The exposed surface is shown in blue and the fixed surface is shown in red. Note that when the skull is exposed, the exposed and the nearby tissues deform. Boundary condition for the displacement parameter. In both models, the exposed surface is assumed free to move and the remaining surfaces are fixed. Therefore, the parameter u is estimated for the exposed surfaces blue surfaces and the surfaces close to it but it is zero for the remaining surfaces red surfaces. In both models, we have conditions for the force F variable rather than the displacement variable. Previous works suggest that this parameter is a constant fixed value for each surface and determine its value for each surface by registering the intra- and pre-operative volumes [ 13 ].
We fix this parameter for the center of the exposed surface and let it change for the remaining exposed surface. Further details can be found in [ 13 ]. In other words, no forces are applied to the fixed surface so the equivalent force for this surface will be zero. The initial value of the parameter F for the center of the exposed surface is set by examining the MR images of six patients. The exact value of F for each part is determined by the optimization process as explained in the next section.
This condition is illustrated in Figure 6. The value of parameter F for the green surface the center of the exposed surface is fixed, for the blue surface the remaining exposed surface is unconstrained, and for the red surface unexposed surface is zero. For both models, there are conditions for the force variable F.
The value of F for the green surface the center of the exposed surface is fixed, for the blue surface the remaining exposed surface is unconstrained, and for the red surface unexposed surface is zero. The parameters of the models change from case to case. Thus, as in our previous work [ 8 ], we use approximated parameters as the initial values and optimize them for each case to maximize the accuracy of the results for the known deformations of each case. As mentioned before, we propose a new approach for determining the parameter F.
The value of this parameter in the center of the exposed surface is also determined in the optimization process. 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 corresponding estimated positions based on the deformation results of applying the two models on the pre-operative images.
Displacements of the landmarks are determined by an expert physician who uses the 3D-Slicer software to mark the corresponding pre- and intra-operative points. The models are then applied to the pre-operative points and their results compared with the corresponding actual results. Figure 7 shows the screen of the software used to define the points. When the expert selects points on the 2 D images of the brain, the coordinates of the point in the 3D space are shown and saved in a text file.
The points are mostly selected near the exposed surface due to larger displacements of the points in this area relative to the other points. Note that the size of the exposed area and the consequent deformation of the brain are not the same in all cases. For instance, if the tumor is large, the surgeon opens a relatively large surface of the brain and therefore, the deformations are large. In this case, the expert selects several points for accurate estimation of the parameters.
In total, our expert has selected about 70 pairs of the corresponding landmarks for each case. One half of the landmarks have been used in the optimization process and the other half in the testing process. We use the Maltlab optimization toolbox fminsearch function to optimize the cost function and find the optimal values of the parameters as explained next.
Illustration of using the 3D-Slicer software for manual selection of the landmarks by expert radiologist. We select points from all 2 D images of the brain. In both models, we do not know the exact value of the force applied to the center of the exposed surface of the brain. The value of this parameter determined in sample cases is used as an initial value and the optimal value is determined by the proposed optimization process. In addition, in the mechanical model, the two parameters Young modulus and Poisson's ratio reported in the literature are not the same for different patients and thus they are also optimized for each case.
For the nonlinear model, in addition to the initial value for F, the parameters listed in Table 1 are used in the optimization process except the parameters of characteristic time. This is because we study the problem in the steady state which is independent of these parameters. Therefore, these values are varied to find the minimum error. For evaluation of our method, we first apply the models on a brain simulation a sphere with the diameter of 22 Cm.
To model the skull opening, we assume that one section of this sphere is exposed and the other sections are fixed. In the meshing process, we use 9, tetrahedral meshes. Figure 8 shows an example of meshing for the spherical brain model. For each model, we use the brain model with specific parameters and boundary conditions. Sphere mesh consisting of 9, tetrahedral elements.
We specify a set of anatomical landmarks for the optimization process and another set for the evaluation of the optimization results. After the optimization, a comparison of the cost function for the evaluation landmarks shows whether the brain deformation is reliably modeled and if the optimization process estimates the model parameters accurately. 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. Figure 9 shows the results of the two models for the sphere.
Note that deformations of the models are smooth and realistic, similar to those of the brain. This is because the models solve their equations assuming that the equivalent work applied to a surface is zero. Although the results are similar but as we will see next, the results of the nonlinear model are more desirable than those of the other model.
For the mechanical and nonlinear models, the mean errors of the points used in the optimization process are 0. The mean errors of the points not used in the optimization process are 0. Therefore, accuracy of the nonlinear model is higher than that of the mechanical model.
This is because the nonlinear model is more flexible than the linear model. In addition, the number of the parameters of the nonlinear model is larger than that of the mechanical model. Consequently, the nonlinear model fits the landmark data more closely than the mechanical model. Deformation results of a sphere as a simple model of the brain. Note that the deformations predicted by the linear mechanical model and nonlinear model are smooth, similar to the brain deformations. Table 2 shows the assumed and estimated parameters of the two models. As seen, the optimization process estimates the parameters of the nonlinear model more closely than those of the mechanical model.
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 nonlinear model is more appropriate than the linear, mechanical model. However, the execution time of the nonlinear model is six times of the mechanical model. The mean execution time is approximately 35 hours for the linear model and hours for the nonlinear model for each case. Both models are implemented on a PC with the 1.
To evaluate the methods on the real data, we have used six image sets each containing 90 slices with 2. Each image set contains both of the pre-operative and intra-operative MRI studies of a brain tumor patient who has undergone surgery. It is commonly acknowledged that tumors are associated with ''stiffer'' tissue relative to the normal tissues. However, the volume of a tumor is usually small relative to the volume of the brain.
Thus, uncertainties about the tumor's mechanical properties do not significantly affect the overall displacement field. Consequently, the tumor was simulated using the same constitutive model as ''healthy'' brain tissue. Also, the parameters of the two models for the tumor are equal to the brain's parameters [ 20 ].
Of course, if specific model parameters are known for the tumor, they can be used in the proposed algorithm. Sample results for the mechanical and nonlinear models are presented in Figure The figure shows 3D results for a representative brain where large deformation is shown in red and small deformation is shown in blue. Note that the brain deformations in the two models are smooth. The results indicate that our simulations are realistic. Note that the maximum deformation is shown in red and the minimum deformation is shown in blue. As seen, the brain deformation in both models is smooth.
Figures 11 a - b compare the 2 D contours of the tumor in the coronal sections obtained from the intra-operative images with the results of the optimization process for the two models. Note that the predicted results of nonlinear model show higher levels of matching than those of the linear model. These results show the tumor slices near the craniotomy surface.
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For the slices deep in the brain or the slices far from the craniotomy, the results of the two models show similar matching; both models follow the deformation quite well. This is because deformations of the brain tissues far from the craniotomy are small. Comparing the 2 D contours of the tumor in the coronal sections obtained from the intra-operative images with the results of the optimization process.
Column a , the results of the nonlinear model, and column b , the results of the linear mechanical model. As seen, the predicted results of the nonlinear model show a higher level of matching.
It must be noted that these results show the tumor slices close to the craniotomy surface of the skull. Figure 12 shows the landmark locations estimated by the models and the corresponding actual results from the intra-operative images. Note that the points estimated by the nonlinear model green points are the closest to the intra-operative points yellow points.
Also, the points estimated by the linear mechanical model pink points are closer to the real results than those estimated by the linear elastic model the elastic model is described in [ 8 ]. The numerical values of the maximum and mean errors of the testing landmarks in six cases are presented in Table 3. Table 4 presents the variations of the estimated parameters of the linear and nonlinear models in six cases relative to the initial values.
The testing landmarks are mostly near the exposed surface. Again, the nonlinear model shows superior performance compared with the linear mechanical model. The errors depend on how much the brain surface is exposed, how much the CSF drains, and in general how much the brain conditions change due to the craniotomy. In addition, the position, the depth, and the size of the tumor affect the results.
Finally, selection of landmarks has an important effect on the results; for a conservative evaluation, critical landmarks with relatively large displacements after opening the skull should be considered. Two groups of resulting points of each model and the corresponding real result from the intra-operative images. The points estimated by the nonlinear model green points are the nearest points to the intra-operative points yellow points while those estimated by the linear mechanical model pink points are superior to those estimated by the linear elastic model.
The execution time of the nonlinear model is approximately six times of the linear mechanical model using a personal computer with a 1. This ratio is an approximation because the execution time depends on the problem complexity, the number of slices, and the mesh resolution that are different for different cases. This method can be used for estimating the deformation of the brain after opening the skull for brain surgery, and calculates the displacements of the anatomical landmarks on the exposed surface of the brain. By optimizing the model parameters for each patient, the prediction accuracy increases.
In addition, devices like neuro-navigators and lasers can be used to determine the coordinates of pre-operative points corresponding to specific intra-operative points. This method does not use intra-operative images. Moreover, by defining a pattern for the force parameter in the proposed models based on specifications like the tumor depth and the exposed surface, approximate parameters of the models can be determined and used in the models to estimate the deformation of the brain without the optimization process.
We have presented a brain shift compensation method based on linear and nonlinear biomechanical models guided by limited intra-operative data. To this end, we have employed finite element methods for descritizing and solving partial differential equations that describe the brain deformation and optimized their parameters for each case for reducing the inaccuracy due to the variations of the parameters from case to case. Also, we have presented a new procedure for defining the force parameter for the models.
To evaluate the proposed method, we have used simulations as well as real MRI data of the brain. Experimental results have shown that both of the linear mechanical and nonlinear models generate shape deformations similar to the brain deformations. In their applications to a simulation study, the nonlinear model generated the most accurate displacements and the linear mechanical model generated more accurate displacements than the linear elastic model.
In addition, in their applications to the real data, the nonlinear model generated the best matching for the tumor while the linear mechanical model outperformed the linear elastic model. The landmarks near the exposed surface showed superiority of the nonlinear model based on the maximum and mean error of the surface landmarks not used in the optimization process. Therefore, depending on the desired levels of speed and accuracy, one of the models can be used.
The results of our study confirm that the brain deformation can be reliably estimated using anatomical landmarks on the exposed surface of the brain that can be easily measured by the neuro-navigators used in the operation rooms. Last but not least, the proposed optimization process eliminates the prediction errors due to the variations of the model parameters from patient to patient. It also confirms the conclusion of [ 23 ] that the results of the linear and nonlinear model are not considerably different and thus, considering the execution speed of the two models, the linear mechanical model may be selected for the modeling of the brain deformation.
Surgical Neurology , — NeuroImage , — Clinical Neurophysiology , — Medical Image Analysis , — Prentice Hall, Englewood Cliffs. New Jersey Journal of Signal Processing Systems , — Medical Imaging Analysis , 6: — Miller K: Method of testing very soft biological tissues in compression.
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Volume Issue 5 Nov in Journal of Neurosurgery. Article Information. Page Count: — Method Results Discussion Summary. View in gallery Flash x-ray taken during the impact of a blow to the head, showing alteration in the position of tags due to brain and brain stem distortion. View in gallery Flash x-ray taken after the blow, showing return of tags to position shown in Fig.
View in gallery Static reference flash x-ray of dog's head with lead tags implanted in the brain, taken before any manipulation. View in gallery Flash x-ray taken after twisting, extending, and flexing the head about the neck, and showing no change in the markers. View raw image Close. Edberg , S. Gurdjian , E. Pudenz , R. PubMed Citation Articles by V. Hodgson Articles by E. Gurdjian Articles by L. Thomas Similar articles in PubMed. Article by V.
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