- Research
- Open Access
Data-guide for brain deformation in surgery: comparison of linear and nonlinear models
- Hajar Hamidian^{1},
- Hamid Soltanian-Zadeh^{1, 2}Email author,
- Reza Faraji-Dana^{3} and
- Masoumeh Gity^{4}
https://doi.org/10.1186/1475-925X-9-51
© Hamidian et al; licensee BioMed Central Ltd. 2010
- Received: 2 January 2010
- Accepted: 15 September 2010
- Published: 15 September 2010
Abstract
Background
Pre-operative imaging devices generate high-resolution images but intra-operative imaging devices generate low-resolution images. To use high-resolution pre-operative images during surgery, they must be deformed to reflect intra-operative geometry of brain.
Methods
We employ biomechanical models, guided by low resolution intra-operative images, to determine location of normal and abnormal regions of brain after craniotomy. We also employ finite element methods to discretize and solve the related differential equations. In the process, pre- and intra-operative images are utilized and corresponding points are determined and used to optimize parameters of the models. This paper develops a nonlinear model and compares it with linear models while our previous work developed and compared linear models (mechanical and elastic).
Results
Nonlinear model is evaluated and compared with linear models using simulated and real data. Partial validation using intra-operative images indicates that the proposed models reduce the localization error caused by brain deformation after craniotomy.
Conclusions
The proposed nonlinear model generates more accurate results than the linear models. When guided by limited intra-operative surface data, it predicts deformation of entire brain. Its execution time is however considerably more than those of linear models.
Keywords
- Mechanical Model
- Nonlinear Model
- Virtual Work
- Exposed Surface
- Biomechanical Model
Background
Medical imaging methods play a key role in localizing tissues and organs during surgery. Pre-operative imaging devices generate high-resolution images of the tissues and organs while intra-operative imaging devices generate their low-resolution images. The pre-operative images however cannot be easily used during surgery since they do not reflect correct anatomy and geometry of tissues and organ intra-operatively. This is due to motions and deformations of soft tissues over time. The end result is that actual positions of the tissues during surgery do not match with those reflected in their preoperative images. To be able to use pre-operative images intra-operatively, they should be deformed based on the tissue geometry reflected in the intra-operative images. However, intra-operative images are low resolution and low quality. To overcome these limitations, intra-operative images are used along with biomechanical models to update pre-operative images such that they reflect the tissue geometry during surgery [1–6]. In this process, the Finite Element Method (FEM) [7] is employed to solve the partial differential equations that govern deformation behavior of soft tissues.
In our previous study [8], we used the finite element method to develop and compare two linear models: mechanical and elastic [9–12] for image-guided neurosurgery. We showed that accurate computation of brain deformation due to craniotomy can be achieved by defining a load through prescribed displacements of the corresponding points in the pre- and intra-operative images. Experimental results showed that the mechanical model was superior to the elastic model; the brain deformation could be estimated by the mechanical model more accurately. The execution time of the mechanical model was however about 50% more than that of the elastic model.
In this paper, a nonlinear model is developed for estimating the brain deformation and compared to the linear mechanical model. The mechanical model [13, 14] 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, the brain deformation is assumed to be infinitesimal, the brain tissue is treated as an elastic material, and the relation between strain and stress is linear. The nonlinear model [15], on the other hand, is based on the equation of equilibrium that relates the covariant differentiation of stress (with respect to the deformed configuration) to the body force per unit mass. In this model, the brain deformation may be large, brain tissue is treated as a hyper visco-elastic material, and the stress-strain behavior of the tissue is non-linear [16, 17].
To solve the equations of the models, actual values of the organ parameters are needed. To this end, we optimize the initial, approximate values to obtain the actual values. The cost function for this optimization is the distance between the estimated positions of the pre-operative anatomical landmarks and their corresponding actual positions in the intra-operative images. One half of these landmarks are utilized in the optimization process and the other half in the evaluation process. We compare the models using their errors on simulated and real data sets, using the corresponding points that are not used in the optimization process.
In the next section, the proposed models, meshing, and boundary conditions are explained. Optimization of the parameters of the models is also described in this section. In Section 3, the results obtained for a test sphere as a model of the brain and real brain extracted from MRI are presented. Finally, Section 4 summarizes the conclusions of the work.
Methods
Construction of 3D Model and Finite Element Mesh
Biomechanical Models of Brain
The biomechanical models guided by low-resolution intra-operative images may be used for updating the high-resolution pre-operative images [18, 19]. These models can be linear or nonlinear. In our previous work [8], we compared two linear models (elastic and mechanical) and found that the mechanical model generated more accurate results. In this paper, we develop a nonlinear model and compare it to the best of the two linear models studied previously, i.e., the mechanical model. Relying on the study of [13], the initial coefficients of the mechanical model are set to Young modules = 3 kPa and Poisson ratio = 0.45. Next, we explain the nonlinear model. The readers are referred to our previous work [8] for the details of the mechanical model.
Non-linear Model
Parameters used for the nonlinear model as the initial values for the optimization process.
Instantaneous Response | Characteristic Time | Characteristic Time |
---|---|---|
C100 = 263 (Pa) | τ_{1} = 0.5 (s) | τ_{2} = 50 (s) |
C010 = 263 (Pa) | ||
C110 = 0 (Pa) | Instantaneous Elasticity | Instantaneous Elasticity |
C020 = 491 (Pa) | g_{1} = 0.450 | g_{2} = 0.365 |
C200 = 491 (Pa) |
Boundary Conditions
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. We also use F = u for the boundary conditions of the fixed boundary nodes because all of the equations lead to the equation Ku = F (K encapsulates all coefficients of the equation) and therefore, according to [13], the non-diagonal elements of the rigidity matrix K for which the deformation is supposed to be known are zero and the diagonal elements are one. Further details can be found in [13].
Optimization Process
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.
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. Also, the parameters g_{1} and g_{2} lead to g_{1} + g_{2} = g in the equation of the steady state. Therefore, these values are varied to find the minimum error.
Results
Simulation Data
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. To implement the models, we have used the COMSOL3.3 software which is based on the finite element methods for solving partial differential equations.
Assumed and estimated parameters and their variations for the models using a sphere.
Mechanical model | Young modulus | Poisson's ratio | Force | |||
---|---|---|---|---|---|---|
Assumed | 0.45 | 3000 | Fx = 1500 | |||
Fy = 1500 | ||||||
Fz = 1500 | ||||||
Estimated | 0.45 ± 0.0056 | 3000 ± 175.6 | Fx = 1500 ± 90.8 | |||
Fy = 1500 ± 87.9 | ||||||
Fz = 1500 ± 93.2 | ||||||
Nonlinear model | C100 | C010 | C200 | C020 | g _{ 1 } +g _{ 2 } | Force |
Assumed | 263 | 263 | 491 | 491 | 0.815 | Fx = -300 |
Fy = 300 | ||||||
Fz = 300 | ||||||
Estimated | 263 ±4.0401 | 263 ±7.8404 | 491 ±14.4139 | 491 ±13.4578 | 0.815 ±0.0083 | Fx = -300 ± 9.1726 |
Fy = 300 ± 10.0278 | ||||||
Fz = 300 ± 8.7998 |
Real Data
To evaluate the methods on the real data, we have used six image sets each containing 90 slices with 2.5 mm thickness and 286x286 pixels with 0.86 mm pixel size. 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.
Maximum and mean errors for the linear mechanical and nonlinear models.
Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | |
---|---|---|---|---|---|---|
Max error of the mechanical model | $\begin{array}{l}\Delta x=3.0mm\\ \Delta y=3.3mm\\ \Delta z=\text{1}\text{.1}mm\end{array}$ | $\begin{array}{l}\Delta x=3.3mm\\ \Delta y=3.2mm\\ \Delta z=1.0mm\end{array}$ | $\begin{array}{l}\Delta x=3.4mm\\ \Delta y=4.4mm\\ \Delta z=\text{0}\text{.3}mm\end{array}$ | $\begin{array}{l}\Delta x=2.9mm\\ \Delta y=3.1mm\\ \Delta z=\text{0}\text{.8}mm\end{array}$ | $\begin{array}{l}\Delta x=2.7mm\\ \Delta y=2.8mm\\ \Delta z=\text{0}\text{.4}mm\end{array}$ | $\begin{array}{l}\Delta x=4.0mm\\ \Delta y=3.4mm\\ \Delta z=1.7mm\end{array}$ |
Mean error of the mechanical model | $\begin{array}{l}\Delta x=1.2mm\\ \Delta y=1.2mm\\ \Delta z=\text{0}\text{.4}mm\end{array}$ | $\begin{array}{l}\Delta x=1.3mm\\ \Delta y=1.0mm\\ \Delta z=\text{0}\text{.3}mm\end{array}$ | $\begin{array}{l}\Delta x=1.3mm\\ \Delta y=1.7mm\\ \Delta z=\text{0}\text{.1}mm\end{array}$ | $\begin{array}{l}\Delta x=0.9mm\\ \Delta y=1.\text{1}mm\\ \Delta z=\text{0}\text{.6}mm\end{array}$ | $\begin{array}{l}\Delta x=0.9mm\\ \Delta y=\text{0}\text{.8}mm\\ \Delta z=\text{0}\text{.1}mm\end{array}$ | $\begin{array}{l}\Delta x=1.0mm\\ \Delta y=0.7mm\\ \Delta z=\text{0}\text{.1}mm\end{array}$ |
Max error of the nonlinear model | $\begin{array}{l}\Delta x=2.5mm\\ \Delta y=3.1mm\\ \Delta z=\text{0}\text{.3}mm\end{array}$ | $\begin{array}{l}\Delta x=3.0mm\\ \Delta y=3.1mm\\ \Delta z=\text{0}\text{.9}mm\end{array}$ | $\begin{array}{l}\Delta x=3.2mm\\ \Delta y=4.0mm\\ \Delta z=\text{0}\text{.3}mm\end{array}$ | $\begin{array}{l}\Delta x=2.9mm\\ \Delta y=2.8mm\\ \Delta z=\text{0}\text{.7}mm\end{array}$ | $\begin{array}{l}\Delta x=2.6mm\\ \Delta y=2.4mm\\ \Delta z=\text{0}\text{.4}mm\end{array}$ | $\begin{array}{l}\Delta x=3.0mm\\ \Delta y=2.9mm\\ \Delta z=\text{1}\text{.1}mm\end{array}$ |
Mean error of the nonlinear model | $\begin{array}{l}\Delta x=0.8mm\\ \Delta y=1.1mm\\ \Delta z=\text{0}\text{.1}mm\end{array}$ | $\begin{array}{l}\Delta x=1.2mm\\ \Delta y=0.8mm\\ \Delta z=\text{0}\text{.2}mm\end{array}$ | $\begin{array}{l}\Delta x=1.1mm\\ \Delta y=1.3mm\\ \Delta z=\text{0}\text{.1}mm\end{array}$ | $\begin{array}{l}\Delta x=0.8mm\\ \Delta y=0.9mm\\ \Delta z=\text{0}\text{.4}mm\end{array}$ | $\begin{array}{l}\Delta x=0.7mm\\ \Delta y=0.7mm\\ \Delta z=\text{0}\text{.1}mm\end{array}$ | $\begin{array}{l}\Delta x=0.8mm\\ \Delta y=0.5mm\\ \Delta z=\text{0}\text{.9}mm\end{array}$ |
Variations of the estimated parameters of the linear mechanical and nonlinear models.
Optimized in six cases | Model | |
---|---|---|
E | 3000 ± 452 | Mechanical |
ν | 0.45 ± 0.03 | |
Resultant Force | 91 -- 710 | |
C _{ 01 } | 253 ± 45 | Nonlinear |
C _{ 10 } | 253 ± 52 | |
C _{ 20 } | 491 ± 73 | |
C _{ 02 } | 491 ± 73 | |
g _{ 1 } +g _{ 2 } | 0.82 ± 0.03 | |
Resultant Force | 121 -- 852 |
The execution time of the nonlinear model is approximately six times of the linear mechanical model using a personal computer with a 1.86 GHz dual-core CPU and 4 GB RAM. 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.
Conclusion
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.
From the computation point of view, the linear mechanical model is about 66% slower than the elastic model and six times faster than the nonlinear model. 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.
Declarations
Acknowledgements
This work was supported in part by a grant from the University of Tehran, Tehran, Iran. Patient-specific geometric data of the brain used in this work were obtained from the pre- and intra-operative MRI studies of the patients who underwent brain tumor surgery at the Department of Neurosurgery (Harvard Medical School, Boston, Massachusetts, USA) and made available to the authors by the Surgical Planning Laboratory, Brigham and Women's Hospital (Harvard Medical School, Boston, Massachusetts, USA). The authors gratefully acknowledge and thank Dr. Ron Kikinis and Dr. Tina Kapur for providing this crucial data.
Authors’ Affiliations
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