1). Mr Image Acquisition

In this study, five patients were processed through the preoperative and intraoperative pipelines. All patients provided written consent prior to imaging for this Vanderbilt Institutional Review Board approved study. For each patient, two sets of T1-weighted MR image volumes, one gadolinium-enhanced and the other non-enhanced, were acquired from a conventional clinical MR scanner (Table 1).

2). Segmentation

To streamline the preoperative pipeline and model generation, a semi-automatic segmentation of the brain was implemented [26]. Briefly described, for each patient, a rigid alignment is performed between patient’s enhanced and non-enhanced image volumes. Once complete, the patient’s non-enhanced image volume is registered to an expertly segmented non-enhanced brain image volume (i.e., atlas volume) first using a mutual information rigid registration followed by a custom-built adaptive bases nonrigid registration algorithm [42]. Once complete, the atlas mask can be transformed such that the contrast-enhanced patient’s can be automatically segmented. We should note that the falx and tentorium have been expertly segmented in the atlas which serves as an automatic approach to deploying the dural septa within our model [27]. The atlas also provides an excellent reference surface set for finite element mesh generation once registered to the patient. We should note that a visual confirmation is performed when complete, and some manual editing of the automatic segmentation is sometimes performed using the open source software ITK-Snap (www.itksnap.org) to correct small discrepancies. At this time, ITK-Snap is also used to manually segment the contrast-enhancing tumor region. We should note that manual methods of tumor segmentation are the standard in commercial guidance systems.

Finally, we should note that in [28], a sensitivity study was performed which compared our brain shift correction results based on models built from our semi-automatic segmentation approach versus those coming an expert manual segmentation approach and no statistical difference between results was found.

3). Surgical Planner

The direction and degree of brain shift are in part dependent on how the head is oriented with respect to gravity, as well as location and the size of the craniotomy. A priori knowledge of these 3 variables helps to limit the size of the atlas of possible deformation solutions and can be provided by the neurosurgeon during preoperative planning. A user-interactive GUI was developed to assist the neurosurgeon in quantifying those variables. Brain and tumor surface meshes were converted from segmented brain and tumor images respectively by using marching cubes and smoothing algorithms [43]. Both surface meshes were rendered in the GUI and the neurosurgeon rotated the brain into the planned position and recorded the transformation. The center of the craniotomy was selected by picking a point on the brain surface and the craniotomy size was determined using the slider tool to adjust the sphere (Figure 2). These 3 variables were used later in defining the boundary conditions of the computational model.

4). Continuum Model

Based on the observation that the brain acts similar to fluid saturated poroelastic medium, Biot’s theory of biphasic consolidation was used to represent the deformation behavior of brain tissue [6], [29]. According to Biot’s theory, the mechanical behavior of a poroelastic medium such as the brain can be described using equations of linear elasticity for solid porous matrix and Darcy’s law for describing the flow of fluid through the porous matrix. Equation (1) represents equilibrium whereby the gradient in interstitial fluid pressure can cause shape change to the solid matrix. In addition, changes in the buoyancy of the surrounding fluid can generate gravity-induced deformations. In equation (2), we see a conservation of fluid mass relationship whereby changes in hydration can effect the the time rate of change of the volumetric strain of the solid matrix. In addition, we also allow for dilatation effects as exchange with capillary beds can occur in response to drugs like mannitol. The model can be described as,

where is the shear modulus defined by with as Young’s modulus and is the Poisson’s ratio. is the displacement vector, is the ratio of fluid volume extracted to volume change of the tissue under compression, is the interstitial pressure, is the tissue density, is the surrounding fluid density, is the gravitational unit vector, is the capillary permeability, is the intracapillary pressure and is the hydraulic conductivity. This constitutive model is a common model and has been used successfully to describe brain shift [17], [18], [27].

5). Computational Biomechanical Model

For each patient, a patient-specific finite element volumetric mesh was generated from the MR images. Briefly, once the patient’s images have been segmented, a marching cubes algorithms can be used to generate bounding surface. A custom-built mesh generator is then used to generate a volumetric tetrahedral mesh [44] with parenchyma, and tumor designated. Parenchyma can be discretized further into white and gray matter elements using an image-to-grid methodology whereby the average image-intensity of voxels within a tetrahedral element can be determined and then used to threshold tissue type [52]. Typically, a brain mesh consists of approximately 100,000 tetrahedral elements (Figure 3).

The boundary conditions applied were determined according to observed conditions commonly associated with brain shift from previous studies [17], [18], [27]. The boundary conditions associated with displacement that were found to produce good estimates of brain shift were as follows: (1) the brainstem area was typically found to be very stable and as a result represent a fixed, i.e., no displacement condition – Figure 4, left, red region, (2) in the region of the craniotomy and surrounding area where the brain can often sag away or shift laterally, the surface was designated as stress free allowing for the brain to fall away from cranial wall – Figure 4, left, green region, (3) the remaining brain surface is bound by the skull such that movement is limited to tangent-to-the-skull motion along the cranial wall only, i.e., a freedom to slip boundary condition – Figure 4, left, black region, (4) slip boundary conditions were also designated for the internal rigid dural septa structures – Figure 4, left, magenta region. As equations (1,2) state, gradients in interstitial pressure can induce deformations and do embody the transient effects of the model. Boundary conditions were either designated at an atmospheric reference pressure in elevations above cerebrospinal fluid (CSF) drainage levels – Figure 4, right, blue region, or non-draining surfaces (i.e. no flux) below drainage levels – Figure 4, right, red region.

6). Atlas Creation

While the above provides a good reference for a single boundary condition set, the surgical environment is quite dynamic. As a result, our strategy is to generate a distribution of possible boundary conditions based on reasonable surgical presentation, a so-called ‘atlas of deformations’. As a result, the boundary conditions in the previous section have been parameterized such that based on minimal preoperative planning a complete deformation atlas can be constructed. The distribution of boundary conditions is based on three mechanisms of brain shift that we have observed to be important : gravity-induced brain shift, brain volume reduction due to administration of hyperosmotic drugs like mannitol, and brain swelling due to edema around the tumor [17], [18]. For gravity-induced deformation, we have varied the atlas to express three different levels of CSF drainage which influences the deployment of pressure-related boundary conditiozns (Figure 4, right). We should also note that with each drainage level, we also account for a distribution of possible head orientations around the estimate from the preoperative plan. While this accounts for inaccuracies in the preoperative plan, it also helps to account for surgical table adjustments during surgery (typically involves a distribution of +/−20 degrees from preoperative estimate, leads to approximately 60 different head orientations). We should note that with each orientation, the boundary condition distribution in Figure 4, left change, i.e., our displacement boundary conditions are parameterized as a function of head orientation. With respect to the influence of hyperosmotic drugs, we have chosen to exploit the second term in equation (2). Our atlas allows for 3 different capillary permeability, i.e., varying , with a fixed intracapillary pressure. Similarly, swelling variations were simulated with 3 different capillary permeability values, and positive intracapillary pressures, however, we did allow for 3 different craniotomy sizes (75%, 100% and 125% of planned size) to account for any deviations from the plan. We should note that boundary swelling, the boundary conditions associated with Figure 4, left, green region are modified to slip-based boundary conditions with stress free in the craniotomy region. For the work here, there were 729 total brain shift solutions contained within our deformation atlas. The different material properties and their varying levels are tabulated in Table 2 and are based on sensitivity studies performed within in vivo porcine brain experiments which we have found to be quite satisfactory for predicting bulk brain shift [45]–[48].

The 729 finite element models were solved by spreading the computation across an 8-node computing cluster to ensure the atlas of solutions was built in time for the day of surgery. The biphasic brain model was solved for displacement and pressure using an open source Portable Extensible Toolkit for Scientific Computation (PETSc) package. A highly automated process of computational model generation, boundary conditions creation and solving for the atlas were developed to streamline and ensure minimal user error. Once complete, the deformation atlas is transferred to the intraoperative data collection and processing computer used in the OR on the day of surgery.

1). Physical to Image Space Registration

The Registration GUI developed is a registration and visualization utility that registers the patient’s physical space to image space using an LRS scan of the face and the corresponding surface from the MR image volume. The face LRS scan was acquired by positioning the LRS directly over face of the patient, making sure to include all if possible, the nose, eye and ear as these structures serve as good landmarks for use in registration. The manual segmentation tool in the LRS acquisition software (Pathfinder Therapeutics, Inc., Nashville, TN) was used to remove extraneous points in the face scan, such as hair, intubation tubes and drapes, that will unnecessarily slow down the registration computation (Figure 5). Once the segmented face LRS scan data is complete, a smoothing process using a commercially available radial basis fitting (RBF) procedure is performed (FarField Technology, Ltd., Christchurch New Zealand). To initialize, three surface fiducials are selected on the LRS data of the patient’s face and the corresponding points are designated on the MR surface counterpart and a rigid registration using Horn’s method [49] is executed. Once complete, the registration is refined using an iterative closest point surface registration [50]. Verification of the registration is confirmed by visual inspection of the overlay of both 3D objects (Figure 6). If the alignment of the two objects is not satisfactory, the user may select new points and execute another registration.

2). Pre-Resection LRS Scan

After the craniotomy and dura is opened, the LRS was moved into place to acquire the exposed cortical brain surface. Care was taken to ensure a direct line of sight between the brain surface and the laser. Once acquired, a simple manual segmentation tool is used to remove extraneous points, isolating just the brain surface (Figure 7). Similar to the face scan, an RBF surface is fit and rigid transformations can be applied to transform the surface to image space. To confirm the positioning accuracy of the optically-tracked LRS, the transformed pre-resection RBF scan was automatically overlaid onto the head surface mesh for visual inspection (Figure 8).

3). Post-Resection LRS Scan

Multiple LRS scans may be taken during the course of the surgery depending on neurosurgeon’s request to track the updated position of the tumor since the amount of brain shift is a function of time. The procedure for all sequential scans is the same. In this study however, only one final cortical brain surface was acquired after tumor resection was thought to be complete and an image update would be useful to confirm complete removal of the tumor in the presence of brain shift. Since time is critical at this juncture as the entire surgical team is waiting for updated images, processing steps were specifically developed to minimize user intervention and computation time. Instead of a full manual segmentation as done previously with the pre-resection LRS scan, a mask of the pre-resection LRS was applied to the post-resection LRS to remove points outside of the craniotomy region. The fully segmented post-resection scan was also automatically fitted with an RBF surface, transformed to image space, and displayed along with the pre-resection RBF for visualization (Figure 9).

4). Homologous Point Pick

Once the pre- and post-resection LRS cortical surfaces were spatially transformed to image space, the Correction GUI is used to determine the driving shift measurements for correcting the image volume for brain shift. To accomplish his, the 2D pre-resection and post-resection bitmaps, i.e., texture information acquired by the LRS unit, were visualized side-by-side. Homologous points were then selected using blood vessel bifurcations as landmarks (Figure 10). These points will produce shift measurements to drive our compensation system.

5). 2D to 3D Correspondence

Once homologous points are selected from the texture information provided by the LRS, they can be related directly to their 3D coordinate positions. We should note that brain shift is possible from the very instant the dura is opened. To accommodate for this initial shift, a correspondence between brain mesh and intraoperative pre-resection LRS-defined features is determined using a closest point operator. Once determined, the shift from pre-resection to post-resection LRS is appended and an entire deformation is complete. In the event that a very limited number of homologous points can be determined, the platform does allow for the calculation to be driven by closest point operators solely with the possibility of weighting from any homologous points that can be determined. For one patient in this study, this feature was used due to a lack of homologous points.

6). Inverse Modeling

With a field of displacements describing cortical surface deformation defined, the correction algorithm is employed that uses a constrained least squares inverse modeling approach based on the atlas and constrained by the measured displacement shift vectors of the cortical surface as well as added constraints to the coefficients of reconstruction. Details of the inverse modeling can be found in previous studies [17], [18], [27]. Briefly, the least-squared errors between the measured shift vectors and predictions from the deformation atlas were minimized by solving the following equations for the weighting coefficients, .

where are the measured shift vectors on the brains surface as determined by the above methods, and is the atlas matrix containing the pre-computed deformation solutions at the selected measurement points on the computer model boundary mesh. The first constraint ensures only positive regression coefficients and the second constraint prevents extrapolations of the solution. The constraints imposed have been shown to successfully predict brain shift [27], [28]. The implementation of the method of Lagrange multiplier in the Optimization Toolbox in MATLAB was used to solve the linear optimization problem, along with the use of the Parallelization Toolbox to improve input/output function speeds. We should note that while other optimization approaches with less constraint can lead to better objective function results, we have found that constraints such as the above are necessary to maintain physically realistic deformations, i.e., a real safety constraint consider the dramatically underdetermined nature of this problem.

7). Deformed Image Update

Once the inverse solution is achieved, a quantitative report is automatically generated based on the optimum solution for assessment, specifically the amounts of shift based on measurements, and remaining error after correction. As equation (3) is solved within the context of matching the sparse measurements at the surface, the coefficients determined are then used to construct a full volumetric deformation field which is subsequently used to deform the preoperative MR images, thus providing an updated image of the deformed brain for use within the neuronavigation system. With respect to image deformation, nodal displacements from the undeformed finite element mesh were trilinearly interpolated onto a regular grid at the same resolution as the preoperative MR images. To ensure there was no extrapolation of displacements outside the brain, the grid of interpolated displacements were multiplied with the binary brain mask. Each undeformed pixel was then transformed to their respective deformed pixel and filled with a trilinearly interpolated pixel intensity value from the undeformed MR images. Since not every deformed brain pixel will be filled, the missing pixels intensity values were interpolated from its surrounding neighbors. The Parallelization Toolbox in MATLAB was used to parallelize and speed up the interpolations of 3 Cartesian components of displacements.

A computer with Intel Quad Core i5 with 16 GB of ram running Windows 7 64bit was used to compute the intraoperative steps. The same computer was also used to acquire the LRS scans.