A deformable template method for describing and averaging the anatomical variation of the human nasal cavity

Anonymized head and neck CT scans from 26 patients were used to create patient specific airway models. The data set included a mix of male and female (9 male, 17 female) subjects aged from 17 to 70 years, with an average age of 52.7 years and a standard deviation of 13 years. The obtained scans were retrospective and anonymized. No pathologies were observed in the airways. Patients were imaged awake, in a supine position. Scans were comprised of stacks of horizontal slices with 0.43 ×0.43 mm planar resolution; slices were spaced 0.6 mm apart vertically.

The segmentation procedure was as follows: First, the airway from the tip of the nose to the trachea was segmented from other structures in the CT images using 3D Slicer software (v4.3.1) [16]. A thresholding procedure was used for the segmentation of air with the threshold varied from -1000 to -400 Hounsfield units (HU); see Fig. 5. The optimal threshold values varied from scan to scan and within a scan for areas with poor contrast or resolution. The latter was used for the interface between the nasal cavity and paranasal sinuses interface due to the small scale and complex geometry of the nasal cavity passages and sinus drainage openings. Manual slice-by-slice editing was used for these problematic areas to obtain anatomically accurate airway segmentation.

In the next step, the segmented image data from all three image planes (axial, coronal and sagittal) was used to create a 3D surface model of the airway using 3D Slicer’s built-in marching cubes algorithm-based ModelMaker module [17]. The procedure was as follows: We applied a default ‘ThresholdEffect’, followed by a ‘PaintEffect’, and then the ‘Unsmooth’ ModelMaker function to generate the surface mesh. The generated model was then exported in an triangular mesh format for editing. MeshLab software (v1.3.3) [18] was then used to remove the sinuses and to smooth the surface of the model (see Fig. 6), using the hole-filling ‘Replace’ module (Smooth mean value coordinates (MVC) method), and the localised robust smoothing, refinement sub-module (default parameters: refinement: 0, reduction: 5.0, smoothing: 5.0). The frontal, maxillary, ethmoidal and sphenoidal sinuses were removed from the nasal cavity by manual editing in MeshMixer software (v10.6.53)¹. To smooth the airway model surface, while preserving its topology and size, a two-step MeshLab Laplacian Smoothing was applied (1D Boundary Smoothing, cotangent weight method). This procedure resulted in smooth 3D representations of the airway of each patient.

There is a large ‘air’ volume in front of the face which is topologically connected to the air inside the nasal cavity. We remove this volume via a morphological opening of the segmented volume [19], in the anterior volume, by a spherical structuring element of radius r. An additional erosion of radius r _ε is used to capture ‘outlier’ areas that spherical structuring elements do not fit e.g. the narrow channel between the nose and the upper lip. The radius r must be set to a value that is larger than the internal wall-to-wall dimensions of the nasal cavity but smaller than the volume outside the face. We used a value of r = 10 mm.

The absolute coordinates of the origin and relative rotation of each individual varies when CT scans are acquired. The first step in the analysis is the alignment of the airways of scans of different subjects so that they are all approximately in the same spatial position and have the same alignment and scale. This is because we ‘slice’ the scan into cross-sections later, and we require the cross-sections to approximately contain the same features in the structure. For alignment, the common approach is to identify a set of ‘landmarks’ on the shape. Landmarks must be “homologous anatomical loci that provide adequate coverage of the morphology, and can be found repeatedly and reliably” [20]. Specifically, landmarks are zero-dimensional points, and we use the term ‘landmark’ as opposed to ‘feature’ for these points, both to make this distinction and for consistency with the field of geometric morphometrics.

Alignment of meshes according to shape similarity requires specification of seven degrees of freedom corresponding to three translation, three rotation, and one scale degree of freedom [11]. We locate a pair of landmarks in each scan. Bone-based landmarks are preferable to tissue-based landmarks as these are less subject to variation based on experimental conditions and nasal cycle phase. In [11], the following landmarks were selected for this purpose:

These landmarks are illustrated in Fig. 7 and specify six degrees of freedom. The final degree of freedom is rotation about the line between the landmarks. In [11], an automatic alignment procedure was used to specify this but there is no need to do this for our method since it is insensitive to rotation about this axis. We simply align the airways such that the AMS is at the origin and the line between the two landmarks is parallel to the y-axis (the z-axis is in the upwards direction towards the top of the head, and the x-axis is in the sideways direction). Additionally, once this alignment has been done, the geometry is uniformly scaled such that the choana lies at a distance of 60 mm from the AMS. This is based on the typical average length of the AMS-Choana distance; see, for example [11].

Coronal cross-sections of the nasal cavity are extracted. This converts the 3D geometry to a set of 2D cross-sections. These cross-sections are processed independently. This is a common approach to studying nasal cavity shape since two-dimensional shape analysis techniques are highly developed [11, 12, 21]. For instance, we can robustly identify the inferior and middle sections of the airway [14]. Sections are shown in Fig. 8. The major sources of variability in the airway shapes are:

We make the observation that, despite the variability in individuals, many airway shapes can be viewed as ‘warped’ or smoothly deformed versions of one another. Such variations are common in biological contexts [23‐25]. Even though large variations are present, the various structures (meatuses, etc.) have approximately the same position and alignment with respect to one another. Thus we describe the cross-sections by first obtaining the medial axis (e.g. Fig. 9).

The medial axis of a shape is the set of all points having more than one closest point on the object’s boundary. In practical terms, the medial axis provides a ‘thinning’ or ‘skeleton’ for a shape. Here the medial axis is computed according to the Zhang-Suen thinning algorithm [26], which provides several desirable properties, for example each pixel on an edge has exactly two neighbors, and each pixel at a bifurcation point has exactly three. The branching of the airway into various meatuses can be seen consistently with this medial axis transformation [14].

Now we envision some ‘reference’ shape template (set of curves) that undergoes a smooth non-intersecting deformation to superimpose on the curves of a desired ‘target’ shape, based on a similarity metric. The deformation process and similarity metric are given in section ‘Matching Procedure’.

In what follows we address the problem of how to deform the reference template to fit target shapes. We desire deformations that are as ‘simple’ as possible in the sense of not requiring complex, convoluted deformations. This is to preserve the spatial relationship of adjacent structures. Also, as we are interested in the position of landmarks, it is desirable to have a deformation formulation that is based on altering the position of (semi)landmarks [20]. That is, we prefer a deformation that can be represented by deforming a small set of ’sample’ points and ’extrapolating’ this deformation to the rest of the space.

To summarize, we require a deformation formulation that has the following features:

Thin-plate splines (TPS) [27] satisfy all of these requirements, and are formulated as follows. Let x _i be a set of control points which are mapped to the points y _i . The thin-plate spline is the unique deformation f that minimizes the following function,

The second term in this equation is the squared curvature of f. λ is a real-valued stiffness parameter that controls the relative weight of stiffness vs. accurate point matching. This deformation f can be found exactly via radial basis functions [28].

The first step of our method is the medial axis transformation, as described. The second step is non-rigid registration of the thinned cross-sections to a ‘template’ consisting of an airway with demarcated sub-regions (using thin-plate splines), using thin-plate splines (see Fig. 10). The template itself is based on the Carleton-Civic airway geometry [11] and is shown in Fig. 11. Even though the Carleton-Civic geometry lacks precision in terms of capturing airway features, for this stage of the process only an approximate skeleton of the airway is required, since we later discard the template’s control point coordinates and use coordinates from the input data.

For finding the optimal deformation, we define the following sets of points:

For each pair of slices, we deform the set of control points, use TPS to extrapolate this deformation to the other points, and try to find the deformation that will optimize the alignment of the target and reference points (see Fig. 10).

Let the set of control points in the reference template be {X _i,j }, where j indexes the cross-sectional image number and i indexes each control point in a cross-sectional image. After non-rigid registration, we obtain the points {Y _i,k,j } where k indexes the target (not reference) cross-section. We compute the average position of each control point \(\bar {Y}_{i,j}=\frac {1}{K}\sum _{k=1}^{K}Y_{i,k,l}\) to obtain a new set of control points. We then align all of the cross-sectional images to these coordinates, to obtain a superimposed image where most of the airways overlap. Finally, the median intensity of this image is determined. This completes the description of our algorithm, however two details remain: The pairwise matching algorithm, and the method used for selecting control points. For the pairwise matching procedure mentioned above, we use the method of deformable robust point matching (RPM), see [28] for further details.

We select control points as follows. There must be enough control points to compactly represent the natural range of deformations that are observed. In theory, if a curve in the reference template can be approximated by a piecewise-linear curve, then the nodes of this curve can be taken as control points and would provide a nearly-complete representation of any possible deformation of the curve that preserves the piecewise-linear structure. Thus, we use the Ramer-Douglas-Puecker (RDP) algorithm [29, 30] on the skeleton to produce a set of control points. The RDP algorithm requires a parameter ε representing the characteristic length scale of the resulting approximation. By varying ε, the number of control points can be varied, with larger ε leading to fewer control points, as shown in Fig. 12.

Our final procedure for obtaining the mean geometry is as follows:

To visualize this procedure, a flowchart is presented in Fig. 13.

We calculate the cross-sectional area (CSA) along the cross-sectional images of the airway, and we also calculate the CSA for the various parts (inferior, middle, superior) of the airway. To do this, the different parts are labelled separately and the number of pixels with each label counted. See Fig. 14.

After this, the procedure is to label all other parts of the skeleton. We must allow for ‘broken’ skeletons as well as multiple extraneous branches. This is done via the following process: consider all non-labeled curves in the skeleton, and pick the one with an endpoint closest to any labeled curve in the skeleton. Then label this curve with the same label. This process is repeated until all curves in the skeleton have been labeled.

The final part of the procedure consists of labeling all pixels in the binary mask that are not in the skeleton. Care must be taken here as we cannot simply assign labels based on proximity to nearest curve in the skeleton. This would give incorrect labels in situations where, for instance, a ‘thick’ curve is in close proximity to a ‘thin’ one. The labeling procedure is similar to the one for curves except it is done based on pixels, i.e. at each step we pick the non-labeled pixel that is closest to the set of labeled pixels and we assign this pixel the same label. This process is repeated until all pixels have been labeled. For the purposes of computational efficiency, an optimized nearest-neighbor search is employed (Fast Library for Approximate Nearest Neighbors (FLANN)) [31].