Abstract
It is imperative that image-guided intervention (IGI) systems provide accurate and precise navigation information to enable the user to trust the system and not place unwarranted confidence in the guidance capabilities of the system. Unfortunately, the actual error associated with the overall targeting capabilities of an IGI system is not readily known. Here we are primarily interested in the application of image-guided surgery in the context of renal interventions. We built a simulation pipeline to study the uncertainty propagation through an optically tracked IGI system to gain insight into the overall accuracy of the system. Our simulation pipeline models several stages, including stylus calibration, tool tracking, patient tracking, and image to patient registration. In the effort to realistically estimate tracking noise and user-associated fiducial localization error (FLE), we conducted several experiments using the optical tracking system. Our simulation suggested that a wider cone angle results in a more accurate tool calibration, which improves further with the collection of additional samples. Furthermore, our simulations also suggested that the image-to-patient registration was the most significant contributor to navigation uncertainty, followed by the fiducial localization error. Lastly, we also observed a 0.72 correlation between the Target Registration Error (TRE) estimated at target fiducials and the distance between the the centroids of the registration and target fiducial landmarks. To validate the simulation predictions, we also conducted several in vitro experiments using a 3D printed patient specific kidney phantom and compared the simulation-based registration predictions with those observed experimentally in vitro. The experiments confirmed the registration metrics (Fiducial Registration Error and TRE) predicted by the simulations, given several specific combinations of fiducial landmarks used to perform the image to patient registration.
Keywords: Image-guided Navigation, Error Analysis, Simulation, Registration
1. INTRODUCTION
Image-guided interventions (IGIs) rely on the integration of pre- and intra-procedural imaging and surgical tracking, as well as the co-registration of the information from these multiple coordinate spaces to the physical patient in order to provide sufficient guidance to reach a desired target. It is imperative that these surgical guidance systems provide accurate and precise navigation information to enable the user to have confidence in the guidance capabilities of the system. Unfortunately, the actual error associated with the overall targeting capabilities of an IGI system is not readily known. Additionally, this error needs to be quantified and presented to the user in a way that facilitates correct, intuitive decision making with respect to the capabilities of the system and its accuracy and precision limitations.
Specifically, we are interested in studying the overall tracking uncertainty and error propagation from surgical tool calibration to the image-to-physical world registration and targeting through simulation and experiments. We are also interested in considerations and limitations users should be aware of when selecting registration fiducials, including appropriate and intuitive error display.
Research has shown how registration impacts the accuracy of surgical guidance and navigation2 and suggested potential ways to visualize the uncertainty.3,4 Moreover, Fitzpatrick et al.1 showed that the fiducial registration error (FRE) and target registration error (TRE) are uncorrelated. This implies that the measured FRE, a commonly used metric in image registration, does not generally predict TRE and can mislead users about the true accuracy of registration, however, it can be used as a checkpoint to ensure data is behaving as expected. Thus, this study uses FRE and TRE correlation as a checkpoint to ensure reliability and relevance of simulation results against in vitro validation results, while displaying the effect of registration on uncertainty using alternative visualizations.
2. METHODOLOGY
To gain a better understanding of the error propagation through an NDI Polaris Spectra optical IGI pipeline, we developed a simulation tool which allowed us to change various parameters and gain insight into their effect on the different stages of the IGI pipeline, from tool calibration to image-to-patient (i.e.virtual-to-physical world) registration. As we are primarily interested in the application of image-guided surgery in the context of renal interventions, we used a 3D printed kidney phantom as a physical model to perform validation experiments to validate the simulation results. All experimental data was collected using the IGT extension for 3D Slicer.5,6 The 3D printed kidney model features 14 fiducials, which were used to define the phantom in the simulation. This enabled us to analyze the registration data in different ways to learn more about how to intelligently select the most appropriate registration fiducials to optimize TRE at the region of interest, as opposed to being misled by the FRE.
2.1. Simulation Overview
The typical workflow associated with an image-guided intervention is shown in Fig. 1. Our simulation tool determines the transforms by adding uncertainty to each model component of the IGI workflow. Hence any location sampled in physical space can be transformed from physical space into model space using the series of transformations in Equation 1:
| (1) |
where Xtip is a 4×1 vector of [x, y, z, 1]T coordinates for a sampled position of the tool tip and Ma–b is a 4×4 matrix of the rotation and translation parameters which describes the transformation from b to a. The distance between the transformed point and actual point is the target registration error (TRE),7 a measure of registration performance.
Figure 1.
Overview of an image-guided tracking system component coordinate frames and transformation flow
As an example, this simulation tool allows us to assess the spatial distribution of the tip location of a tracked surgical instrument after calibration, while also taking into account tracking noise and variations associated with the fiducial localization error (FLE), as shown in Fig. 2.
Figure 2.
The 95th percentile confidence ellipsoids showing the uncertainty distribution associated with the location of the tip of the tracked tool at each stage (Tool, Tracker, Physical, and Model) in the IGI pipeline, it is reasonable for a user to assume that the tool tip will be within the corresponding ellipsoid 95% of the time. This shows the accumulation of uncertainty after each transformation.
2.2. Uncertainty Sources: Tracker Noise and Location Error
The uncertainty in optical tracking systems arises primarily from tracker read noise, as well as error associated with tool placement, i.e.fiducial localization error (FLE). To address the tracker noise, we recorded 10,000 data frames of a stationary stylus using an NDI Polaris Spectra optical tracker. We then computed a running average of 40 samples and subtracted the average from the raw samples to generate a noise sample using the following equation: , where i starts at 20 and goes to 9,980. The average subtracted samples were stored in a database to be used by the simulation. Moreover, to account for slight hand movements on the position of the tool (i.e., jitter), we recorded the pose information as well. Since this is a large sample set, it closely resembles the real distribution of tracker noise.
To model localization error, we used an isotropic Gaussian distribution in the x, y, and z direction with a single variance parameter. Since the surface normal changes at each point and therefore each point will have a slightly different amount of variability in each direction, using an isotropic distribution is a reasonable approximation. Nevertheless, since the tool tip is placed on a rigid surface in our experiments, the variability along the direction perpendicular to the surface is inherently reduced.
2.3. Pivot Calibration: MTool–Tip
The surgical instrument is first calibrated to determine the transformation MTip–Tool that describes the position of the tool tip relative to the retro-reflective markers on the dynamic reference frame (DRF) attached to the instrument. To determine the tool tip transformation, we used an algebraic formulation for the pivot calculation, as detailed in Yaniv.8 To generate the pivot calibration, we simulated the rotation of the tool around the tool’s longitudinal axis, mimicking the pivoting the tool at a given angle, then added randomly selected noise values to the points representing the tool from the noise database. Since it is a virtual DRF, we can control the variables number of tool poses recorded, tool tip location relative to the DRF (i.e., length of the tool), and angle to assess how distance from the tool tip to the tool-mounted DRFs affects the tool calibration.
2.4. Transformation Matrix Calculation MTracker–Tool & MTracker–Physical
The markers on the DRF, attached to the surgical instrument and also to the physical model (i.e., patient), are “read” by the tracker; the position of the tool and physical model relative to the tracker are reported as two transformations: MTool–Tracker and MTracker–Physical. To determine the tool-to-tracker transform, noise values are again randomly selected from the noise database, along with a jitter value added to the tool DRF model, and the transform calculated using a least squares fit solution. The simulation can also vary the number of readings used at this step, which enables is us to study the effect of multiple averaged readings as a means to reduce in the tool position.
To determine the tracker-to-physical space transformation, we assumed an anisotropic Gaussian distribution in the x, y, and z directions for the uncertainty.9 The standard deviations are based on the distribution of the noise samples collected using the stylus. Jitter is not added to the reference frame, since it is rigidly attached to the body.
2.5. Physical to Model Space Registration Transformation MPhysical–Model
The registration transformation between the physical and model space (MPhysical–Model) is determined by sampling fiducial locations in physical space that correspond to landmarks in the virtual / image space. This transformation includes all the errors and uncertainties that were introduced in the previous steps and additionally incorporates FLE. We also calculate the FRE as a reference value.
To estimate fiducial localization error (FLE), we sampled a fiducial landmark on the surface of the kidney phantom 100 times using the tracked pointer and determined the variance in the x, y, z directions. We repeated the experiment for several other fiducial landmarks on the kidney phantom.
2.6. Experimental Validation of Simulation Results
Using our simulation pipeline, including pointer calibration with measured tracker noise and FLE uncertainty, we computed the FRE and TRE achieved using all combinations of six landmarks (out of the total of fourteen surface landmarks) for physical to image space registration. We then selected the sets of six landmarks that yielded the highest TRE (i.e., worst registration) and lowest TRE (i.e., best registration) according to the simulation results, and conducted the same registration experimentally, in vitro, using the 3D printed kidney phantom, then assessed the experimental FRE and TRE for these registration cases. Each experimental registration was conducted nine times for each set of six registration fiducials.
One objective was to assess whether there was a correlation between TRE and the distance between the centroid of the registration fiducials and the target fiducial(s). To assess these effects, we used simulated fiducial landmarks both on the kidney surface not used for the registration, as well as sub-surface landmarks. The TRE associated with the targets inside the kidney cannot be experimentally determined, but we could verify the trend based on the TRE associated with the surface target landmarks. Additionally, to study the effect of the choice of the registration fiducials on the TRE achieved in a specific region of interest, we selected sub-surface kidney target landmarks that were closer to the centroid of the registration landmarks than the kidney surface landmarks, therefore mimicking both a proximal and distal region of interest.
3. RESULTS
The noise from the tracking system estimated using experimental data is shown in Table 1. The “Tracker Noise” entry is an estimate from the 10,000 data points recorded using a rigid stylus. This error is present in all measurements, but it is small.
Table 1.
Localization error from stationary Dynamic Reference Frame and placing tool tip on several targets.
| Tracker Noise Error (mm) | |||
|---|---|---|---|
| σx | σy | σz | σd |
| 0.01 | 0.01 | 0.04 | 0.02 |
| Fiducial Localization Error (mm) | ||||
|---|---|---|---|---|
| σx | σy | σz | σd | |
| Flat Target | 0.12 | 0.72 | 0.65 | 0.46 |
| Divot A | 0.42 | 0.20 | 0.34 | 0.45 |
| Divot B | 0.27 | 0.13 | 0.18 | 0.19 |
| Divot C | 0.07 | 0.17 | 0.23 | 0.19 |
The “Flat Target” entry is an estimate of the error associated with a user targeting a fixed point in space i.e., a measure of the FLE. The smaller error in the x direction for the “Flat Target” is a consequence of the target being located on a flat surface, perpendicular to the x-axis of the tracking system, and hence serves as a critical checkpoint for our experiment.
The “Divot A, B, and C” entries represent the FLE variance at three landmarks located on the kidney surface. The differences in the divot error arise primarily from 3D printing imperfections across different divots. Visually divots B and C seemed to have a smoother “fit” and 3D printing characteristics than divot A. The directional standard deviation (σx, σy, σz) represents inherent 3D printing uncertainties along with the greater uncertainty in the plane perpendicular to the surface normal, while σd represents the standard deviation of the overall error.
Using the simulation tool, we calculated the FRE and TRE associated with all 3003 possible combinations of six (out of fourteen) registration landmarks. In Fig. 3 we show a plot for all combinations of TRE vs. FRE. This agrees with the findings of Fitzpatrick1 that FRE and TRE are uncorrelated.
Figure 3.
Target Registration Error versus Fiducial Registration Error showing no correlation (r = 0.01)
The instrument calibration simulation (Fig. 4) indicated that a wider cone angle and additionally sampled poses reduce the overall instrument calibration error. Although the calibration error is not particularly large, it can be significantly reduced (by as much as 50%) by describing a wider cone and increasing the number of sample poses to at least 200.
Figure 4.
Localization error after pivot calibration as a function of the cone angle (30°, 40°, 50°, and 60°) and number of tool poses used during calibration.
Fig. 5 shows the probability of the tip of a tracked instrument being within a certain distance from a desired target during image-guided navigation using this platform. This plot is similar to Fig. 2, but shows the one-dimensional probability. It should be noted the probability is not isotropic, but will follow some distribution as in Fig. 2. The plot shows how the cumulative distribution function (CDF) (i.e., the probability) is affected by each stage of the pipeline, specifically tool calibration (Tool), tracking noise (Tracker), tracking noise from physical space (Physical), and FLE and registration error (Model). Fig. 5 illustrates the propagation of uncertainty associated with tracked tool navigation. The most significant contributors are fiducial localization and the physical to image/model space registration, which is portrayed here for both best and worst case registrations. Another method of using this plot is to determine if the probability of a maximum distance from a target is sufficient to proceed with the procedure or if the registration needs to be adjusted.
Figure 5.
Cumulative Distribution Function for the probability that the tip will be reported at a given distance from the target. The CDF for each stage as well as the model space CDF for the fiducial combination giving the best and worst TRE values for a specific target is shown.
Table 2 shows the two best and two worst TRE values at three levels of assumed localization error. The amount of localization error is an indication of how far the tool is from the tracker. Our experimental results of measuring the TRE compared to the simulation results showed similar trends. However, the error from the second worst case combination was greater than the predicted worst case in the experimental data. This is likely due to higher variance in the experimentally obtained samples, due to user error and a relatively small sample size. We also conducted simulations with different amounts of localization error (σ), which is a indirect measure of the distance of a tool from the tracking system. This was compared to the experimental results to get an approximation of the amount of localization error present. After several iterations with different values of fiducial localization variance (σ), we found that a σ = 0.5 yielded TRE values which correspond closest to the experimental TRE values. This could be further refined with additional iterations and more experimental data, but it does demonstrate effect FLE has on the overall uncertainty. However, since FLE is dependent upon the working distance from the tracker, it varies based on experimental setup.
Table 2.
Comparison of simulated and experimentally recorded maximum and minimum predicted TRE values from different registration fiducial combinations. σ is the standard deviation of the FLE.
| TRE Prediction Combination: | RMS Target Registration Error (mm) | |||
|---|---|---|---|---|
| Worst | 2nd Worst | 2nd Best | Best | |
| Simulation (FLE σ=0.1) | 0.23 | 0.21 | 0.14 | 0.14 |
| Simulation (FLE σ=0.5) | 0.88 | 0.77 | 0.30 | 0.30 |
| Simulation (FLE σ=1.0) | 1.78 | 1.50 | 0.55 | 0.55 |
| Experimental | 0.97 | 1.55 | 0.57 | 0.58 |
We predicted that the best combination of fiducials to yield a minimum TRE at a target point should be the one whose centroid is near the target point, while the worst would be a combination with a centroid that is displaced significantly. We plotted TRE versus the distance between the centroid of the registration fiducials and the centroid of target fiducials (Fig. 6), which shows correlation in how far they are from each other, corroborating our hypothesis. This makes sense for a least squares solution, as the location with the lowest error is the center of rotation. Therefore, while FRE does not predict TRE, the location of the fiducials can provide insight into TRE.
Figure 6.
TRE vs. distance from the centroid of the registration fiducials to the target fiducials showing improved registration at the targets located within a region of interest proximal to the registration fiducials (0.72 correlation).
4. CONCLUSION
This work addresses several challenges users should be aware of when employing image-guided navigation systems, and provides several “lessons learned” to help the user reduce uncertainty. As such, this study can serve as a baseline for applications relying on image-guided navigation.
Our simulations confirmed there is no correlation between TRE and FRE. We found that larger cone angles and increasing the number of sample poses reduces instrument calibration error. A comparison between simulation and experimental registration showed agreement between the achieved TRE for both best- and worst-case registration scenarios, using an estimated FLE standard deviation of 0.5 mm, which closely matches the experimentally estimated FLE standard deviation of 0.46 mm.
The simulation indicates that the best registration fiducials for a specific target should have a centroid as close as possible to the centroid of the target(s) which agrees with previous work.10 Similarly, for tools it is advised to keep the distance from DRF markers to tool tip as short as possible since error increases with distance from the rotation point or centroid. The experimental data for best and worst case TRE show agreement with the simulation predictions, and an even stronger agreement is expected, provided a larger experimental sample size is available.
Future work entails the incorporation of non-rigid phantoms with non-rigid registration and understanding how the error changes with deformation. Moreover, additional experiments involving several users and additional measurements would help observe and better understand the statistical variance across operators. Lastly, the use of phantoms equipped with accessible interior targets would enable us to experimentally validate the trends observed when experimentally assessing TRE at locations within the kidney phantom.
ACKNOWLEDGMENTS
Research reported in this publication was supported in part by the National Institute of General Medical Sciences of the National Institutes of Health under Award No. R35GM128877
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