The Mathematics of Three N-Localizers Used Together for Stereotactic Neurosurgery

The N-localizer enjoys widespread use in image-guided stereotactic neurosurgery and radiosurgery. This article derives the mathematical equations that are used with three N-localizers and provides analogies, explanations, and appendices in order to promote a deeper understanding of the mathematical principles that govern the N-localizer.


Introduction
The N-localizer is a device that may be attached to a stereotactic frame ( Figure 1) in order to facilitate image-guided neurosurgery and radiosurgery using tomographic images that are obtained via computed tomography (CT), magnetic resonance (MR), or positron-emission tomography (PET) [1]. The mathematics of the N-localizer have been discussed previously [2].

FIGURE 1: Three N-Localizers Attached to a Stereotactic Frame
Three N-localizers are attached to this stereotactic frame and are merged end-to-end such that only seven rods are present. The vertical rod at the right rear of the frame is larger in diameter than the other rods. This large rod facilitates unambiguous interpretation of the fiducial circles and ellipses that the seven rods create in a tomographic image, as explained in the legend to Figure 5.

Technical Report
The N-localizer comprises a diagonal rod that extends from the top of one vertical rod to the bottom of another vertical rod ( Figure 2). Assuming for the sake of simplicity that the two vertical rods are perpendicular to the tomographic section, the cross section of each vertical rod creates a fiducial circle and the cross section of the diagonal rod creates a fiducial ellipse in the tomographic image, as shown in Figure 2b. As the tomographic section moves from the top of the N-localizer towards the bottom of the N-localizer, i.e. towards its point of attachment to the stereotactic frame (Figure 1), the ellipse will move away from circle and toward circle . The relative spacing between these three fiducials permits precise localization of the tomographic section with respect to the N-localizer. The distance between the centers of circle and ellipse , and the distance between the centers of circles and are used to calculate the ratio . This ratio represents the fraction of diagonal rod that extends from the top of vertical rod to the point of intersection of rod with the tomographic section. These linear geometric relationships exist due to the properties of similar triangles and are valid even if the vertical rods are not perpendicular to the tomographic section [3]. Tomographic image. The intersection of the tomographic section with the rods , , and creates fiducial circles and and fiducial ellipse in the tomographic image. The distance between the centers of circle and ellipse and the distance between the centers of circles and are used to calculate the ratio . This ratio represents the fraction of diagonal rod that extends from the top of rod to the point of intersection of rod with the tomographic section.
It is convenient to ignore the thickness of the tomographic section and to approximate the tomographic section as an infinitely thin plane. This "central" plane lies midway between the top and bottom halves of the tomographic section, analogous to the way that a slice of cheese is sandwiched between two slices of bread. The central plane approximation is susceptible to error because of the partial volume effect that derives from the several-millimeter thickness of the tomographic section [4][5]. The partial volume effect prevails because any structure that passes partially into the tomographic section, but does not span the full thickness of that section, may be visible in the tomographic image. Hence, the position of that structure is determined to only a several-millimeter error that is a well-known limitation of tomographic imaging. In the following discussion, the term "tomographic section" will be used as an abbreviation for the term "central plane of the tomographic section." The fraction is used to calculate the coordinates of the point of intersection between the long axis of rod and the tomographic section ( Figure 3).  which lies on the long axis of rod in the three-dimensional coordinate system of the N-localizer, corresponds to the analogous point , which lies at the center of ellipse in the twodimensional coordinate system of the tomographic image ( Figure 2b). Hence, there is a one-to-one linear mapping between a point from the N-localizer and a point from the tomographic image.
The attachment of three N-localizers to a stereotactic frame permits calculation of the , , and coordinates for the three respective points, , , and , in the three-dimensional coordinate system of the stereotactic frame ( Figure 4). These three points correspond respectively to the three analogous points, , , and , in the two-dimensional coordinate system of the tomographic image. In the following discussion, the symbols, , , and , will be used as a shorthand notation for , , and . The symbols, , , and , will be used as a shorthand notation for , , and .

Three-Dimensional Coordinate System of the Stereotactic Frame
The quadrilateral represents the tomographic section. The large oval depicts the circular base of the stereotactic frame (in perspective). The vertical and diagonal lines that are attached to the large oval represent the nine rods. The centers of the six fiducial circles and the three fiducial ellipses that are created in the tomographic image by these nine rods are shown as points that lie in the tomographic section. The tomographic section intersects the long axes of the three diagonal rods at the points , , and that coincide with the respective centers , , and of the three ellipses ( Figure 6). The , , and coordinates of the respective points of intersection , , and are calculated in the three-dimensional coordinate system of the stereotactic frame using Equations 1 and 2. Because these three points determine the spatial orientation of a plane in three-dimensional space, the spatial orientation of the tomographic section is determined with respect to the stereotactic frame. The target point lies in the tomographic section. The coordinates of this target point are calculated in the three-dimensional coordinate system of the stereotactic frame using Equation 6.  The three points,   ,  , and  , lie on the three respective diagonal rods,  ,  , and  , and   have respective  coordinates,  , , and , in the three-dimensional coordinate system of the stereotactic frame ( Figure 4). The analogous  three points,  ,  , and  , lie at the centers of the three respective ellipses,  ,  , and  ,  and have coordinates, , , and , in the two-dimensional coordinate system of the tomographic image ( Figures 5-6).  -coordinate may be set arbitrarily to any non-zero value, e.g., 1, so long as same value of is used for each of the three -coordinates. The equations that are presented in the remainder of this article assume that a value of has been used to project the , , and coordinates. If a value of were used instead of to project these coordinates, the equations that are presented in the remainder of this article would no longer apply and would require revision so that the calculations that these equations describe may produce correct results.
Because three points determine the orientation of a plane in three-dimensional space, the three coordinates, , , and , together with the three coordinates, , , and , determine the spatial orientation of the tomographic section with respect to the stereotactic frame. This spatial orientation or linear mapping is specified by the matrix elements through in the matrix equation . For this reason, the Brown-Roberts-Wells (BRW) stereotactic frame [7] that was used initially with computed tomography (CT) required modification to eliminate nonlinear distortion of MR images. The CT-compatible BRW frame comprised an aluminum ring in which the magnetic field that the MR scanner generated to acquire MR images induced eddy currents. Those eddy currents distorted the MR images. Replacing one section of the aluminum ring with a nonmetallic insert prevented magnetically induced circumferential eddy currents and eliminated nonlinear distortion of the MR images [8].
An analogy provides insight into how the transformation of Equation 3 operates. Consider the tomographic image to be an elastic membrane. The transformation describes the process of stretching the membrane in the plane of the tomographic image, rotating the membrane about an axis that is normal to the plane of the tomographic image, tilting the membrane, if necessary, so that it is not parallel to the base of the stereotactic frame, and lifting the membrane into place upon the scaffold of the three N-localizers, such that the three points, , , and , from the tomographic image precisely coincide with the respective three points, , , and , from the stereotactic frame. Then, any other point that lies on the membrane, e.g., the target point , is transformed by the same stretching, rotating, tilting, and lifting processes that transformed the three points, , , and . In this manner, the coordinates of the target point may be transformed from the two-dimensional coordinate system of the tomographic image into the three-dimensional coordinate system of the stereotactic frame to produce the coordinates of the analogous target point .
In Equation 4, represents the matrix of , , and coordinates in the coordinate system of the stereotactic frame. represents the matrix of , , and coordinates in the coordinate system of the tomographic image. represents the matrix of elements, through , that defines the transformation from the two-dimensional coordinate system of the tomographic image to the three-dimensional coordinate system of the stereotactic frame.  (Figure 7). In this case, and , so neither nor appears in the intermediate tomographic image.
However, in this case, the neurosurgeon may wish to know where the intended probe trajectory would intersect the intermediate tomographic section. In order to provide this information, the points and are used to define the vector from to . This vector is then used to calculate the coordinates of a third point for which (Figure 7). Because to in Order to Obtain the Point that Appears in the Tomographic Image"> to in Order to Obtain the Point that Appears in the Tomographic Image" itemprop="image" that Appears in the Tomographic Image"> to in Order to Obtain the Point that Appears in the Tomographic Image"> to in Order to Obtain the Point that Appears in the Tomographic Image" itemprop="image" that Appears in the Tomographic Image"> In Equation 13, the third row of the transformation matrix includes elements , , and and the fourth row includes elements , , and . This non-standard numbering convention for these matrix elements is convenient to the remainder of this derivation of Equation 3. Also, the matrix elements in the fourth column of this transformation matrix have the values of 0, 0, 0 and 1 because Equation 13 expresses an affine transformation that comprises only scale, rotate and translate operations [9]. These operations accomplish the stretching, rotating, tilting and lifting processes that were described for the membrane analogy in association with Equation 3. There is a significant difference between Equations 3 and 13. None of the matrices in Equation 13 have an inverse because neither nor is a square matrix. In contrast, the matrices , , and in Equation 3 potentially have inverses because these matrices are square matrices. Equations 5, 7, and 8 require that these matrices have inverses. Hence, in order to express the transformation from the two-dimensional coordinate system of the tomographic image to the three-dimensional coordinate system of the stereotactic frame and vice versa, Equation 3 must be used instead of Equation 13.

Three-Dimensional Coordinate System of the Stereotactic Frame
The quadrilateral represents the tomographic section. The large oval depicts the circular base of the stereotactic frame (in perspective). The vertical and diagonal lines that are attached to the large oval represent the nine rods. The centers of the six fiducial circles and the three fiducial ellipses that are created in the tomographic image by these nine rods are shown as points that lie in the tomographic section. The tomographic section intersects the long axes of the three diagonal rods at points , , and that coincide with the respective centers , , and of the three ellipses ( Figure 10). The target point lies in the plane of the triangle . Hence, its coordinates may be expressed as a linear combination of the , , and coordinates of the points , , and using barycentric coordinates as indicated by Equation 27.
Similarly, with reference to Figure 10, the analogous target point may be represented as a linear combination of the three points , , and .