Abstract
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.
Keywords: stereotactic surgery, stereotactic radiosurgery, computed tomography, magnetic resonance imaging, n-localizer, sturm-pastyr localizer, monte carlo, image-guided surgery, image-guided radiosurgery, deep brain stimulation
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 three-dimensional 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].
 |
In this equation, 
and 
are distances measured in the 
coordinate system of the medical image, 
is the 
-coordinate of the top of rod 
, and 
is the 
-coordinate of the bottom of rod 
. The numeric values for 
and 
are established by the manufacturing specifications for the N-localizer. The fraction 
is dimensionless, and hence the units of 
are the units of 
and 
that are specified by the manufacturer. For this reason, calculations for the N-localizer do not require the specification of the pixel size for the medical image [3,4].
Figure 1. The N-Localizer and its Intersection with a 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).
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 
.
3. The 
pair is used to calculate the 
, 
, and 
coordinates of the fiducial points 
, 
, and 
, respectively, in millimeters.
4. The 
, 
, and 
coordinates are perturbed via random numbers [7,8] in the range 
mm via 
iterations to create 
sets of perturbed 
, 
, and 
coordinates, where the superscript 
designates the 
-th perturbed coordinate.
5. Each set of perturbed 
, 
, and 
coordinates is used to calculate a set of perturbed distances 
, 
, and 
via the Pythagorean distance equation.
6. Each set of perturbed distances 
, 
, and 
is used to calculate a perturbed 
-coordinate.
7. The 
perturbed 
-coordinates are used to calculate the root mean square (RMS) error 
.
8. A new 
pair is chosen and steps 3-7 are repeated.
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 tomographic section, for which 
, 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.
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 reveals that triangles 
and 
are both right triangles, thus
 |
Hence, the 
, 
, and 
coordinates of the fiducial points 
, 
, and 
are
 |
Steps 4-7 of the Monte Carlo algorithm then proceed as follows. The 
, 
, and 
coordinates of the fiducial points 
, 
, and 
are perturbed 
times by random numbers to obtain 
perturbed 
, 
, and 
coordinates, from which 
perturbed distances 
and 
are calculated, from which 
perturbed 
-coordinates are calculated via Equation (1) and used to calculate the RMS error 
. Then a new 
pair is chosen and steps 3-7 of the Monte Carlo algorithm are repeated.
Monte Carlo simulation for the Sturm-Pastyr localizer
For step 3 of the Monte Carlo algorithm applied to the Sturm-Pastyr localizer, calculation of the 
, 
, and 
coordinates for a 
pair begins with calculation of the distances 
and 
. For this calculation, Equations (A1, A2) of the Appendix are solved for 
and 
to obtain
 |
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
 |
Steps 4-7 of the Monte Carlo algorithm then proceed as follows. The 
, 
, and 
coordinates of the fiducial points 
, 
, and 
are perturbed 
times by random numbers to obtain 
perturbed 
, 
, and 
coordinates, from which 
perturbed distances 
and 
are calculated, from which 
perturbed 
-coordinates are calculated via Equation (2) and used to calculate the RMS error 
. Then a new 
pair is chosen and steps 3-7 of the Monte Carlo algorithm are repeated.
Discussion
Figure 4 shows the results of Monte Carlo simulation for the N-localizer and the Sturm-Pastyr localizer. The RMS error in 
for the Sturm-Pastyr localizer approaches the smaller RMS error for the N-localizer at only large values of 
and tilt angle 
. For all other values of 
and 
, the Sturm-Pastyr localizer incurs significantly more RMS error than the N-localizer.
Figure 4. RMS Error in 
Plotted vs. 
for the N-Localizer and 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. This figure plots the RMS and maximum errors for both localizers at 
mm and 
degrees vs. the maximum perturbation for the following continuous ranges of white noise: 
, 
, 
, 
, and 
mm. The RMS and maximum errors for the N-localizer scale linearly with the maximum perturbation: the slope and correlation coefficient of a linear least-squares fit to the RMS-error data are 0.76 and 0.999991, respectively; the slope and correlation coefficient of a linear least-squares fit to the maximum-error data are 2.21 and 0.9998, respectively. As can be appreciated from Figure 6, the RMS and maximum errors for the Sturm-Pastyr localizer scale slightly super-linearly, as demonstrated by the slight upward concavity of the Sturm-Pastyr curves. The combination 
mm and 
degrees pertains to a medical image that is obtained near the base of the stereotactic frame and almost parallel to the base of the frame. Such an image would be acquired for functional neurosurgery of the basal ganglia or for the insertion of deep brain stimulation implants.
Figure 6. RMS and Maximum Errors vs. Maximum 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.
Acknowledgments
The authors thank John A. Robinson for helpful comments.
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 
.
Understanding the derivation of Dai, et al. requires familiarity with only trigonometry and algebra. However, because Dai, et al. omitted several intermediate steps from their derivation, it is unnecessarily obscure. The intermediate steps are provided below and in addition, the result reported by Dai, et al. is extended to yield an expression that contains no trigonometric functions.
The derivation of Dai, et al. produces an equation for the angle 
in terms of the distances 
and 
as follows.
Application of the law of sines to triangle 
of Figure 7 yields
 |
In this equation, 
and 
.
Similarly, application of the law of sines to triangle 
yields
 |
In this equation, 
and 
.
Dividing Equation (A1) by Equation (A2) eliminates 
and 
to yield
 |
Applying the rule for the sine of the sum of two angles followed by the rule for the cosine of the sum of two angles produces
 |
Cross multiplication by the denominators and then factoring in 
and 
yields
 |
Substituting 
into Equation (A5) yields
 |
Solving Equation (A6) for 
completes the derivation of Dai, et al [6].
 |
Solving Equation (A1) for 
yields
 |
Computing 
via Equation (A7) and substituting 
into Equation (A8) calculates 
in terms of 
and 
when the tomographic section is not perpendicular to vertical rod 
.
An expression for 
in terms 
and 
that does not involve trigonometric functions is obtained by first constructing expressions for 
and 
via inspection of Equation (A6)
 |
Applying the rules for the sine and cosine of the sum of two angles to Equation (A8) yields
 |
Substituting 
into Equation (A10) yields
 |
Substituting Equation (A9) into Equation (A11) yields a non-linear expression for 
in terms 
and 
that does not involve trigonometric functions [5]
 |
When 
, Equation (A12) reduces to Equation (3).
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Consent was obtained by all participants in this study
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