ROWAN UNIVERSITY

Table of Contents
1) Design Objectives ------------------------------------------------------------------------------ 3
2) Schematic of Carotid Artery ------------------------------------------------------------------ 3
3) Governing Equations -------------------------------------------------------------------------- 4
4) Boundary Conditions -------------------------------------------------------------------------- 4
5) First Simplified Model: Steady State Flow ------------------------------------------------- 5 a.
Problem Type Specification b.
Geometry c.
Mesh d.
Subdomain Settings e.
Boundary Settings f.
Solver Parameters g.
Postprocessing
6) Second Simplified Model: Pulsatile Flow -------------------------------------------------- 8 a.
Problem Type Specification b.
Geometry c.
Mesh d.
Subdomain Settings e.
Boundary Settings f.
Solver Parameters g.
Postprocessing
7) Carotid Artery with Bifurcation: Steady State Model ----------------------------------- 10 a.
Problem Type Specification b.
Geometry
8) Next Steps ------------------------------------------------------------------------------------- 11
9) References ------------------------------------------------------------------------------------- 12
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1.
Design Objectives: a.
Atherosclerosis is a disease in which plaque builds up in the carotid artery. It is directly related to low shear stress regions in the carotid [1]. b.
Therefore, we want to model the blood flow through this artery, where the bifurcation is of particular interest due to the fact that this is where the majority of the low shear stress regions are located [1]. c.
We will model blood flow in the carotid artery using two different models: a.
Men under 30 b.
Men over 50 d.
As can be seen in the measurements given below, younger and older men have significantly different values for the diameters of their carotid arteries. Therefore, we want to discover how this change in diameter affects shear stress. e.
We will evaluate shear stress at the bifurcation in both models. f.
The carotid will be made into a cylindrical 3D model in Solidworks and imported into COMSOL Multiphysics.
2.
Schematic for Carotid Artery for an Under 30 Male
177 mm 177 mm
208 mm
Figure 1. Carotid Artery Model for Under 30 Male
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Age Under 30
IMT Thickness (mm) [2]. 0.43
CCA Diameter (mm) [2]. 5.80
Over 50
0.52
6.33
ECA Diameter (mm) [1]. 22.2
ICA Diameter (mm) [1]. 31.1
CCA Length (mm) [1]. 208
24.23*
33.94*
227*
ECA Length (mm) [1]. 177
ICA Length (mm) [1]. 177
193.17*
193.17*
Table 1. Measurements in Carotid Artery
Note:
IMT Thickness = Wall thickness
CCA = Common Carotid Artery
ECA = External Carotid Artery
ICA = Internal Carotid Artery
* Values were estimated using the proportion from values found for CCA diameter.
The CCA diameter increased by a factor of 1.09138 from under 30 males to over 50 males. Since no information was available for over 50 males other than CCA diameter and IMT thickness, these values were estimated using this factor.
3.
Governing Equations: a.
Blood Flow: i.
Continuity: ∇ ∙ 𝑢 = 0 ii.
Navier-Stokes: 𝜌 (
𝜕𝑢
𝜕𝑡 b.
Shear Rate:
+ 𝑢 ∙ ∇𝑢) = −∇𝜌 + 𝜇∇ i.
𝛾̇ =
8𝑣 𝑑 a.
𝛾̇ is the shear rate (1/s)
2 𝑢 b.
v is the fluid velocity (m/s) c.
d is the inner diameter (m) c.
Shear Stress i.
𝜏 𝑤
= 𝛾̇𝜇 a.
𝜇 is viscosity (Pa*s)
4.
Boundary Conditions a.
Wall Shear: 𝜏 𝑤
= 𝜏(𝑧 = 0) = 𝜇
𝜕𝑢
𝜕𝑧
| 𝑧=0 i.
𝜇 is dynamic viscosity (Pa*s) ii.
𝑧 is height (m) iii.
𝑢 is velocity of the fluid (m/s) b.
Velocity profile: 𝑣(𝑟) = 𝑣 𝑜
(1 − 𝑟
2
𝑅 2
) i.
v(r) is velocity r distance from center (m) ii.
𝑣 𝑜
is velocity at center of the carotid inlet (m/s) iii.
𝑅 is the radius of the common carotid (m)
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c.
Center of common carotid velocity and average velocity relation:
= 2𝑣 𝑎𝑣 𝑣 𝑜
5.
First Model: Steady State Flow Through a Pipe a.
Problem Type Specification i.
This problem is a 2D Axisymetric Steady State Incompressible
Navier-Stokes Fluid Dynamics problem b.
Geometry i.
The geometry for a carotid artery consists basically of one straight tube which then bifurcates into two tubes. ii.
In order to make a preliminary model, we simply modeled a straight tube in COMSOL Multiphysics, representing the common carotid artery. a.
A 2-Dimensional model was used because the CCA can be assumed to be a symmetric cylinder. Therefore, making the model 2D would significantly reduce the computational domain while minimally affecting accuracy. iii.
The dimensions used are the same dimensions found in the CCA for males under 30. c.
Mesh i.
Since this is a simple 2D problem, a structured mesh was used. ii.
Mesh Statistics a.
Number of mesh points: 1811 b.
Number of elements: 3208 d.
Subdomain Settings i.
We need to define property values for both the density and viscosity of the fluid which will be traveling through the pipe.
Since in the actual model blood will be the fluid, it will also be used here. a.
𝜌 = 1055 𝑘𝑔/𝑚 3
[3]. b.
𝜇 = 0.0035 𝑃𝑎 𝑠 [3]. ii.
The initial value used for velocity will be 0.4 m/s. This value comes from the average value for pulsatile flow in a CCA. [4]. e.
Boundary Settings i.
There are a total of four boundaries in the simplified model. These are shown in Table 2 and Figure 2.
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Boundary
Number
1
2
3
4
Boundary
Type
Wall
Inlet
Outlet
Wall
Boundary
Condition
No-slip
Velocity
Pressure
No-slip
Quantity
N/A
𝑈
𝑃
𝑂
𝑂
N/A
Table 2. Boundary Conditions for First Simplified Model
Value
N/A
0.4 m/s
101000 Pa
N/A z
BC1) Wall r
BC3) Outlet
BC2) Inlet
BC4) Wall
Figure 2. Schematic of Boundary Conditions for First Model

Note: Schematic not drawn to scale for spatial reasons. ii.
Boundaries 1 and 4 correspond to the walls, which are specified to have the no-slip boundary condition in COMSOL. iii.
Boundary 2 is specified as the inlet under boundary type. This boundary condition is a velocity condition where the value is 0.4 m/s. iv.
Boundary 3 is specified as the outlet boundary type. The boundary condition here is pressure. The value of pressure is 101000 Pa, or atmospheric pressure. f.
Solver Parameters i.
The Direct (UMFPACK) solver was used for this solution. g.
Postprocessing i.
This model can be validated by using equations from simple fluid dynamics. ii.
Volumetric flow rate:
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a.
𝑉̇ = 𝑉 𝑎𝑣𝑔
𝐴 𝑐
= (0.305
𝑚
) ∗ (5.8 𝑚𝑚 ∗ 208 𝑚𝑚) 𝑠 b.
𝑉̇ = 0.000368
𝑚
3 𝑠 iii.
Pressure drop a.
∆𝑃 = 𝑃
1
− 𝑃
2
=
8𝜇𝐿𝑉 𝑎𝑣𝑔
𝑅 2 b.
∆𝑃 = 211.2 𝑃𝑎 c.
From figure 3, it can be seen that the pressure drop is approximately 200 Pa, a value extremely similar to the analytical results obtained above.
Figure 3. Pressure Drop for Simplified Model 1
7

Figure 4. Velocity Field for Simplified Model 1
6.
Second Model: Pulsatile Flow Through a Pipe a.
Problem Type Specification i.
This problem is a 2D Axissymetric Transient Incompressible
Navier-Stokes Fluid Dynamics problem b.
Geometry i.
The same geometry will be used in this model as was in the first simplified model. c.
Mesh i.
Since this is a simple 2D problem, a structured mesh was used. ii.
Mesh Statistics a.
Number of mesh points: 1881 b.
Number of elements: 3328 d.
Subdomain Settings i.
The values for density and viscosity of blood stay the same. a.
𝜌 = 1055 𝑘𝑔/𝑚 3
[3]. b.
𝜇 = 0.0035 𝑃𝑎 𝑠 [3]. ii.
The initial value used for velocity will be sin(0). Since the input velocity is sinusoidal, the initial value will be sin(t) at t=0.
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e.
Boundary Settings i.
The boundary conditions are consistent with our first simplified model except for our inlet velocity. This will be changed to a sinusoidal input to model pulsatile flow.
Boundary
Number
Boundary
Condition
Quantity Value
1
2
3
4
Boundary
Type
Wall
Inlet
Outlet
Wall
No-slip
Velocity
Pressure
No-slip
N/A
𝑈
𝑃
𝑂
𝑂
N/A
N/A
Sin(t)
101000 Pa
N/A
Table 3. Boundary Conditions for Second Simplified Model f.
Solver Parameters i.
Since this is a transient problem, a time step will need to be used. ii.
The range of times will be 0 to 0.75 seconds with a time step of
0.001 seconds. iii.
The Direct (UMFPACK) solver was used for this solution. g.
Postprocessing i.
The graphs included for this model are for velocity and pressure distribution.
Figure 5. Velocity Field for Simplified Pulsatile Flow
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Figure 6. Pressure Drop for Simplified Pulsatile Flow
7) Carotid Artery with Bifurcation: Steady State Model a.
Problem Type Specification i.
This problem is a 3D Steady State Incompressible Navier-Stokes Fluid
Dynamics problem. b.
Geometry i.
The geometry here is the actual geometry which will be used in our under
30 male model. ii.
The dimensions used here are the same as those specified in section 2. iii.
Figure 7 shows the layout of the carotid in COMSOL Multiphysics.
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Figure 7. 3D Carotid Model in COMSOL
8) Next Steps a.
Finish model for steady state bifurcation analysis. b.
Solve model and verify through already solved models. c.
Create a model for pulsatile flow with carotid artery bifurcation. d.
Solve model. Verify through other models found in papers. e.
Perform sensitivity analysis on final model to get better results.
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References
[1]. Urquiza, S. A., P. J. Blanco, M. J. Venere, and R. A. Feijoo. Multidimensional
Modelling for the Carotid Artery Blood Flow . Thesis. Univesidad Nacional De Mar Del
Plato, 2005. Print.
[2]. Denarie, N., J. Gariepy, G. Chironi, M. Massonneau, F. Laskri, J. Salomon, J.
Levenson, and A. Simon. Distribution of Ultrasonographically-assessed Dimensions of
Common Carotid Arteries in Healthy Adults of Both Sexes . Thesis. Paris, France, 1999.
Print.
[3]. Gay, Mickael, and Lucy T. Zhang. "Numeric Studies of Blood Flow in Healthy,
Stenosed, and Stented Carotid Arteries." International Journal for Numerical Methods in
Fluids (2008). Print.
[4]. Datta, Ashim K., and Vineet Rakesh. "Flow in Human Carotid Artery Bifurcation."
An Introduction to Modeling of Transport Processes: Applications to Biomedical
Systems . Cambridge, UK: Cambridge UP, 2010. 262. Print.
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