## Abstract

### Purpose

The accuracy of pedicle screw placement during image-guided spine surgery (IGSS) can be characterized by estimating the target registration error (TRE). The major factors that influence TRE were identified, minimized, and verified with in vitro experiments.

### Materials and methods

Computed-tomography-compatible markers are placed over anatomical landmarks of lumbar vertebral segments in locations that are feasible and routinely used in surgical procedures. TRE was determined directly for markers placed on the pedicles of vertebra segments. First, optimum selections of landmarks are proposed for different landmarks according to the minimum achievable TRE values in different configurations. These anatomical landmarks are feasible and accessible to overcome constraints that may be imposed during surgical procedures. Second, the effect of fiducial weighting on corresponding points to overcome anisotropic localization error based on maximum likelihood approach is evaluated. Third, an experimental model for fiducial localization error (FLE) is derived to obtain the weights. At the end, an error zone was obtained for each marker to indicate the possible acceptable deviation from the marker’s exact location in practice. This study was performed in vitro on a spine phantom.

### Results

Optimal landmark selection led to a 30 % reduction in TRE. In addition, optimum weighting of the fiducials in an FLE model that incorporates anisotropic localization error in the registration algorithm led to a 28 % reduction in the TRE.

### Conclusion

Landmark configuration, transformation parameters, and fiducial localization error are factors that significantly affect the total TRE. These factors should be optimized to minimize the TRE. Both the optimum configuration of landmarks and the anisotropic weighing of fiducials have significant impact on the registration accuracy for IGSS.

## Introduction

Image-guided interventions have increased the accuracy of surgical procedures to a great extent. A key step in image-guided interventions is the registration of preoperative images to the intra-operative physical anatomy of the patient, which can be performed by matching corresponding landmarks in both spaces. Two types of landmarks can be used for aligning the two coordinate systems. Fiducial markers which are attached to specified locations on the anatomy, and anatomical landmarks which are distinct points on the anatomy. In neurosurgery, the alignment of the two coordinate systems can be done by using a combination of anatomical landmarks and attached markers to the head before imaging and throughout the procedure. A rigid registration is performed to bring the sets of corresponding points into one coordinate system. In spine surgery, anatomical bony landmarks are used for initially aligning the two point sets. The accuracy of pedicle screw placement in spine surgery is of great importance due to the crucial neurovascular structures surrounding the spine [1]. Image-guided spine surgery (IGSS) has improved this accuracy to a great extent.

Several studies have been performed to determine the effect of fiducial configuration on different surgical applications. This effect has been widely investigated in neurosurgery. West et al. considered two clinical situations, including cranial neurosurgery and pedicle screw placement. They applied theoretical results of target registration error (TRE) prediction to three configurations for cranial neurosurgery and a common configuration for pedicle screw placement. They proposed that the fiducial configuration significantly affected the TRE [2]. Shamir et al. [3] proposed optimal selection of anatomical landmarks and placement of fiducials for minimum TRE in image-guided neurosurgery. They also performed a study on localization and registration accuracy in image-guided neurosurgery [4]. In another study, they proposed new methods for optimal selection of anatomical landmarks and placement of fiducial markers based on empirical simulation-based TRE estimation built on actual fiducial localization error (FLE) data [5]. Wang and Song performed three studies on improving the accuracy of TRE, proposing guidelines, and distributing templates for fiducial placement in image-guided neurosurgery [6–8]. The effect of landmark configuration on TRE in craniomaxillofacial surgery was investigated in [9].

TRE is used to assess the accuracy of image-guided interventions. Because the target is usually deep within the anatomy, TRE cannot be directly measured in clinical procedures, and hence, its estimation is required. Fiducial registration error (FRE), which indicates the registration accuracy of fiducials, is not a good estimation of TRE. As shown in [10], these two quantities are not correlated. Therefore, TRE estimation methods have been proposed to provide the surgeon with a quantitative measure of the navigation system’s accuracy in the anatomical target location. Fitzpatrick presented an estimation of TRE that considers FLE with isotropic distribution [11]. Moghari and Abolmaesumi proposed the maximum likelihood approach for TRE estimation that takes into account anisotropic noise for FLE [12]. They also gave a report on the comparison of different TRE estimation methods. They indicated that all algorithms derive the same distribution of TRE at a target location when FLE has identical and isotropic zero-mean Gaussian noise [13].

Several studies indicate that the 3 main factors—fiducial configuration relative to the target position, accuracy of rigid registration transformation obtained by matching fiducial pairs in both sets, and value of FLE associated with each fiducial—affect the total TRE value significantly.

Estimating the transformation parameters from the rigid registration can be performed by minimizing the RMS distance between the corresponding fiducials in both point sets. These least square problems have closed-form solutions [14–17]. These solutions ignore localization anisotropy and are available only for isotropic weighting [18]. Another alternative for the least squares problem is the maximum likelihood approach, which derives an optimum estimation of the transformation parameters by incorporating the anisotropy in fiducial localization [19, 20]. This maximum likelihood approach introduces a weighing to the RMS distance of point pairs known as ideal weighting. Minimizing this weighted RMS distance for fiducial points results in optimum parameters for the rigid registration transformation. An iterative solution for solving the rigid point-based registration with anisotropic weighting was introduced by Balachandran and Fitzpatrick [18], which is used in this study. This weighting has been used to derive the anisotropic iterative closest point algorithm (A-ICP) [21].

TRE is highly affected by the configuration of landmarks used in point-based registration to align the preoperative images and the intra-operative physical space. Therefore, determining an optimal configuration of landmarks to minimize the TRE is crucial in image-guided interventions, including IGSS. In this study, we investigated the effect of different landmarks and their configurations on the TRE for a vertebra in IGSS. To the best of our knowledge, this is the first study to investigate the effect of different anatomical landmark configurations on vertebra. This study introduces a framework to select the optimal number and configuration of points to obtain the ground truth transformation for vertebra registration. The gold standard can be obtained by placing markers on the proposed anatomical landmarks.

As mentioned by Shamir et al. [22], four factors—accurate localization of fiducials and targets in preoperative images and intra-operative physical anatomy, accurate rigid registration transformation, and accuracy of the tracking system—affect the accuracy of image-guided systems. To ensure an optimum framework and minimization of these effects, we used markers on anatomical landmarks and target locations to minimize the localization errors. However, to compensate for the anisotropy in the FLE imposed by the tracker to the acquired physical points, a weighted RMS distance between corresponding points was used to estimate the transformation parameters. We also used an experimental model for FLE that considers the anisotropy imposed on the physical points acquired by the tracker. We measured the FRE, estimated TRE using Fitzpatrick’s TRE estimation formula (FTRE), estimated TRE using Danilchenko’s method (DTRE), and TRE for unweighted and anisotropic weighted fiducial points (MTRE and MTREa).

## Materials and methods

### Setup for phantom and markers

For the registration process, it is necessary to match corresponding points in the CT space and physical space of the vertebra. Moreover, these markers are necessary for the gold standard. Therefore, having some special markers available both in CT images and in real space is crucial. To perform this, we designed and manufactured some particular markers and used them to mark each vertebra in the spine phantom. The design of this marker (Fig. 1a, b) was crucial because it could directly affect the registration accuracy and hence the whole test results. The marker comprises a cross with a hole in its center, which is mounted on a circular base. This special design not only makes the marker traceable in CT images but also simplifies the determination of its exact center both in CT images and in real space. The center of the marker is available in CT images as the junction of two lines of the cross, and the small hole in the center of the cross guides the pointer toward the center of the marker in real space.

The CT compatibility of the material used in the production of the phantom is essential. For the markers, we used VeroWhite-FullCure830 material and rapid prototyping method to maintain the CT compatibility and accuracy of the markers.

A set of thirteen markers were placed on each vertebra; of these, nine were placed on the posterior surface of the vertebra and were used for registration purpose and four were placed on the body and were used for calculating the TRE for evaluating the registration accuracy. In this study, one of the markers on the pedicle was used as the target in order to evaluate the accuracy in pedicle screw placement in image-guided spine surgery. The posterior surface markers were placed on anatomical landmarks, including the tip of the spinous process, the left and right transverse processes, the left and right superior and inferior articular processes, the junctions of the lamina, and the side of the spinous process. The markers attached to the vertebrae are shown in Fig. 1e.

### CT imaging

The CT scans were obtained using the preoperative spine neurosurgery protocol at the Imam Khomeini Complex Hospital. Axial slices were acquired with a voxel size of \(0.625 \times 0.625 \times 0.625\,\text{ mm }^{3}\).

### Physical surface and CT point acquisition

Surface points were acquired by touching each point using a pointer equipped with camera-detectable markers. Micron tracker (an optical tracker) detects the checker board markers installed on the pointer and gives the coordinate of the tip of the pointer that indicates the exact location of the point touched by the pointer. Parsiss\(^{\copyright }\) Image Vision^{Footnote 1} navigation system was used as an interface between the tracker and the computer to record the locations of the points.

The exact locations of these fiducials in the 3D reconstructed surface from CT images were acquired using Image Vision navigation system (Fig. 2a). Their location in the world coordinate was acquired by pointing a tracked pointer tip to the center of these markers (Fig. 2b).

### Point registration

FLE has a direct impact on FRE. Hence, incorporating FLE in FRE computation can result in more accurate registration. However, this can be performed by associating anisotropic weights to each fiducial used in the registration.

The weighted RMS distance between corresponding points in the two point sets was minimized to obtain the rigid registration parameters. The weighting, which provides maximum likelihood solution [23] for solving the rigid registration problem, takes into consideration the anisotropy and inhomogeneity of the FLE in both point sets.

The problem of rigid fiducial registration is to find the rotation (R) and translation (t) matrices that minimize this equation.

This can be rewritten as

where

where \(\sum _{Xi}\) and \(\sum _{Yi}\) are the covariance matrices associated with each point in the two point sets to be registered [23].

Transformation parameters were obtained by minimizing the RMS distance between selected fiducials once, by using uniform weighting followed by anisotropic weighting.

The covariance matrix used in the ideal weighting was calculated for the physical points as described in the “Results” section. The major anisotropy in the physical points is induced by the tracker as it experiences more localization uncertainty in the direction from the camera to the target compared to the two perpendicular directions. In addition, the use of the reference frame, which allows the motion of the patient, makes the anisotropy even worse [18].

FRE and TRE were calculated for marker-based registration with each set of transformation parameters obtained as discussed above. These parameters were applied to the intra-operative location of the markers to map them to their corresponding locations in the preoperative images. The measured TRE calculated using the transformation parameters obtained from uniform weighting is referred to as MTRE, and the one calculated using the transformation parameters obtained from anisotropic weighting is referred to as MTREa.

### FLE model

To obtain a reliable model for FLE, the anisotropy and inhomogeneity of the FLE distribution were taken into consideration [5]. Some studies have been performed to address these issues. In our study, we used a simple empirical model for FLE that considers the anisotropy and inhomogeneity of localization uncertainty for each point. The fiducial localization imprecision (FLI) for each point was calculated by computing the measured distances between repeated selections of the same landmark in 15 trials. The covariance matrix of the FLI (Sigma (FLI)) was then calculated by computing the variance of the measurements along each principle axis. Because of the specific structure of the markers used in this study, we found that the shape of Sigma (FLE) was similar to that of Sigma (FLI), which allowed FLI to be used in calculations instead of FLE.

### Landmark configurations

It can be inferred from Fitzpatrick’s equation that TRE not only is determined by the accuracy in locating the fiducials but also is influenced by their number and distribution. There have been recommendations for lower TRE values, including spreading the fiducial placement around the target, keeping the center of gravity more close to the target, employing more fiducials, and arranging fiducials nonlinearly. However, these hints are very general to follow and do not provide the surgeons or OR technicians with specified rules to achieve the best possible selection of fiducials.

We performed extensive experiments to consider different marker configurations on the posterior surface of a vertebra to estimate the TRE for pedicle screw insertion in which the target was placed on the pedicle of the vertebra.

Different sets of landmarks were considered; these selections differ in the number of landmarks and their arrangements. For each set, the TRE values were obtained directly by touching the targets using the camera’s detectable probe.

Four different optimum configurations were proposed, including 4, 5, 7, and 9 numbers of landmarks. They were chosen according to the above-mentioned rules for landmark selection. In addition, their feasibility and accessibility during surgical intervention were considered while selecting these sets.

## Results

In this study, a marker was placed in the pedicle of the vertebra as the target to estimate the registration accuracy in the pedicle screw insertion in IGSS.

We conducted three experiments to evaluate the different factors affecting TRE for the specified target in vertebra. (1) Covariance matrices for each point in physical space were acquired through an empirical model. (2) Transformation parameters for the rigid registration were obtained using uniform and anisotropic weighting. (3) Different configurations of anatomical landmarks were tested and optimum configurations were proposed.

### Calculating FLI covariance matrix for the acquired physical points

The FLI for each attached marker was calculated in the physical space from 15 trials. To calculate the covariance of FLI for indicating its anisotropy for each measured fiducial location in physical space, we defined the principle axis for each point and calculated the variance of FLI along that axis. Because the distance from the camera to the markers was much greater than the distance between the markers themselves, the principle axis for every fiducial was considered the same and was considered the coordinate system of the camera [18]. The covariance matrix for each point P was defined as \(\Sigma _p =V_p S_p^2 V_p^{\prime }\), where the columns of \(V_p\) are the principal axes of the localization imprecision and \(S_p =diag(\sigma _{p1} ,\sigma _{p2} ,\sigma _{p3})\) is a diagonal matrix, with \(\sigma _{pj}\) representing the standard deviation along the principal axis j [24].

To calculate the variance along each principle axis, the coordinates (x, y, z) of each point were measured in 15 trials. The average of the measured distance between repeated selections of the same landmark was calculated along each coordinate axis and the variance along each axis was calculated accordingly. This resulted in different variances corresponding to each point along each direction, which in fact has modeled the inhomogeneous and anisotropic noise in these physical point sets acquired by the camera.

These measurements indicated the FLI in each point; however, as already mentioned, because of the special structure and dimension of the markers (Fig. 1a, b) designed in this study, the FLI values were very close to the actual FLE values. This specific design has caused a small deviation of maximum 0.3 mm for the tool tip movement during different measurements (Fig. 1c, d), thus enabling us to safely replace FLE values with FLIs in TRE equations.

### Effect of fiducial weighting on TRE

The corresponding point registration problem (finding parameters R and T) was performed using the weighted FRE in equation (1), which was solved using the iterative solution proposed by Balachandran and Fitzpatrick [18]. This accounted for the anisotropic and inhomogeneous noise in the FLE in the physical points acquired by the tracker, which was not considered previously in the closed-form solution of the least squares approach. MTREa is calculated using these parameters. The estimated values of TRE are also obtained using Danilchenko’s algorithm (DTRE), which includes the anisotropy of the localization error. These results are indicated in Table 2. Because of the similarity in the shape of Sigma (FLE) and Sigma (FLI) as discussed in the previous section, Balachandran’s algorithm performed identically regardless of which Sigma was used.

The mean, standard deviation, and maximum value for the difference and the correlation coefficient of the 2 variables for each measuring method are calculated and indicated in Table 1. Figure 3 shows the histograms comparing the difference between Fitzpatrick’s estimation method (FTRE) and the 2 TRE measuring methods (MTRE and MTREa). The mean, standard deviation, and maximum value for the difference and the correlation coefficient by using Pearson correlation coefficient of the two variables for each measuring method were computed. The correlations are shown in Fig. 4. As indicated by the results, the correlation between TRE values obtained from Fitzpatrick’s estimation formula (FTRE) and the measured TRE obtained for transformation parameter estimation with uniform weighting of fiducials (MTRE) was higher than its correlation with TRE obtained from anisotropic weighting of fiducials (MTREa). This was expected because Fitzpatrick’s estimation formula does not incorporate the anisotropy of the fiducial localization. Thus, a TRE estimation method which incorporates the anisotropy of the points, like Danilchenko’s estimation, should be used.

### Effect of different landmark configurations on TRE

We considered different numbers and configurations of the attached markers on the posterior surface of a vertebra to estimate the TRE for pedicle screw insertion for which the target is in the pedicle of the vertebra.

Different sets of landmarks were considered; these selections differed in the number of landmarks and their arrangements. For each set, the TRE values were measured directly by touching the targets and by using the camera’s detectable probe. Four optimum sets of landmarks were proposed, including 4, 5, 7, and 9 combinations of points.

Table 2 illustrates an optimum selection of 4, 5, 7, and 9 anatomical landmarks according to the constraints mentioned above. It also indicates the values of the measured TRE obtained using uniform and anisotropic weightings and the values for the estimated TRE obtained using Danilchenko’s method.

The column 2 in Table 3 shows that the tip of the left transverse process is replaced by the left inferior articular process, which, as the result indicates, does not affect the registration accuracy significantly; moreover, because it is more feasible in clinical condition, it can be used instead. Column 3 shows a good distribution of landmarks around the target; however, because it does not cover the whole surface to be registered, it does not indicate a good initial alignment for surface registration.

### Markerless registration

It is quite obvious that markers used in this study are not available in practical situation. Thus, we performed several experiments to obtain an error zone corresponding to each marker that indicated the percentage increase in error by moving away from the exact location of the marker. This makes our results more consistent with practical conditions. Figure 5 shows the corresponding error region for one marker in which selecting a point in the blue region results in a 10 % increase in the TRE for the specified target. As it shows by moving away further from the marker, the TRE value increases accordingly. This process has been done for all nine markers to obtain the corresponding error zones. This error map could be very useful in navigation systems for guiding the surgeon to select the proper points inside the safe zone. The effect of a proper and an improper selection of one landmark (with the selected landmark inside and outside of the safe zone, respectively) is indicated in Table 4.

## Discussion

Our results showed that optimal anatomical landmark selection reduced the TRE by 30 % from 1.51 mm in the worst selection to 1.06 mm in the best selection.

However, association of anisotropic weights to fiducials in the registration process resulted in a 28 % reduction in TRE even further to 0.83 mm in the best selection.

The results indicate that when utilizing uniform weighing for calculating the transformation parameters, increasing the number of fiducials does not necessarily lead to a decrease in TRE. This is because the least square registration method assumes an isotropic and homogenous localization error for all fiducials, which is not the case in surgical navigation systems. As shown in the empirical method performed for calculating Sigma (FLI) for each point, which as discussed before is similar to Sigma (FLE), these values are anisotropic and differ from point to point. However, the use of more accurate registration methods, which incorporate the FLE’s anisotropy and inhomogeneity, can lead to a decrease in TRE.

In addition, the configurations of landmarks also play an important role in the TRE. As intuitively expected, a good configuration with possible small number of fiducials can lead to lower TRE than that observed with improper configuration with large number of fiducials. Based on our extensive experiences, it was observed that increasing the number of fiducial points not only lengthened the process of registration before performing surgery but also produced outliers in some cases.

Fiducial markers were attached to the anatomical landmarks and the target position before imaging. The exact location of the target was obtained in our study. However, these markers are not available in practice because of which, bony anatomical landmarks should be selected. Because the exact locations of these landmarks were not available practically, we obtained an error zone for each marker, which indicated the percentage increase in error by deviating from the exact location of the marker.

## Conclusion

We investigated the important factors that affect the TRE. This experimental study was performed on a vertebra phantom for a specific target placed in the pedicle of the vertebra for evaluating the TRE in pedicle screw insertion. The configuration and number of landmarks, and the accuracy of the rigid registration were studied. In addition, a FLE model was considered to incorporate the localization uncertainty of physical points.

To the best of our knowledge, this is the first study to evaluate the effect of landmark configuration on TRE on vertebra in spine surgery. We proposed optimum selections of landmarks for pedicle screw placement in IGSS. The landmarks were selected to obtain the best alignment of the vertebra while having the optimum TRE in the pedicle being feasible in clinical conditions. The proposed configurations can be easily adapted to clinical conditions during surgical procedures. The results showed a significant effect of the selected landmarks and their configuration on the TRE; moreover, the proposed configurations showed an improvement in the registration accuracy.

We experimentally measured the TRE for the specified target using 2 different sets of transformation parameters obtained from isotropic and anisotropic weighting of the points’ RMS distances. These values were compared with the TRE values obtained using Fitzpatrick’s and Danilchenko’s TRE estimation methods.

In this study, the general rules from Fitzpatrick’s TRE estimation for optimum landmark selection were used for the landmark configurations around a specified target.

The results showed that the accuracy of the rigid registration in finding the transformation parameters to match the corresponding points affected the TRE. Therefore, in navigation systems with optical tracking systems that experience anisotropic localization uncertainty along the camera’s depth direction, the maximum likelihood approach for solving the rigid registration problem as well as for employing the ideal weighting of fiducials for the weighted RMS distance instead of commonly used uniform weighting should be considered. It appears that the ML estimation has performed quite well among other estimation models. Because the ideal weighting incorporates the covariance matrices of FLEs in both point sets, an accurate model for FLE is required. The importance of fiducials’ configuration on TRE is also a well-known factor that should be considered for reducing the TRE. Overall, to achieve the minimum possible TRE value, a proper landmark configuration should be practically performed and a theoretically accurate rigid registration should be considered by incorporating FLE in the derivation of the transformation parameters.

## Notes

- 1.
Parseh Intelligent Surgical Systems, www.parsiss.com.

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## Acknowledgments

The authors would like to thank the reviewers for carefully reading the paper and providing valuable comments and suggestions. The first author would also like to thank Professor J. Michael Fitzpatrick for helpful communication.

### Conflict of Interest

Marzieh Ershad, Alireza Ahmadian, Nassim Dadashi Serej, Hooshang Saberi, and Keyvan Amini Khoiy declare that they have no conflict of interest

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Ershad, M., Ahmadian, A., Dadashi Serej, N. *et al.* Minimization of target registration error for vertebra in image-guided spine surgery.
*Int J CARS* **9, **29–38 (2014). https://doi.org/10.1007/s11548-013-0914-7

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### Keywords

- Target registration error
- Fiducial localization error
- Rigid registration
- Image-guided surgery