass Me matrices are known.
The major requirements to be met are stable and computationally as inexpensive as possible computation that yields deformation behavior plausible to human perception. It should be emphasized that the linear finite element method handles arbitrary large translations of elements with ease. However, already moderately large rotations result in artificial enlargement of the model and deteriorate the obtained results beyond the point of visually acceptable, i.e. plausible behaviour. The formulation proposed below aims at resolving this issue in an efficient manner.
The very essence of the formulation is described by means of Fig. 1. The idea is that each element is assigned a local reference frame which is attached to the element and
Original element rotated to deformed configuration
Deformed element
Original element configuration
Deformed element rotated back to original configuration
Fig. 1. Decomposition of element motion into rigid-body and deformable motion
performs the same rigid-body motion as the element itself. This allows decomposition of the overall motion into a rigid-body motion, described by the rotational matrix Re, and a deformation-induced motion. The deformation behavior is characterized by means of the element linear stiffness matrix Ke with respect to the local reference frame.
The fact that the behavior of the element remains linear with respect to the local reference frame has a crucial impact on general characteristics of the approach. It will be seen below that this fact enables high numerical efficiency compared to the rigorous geometrically nonlinear finite element method analysis. At the same time, the approach is characterized by very high stability as the condition number of the system of equations does not change dramatically as deformation proceeds. The rigorous geometrically nonlinear finite element method is notorious for stability issues requiring iterations and cuts in increments used in the computation.
The idea of the corotational formulation that was described above can be summarized into a relatively simple equation, which computes the internal elastic forces of the element at time t as:
'Fe = 'Re °KeCRe1 'xe " °xe)> (1)
where xe denotes the configuration (nodal coordinates) and the left superscript denotes the moment in time at which the quantity is taken, hence 0 and t refer to the initial and current configuration, respectively. The first term in the parenthesis on the right-hand side of the equation yields the current element configuration rotated back to the original element orientation. It is given in Fig. 1 in dotted lines close to the original configuration by rotating the deformed element backwards through Re1. Hence, the entire term in the parenthesis defines the rotation-free displacements, which are, furthermore, multiplied with the element stiffness matrix to yield the internal elastic forces in the original element configuration. The forces are finally rotated from the initial to current element configuration, i.e. through R e. The equation can be rewritten in the following form:
t F - tR tR -1 tx tR -Fe - Re Ke Re Xe - Re Ke Xe -
- t Ke tXe - tRef0e, (2)
where Ke is the rotated element stiffness matrix and f0e is a contribution to the vector of internal forces that can be precomputed. This form reveals where the efficiency of the formulation resides. Namely, in a step prior to interactive simulation, it is necessary to compute the linear stiffness matrix K e and vector f0e for each single element. Over the course of simulation, the element rotation matrix Re is computed based on the initial and current element configurations and further used to rotate the element stiffness matrix and vector f0e. The structural stiffness matrix is together with the vector of internal forces reassembled and the resulting system of equations solved.
The extraction of the rotational part of element motion is an important part of this formulation. The linear tetrahedron belongs to the family of isoparametric elements and, hence, its geometry is described by the same linear shape functions used for the description of the displacement field. As a consequence, the Jacobian matrix, which relates Cartesian and natural coordinates of the element, and the deformation gradient matrix are constant over the entire domain of the element. Polar decomposition of the deformation gradient matrix yields the rotation matrix.
3. Time integration and solver for numerically efficient geometrically nonlinear dynamics
In order to resolve a dynamic response, time discretization is applied and one of the two types of time-integration schemes may be used for dynamic computations—explicit and implicit.
For an explicit solver, in order to determine the configuration at time t + At, where At denotes the time increment, the dynamic finite element method system of equations is written at time t:
M tii -tFeXt -tFmt, (3)
where M is the structural mass matrix, u is the vector of nodal displacements and the dots above the symbol denote the time derivatives—hence, two dots denote accelerations, while Fext and Fint are the vectors of internal and external nodal forces, respectively. The vector of internal forces comprises the elastic forces and, if the structural damping is accounted for in the simulation, the corresponding forces may also be added to this vector. The central difference method is the most commonly applied explicit time-integration scheme. Upon diagonalization of the mass matrix (lumped mass matrix), the explicit time integration decouples the system of equations thus enabling a great efficiency in computing a single time step. However, since it represents a prediction of the future configuration (at time t + At) based on the previous two determined configurations (at time t - At and t) and the dynamic equilibrium at time t,
it is only conditionally stabile, thus requiring a relatively small time step. The size of the critical time step depends on the highest eigenfrequency of the finite element method model and it represents a critical demand with respect to the objective of real-time computation, even for the considered type of materials which are not stiff. The presence of structural damping further decreases the critical time step of an explicit solver dramatically.
On the other hand, the implicit time-integration schemes use the dynamic equilibrium at time t + At to determine the configuration at time t + At, hence the dynamic finite element method equation reads:
M t+Atuu (k) + tC t+AtU (k) + tKt Au(k) =
_ t+At F _ t+A F (k-1) (4)
_ ext rint , (4)
where K and C are the structural stiffness and damping matrices, respectively, a dot above u denotes velocities and since the solution of the nonlinear problem proceeds itera-tively until the converged solution has been reached, (k) in the right subscript denotes the iteration number. The major advantage of implicit time integration is the unconditional stability, which allows the usage of much larger time steps in the simulation. Since it is the human perception that plays the major role in surgical simulations and the human eye may register a limited number of frames per second, "relatively slow dynamics" is of primary interest. Even if "high frequency dynamics" was involved in the considered dynamic deformational behaviour, the requirements of surgical simulation permit to simply filter it out, which is effectively done by using sufficiently large time steps. Although an implicit time-integration scheme is more expensive in terms of the necessary computational effort for the computation of a single time step, the time step of an implicit timeintegration scheme can be sufficiently larger than that of an explicit time-integration scheme so that the overall simulation is more efficient with the implicit time integration. The authors use the Newmark time-integration scheme [19].
As obvious from Eq. (4), the implicit time integration schemes keep the finite element system of equations coupled. Solving this system is numerically the most demanding task within the framework of the finite element method. The authors use the preconditioned conjugate gradient method as the major iterative method. This solver offers certain advantages with respect to the needs of sur-
gical simulators. The considered computation (Eq. (4)) requires the update of the system matrix in each time step due to the geometrical nonlinearities and the right-hand side of the equations, i.e. the excitation, is also variable during the simulations. With these simulation characteristics, the advantages typically offered by a direct solver could not come to the fore. The iterative solver, on the other hand, provides a very easy way of performing a trade-off between the solution accuracy and computational effort by limiting the number of performed iterations. Furthermore, the efficiency of the iterative solver can be noticeably improved by a reasonable choice of the starting vector of the iterative process. One should notice that the nodal velocities do not change dramatically within a time step, especially not in applications such as surgical simulations, where "slow dynamics" is of primary interest. This fact can be used to have a rather good starting vector, which improves the numerical efficiency of simulation as fewer iteration steps are needed to arrive to the solution. Finally, the iterative solver allows the usage of sparse form of the system matrix, thus requiring only a fraction of memory used by direct solvers.
4. Coupled-mesh technique and finite element meshing
Using hardware of 'an average PC configuration', the presented corotational formulation enables simulation of dynamic behaviour of finite element method models with several thousand tetrahedral elements at interactive frame rates. This depends not only on the hardware configuration, but also on specific simulation parameters, such as the chosen time step. The authors mostly use the time step of 102 s, which offers acceptable and stable solutions for the material properties typical for soft tissues. However, a volumetric mesh of several thousand elements can rarely offer an appealing representation of internal organ surface due to their relatively complex geometry. The idea on how to conciliate the request for numerically very efficient simulation with the request for high-quality graphics that implies realistic appearance of the objects consists in introducing a coupled-mesh technique.
The coupled-mesh technique implies parallel usage of two meshes—a volumetric finite element mesh, which is used to compute deformations, and a detailed triangulated
m
Fig. 2. Vertex (2598 vertices) (a) and surface (5192 faces) (b) representations of a liver
Fig. 3. Liver model: geometry (a), finite element method mesh 1 (846 nodes, 2945 elements) (b) and mesh 2 (660 nodes, 2640 elements) (c)
surface mesh that represents the actual, complex geometry of internal organs. The detailed mesh can be obtained by CT scans [20]. It contains a great number of vertices thus providing a detailed representation of the surface geometry. A topology that interconnects the vertices into triangles is also defined to enable the surface representation by means of primitives. This is illustrated in Fig. 2, which depicts the vertex and triangulated surface representations of a liver.
As a prestep of interactive simulation, the finite element mesh and the triangulated surface mesh are coupled to each other. For each vertex of the surface mesh, a corresponding finite element is found and the surface vertex is assigned to this element. The criterion to find the corresponding element is based on the local coordinates of the vertex with respect to the elements. The relation between the vertex global coordinates, {x, y, z}T, and vertex local coordinates, %2, %3, %4}T, with respect to an element whose global nodal coordinates are {x,y, zi}T, i = 1, 2, 3, 4 (linear tetrahedron), reads:
X " X1 X2 X3 X4 V %1'
y _ yi y 2 y3 y4 % 2 ._ A- % 2
z z1 z 2 z3 z4 %3 %3
1 1 1 1 1 % 4 % 4
It should be noticed that the sum of all local coordinates is always 1. Now, based on Eq. (5), the vertex local coordinates with respect to the element are obtained in a straightforward manner by means of A-1 and vertex global coordinates. If all four local coordinates are in the interval [0, 1] the vertex is inside the element and, hence, the vertex is assigned to that element.
The finite element meshing is another very important aspect. The principles of finite element meshing in typical
engineering applications are well established. Since it is the accuracy that has the priority in engineering computations, the finite element mesh is set up so that it describes the actual geometry pretty accurately, poorly shaped elements are avoided and the mesh is finer in the regions where greater gradients of the quantities of interest are expected— to name but a few important criteria. One may perform meshing of the geometry of internal organs based on these principles. This, however, would result in a great number of elements and the mesh would typically consist of elements of various sizes, depending on the meshed region. Both aspects have a negative impact on the numerical efficiency of the finite element method model, as the former increases the number of degrees of freedom, while the latter deteriorates the condition number of the system of equations, thus requiring a greater number of iterations within the iterative solver to get the solution. But as it has been already stressed out, the accuracy of deformational behaviour in surgical simulators has a priority of lower rank, as plausible deformation behavior is set as an objective that is sufficient for the purpose. This fact allows meshing based on different principles. The mesh can be set so as to just roughly resemble the actual geometry and it can be actually somewhat larger or smaller than the geometry itself. This further enables the usage of elements of the same or very similar size. The geometry of the liver given in Fig. 2 has been meshed using both approaches and the resulting finite element meshes are given in Fig. 3 (different perspective compared to Fig. 2) together with the vertex representation of the geometry.
The approach that is based on less precise finite element method mesh appears to be a promising alternative for the purpose of surgical simulation, and the authors used this
Fig. 4. Stomach model: geometry (994 vertices) (a), finite element model (269 nodes, 726 elements) (b), coupled meshes (surface with 1984 faces) (c)
Fig. 5. Colon model: geometry (1122 vertices) (a), finite element model (315 nodes, 1016 elements) (b), coupled meshes (surface with 2240 faces) (c)
Fig. 6. Spleen model: geometry (962 vertices) (a), finite element model (552 nodes, 2226 elements) (b), coupled meshes (surface with 1920 faces) (c)
approach to mesh the geometry of several more internal organs, such as stomach (Fig. 4), colon (Fig. 5) and spleen (Fig. 6).
5. Plausibility of deformation behavior and numerical efficiency
The presented simplified corotational formulations neglects several aspects of the rigorous geometrically nonlinear analysis in such a manner that geometrical nonli-nearities are still included up to a great extent but with significantly improved numerical efficiency. The behavior of each finite element remains purely linear with respect to the local reference frame, but the rotations of finite elements can be arbitrarily large. As long as the geometrical nonli-nearities are dominated by large local rotations, this formulation yields results that are a rather good approximation of those obtained by the rigorous geometrically nonlinear formulation. The aspect of achievable accuracy in terms of
numbers with the presented corotational finite element method was considered by Marinkovic et al. [21] for finite element models with solid elements and by Zehn et al. [22] for finite element models with shell elements, i.e. for thin-walled structures. The formulation can also be extended by the so-called projector matrix [23] in order to improve the accuracy with a relatively small additional numerical effort [24].
For the purpose of surgical simulator, it suffices to have models which yield deformational behaviour that appears realistic to human perception. Figure 7 demonstrates that this objective can be met by the presented formulation, despite the simplifications used in the development. It shows a several snapshots from an interactive simulation involving the liver model shown in Fig. 2 and using the finite element method mesh 2 given in Fig. 3.
It was already pointed out that the combination of the presented finite element method formulation and applied
Table 1
Numerical efficiency of different hardware configurations with the liver model
5192 faces
Graphics
^ 2598 vertices Finite element mesh Vertices Surface
AMD S140 (2.7 GHz), NVidia 7025 Ratio 0.61 0.66 0.57
Frames/s 12 13 11
Intel E6750 (2.66 GHz), NVidia8600GT-OC Ratio 1.11 1.14 1.07
Frames/s 22 23 21
Intel E8500 (3.16 GHz), NVidia 8600GT Ratio 0.86 0.87 0.84
Frames/s 18 18 17
Intel E8500 (3.16 GHz), NVidia 8800GT Ratio 1.35 1.39 1.30
Frames/s 27 28 26
AMD II X2 250 (3.0 GHz), NVidia 8600GT Ratio 0.82 0.77 0.82
Frames/s 17 16 17
Intel i7-870 (2.93 GHz), NVidia 550 GTI Ratio 1.48 1.54 1.38
Frames/s 30 31 28
Intel i3-2120 (3.3 GHz), NVidia 750 GTI Ratio 1.68 1.79 1.49
Frames/s 34 36 30
Hardware configuration
(processor, graphic card)
solver with 'an average PC configuration' allows interactive simulation (real time or nearly real time) with finite element method models of up to several thousand elements. To provide a better assessment of this statement, the liver model has been tested on several available hardware configurations and the results are summarized in Table 1. As comparison criteria, the ratio between the virtual and real time ('ratio' in the table; greater than 1 means 'faster than real time'), as well as the number of frames per second are chosen. Processors and graphic cards are given as components with the greatest influence on the achieved result. It should be emphasized that, in each case of graphical representation (finite element mesh, vertex or surface), the simulation is performed so that 5 time steps, each 10-2 s, are first computed before the current configuration is depicted on the screen. A significant difference in efficiency can be noticed between the two configurations that use the same Intel E8500 processor. The configuration with NVidia 8800GT is approximately 50% more efficient than the one with NVidia 8600GT. Similarly, the Intel E6750 processor (with less computational power than Intel E8500) yields better results with the overclocked NVidia 8600GT-0C than the Intel E8500 processor with the NVidia 8600GT that is not overclocked. This clearly illustrates the significance of graphic card performance in virtual-reality applications. Although some processors in the considered configurations are multicore processors, the advantage of parallel computing is not used, i.e. only one core is used for the computation.
6. Conclusions
VR-based surgical simulation uses computer generated environment for training various surgery scenarios, among which laparoscopic and endoscopic scenarios are most frequent because, in reality as in simulation, they comprise a 2D visual system with limited haptic interaction. The paper offers the corotational finite element method formulation enriched by additional features in order to provide a solid basis for the background physics of surgery training devices. The presented finite element method formulation allows a significant deal of computation to be done prior to interactive simulation, thus offering high numerical efficiency and therewith the possibility of performing simulations at interactive frame rates. The applied solutions also offer the flexibility of performing even a dynamical (i.e. during simulation) trade-off between the numerical efficiency and accuracy. The plausibility of deformational behaviour is retained even with rather large deformations, which is the objective from the point of accuracy. The applied solver type has a great parallelization potential, which gains even more in importance due to the possibility of shifting the computation from CPU to GPU (graphics processing unit). A new concept of using general purpose GPU as a modified form of stream processor gives a massive floating-point computational power (for the state-of-the-art GPUs expressed in teraflops, i.e. 1012 floating-point operations per second).
In the future work, the corotational finite element method formulation should be extended to cover further func-
tionalities which are of great importance for surgical simulation. Contact is one of the primary objectives as it occurs not only between organs (including self-contact), but also any interaction of a surgeon and an organ is actually realized through contact between an instrument held by the surgeon and the organ. For the considered type of material, i.e. soft tissue of internal organs, the method of dimensionality reduction (MDR) [25, 26] appears to be a very promising approach as it offers means to retain high computational efficiency, which is the primary objective. Also, inclusion of material nonlinearities and tearing belongs to further objectives of great importance for surgical simulation. These objectives should particularly benefit from the possibility of computational parallelization.
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Поступила в редакцию 09.02.2017 г.
Сведения об авторах
Dragan Marinkovic, Dr.-Ing., Technische Universität Berlin, Germany, dragan.marinkovic@tu-berlin.de Manfred Zehn, Prof. habil. Dr.-Ing., Technische Universität Berlin, Germany, manfred.zehn@tu-berlin.de
