Stereotactic Localization: From Single-Slice to Multi-Slice Registration Including a Novel Solution for Parallel Bipanels

Abstract

Frame-based stereotactic localization generally assumes that all required fiducials are present in a single-slice image which can then be used to form targeting coordinates. Previously, we have published the use of novel localizers and mathematics that can improve stereotactic localization. As stereotactic procedures include numerous imaging slices, we sought to investigate, develop, and test techniques that utilize multiple slices for stereotactic localization and provide a solution for a parallel bipanel N-localizer. Several multi-slice equations were tested. Specifically, multi-slice stereotactic matrices (ms-SM) and multi-slice normal to parallel planes (ms-nPP) were of particular interest. Bipanel (2N) and tripanel (3N) localizer images were explored to test approaches for stereotactic localization. In addition, combination approaches using single-slice stereotactic matrices (ss-SM) and multi-slice methods were tested. Modification of ss-SM to form ms-SM was feasible. Likewise, a method to determine ms-nPP was developed. For the special case of the parallel bipanel N-localizer, single-slice and multi-slice methods fail, but a novel non-linear solution is a robust solution for ms-nPP. Several methods for single-slice and multi-slice stereotactic localization are described and can be adapted for nearly any stereotactic system. It is feasible to determine ms-SM and ms-nPP. In particular, these methods provide an overdetermined means to calculate the vertical z, which is determined for a tripanel system using single-slice methods. In addition, the multi-slice methods can be used for extrapolation outside of the localizer space. Importantly, a novel non-linear solution can be used for parallel bipanel N-localizer systems, where other methods fail. Finally, multi-slice stereotactic localization assumes strict patient and imaging system stability, which should be carefully assessed for each case.

Keywords: frame-based stereotactic surgery; functional neurosurgery; localization; stereotactic and functional neurosurgery; stereotactic frame.

Figure 1. A stereotactic volume of image slices

Stereotactic imaging includes the use of computed tomography (CT) and magnetic resonance imaging (MRI) for image acquisition. Each one of these images can be considered a mathematical plane (black). These images, in frame-based stereotaxis, include fiducials , where the vertical rods are represented (blue). Using these fiducials, such as for N-localizer or Sturm-Pastyr localizers, the normal vector to the plane can be computed (green). Each image in the stack is separated by a distance from plane-to-plane (yellow), which may be assumed to be a constant. Finally, errors (red) can be seen when these assumptions are not correct, causing displacement of , and/or . This illustrative image exaggerates and may not fully represent potential problems with image slices.

Figure 2. Graphical interpretation of and for multi-slice normal to parallel planes computation relative to x-, y-, and z-axes

The value of are values along each image slice (black vertical lines), whereas are values along each image slice. When there is a rotation angle such as frame tilt , then varies differently than . When there is no rotation, then is equal to .

Figure 3. Multi-slice normal to parallel planes (ms-nPP) solution for a parallel N-localizer

Here we analyzed the non-linear solution using Monte Carlo simulations and computed Root Mean Square errors (RMSe). We find that the error rates along the z-axis were low throughout the volume. Tested locations include (x,y) positions that are in the center (100,100) (top left), right (50, 100) (top middle), left (150,100) (top right), anterior (100,150) (bottom left), posterior (100,50) (bottom middle), anterior-right (40, 215) (bottom right).

Figure 4. Utilization of the multi-slice stereotactic matrices (ms-SM) for three adjacent slices in a 3N localizer system

Root Mean Square error (RMSe) is computed for all points in the middle slices by using the single-slice stereotactic matrix (ss-SM) and then comparing those results to the ms-SM. The results demonstrate a consistently low error (<0.4mm). Therefore, multi-slice computations can be adopted to subsets of slices, adjacent slices, or for the whole volume of slices.

Figure 5. Example of single case studies using single-slice, multi-slice, or combination solutions

The results of the tests were compared to ss-SM and Root Mean Square error (RMSe) was computed for each slice in the volume. For 2N systems, six points per slice were used to compute RMSe. For 3N systems, nine points per slice were used to compute RMSe. Multi-slice stereotactic matrix (ms-SM) can be used to compute XYZ data throughout the volume, here demonstrating an oscillation (A). Also, ms-SM can be used to compute XY and this can be combined with multi-slice normal to parallel planes (ms-nPP) for Z (B), which has slightly less error than A. Alternatively, single-slice stereotactic matrices (ss-SM) can be used on each slice for XY and combined with ms-SM (C) or ms-nPP (D) for Z. Also, ss-SM or ms-SM for XY can be used with non-linear ms-nPP for Z on parallel bipanel 2N systems (E-F).

Figure 6. Representation of parallel (left) versus anti-parallel (right) bipanel N-localizer

Points (red dots) along the diagonal bars can be followed in the vertical direction. The parallel bipanel contains diagonal bars that are parallel, and the points form lines that are also parallel and all coplanar. The anti-parallel bipanel contains points that form bars and lines that are not parallel and non-coplanar. Because of this relationship, the parallel bipanel requires a non-linear solution for multi-slice normal to parallel planes (ms-nPP), whereas the anti-parallel bipanel can be solved via linear solutions for ms-SM and ms-nPP.

Figure 7. Anti-parallel bipanel phantom testing using multi-slice linear techniques

Here, we utilized both multi-slice stereotactic matrix (ms-SM) and multi-slice normal to parallel planes (ms-nPP) and computed Root Mean Square error (RMSe) compared to single-slice stereotactic matrices (ss-SM) results.  We can observe a relatively low RMSe (<0.15mm) demonstrating good functionality of these techniques in the setting of the anti-parallel bipanel.