Abstract: Computational Fluid Dynamic simulations are performed in real patient individual pharynx geometries of an Obstructive Sleep Apnea patient. The Navier-Stokes equations as well as the Reynolds Averaged Navier-Stokes equations and k − ∊ and k −ω turbulence models are used. The velocity profile and pressure distribution of the patient without any treatment and the patient wearing a mandibular advancement appliance are compared to each other. The simulation results for the different model conditions all lead to similar results showing the robustness of the numerical solutions. The pressure loss along the pharynx is lower in the presence of a mandibular appliance, which can indicate the reduction of OSAHS severity.
The Obstructive Sleep Apnea Hypopnea Syndrome (OSAHS) is a sleep related breathing disorder that leads to cessations in respirations due to reversible soft tissue deformations in the area of the human pharynx [1–4]. OSAHS is connected to a higher risk of stroke, cardiopulmonary death, hypertension as well as a reduction in life quality and depression [5–8]. In the understanding of the abnormal flow situation within the patient’s pharynx numerical simulation of the airflow in digital models of patient individual airway geometries may be a useful tool for diagnosis as well as therapy. In  an experimental comparison study is performed that uses an idealized geometry of a pharynx, which is a circular tube with a constricted area. Computational Fluid Dynamic (CFD) calculations are compared to the experimental measurement of velocity profiles in that tube. It could be shown that using computations based on the Navier-Stokes equations can simulate the true velocity profile quite well. The forgoing study is now extended to real patient data. The velocity profile as well as the pressure distribution in a real patient’s pharyngeal geometry obtained from medical image data is computed and compared to the same patient’s airway, when he is wearing a mandibular advancement appliance (MAA), which is a therapy possibility for OSAHS. The idea to reposition the mandibular with a dental appliance is to also reposition the attached soft tissue and remodel the pharyngeal lumen to improve the awkward pressure distribution. Within this study it is examined whether the numerical simulation shows up a change in the pressure or the velocity in the human pharynx due to a mandibular advancement. Despite that different simulation conditions are examined: the usage of the Navier-Stokes equations as well as their Reynolds Averaged parts together with two different turbulence models, k − ∊ and k − ω, are tested to validate the robustness of CFD calculations.
2 Material and Methods
Numerical simulations are performed using real patient’s data. Two different situations are compared: The patient’s natural airway geometry without any treatment and the same patient’s pharynx, where the patient is wearing a MAA. In order to prepare the real data for numerical simulations digital models have to be created.
2.1 Creation of Digital Models
The geometry of a patient’s pharynx suffering from OSAHS is extracted from a cone beam CT image by a watershed segmentation. Based on the segmentation the surface topology can be reconstructed using a Marching Cubes Algorithm and Laplacian Smoothing to gain a realistic smooth pharyngeal surface. The three-dimensional surface representation can be used as computational domain for the numerical simulation.
2.2 Numerical Simulations
The flow of the air in a human pharynx can be described by the stationary Navier-Stokes equations, which are a set of nonlinear partial differential equations of second order prescribing mass and momentum conservation of a fluid
where u is the fluid’s velocity vector, p the pressure, ρ the fluid’s density and μ its dynamic viscosity. I denotes the identity matrix. The values for viscosity and density are chosen to be μ = 18.82 · 10−6 Pa s and ρ = 1.15 kgm−3 matching the physical properties of air with a temperature of T = 34° C, which was measured as mean temperature value in the nasopharynx in 50 volunteers .
To deal with turbulence effects that may occur, the Navier-Stokes equations are averaged over time to get the Reynolds Averaged Navier-Stokes (RANS) equations
where U and P denote the time averaged values of u and p. The RANS equations introduce additional unknowns τ = −ρu′ × u′, which are called Reynolds stresses. The Reynolds stresses are the time averaged cross correlation of the turbulent fluctuations u′ multiplied by ρ. To deal with this closure problem of having more unknowns than equations, auxiliary equations are introduced that model the transport of turbulent parameters. The Reynolds stresses are therefore replaced by the so called turbulent viscosity
where k is the turbulent kinetic energy, ∊ is the dissipation, ω is the specific dissipation rate and Cμ = 0.09 is an experimentally determined constant. By using model equations for the kinetic energy k and either modeling equations for the dissipation ∊ or the specific dissipation rate ω solutions for the time averaged values for velocity U and P pressure can be obtained. The free stream behavior of a flow is quite different to the near-wall flow, hence the modeling equations for the turbulent parameters are incorrect in the near-wall regime. Therefore wall functions are used that estimate the near-wall flow based on a logarithmic profile. The k − ∊ turbulence model by Launder and Sharma  as well as the k − ω turbulence model by Wilcox  where used within this study. For all different simulation conditions and domains (with and without MAA) an inlet volume flow of uin = 400 ml s−1 was chosen and zero pressure at the outlet. All other boundaries were chosen to be no-slip walls. The simulations were performed with the commercially available simulation environment COMSOL Multiphysics©.
Within this work three different simulation conditions were tested:
the simulation without a turbulence model using the Navier-Stokes equations,
the simulation with the k−∊ turbulence model and the RANS equations and
the simulation with the k − ω turbulence model and the RANS equations.
Simulation resultswere obtained for velocity and pressure in the pharynx without a MAA as well as the same patient’s pharynx with MAA. Figure 1 shows the simulation results for the pharynx without MAA and with MAA obtained with the k − ∊ turbulence model. The velocity is increased after the most constricted part of the pharynx in both cases with and without MAA. But, it is of significance that the highest velocity in the pharynx without MAA is two times higher than with MAA. A higher velocity is caused by a loss in pressure, which could be the reason for the collapse of a pharynx. The same result shows up in figure 2, where mean values the dimensionless Reynolds number Re and the mean pressure are presented in axial slices with different distance z+ to the inlet. The increase of the Reynolds number is smaller in the case of the patient wearing the MAA, which is caused by the smaller velocities. Also the minimal mean pressure is smaller for the simulation without MAA and a reduction in pressure loss along the pharynx can be observed with the patient wearing the MAA. Figure 2 shows also that the results obtained under different simulation conditions (no turbulence model, k − ∊ model, k − ω model) are very similar to each other.
The fact that the three different simulation types all result in very similar values for the velocity and pressure is a good proof of the robustness of CFD as a tool in the patient individual treatment of OSAHS. The results obtained without any turbulence model often restricted to laminar flows is also similar to the results obtained using turbulence models, which may be a hint that turbulence e_ects are of less importance to the pharyngeal airflow.
5 Conclusion and Outlook
In diagnosis and therapy of OSAHS CFD are a powerful tool, because they enable the possibility to examine the abnormal fluid situation, velocity profile and pressure distribution in the patient’s individual pharynx. Within this study three main points show up:
simulation results obtained without turbulence model, with k − ∊ and k − ω model are very similar to each other,
the Reynolds number and the maximal velocity is decreased in the presence of a MAA and
the pressure loss is decreased in the presence of a MAA.
Pharyngeal collapses may result of an awkward pressure distribution, e.g. a low pressure behind a constriction that deforms the soft tissue. Decreased pressure lossmay therefore be a sign for an improvement of the disease. Hence, the decreased pressure loss in the presence of a MAA is exactly what is expected from the MAA as a treatment of OSAHS and confirmed by the simulation results. For a future task it is important to validate this results with a larger scale of patient data, so it may be possible to even correlate quantitatively computed values of pressure and velocity with a reduction of OSAHS severity and use them as patho-physiological parameters for diagnosis and patient individual therapy evaluation.
A further future task is to include a turbulence model with a better near-wall treatment in the examination, because in the experimental comparison study  the k − ∊ turbulence model suited on low Reynolds numbers without wall functions performed even superior to the simulation without turbulence model.
This work is supported by the SICAT GmbH & Co. KG, Bonn (www.sicat.de) and the Graduate School for Computing in Medicine and Life Science funded by Germany’s Excellence Initiative (DFG GSC 235/1).
Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent is not applicable. Ethical approval: The conducted research is not related to either human or animals use.
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