Estimating Brain Deformation During Surgery Using Finite Element Method: Optimization and Comparison of Two Linear Models

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1 J Sign Process Syst (2009) 55: DOI /s Estimating Brain Deformation During Surgery Using Finite Element Method: Optimization and Comparison of Two Linear Models Hajar Hamidian & Hamid Soltanian-Zadeh & Reza Faraji-Dana & Masoumeh Gity Received: 1 January 2008 /Revised: 19 March 2008 /Accepted: 4 April 2008 /Published online: 10 May 2008 # 2008 Springer Science + Business Media, LLC. Manufactured in The United States Abstract This paper addresses estimation of brain deformation during craniotomy using finite element modeling. Two mechanical models are optimized and compared for this purpose: linear solid-mechanic model and linear elastic model. Both models assume the realistic finite deformation of the brain after opening the skull. In this study, we use pre-operative and intra-operative magnetic resonance images (MRI) of five patients undergoing brain tumor surgery. Anatomical landmarks are identified by an expert radiologist on MRI and used for the method development and comparison studies. We use tetrahedral finite element meshes and optimize model parameters by minimizing the mean distance between the predicted locations of the anatomical landmarks using the pre-operative images and H. Hamidian : H. Soltanian-Zadeh Control and Intelligent Processing Center of Excellence (CIPCE), School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran H. Hamidian h.hamidian@ece.ut.ac.ir H. Soltanian-Zadeh (*) Radiology Image Analysis Laboratory, Henry Ford Hospital, One Ford Place, 2F, Detroit, MI 48202, USA hamids@rad.hfh.edu hszadeh@ut.ac.ir R. Faraji-Dana School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran reza@ut.ac.ir M. Gity Department of Radiology, University of Tehran Medical School, Tehran, Iran p_gity@yahoo.com their actual locations on the intra-operative images. Evaluation of the objective function using a second set of landmarks not used in the optimization process suggests that accuracy of the solid mechanic model is higher than that of the elastic model for our application. Visual inspection of the results confirms this conclusion. The proposed method along with the location information of the surface landmarks measured in the operating room and marked on the pre-operative images can be used to estimate the brain deformations without needing intra-operative images. In this case, since the parameters of the brain tissue are not the same for different patients, the proposed optimization process is crucial for obtaining accurate results. Keywords Brain deformation. Finite element method. Function optimization. Mechanical modeling. Magnetic resonance imaging. Neurosurgery 1 Introduction Mechanical properties of soft tissue such as brain, liver, and kidney have been studied in recent years for applications such as robotic surgery systems [1], surgical operation planning, and surgery training systems based on the virtual reality techniques. However, in a common neurosurgical procedure, the brain deforms after opening the skull, causing misalignment of the subject to the preoperative images such as magnetic resonance images (MRI) or computed tomography (CT) images [2, 3] (Fig. 1). This happens because of cerebrospinal fluid (CSF) leakage, dura opening, anaesthetics and osmotic agents, as well as other surgical conditions that are different from the normal state. Opening the scalp and CSF leakage cause the gravitational shift of the tissue due to disappearance of tension and

2 158 H. Hamidian et al. Figure 1 MRI images a) before opening the skull, b) after opening the skull. Note the brain deformation after opening the skull due to the changes from the normal state. Also, note that the resolution of the intraoperative image is lower than the pre-operative images. The intact tumor is seen in the intraoperative image since the intraoperative image was acquired before the tumor excision. pressure forces on the boundary conditions of the brain [4, 5]. While the intra-operative imaging is the best way to determine this deformation, intra-operative images are costly and suffer from the constraints of the operating room. Spatial resolution and contrast of intra-operative images are typically inferior to those of pre-operative ones [6]. These problems can be avoided by using biomedical models of the brain and numerical calculations of the expected deformations. In this paper, two models are used as described next. The first model is based on the biphasic soft-tissue [7] that assumes the brain tissue behaves as a linear elastic material and indicates that the stress can be related to strain by Hook s law. The brain deformation is supposed to be infinitesimal, and the response of loading influenced by both elastic behavior of the tissue and the pressure of the extra-cellular fluid [8, 9]. The second model is based on the principle that the sum of the virtual work from the internal strains is equal to the work from the external loads. In this formulation, brain deformation is supposed to be infinitesimal, brain tissue is treated as an elastic material, and the relation between strain and stress is linear [10, 11]. In most practical cases, such models utilize the finite element methods [12] to solve sets of partial differential equations governing the deformation behavior of the tissue. For solving these models, the actual values of the brain s parameters are needed. Previous works used approximate values of the brain parameters, which we also use in this work as initial values. However, we optimize these parameters by minimizing the distance between the estimated positions of the anatomical landmarks and their actual positions in the intra-operative images. We then compare the two models using their resulting errors on a simulated model and real brain images. This method finds the optimal parameters of the brain in different patients and estimates the deformation of the brain based on the displacements of the anatomical landmarks. In the next section, the mechanical models, meshing, and boundary conditions are explained for solving the problem using finite element method. Optimization of the models parameters is also described. In Section 3, the results of our implementation on a test sphere as a model of the brain and real brain extracted from MRI are presented. Section 4 summarizes conclusions of the work. 2 Materials and Methods Patient-specific geometric data are obtained from a set of five pre-operative and intra-operative MRI of patients undergoing brain tumor surgery. In order to specify the displacements of the brain tissue, anatomical landmarks are defined in both of the pre- and intra-operative images that are registered rigidly using the ITK software (open-source toolkit for performing registration and segmentation). The displacement of the landmarks can be defined by medical experts, imaging device such as MRI, CT, laser devices, or spectroscopic camera. As mentioned previously, the resolution of the intra-operative tomography images is low and thus it may be superior to image the exposed surface of the brain or tissues that are important for surgery like the tumor. In order to distinguish between the brain parenchyma and tumor, the images are segmented using the 3D-Slicer (open-source software for visualization, registration, segmentation and quantification of medical data); see Fig. 2. After the segmentation, three-dimensional models of the surfaces of the brain parenchyma and tumor are created using the COMSOL3.3 software, which is based on the finite element methods for solving partial differential equations, visualization, and meshing and has strong post processing modules. For creating the 3D model, the slices that are near the craniotomy surfaces, have higher resolutions than the slices far from the surface. After creating the 3D models, automatic 4-noded tetrahedral meshes with

3 Estimating brain deformation during surgery using finite element method 159 Lagrange shape functions are generated for both of the parenchyma and the tumor using the COMSOL3.3 software. An example of the mesh generated by the software is shown in Fig. 3, which consists of 15,164 tetrahedral elements. 2.1 Biomedical Models of the Brain As explained in Section 1, for determining the deformation of the brain, a model of the brain may be used. Such a model provides numerical formulations that describe the behavior of the brain tissue. These formulations can be linear or non-linear. Linear models are simpler to implement and run relatively fast. Nonlinear models on the other hand are relatively complicated and computationally intense although they may generate more accurate results [13, 14]. In this paper, we optimize and compare two linear models, since the execution speed is critical for surgical applications Linear Solid-Mechanic Model In this model, the body is assumed to be a linear elastic continuum with no initial stresses or strains. The energy of the body s deformation caused by externally applied forces can be expressed as Eq. 1 [10]. E ¼ 1 Z Z σ T " dω þ F T udω; ð1þ 2 Ω Ω where F=F(x,y,z) is the total force applied to the elastic body, Ω is the elastic body, u is the displacement vector, and ɛ is the strain vector that can be defined as Eq. 2 [10]. @x T ¼ Lu Also, σ is the stress vector and in the case of linear elasticity, with no initial stresses or strains, relates to the Figure 2 Segmented MRI of the head used when building the patient specific brain mesh. a) The tumor segmentation is indicated by pink in the pre-operative image; b) The tumor segmentation is indicated by pink in the intra-operative image, c) Segmentation of the whole brain. The pre-operative image is used to build a computational model of the brain and tumor, and the intra-operative image is used to measure the displacement of anatomical landmarks. Figure 3 Patient-specific brain mesh consisting of 15,164 tetrahedral elements.

4 160 H. Hamidian et al. strain vector by the linear equation σ=dɛ where D is the elasticity matrix describing the material properties and is described as Eq. 3 [10]. The value of D depends on two material parameters: the Young modules (E) that relates tension and the stretch in the main orthogonal directions and the Poisson ratios (ν) which represent the ratio of the lateral contraction due to longitudinal stress in a given plane. 2 Eð1 νþ D ¼ ð1 þ νþð1 2νÞ 6 4 ν ν 1 ν ν 1 ν ν ν ð Þ ν ð Þ 0 1 2ν ð Þ ð3þ Volumetric deformation of the brain is founded by solving Eq. 1 for the displacement vector u, which minimizes the energy function E. Numerical solution to these equations could be written as the global linear Eq. 4 [10]. Ku ¼ F ð4þ The solution of Eq. (4) defines the deformation field that results from the forces applied to the body. We rely on the study of Ferrant et al. in [10] and choose our initial coefficients accordingly (Young modules = 3 kpa, Poisson ratio = 0.45) Linear Elastic Model This model assumes that the deformation of the brain tissue as a poroelastic material occurs because of its elastic behavior and the pressure of extracellular fluid [12]. The important body forces that affect the brain deformation are gravity (ρ t g), where ρ t is the brain density, the buoyancy of the surrounding fluid (ρ f g), where ρ f is the density of the surrounding fluid, and the pressure gradient of the exteracellular fluid ( p). Based on these assumptions, the model can be written as [15]: r:gru þr G 1 2u ðr:uþ arp ¼ r t r f g ð5þ where G is the shear modulus, υ is the Poisson s ratio, u is the displacement vector, p is the pore fluid pressure, and p is the ratio of the fluid volume extracted to deform the tissue under compression. Equation 5 relates the mechanical property of the tissue to the gradient of the fluid pressure across the medium. This equation considers three scalar equations involving four dependant variables: u x, u y, u z, and p. To solve the problem, we need a fourth equation that relates the deformation of the tissue to the drainage of the fluid during craniotomy. It can be described as Eq. 6 [16]. ðr:uþ r:krp þ S ¼ Ψ ð6þ where k is hydraulic conductivity, 1/S is the amount of fluid, which can be forced into the tissue under constant volume, and Ψ is the pressure source strength. The initial values of the parameters are listed in Table 1. These equations describe the behavior of the brain s tissue as linear elastic material and the pore fluid as incompressible material. Generally, the brain can be considered as a saturated material thus omitting the derivation of the pressure with respect to the time in Eq. 6 (i.e., α=1, 1=S ¼ 0). This model allows accessing more elastic boundary conditions of the intracranial pressure by modeling CSF drainage while maintaining linearity. That is an important computational advantage. 2.2 Boundary Conditions We need boundary conditions to solve the equations. In both models, we assume the exposed surface is free to move and the remaining surface is fixed. For the first model, we have conditions for the displacement variable and force per unit (F). We use F=u for the boundary conditions of the fixed boundary nodes because the elements of the rigidity matrix K in Eq. 4 for which the deformation is supposed to be known need to be set to zero and the diagonal elements to one. More details can be found in [10]. For the exposed surface, we assume its nodes are free to change. The initial value of the parameter F is set by examining the MR images of five different patients. The exact value of F for each part is determined by the optimization process. Table 1 Parameters used for the second model as initial values for the optimization process. Parameter symbol Description Value E Young modulus 2100 Pa ν Poisson s ratio 0.46 ρ t Density of brain tissue 1,000 kg/m 3 ρ f Density of surrounding tissue 1,000 kg/m 3 K Hydraulic conductivity 1e-7m 3 s/kg α Ratio of fluid volume extracted 1 to volume change of the tissue under compression 1/S Amount of fluid which can be 0 forced into the tissue under constant volume Ψ Magnetic dipole moment Pa/s G Shear modulus E/2(ν+1) Pa

5 Estimating brain deformation during surgery using finite element method 161 For the second model, the boundary condition for pressure (p) instead of the displacement variable is needed. The corresponding nodes in the mesh lying above the level of intra-operative CSF drainage are assumed to reside at the atmospheric pressure (Dirichlet condition in pressure), while those that do not, are the non-draining regions of the brain (Neumann condition in pressure). 2.3 Optimization Process The parameters of the brain in each model are not the same for different patients and thus usually approximated parameters are used. In this paper, we use an optimization process to optimize these parameters and achieve most accurate results with respect to the real deformations. To this end, we choose a cost function defined as the sum of the distances between the actual positions of the anatomical landmarks in the intra-operative images and their estimated positions based on the deformations of the pre-operative images by the two models. Displacements of the landmarks can be determined by an expert physician, an imaging device such as MRI, CT, or a spectroscopic camera. We use the Maltlab optimization toolbox for to optimize the cost function and find the optimal values of the parameters as explained next. In the first model, we do not know the exact value of the force applied to the center of the exposed surface of the brain. This parameter is found by the optimization process. In addition, the two parameters (Young modulus and Poisson s ratio) reported in the literature are not the same for different patients and thus they will be also optimized. For the second model, the parameters in Table 1 except for α, 1/S are optimized. This is because, as mentioned in Section 2.1.2, we model the brain as a saturated material for which the derivative of the pressure with respect to time is omitted in Eq. 6 (i.e., α=1, 1=S ¼ 0). We determine the brain s deformation in the steady state, meaning that we do not consider the transient changing of p in Eq. 6 and thus Ψ/k is optimized. In this study, we have used 10 points of the sphere for the optimization process and another 10 points for the evaluation of the results. To implement the models, we have used the COMSOL3.3 software, which is based on the finite element methods for solving partial differential equations. Figure 4 shows the results of the two models for the sphere. As seen, the deformation of the first model is more realistic (similar to the brain deformation) because its deformation is smooth. The result of the second model has some outstanding points and the deformation is not smooth. In addition, in the first model, the mean error of the points used in the optimization process is mm, and the mean error of the points that are not used in the optimization process is mm. For the second model, 3 Results For modeling the brain, we have used a sphere with the diameter of 22 cm, which is approximately the size of the brain. To model the skull opening, we have assumed that one section of this sphere is exposed and the other sections are fixed. We have assumed a model with specific parameters. We have specified a set of anatomical landmarks for use in the optimization process and another set for the evaluation of the optimization results. After the optimization, a comparison of the cost function for the two sets of the anatomical landmarks shows if the brain deformation can be reliably modeled by any of the models. Figure 4 Deformation results of a sphere as a simple model of the brain a) using the first model and b) using second model. Note that the deformation predicted by the first model is smooth but the one predicted by the second model has outstanding points and is not smooth. Therefore, the deformation predicted by the first model is more similar to that of a real brain the second model.

6 162 H. Hamidian et al. these values are and mm, respectively. Therefore, accuracy of the first model is higher than the second model. Table 2 compares the assumed and estimated parameters of the two models. It includes the means and standard deviations of the estimated values of the two parameters that are used in both models: Young modulus and Poisson s ratio. Note that the first model estimates these parameters more accurately than the second model. For more accurate estimation of the parameters, we must consider the tolerance of the parameters with respect to the initial values of parameters and how much the model is sensitive to the fluctuations of the parameters. Note that the optimization process has estimated the Young modulus and Poisson s ratio for the first model more accurately than the second model. In addition, the second model is very sensitive to the variation of the parameter ρ t ρ f such that its slight change causes a dramatic change in the results of the model. On the other hand, a slight change in the parameter F in the first model causes a slight change in the results. The sensitivity of the first model to the parameter F is approximately equal to the sensitivity of the second model to the parameter Ψ/k. Comparing the final values of the cost function for the points used in the optimization process and those not used in the optimization process, we conclude that the parameters of the first model can be estimated more reliably than the second model and thus the first model is a more realistic model for our application. To evaluate the methods on the real data, we have used five pre-operative and post-operative MRI studies of different patients undergoing brain tumor surgery. The pre-operative and intra-operative images have been registered rigidly using ITK, and then anatomical landmarks have been determined by our expert radiologist in the preand intra-operative images. For each patient study, the expert has selected about 60 pairs of corresponding landmarks. One half of the landmarks have been used in the optimization process and the other half have been used for testing of the methods. As mentioned before, the brain and the tumor have been segmented using the 3D-Slicer software and 3D models of the brain and the tumor have been created using the COMSOL3.3 software. For mesh generation and numerically solving the models equations, we have used the COMSOL3.3 software. For the optimization process, we have used the MATLAB optimization toolbox. By comparing the matching error of the training landmarks and that of the testing landmarks, capability of each model for accurate estimation of the brain deformations has been evaluated. Sample results for the first and second model are presented in Figures 5 and 6. Figures 5a and 6a show 3D results for a representative brain deformation. Note that maximum deformation is shown in red and the minimum deformation is shown in blue. The brain deformation in this model is smooth. Figures 5b and 6b show the 3D results of the brain deformation with arrows showing the deformation of each point inside the brain. Figures 5c d and 6c d compare the tumor in 3D and its 2D contours in the coronal sections obtained from the intra-operative images with the results of the optimization process for the first and second models, respectively. Note that the predicted results of both models are in a good agreement with the actual intraoperative results for the tumor in two and three dimensions. However, the results of the first model show a higher level of matching in the two dimensional views. Table 2 Assumed and estimated parameters and their variations for the two models using a sphere as a simple model of the brain. Assumed Estimated Model Young modulus (E) ±176 First model s parameter Poisson s ratio (n) ±0.01 First model s parameter Force Fx=1500 Fx=1500±91 First model s parameter Fy=1500 Fy=1500±88 Fz=1500 Fz=1500±93 Young modulus (E) ±195 Second model s parameter Poisson s ratio (ν) ±0.07 Second model s parameter ρ t ±60 Second model s parameter ρ f ±40 Second model s parameter Ψ/k 1.01e4 1.01e4±0.08e4 Second model s parameter Specific parameters are assumed and the resulting deformations of specific points are used in the optimization process. Other points are used for testing of the optimization process. Then, the parameters are changed and the optimization process is repeated to estimate the model parameters again. The results shows if the brain deformation can be modeled by any of the proposed models and if the proposed optimization method can accurately estimate that model. For each model, we have done this process and have compared the accuracy of models using the matching error of the points used in the optimization process and those not used. Note that the Young modulus and Poisson s ratio of the first model are estimated more accurately than those of the second model. In addition, the second model is very sensitive to the variation of parameter ρ t ρ f and its slight change affects the results more than the parameter F in the first model. Finally, the optimization process estimates the parameter F more accurately than Ψ/k.

7 Estimating brain deformation during surgery using finite element method 163 Figure 5 Results of optimizing the parameters of the first model for the real brain data. a) 3D result of the brain deformation where maximum deformations are shown in red and minimum deformations are shown in blue. The brain deformation in this model is smooth. b) 3D result the brain deformation with arrows showing deformation of each point inside the brain. c) Comparison of the predicted and actual tumor in 3D and d) Comparison of the 2D contours of the tumor obtained from the intra-operative images with those predicted by the first model. Note the good matching of the results in both two and three dimensions. The estimated parameters for the first and second models and the tolerance of the maximum and mean error for the testing landmarks in five cases are presented in Tables 3 and 4. The testing landmarks are mostly near the exposed surface and the accuracy of both models is acceptable for them. However, the accuracy of the first model in terms of both the maximum and the mean error is superior to the second model.

8 164 H. Hamidian et al. Figure 6 Results of optimizing the parameters of the second model for the real brain data. a) 3D result of the brain deformation where maximum deformations are shown in red and minimum deformations are shown in blue. The brain deformation in this model is not smooth. b) 3D result the brain deformation with arrows showing deformation of each point inside the brain. c) Comparison of the predicted and actual tumor in 3D and d) Comparison of the 2D contours of the tumor obtained from the intra-operative images with those predicted by the first model. Note inaccurate matching of the contours. The impact of the proposed randomized landmark selection process for the optimization and evaluation processes on the final results has been studied by repeating the experiments after the roles of the landmarks are swapped, i.e., those used initially in the optimization process are used in the testing phase and those initially used in the testing phase are used in the optimization process. The final results are compared in Table 3. Note that

9 Estimating brain deformation during surgery using finite element method 165 Table 3 Maximum and mean errors for real data. Case 1 (mm) Case 2 (mm) Case 3 (mm) Case 4 (mm) Case 5 (mm) Maximum error for first model Δx=3.4 Δx=2.9 Δx=4.0 Δx=3.1 Δx=2.7 Δy=4.4 Δy=3.1 Δx=3.6 Δx=3.0 Δx=2.8 Δz=0.3 Δz=0.8 Δx=0.6 Δx=0.8 Δx=0.4 Mean error for first model Δx=1.3 Δx=0.9 Δx=1.6 Δx=1.2 Δx=0.9 Δy=1.7 Δy=1.1 Δy=1.4 Δy=1.0 Δy=0.8 Δz=0.1 Δz=0.6 Δz=0.3 Δz=0.3 Δz=0.1 Maximum error for second model Δx=3.8 Δx=3.1 Δx=3.4 Δx=3.3 Δx=3.0 Δy=4.4 Δy=3.2 Δy=3.9 Δy=2.8 Δy=2.9 Δz=0.4 Δz=1.0 Δz=0.8 Δz=0.5 Δz=0.7 Mean error for second model Δx=1.4 Δx=1.2 Δx=1.5 Δx=1.4 Δx=1.1 Δy=1.8 Δy=1.5 Δy=1.6 Δy=1.0 Δy=1.2 Δz=0.2 Δz=0.6 Δz=0.5 Δz=0.2 Δz=0.4 Maximum error for first model Δx=3.4 Δx=3.1 Δx=3.6 Δx=2.9 Δx=3.0 Δy=4.6 Δy=3.2 Δy=3.9 Δy=3.1 Δy=3.0 Δz=0.2 Δz=1.0 Δz=0.8 Δz=0.7 Δz=0.6 Mean error for first model Δx=1.2 Δx=1.0 Δx=1.4 Δx=1.0 Δx=1.2 Δy=1.7 Δy=1.1 Δy=1.5 Δy=1.0 Δy=1.0 Δz=0.1 Δz=0.5 Δz=0.4 Δz=0.3 Δz=0.2 Maximum error for second model Δx=3.8 Δx=3.5 Δx=3.3 Δx=3.3 Δx=3.3 Δy=4.7 Δy=3.5 Δy=4.0 Δy=3.0 Δy=3.1 Δz=0.3 Δz=0.8 Δz=1.0 Δz=0.6 Δz=0.6 Mean error for second model Δx=1.5 Δx=1.6 Δx=1.5 Δx=1.3 Δx=1.4 Δy=2.0 Δy=1.6 Δy=1.7 Δy=1.1 Δy=1.2 Δz=0.1 Δz=0.4 Δz=0.6 Δz=0.2 Δz=0.3 Here, pre-operative and intra-operative MRI studies of five patients undergoing brain tumor surgery are used. The pre- and intra-operative images are registered rigidly, and then pairs of anatomical landmarks are determined by an expert radiologist in the corresponding images. One half of the landmarks are used in the optimization process and the other half are used for the testing. Based on the error of the testing landmarks, ability of each method to estimate real brain deformations is evaluated. Note that the accuracy of both models are acceptable but the accuracy of the first model in terms of the maximum and mean errors is higher than the second model. The experiments have repeated for landmarks by changing their roles in optimization and testing processes. The results are shown in the second part of the table. there are minor differences between the two results confirming suitability of the proposed landmark selection and the optimization process. The tolerance of the parameters and the error depend on how much of the brain surface is exposed, how much of the CSF drains, and how much the brain conditions change due to the surgery. In addition, the position, the depth, and the Table 4 Estimated parameters for the two models. Optimized in five cases Model Young modulus 0.45±0.03 First model s parameter Poisson s ratio 3000±269 First model s parameter Resultant Force ±549.6 First model s parameter E 2100±205 Second model s parameter ν 0.45±0.02 Second model s parameter ρ t 1000±41 Second model s parameter ρ f 1000±13 Second model s parameter Ψ/k 1.01e4±0.10e4 Second model s parameter size of the tumor relative to the brain affect the results. When the tumor is located in depth, the deformation of the tumor due to the opening of the skull will be small and thus the matching of the actual and predicted locations of the tumor will be better than a superficial tumor. Also, selection of landmarks has an important role in the results. Thus, the expert should define important landmarks with large displacements after opening the skull. The execution time of the first model is approximately 1.5 times of the second model using a personal computer with a 3.6 GHz dual-core central processing unit (CPU) and 4 GB random access memory. Both models can be used for estimating the brain deformation after opening the skull using the displacements of the anatomical landmarks on the exposed surface of the brain. Previous work used approximations for the models parameters. To achieve more accurate prediction of the brain deformation, these parameters are optimized in this paper for each patient. Also, previous work used intra-operative images to determine the boundary condition of the force parameter in the mechanical model. We determine this boundary condition through an

10 166 H. Hamidian et al. optimization process. Consequently, the proposed method estimates the brain deformation without using the intraoperative images. It uses the positions of a few landmarks on the surface of the brain which can be easily estimated after the skull is opened using intra-operative equipment such as neuronavigators or optical positions systems. 4 Conclusion In estimating the brain deformations, model selection is an important step for obtaining accurate and reliable results. To this end, we studied two linear mechanical models for describing the mechanical properties of the brain based on finite deformation and optimized their parameters for a simple sphere and real models of the brain extracted from MRI. The first model is based on the virtual works, applied to the inside and outside of the brain, but the second model is based on the equations that relate the displacement of the volume to the pressure of the fluid. The first model generated shape deformations similar to the brain deformations. Also, its application to a sphere generated more accurate displacements than the second model. The application of both models to real data have has a good matching for the tumor in two and three dimensions. However, the first model predicts the tumor location more accurately in two dimensions. The landmarks near the exposed surface show improved matching by the first model based on the maximum and mean of the error of the surface landmarks that are not used in the optimization process. On the other hand, the first model is about 50% slower than the second model, yet, using a high speed CPU, the difference will not be significant. The results of our study confirm that the brain deformation can be reliably estimated using anatomical landmarks on the exposed surface of the brain whose locations can be easily measured by the neuro-navigators used in the operation rooms. Also, the proposed optimization process will eliminate the prediction errors due to the variations of the model parameters from patient to patient. However, the optimization process does not affect the mesh of the next iteration; a new mesh is generated in each iteration of the algorithm. Therefore, the degeneration problem does not occur in our implementation. Acknowledgement Patient-specific geometric data for the brain were obtained from pre- and intra-operative MRI studies of patients undergoing brain tumor surgery at the Surgical Planning Laboratory, Brigham and Women s Hospital, Harvard Medical School, Boston, MA, USA. The authors gratefully acknowledge and thank Prof. Ron Kikinis and Dr. Tina Kapur for providing this crucial data. References 1. Wittek, A., Miller, K., Laporte, J., Kikinis, R., & Warfield, S. (2004). Computing reaction forces on surgical tools for robotic neurosurgery and surgical simulation. 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11 Estimating brain deformation during surgery using finite element method 167 Foundation (NSF), American Institute of Biological Sciences (AIBS), and international funding agencies. Hajar Hamidian M.S. was born in Shiraz, Iran in She received B.S. degree in Biomedical Engineering from Shahed University, Tehran, Iran in 2005 and M.S. degree in Biomedical Engineering from the University of Tehran, Tehran, Iran in Her research interests include signal and image processing and analysis, neurosurgery, computational modeling, and their medical applications. Hamid Soltanian-Zadeh, Ph.D. received B.S. and M.S. degrees in Electrical Engineering: electronics (with honors) from the University of Tehran, Tehran, Iran in 1986 and M.S.E. and Ph.D. degrees in Electrical Engineering: systems and bioelectrical sciences from the University of Michigan, Ann Arbor, Michigan, USA in 1990 and 1992, respectively. He was a post-doctoral research fellow and an Assistant Scientist at Henry Ford Hospital, Detroit, Michigan, USA, from 1992 to 1994 and an associate scientist and senior scientist afterwards. Since 1994, he has also been with the Department of Electrical and Computer Engineering, the University of Tehran, Tehran, Iran, where he is currently a full Professor and director of the Control and Intelligent Processing Center of Excellence (CIPCE). Prof. Soltanian-Zadeh directs research projects in the Department of Radiology, Henry Ford Hospital, Detroit, Michigan, USA and collaborates with the Institute for studies in theoretical Physics and Mathematics (IPM), Tehran, Iran. His research interests include medical imaging, signal and image processing and analysis, pattern recognition, and neural networks. He has supervised or advised 80 M.S. theses and 12 Ph.D. dissertations and co-authored over 500 papers in journals and conference records. Several of his presentations received honorable mention awards at the SPIE and IEEE conferences. In addition to the IEEE, where he is a senior member and vice president of the Iran Section, he is a member of the SPIE, SMI, and ISBME and has served on the scientific committees of several national and international conferences and review boards of about 40 scientific journals. Prof. Soltanian-Zadeh is a member of the Iranian Academy of Sciences, has been recognized as an outstanding researcher by the national and international organizations, and has served on the study sections of the National Institutes of Health (NIH), National Science Reza Faraji-Dana, Ph.D. received B.S. degree (with honors) from the University of Tehran, Tehran, Iran in 1986 and M.A.Sc. and Ph.D. degrees from the University of Waterloo, Waterloo, ON, Canada in 1989 and 1993, respectively, all in Electrical Engineering. He was a Postdoctoral Fellow with the University of Waterloo for one year. In 1994, he joined the School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran where he is currently a Professor. He has been engaged in several academic and executive responsibilities, among which was his deanship of the Faculty of Engineering for more than four years, up until summer 2002, when he was elected as the University President by the university council. He was the President of the University of Tehran until December He is the author of several technical papers published in reputable international journals and refereed conference proceedings. Prof. Faraji-Dana has been the Chairman of the IEEE-Iran Section since March He received the Institution of Electrical Engineers Marconi Premium Award in Masoumeh Gity, M.D. was born in Daregaz, Iran in She graduated from the University of Tehran Medical Sciences as a Medical Doctor in 1987 and finished the Radiology postgraduate course in 1990 with the first degree in the national board exam. She has been with the University of Tehran Medical Sciences Radiology Department since then. She is now an Associate Professor and is involved in different fields of Diagnostic Radiology mainly in Breast Imaging and Neuroradiology. Dr. Gity is interested in medical image analysis and computer-aided-diagnosis (CAD) and has collaborated with several research projects in this field. She is the author or coauthor of three books and more than 60 articles.