ABAQUS (HKS Inc) [9], the mesh was verified to meet the geometric and topological compatibility requirements for finite element solvers. The resulting mesh is ...
Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai, China, September 1-4, 2005
A Finite Element Method Based Deformable Brain Atlas Suited for Surgery Simulation Chunping Gao 1,2, Francis E. H. Tay 1, Wieslaw L. Nowinski2
1
Department of Mechanical Engineering, National University of Singapore 2 Biomedical Imaging Lab, Singapore
Abstract—Brain Atlases are valuable tools that assist neurosurgeons during the planning of an intervention. Since current printed and digitalized brain atlas have several disadvantages, including lack of physical descriptions of brain structures and incompatibility with finite element analysis, we have developed a meshed brain atlas for finite element analysis of surgical procedures. Our meshed atlas is a multiple-object FE model integrated with knowledge of mechanical properties of brain tissue with detailed anatomical information. A nonlinear hyper-viscoelastic mechanical property typically suited for neurosurgery simulation is incorporated. We then partially validated the utility of this model by numerical simulation of a gravity-induced brain deformation scenario. This meshed atlas is able to account for the deformation of both the whole brain and individual structures. It can be applied in large scale finite element simulations and, therefore, offers the possibility of developing bio-mechanical surgical planning and training systems. Keywords— Brain atlas, modeling, simulation I. INTRODUCTION
Neurosurgery particularly requires using explicit medical images as the brain is the most complicated object in human anatomy. To identify important structures in the brain and bear them in mind for the surgical strategy, neurosurgeons frequently consult brain atlases during the planning of operations to improve their orientation. Brain atlases are usually printed books [1] or computerized system [2]. These conventional brain atlases only provide anatomical information of brain, whereas the physical properties of the brain are not addressed. When brain tissue encounters finite deformation, such as tumor growth and hydrocephalus, the information transfer from the brain atlas to the deformed brain structures under surgical load has to happen solely in the mind of the surgeon, requiring a long time experience. To address this problem, there is an increasing demand of surgery planning and simulation system. A biomechanical model typically suited for simulating neurosurgery procedures is essential for developing accurate simulation systems. Finite Element (FE) models, in conjunction with description of mechanical behavior, are becoming more and more important in the application of neurosurgery simulation. It produces more accurate simulations and more reliable behavior than other types of physical based models [3]. The FE model of the brain is required to contain detailed anatomical information and boundary conditions to account for its complexity.
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Additionally the mechanical behavior of the brain tissue is demanding and complicated. Depending on the application, viscoelastic, poroelastic and pure linear elastic models have been used in different analyses [4]. Presently the brain atlas models are purely geometric and not FEM-based. It requires substantial manual efforts to make an atlas model compatible with FEM analysis. Meanwhile existing biomechanical brain models are built as a whole or only with a few structures, often consisting of tetrahedral elements, regarded as homogeneous linear elastic bodies [5]. The complex geometric information of the brain, such as sub-cortical structures and sulci, was much simplified. However in neurosurgery, as the brain is a complex organ with multiple structures, some structures, such as corpus callosum, behave much differently from the rest of brain [6], which indicates a need for modeling the brain structures separately. In addition, mechanical properties of soft tissues have been shown to be clearly nonlinear [7] and strain rate-dependant [4]. The assumption of linear elasticity is only valid in small deformation cases, which is not always the case in surgery simulations. In this paper we proposed a Finite Element Method (FEM) based brain atlas model, incorporated with a hyperviscoelastic material property typically suited for neurosurgery simulations. Our FE atlas distinguishes ourselves as: 1) it is the first FE based deformable brain atlas, employing a non-linear hyper-viscoelastic material property of the brain tissue typically suited for surgery simulation. An accurate deformation prediction of each structure of the brain is possible; 2) it is comprised of multiple brain structures, preserving salient anatomical information by using an automatic multi-material mesh generator; 3) it consists of all hexahedral elements, which is much preferable for biomechanical simulation. II. METHODOLOGY
In view of the fact that the well-established stereotaxic brain atlas of Talairach and Tournoux (TT 1988) [1] is used by many neurosurgical groups for planning of interventions, we chose TT atlas as anatomic basis. The geometric and anatomical identity information provided by the TT brain atlas is fully modeled by using this method. 2.1. Brain Atlas Computerization
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The computerization of TT brain atlas has been developed [2]. In order to construct the computerized version of the TT atlas, the printed atlas plates were digitized with high resolution and the digitized plates were extensively preprocessed, enhanced and extended. The electronic images were fully segmented and labeled with subcortical structures and cortical regions including Brodmann’s areas and gyri. Three dimensional polygonal models of the subcortical structures and Brodmann’s areas were also constructed.
neighboring elements, indicating its structure identity. Those elements with all the same structure labels were defined as internal elements, while those with different structure labels were defined as surface elements. The corresponding structure surfaces were then extracted. Due to the sparseness of the TT brain atlas and the variable distance between each pair, a weighted smoothing algorithm in combination with Laplacian smoothing algorithm was performed to guarantee a good quality of surface mesh as well as volumetric mesh.
2.2. Multi-Object Mesh Generation For mesh generation of FEM, tetrahedral meshes are generated as opposed to an all hexahedral mesh. In complex biomedical structures, the 3D domain can be more easily decomposed by tetrahedral elements than hexahedral elements. However, hexahedral elements have been generally favored in biomechanics domain, because they are well known to have many favorable characteristics over tetrahedral elements, such as accuracy and speed [8]. We have chosen hexahedral elements for their accuracy and robustness. As most of available hexahedral mesh generators do not allow meshing of multiple objects in one process [9], and are usually designed for regular and convex objects, which is often not the case for anatomical structures, none of them addressed constructing integral models of multi-material domains without resorting to manual intervention. We developed an automatic multiobject hexahedral mesh generator in order to generate an accurate model with detailed anatomy.
By using verification tools in a commercial package ABAQUS (HKS Inc) [9], the mesh was verified to meet the geometric and topological compatibility requirements for finite element solvers. The resulting mesh is shown in Fig.1 (bottom) and Fig. 2 (centre).
Fig.2 Detailed brain mesh (42,913 elements, 51,554 nodes); And 44 Subcortical structures extracted
For geometric validation of our meshed atlas, we have developed both qualitatively anatomical validation tools and quantitative analysis. Fig. 3 shows the FE Mesh toolkit implemented in JAVA to validate and edit the atlas mesh. For each surface node, RMS error in coordinates was computed as RMSerror
1 n
n
¦X
mesh i
X iatlas
2
,
i 1
where X imesh was the ith surface nodes of generated atlas mesh and X iatlas was the corresponding surface node of the original atlas. The computed RMS error was less than 0.2mm by adjusting the smoothing algorithm parameters.
Fig.1 Brain atlas based hexahedral mesh; (Top) Original electronic atlas [2]; (Below) 3D hexahedral mesh
In our modeling, the 27 axial slices of digitalized TT atlas with an inter-slice distance of 2-5 mm were used. These atlas images were fully segmented and labeled by color code. The images were sampled at a user-specified resolution, enabling a multi-resolution control to meet the requirements of applications. The resulting grids were connected to form a 3D frame structures. A voxel-like [10] gross hexahedral model was constructed where the dimensions for different element were not constant. For each element the corresponding label was read from atlas images and assigned to its nodes. These node labels were checked and grouped into specified arrays. Therefore, each hexahedral element has 8 labels corresponding to the 8
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Fig.3 Meshed Brain Atlas Validation Tool; Overlaid with orginal atlas slice.
By using this mesh generator, a detailed FE brain mesh was built from the atlas images. The fully hexahedral brain mesh with 44 sub-cortical structures mesh parts was constructed automatically. The deformation of each structure can be modeled and simulated separately. III. BIOMECHANICAL MODELING
material coefficient, which can assume any real value without restrictions. The complete list of material constants for brain tissue is given in Table 1. This model is suited for brain tissue behaviour in compression and extension for strains up to 30% and strain rates ranging over a few orders of magnitude, which includes the typical strain rate relevant to surgical procedures (0.01-1/s) [4]. TABLE 1 List of material constants for constitutive model of brain tissue, Equations 1 and 2, n=2 Instantaneous response k=1 k=2
The mechanical properties of brain tissue exhibit different behaviour characteristics at different scenarios. In surgical procedures, the brain tissue can be considered as single-phase continuum due to the fact that the time scale of the interstitial fluid flow is quite small. 3.1. Assumptions 1) Relatively simple model: according to recent work by [11], there is no significant difference between white and grey matter elasticity, and so homogeneity and isotropy are assumed for the entire brain for simple surgery simulations. 2) Quasi-Static model: as brain deformation during surgery is a relatively slow process with negligible dynamic components, we use a quasi-static model. 3.2. Hyper-viscoelastic Biomechanical model For large deformation cases, linear elastic material properties are not valid and nonlinear properties are used [8]. A recent biomechanical experiment made on brain is the study conducted in several in-vivo and in-vitro experiments on pig brains [4]. This work shows that brain tissue can be modeled with a homogeneous hyperviscoelastic material. The brain also exhibits non-linear stress-strain behaviour and stress-strain rate dependency. In finite deformation cases, an accepted way of describing mechanical properties of materials is by using a suitable strain energy function, W. To determine the Lagrange stress Tij as a function of the deformation gradient Fij, the strain energy function W was differentiated with respect to the appropriate deformation gradient, wW (1) Tij wFij The following Ogden-type form of the energy function for very soft biological tissues was used [4]. W
2
D
2
t
d
³ [P(t W ) dW (OD OD OD 3)]dW 1
0
2
3
(2)
The strain energy function is represented in the form of a convolution integral n
P
P0 [1
¦ g (1 e k
t
Wk
)]
(3)
k 1
describes the relaxation of the shear modulus of the tissue. W is a potential function, ODi are principal stretches, P 0 is the instantaneous shear modulus in undeformed state, IJk is characteristic time, gk are relaxation coefficients, and Į is a
P 0 =842 Pa;
Į =-4.7
t1=0.5 s; g1=0.450
t2=50 s; g2=0.365
IV. FINITE ELEMENT SIMULATION
This FE-based atlas model combined detailed anatomic information with material properties suited for a range of brain deformation cases in surgery. We simulated a simple scenario of brain deformation induced by gravity for a partial validation. A commercially available FEM package ABAQUS (HKS Inc) [9] were used. The meshed atlas was firstly generated by the above mentioned method. For computational efficiency while keeping the salient shape features, the element size was set to approximate 3×3×4 mm3 with the total element number of 42,913. The resultant hexahedral mesh with 44 subsets of cortical and intra-cortical structure was imported into ABAQUS 6.4 (HKS Inc). All tissues were modeled using 8 node hexahedral hybrid element C3D8H suited for nonlinear analysis. Non-linear geometrically procedures were applied. The hyper-viscoelastic constitutive model in 3.2 was used to define material properties of brain tissue. At this stage of work we regarded the whole brain tissue as a homogenous isotropic material, while the meshed individual structures provide possibility for future applications of heterogeneity. We selected to simulate gravity influence on brain deformation during surgeries. Before the cranium is opened, gravity acts uniformly on the brain. However, the outer surface of the brain is tethered to the inner surface of the skull by the arachnoid strands, which effectively prevents the brain from moving under the influence of gravity. Once the cranium is opened, however, the arachnid strands under the craniotomy region are severed and the brain is free to move and deform under gravity. To simulate this surgical situation, we firstly fixed the position of all external points on the grey matter surface, simulating the connection of the brain to the inside of the skull by the arachnoid strands as the pre-operative situation. We then applied gravity to the model to simulate the brain sagging when the cranium was opened. We have kept vertices at the basis of occipital lob (on 15% of the total height) fixed during the whole process, in consistency with the simulation in [6]. The deformation of the whole brain and sub-
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structures were simulated in the same round. The displacement fields of the whole brain and corpus callosum are shown in Fig.4.
Fig.4 Displacement field of the atlas brain and corpus callosum
A convergence study was conducted by generating a denser mesh and coarser mesh (Fig.5). The difference between results obtained is within 5%, giving evidence that the finite element approximation has converged.
matter, sulci, cortical and sub-cortical structures, etc. Secondly, we employed a nonlinear hyper-viscoelastic biomechanical model which accounts specifically for surgery simulation. Lastly our model is able to distinguish individual structures, apply different boundary conditions to structures, and simulate their different behaviours during surgery procedures, provided that more information, such as material properties for brain structures, is available. Our proposed approach is general to model and simulate complex brain deformation during neurosurgery, which will be our future work. Possible applications for this finite element atlas model include neurosurgery training, simulation, and deformation prediction. Future work includes study of boundary conditions and heterogeneity of brain tissue with respect to individual structures, registration from the atlas model to patient specific model, simulation of brain deformation where intracranial fluid pressure and bi-phase models should be taken into account. ACKNOWLEDGMENT
Fig.5 Coarse mesh (42913 elements) and dense mesh (172058 elements)
The observation of the average computed displacement field for frontal cuts along the gravity direction is 5.546 mm (Fig.6), in consistency with the result in [6], which indicates that the global displacement function is well transcribed.
The first author would like to thank A/Prof. K. Miller of Department of Mechanical Engineering, University of Western Australia, for discussions on biomechanics. REFERENCES [1] J.Talairach, P.Tournoux, Co-Planar Stereotaxic Atlas of the Human Brain (Thieme, New York, 2003)
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[2] W.L.Nowinski, A. Thirunavuukarasuu, and A.L. Benabid, The Cerefy Clinical Brain Atlas (Thieme Medical Publishers, Inc., 2004)
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[3] H.Delinguette, Toward Realistic Soft-tissue modeling in medical simulation, Proceedings of the IEEE 86 (3), 1998, 512-523
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Fig.6 Displacement along the gravity direction (unit: mm)
[4] K.Miller, Biomechanics of Brain for computer Integrated Surgery (Publishing House of Warsaw Univ. of Technology. 2002)
V. DISCUSSION AND CONCLUSION
In this study, we have developed a novel FEM-based brain atlas model, including the gross brain and individual brain structures. Detailed anatomic information from atlas was integrated with nonlinear hyperelastic material property of brain tissue to form a novel physical-based deformable brain atlas. Individual structure, such as corpus callosum, was isolated and analyzed. For partially validation we simulated a simple deformation scenario by gravity. With no a-priori conclusive information about sub-cortical structure’s material properties and boundary contiditions, we assumed a homogenous material model with displacement constraints only applied on surface grey matter at this stage of our work. The preliminary results under this assumption are promising. The novelties of our work are: firstly, we have implemented a finite element meshed brain atlas, comprising anatomical information of white matter, grey
[5] M.Ferrant, A.Nabavi, B. Macq, et al, Registration of 3D intraoperative MR images of the brain using a finite element biomechanical model, MICCAI Proceeding 2000, 19-28 [6] O. Clatz , H. Delingeutte, et al. Patient-specific biomechanical model of the brain: application to Parkinson’s disease, IS4T Proc. 2003, 321-331 [7] Y.C.Fung, Biomechanics: Mechanical Properties of Living Tissues. (Springer-Verlag, New York, USA. 1981) [8] T.J.R.Hughes, The finite element methods, linear static and dynamic finite element Analysis. (Englewood Cliffs, NJ: Prentice-Hall, 1987) [9] ABAQUS Theory Manual Ver. 6.4 (Hibbit, Karlsson & Sorrenson, Inc.2001) [10] J.H.Keyak, Improved Prediction of Proximial Femoral Fracture load using nonlinear finite element models, Medical Engineering and physics, Vol.23, 2001, 165-173 [11] H.Ozawa, et al. Comparison of spinal cord gray matter and white matter softness: measurement by pipette aspiration method. J. Neurosurgery 95, 2001, 221-4
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