Effects of the Nasal Cavity Complexity on the Pharyngeal Airway Fluid Mechanics: A Computational Study

Abstract

The impact of the human nasal airway complexity on the pharyngeal airway fluid mechanics is investigated at inspiration. It is the aim to find a suitable degree of geometrical reduction that allows for an efficient segmentation of the human airways from cone-beam computed tomography images. The flow physics is simulated by a lattice Boltzmann method on high-performance computers. For two patients, the flow field through the complete upper airway is compared to results obtained from three surface variants with continuously decreasing complexity. The most complex reduced airway model includes the middle and inferior turbinates, while the moderate model only features the inferior turbinates. In the simplest model, a pipe-like artificial structure is attached to the airway. For each model, the averaged pressure is computed at different cross sections. Furthermore, the flow fields are investigated by means of averaged velocity magnitudes, in-plane velocity vectors, and streamlines. By analyzing the averaged pressure loss from the nostrils to each cross section, it is found that only the most complex reduced models are capable of approximating the pressure distribution from the original geometries. In the moderate models, the geometry reductions lead to overpredictions of the pressure loss in the pharynx. Attaching a pipe-like structure leads to a higher deceleration of the incoming flow and underpredicted pressure losses and velocities, especially in the upper part of the pharynx. Dean-like vortices are observed in the moderate and pipe-like models, since their shape comes close to a ${90}^{\circ }$-bend elbow pipe.

Introduction

In computational fluid dynamics (CFD), complex mathematical models and numerical methods are used to simulate fluid motion in and around arbitrary geometries [1, 2]. Because of its noninvasive nature, CFD has obtained a significant interest in the medical and engineering community. It is a validated method for the evaluation of the airflow in human upper airway models reconstructed from magnetic resonance (MR) or computed tomography (CT) imaging data [3–5].

To simulate the flow in the human airways, it is necessary to generate a corresponding three-dimensional model of the airway [3, 6]. A crucial step of this process is the segmentation of medical image data into airway and non-airway regions, and the identification of the air-tissue interface. Depending on the model generation method and the complexity of the geometry, the segmentation process may consume a considerable amount of time. The nasal airway is a complex anatomical structure with narrow, convoluted and intricate passages and is surrounded by the paranasal sinuses [7, 8]. It is hence quite tedious and time consuming to perform the segmentation manually. Fortunately, CT images feature sufficiently high contrasts that capture thin channels. They allow for an automatic segmentation by thresholding all areas that represent air in an image [3, 6, 9]. Despite its advantages, the CT imaging process entails high radiation doses leading to an increased risk for a patient’s health. Unlike CT imaging procedures, cone-beam computed tomography (CBCT) techniques feature short acquisition times and low radiation doses [10]. It is hence not surprising that they have gained significant popularity. Unfortunately, CBCT data have a higher noise ratio and lower contrasts compared to CT data, which may lead to uncertainty in the identification of narrow passages. Segmentation then needs to be performed semi-automatically, which makes it more difficult and again more time consuming.

Based on the extracted model, a computational mesh is usually generated that is input to a numerical method that simulates the flow numerically on computers. In [4, 6, 11–13], a lattice Boltzmann (LB) method is used to analyze flow in nasal airways with corresponding models stemming from CT images. The LB method has shown to be a fast and accurate method to simulate complex flows in intricate geometries [14, 15]. In [6], a general analysis of the respiratory flow in a single nasal cavity with a focus on secondary flow phenomena is investigated for inspiration and expiration. In two studies [4, 12], the results of a previous study [6] are extended towards a classification into different respiratory ability groups, which supported a-priori decision-making in surgical interventions. In the study of Waldmann et al. [13], effects of miniscrew-assisted rapid maxillary expansion (MARME) treatments on the respiratory flow are investigated. The findings reveal that respiratory resistance and the averaged wall-shear stress decreased after MARME treatment, whereas the heating capability deteriorated. The studies in [11] concentrate on particle simulations and evaluate different parallel approaches for coupled LB-Lagrangian simulations. Hörschler et al. [3] use a finite volume method to simulate the flow in a model of the nasal cavity at low Reynolds numbers Re and juxtapose numerical findings to experimental results. They show that for $Re<500$ the flow can be considered laminar.

In extension to the nasal cavity, the pharynx also plays a crucial role in respiration diagnostics, e.g., in the analysis of the obstructive sleep apnea syndrome (OSAS) [16], which is a disorder that is characterized by intermittent closure of the pharyngeal airway. Mihaescu et al. [17] and Mylavarapu et al. [18] investigate the flow in realistic airway models reconstructed from CT and MR images. In [18], the model with a maximum narrowing at the retropalatal pharynx is used and the oral and nasal cavity are omitted. The investigations focused on steady flows and compared large-eddy simulation (LES) and Reynolds-averaged Navier–Stokes (RANS) simulation results. In [19], steady-state RANS and LES computations were performed for inspiratory and expiratory flow analysis, and the wall static pressure and the mean axial velocity were evaluated. Medical treatment increases the cross-sectional area in the whole airway and reduces local velocities, pressure loss, and wall-shear stress. Ito et al. [20] reconstruct the surface of the nasal and pharyngeal airway from coarse CT data for 23 patients. Their results agree with the conclusions in [19]. Hur et al. [21] conduct RANS simulations of the OSAS and use airway models based on CBCT data. Considering fluid-structure interactions, they show how a miniscrew-assisted rapid palatal expansion (MARPE) improves the airflow and decreases resistance in the upper airway. Chang et al. [22] reconstruct airway models from CBCT images before and after maxillomandibular advancement (MMA) and reported that less exertion was needed to breath after surgery. They consider fluid–structure interactions and used RANS simulations for airflow predictions. Hagen et al. [23] simplified the nasal airway in OSAS cases by using a pipe-like structure. The pressure distributions in the pharynx of the original and the simplified models were in good agreement. Unfortunately, only the impact of a single simplification is tested. The aforementioned studies indicate that CBCT data can be successfully used for CFD analyses.

The purpose of this study is to evaluate the influence of various nasal airway simplifications on the accuracy of CFD analysis in the pharynx region. Therefore, the upper respiratory tract from the nasal cavity, its simplification in three stages, and the pharyngeal airway with nasopharynx, velopharynx, oropharynx, and hypopharynx, see Fig. 2, are included. The first stage includes the middle and inferior turbinates. The second stage only includes the inferior turbinates. In the third stage, a pipe-like structure replaces the frontal part of the nasal cavity. An overview of the surface variants is given in “Surface Variants”.

Fig. 2.

Geometry and anatomical terminology of the airway of patient A. Furthermore, the coordinate system, the length scales $L_c$ and L, and the locations of slices $C{S}_0$ and $C{S}_9$ are introduced. The inlet and outlet areas are highlighted in red/blue and green

The materials and methods are presented in “Materials and Methods”. The results based on various anatomical changes affecting the nasal cavity complexity are presented in “Results”. Finally, a summary and conclusions are provided in “Summary and Conclusions”.

Materials and Methods

The flow field in the upper airways is simulated by an LB method. The corresponding surfaces are extracted from CBCT images. In the following, details on the medical images used in this study are given in “Medical Imaging”, followed by a description of the surface extraction process in “Surface Extraction”. “Surface Variants” explains from a medical point of view which surface variants of the upper airway are chosen. Subsequently, “Numerical Methods” presents the numerical methods that are used to conduct the simulations.

Medical Imaging

The original data for the surface extraction of a patient’s upper airway stem from n=460 two-dimensional CBCT images that are taken in transverse planes. A combination of all 2D images results in a three-dimensional density representation of the scanned patient, denoted as a. The CBCT images are obtained from an Alphard Vega CBCT scanner (Asahi Roentgen Co., Kyoto, Japan) under the following conditions: 80 kV, 5 mA, $0.39\times 0.39\times 0.39mm$ voxel size, and a $200\times 179mm$ field of view. Each image contains information on the density field given in Hounsfield units HU, with $HU\in [-1024,3071]$. Different HU values and ranges correspond to different material properties, e.g., at an HU value of around $HU=-1000$ air is found and at HU=0 water is present.

The CBCT scans are obtained on the basis of their clinical orthodontic indication. The patients are scanned in natural head position as described by Hwang et al. [24]. Before using the scans for segmentation, the patient’s personalized data is removed and facial structures are blurred to guarantee data protection. In this study, CBCT scans of two patients are used. In the following, they are referred to as models A and B.

Surface Extraction

The surface geometry of the upper airway is extracted from the CBCT images in multiple steps as described previously [4, 6]. In the first step, the raw CBCT images are filtered using a convolutional filter (CF) [?] and a gradient anisotropic diffusion filter (GADF) [?] to generate a higher contrast between voxels representing air and other materials. The CF increases the HU gradients at the air–tissue interface at cell locations a(i,j,k), $i,j\in 1,...,m$ in the k-th CBCT image, $k\in 1,...,n$, to $a_{\mathit{CF}}(i,j,k)$ using a convolution operation, i.e., applying

$$\begin{array}{c}\hfill a_{\mathit{FC}}(i,j,k)=\sum _{l=-1}^1\sum _{n=-1}^{1}a(i+l,j+n,k)·w(l,n).\end{array}$$ 1

The kernel w is a $3\times 3$ matrix that multiplies weights with values in the surrounding cells of a(i,j). Preliminary investigations have shown the following kernel to yield the best results:

$$\begin{array}{c}\hfill w=\left\{\begin{array}{cc}10\hfill & ,l=n=0\hfill \\ -1\hfill & ,l\ne 0\wedge n\ne 0.\hfill \end{array}\right)\end{array}$$ 2

To employ the CF at boundaries of CBCT images, missing cells are artificially created outside the image domain. They are filled by the averaged HU value of the corresponding image. A deficit of solely using the CF is that not only gradients at air–tissue interfaces are increased but also smaller gradients at other material interfaces. To overcome this issue, the GADF smoothens unwanted gradients by using the classic Perona–Malik, gradient magnitude-based equation to reduce image noise without removing edges [25].

Figure 1a shows a CBCT image of a coronal cross section of a patient’s head through the nasal airway. Figure 1b compares HU values along the vertical red line depicted in Fig. 1a for the original CBCT, after application of solely the CF, and using a combination of both the CF and GADF. Obviously, gradients are increased by the CF whenever there is a transition from air to other materials, e.g., at location $i=150$. Noise is then reduced by the GADF, e.g., in the range of $i=150,\dots ,175$.

Fig. 1.

Hounsfield values HU from CT recordings are compared to the same data filtered by a convolutional filter (CF) and by a combination of the CF and a gradient anisotropic diffusion filter (GADF). (a) Base CT image; (b) the original Hounsfield values along the red line in the (a)

After filtering, the upper airway is segmented by a region growing algorithm using a fixed threshold of $\mathrm{\Gamma }=-500HU$, which lies in the middle of a range of values recommended by Kabuliak et al. [9] for segmentation of the upper airway from CBCT images. Thus, the CBCT voxels with $a_{\mathit{CF}}(i,j,k)\le \mathrm{\Gamma }$ are marked as airway voxels, while all others with $a_{\mathit{CF}}(i,j,k)>\mathrm{\Gamma }$ are considered external to the airway, i.e., a binary image s(i, j, k) is created by

$$\begin{array}{c}\hfill s(i,j,k)=\left\{\begin{array}{cc}1\hfill & ,a_{\mathit{CF}}(i,j,k)\le \mathrm{\Gamma }\hfill \\ 0\hfill & ,a_{\mathit{CF}}(i,j,k)>\mathrm{\Gamma }.\hfill \end{array}\right)\end{array}$$ 3

In Fig. 1b, the threshold at $\mathrm{\Gamma }=-500$ is shown as a black horizontal dashed line. As mentioned in “Introduction”, the CBCT data are known to have lower contrasts compared with the CT data, with a relatively lower radiation dose. Despite filter application, some parts of the nasal cavity in CBCT images may not be correctly identified, e.g., at the location range $i=50,\dots ,75$ in Fig. 1b. Such portions are re-segmented manually using the open-source software 3D Slicer [26].

The inflow areas at the nostrils (inlets) and the outflow area at the lower pharynx (outlet) are defined to exhibit face normals pointing in the direction of the main flow; see Fig. 2. With the help of a slicing operation provided by 3D Slicer the segmentation in s(i, j, k) is reduced to $s_{\mathit{red}}(i,j,k)$ by

$$\begin{array}{c}\hfill s_{\mathit{red}}(i,j,k)=\left\{\begin{array}{cc}1\hfill & ,(i,j,k)\text{in the upper airway}\hfill \\ 0\hfill & ,(i,j,k)\text{in the remaining part.}\hfill \end{array}\right)\end{array}$$ 4

The array $s_{\mathit{red}}$ is then modified to $s_{\mathit{mod}}$ on the following criteria

$$\begin{array}{c}\hfill s_{\mathit{mod}}(i,j,k)=\left\{\begin{array}{cc}H{U}_{\mathit{min}}\hfill & ,s_{\mathit{red}}(i,j,k)=0\hfill \\ H{U}_{\mathit{max}}\hfill & ,s_{\mathit{red}}(i,j,k)=1\hfill \\ HU\hfill & ,(s_{\mathit{red}}(i,j,k)=0\wedge \hfill \\ \hfill & \exists s_{\mathit{red}}^N(i,j,k)=1),\hfill \end{array}\right)\end{array}$$ 5

where $s_{\mathit{red}}^N(i,j,k)$ includes all neighboring voxels of $s_{\mathit{red}}(i,j,k)$ and $H{U}_{min,max}$ are the minimum and maximum HU values.

The subsequent steps are inspired by the processing pipeline described previsouly in [4, 6]. First, the Marching cubes algorithm [27] is applied to $s_{\mathit{mod}}$ with the threshold $\mathrm{\Gamma }$. This results in a three-dimensional triangulated model of the upper human airway. Such a model is based on bilinear interpolation and a triangle look-up table. However, the surface is not smooth enough and contains aliasing effects due to the cubic structure of the voxels [6]. Therefore, it is smoothed by a windowed sync filter (WSF) [28]. The WSF is chosen because it does not shrink the surface, compared to, e.g., a Laplacian filter [6]. The WSF interprets changes in neighboring vertices in terms of frequencies. A pass-band for low frequencies ($pb=0.2$), which is then used representing the general structure of the nasal cavity shape, is applied to obtain sufficiently smooth and anatomically realistic geometries. In more detail, if a surface features small scale fluctuations with higher frequencies, the WSF redistributes vertices more evenly for fluctuations above pb without changing large-scale variations at lower frequencies below pb. The filter operates iteratively. 100 iterations have shown to yield a sufficiently smooth surface for the purpose of this work.

Finally, the generated surface is subdivided into sub-surfaces, i.e., into the left and right nostrils, the lower end of the pharynx, and the remaining surface of the upper airway. This subdivision is in a later stage necessary to prescribe the correct boundary conditions for the flow simulation, cf. “Numerical Methods”. Figure 2 exemplarily shows the subdivided geometry for patient A. The figure also introduces axes of the coordinate system that is used to define planes for analyzing the flow field in “Results”. Planes spanned by the x- and y-axis correspond to the transverse view of the CBCT data and planes spanned by y- and z-axis lie in the coronal view.

Surface Variants

Four different models are analyzed for each patient A and B:

• the original surface (I),

• the complex surface with middle and inferior turbinates (II),

• the moderate surface with inferior turbinates (III),

• and the surface containing a pipe-like structure (IV).

Surfaces of all models, A(g) and B(g) with $g\in \{I,II,III,IV\}$ are shown in Figs. 3 and 4.

Fig. 3.

Surface variations for patient A. (a), Model A (I); (b), Model A (II); (c), Model A (III); (d) Model A (IV)

Fig. 4.

Surface variations for patient B. (a), Model B (I); (b), Model B (II); (c), Model B (III); (d) Model B (IV)

The analyses presented in “Analysis of the Total Pressure Loss” are based on area-averaged quantities that are computed in cross sections $C{S}_0,\dots ,C{S}_9$, which are equally distributed along the pharynx. For each patient, the upper cross section $C{S}_0$ is defined as a plane passing through the posterior nasal spine. The lower cross section $C{S}_9$ is defined as a plane passing through the most anteroinferior point of the fourth cervical vertebrae. All cross sections are parallel to the x-/z-plane. The positions of the first and the last cross sections $C{S}_0$ and $C{S}_9$ are illustrated in Fig. 2.

The shape of the pipe-like structure in model (IV) is generated in four steps. The dashed lines $d_1,\dots ,d_4$, exemplarily shown in Fig. 3d for model A(IV), highlight these steps. They are positioned along the model length L, which is measured in the positive x-direction from the nostrils to the complete anterior part of the airway, as shown in Fig. 2. At location $d_1$, the original surface of the airway is cut at the level of a plane passing from the sella to the posterior nasal spine, which is perpendicular to the midsagittal plane. The remaining boundary edges are extruded in the negative x-direction toward location $d_2$, which is positioned at (3/4)L. At $d_3$, which is located at L/2, the cross section forms a circle connecting to the boundary edges at $d_2$. From $d_3$ to $d_4$, located at 0L, the circumference of the pipe-like structure successively decreases toward a single inflow area with a size equal to the summed area of the left and right nostrils of models $A(I)\dots ,A(III)$.

Numerical Methods

The numerical simulations are conducted using an LB method that operates on three-dimensional unstructured Cartesian meshes. The LB method is described in “Lattice-Boltzmann Method”. The subsequent “Computational Mesh and Simulation Setup” presents the grid generation process, the simulation setup, and contains a description of the initial and boundary conditions.

Lattice Boltzmann Method

To compute the flow field in the upper airways an LB method as part of the multiphysics Aerodynamisches Institut Aachen (m-AIA) simulation framework is used. The governing equation is the Boltzmann equation

$$\begin{array}{c}\hfill \frac{\partial f}{\partial t}+{\xi }_h\frac{\partial f}{\partial x_h}+\frac{F_h}{m}\frac{\partial f}{\partial {\xi }_h}=\\ \hfill {\int }_{{\mathit{\xi }}_{\mathit{\beta }}}{\int }_A{\xi }_r(f^{\prime }{f}_{\beta }^{\prime }-f{f}_{\beta })dAd{\mathit{\xi }}_{\mathit{\beta }},\end{array}$$ 6

where f and $f_{\beta }$ are the particle probability distribution functions (PPDFs) before ( f and $f_{\beta }$) and after ($f^{\prime }$ and $f_{\beta }^{\prime }$) collision, t is the time, $\xi$ is the molecular velocity, F denotes an external force, m represents the molecular mass, ${\xi }_r=\xi -{\xi }_{\beta }$ is the relative velocity between two colliding particles with velocities $\xi$ and ${\xi }_{\beta }$, and dA is the differential cross section. The subscript $h\in \{x,y,z\}$ represents the three spatial coordinates. Bhatnagar, Gross, and Krook [29] proposed the BGK model to simplify the non-linear integro-differential part on the right-hand side of Eq. 7. Neglecting forcing terms, the BGK equation reads

$$\begin{array}{c}\hfill \frac{\partial f}{\partial t}+{\xi }_h\frac{\partial f}{\partial h}=-\frac{1}{\tau }(f-f^{\mathit{eq}}).\end{array}$$ 7

The collision process is governed by the relaxation parameter $1/\tau$ with relaxation time $\tau$. In the limit, the PPDFs approach the Maxwellian equilibrium state $f^{\mathit{eq}}$. Additional details on the relaxation process can be found in [15]. To simulate the flow, Eq. 7 is discretized to yield the lattice BGK equation

$$\begin{array}{c}\hfill f_r(\mathit{x}+{\mathit{\xi }}_r·\delta t,t+\delta t)=\\ \hfill f_r(\mathit{x},t)-\frac{1}{\tau }(f_r(\mathit{x},t)-f_r^{\mathit{eq}}(\mathit{x},t)),\end{array}$$ 8

where $\mathit{x}$ denotes the location in space, $\delta t$ is the time increment per simulation iteration, and r is the PPDF direction. In this work, the space is discretized by a hierarchical Cartesian mesh [30] and the three-dimensional discretization model D3Q27 [31] is used. That is, the Cartesian directions, the cubic edge-diagonal directions, and the cubic space-diagonal directions are considered for the propagation of the PPDFs. With each direction r, a PPDF $f_r(\mathit{x},t)$ and a molecular velocity vector ${\mathit{\xi }}_r$ is associated. The macroscopic variables, i.e., the density $\rho$ and the velocity vector $\mathit{u}$ can be obtained from the moments of the PPDFs

$$\begin{array}{cc}\hfill \rho =& \sum _r{f}_r\hfill \end{array}$$ 9

$$\begin{array}{c}\hfill \mathit{u}={\rho }^{-1}\sum _r{\mathit{\xi }}_r{f}_r.\end{array}$$ 10

In the LB context, the spatial and temporal spacing are set to $\mathrm{\Delta }x=\mathrm{\Delta }t=1.0$ such that the speed of sound $c_s$ yields

$$\begin{array}{c}\hfill c_s=\frac{1}{\sqrt{3}}.\end{array}$$ 11

The pressure can then be expressed by a simplified form of the ideal gas law

$$\begin{array}{c}\hfill p=c_s^2\rho .\end{array}$$ 12

Computational Mesh and Simulation Setup

As mentioned above, the space in the airways needs to be discretized prior to a simulation. Therefore, a massively parallel grid generator is applied to generate a uniform hierarchical unstructured Cartesian mesh.

The grid generation starts with a single cube of edge length $L_c$, cf. Fig. 2. Note that cubes are in the following also referred to as cells as this is a common terminus in the CFD community. The initial cell contains the whole upper airway geometry and is continuously subdivided into smaller cells constituting an octree with parent-child relationships. At each iteration, a parent cell is subdivided into eight smaller cells and cells outside the airway geometry are removed. The subdivision process is repeated until a predefined octree hierarchy level (OHL) is reached. The OHL in this study is chosen in such a way that for all models the grid spacing $\delta x=L_c/2^{\mathit{OHL}}$ is $5·{10}^{-5}m$. Therefore, the initial cell for refinement is scaled accordingly. Such small grid spacings are necessary to resolve thin channels and to accurately resolve boundary layers [12, 32].

The number of cells $\mathcal{N}$ for each model A(g) and B(g) are listed in Table 1. Obviously, reducing the complexity also reduces the number cells required for simulation. From model (I) to model (II) the number $\mathcal{N}$ is roughly bisected. Model (III) further reduces this number. The only exception is model (IV), which features in contrast to (II)) and (III) again an increased volume that needs to be meshed. Considering furthermore the good scalability of the simulation code [14], it is expected that the computational cost almost decreases linearly with $\mathcal{N}$.

Table 1.

Number of cells $\mathcal{N}$ for models A(I) - A(IV) and B(I) - B(IV). Furthermore the hydraulic diameter $d_p$, the surface area $A_p$, the circumference $c{f}_p$, and the Reynolds number Re at the outlet for models A and B are given

Model	A(I)	A(II)	A(III)	A(IV)	B(I)	B(II)	B(III)	B(IV)
$\mathcal{N}$ $[·{10}^6]$	470.0	180.0	140.0	255.0	500.0	280.0	180.0	325.0
$d_p[mm]$	12.40				13.03
$A_p[m{m}^2]$	244.00				234.00
$c{f}_p[mm]$	76.95				71.82
Re	815.80				856.30

The Reynolds number in the pharynx is defined by $Re=(u_{\mathit{ref}}{d}_p)/\nu$, where $u_{\mathit{ref}}$ is the reference velocity that corresponds to a volume flux of $\dot{V}=250ml/s$ for respiration at rest [12], $d_p=4A_p/c{f}_p$ is the hydraulic diameter of the outlet geometry with surface area $A_p$ and circumference $c{f}_p$, and $\nu$ is the kinematic viscosity of air. Table 1 lists the hydraulic diameters, outlet areas, circumferences, and Reynolds numbers per simulation setup. Three types of boundary conditions are imposed. At the outlet, an adaptable Dirichlet condition for the pressure is prescribed [4, 12]. This condition continuously decreases the pressure until Re is reached at the outlet. Additionally, a von Neumann condition is used for the velocity at the outlet. At the inlet, a Saint–Venant pressure boundary condition [4, 12] is used, which extrapolates the momentum. In the end, a pressure difference between the inlet(s) and the outlet drives the flow. On the remaining surface a no-slip condition is imposed. It uses the second-order accurate interpolated bounce-back condition from Bouzidi et al. [33]. The flow field is initialized with $\rho =1.0$, $\mathit{u}={(0,0,0)}^T$, and with the corresponding equilibrium PPDFs.

The simulations are run on the JURECA [27] and the CLAIX high-performance computing (HPC) systems at Forschungszentrum Jülich and at RWTH Aachen University.

Results

The flow in the various geometries is analyzed according to the total pressure loss along the airway in “Analysis of the Total Pressure Loss”. Subsequently, in “Analysis of the Velocity Magnitude” and “Streamline Analysis of the Flow” the shape-induced change in velocity magnitude and streamlines are discussed. In the latter two investigations, the models A and B deliver similar results. Therefore, only the results for patient A are presented.

Analysis of the Total Pressure Loss

Pressure fields are analyzed in terms of the area-averaged total pressure $P_{w(g)}$, $w\in \{A,B\}$, $g\in \{I,II,III,IV\}$ at cross sections $C{S}_0,...,C{S}_9$, including the static and dynamic pressure, and the total pressure difference, i.e., the total pressure loss $P{L}_{w(g)}$ between cross sections $C{S}_0,...,C{S}_9$ and at the inlets. Figure 5a, b shows for models A and B the streamwise development of the total pressure in the four models (I)-(IV) with the red curve representing the reference model (I). The error to model (I) of the total pressure loss of the three simplified models (II) - (IV) is shown in Fig. 5c, d. It is defined by

$$\begin{array}{c}\hfill E_{w(q)}=\frac{P{L}_{w(q)}-P{L}_{w(I)}}{P{L}_{w(I)}}·100\%,\end{array}$$ 13

with $q\in \{II,III,IV\}$. It should be noted that despite $E_{w(q)}$ reaches quite high values $>40\%$ in the following analysis, the difference in Pascal [Pa] is quite small. Table 2 provides the corresponding quantitative results and additionally shows the absolute error in Pascal, i.e.,

$$\begin{array}{c}\hfill E_{w(q),abs}=P{L}_{w(q)}-P{L}_{w(I)}.\end{array}$$ 14

Fig. 5.

Area-averaged total pressure $P_{w(g)}$ in [Pa], and error $E_{w(q)}$ in percent at cross sections $C{S}_0,...,C{S}_9$. (a), Total pressure in model A; (b), Total pressure in model B; (c), error in model A; (d), error in model B

Table 2.

Area-averaged total pressure $P_{w(g)}$ given in X+101,322[Pa], where X is the corresponding value in the table, and errors $E_{w(q)}$ and $E_{w(q),abs}$ at cross sections $C{S}_0,...,C{S}_9$

0	2.62	2.61	2.43	2.74	2.45	2.45	2.15	2.61	3.6	47.0	-33.0	0.4	55.6	-29.1
									(0.014)	(0.183)	(-0.128)	(0.002)	(0.306)	(-0.161)
1	2.60	2.58	2.40	2.70	2.42	2.40	2.09	2.55	4.0	47.0	-25.9	3.5	57.1	-22.8
									(0.016)	(0.190)	(-0.105)	(0.020)	(0.331)	(-0.132)
2	2.54	2.52	2.33	2.65	2.40	2.39	2.08	2.50	5.0	45.2	-23.5	1.1	52.5	-16.5
									(0.023)	(0.207)	(-0.108)	(0.007)	(0.315)	(-0.099)
3	2.49	2.41	2.30	2.54	2.36	2.37	2.05	2.46	15.6	36.2	-1.0	-1.0	47.7	-16.5
									(0.080)	(0.185)	(-0.051)	(-0.007)	(0.307)	(-0.106)
4	2.23	2.25	2.16	2.47	2.33	2.28	2.02	2.35	3.0	9.8	-31.3	7.1	46.2	-3.2
									(0.023)	(0.075)	(-0.241)	(0.047)	(0.311)	(-0.022)
5	1.90	1.91	1.81	2.13	2.27	2.27	1.95	2.32	1.1	18.9	-13.8	-0.9	43.8	-6.5
									(0.011)	(0.190)	(-0.139)	(-0.007)	(0.319)	(-0.047)
6	1.83	1.90	1.79	2.02	2.25	2.26	1.93	2.27	-6.0	3.2	-16.6	-4.1	42.5	-2.0
									(-0.070)	(0.038)	(-0.194)	(-0.031)	(0.320)	(-0.015)
7	1.57	1.40	1.78	1.85	2.24	2.13	1.87	2.26	12.1	-14.7	-19.5	15.1	49.5	-1.7
									(0.173)	(-0.210)	(-0.279)	(0.115)	(0.376)	(-0.013)
8	1.29	1.14	0.96	1.32	2.01	2.07	1.83	2.25	9.2	19.0	-2.0	-6.1	17.4	-24.4
									(0.157)	(0.326)	(-0.034)	(-0.061)	(0.173)	(-0.242)
9	0.78	0.92	0.47	0.76	1.98	1.77	1.57	1.79	-6.1	13.9	1.2	21.4	41.1	18.5
									(-0.135)	(0.309)	(0.026)	(0.217)	(0.418)	(0.188)

Based on Fig. 5a, b, it is apparent that the total pressure curves of models (I) - (IV) show similar trends. For both patients, the pressure curves of models (II) are closest to the pressure curve of their corresponding models (I). Figure 5a, c and the values in Table 2 reveal that for patient A the pressure $P_{A(IV)}$ is overpredicted and the pressure loss $P{L}_{A(IV)}$ is underpredicted. At locations upstream of $C{S}_5$, the error $E_{A(IV)}$ ranges from $-31.3\%$ to $-33.0\%$. Considering model A(III), large overpredictions of $P{L}_{A(III)}$ are observed in the upper part of the pharynx from $C{S}_0$ to $C{S}_3$, with errors $E_{A(III)}>36.2\%$ models is due to the difference of the outlet pressures, cf. Fig. 5a, inherited from the adaptive boundary condition satisfying the Reynolds number condition, see “Computational Mesh and Simulation Setup”. Similar to patient A, the pipe-like structure of B(IV) also leads to reduced losses compared with model B(I), except for an increase at $C{S}_9$, as shown in Fig. 5b. In all cross sections of model B(III) higher errors $E_{B(III)}$ are obtained. Excluding the loss at $C{S}_8,E_{B(III)}>40\%$ for all the errors. In model B(II), the error of the losses is below $E_{B(II)}<7\%$ except at $C{S}_7$ and $C{S}_9$. Similar to model A, the pressure fluctuations near the outflow region are due to pressure differences that are prescribed to yield flows of the required Reynolds numbers.

Analysis of the Velocity Magnitude

Based on the following analysis, the results for model B are similar to those found for A. Therefore, only the results for patient A are discussed. In Fig. 6, velocity fields in the pharynx of patient $A$ are shown. In the x-/z-plane involving the middle of left and right nostrils, plots of the velocity magnitude are superimposed with in-plane velocity vectors. The oropharynx and hypopharynx of models A(I)–A(IV) reveal few qualitative differences in the velocity field. However, in the nasopharynx, various flow phenomena can be observed. Models A(I) and A(II) reveal two main streams of inflow, highlighted by the red arrows in Fig. 6a, b. The first stream originates directly from the inferior turbinate while the second stream passes through the middle turbinate and is then deflected downwards by the fornix of the pharynx. Consistent with the findings of the pressure loss, omitting the superior turbinate has no apparent impact on the fluid flow in the pharynx. Unlike models A(I) and A(II), the models A(III) and A(IV) have only a single main stream of incoming flow, as indicated by the red arrows in Fig. 6c, d. In model A(III), the complete flow passes through the inferior turbinate, which increase the magnitude of velocity in the nasopharynx related to the high pressure losses shown in Fig. 5. Conversely, the relatively large volume and the diverging form of the pipe-like structure decelerate the influx of fluid, resulting in the aforementioned underprediction of the total pressure loss. Furthermore, the recirculation zone highlighted by a green oval in Fig. 6d contributes to the deceleration of the flow.

Fig. 6.

Velocity magnitude in the pharynx of patient $A\mathit{in}x$-/$z$-planes in the middle of left and right nostrils superimposed by in-plane velocity vectors (a), Model A (I); (b), Model A (II); (c), Model A (III); (d) Model A (IV)

Considering the velocity magnitudes and in-plane velocity vectors in the $x$-/$y$-plane at $C{S}_0$ in Fig. 7, the flow fields in A(I) and A(II) are quite similar. The same is true for models A(III) and A(IV). From A(II) to A(III) a major change in the flow behavior is visible. In models A(III) and A(IV), two counter-rotating vortices are observed, see Fig. 7c, d. These vortices arise from the bend structure and feature a Dean-vortex-like behavior, which is usually observed in flow through ${90}^{\circ }$-bend elbow pipe geometries. This comes from the fact that the shapes in models (III) and (IV) come close to such of an elbow pipe. The vortical behavior of the flow is in the following further analyzed by means of streamlines in the next section.

Fig. 7.

Velocity magnitude in the pharynx of patient $A\mathit{in}x$-/$y$-planes at $C{S}_0$ superimposed by in-plane velocity vectors (a), Model A (I); (b), Model A (II); (c), Model A (III); (d) Model A (IV)

Streamline Analysis of the Flow

Figures 8 and 9 provide further details of the role of different surface variants in the development of different flow phenomena. In models A(I), A(II), and A(III), blue and red streamlines represent the flow entering the left and right nostrils, respectively. Flow entering the pipe-like structure A(IV) is visualized by green streamlines in Fig. 9b. The streamlines start in the corresponding inlet areas. The position of the cuts is shown along with cut or slightly opaque displayed geometries.

Fig. 8.

Streamlines entering left (blue) and right (red) nostrils for models A (I) and A (II). (a) Streamlines of model A (I); (b), streamlines of model A (II)

Fig. 9.

Streamlines entering left (blue) and right (red) nostrils for model A (III) (a) and the pipe-like structure (green) for model A (IV) (b)

In model A(I), fluid enters the nostrils and is split in both cavities into two main streams, following the lower and middle turbinate channels, as shown in Fig. 8a. The streams reunite again before entering the pharynx. Streamlines through the frontal, ethmoidal, and maxillary sinuses are not observed. The flow field in A(II), as shown in Fig. 8a, is similar to that of model A(I), except for as lightly compressed distribution of streamlines at the interface between middle and superior turbinate, where model A(II) has been cut. Unlike in the aforementioned cases, the two mainstreams are not given as much space to develop in model A(III). Upstream of the merging zone in the pharynx, the two counter-rotating vortices are shown in Fig. 9a. They correspond to the Dean-vortex-like phenomena described in “Analysis of the Velocity Magnitude”. The spinning motion of the vortices is caused by airstreams along the outer part of the inferior turbinate and the airflow through the inner part, which is then deflected at the nasopharynx. These phenomena are observed for incoming air through both the left and right channels. A similar pair of vortices is visible in Fig. 9b in model $A(IV)$. Here, obviously the streamlines are not compressed by narrow channels. Instead, the fluid flows uniformly through the pipe-like structure until it is deflected downwards.

Summary and Conclusions

The accurate prediction of the respiratory fluid mechanics via numerical simulative methods requires a precise representation of the human upper airway geometry. The geometry extraction process can be fully automated for CT data. Recording of CT data entails, however, the exposure of a patient to unnecessary radiation. In contrast, CBCT data can be generated rapidly using a substantially lower dose of radiation. Unfortunately, due to their high noise ratio and low contrasts, the segmentation of CBCT data is a challenge. The overhead of manual segmention of medical components can be reduced by decreasing the complexity of the nasal geometry.

To understand what degree of complexity reduction is necessary to yield simulation results that come with a tolerable error, highly resolved lattice Boltzmann simulations with the simulation framework ZFS have been performed on HPC systems for two patients with four stages of complexity reduction: (I) the original geometry, (II) a geometry including the middle and inferior turbinates, (III) a geometry including only the inferior turbinates, and (IV) a case where the nasal cavity has been replaced by a pipe-like structure. With each level of reduction from (I) to (III) the number of computational cells and hence the computational effort is reduced. The mesh of model (IV) consists again of more cells due to a volume increase.

A novel extraction pipeline that includes pre-filtering of CBCT data to enhance the image quality has been presented. The pre-filtering combines a convolutional and a gradient anisotropic diffusion filter to reduce the manual effort for segmenting the CBCT data.

The change in the area-averaged pressure shows that all models follow the original geometry (I) adequatly with model (II) via maximum approximation. The total pressure loss along the airway in the pharynxtolarynx region, as an indicator of the respiratory efficiency at inspiration, has also been analyzed for all reduced models. The results have been juxtaposed with the pressure loss values of the original geometry (I). They reveal that the relative error for only model (II) is sufficiently small to be usable in CFD computations. Considering the computational cost, the reduction to model (II) roughly halved the number of computational cells. To further corroborate these findings, the velocity magnitude has been analyzed in various cross sections along the pharynxtolarynx. While in models (I) and (II) the pharyngeal center is fed by two incoming streams from the nasal cavity, the upper stream is missing in models (III) and (IV). In the pipe-like geometry, a recirculation zone appears in this region. The missing upper stream also has an impact on the in-plane velocity magnitude behavior at the outlet. In both models (III){ and}(IV), Dean-vortex-like structures are found, which are obviously generated by the ${90}^{\circ }$ bent shape of the whole setup. These vortices are not detected in models (I) and (II). Finally, secondary flow phenomena are further analyzed based on the streamlines in the upper airways. They show how incoming fluid spreads through the complete middle and inferior turbinate in models (I) and (II), leading to a development of the aforementioned two streams. Additionally, they provide information on how the Dean-vortex-like structures arise.

To summarize, reducing the shape of the nasal cavity to less than the middle and inferior turbinates massively accelerates the segmentation. However, such a simplification results in unacceptable errors for the simulation. Yet, excluding the superior turbinate and keeping the middle and inferior turbinates still allows for an accelerated segmentation but with acceptable errors. The computational cost is then approximately bisected compared to the original geometry.

Numerical flow simulations are on the verge of becoming a common diagnostic tool for analyzing changes of the breathing capability after surgical interventions. Ideally, surgeries are conducted virtually and such an analysis can be done without an actual operation. To use the tool on a frequent basis, a large number of simulations is needed and efficient segmentation and computation techniques, as proposed in the present study, are mandatory.

Author Contributions

HA and MR performed the semi-manual segmentation. MR conducted numerical flow simulations. MR and AL extracted and interpreted simulation results. KCL designed the conception of the study. KCL, HA, MR, and AL drafted the article. WS was involved in manuscript editing. HA and MR revised the article critically for important intellectual content. All authors approved the version to be submitted. *, † : Contributed equally.

Declarations

Conflicts of interest

The authors declare that they have no conflict of interest.