High quality statistical shape modelling of the human nasal cavity and applications
The human nose is a complex organ that shows large morphological variations and has many important functions. However, the relation between shape and function is not yet fully understood. In this work, we present a high quality statistical shape model of the human nose based on clinical CT data of 46 patients. A technique based on cylindrical parametrization was used to create a correspondence between the nasal shapes of the population. Applying principal component analysis on these corresponded nasal cavities resulted in an average nasal geometry and geometrical variations, known as principal components, present in the population with a high precision. The analysis led to 46 principal components, which account for 95% of the total geometrical variation captured. These variations are first discussed qualitatively, and the effect on the average nasal shape of the first five principal components is visualized. Hereafter, by using this statistical shape model, two application examples that lead to quantitative data are shown: nasal shape in function of age and gender, and a morphometric analysis of different anatomical regions. Shape models, as the one presented here, can help to get a better understanding of nasal shape and variation, and their relationship with demographic data.
1. Introduction
The human nose is an important and complex organ having many functions, including ventilation, filtration and olfaction. People who suffer from a nasal function impairment have a reduced overall quality of life [1]. Improving our knowledge of the key structures and key features of the nasal cavity is therefore important. The human nose shows large variation in shape between different individuals, for example variations of the bone structure [2,3] or temporal variations in the nasal cycle [4]. These variations alter the airflow through the nasal cavity, and can thereby affect the functioning of the nose [5]. To get a better understanding of normal and pathological functioning of the nasal cavity, it is important to study the effects of these variations. Nowadays, surface models of the nasal cavity are typically based on tomographic data coming from computed tomography (CT) scans. A whole range of studies have been conducted in the past, using data coming from a single or few patients [6,7], to studies where data of a much larger population is used [8–13]. The approach in this work is different from the previous in that it is not directly limited to the shapes obtained from the tomographic data. From this point of view, it is much more related to the work done by Gambaruto et al. [14] about nasal airflow modelling using the representation of a nasal passage based on a Fourier descriptor.
We propose to use the statistical shape modelling (SSM) technique to capture the shape variations [15]. In this way, we avoid a priori assumptions about morphology. SSM is a widely used tool to capture the morphological variability present in a population. A main application is their use as prior knowledge in model-based automated image segmentation. They were popularized for this purpose by Cootes et al. [16]. An overview of their use in medical image segmentation can also be found in Heimann & Meinzer [17]. In the present work, clinical CT data is used to capture the nose in 46 patients, and from this data a high quality statistical shape model is created. In literature, different techniques (e.g. level sets, Fourier descriptors) have already been used to create a shape model of the nose [18–22]. In Liu et al. [20] and Sun et al. [22] image alignment and image processing of cross-sections are used to create a standardized geometrical model. In the latter work, a full upper respiratory airway model is created, containing less details of the nasal complex. A similar approach is followed in Nejati et al. [21], where cross-sections of the segmented mesh are taken and processed. Thinned representations of the nasal cavity are then computed, and a reference template is deformed such that it matches this thinned representation. Finally, all deformations are averaged to obtain an average geometry. They reported an increase in tolerance to a wider variety of nasal geometries than earlier methods. However, none of the methods discussed above use the direct three-dimensional information coming from training shapes. In Huang et al. [19] level sets are used to construct a shape model. Their model and the shape instances it produces for different positions along the PC axes show artefacts, i.e. left and right nasal channels are fused at the top of the nasal cavity, and different turbinates are merged together.
The technique used in the present work, based on cylindrical parametrization, is highly suitable for creating a nasal shape model because of the shape of the nasal cavity. More precisely, all nasal shapes of the population were closed at the post-nasal region in such a way that the resulting nasal cavity can be seen as a U-bended cylinder. For more information, the reader is referred to §2.2. We will show that this approach not only allows capturing and qualitatively reporting the natural anatomical variations present, but also more importantly, enables to relate shape variations with other features. In the following, this is called the quantitative analysis of the nasal cavity. In this paper, we show two quantitative applications of the SSM: we look at how age and gender are related with nasal shape variations, and we apply the shape model to study and discuss the influence of shape variation on the volume, surface and surface-to-volume ratio (SVR) of certain anatomical regions. These regions are the nasal valve and nasal vestibule region, the respiratory region, the olfactory region and the post-nasal region. The anatomy of these regions is important for the well-functioning of the nose, e.g. respiratory region for heating and humidifying the inhaled air. For sake of clarity, it is defined what is meant with ‘shape’ in this work. Strictly speaking, when two objects fall perfectly together after alignment (due to rotation and translation), they are said to have the same ‘form’. When these two objects are additionally allowed to have a different relative scale, they are said to have the same ‘shape’. In this work, however, we consistently use the term ‘shape’ instead of ‘form’, due to its connotation with ‘statistical shape model’, although we did not correct for scale in the alignment.
2. Material and methods
2.1. Data acquisition
Clinical cone-beam CT (CBCT) scans of six patients and helical CT scans of 40 patients were analysed. Data were obtained from patients with very different nasal or sinus related complaints, where a CT scan was requested by the treating ear nose throat (ENT) physician. The spatial resolution of the scans ranges from 250 to 430 µm perpendicular to the axial scan direction, and from 250 to 600 µm in the axial direction. The group included 29 females and 17 males. The average male subject age was 37 years (minimum age is 17, maximum is 60), and the average female subject age was 46 years (minimum age is 20, maximum is 84).
2.2. Individual model generation
First, the different steps are given that were followed to generate 46 different surfaces of the nasal cavity based on CT data. A classical and well-known approach was used in this work. For each CT scan a three-dimensional geometrical model was built by segmentation using Amira v. 6.0.1. On CT slices of the nose one can observe air, soft tissue and bone. Greyscale thresholding was used to determine the nasal passages, because it is the simplest way of segmentation and very suitable to find air spaces in the skull. This, however, led to inclusion of the sinuses, which are connected to the nasal cavity. Manual separation of these sinuses was overseen by an ENT physician. This separation, and by extension all steps described below, was done by the same operator for all 46 CT scans to avoid inter-operator generated variations between segmentations. At this stage and where necessary, some small corrections were made in the file containing the result of the segmentation, from now on called ‘label file’. Eventually, the goal is to represent each nasal shape with the same amount of points, where those points are located at the same anatomical position. This is done in the surface correspondence step, discussed below, but demands that all the training surfaces have the topology of a sphere (genus-0). To meet this requirement, holes in the label files were therefore filled such that the eventual geometrical surface extracted from the label file is genus-0. The field of view also varies for each scan, so that the end of the post-nasal region is different for each nasal shape. For this reason, the labels cannot be used as is. This would introduce a large non-anatomical variation in the shape model, as variation would be the extent of the region over which the scan was taken. Therefore, the labels were cut perpendicular to the axial scan direction at the floor of the inferior turbinates (figure 1).
The generated labels are used to build a triangulated surface geometry by applying the marching cubes algorithm [23]. As a next step, these surface models were smoothed, taking care that potential surface shrinkage was minimal. Algorithms exist that can be used for this (for example, the surface smoothing algorithm in Amira v. 6.0.1 is based on the paper ‘Curve and surface smoothing without shrinkage’ by Taubin [24]). At the end, the surface models were still quite rough when, for example, comparing them to those used in computational fluid dynamics. Nonetheless, further smoothing was unnecessary because of the smoothing effect of averaging when deriving shape model instances. The number of triangles was not decimated, because this is not necessary for the correspondence procedure.
All surfaces were exported in the stereo lithographic (STL) file format, and tiny holes were cut in both nostrils at the same anatomical position for all shapes. This was the final prerequisite of the correspondence algorithm. To increase the amount of morphological variation captured by the statistical shape model, mirrored versions of all nasal cavities are generated. Paraview v. 5.4.0 was used to generate these mirrored shapes, hereby doubling the size of the training data for the statistical shape model. The average shape obtained from principal component analysis (PCA) will be symmetrical because of this step.
2.3. Surface correspondence
Surface correspondence defines a dense one-to-one map between the surfaces in the population. In this paper, the correspondence method of Huysmans et al.[15] was modified to guarantee a uniform surface sampling. The method proceeds in four main steps. In the first step, each surface was cylindrically parametrized, using the method of Huysmans et al. [25], equipping the vertices of each surface with
It is a function of the shape mode variances
2.4. Statistical shape modelling
Using the correspondence calculated in the previous step, each shape is now represented by a set of 100 K corresponding landmark points distributed over the boundary surface of the nasal cavity and part of the post-nasal cavity. A SSM was built from the 92 corresponded nasal shapes by applying PCA on the vertices of the shapes. From a mathematical standpoint, a three-dimensional shape with n landmarks
For all 46 training examples and 46 mirrored counterparts, such a vector was generated. An important step is to ensure that all the shapes are optimally aligned. A nasal shape was considered to be independent of position and orientation, but not of scale. To determine the optimal poses, the Procrustes analysis was used. In this analysis, all shapes are aligned with a reference shape. In the first iteration no average shape is yet calculated, and cannot be used as reference shape. For this reason, a shape from the training set is chosen as reference shape to start the analysis. Because no scaling had to be used, the analysis came down to two steps. In the first part, differences due to translation were removed by calculating the centroid of each shape and translating them to the origin. In the second part, quaternions were used to compute the rotation matrix between each shape of the training set and the reference shape.
In a next step, PCA was applied, delivering an average nasal shape
P contains the eigenvectors of the covariance matrix and b is a vector given by equation (2.4). It is common to limit the possible value of
In figure 3, the normalized cumulative sum of all
2.5. Statistical analysis of shape
Because 95% of the total variation captured by the shape model is described by 46 shape modes, only these modes are considered in the following. This allows expressing the shapes of the training population in a useful way for further statistical analysis. More specifically, there are 46 values for each
The total population was divided according to gender. Next, for each parameter
This allows creating a visual map of how the shape of the nasal cavity is correlated with age. For the sake of clarity only the regression models of the female nasal shapes will be considered; the procedure is identical for the male shapes. As input, a lower age of 20, and an upper age of 80, is given to the regression model. The model outputs
Other approaches similar to the one above exist, e.g. linear discriminant analysis, but for the sake of brevity we will not go deeper into them.
For the relation between nasal shape and gender a similar approach to the one above was used. For each corresponding male and female set of
2.6. Volume partitioning
A simple approach was followed to partition the total volume of each shape. The average nasal cavity was divided into four anatomical regions (figure 4a–c) based on [30–33]. The division was done using a bounding box in Paraview, and by selecting those triangular faces on the surface that comprise the different volumes of interest. This puts a constraint on the way the anatomical regions can be defined. A trade-off exists; lowering the box will at the same time add unwanted parts to the region. Therefore, the regions definition was overseen by an ENT physician and the current division made an optimal compromise. The nasal valve and nasal vestibule region compose a first region (green, NVVR). The respiratory region is shown in (blue, RR), the olfactory region on top in (yellow, OR) and posteriorly the post-nasal region in (pink, PNR). Because of the way the different regions are defined, the nasal valve (separating the NVVR and RR) is implicitly represented by a plane. In reality, the nasal valve has a non-planar shape. However, in literature it is not uncommon to make such an approximation because it allows a precise definition, e.g. for modelling purposes [32]. A fairly good agreement exists when comparing the anatomical regions with definitions in existing literature [31–33]. Small differences are not unusual because of a difference in nasal shape, and variation also exists between literature. For example, the definition of the olfactory region used in this work, has a better agreement with Croce [31] than with Netter [33]. As a last step the NVVR, OR and PNR were closed by defining a plane for each region connecting the boundary triangles, i.e. those triangles with neighbours at only one side. The tiny holes made for the correspondence step were also closed at this point. These steps made it possible to define a volume for all anatomical regions.
Finally, the volume of the respiratory region was calculated by subtracting the volume of the total nasal shape from that of the other three previously mentioned. The ten first shape modes are consecutively applied to the average shape with
3. Results
3.1. Qualitative visualization
The different shape modes can be visualized by applying them to the average nasal shape using equation (2.3). In figure 5 this is shown for the first five modes, with values for
The first eigenmode applies a general scaling to average nasal shape. It is not a perfect isotropic scaling, but nonetheless there is a clear size increase/decrease in all three dimensions. This could be expected because in the alignment of the shapes only rotation and translation were used and no scaling. The second mode predominantly results in the contraction or expansion of the nasal shape along the anterior–posterior axis, and can be clearly observed in the region behind the choana. The expansion of the shape along the anterior–posterior axis goes hand in hand with a, be it smaller, expansion in the perpendicular plane, making the nose wider. The third eigenmode has a large effect on the curvature of the line connecting the anterior tip of the nose and the end of the post-nasal region. More precisely: the anterior tip of the nose drops and the region behind the choana flattens with decreasing values for
3.2. Quantitative applications
Figure 6 shows the result of the statistical analysis described in the ‘Material and methods’ section. The analysis gave
3.3. Morphometric analysis of anatomical regions
3.3.1. Nasal volume
As a second application example for the nasal shape model, the surface, volume and their ratio was studied for the different anatomical regions under the influence of each of the first 10 shape modes. These first 10 modes represent 69% of the total variation captured by the shape model. Not more are studied because of the extensive manual work involved per eigenmode.