### Abstract

This paper is a supplement to "The Origin of the N-Localizer for Stereotactic Neurosurgery" [1] and "The Mathematics of the N-Localizer for Stereotactic Neurosurgery" [2]. It clarifies the early history of the N-localizer and presents further details of the mathematics of the N-localizer.

### Introduction

The N-localizer has become an important neurosurgical tool that has achieved widespread use in modern stereotactic neurosurgery and radiosurgery. The N-localizer produces two circles and one ellipse in tomographic, sectional images that are obtained via computed tomography or magnetic resonance imaging (Figure *1*). The relative spacing between the ellipse and the two circles precisely determines the location of the tomographic section relative to the N-localizer [3-4].

Russell A. Brown invented the N-localizer in May 1978 when he was a third-year medical student at the University of Utah and during a research elective in the laboratory of James A. Nelson. In August 1978, Brown built the first CT-compatible stereotactic frame in order to test the concept of the N-localizer (Figure *2*). In October 1978, he presented this frame at a joint meeting of the Western Neurological Society and the American Academy of Neurological Surgeons [5]. Following this presentation, additional presentations [6], and disclosures [7], the concept of the N-localizer spread rapidly and inspired its incorporation into three different CT-compatible stereotactic frames that were built in 1979 and reported in 1980: The Brown-Roberts-Wells frame [8], a Leksell frame that was modified to render it CT-compatible [9], and a frame that resulted from a collaboration between Pfizer Medical Systems and the University of Pittsburgh Medical Center (the “UPMC” frame) [10].

During the 35 years since the invention of the N-localizer, some misconceptions have arisen concerning its history in relation to subsequent developments in image-guided stereotactic surgery. Kondziolka and Lunsford of the University of Pittsburgh School of Medicine have claimed [11], "At our center, the first CT compatible stereotactic head frame, in collaboration with industry, was constructed in 1978 and utilized in 13 patients [10, 12]. [...] During this interval, the newly redesigned Leksell CT compatible stereotactic head frame [9] was used for dedicated brain biopsies under the direction of its inventor, Lars Leksell. Several groups were working on devices to allow accurate CT based stereotactic surgery [13]."

The above statement is not the only instance wherein Lunsford and Kondziolka have inaccurately interpreted the history of image-guided stereotactic surgery; we have discussed another instance previously [1]. But in particular, the above statement presents an erroneous chronology and disregards the fact that the adoption of the N-localizer for the Leksell and UPMC stereotactic frames was derivative. For both of these frames, the knowledge of the N-localizer originated from Brown's prior invention of the N-localizer [1, 9-10]. The UPMC frame was not the first CT-compatible stereotactic frame. And the UPMC frame was built in 1979, not in 1978. The documents that corroborate these facts have remained preserved in the archives of the U.S. Patent and Trademark office for nearly 30 years. The following technical report, which is based on these archives, recounts the development of image-guided stereotactic surgery at the University of Pittsburgh Medical Center and relates the events that led to the construction of the UPMC frame.

### Technical Report

Prior to the invention of the N-localizer, a method had been reported for estimating the location of a tomographic image relative to patient anatomy [14-15]. This method involved a plate into which were milled vertical slots whose tops lay along a diagonal line (Figure *3*). Because this slotted plate required counting numerous notches that were visible in the tomographic image, it was cumbersome to use and susceptible to human error [1, 16]. Despite these limitations, in the summer of 1978, Dade Lunsford modified a Leksell Stalix frame via the addition of two aluminum plates having milled vertical slots in the hope of creating a CT-compatible stereotactic frame [17].

In the autumn of 1978, Lunsford, et al. began to collaborate with John Perry of Pfizer Medical Systems [1, 17]. By mid-January of 1979, Perry, Lunsford, et al. had added a third aluminum plate to the Stalix frame and Perry had developed computer software that used information from the three slotted plates to transform coordinates from a CT scan image into the coordinate system of the Stalix frame [16, 18-19]. In a letter dated January 15, 1979 and addressed to Dade Lunsford, Arthur Rosenbaum and David Zorub of the University of Pittsburgh School of Medicine, Perry provided detailed instructions that explained the procedure required for obtaining CT scan images of the modified Stalix frame and for using computer software to transform coordinates from a CT scan image into the coordinate system of the Stalix frame [16].

Consequently, in early 1979, Perry, Lunsford, et al. appeared to be at the point of performing clinical cases with the modified Stalix frame. However, according to Lunsford, the modified Stalix frame was never used clinically because the aluminum composition of the frame and attached slotted plates created artifacts in the CT scan image [18]. Lunsford’s account is corroborated by Perry, et al. who reported that the UPMC “frame was made after attempts to modify the Leksell frame proved difficult” [10].

Lunsford has asserted that the Stalix frame was abandoned in the fall of 1978 and that thereafter the UPMC frame was constructed [18]. However, he appears to have forgotten the contents of the January 1979 letter from Perry to Lunsford, et al., a copy of which remains preserved in the archives of the U.S. patent office [16]. This letter refutes Lunsford's assertion and clearly demonstrates that in January 1979, Perry, Lunsford, et al. were preparing to use the modified Stalix frame for clinical cases but they had not yet done so. It was only after they had attempted to use the modified Stalix frame that they understood the severity of the artifacts created by the aluminum and subsequently abandoned the Stalix frame [10, 18]. And it was only after Brown had disclosed the N-localizer to Perry in early 1979 that Perry understood the concept of the N-localizer and subsequently adopted the N-localizer for the UPMC frame [1, 7, 10].

The second [4] of Brown's two publications that announced the N-localizer and reported evaluations of the first CT-compatible stereotactic frame [4-5] was submitted for publication in January 1979. The first [5] of these publications affirms that Brown presented this stereotactic frame at a joint meeting of the Western Neurological Society and the American Academy of Neurological Surgeons in October 1978. These and other publications [4-5, 10], together with historical documents [16, 19] from the archives of the U.S. patent office, confirm that the UPMC frame was not the first CT-compatible stereotactic frame and further confirm that the UPMC frame was constructed in 1979, not in 1978. The relevant historical documents [16, 19] are a matter of public record and may be obtained from the U.S. Patent and Trademark Office by requesting a copy of the folder for Interference No. 101267. In order to facilitate access to these documents, we have included copies as the appendices (labeled as "figures") to this paper.

### Discussion

The mathematics of the N-localizer have been discussed previously [4] and in considerable detail [2]. We review the mathematics below in preparation for presenting further details. The reader who is not interested in the details of the mathematics may choose to skip directly to the Conclusions.

#### Review of the mathematics of the N-Localizer

The N-localizer comprises a diagonal rod that extends from the top of one vertical rod to the bottom of another vertical rod (Figure *4*). In the tomographic image, each of the two vertical rods creates a fiducial circle and the diagonal rod creates a fiducial ellipse. The ellipse moves away from one circle and towards the other circle as the position of the tomographic section moves upward with respect to the N-localizer. The relative spacing between the three centroids permits precise localization of the tomographic section relative to the N-localizer. The distance \(d_{AB}\) between the centroids of circle A and ellipse B and the distance \(d_{AC}\) between the centroids of circles A and C are used to calculate the ratio \(f=d_{AB}/d_{AC}\). This ratio represents the fraction of the diagonal rod B that extends from the top of vertical rod A to the point of intersection of rod B with the tomographic section.

The fraction \(f\) is used to calculate the \(\left(x,y,z\right)\) coordinates of the point of intersection \({P}'_{B}\) between rod B and the tomographic section (Figure *5*). In this figure, points \({P}'_{A}\) and \({P}'_{C}\) represent the beginning and end, respectively, of the vector that extends from the top of rod A to the bottom of rod C. This vector coincides with the long axis of rod B. The \(\left(x_{A},y_{A},z_{A}\right)\) coordinates of the beginning point \({P}'_{A}\) and the \(\left(x_{C},y_{C},z_{C}\right)\) coordinates of the end point \({P}'_{C}\) are known from the geometry of the N-localizer. Hence, linear interpolation may be used to blend points \({P}'_{A}\) and \({P}'_{C}\) to obtain the \(\left(x_{B},y_{B},z_{B}\right)\) coordinates of the point of intersection \({P}'_{B}\) between the long axis of rod B and the tomographic section, as follows$${P}'_{B}={P}'_{A}+f\left({P}'_{C}-{P}'_{A}\right)=f{P}'_{C}+\left(1-f\right){P}'_{A}\;\;\;\;\;\;\;\;\;\;\left(1\right)$$The vector form of Equation 1 shows explicitly the \(\left(x,y,z\right)\) coordinates of points \({P}'_{A}\), \({P}'_{B}\) and \({P}'_{C}\)$$\begin{bmatrix}x_{B}\;y_{B}\;z_{B}\end{bmatrix} =f \begin{bmatrix}x_{C}\;y_{C}\;z_{C}\end{bmatrix} +\left(1-f \right) \begin{bmatrix}x_{A}\;y_{A}\;z_{A}\end{bmatrix}\;\;\;\;\;\;\;\;\;\;\left(2\right)$$Equation 1 or 2 may be used to calculate the \(\left(x_{B},y_{B},z_{B}\right)\) coordinates of the point of intersection \({P}'_{B}\) between rod B and the tomographic section. The point \({P}'_{B}\), which lies on rod B in the three-dimensional coordinate system of the N-localizer, corresponds to an analogous point \(P_{B}\), which lies at the centroid of ellipse B in the two-dimensional coordinate system of the tomographic image. Hence, there is a one-to-one correspondence between a point from the N-localizer and a point from the tomographic image.

The attachment of three N-localizers to the stereotactic frame permits calculation of the \(\left(x_{B_{1}},y_{B_{1}},z_{B_{1}}\right)\), \(\left(x_{B_{2}},y_{B_{2}},z_{B_{2}}\right)\) and \(\left(x_{B_{3}},y_{B_{3}},z_{B_{3}}\right)\) coordinates for the three respective points \({P}'_{B_{1}}\), \({P}'_{B_{2}}\) and \({P}'_{B_{3}}\) in the three-dimensional coordinate system of the stereotactic frame. These three points correspond respectively to the three points \(P_{B_{1}}\), \(P_{B_{2}}\) and \(P_{B_{3}}\) in the two-dimensional coordinate system of the tomographic image. In the following discussion, the symbols \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) will be used as a shorthand notation for \({P}'_{B_{1}}\), \({P}'_{B_{2}}\) and \({P}'_{B_{3}}\). The symbols \(P_{1}\), \(P_{2}\) and \(P_{3}\) will be used as a shorthand notation for \(P_{B_{1}}\), \(P_{B_{2}}\) and \(P_{B_{3}}\).

The three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) lie on the three respective diagonal rods \(B_1\), \(B_2\) and \(B_3\) and have respective \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates in the three-dimensional coordinate system of the stereotactic frame (Figure *6*). The analogous three points \(P_{1}\), \(P_{2}\) and \(P_{3}\) lie at the centroids of the three respective ellipses \(B_1\), \(B_2\) and \(B_3\) and have \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates in the two-dimensional coordinate system of the tomographic image (Figure *7*). Because three points determine a plane in three-dimensional space, the three coordinates \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) together with the analogous three coordinates \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) determine the spatial orientation of the tomographic section relative to the stereotactic frame. This spatial orientation permits calculation of the \(\left(x_{T},y_{T},z_{T}\right)\) coordinates of the target point \({P}'_{T}\) in the three-dimensional coordinate system of the stereotactic frame, given the \(\left(u_{T},v_{T}\right)\) coordinates of the analogous target point \(P_{T}\) in the two-dimensional coordinate system of the tomographic image.

In order to facilitate calculation of the \(\left(x_{T},y_{T},z_{T}\right)\) coordinates of the target point \({P}'_{T}\), it is convenient to project the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates of the three respective points \(P_{1}\), \(P_{2}\) and \(P_{3}\) onto the \(w = 1\) plane in three-dimensional space by appending a third coordinate \(w = 1\) to create \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates. The \(w\)-coordinate may be set arbitrarily to any non-zero value, e.g., one, so long as same value of \(w\) is used for each of the three \(w\)-coordinates. The equations that are presented in the remainder of this paper assume that a value of \(w = 1\) has been used to project the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates. If a value of \(w \neq 1\) were used instead of \(w = 1\) to project the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates, the equations that are presented in the remainder of this paper would no longer apply and would require revision so that the calculations that these equations describe may produce correct results.

The correspondence, or transformation, between the two-dimensional coordinate system of the tomographic image and the three-dimensional coordinate system of the stereotactic frame may be represented using the three \(\left(x,y,z\right)\) and \(\left(u,v,1 \right)\) coordinate pairs \(\left(x_{1},y_{1},z_{1}\right)\) and \(\left(u_{1},v_{1},1\right)\); \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(u_{2},v_{2},1\right)\); and \(\left(x_{3},y_{3},z_{3}\right)\) and \(\left(u_{3},v_{3},1\right)\) in the following matrix equation$$\begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{bmatrix} = \begin{bmatrix} u_1 & v_1 & 1 \\ u_2 & v_2 & 1 \\ u_3 & v_3 & 1 \end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\left(3\right)$$Equation 3 represents concisely a system of nine simultaneous linear equations that determine the spatial orientation of the tomographic section relative to the stereotactic frame. This equation transforms the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates from the two-dimensional coordinate system of the tomographic image to create \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates in the three-dimensional coordinate system of the stereotactic frame.

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 processes 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 lastly, lifting the membrane into place upon the scaffold of the three N-localizers such that the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\) from the tomographic image precisely coincide with the respective three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) from the stereotactic frame. Then *any* other point that lies on the membrane, for example, the target point \(P_{T}\), is transformed by the same stretching, rotating, tilting and lifting processes that transformed the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\). In this manner, the \(\left(u_{T},v_{T}\right)\) coordinates of the target point \(P_{T}\) 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 \(\left(x_{T},y_{T},z_{T}\right)\) coordinates of the analogous target point \({P}'_{T}\).

Equation 3 may be written in more compact form as$$\mathbf{F}=\mathbf{S}\mathbf{M} \;\;\;\;\;\;\;\;\;\;\left(4\right)$$In Equation 4, \(\mathbf{F}\) represents the matrix of \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates from the coordinate system of the stereotactic frame. \(\mathbf{S}\) represents the matrix of \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates from the coordinate system of the tomographic image. \(\mathbf{M}\) represents the matrix of elements \(m_{11}\) through \(m_{33}\) that defines the transformation from the two-dimensional coordinate system of the tomographic image into the three-dimensional coordinate system of the stereotactic frame.

Solving Equation 4 for \(\mathbf{S}\) yields$$\mathbf{S}=\mathbf{F}\mathbf{M}^{-1} \;\;\;\;\;\;\;\;\;\;\left(5\right)$$

Equation 5 transforms the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates from the three-dimensional coordinate system of the stereotactic frame to create \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates in the two-dimensional coordinate system of the tomographic image.

The above review of the mathematics of the N-localizer is sufficient to permit the presentation of further details below.

#### Requirements for the matrices \(\mathbf{F}\) and \(\mathbf{S}\)

It is possible to solve Equation 4 for the transformation matrix \(\mathbf{M}=\mathbf{S}^{-1}\mathbf{F}\) and this solution implies that \(\mathbf{M}\) exists so long as the inverse matrix \(\mathbf{S}^{-1}\) exists. It is possible to solve Equation 5 for the inverse transformation matrix \(\mathbf{M}^{-1}=\mathbf{F}^{-1}\mathbf{S}\) and this solution implies that \(\mathbf{M}^{-1}\) exists so long as the inverse matrix \(\mathbf{F}^{-1}\) exists.

From the above considerations, it is apparent that Equations 4 and 5 require that the inverse matrices \(\mathbf{S}^{-1}\) and \(\mathbf{F}^{-1}\) exist. \(\mathbf{S}^{-1}\) exists as long as the points \(P_{1}\), \(P_{2}\) and \(P_{3}\) are not collinear and no column of \(\mathbf{S}\) contains three zeros. \(\mathbf{F}^{-1}\) exists as long as the points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) are not collinear and no column of \(\mathbf{F}\) contains three zeros.

The collinearity requirement is satisfied by the positions of the three N-localizers relative to the stereotactic frame. Because for contemporary stereotactic frames the three N-localizers are positioned either around the circumference of a circle or on the faces of a cube, neither the points \(P_{1}\), \(P_{2}\) and \(P_{3}\) nor the points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) can possibly be collinear.

The requirement that no column of \(\mathbf{S}\) contain three zeros is satisfied by choosing \(w=1\) and not \(w=0\) when projecting the the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates to create \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates.

The requirement that no column of \(\mathbf{F}\) contain three zeros may be satisfied by defining the three-dimensional coordinate system of the stereotactic frame such that the \(z\)-coordinate does not equal zero anywhere along the diagonal rods. Figure *5* demonstrates that Equations 1 and 2 will never produce \(z=0\) along the diagonal rods so long as the interval from \(z_A\) to \(z_C\) does not contain zero. One way to satisfy this requirement is to define the origin of the three-dimensional coordinate system of the stereotactic frame to be the corner of a cube that surrounds the stereotactic frame, such that the \(z\)-coordinate of the origin is less than the \(z_C\)-coordinate of point \(P_C\).

When all three of the above requirements are satisfied, the matrix \(\mathbf{M}\) will correctly transform the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates of points \(P_{1}\), \(P_{2}\) and \(P_{3}\) from the two-dimensional coordinate system of the tomographic image to create the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates of points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) in the three-dimensional coordinate system of the stereotactic frame. And the inverse matrix \(\mathbf{M}^{-1}\) will correctly perform the inverse transformation.

#### Transformation of the target point \(P_T\) and the analogous target point \({P}'_T\)

The above analysis addresses the transformation of points \(P_{1}\), \(P_{2}\) and \(P_{3}\) and analogous points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\). However, the above discussion of the membrane analogy claims that *any* point that lies in the plane of points \(P_{1}\), \(P_{2}\) and \(P_{3}\) may be transformed from the two-dimensional coordinate system of the tomographic section into the three-dimensional coordinate system of the stereotactic frame via the same transformation that transforms points \(P_{1}\), \(P_{2}\) and \(P_{3}\). The following analysis proves this assertion.

Equation 3 transforms en masse the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates of the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\) to create the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates of the three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\). An alternative is to transform separately the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates of the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\)

$$ {P}'_1 = \begin{bmatrix}x_1\;y_1\;z_1\end{bmatrix} = \begin{bmatrix}u_1\;v_1\;1\end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} = P_1\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(6a\right) $$

$$ {P}'_2 = \begin{bmatrix}x_2\;y_2\;z_2\end{bmatrix} = \begin{bmatrix}u_2\;v_2\;1\end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} = P_2\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(6b\right) $$

$$ {P}'_3 = \begin{bmatrix}x_3\;y_3\;z_3\end{bmatrix} = \begin{bmatrix}u_3\;v_3\;1\end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} = P_3\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(6c\right) $$

Equation 6 produces the same result for the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates of points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) as does Equation 3.

It is possible to represent any point that lies in the plane defined by three other points as a linear combination of those three points. For example, with reference to Figure *6*, the target point \({P}'_T\) may be represented as a linear combination of the three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\)

$${P}'_T = b_1{P}'_1 + b_2{P}'_2 + b_3{P}'_3 \;\;\;\;\;\;\;\;\;\; \left(7\right) $$

Similarly, with reference to Figure *7*, the analogous target point \(P_T\) may be represented as a linear combination of the three points \(P_1\), \(P_2\) and \(P_3\).

$$P_T = b_1P_1 + b_2P_2 + b_3P_3 \;\;\;\;\;\;\;\;\;\; \left(8\right) $$

As used in Equations 7 and 8, the terms \(b_1\), \(b_2\) and \(b_3\) are known as *barycentric* coordinates and satisfy the condition \(b_1 + b_2 + b_3 = 1\) [20]. Because the matrix \(\mathbf{M}\) that transforms \(\mathbf{S}\) into \(\mathbf{F}\) describes a linear transformation, the same barycentric coordinates \(b_1\), \(b_2\) and \(b_3\) apply to either \(P_T\) or \({P}'_T\) and hence may be used in both Equation 7 and Equation 8. These equations describe interpolation in a plane that is analogous to the interpolation along a line that is expressed by Equation 1.

Using matrix \(\mathbf{M}\) to transform point \(P_T\) as suggested by Equation 4 and substituting Equations 6-8 yields

$$\begin{align*}P_T\:\mathbf{M} & = \left(b_1P_1 + b_2P_2 + b_3P_3\right)\mathbf{M} \\ & = b_1P_1\mathbf{M} + b_2P_2\mathbf{M} + b_3P_3\mathbf{M} \\ & = b_1{P}'_1 + b_2{P}'_2 + b_3 {P}'_3 \\ & = {P}'_T\end{align*} \;\;\;\;\;\;\;\;\;\; \left(9\right) $$

Eliminating the intermediate steps from Equation 9 and showing explicitly the \(\left(x_T,y_T,z_T\right)\) coordinates of \({P}'_T\) and the \(\left(u_T,v_T\right)\) coordinates of \(P_T\) yields

$${P}'_T = \begin{bmatrix}x_T\;y_T\;z_T\end{bmatrix} = \begin{bmatrix}u_T\;v_T\;1\end{bmatrix}\mathbf{M} = P_T\:\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(10\right) $$

Equation 10 proves that any point \(P_T\) that lies in the tomographic image may be transformed from the two-dimensional coordinate system of that image into the three-dimensional coordinate system of the stereotactic frame to obtain the analogous point \({P}'_T\).

Using matrix \(\mathbf{M}^{-1}\) to transform point \({P}'_T\) as suggested by Equation 5 and substituting Equations 6-8 yields

$$\begin{align*}{P}'_T\,\mathbf{M}^{-1} & = \left(b_1{P}'_1 + b_2{P}'_2 + b_3 {P}'_3\right)\mathbf{M}^{-1} \\ & = b_1{P}'_1\,\mathbf{M}^{-1} + b_2{P}'_2\,\mathbf{M}^{-1} + b_3 {P}'_3\,\mathbf{M}^{-1} \\ & = b_1P_1 + b_2P_2 + b_3P_3 \\ & = P_T\end{align*} \;\;\;\;\;\;\;\;\;\; \left(11\right)$$

Eliminating the intermediate steps from Equation 11 and showing explicitly the \(\left(x_T,y_T,z_T\right)\) coordinates of \({P}'_T\) and the \(\left(u_T,v_T\right)\) coordinates of \(P_T\) yields

$$P_T = \begin{bmatrix}u_T\;v_T\;1\end{bmatrix} = \begin{bmatrix}x_T\;y_T\;z_T\end{bmatrix}\mathbf{M}^{-1} = {P}'_T\:\mathbf{M}^{-1} \;\;\;\;\;\;\;\;\;\; \left(12\right) $$

Equation 12 proves that any point \({P}'_T\) that lies in the plane of the tomographic section may be transformed from the three-dimensional coordinate system of the stereotactic frame into the two-dimensional coordinate system of the tomographic image to obtain the analogous point \(P_T\).

Solving Equation 4 for \(\mathbf{M}\) yields$$\mathbf{M}=\mathbf{S}^{-1}\mathbf{F} \;\;\;\;\;\;\;\;\;\;\left(13\right)$$

Substituting Equation 13 into Equation 10 and showing explicitly the elements of matrices \(\mathbf{S}\) and \(\mathbf{F}\) from Equation 3 yields the transformation equation that accompanied the introduction of the Brown-Roberts-Wells stereotactic frame [8]

$$\begin{bmatrix}x_T\;y_T\;z_T\end{bmatrix} = \begin{bmatrix}u_T\;v_T\;1\end{bmatrix}\begin{bmatrix} u_1 & v_1 & 1 \\ u_2 & v_2 & 1 \\ u_3 & v_3 & 1 \end{bmatrix}^{-1}\:\begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{bmatrix} \;\;\;\;\;\;\;\;\;\; \left(14\right)$$

### Conclusions

The first CT-compatible stereotactic frame, and the only CT-compatible stereotactic frame that was constructed in 1978, is the frame that Brown built in order to test the concept of the N-localizer. Dissemination of the concept of the N-localizer inspired its incorporation into three different CT-compatible stereotactic frames that were constructed in 1979 and reported in 1980: the Brown-Roberts-Wells frame, a Leksell frame that was modified to render it CT-compatible, and the UPMC frame. The Brown-Roberts-Wells frame, its successor the Cosman-Roberts-Wells frame, and the Leksell frame all achieved widespread clinical use. The UPMC frame was utilized in 13 patients and then abandoned.

The N-localizer permits determination of the spatial orientation of a tomographic section relative to the stereotactic frame. When used to the full extent of its capability, the N-localizer allows calculation of the spatial orientation of a tomographic section that is arbitrarily oriented relative to the frame. This method obviates the need to affix the frame to the tomographic scanner bed or to align the frame to the scanner [3-6, 8, 13, 21]. It is possible to forgo these advantages and instead employ the N-localizer in a restricted manner that requires fixation of the frame to the scanner bed and also requires careful alignment of the frame to ensure that the tomographic section is parallel to the base of the frame [9].

The impetus for this restricted use of the N-localizer appears to have been concern over the N-localizer's requirement for "advanced computer technology" [9]. However, 30 to 35 years ago, exploiting the full capability of the N-localizer required only the CT scanner computer [4-6, 8, 13] or even a mere hand-held, programmable calculator [13, 21] to perform the calculation indicated by Equation 14. At that time, neither the CT scanner computer nor the hand-held calculator exemplified advanced computer technology. In this era of modern neurosurgery, the computation required to plan stereotactic radiosurgery far exceeds the minuscule computation required to harness the full capability of the N-localizer.

### References

- Brown RA, Nelson JA: The origin of the N-localizer for stereotactic neurosurgery. Cureus. 2013, 5:e140. http://www.cureus.com/articles/2335-the-origin-of-the-n-localizer-for-stereotactic-neurosurgery.
- Brown RA: The mathematics of the N-localizer for stereotactic neurosurgery. Cureus. 2013, 5:e142. http://www.cureus.com/articles/2336-the-mathematics-of-the-n-localizer-for-stereotactic-neurosurgery.
- Brown RA, Nelson JA: Invention of the N-localizer for stereotactic neurosurgery and its use in the Brown-Roberts-Wells stereotactic frame. Neurosurg. 2012, 70:173-176. 10.1227/NEU.0b013e318246a4f7
- Brown RA: A stereotactic head frame for use with CT body scanners. Investigative Radiology. 1979, 14:300-304. 10.1097/00004424-197907000-00006
- Brown RA: A computerized tomography-computer graphics approach to stereotaxic localization. J Neurosurg. 1979, 50:715-720. 10.3171/jns.1979.50.6.0715
- Brown RA, Roberts TS: A stereotaxic frame and computer software for use with CT body scanners. Stereotactic Cerebral Irradiation. Szikla G (ed): Elsevier/North-Holland Biomedical Press, Amsterdam, the Netherlands; 1979. 25-27.
- Brown RA: Personal communication with Perry JH. January-February 1979,
- Brown RA, Roberts TS, Osborn AE: Stereotaxic frame and computer software for CT-directed neurosurgical localization. Invest Radiol. 1980, 15:308-312. 10.1097/00004424-198007000-00006
- Leksell L, Jernberg B: Stereotaxis and tomography. A technical note. Acta Neurochir (Wien). 1980, 52:1–7. 10.1007/BF01400939
- Perry JH, Rosenbaum AE, Lunsford LD, Swink CA, Zorub DS: Computed tomography-guided stereotactic surgery: Conception and development of a new stereotactic methodology. Neurosurg. 1980, 7:376-381. 10.1227/00006123-198010000-00011
- Kondziolka DS, Lunsford LD: CT and image-guided surgery. Textbook of Stereotactic and Functional Neurosurgery. Lozano AM, Gildenberg PL, Tasker RR (ed): Springer Verlag, New York; 2008. 619-629.
- Rosenbaum AE, Lunsford LD, Perry JH: Computerized tomography guided stereotaxis: A new approach. Appl Neurophysiol. 1980, 43:172-173. 10.1159/000102252
- Roberts TS, Brown R: Technical and clinical aspects of CT-directed stereotaxis. Appl Neurophysiol. 1980, 43:170-171. 10.1159/000102251
- Lee SH, Villafana T, Lapayowker MS: CT intracranial localization with a new marker system. Neuroradiol. 1978, 16:570-571. 10.1007/BF00395364
- Villafana T, Lee SH, Lapayowker MS: A device to indicate anatomical level in computed tomography. J Comput Assist Tomogr. 1978, 2:368-371. 10.1097/00004728-197807000-00028
- Perry JH: Letter to Lunsford D, Rosenbaum A, and Zorub D. Available in Appendix 1 and U.S. Patent and Trademark Office Interference folder 101267. 1979, 1-3.
- Lunsford LD: Email to Brown RA. August 9, 2012,
- Lunsford LD: Email to Brown RA. August 10, 2012,
- Matthews RS: Letter to Nixon LS. Available in Appendix 2 and U.S. Patent and Trademark Office Interference folder 101267. 1979, 1-7.
- Barnhill RE: Representation and approximation of surfaces. Mathematical Software III. Rice JR (ed): Academic Press, New York; 1977. 69-120.
- Brown RA, Roberts TS, Osborn AG: Simplified CT-guided stereotaxic biopsy. Am J Neuroradiol. 1981, 2:181-184.

### Appendices

### Related articles

#### The History and Mathematics of the N-Localizer for Stereotactic Neurosurgery

###### Ethics Statement and Conflict of Interest Disclosures

**Conflicts of interest:** The authors have declared that no conflicts of interest exist.

### Article Information

###### DOI

10.7759/cureus.156

###### Cite this article as:

Brown R A, Nelson J A. (January 16, 2014) The History and Mathematics of the N-Localizer for Stereotactic Neurosurgery. Cureus 6(1): e156. doi:10.7759/cureus.156

###### Publication history

Received by Cureus: December 28, 2013

Peer review began: December 30, 2013

Published: January 16, 2014

###### Copyright

**© Copyright **2014

Brown et al. This is an open access article distributed under the terms of the Creative Commons Attribution License CC-BY 3.0., which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

###### License

This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

#### The History and Mathematics of the N-Localizer for Stereotactic Neurosurgery

### Figures etc.

##
**Retracted:**
The History and Mathematics of the N-Localizer for Stereotactic Neurosurgery

### Abstract

This paper is a supplement to "The Origin of the N-Localizer for Stereotactic Neurosurgery" [1] and "The Mathematics of the N-Localizer for Stereotactic Neurosurgery" [2]. It clarifies the early history of the N-localizer and presents further details of the mathematics of the N-localizer.

### Introduction

The N-localizer has become an important neurosurgical tool that has achieved widespread use in modern stereotactic neurosurgery and radiosurgery. The N-localizer produces two circles and one ellipse in tomographic, sectional images that are obtained via computed tomography or magnetic resonance imaging (Figure *1*). The relative spacing between the ellipse and the two circles precisely determines the location of the tomographic section relative to the N-localizer [3-4].

Russell A. Brown invented the N-localizer in May 1978 when he was a third-year medical student at the University of Utah and during a research elective in the laboratory of James A. Nelson. In August 1978, Brown built the first CT-compatible stereotactic frame in order to test the concept of the N-localizer (Figure *2*). In October 1978, he presented this frame at a joint meeting of the Western Neurological Society and the American Academy of Neurological Surgeons [5]. Following this presentation, additional presentations [6], and disclosures [7], the concept of the N-localizer spread rapidly and inspired its incorporation into three different CT-compatible stereotactic frames that were built in 1979 and reported in 1980: The Brown-Roberts-Wells frame [8], a Leksell frame that was modified to render it CT-compatible [9], and a frame that resulted from a collaboration between Pfizer Medical Systems and the University of Pittsburgh Medical Center (the “UPMC” frame) [10].

During the 35 years since the invention of the N-localizer, some misconceptions have arisen concerning its history in relation to subsequent developments in image-guided stereotactic surgery. Kondziolka and Lunsford of the University of Pittsburgh School of Medicine have claimed [11], "At our center, the first CT compatible stereotactic head frame, in collaboration with industry, was constructed in 1978 and utilized in 13 patients [10, 12]. [...] During this interval, the newly redesigned Leksell CT compatible stereotactic head frame [9] was used for dedicated brain biopsies under the direction of its inventor, Lars Leksell. Several groups were working on devices to allow accurate CT based stereotactic surgery [13]."

The above statement is not the only instance wherein Lunsford and Kondziolka have inaccurately interpreted the history of image-guided stereotactic surgery; we have discussed another instance previously [1]. But in particular, the above statement presents an erroneous chronology and disregards the fact that the adoption of the N-localizer for the Leksell and UPMC stereotactic frames was derivative. For both of these frames, the knowledge of the N-localizer originated from Brown's prior invention of the N-localizer [1, 9-10]. The UPMC frame was not the first CT-compatible stereotactic frame. And the UPMC frame was built in 1979, not in 1978. The documents that corroborate these facts have remained preserved in the archives of the U.S. Patent and Trademark office for nearly 30 years. The following technical report, which is based on these archives, recounts the development of image-guided stereotactic surgery at the University of Pittsburgh Medical Center and relates the events that led to the construction of the UPMC frame.

### Technical Report

Prior to the invention of the N-localizer, a method had been reported for estimating the location of a tomographic image relative to patient anatomy [14-15]. This method involved a plate into which were milled vertical slots whose tops lay along a diagonal line (Figure *3*). Because this slotted plate required counting numerous notches that were visible in the tomographic image, it was cumbersome to use and susceptible to human error [1, 16]. Despite these limitations, in the summer of 1978, Dade Lunsford modified a Leksell Stalix frame via the addition of two aluminum plates having milled vertical slots in the hope of creating a CT-compatible stereotactic frame [17].

In the autumn of 1978, Lunsford, et al. began to collaborate with John Perry of Pfizer Medical Systems [1, 17]. By mid-January of 1979, Perry, Lunsford, et al. had added a third aluminum plate to the Stalix frame and Perry had developed computer software that used information from the three slotted plates to transform coordinates from a CT scan image into the coordinate system of the Stalix frame [16, 18-19]. In a letter dated January 15, 1979 and addressed to Dade Lunsford, Arthur Rosenbaum and David Zorub of the University of Pittsburgh School of Medicine, Perry provided detailed instructions that explained the procedure required for obtaining CT scan images of the modified Stalix frame and for using computer software to transform coordinates from a CT scan image into the coordinate system of the Stalix frame [16].

Consequently, in early 1979, Perry, Lunsford, et al. appeared to be at the point of performing clinical cases with the modified Stalix frame. However, according to Lunsford, the modified Stalix frame was never used clinically because the aluminum composition of the frame and attached slotted plates created artifacts in the CT scan image [18]. Lunsford’s account is corroborated by Perry, et al. who reported that the UPMC “frame was made after attempts to modify the Leksell frame proved difficult” [10].

Lunsford has asserted that the Stalix frame was abandoned in the fall of 1978 and that thereafter the UPMC frame was constructed [18]. However, he appears to have forgotten the contents of the January 1979 letter from Perry to Lunsford, et al., a copy of which remains preserved in the archives of the U.S. patent office [16]. This letter refutes Lunsford's assertion and clearly demonstrates that in January 1979, Perry, Lunsford, et al. were preparing to use the modified Stalix frame for clinical cases but they had not yet done so. It was only after they had attempted to use the modified Stalix frame that they understood the severity of the artifacts created by the aluminum and subsequently abandoned the Stalix frame [10, 18]. And it was only after Brown had disclosed the N-localizer to Perry in early 1979 that Perry understood the concept of the N-localizer and subsequently adopted the N-localizer for the UPMC frame [1, 7, 10].

The second [4] of Brown's two publications that announced the N-localizer and reported evaluations of the first CT-compatible stereotactic frame [4-5] was submitted for publication in January 1979. The first [5] of these publications affirms that Brown presented this stereotactic frame at a joint meeting of the Western Neurological Society and the American Academy of Neurological Surgeons in October 1978. These and other publications [4-5, 10], together with historical documents [16, 19] from the archives of the U.S. patent office, confirm that the UPMC frame was not the first CT-compatible stereotactic frame and further confirm that the UPMC frame was constructed in 1979, not in 1978. The relevant historical documents [16, 19] are a matter of public record and may be obtained from the U.S. Patent and Trademark Office by requesting a copy of the folder for Interference No. 101267. In order to facilitate access to these documents, we have included copies as the appendices (labeled as "figures") to this paper.

### Discussion

The mathematics of the N-localizer have been discussed previously [4] and in considerable detail [2]. We review the mathematics below in preparation for presenting further details. The reader who is not interested in the details of the mathematics may choose to skip directly to the Conclusions.

#### Review of the mathematics of the N-Localizer

The N-localizer comprises a diagonal rod that extends from the top of one vertical rod to the bottom of another vertical rod (Figure *4*). In the tomographic image, each of the two vertical rods creates a fiducial circle and the diagonal rod creates a fiducial ellipse. The ellipse moves away from one circle and towards the other circle as the position of the tomographic section moves upward with respect to the N-localizer. The relative spacing between the three centroids permits precise localization of the tomographic section relative to the N-localizer. The distance \(d_{AB}\) between the centroids of circle A and ellipse B and the distance \(d_{AC}\) between the centroids of circles A and C are used to calculate the ratio \(f=d_{AB}/d_{AC}\). This ratio represents the fraction of the diagonal rod B that extends from the top of vertical rod A to the point of intersection of rod B with the tomographic section.

The fraction \(f\) is used to calculate the \(\left(x,y,z\right)\) coordinates of the point of intersection \({P}'_{B}\) between rod B and the tomographic section (Figure *5*). In this figure, points \({P}'_{A}\) and \({P}'_{C}\) represent the beginning and end, respectively, of the vector that extends from the top of rod A to the bottom of rod C. This vector coincides with the long axis of rod B. The \(\left(x_{A},y_{A},z_{A}\right)\) coordinates of the beginning point \({P}'_{A}\) and the \(\left(x_{C},y_{C},z_{C}\right)\) coordinates of the end point \({P}'_{C}\) are known from the geometry of the N-localizer. Hence, linear interpolation may be used to blend points \({P}'_{A}\) and \({P}'_{C}\) to obtain the \(\left(x_{B},y_{B},z_{B}\right)\) coordinates of the point of intersection \({P}'_{B}\) between the long axis of rod B and the tomographic section, as follows$${P}'_{B}={P}'_{A}+f\left({P}'_{C}-{P}'_{A}\right)=f{P}'_{C}+\left(1-f\right){P}'_{A}\;\;\;\;\;\;\;\;\;\;\left(1\right)$$The vector form of Equation 1 shows explicitly the \(\left(x,y,z\right)\) coordinates of points \({P}'_{A}\), \({P}'_{B}\) and \({P}'_{C}\)$$\begin{bmatrix}x_{B}\;y_{B}\;z_{B}\end{bmatrix} =f \begin{bmatrix}x_{C}\;y_{C}\;z_{C}\end{bmatrix} +\left(1-f \right) \begin{bmatrix}x_{A}\;y_{A}\;z_{A}\end{bmatrix}\;\;\;\;\;\;\;\;\;\;\left(2\right)$$Equation 1 or 2 may be used to calculate the \(\left(x_{B},y_{B},z_{B}\right)\) coordinates of the point of intersection \({P}'_{B}\) between rod B and the tomographic section. The point \({P}'_{B}\), which lies on rod B in the three-dimensional coordinate system of the N-localizer, corresponds to an analogous point \(P_{B}\), which lies at the centroid of ellipse B in the two-dimensional coordinate system of the tomographic image. Hence, there is a one-to-one correspondence between a point from the N-localizer and a point from the tomographic image.

The attachment of three N-localizers to the stereotactic frame permits calculation of the \(\left(x_{B_{1}},y_{B_{1}},z_{B_{1}}\right)\), \(\left(x_{B_{2}},y_{B_{2}},z_{B_{2}}\right)\) and \(\left(x_{B_{3}},y_{B_{3}},z_{B_{3}}\right)\) coordinates for the three respective points \({P}'_{B_{1}}\), \({P}'_{B_{2}}\) and \({P}'_{B_{3}}\) in the three-dimensional coordinate system of the stereotactic frame. These three points correspond respectively to the three points \(P_{B_{1}}\), \(P_{B_{2}}\) and \(P_{B_{3}}\) in the two-dimensional coordinate system of the tomographic image. In the following discussion, the symbols \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) will be used as a shorthand notation for \({P}'_{B_{1}}\), \({P}'_{B_{2}}\) and \({P}'_{B_{3}}\). The symbols \(P_{1}\), \(P_{2}\) and \(P_{3}\) will be used as a shorthand notation for \(P_{B_{1}}\), \(P_{B_{2}}\) and \(P_{B_{3}}\).

The three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) lie on the three respective diagonal rods \(B_1\), \(B_2\) and \(B_3\) and have respective \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates in the three-dimensional coordinate system of the stereotactic frame (Figure *6*). The analogous three points \(P_{1}\), \(P_{2}\) and \(P_{3}\) lie at the centroids of the three respective ellipses \(B_1\), \(B_2\) and \(B_3\) and have \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates in the two-dimensional coordinate system of the tomographic image (Figure *7*). Because three points determine a plane in three-dimensional space, the three coordinates \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) together with the analogous three coordinates \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) determine the spatial orientation of the tomographic section relative to the stereotactic frame. This spatial orientation permits calculation of the \(\left(x_{T},y_{T},z_{T}\right)\) coordinates of the target point \({P}'_{T}\) in the three-dimensional coordinate system of the stereotactic frame, given the \(\left(u_{T},v_{T}\right)\) coordinates of the analogous target point \(P_{T}\) in the two-dimensional coordinate system of the tomographic image.

In order to facilitate calculation of the \(\left(x_{T},y_{T},z_{T}\right)\) coordinates of the target point \({P}'_{T}\), it is convenient to project the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates of the three respective points \(P_{1}\), \(P_{2}\) and \(P_{3}\) onto the \(w = 1\) plane in three-dimensional space by appending a third coordinate \(w = 1\) to create \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates. The \(w\)-coordinate may be set arbitrarily to any non-zero value, e.g., one, so long as same value of \(w\) is used for each of the three \(w\)-coordinates. The equations that are presented in the remainder of this paper assume that a value of \(w = 1\) has been used to project the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates. If a value of \(w \neq 1\) were used instead of \(w = 1\) to project the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates, the equations that are presented in the remainder of this paper would no longer apply and would require revision so that the calculations that these equations describe may produce correct results.

The correspondence, or transformation, between the two-dimensional coordinate system of the tomographic image and the three-dimensional coordinate system of the stereotactic frame may be represented using the three \(\left(x,y,z\right)\) and \(\left(u,v,1 \right)\) coordinate pairs \(\left(x_{1},y_{1},z_{1}\right)\) and \(\left(u_{1},v_{1},1\right)\); \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(u_{2},v_{2},1\right)\); and \(\left(x_{3},y_{3},z_{3}\right)\) and \(\left(u_{3},v_{3},1\right)\) in the following matrix equation$$\begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{bmatrix} = \begin{bmatrix} u_1 & v_1 & 1 \\ u_2 & v_2 & 1 \\ u_3 & v_3 & 1 \end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\left(3\right)$$Equation 3 represents concisely a system of nine simultaneous linear equations that determine the spatial orientation of the tomographic section relative to the stereotactic frame. This equation transforms the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates from the two-dimensional coordinate system of the tomographic image to create \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates in the three-dimensional coordinate system of the stereotactic frame.

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 processes 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 lastly, lifting the membrane into place upon the scaffold of the three N-localizers such that the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\) from the tomographic image precisely coincide with the respective three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) from the stereotactic frame. Then *any* other point that lies on the membrane, for example, the target point \(P_{T}\), is transformed by the same stretching, rotating, tilting and lifting processes that transformed the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\). In this manner, the \(\left(u_{T},v_{T}\right)\) coordinates of the target point \(P_{T}\) 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 \(\left(x_{T},y_{T},z_{T}\right)\) coordinates of the analogous target point \({P}'_{T}\).

Equation 3 may be written in more compact form as$$\mathbf{F}=\mathbf{S}\mathbf{M} \;\;\;\;\;\;\;\;\;\;\left(4\right)$$In Equation 4, \(\mathbf{F}\) represents the matrix of \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates from the coordinate system of the stereotactic frame. \(\mathbf{S}\) represents the matrix of \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates from the coordinate system of the tomographic image. \(\mathbf{M}\) represents the matrix of elements \(m_{11}\) through \(m_{33}\) that defines the transformation from the two-dimensional coordinate system of the tomographic image into the three-dimensional coordinate system of the stereotactic frame.

Solving Equation 4 for \(\mathbf{S}\) yields$$\mathbf{S}=\mathbf{F}\mathbf{M}^{-1} \;\;\;\;\;\;\;\;\;\;\left(5\right)$$

Equation 5 transforms the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates from the three-dimensional coordinate system of the stereotactic frame to create \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates in the two-dimensional coordinate system of the tomographic image.

The above review of the mathematics of the N-localizer is sufficient to permit the presentation of further details below.

#### Requirements for the matrices \(\mathbf{F}\) and \(\mathbf{S}\)

It is possible to solve Equation 4 for the transformation matrix \(\mathbf{M}=\mathbf{S}^{-1}\mathbf{F}\) and this solution implies that \(\mathbf{M}\) exists so long as the inverse matrix \(\mathbf{S}^{-1}\) exists. It is possible to solve Equation 5 for the inverse transformation matrix \(\mathbf{M}^{-1}=\mathbf{F}^{-1}\mathbf{S}\) and this solution implies that \(\mathbf{M}^{-1}\) exists so long as the inverse matrix \(\mathbf{F}^{-1}\) exists.

From the above considerations, it is apparent that Equations 4 and 5 require that the inverse matrices \(\mathbf{S}^{-1}\) and \(\mathbf{F}^{-1}\) exist. \(\mathbf{S}^{-1}\) exists as long as the points \(P_{1}\), \(P_{2}\) and \(P_{3}\) are not collinear and no column of \(\mathbf{S}\) contains three zeros. \(\mathbf{F}^{-1}\) exists as long as the points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) are not collinear and no column of \(\mathbf{F}\) contains three zeros.

The collinearity requirement is satisfied by the positions of the three N-localizers relative to the stereotactic frame. Because for contemporary stereotactic frames the three N-localizers are positioned either around the circumference of a circle or on the faces of a cube, neither the points \(P_{1}\), \(P_{2}\) and \(P_{3}\) nor the points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) can possibly be collinear.

The requirement that no column of \(\mathbf{S}\) contain three zeros is satisfied by choosing \(w=1\) and not \(w=0\) when projecting the the \(\left(u_{1},v_{1}\right)\), \(\left(u_{2},v_{2}\right)\) and \(\left(u_{3},v_{3}\right)\) coordinates to create \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates.

The requirement that no column of \(\mathbf{F}\) contain three zeros may be satisfied by defining the three-dimensional coordinate system of the stereotactic frame such that the \(z\)-coordinate does not equal zero anywhere along the diagonal rods. Figure *5* demonstrates that Equations 1 and 2 will never produce \(z=0\) along the diagonal rods so long as the interval from \(z_A\) to \(z_C\) does not contain zero. One way to satisfy this requirement is to define the origin of the three-dimensional coordinate system of the stereotactic frame to be the corner of a cube that surrounds the stereotactic frame, such that the \(z\)-coordinate of the origin is less than the \(z_C\)-coordinate of point \(P_C\).

When all three of the above requirements are satisfied, the matrix \(\mathbf{M}\) will correctly transform the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates of points \(P_{1}\), \(P_{2}\) and \(P_{3}\) from the two-dimensional coordinate system of the tomographic image to create the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates of points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) in the three-dimensional coordinate system of the stereotactic frame. And the inverse matrix \(\mathbf{M}^{-1}\) will correctly perform the inverse transformation.

#### Transformation of the target point \(P_T\) and the analogous target point \({P}'_T\)

The above analysis addresses the transformation of points \(P_{1}\), \(P_{2}\) and \(P_{3}\) and analogous points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\). However, the above discussion of the membrane analogy claims that *any* point that lies in the plane of points \(P_{1}\), \(P_{2}\) and \(P_{3}\) may be transformed from the two-dimensional coordinate system of the tomographic section into the three-dimensional coordinate system of the stereotactic frame via the same transformation that transforms points \(P_{1}\), \(P_{2}\) and \(P_{3}\). The following analysis proves this assertion.

Equation 3 transforms en masse the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates of the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\) to create the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates of the three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\). An alternative is to transform separately the \(\left(u_{1},v_{1},1\right)\), \(\left(u_{2},v_{2},1\right)\) and \(\left(u_{3},v_{3},1\right)\) coordinates of the three points \(P_{1}\), \(P_{2}\) and \(P_{3}\)

$$ {P}'_1 = \begin{bmatrix}x_1\;y_1\;z_1\end{bmatrix} = \begin{bmatrix}u_1\;v_1\;1\end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} = P_1\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(6a\right) $$

$$ {P}'_2 = \begin{bmatrix}x_2\;y_2\;z_2\end{bmatrix} = \begin{bmatrix}u_2\;v_2\;1\end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} = P_2\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(6b\right) $$

$$ {P}'_3 = \begin{bmatrix}x_3\;y_3\;z_3\end{bmatrix} = \begin{bmatrix}u_3\;v_3\;1\end{bmatrix} \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} = P_3\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(6c\right) $$

Equation 6 produces the same result for the \(\left(x_{1},y_{1},z_{1}\right)\), \(\left(x_{2},y_{2},z_{2}\right)\) and \(\left(x_{3},y_{3},z_{3}\right)\) coordinates of points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\) as does Equation 3.

It is possible to represent any point that lies in the plane defined by three other points as a linear combination of those three points. For example, with reference to Figure *6*, the target point \({P}'_T\) may be represented as a linear combination of the three points \({P}'_{1}\), \({P}'_{2}\) and \({P}'_{3}\)

$${P}'_T = b_1{P}'_1 + b_2{P}'_2 + b_3{P}'_3 \;\;\;\;\;\;\;\;\;\; \left(7\right) $$

Similarly, with reference to Figure *7*, the analogous target point \(P_T\) may be represented as a linear combination of the three points \(P_1\), \(P_2\) and \(P_3\).

$$P_T = b_1P_1 + b_2P_2 + b_3P_3 \;\;\;\;\;\;\;\;\;\; \left(8\right) $$

As used in Equations 7 and 8, the terms \(b_1\), \(b_2\) and \(b_3\) are known as *barycentric* coordinates and satisfy the condition \(b_1 + b_2 + b_3 = 1\) [20]. Because the matrix \(\mathbf{M}\) that transforms \(\mathbf{S}\) into \(\mathbf{F}\) describes a linear transformation, the same barycentric coordinates \(b_1\), \(b_2\) and \(b_3\) apply to either \(P_T\) or \({P}'_T\) and hence may be used in both Equation 7 and Equation 8. These equations describe interpolation in a plane that is analogous to the interpolation along a line that is expressed by Equation 1.

Using matrix \(\mathbf{M}\) to transform point \(P_T\) as suggested by Equation 4 and substituting Equations 6-8 yields

$$\begin{align*}P_T\:\mathbf{M} & = \left(b_1P_1 + b_2P_2 + b_3P_3\right)\mathbf{M} \\ & = b_1P_1\mathbf{M} + b_2P_2\mathbf{M} + b_3P_3\mathbf{M} \\ & = b_1{P}'_1 + b_2{P}'_2 + b_3 {P}'_3 \\ & = {P}'_T\end{align*} \;\;\;\;\;\;\;\;\;\; \left(9\right) $$

Eliminating the intermediate steps from Equation 9 and showing explicitly the \(\left(x_T,y_T,z_T\right)\) coordinates of \({P}'_T\) and the \(\left(u_T,v_T\right)\) coordinates of \(P_T\) yields

$${P}'_T = \begin{bmatrix}x_T\;y_T\;z_T\end{bmatrix} = \begin{bmatrix}u_T\;v_T\;1\end{bmatrix}\mathbf{M} = P_T\:\mathbf{M} \;\;\;\;\;\;\;\;\;\; \left(10\right) $$

Equation 10 proves that any point \(P_T\) that lies in the tomographic image may be transformed from the two-dimensional coordinate system of that image into the three-dimensional coordinate system of the stereotactic frame to obtain the analogous point \({P}'_T\).

Using matrix \(\mathbf{M}^{-1}\) to transform point \({P}'_T\) as suggested by Equation 5 and substituting Equations 6-8 yields

$$\begin{align*}{P}'_T\,\mathbf{M}^{-1} & = \left(b_1{P}'_1 + b_2{P}'_2 + b_3 {P}'_3\right)\mathbf{M}^{-1} \\ & = b_1{P}'_1\,\mathbf{M}^{-1} + b_2{P}'_2\,\mathbf{M}^{-1} + b_3 {P}'_3\,\mathbf{M}^{-1} \\ & = b_1P_1 + b_2P_2 + b_3P_3 \\ & = P_T\end{align*} \;\;\;\;\;\;\;\;\;\; \left(11\right)$$

Eliminating the intermediate steps from Equation 11 and showing explicitly the \(\left(x_T,y_T,z_T\right)\) coordinates of \({P}'_T\) and the \(\left(u_T,v_T\right)\) coordinates of \(P_T\) yields

$$P_T = \begin{bmatrix}u_T\;v_T\;1\end{bmatrix} = \begin{bmatrix}x_T\;y_T\;z_T\end{bmatrix}\mathbf{M}^{-1} = {P}'_T\:\mathbf{M}^{-1} \;\;\;\;\;\;\;\;\;\; \left(12\right) $$

Equation 12 proves that any point \({P}'_T\) that lies in the plane of the tomographic section may be transformed from the three-dimensional coordinate system of the stereotactic frame into the two-dimensional coordinate system of the tomographic image to obtain the analogous point \(P_T\).

Solving Equation 4 for \(\mathbf{M}\) yields$$\mathbf{M}=\mathbf{S}^{-1}\mathbf{F} \;\;\;\;\;\;\;\;\;\;\left(13\right)$$

Substituting Equation 13 into Equation 10 and showing explicitly the elements of matrices \(\mathbf{S}\) and \(\mathbf{F}\) from Equation 3 yields the transformation equation that accompanied the introduction of the Brown-Roberts-Wells stereotactic frame [8]

$$\begin{bmatrix}x_T\;y_T\;z_T\end{bmatrix} = \begin{bmatrix}u_T\;v_T\;1\end{bmatrix}\begin{bmatrix} u_1 & v_1 & 1 \\ u_2 & v_2 & 1 \\ u_3 & v_3 & 1 \end{bmatrix}^{-1}\:\begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{bmatrix} \;\;\;\;\;\;\;\;\;\; \left(14\right)$$

### Conclusions

The first CT-compatible stereotactic frame, and the only CT-compatible stereotactic frame that was constructed in 1978, is the frame that Brown built in order to test the concept of the N-localizer. Dissemination of the concept of the N-localizer inspired its incorporation into three different CT-compatible stereotactic frames that were constructed in 1979 and reported in 1980: the Brown-Roberts-Wells frame, a Leksell frame that was modified to render it CT-compatible, and the UPMC frame. The Brown-Roberts-Wells frame, its successor the Cosman-Roberts-Wells frame, and the Leksell frame all achieved widespread clinical use. The UPMC frame was utilized in 13 patients and then abandoned.

The N-localizer permits determination of the spatial orientation of a tomographic section relative to the stereotactic frame. When used to the full extent of its capability, the N-localizer allows calculation of the spatial orientation of a tomographic section that is arbitrarily oriented relative to the frame. This method obviates the need to affix the frame to the tomographic scanner bed or to align the frame to the scanner [3-6, 8, 13, 21]. It is possible to forgo these advantages and instead employ the N-localizer in a restricted manner that requires fixation of the frame to the scanner bed and also requires careful alignment of the frame to ensure that the tomographic section is parallel to the base of the frame [9].

The impetus for this restricted use of the N-localizer appears to have been concern over the N-localizer's requirement for "advanced computer technology" [9]. However, 30 to 35 years ago, exploiting the full capability of the N-localizer required only the CT scanner computer [4-6, 8, 13] or even a mere hand-held, programmable calculator [13, 21] to perform the calculation indicated by Equation 14. At that time, neither the CT scanner computer nor the hand-held calculator exemplified advanced computer technology. In this era of modern neurosurgery, the computation required to plan stereotactic radiosurgery far exceeds the minuscule computation required to harness the full capability of the N-localizer.

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