Reconstructing the nasal septum from instrument motion during septoplasty surgery

Abstract.

Purpose: Surgery involves modifying anatomy to achieve a goal. Reconstructing anatomy can facilitate surgical care through surgical planning, real-time decision support, or anticipating outcomes. Tool motion is a rich source of data that can be used to quantify anatomy. Our work develops and validates a method for reconstructing the nasal septum from unstructured motion of the Cottle elevator during the elevation phase of septoplasty surgery, without need to explicitly delineate the surface of the septum.

Approach: The proposed method uses iterative closest point registration to initially register a template septum to the tool motion. Subsequently, statistical shape modeling with iterative most likely oriented point registration is used to fit the reconstructed septum to Cottle tip position and orientation during flap elevation. Regularization of the shape model and transformation is incorporated. The proposed methods were validated on 10 septoplasty surgeries performed on cadavers by operators of varying experience level. Preoperative CT images of the cadaver septums were segmented as ground truth.

Results: We estimated reconstruction error as the difference between the projections of the Cottle tip onto the surface of the reconstructed septum and the ground-truth septum segmented from the CT image. We found translational differences of $2.74(2.06-2.81)\mathrm{mm}$ and a rotational differences of $8.95(7.11-10.55)\mathrm{deg}$ between the reconstructed septum and the ground-truth septum [median (interquartile range)], given the optimal regularization parameters.

Conclusions: Accurate reconstruction of the nasal septum can be achieved from tool tracking data during septoplasty surgery on cadavers. This enables understanding of the septal anatomy without need for traditional medical imaging. This result may be used to facilitate surgical planning, intraoperative care, or skills assessment.

1. Introduction

Surgery involves manipulating patient anatomy to achieve a clinical goal. In fact, many surgical procedures are indicated by pathological alterations of geometry of anatomical structures. For example, deformities in the nasal septum lead to difficulty breathing. Surgery to treat deformities involves correcting the geometry of the anatomical structures. It follows that the extent of deformity affects difficulty or complexity of the surgical intervention and influences its outcomes. Clearly, quantifying geometry of anatomical structures can have a significant impact on surgical care, for example, by facilitating surgical planning, supporting intraoperative decision-making, and predicting outcomes.¹

Reconstructing anatomy is necessary to quantify change in geometry due to surgery. Most research on reconstructing anatomy has relied upon videos of the surgical field; however, little research exists on reconstructing anatomy using instrument motion data. For the large part, during surgery, instruments constantly interact with anatomical surfaces, and thus the point cloud formed by the instrument tip positions is a rich source of data to reconstruct anatomy. In fact, instrument motion is known to be an effective data source to assess surgical skill.^2–4 Existing methods primarily reconstruct anatomy from a highly structured point clouds obtained in well controlled experimental settings. To our knowledge, there is no method developed to reconstruct geometry using unstructured point clouds obtained from routine surgical care in the operating room.

1.1. Anatomical Surface Reconstruction

Several previous works have attempted to reconstruct anatomical surfaces from point cloud data systematically sampled on the surface of the anatomy. These methods can be distinguished based on whether they use a priori statistical knowledge about the shape of the surface.⁵ Hoppe et al.⁶ proposed one of the first and most common methods on surface reconstruction without a prior. One of the earliest and most common methods for surface reconstruction with a prior is using statistical shape models by Cootes et al.⁷ A review of surface reconstruction techniques from structured or unstructured point clouds is provided by Lim and Haron.⁸

1.2. Nasal Septoplasty Surgery

Nasal septoplasty is a common head and neck surgery to correct a structural deviation in the nasal septum in patients presenting with difficulty in breathing.⁹ The shape of the septum alters geometry of the nasal airway, resulting airflow, and consequently causes patient symptoms. From a surgical perspective, septum shape affects technical complexity of the surgery, choice of surgical approach, and the likelihood that surgery improves patient outcomes.

Despite being critical for surgical care and outcomes, surgeons largely operate without adequate information on shape of the septum for routine septoplasty procedures. Although patients may undergo anterior rhinoscopy or nasal endoscopy in the clinic before surgery,¹⁰ these investigations do not allow full reconstruction of the nasal septum. Although CT imaging allows reconstruction of the nasal septum, it is not routinely indicated unless there is concomitant pathology such as chronic rhinosinusitis requiring surgery.^11,12 Furthermore, intraoperative endoscopy can allow video reconstruction of the septum shape but endoscopic surgery is also not routinely indicated or reimbursed for septoplasty. Finally, even when CT imaging or endoscopic video is available, they do not allow assessment of intraoperative evolution of the surgical procedure.

We suggest that the shape of the septum and its deviation can be understood by intraoperatively analyzing motion data from tracked instruments used in septoplasty surgery. More specifically, this work considers Cottle motion during the flap elevation phase of septoplasty. The flap elevation phase in septoplasty is a critical phase involving the separation of the mucosal flap from the nasal septum using the Cottle elevator. Our clinical experience indicates that this is one of the most difficult phases of septoplasty. Details on flap elevation are provided by Fettman et al.¹³

1.3. Objective

This work develops and validates a method for reconstructing the nasal septum from the motion of the surgical instrument tip during the elevation phase of septoplasty surgery. The proposed methods do not require the septum’s surface to be explicitly delineated but rather use tool motion data passively acquired during surgery. The proposed methods automatically recover geometry of anatomy in the surgical field, which informs disease severity, technical complexity, and extent of surgery as well as their association with postoperative outcomes.

2. Methods

2.1. Experimental Setup

We follow the same data collection protocol by Ahmidi et al.² Instruments were tracked during the surgery using an NDI Aurora electromagnetic tracking system (Northern Digital Inc.) with 6 DOF pose sensors (root-mean-square position accuracy 0.70 mm and orientation accuracy 0.30 deg). A sensor was affixed to the patient’s forehead as a reference; sensors were affixed to all applicable surgical instruments, including the Cottle elevator (Fig. 1) used during elevation of the mucosal flap. Sensors were affixed using specially designed instrument mounts to minimize the effects of electromagnetic field distortions due to the ferromagnetic instruments (Fig. 2).²

Fig. 1.

Photograph of a Cottle elevator used for flap elevation during septoplasty surgery. Both tips can be used for elevation.

Fig. 2.

(a) Photograph of ongoing tracked septoplasty surgery and (b) diagram of data collection setup.

Pivot calibration was performed separately for each tool, to determine the tool’s tip. Prior to each septoplasty, the operator used the Cottle to draw a circle around the patient’s nose and a line along the dorsum of the nose.

During surgery, the ongoing phase and instrument in use is manually annotated by a trained observer. These annotations are revised postoperatively by the same observer to ensure quality. In particular, the beginning and end of the nose circle, nose line, and flap elevation phase are delineated.

2.2. Fitting the Shape Model to Trajectory Data

The following steps provide a high-level overview of the proposed method for reconstructing the nasal septum (Fig. 3). Details of the method are presented in the subsequent sections.

Fig. 3.

Block diagram illustration of the proposed method for reconstructing the nasal septum.

• 1.Computation of the statistical shape model of the septum.
• 2.Initial alignment of the mean septum via iterative closest point registration.
• 3.Computation of the orientation of points on the instrument’s trajectory.
• 4.Combined registration and shape modeling of the septum using iterative most likely oriented point (IMLOP) registration with regularization.

2.2.1. Statistical shape model of the septum

We built a statistical shape model of the nasal septum following the work of Sinha et al.¹⁴ We used a pre-existing database of 50 CT images of the head. First, we generated a template CT image of the head by iteratively registering all of the images to a target image and computing the mean displacement field. The method is based on the ANTs software;¹⁵ it uses a non-linear deformation with cross correlation as the similarity metric. Subsequently, we segmented the septum of the template CT image and computed the segmented septums in all CT images by applying the displacement field to it. Finally, we generated a statistical shape model for the septum from the 50 segmented septums.

The number of modes of variation in our shape model of the septum $M$ is a parameter that may be adjusted. A total of 7140 points $r_i$ were sampled on the shape. We define $\overline{r}$ to be the mean shape, $E=[e_1,\dots ,e_M]$ to be eigenvectors of the shape model, and ${\lambda }_1,\dots ,{\lambda }_M$ to be the eigenvalues of the shape model.

2.2.2. Initial registration of the septum

As an initial step, we roughly register the template CT image to the reference sensor on the patient’s forehead in the following way [Eq. (1)]. First, in the template CT image, we manually select fiducial points evenly distributed around the nose on a circle ${\mathrm{NC}}^{\mathrm{temp}}$ and along the dorsum of the nose ${\mathrm{NL}}^{\mathrm{temp}}$ with sufficient density to capture local features (see Fig. 4). Subsequently, we register these fiducials to the Cottle tip trajectory during the drawing of the nose circle ${\mathrm{NC}}^{\mathrm{ref}}$ and nose line ${\mathrm{NL}}^{\mathrm{ref}}$ on the dorsum using a multi-part ICP method. In our multi-part ICP, we only permit correspondences between points on the nose circles and between points two nose lines on the dorsum (i.e., points on the nose circle are not permitted to correspond to points on the nose line). We perform this multi-part ICP over 312 initial rotations¹⁶ and select the registration with the smallest fiducial registration error. This provides an initial estimate of the pose of the nasal septum aligned with the Cottle tip motion during elevation in the reference sensor coordinate frame:

$$T_{\mathrm{init}}^{\mathrm{ref}\to \mathrm{temp}}=\mathrm{ICP}(({\mathrm{NC}}^{\mathrm{ref}},{\mathrm{NL}}^{\mathrm{ref}}),({\mathrm{NC}}^{\mathrm{temp}},{\mathrm{NL}}^{\mathrm{temp}})).$$ (1)

Fig. 4.

(a) Sagittal and (b) coronal views of fiducial points selected on nose circle (red) and nose line (green) on the dorsum in the template CT image.

2.2.3. Computing trajectory points orientation

We fit the shape model to the trajectory data from the Cottle tip during the elevation phase, where the start and stop times of the phase is manually annotated by the trained observer. Consider the sequence of tool trajectory points $p_i^{\mathrm{ref}}$, $i=1,\dots ,T$, as manually annotated by the trained observer. We computed orientation for trajectory points on the surface of the septum in the following ways.

• 1.Determine which points are associated with the surface of the septum.
• 2.Approximate the orientation for each trajectory point $p_i^{\mathrm{ref}}$ to associate it with one side of the septum.

First, we discard outlying points during elevation (e.g., points where the Cottle is outside the nasal cavity) via a nearest neighbor approach [Eq. (2)]. Those points $p_i^{\mathrm{ref}}$ whose mean distance ${\mathrm{MND}}_i$ to the nearest neighbors $N(p_i^{\mathrm{ref}})$ was more than one standard deviation ${\sigma }_{\mathrm{MND}}$ greater than the mean of the mean distances $\overline{\mathrm{MND}}$ to neighbors was removed. We used 12 nearest neighbors. This eliminates the sparse extraction motions observed outside the nasal cavity:

$$\begin{gathered} {\mathrm{MND}}_i=\frac{1}{|N(p_i^{\mathrm{ref}})|}\sum _{q\in N(p_i^{\mathrm{ref}})}|p_i^{\mathrm{ref}}-q| \\ {\mathrm{MND}}_i\{\begin{array}{l}<\overline{\mathrm{MND}}+{\sigma }_{\mathrm{MND}}\Rightarrow \text{keep}\\ >\overline{\mathrm{MND}}+{\sigma }_{\mathrm{MND}}\Rightarrow \text{discard}\end{array}. \end{gathered}$$ (2)

Second, to estimate the local septum normal at each point on the trajectory, we observe that the Cottle is rotated 180 deg about its axis when operating on the other side of the nose. We compute the direction vector the spoon tip is facing for each point $s_i$, and the signed orthogonal distance from each point $d_i$ to the initial estimate of the septal plane $\widehat{n}$, $x_0$ (see Sec. 2.2.2). We use a Gaussian mixture model on the combined orientation and distance to determine two clusters of poses. The mixture model is initialized based on the signs of the distances. We assume these clusters $c_i$ represent motion on the two sides of the septal plane. We assign these points to have orientation $v_i^{\mathrm{ref}}$ normal to the initially estimated septum:

$$\begin{gathered} c_i=\mathrm{GMM}([s_i,d_i]) \\ {v}_i^{\mathrm{ref}}=\{\begin{array}{ll}\widehat{n},& \text{if}{c}_i=0\\ -\widehat{n},& \text{if}{c}_i=1\end{array}. \end{gathered}$$ (3)

Additionally, we randomly resample the points on the trajectory $p_i$ with replacement to get an equal number of points on each side of the initial estimate of the septal plane. For each side, the number of points after resampling was equal to the number of points on the side of the initial septum originally with fewer points.

2.2.4. Combined registration and shape modeling

Once we have the oriented points within the nasal cavity from the trajectory, we seek to fit the septum shape model to these oriented points. To this end, we iteratively update the transformation $T_i^{\mathrm{ref}\to \mathrm{temp}}$ via the IMLOP registration protocol¹⁷ and update the shape model parameters, in the following ways (details provided subsequently).

• 1.Register the oriented septum surface with the oriented trajectory points via IMLOPs registration.
• 2.Fit the shape model to the trajectory points via active shape modeling.
• 3.Iterate.

We register the oriented septum surface $r_i^{\mathrm{temp}}$, $u_i^{\mathrm{temp}}$, where the orientation is taken as the surface normals, with the oriented trajectory points $p_i^{\mathrm{ref}}$, $v_i^{\mathrm{ref}}$, where orientation is computed as described already. We use the standard IMLOP registration [Eq. (4)]¹⁷ with inlier ratio ${\lambda }_i$:

$$T^{\mathrm{ref}\to \mathrm{temp}}=\mathrm{IMLOP}(\{[p_i^{\mathrm{ref}},v_i^{\mathrm{ref}}]\},\{[r_i^{\mathrm{temp}},u_i^{\mathrm{temp}}]\};{\lambda }_i).$$ (4)

We fit the septum shape model to the trajectory points using standard active shape modeling methods.^7,18 Correspondences between shape model points and trajectory points are determined from IMLOP registration. We clip the shape model weights to be within three standard deviations:¹⁹

$$\begin{gathered} b=E^T(T^{\mathrm{ref}\to \mathrm{temp}}{p}_i^{\mathrm{ref}}-{\overline{r}}^{\mathrm{temp}}) \\ \text{clip}:-3\sqrt{{\lambda }_j}\le b_j\le 3\sqrt{{\lambda }_j} \\ \begin{array}{c}{r}_i^{\mathrm{temp}}={\overline{r}}^{\mathrm{temp}}+Eb\end{array}. \end{gathered}$$ (5)

These two steps are performed iteratively until a stopping criteria is reached. We iterated until the change in the root-mean-square error (which is a weighted sum of position and orientation error) between corresponding points during the IMLOP phase dropped below a predefined threshold of 0.01 (Ref. 17) (Fig. 5).

Fig. 5.

Initial (blue) and full (green) reconstructions of the septum from Cottle tip trajectory, with the ground-truth septum (red) for the (a) axial, (b) sagittal, and (c) coronal views. Points along the trajectory with their estimated orientation are illustrated (cyan and black). Nose circle (yellow) and nose line (magenta) illustrated.

2.2.5. Regularization of the fitting

We hypothesize that our initial rough estimation of the septum could be an accurate fit. Thus we introduce two methods for regularization in our methods: (1) regularization of the IMLOP registration and (2) regularization of the shape model parameters.

To regularize the registration, we sample a small subset of oriented points on the septum surface. We add these points to the set of oriented trajectory points. Thus the IMLOP registration is biased toward the initial registration. We control regularization by tuning the proportion ${\lambda }_t$ of the sample used.

To regularize the shape model parameters, we use the standard L2 regularization (i.e., ridge regularization). This regularization is applied just prior to clipping. It biases the shape model toward the mean septum shape in the absence of otherwise compelling evidence. The shape model regularization is taken to be proportional to the dataset size and is controlled by tuning the regularization parameter ${\lambda }_r$:

$$b_j=\frac{b_j}{1+{\lambda }_r\frac{|r_i|}{|p_i|}}.$$ (6)

2.3. Validation Study on Cadaver Septums

We validated the accuracy of the proposed methods for septum reconstruction on a set of 12 cadaver septums. Participants performed septoplasty surgeries on cadaver heads using standard procedures over the course of 10 months at Johns Hopkins Hospital. Seven of the septoplasties were performed by two expert surgeons; five of the septoplasties were performed by medical or engineering personnel.

Prior to each septoplasty surgery, four to seven fiducial screws were affixed to the skull. Fiducials were placed at the temporal, frontal, and maxillary regions of the skull on both the left and right sides. Subsequently, a CT image of the cadaver head was acquired and a trained human manually segmented the nasal septum from the CT image. Prior to each septoplasty surgery, a tracked pointer tool was pivoted on each screw head to allow for rigid fiducial registration between the CT images and the reference tracker attached to the skull. By registering the segmented septum into the reference tracker coordinate frame, we get a ground-truth estimate of the nasal septum.

2.3.1. Measures of reconstruction accuracy

First, we compute the topological accuracy of the reconstruction. We project the Cottle tip at each timestamp onto the closest point on both the reconstructed septum and ground-truth septum. We filter out motions off the septum’s surface by a distance threshold. We report the distance between these projections (projection position differences) as well as the angle between the smoothed local normals at these projections (projection orientation differences). Large differences in angles ($>90\mathrm{deg}$) due to errors in translation are filtered out. Subsequently, we consider the overall accuracy of the pose of the reconstructed septum.

Next, we measure the overall error in pose of the reconstructed septum. We compute the distance between the centroids of the reconstructed and ground-truth septums (translational distance), and we compute the angles between the principal components of the ground-truth and reconstructed septums (rotational distance). We also report these measures broken down into their in-plane and out-of-plane components (computed using the normal vector to the planar approximation of the ground-truth septum).

Finally, we report two standard measures of similarity: the median Hausdorff distance and the Dice similarity coefficient.

2.3.2. Hyperparameter search

We performed a grid search over the space of hyperparameters using a leave-one-septum-out approach. That is, we reconstructed each septum using the set of hyperparameters achieving the best performance on all other septums. The adjusted hyperparameters were the number of modes of variation in the shape model $M$, the inlier ratio in IMLOP ${\lambda }_i$, the registration regularization ${\lambda }_t$, and the shape model regularization ${\lambda }_r$ [Eq. (7)]. The search space for these hyperparameters was determined manually by empirical testing on the dataset:

$$\begin{gathered} M=\{\mathrm{10,20},50\}, \\ {\lambda }_i=\{0.6,1.0\}, \\ {\lambda }_t=\{{10}^{-2},{10}^{-1.5},{10}^{-1}\}, \\ {\lambda }_r=\{{10}^1,{10}^2,{10}^3\}. \end{gathered}$$ (7)

As a comparison, we report the same measures for septums estimated as an infinite plane using the method proposed by Ahmidi et al.² and septums reconstructed using just the initial registration phase (see Sec. 2.2.2).

2.3.3. Confounding factors

We investigated the effect of two confounding factors on the reconstructions: septal deflection and operator skill levels. Septal deflection was determined from the manual segmentations of the septums from CT images. It was computed as the 95th percentile distance between a planar estimate of the septum and the medial point of the segmentation. Operator skill level was determined by appointment status.

To determine the effect of the confounding factors, we compute the rank correlation (with $p$-value) of the measures of reconstruction accuracy against the septal deflection and operator skill level.

3. Results

Septal reconstruction succeeded in all cases with complete datasets (Fig. 6). Due to technical difficulties with the tool tracking equipment; however, complete datasets for septoplasty surgery were only available in 10 out of 12 cases. Data were assessed for normality using Jarque–Bera test and were found to be non-normally distributed. Data are thus reported as median (interquartile range).

Fig. 6.

(a) Axial, (b) sagittal, (c) coronal, and (d) 3D views of the reconstructed septum compared to the ground-truth septum in CT. CT image is rendered in 3D view.

The fiducial registration error for the CT to reference tracker registration (see Sec. 2.3) was $7.92(6.52-9.73)\mathrm{mm}$, which corresponds to approximate target registration error of $4.19(3.70-4.98)\mathrm{mm}$ given the fiducial configuration used.²⁰

Overall, the full reconstruction of the septum achieved projected position differences of $2.74(2.06-2.81)\mathrm{mm}$ and projected orientation differences of $8.95(7.11-10.55)\mathrm{deg}$ compared to the ground-truth septums, given the optimal hyperparameters from the other septums.

Initial registration (see Sec. 2.2.2) achieved projected position differences of $2.80(2.30-3.11)\mathrm{mm}$ and projected orientation differences of $9.35(7.35-11.54)\mathrm{deg}$. The root-mean-square error for initial registration was $5.91(5.53-7.15)\mathrm{mm}$.

The differences between the reconstructed septums and the ground-truth septums are reported in Table 1 for the hyperparameters achieving the best performance on all other septums. Table 1 also includes results using planar fit² and the initial registration.

Table 1.

Accuracy metrics for septum reconstruction. Results from the planat fit (planar),² initial registration (initial), and for the full reconstruction (full). Values are reported as median (interquartile range). The best performing method for each metric is indicated in bold.

Metric	Planar2	Initial	Full
Projected position differences (mm)	$2.85(2.56-3.45)$	$2.80(2.30-3.11)$	$\mathbf{2.74}(\mathbf{2.06}\mathbf{-}\mathbf{2.81})$
Projected orientation differences (deg)	$11.86(6.64-16.13)$	$9.35(7.35-11.54)$	$\mathbf{8.95}(\mathbf{7.11}\mathbf{-}\mathbf{10.55})$
Translational distance (mm)	—	$\mathbf{7.56}(\mathbf{5.15}\mathbf{-}\mathbf{15.22})$	$11.22(6.73-15.19)$
In-plane translational distance (mm)	—	$\mathbf{6.91}(\mathbf{4.06}\mathbf{-}\mathbf{15.21})$	$9.56(4.95-12.77)$
Out-of-plane translational distance (mm)	—	$2.76(1.86-3.17)$	$\mathbf{1.81}(\mathbf{0.70}-\mathbf{3.16})$
Rotational distance (deg)	—	$16.36(12.32-18.95)$	$\mathbf{15.88}(\mathbf{13.38}\mathbf{-}\mathbf{19.78})$
In-plane rotational distance (deg)	—	$\mathbf{14.80}(\mathbf{9.55}\mathbf{-}\mathbf{17.99})$	$16.75(12.92-23.82)$
Out-of-plane rotational distance (deg)	—	$5.24(1.25-5.76)$	$\mathbf{3.80}(\mathbf{1.54}\mathbf{-}\mathbf{9.22})$
Median Hausdorff distance (mm)	—	$2.93(2.45-4.43)$	$\mathbf{2.78}(\mathbf{2.65}\mathbf{-}\mathbf{4.34})$
Dice similarity coefficient	—	$0.22(0.12-0.32)$	$\mathbf{0.23}(\mathbf{0.07}\mathbf{-}\mathbf{0.34})$

Full reconstruction of the nasal septum took, on average, 2 min and 37 sec.

The effects of the potential confounding variables (i.e., septal deflection and operator skill level) are reported in Table 2 and Fig. 7. Of the 10 cases with complete data, the median (interquartile range) deflection in the cadaver septums was $1.82(1.44-2.42)\mathrm{mm}$. Six trials were performed by experts, and four trials were performed by novices. The only correlation that was statistically significant was skill and in-plane rotational distance (less error was associated with less skilled operators); none of the other correlations were statistically significant.

Table 2.

Rank correlation between metrics for septum reconstruction with septal deflection (deflection) and operator skill level (skill). Results are from the full reconstruction method.

Metric	Deflection	Skill
Projected position differences (mm)	0.43	0.07
Projected orientation differences (deg)	0.10	$-0.14$
Translational distance (mm)	0.20	0.28
In-plane translational distance (mm)	$-0.36$	0.28
Out-of-plane translational distance (mm)	0.43	0.07
Rotational distance (deg)	$-0.21$	0.42
In-plane rotational distance (deg)	0.09	0.85
Out-of-plane rotational distance (deg)	$-0.13$	$-0.28$
Median Hausdorff distance (mm)	0.56	0.57
Dice similarity coefficient	0.00	$-0.28$

Fig. 7.

(a) Projected position differences versus septal deflection and (b) projected position differences versus operator skill level for the reconstructed septums.

4. Discussion

The proposed method for septum reconstruction succeeded in all cases where there was a complete dataset available. The reconstruction is performed from data acquired during the surgery, and it does not require explicit delineation of the septum’s surface.

The local position error in the proposed reconstructions is $2.74(2.06-2.81)\mathrm{mm}$ on average, given the optimal regularization. This is comparable to accuracy found in surgical navigation systems (and thus suitable for navigation), and it is typical accuracy in co-registration of CT and MRI.²¹ Some of this error can be attributed to the CT to reference tracker registration used to compare the reconstructions to the manually segmented septums, which had an fiducial registration error of $7.92(6.52-9.73)\mathrm{mm}$. The large fiducial registration error is likely due to magnetic field distortions due to ferromagnetic materials in the operating environment. The target registration error at the centroid of the septum, however, was $\sim 4.19(3.70-4.98)\mathrm{mm}$ given the fiducial configuration.²⁰ Furthermore, we performed a simulation experiment perturbing the fiducial points according to the fiducial localization error²⁰ and observed target registration error $4.41(3.59-5.14)\mathrm{mm}$, where the magnitude of error out-of-plane was $2.41(2.00-2.91)\mathrm{mm}$. Further error could be due to the statistical shape model; it is computed through automated non-linear registration, which may introduce some error in the shape model’s modes of variation.

The local orientation error is on average $8.95(7.11-10.55)\mathrm{deg}$ in the proposed reconstructions, given the optimal regularization. This exceeds the performance achieved for the optimal planar estimate of the septum (fitted via prinipcal components analysis to the ground-truth septum with thickness equal to the mean thickness of the ground-truth septum), where we observed local orientation errors of $9.87(6.35-11.21)\mathrm{deg}$. We note that the local orientation errors are affected by position errors due to the way corresponding points on the reconstructed and ground-truth septums are found. Although we filter out instances where the orientation difference in projections is $>90\mathrm{deg}$ (which indicates corresponding points are on different sides of the septum due to position error), we believe the reported orientation error considerably overestimates the true orientation error.

The full reconstruction from elevation data did not significantly improve upon the initial reconstruction and had small effect size, but descriptive statistics indicate some benefit to the proposed reconstruction. We hypothesize the lack of observed improvement is because the cadavers’ septums did not have large deflections [$1.82(1.44-2.42)\mathrm{mm}$], and thus the added value of the shape model is limited. Indeed, by optimally registering the mean septum to the ground-truth septums, we found local position error of $1.78(1.62-2.30)\mathrm{mm}$ and local orientation error $7.03(3.38-8.27)\mathrm{deg}$. These values provide loose upper bounds on the added value of the shape modeling for the septums in our dataset. The results are also limited by the small sample size due to expense and limited availability of cadavers.

We used the position and orientation difference between corresponding points as the primary measures of accuracy in this work. This is because instrument motion only occurs in one region of the septum and reconstruction will be most valuable in that region. Thus local measures of accuracy better quantify the utility of the reconstruction. Traditional measures of closed surface comparison and overall measures of pose accuracy are not as useful in this application. In particular, we observe that the Dice similarity coefficient is not well-suited for this application due to the thin shape of the septum, which makes it very sensitive to small errors in translation.

Initial reconstruction achieves the greatest accuracies for some global reconstruction metrics (i.e., overall translational and rotational distance), as this is exactly the quantity that the ICP registration minimizes. The full reconstruction optimizes the local fits, at the expense of the global fit. We observe, however, that global pose error is primarily due to in-plane error. Because the septum is nearly planar, there is a lack of constraints on in-plane translation and rotation. We propose that further annotation could be used to better constraint in-plane translation and rotation, making the problem better posed.

The validation study has demonstrated the accuracy of reconstructing the septum in cadavers. The septal deflections observed in our dataset were $1.82(1.44-2.42)\mathrm{mm}$, whereas the deflections observed in septums generated randomly from our statistical shape model derived from a general population were $1.44(1.14-1.85)\mathrm{mm}$. Nasal septoplasty is commonly performed on skull-base surgery or cosmetic surgery patients, whose septal deflections follow the same distribution as the general population. We thus anticipate our results on this cadaver dataset will apply to such septoplasty cases. Further issues due to different distributions in septal deflections may be addressed by selecting a larger CT image database, which contains data from septoplasty candidates with a greater diversity of pathologies or deflections and building a shape model using from a more representative set of septums. Tuning of the regularization parameters may also be necessary to capture these greater deviations. Furthermore, we have shown the effect of both deflection and operator skill level on reconstruction accuracy. Our dataset is heterogeneous with respect to operator skill level, like encountered in patient data where surgeries may be performed by attending surgeons or residents of varying skill levels.

The proposed methods for septum reconstruction may generalize to anatomical surface reconstruction in other hard tissue applications, with some caveats. One of the challenges of reconstructing the septum from tool tip data is that the thickness of the septum (2 mm at its thinnest) is approximately equal to the accuracy of the tracking system. This necessitates the use of orientation information to distinguish tool tip motion on either side of the septum. The proposed method for determining the orientation of trajectory points is specific to reconstruction of the nasal septum. Furthermore, the initial registration phase requires a reliable set of landmarks rigidly connected to the relevant anatomy. This may not be possible to acquire in all interventions.

Monitoring instrument trajectories during septoplasty surgery allow reconstruction of the nasal septum at the conclusion of the elevation phase without the need for CT. This enables objective determination of the shape of the septum, which has application in diagnosis of pathology, clinical decision support, and documentation of nasal obstruction for insurance purposes. This also enables real-time virtual endoscopy or other surgical navigation modalities during portions of the surgery after elevation by tracking the instruments relative to the reconstructed septum, which has been shown to have added value in septoplasty.²² Finally, it enables surgical skills assessment in septoplasty by analyzing instrument motion relative to local anatomy and septum shape, which are considerable confounding factors in the performance of septoplasty surgery. Indeed, analyzing tool motion relative to anatomy has been shown to have added value in other applications.²³

5. Conclusion

We have proposed a method for reconstructing the nasal septum from instrument motion data during the elevation phase of septoplasty surgery. The method does not require explicit delineation of the septum. The proposed method succeeded in all cases with complete datasets. We have measured the accuracy of the proposed methods using a dataset of tracked septoplasty surgeries performed on cadavers and found our reconstructions to be within $2.74(2.06-2.81)\mathrm{mm}$ and $8.95(7.11-10.55)\mathrm{deg}$ of the true septum, given the optimal regularizations. Our method outperforms estimates of the septal plane used in the prior work.² This work enables computation of the septal anatomy in real time. This will enable objective computation of septal deviation, surgical navigation, and performance assessment.

Biographies

Matthew S. Holden is an assistant professor in the School of Computer Science at Carleton University. Previously, he was a postdoctoral fellow at the Malone Center for Engineering in Healthcare at Johns Hopkins University. His primary research interest is in surgical data science, where he investigates machine learning methods for time series data collected in the operating environment, with the goal of improving patient outcomes.

Lisa Ishii, MD, MHS, is Professor of Otolaryngology, Head & Neck Surgery, Johns Hopkins University, and Senior Vice President, Operations, Johns Hopkins Health System. Her research interest includes the development of objective assessment of surgical skill.

Gregory Hager is the Mandell Bellmore Professor of Computer Science and the founding Director of the Malone Center for Engineering in Healthcare at Johns Hopkins University. His research interests include computer vision and vision-based and collaborative robotics with applications to medicine and manufacturing. He is a fellow of the ACM, IEEE, AAAS, the MICCAI Society, and AIMBE, for his contributions to vision-based robotics, medical imaging and his work on the analysis of surgical technical skill.

Biographies of the other authors are not available.

Disclosures

The authors have no conflicts of interest to declare.