Comparative Accuracies of the N-Localizer and Sturm-Pastyr Localizer in the Presence of Image Noise

The N-localizer and the Sturm-Pastyr localizer are two technologies that facilitate image-guided stereotactic surgery. Both localizers enable the geometric transformation of tomographic image data from the two-dimensional coordinate system of a medical image into the three-dimensional coordinate system of the stereotactic frame. Monte Carlo simulations reveal that the Sturm-Pastyr localizer is less accurate than the N-localizer in the presence of image noise.


Introduction
The N-localizer was introduced in 1979 [1], and the Sturm-Pastyr localizer was introduced in 1983 [2]. Both localizers enable the geometric transformation of tomographic image data from the two-dimensional coordinate system of a medical image into the three-dimensional coordinate system of the stereotactic frame. Geometric transformation requires calculations that differ substantially between the two localizers in ways that impact the accuracy of the calculations when the effects of image noise are considered.

Technical Report
Geometric transformation requires the calculation of coordinates in the threedimensional coordinate system of the stereotactic frame. The following presentation discusses the calculation of only the -coordinate because the calculation of the coordinates is trivial due to features of the N-localizer and Sturm-Pastyr localizer. Specifically, the N-localizer includes two vertical rods that have fixed values of and , and the Sturm-Pastyr localizer includes one vertical rod that has fixed values of and . Figure 1 depicts the N-localizer that comprises two vertical rods and one diagonal rod. For the N-localizer, calculation of the -coordinate of the point of intersection of the cylindrical axis of rod with the tomographic section is performed via linear interpolation between the two ends of rod according to the following equation [3].

Tomographic Section
Side view of the N-localizer. A tomographic section intersects rods , , and . Tomographic image. The intersection of the tomographic section with rods , , and creates fiducial circles and and fiducial ellipse in the tomographic image. The distance between the centers of ellipse and circle and the distance between the centers of circles and are used to calculate the -coordinate of the point of intersection of the cylindrical axis of rod with the tomographic section [3]. Figure 2 depicts the Sturm-Pastyr localizer that comprises two diagonal rods and one vertical rod. For the Sturm-Pastyr localizer, calculation of the -coordinate of the point of intersection of the cylindrical axis of rod with the tomographic section is performed via the following non-linear equation that is derived in the Appendix [5].
In this equation, and are distances measured in the coordinate system of the medical image. At the bottom of rod , i.e., at the apex of the V-shaped Sturm-Pastyr localizer, . When vertical rod is perpendicular to the tomographic section, i.e., when the tomographic section is parallel to the base of the stereotactic frame, Equation (2) reduces to This equation applies because the Sturm-Pastyr localizer is manufactured such that the angle between rods and , and the angle between rods and , are both [6].

FIGURE 2: The Sturm-Pastyr Localizer and its Intersection with a Tomographic Section
Side view of the Sturm-Pastyr localizer. A tomographic section intersects rods , , and . Tomographic image. The intersection of the tomographic section with rods , , and creates fiducial ellipses and and fiducial circle in the tomographic image. The distance between the centers of ellipse and circle and the distance between the centers of circle and ellipse are used to calculate the -coordinate of the point of intersection of the cylindrical axis of rod with the tomographic section [6].

Equation (
3) requires specification of the pixel size for the medical image to permit conversion of the distances and to millimeters. Equation (2) also requires specification of the pixel size because the units of calculated by Equation (2) are the units of and , as demonstrated by dimensional analysis of Equation (2). This requirement, which does not apply to the N-localizer, renders the Sturm-Pastyr localizer susceptible to error. An erroneous value of will be calculated via Equations (2,3) if the pixel size is specified incorrectly via user input, or computed incorrectly from fiducials in the medical image [6], or recorded incorrectly in medical image metadata that require frequent calibration of the imaging system to guarantee correct pixel size. Figures 1, 2 demonstrate that the tomographic section of a medical image has a finite thickness. It is convenient to ignore this thickness and to approximate a 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. In the following presentation, the term "tomographic section" will be used as an abbreviation for the term "central plane of the tomographic section." Similarly, it is convenient to ignore the diameter of rods , , and in Figures 1, 2 and to approximate each rod as an infinitely thin cylindrical axis. In the following discussion, the term "rod" will be used as an abbreviation for the term "cylindrical axis of a rod." Hence, in the following presentation, the intersection of a "rod" with a "tomographic section" is equivalent to the intersection of a line with a plane and defines a point.

Monte Carlo algorithm
The accuracies of the N-localizer and Sturm-Pastyr localizer are compared via Monte Carlo simulation that is performed using the following algorithm.

1.
A -coordinate is chosen to express the height above the base of the stereotactic frame, i.e., above the base of the localizer.
2. An angle is chosen to express the angle by which the tomographic section is tilted with respect to the localizer such that line is tilted relative to the base of the stereotactic frame (see Figures 3, 7).

The
pair is used to calculate the , , and coordinates of the fiducial points , , and , respectively, in millimeters.

Monte Carlo simulation for the N-localizer
Step 3 of the Monte Carlo algorithm requires calculation of the , , and coordinates for a pair. To promote clarity, the calculation for a pair, for which , is discussed first. Figure 3 depicts an N-localizer wherein rods , , and intersect both a non-tilted , and a tilted tomographic section, for which . For the non-tilted section, calculation of the , , and coordinates of the respective fiducial points , , and begins with calculation of the distances and .
The assumption that vertical rods and are separated by mm yields mm. The assumption that vertical rods and are mm high yields mm. Making the simplification that then yields per Equation (1), where is specified in millimeters.

FIGURE 3: Depiction of the N-Localizer
The N-localizer is depicted by rods , , and that intersect a non-tilted tomographic section at fiducial points , , and . The rods also intersect a tomographic section that is tilted by the angle at fiducial points , , and . The distance between points and is . The distance between points and is .
Given the distances and , it is possible to assign values to the , , and coordinates of the fiducial points , , and . Making the simplification that the fiducial points lie along the -axis, a simple assignment is For the tilted section, calculation of the , , and coordinates begins with calculation of the distances and . Figure 3  In these equations, for the Sturm-Pastyr localizer [6]. Hence, and are functions of only and .
Given the distances and , it is possible to assign values to the , , and coordinates of the fiducial points , , and . Making the simplification that the fiducial points lie along the -axis, a simple assignment is the Sturm-Pastyr localizer incurs significantly more RMS error than the N-localizer.

Sturm-Pastyr Localizer
The RMS error in is plotted vs. for the N-localizer (solid curves) and the Sturm-Pastyr localizer (dashed curves). Each curve is generated using the value of that is specified in degrees to the right of the curve.

RMS: root mean square
The RMS error for the Sturm-Pastyr localizer increases as decreases and as increases. These trends may be understood by inspecting Equation (7), which shows that is directly proportional to and inversely proportional to ; this sine term is maximized when degrees. These trends may also be understood by inspecting Figure 7, which shows that is minimized for a given value of when line segment is perpendicular to line segment , i.e., when . Thus, an increase in in the range degrees or a decrease in decreases and consequently, the random perturbations in the range mm become more significant relative to and thereby increase the RMS error.
Equation (7) also shows that is inversely proportional to and hence increases monotonically as increases in the range degrees, where .
And Equation (2) shows that depends on and in a non-linear manner. Figure 5 demonstrates the effect of this non-linearity on the RMS error in for the Sturm-Pastyr localizer and reveals that the maximum RMS error occurs near degrees.

FIGURE 5: RMS Error in Plotted vs. for the Sturm-Pastyr Localizer
The RMS error in is plotted versus for the Sturm-Pastyr localizer. Each curve is generated using the value of that is specified in millimeters to the left of the curve. The curves for mm are similar to the curve for mm and are omitted.

RMS: root mean square
The RMS error for the N-localizer decreases as increases. This trend may be understood by inspecting Equation (6), which shows that the unperturbed , , and coordinates are inversely proportional to . Hence, as increases in the range degrees, the unperturbed coordinates increase as well and in consequence, the random perturbations in the range mm become less significant relative to the magnitudes of the unperturbed coordinates and thereby decrease the RMS error.
Random perturbations in the range mm are used for the Monte Carlo algorithm due to the following considerations. A typical field of view (FOV) for a medical image that is used for planning stereotactic surgery lies in the range mm and comprises 512x512 pixels. Hence, the pixel size for such an image is in the range mm. A conservative estimate that the center of each fiducial circle or ellipse is displaced at most two pixels by random noise yields the perturbation range mm.
The effect of various perturbation ranges on the errors incurred by the N-localizer and Sturm-Pastyr localizer is shown in Figure 6.

Perturbation for the N-Localizer and Sturm-Pastyr Localizer at mm and Degrees
The RMS and maximum errors are plotted vs. the maximum perturbation for the N-localizer (solid and dot-dashed curves) and the Sturm-Pastyr localizer (dashed and long-dashed curves) at mm and degrees.
RMS: root mean square

Conclusions
The Sturm-Pastyr localizer was originally intended for use with a medical image that is parallel to the base of the stereotactic frame, as depicted in Figure 2, wherein vertical rod is perpendicular to the tomographic section. Obtaining such a parallel image is difficult because it requires precise alignment of the patient. The equations presented in the Appendix extend this localizer for use with a medical image that is not parallel to the base of the stereotactic frame. But these equations cannot surmount the V-shape of the Sturm-Pastyr localizer that hampers its accuracy for a non-parallel image. And, even for a parallel image, the accuracy of this localizer degrades substantially near the apex of the V, i.e., near the base of the stereotactic frame. This decreased accuracy may hinder the effectiveness of the Sturm-Pastyr localizer for targets deep in the brain, e.g., for functional neurosurgery of the basal ganglia or for insertion of deep brain stimulation implants.
In contrast to the Sturm-Pastyr localizer, the N-localizer is intended for use with a medical image that is not perforce parallel to the base of the stereotactic frame. Hence, there is no requirement to precisely align the patient to obtain a parallel image. In fact, the accuracy of the N-localizer increases for a non-parallel image. And for either parallel or non-parallel images, the N-localizer is more accurate than the Sturm-Pastyr localizer. An additional advantage of the N-localizer compared to the Sturm-Pastyr localizer is that the N-localizer does not require specification of the pixel size for a medical image.

Appendices
The Sturm-Pastyr localizer is designed to provide the -coordinate when vertical rod of the localizer is perpendicular to the tomographic section, i.e., when the tomographic section is parallel to the base of the stereotactic frame. This idealized case is depicted in Figure 2 but not in Figure 7. In the idealized case, and because angles and shown in Figure 7 are both degrees [6]. However, achieving the idealized case is impractical due to the difficulty of precisely aligning the patient such that the tomographic section is perpendicular to vertical rod . Moreover, image noise perturbs the distances and such that even if the patient is precisely aligned. For these reasons, Dai et al. have derived equations that permit calculation of from and when the tomographic section is not perpendicular to vertical rod [6].

FIGURE 7: Depiction of the Sturm-Pastyr Localizer
The Sturm-Pastyr localizer is depicted by rods , , and that intersect the tomographic section at fiducial points , , and . The tomographic section is tilted by degrees. The distance between points and is . The distance between points and is . The distance between points and is . Because angle is a constant for the Sturm-Pastyr localizer, i.e., degrees [6], angles , , , and are functions of only angle , e.g., angle . Application of the law of sines to triangle of Figure 7 yields

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Human subjects: All authors have confirmed that this study did not involve human participants or tissue. Animal subjects: All authors have confirmed that this study did not involve animal subjects or tissue. Conflicts of interest: In compliance with the ICMJE uniform disclosure form, all authors declare the following: Payment/services info: All authors have declared that no financial support was received from any organization for the submitted work. Financial relationships: All authors have declared that they have no financial relationships at present or within the previous three years with any organizations that might have an interest in the submitted work. Other relationships: All authors have declared that there are no other relationships or activities that could appear to have influenced the submitted work.