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Fluid Dynamic Model Analysis of Velocity and Pressure Changes in

Varying Degrees of Tracheal Stenosis

Truong Mai, Sarah Kazmi, Vincent Mah, Sean EsquedaDepartment of Bioengineering, University of California, Riverside, Riverside, CA, U.S.A

Abstract - Tracheal stenosis patients are normally asymptomatic until a significant amount of narrowing occurs which tends to require immediate attention. Understanding the air fluid dynamics can assist in early detection of tracheal stenosis. The purpose of our project was to utilize computerized software to visualize the air fluid dynamics of varying degrees of tracheal stenosis at the narrowest point of stenosis. We based our model on the works that were done by Brouns in which they utilized CT scans in order to simulate more realistic models of the trachea. Our main goal was to replicate their models and explore the similarities and differences along with the benefits and disadvantages that the model possessed. We analyzed the behavior of air flow my finding the velocity and pressure drops in varying degrees of tracheal stenosis (0%, 75%, 90%). (INSERT OVERALL CONCLUSION HERE)

Nomenclature

COMSOL – COMSOL Multiphysics© software

Introduction

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When a patient undergoes a condition of tracheal stenosis, the windpipe experiences a narrowing that prevents an individual from breathing normally. The causes of this are trauma, inflammatory diseases, and collagen vascular diseases[1]. Stenosis impedes the air to the lungs and causes abnormal flow and pressure. However, often times symptoms do not arise until the condition becomes so severe that immediate attention is required. Patients do not usually experience breathing trouble until the stenosis takes over around seventy-five percent of the trachea. This leads to the question of why breathing troubles are reported well after the stenosis has appeared. In order to answer this question, it seems most apparent to look at the air flow in the trachea, around the stenosis area.

Because of the intricacy of computing the air flow field has been very challenging for scientists to do using experimental measurements, research is slow for this area. However, the understanding of fluid dynamic studies in the upper airways has regained a new sense of interest because of the vast possibilities offered by computational fluid dynamics applied to this area[5]. Computational fluid dynamics allows you to use the Navier-Stokes equations in order analyze and solve problems related to fluid flow. Recently, Brouns & Jayaraju modeled the airflow characteristics on tracheal flow, mainly using Reynolds-averaged Navier-Stokes modeling methods because of its low computational costs. He noticed that the pressure drop in the trachea is dominated by the cross-sectional area that the stenosis occupies.

Our report will simulate several models of air flowing downwards through simplified tracheal models. We will first start with a normal trachea without a stenosis, then move on to stenosis of varying sizes applied to the trachea model. The velocity and pressure will be computed for the air flowing through the stenosis, as well as the vorticity each size of the stenosis creates. The overall purpose of this experiment is to explain why patients only state breathing troubles starting at a certain percentage that the stenosis populates in the trachea.

Method

Using the Navier-Stokes equation incorporated in COMSOL, we planned our experiment around finding the velocity profile and pressure drop downwards through the trachea with and without the stenosis.

We are able to model the velocity field and the pressure in our model because these values are located within the Navier-Stokes equation, as shown here:

It can be assumed that this model is not time dependent, making the model much simpler and making it faster to compute and solve.

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Along with the Navier-Stokes equation, we use the continuity equation in order to determine the boundary conditions and to show that our fluid model is incompressible:

Because we are defining the relationship between the velocity and pressure, we will also include Bernoulli’s equation to show the connection:

The last equation we will use for this model is Reynold’s equation:

We will use Reynold’s number to determine the kind of flow along a certain point within the system at varying sizes of stenosis.

Assumptions: The assumptions that we were required for us to make the model on COMSOL simplify the human trachea because an actual trachea isn’t perfectly cylindrical. Our model consisted of five varying blocks that were smoothed out to replicate the bronchi and the cartilage rings. These blocks are lined along the rotating axis which will later create a 3-dimensional mode. A trachea of this kind will not physiologically be correct because a trachea isn’t a straight tube. The trachea has a flat back and it has slight curves rather than being completely straight as in our model. To simplify the Navier-Stokes equation, we conducted a stationary model to simulate a steady-state model. We also assumed that the fluid was Newtonian and incompressible so that the fluid – in our case air – has a constant viscosity and density respectively. The flow was also assumed to be laminar rather than turbulent due to a limitation in COMSOL, and the walls are taken with a no-slip condition. Geometry: The dimensions for our 2-dimensional model are taken from the upper limit a normal human male trachea of with a radius 1.35 cm (Breatnach et al.). This radius was used to depict the bronchi and the cartilage ring was found to be 1.25 cm according to Patel et al. Rather than looking at a full length trachea, we analyzed a segment of the trachea with a 6.5 cm length to further simplify the anatomy of the trachea. The three bronchi “rings” have a height of 1.5 cm and the height for the two cartilage rings are 1 cm. For the normal trachea with 0% stenosis, the five “rings” are combined together through a union. A circle with its center placed along the bronchi wall at 3.25 cm is made and subtracted to create a stenosis. The radii of the circle is adjusted and subtracted from the tracheal wall to simulate various percentages of stenosis. The geometry of this model is shown at Figure 1.

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Subdomain: The density of air was found to be 1.177 kg·m-3 (“Air Properties Definitions”). Assuming air as a Newtonian Fluid is significantly important in our model because the viscosity of air will be held constant at 0.00002 Pa·s (“Viscosity”). Boundary: The walls of the trachea were set to be no-slip and have inlet velocity at the top of the trachea with 0.87 m/s. The inlet velocity used was calculated by using a volumetric flow rate of 30 L/min which was provided by Jayaraju et al. Knowing the cross-sectional area of the inlet, we divided the volumetric flow rate with cross-sectional area of the inlet and was able to calculate the inlet velocity. COMSOL could not produce an answer without an outlet value so we set the outlet to have a velocity of 0.5 m/s.

Figure 1: The diagram above depicts the dimensions of the trachea on 2D axi-symmetrical over r-axis of our COMSOL model. The air will flow from the left to right.

This is a 2D symmetrical model over z-axis, so it is not necessary to make 3D model. This is There is airflow from the left and air out on the right side of the model. Initial velocity is 0.8732m/s and zero pressure. Some assumption is incompressible, no slip, gravitational force neglected, and velocity on z-direction only and at steady state. Our geometry is RCCS. Boundary condition is at x = 0, v(z) is finite; at x=r, v(z) is zero.

Table 1: Material properties of air.

Density 1.177 kg/m3

Viscosity 0.00002 Pa·s

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Figure 2: This figure illustrates the finer mesh that was applied to our model to obtain more accurate results.

Results:Table 2: Comparison of initial conditions from literature and the values applied to our model.

Literature Values Our Values

Stenosis Size Depends on Degree of Severity

1 mm (100% stenosis) Radius of 1.35 cm (100% stenosis)

Stenosis Shape Trapezoid Semi-circle

Inlet Velocity 0.8732 m/s 0.8732 m/s

Mesh 750,000 cells, hexahedral shaped

Finer

Table 3: This table shows the values of velocity at the point of interest (r,z) = (0, 3.25).

Percent Stenosis Velocity (m/s) Percent Stenosis Velocity (m/s)

0 0.8732 70 9.8000

25 1.5000 75 14.0000

50 3.4000 80 22.0000

55 4.2000 85 39.5000

60 5.3500 90 88.0000

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65 7.2300

Figure 3: Velocity field at zero percent stenosis (m/s). Goes straight downwards with laminar flow. This is our control variable for the report.

Figure 4: The figure above displays the pressure at 0% stenosis (Pa). This is our control variable for the report.

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Graph 5: This graph shows the values of the small pressure drop in the stenosis from top to bottom for 0% stenosis. Control variable.

Figure 6: Velocity is at 75% stenosis (m/s). Velocity after stenosis become bigger because the pressure drops after stenosis larger.

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Figure 7: Pressure’s trend at 75% stenosis. At the z= 0.75, pressure’s trend is curl shape, this predicts that the velocity will move in different directions.

Figure 8: Pressure at 75% stenosis (Pa). There is a lot pressure before the stenosis and drops significantly after stenosis.

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Graph 9: Graph for the pressure at 75% stenosis (Pa). The pressure begins from the right to the left because the airflow from top to bottom of trachea. Pressure drops show

the flow from high pressure to low pressure. Then pressure stabilizes from 3 to 0.

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Figure 10: Velocity at 90% stenosis (m/s). As explanation in figure 6, the velocity is higher after stenosis. Velocity becomes weaker from z=0 to 1.35 and expresses the no-

slip condition. The vorticity is developed at the end of the trachea.

Figure 11: Pressure’s trend at 90% stenosis. In this figure, it is much easier to see the vorticity at the bottom of the trachea

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Figure 12: Pressure at 90% stenosis (Pa).

Graph 13: Pressure at 90% stenosis (Pa).

Figure 14: The figure shows velocity increases proportional to the percentage of stenosis at point (0,3.25)

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Table 4: Velocity from our Comsol model and from literature data

Percent Stenosis of our data (%)

Our velocity at stenosis area (m/s)

Percent Stenosis from literature data (%) [15]

Velocity at cross-section D in literature (m/s) [15]

50 1.62 49 7.5

75 3.37 75 10.5

85 5.94 84 12.5

90 8.7 91 21.5

Table 5: Percent error between velocity of our data and literature data

Percent Stenosis (%) Percent Error (%)

50 78.3

75 67.9

85 52.5

89 59.5

Table 6: Difference pressure of our data and literature data and percent error

Our difference pressure (Pa)

Literature difference pressure (Pa)

Percent Error (%)

Percent Stenosis (%)

75 7 51 86.27

90 57 680 91.61

Discussion (Sarah and Vincent) - why these results could be significant (what the reasons might be for the patterns found or not found)Velocity Analysis:

Looking at our 2-D axi-symmetric model, we were able to show that the magnitude of the velocity increased as the degree of stenosis increased. Comparing our velocity values for varying sizes of stenosis with the literature values, we were not able to reproduce the results of Jayaraju’s simulation. For Jayaraju’s simulation, he got a max velocity of 91 m/s for 90% stenosis at a volumetric flow rate of 30 L/min, while we got a max velocity of 9.63 m/s at the same volumetric flow rate and stenosis. Even

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though we could not get the same values, our model at least followed the same trend for increasing velocity as Jayaraju’s model. In our study, there was only a slight increase in velocity for the 0-75% stenosis cases. However, after 75%, there were significant increases in velocity for the model. This also occurs in Jayaraju’s study, showing that our model can accurately represent the velocity field through the trachea.

Vorticity Analysis:In laminar flow, the vorticity will be 0 at the axis. Our model with a 75% stenosis

shows a vorticity below the stenosis which is indicated by the stream lines. In the trachea model with a 90% stenosis, the streamlines indicated that there was also vorticity. The further away from the axis, the higher the vorticity will be. The vorticity found does not mean there is turbulent flow as swirling can happen in laminar flow.

Pressure Analysis:The pressure drops we found at a 75% stenosis and 90% stenosis was 7 Pa and

57 Pa respectively. Jayaraju found that at a 75% stenosis the pressure drop was 51 Pa and a 90% stenosis produced a 680 Pa pressure drop from the inlet to the outlet. The small change in pressure relates to why our velocity values at these stenosis percentages are lower compared to the values found by Jayaraju et al. The fluid requires a pressure drop in order to move down the trachea. With low pressure drops, the velocity of the fluid will be lower. Our small pressure drops may have been due to a lower inlet pressure. The size of the stenosis is the cause in the pressure drops, but if the inlet pressure isn’t high enough, then the pressure drop cannot replicate the values found from Jayaraju et al. In the study by Jayaraju et al, the pressure drop significantly increased between 75% and 90%. Though not as significant of a change, we found that the pressure drop in our 75% and 90% stenosis model showed a sizeable increase.

Error AnalysisThere is error in our velocity and pressure difference due to limitations. The

method used to find our values at the point of interest (r,z) of (0,3.25) was picked by clicking on the surface of the velocity graph on comsol which hold to be very accurate. The errors found in with these values can be due to the geometry of our model. Our model is very simplified compared a real human trachea. The literature model we compared to was created using a CT scan of the upper respiratory tract. Unlike the CT scanned model, our model is more of a straight cylinder with ridges. Another reason is that COMSOL is limited to only using a Laminar Flow Study. According to Jayaraju et al., a trachea with a volumetric flow rate of 30 L/min will already have a turbulent flow before the air reaches the stenosis. Because of this COMSOL limitation, the values obtained from COMSOL may have been affected. Since we cannot observe the turbulent flow in a laminar flow study, we examined the vorticity after the point of stenosis. A Time-Dependent Study may give us a more accurate pressure drop

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because the pressure may change over time. Also, we didn’t corporate the Young’s Modulus and Poisson’s Ratio to our model, so our model lost viscoelastic properties. The trachea in both our model and literature model didn’t simplify the surface tension because of assuming the trachea as a dry airflow. Lastly, the accuracy of our values are mesh dependent. The finest mesh we were able to run our model simulation was a finer mesh. Coarser meshes produced lower velocity and pressure values.

ConclusionIn this report, it has been determined that:(1) As velocity increased, there was a pressure drop within the trachea. This is known to be due to their relationship in the Bernoulli equation.

(2) The vorticity in the trachea after the stenosis increased as the stenosis was made bigger

Some of our values compared to literature are different in the model, however they both share the same trend that we were looking for. If we were able to modify our model in Comsol to be a little more realistic of the trachea, it can represent an efficient way of studying how a stenosis affects airflow.

AcknowledgementsWe would like to thank Dr. V.G.J. Rodgers for consistently supporting and motivating us to perform at our highest potential.

References[1] Coombs, Bernard D., Richard Webb, and Robert Krasny." Tracheal Stenosis Imaging. Medscape, 7 Nov. 2013. Web. 14 Mar. 2015.[2]Air Properties Definitions. N.p., n.d. Web. 14 Mar. 2015.[3] "Bernoulli Equation." Pressure. N.p., n.d. Web. 13 Mar. 2015.[4] Breatnach, E., Gc Abbott, and Rg Fraser. "Dimensions of the Normal Human Trachea." American Journal of Roentgenology 142.5 (1984): 903-06. Web.[5] Brouns, Mark. Jayarayu, Santhosh T. Lacor, Chris. Mey, Johan De. Noppen, Marc. Vincken, Walter. Verbanck, Sylvia. Tracheal Stenosis: A Flow Dynamics Study. 28. November. 2006. Print. [6] Malvè M., del Palomar A. Pérez, Chandra S., et al. FSI Analysis of a Human Trachea Before and After Prosthesis Implantation J Biomech Eng 133(7), 071003 (2011) (12 pages); doi:10.1115/1.4004315[7] Butler, JP., M. Nakamura, H. Sasaki, T. Sasaki, and T. Takishima. "Result Filters." National Center for Biotechnology [8] N. Viscosity Handout (n.d.): n. pag. Web.[9] Patel, N., RP. Mahajan, and FR. Ellis. "Result Filters." National Center for Biotechnology Information. U.S. National Library of Medicine, Jan. 1993. Web. 16 Feb. 2015

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[10] "Reynolds Number." Reynolds Number. N.p., n.d. Web. 14 Mar. 2015.[11] Tang, Julian W., Andre D. Nicolle, Christinan A. Klettner, Jovan Pantelic, Liangde Wang, Amin Bin Suhaimi, Ashlynn Y. L. Tan, Grarrett W. X. Ong, Ruikun Su, Chandra Sekhar, David D. W. Cheong, and Kwok Wai Tham. "Airflow Dynamics of Human Jets: Sneezing and Breathing - Potential Sources of Infectious Aerosols." N.p., 1 Apr. 2013. Web. 16 Feb. 2015.[12] 2.9 The Vorticity Equation (n.d.): n. pag. Web.[13] Tang, Julian W., Andre D. Nicolle, Christinan A. Klettner, Jovan Pantelic, Liangde Wang, Amin Bin Suhaimi, Ashlynn Y. L. Tan, Grarrett W. X. Ong, Ruikun Su, Chandra Sekhar, David D. W. Cheong, and Kwok Wai Tham. "Airflow Dynamics of Human Jets: Sneezing and Breathing - Potential Sources of Infectious Aerosols." N.p., 1 Apr. 2013. Web. 16 Feb. 2015. (velocity of air in bronchi)

[14] Warhaft, Z. "Transition and Turbulance." The Engine and the Atmosphere: An Introduction to Engineering. N.p.: Cambridge UP, 1997. N. pag. Princeton University. Web. 14 Mar. 2015.

[15] Jayaraju, Santhosh T. Brouns, Mark. Lacor, Chris. Verbanck, Sylvia. Mey, Johan De. Effects of Tracheal Stenosis on Flow Dynamics in Upper Human Airways. 2006. Print.

Appendix1) Calculation

a) Inlet velocityQ= 30L/min = 0.0005m3/s [10]r=1.35cm = 0.0135m

This value makes sense because the real value from literature is about 0.8 to 1.3m/s [12]

b) Difference pressure

2) Reynold number [10]

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3) Bernoulli’s Equation [3]

4) Vorticity equation [12]

5) Navier stoke equation

6) Continuity

(assuming the velocity is function of z only)7) Percent Error (%)

In figure 3, the maximum velocity is about 1m/s and uniform in the whole trachea. this happens because of two reasons. The inlet velocity is 0.8732m/s. There is an increase in velocity because the radius of the bronchi is larger than cartilage. As the result, the cross-section area increase back and forth.

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In figure 4, there is a