COMSOL 6.4 - Blood Flow in a Stenosed Pulmonary Artery

Blood Flow in a Stenosed Pulmonary Artery

Modeling blood flow through a stenosed pulmonary artery is crucial for diagnosing and treating cardiovascular diseases. Simulations can help predict how arterial stenosis impacts blood circulation and guide clinical interventions.

In this example, a Carreau fluid is used to capture the non-Newtonian behavior of blood. The stenosis is modeled as a porous medium to capture the resistance and flow disturbances caused by the narrowing.

Although this example does not provide real data or results, the demonstrated approach offers valuable insights into blood flow dynamics, which are crucial for improving diagnostic tools and treatment strategies for pulmonary artery stenosis.

Model Definition

The modeled geometry is shown in Figure 1.

Figure 1: Modeled geometry with stenosis (dark gray)

The idea for this model comes from Ref. 1. The authors studied the hemodynamics of chronic thromboembolic pulmonary hypertension (CTEPH), comparing the effects of Newtonian and non-Newtonian fluid models on blood flow and pressure within stenosed pulmonary arteries. The authors did not provide a detailed mathematical description of the stenosed area. In this model, a is used to simulate blood flow through the stenosis. The blood is modeled as a Carreau fluid, similar to the approach used by the authors. The apparent viscosity, denoted as μapp, of a Carreau fluid is defined as follows:

(1)

with the viscosity μinf at infinite shear rate, the relaxation time λ and the power index n and the shear rate

. To account for the effect of stenosis on the non-Newtonian behavior of blood, the apparent shear rate method is employed in Equation 1 using the capillary bundle approach. For the porous domain, an apparent shear rate is defined, which considers the influence of the porous material on the fluid’s non-Newtonian properties.

This is done by incorporating parameters such as the permeability κ and porosity εp of the stenosed region, the velocity magnitude |u|, and a correction factor α that accounts for the shape of the pores (for example, tortuosity). The correction factor needs to be determined experimentally or through numerical simulations at the pore scale. An approximation can be made using the capillary bundle approach:

with the tortuosity factor C. The parameters used in this model are summarized in Table 1 The Carreau parameters for blood are taken from Ref. 1. The parameters describing the porous medium are estimated values; one approach to determining these parameters involves modeling the stenosed area at the pore level and computing the permeability based on the pressure drop (Ref. 2).

Table 1: Modeling Parameters.
Quantity	Value	Description
εp	0.5	Porosity
κ	10-7 m2	Permeability
ρ	1120 kg/m3	Density of blood
μ0	0.056 Pa·s	Zero shear rate viscosity
μinf	0.00345	Infinite shear rate viscosity
λ	3.313	Relaxation time
n	0.3568	Power index

One measurement of the severity of the stenosis is the Quantitative Pulmonary Pressure Ratio (QPPR) which relates the pressures at both ends of the stenosis as follows

(2)

For this model, QPPR is determined for the maximum systolic pressure.

Results and Discussion

The velocity distribution at the end of the simulation is depicted in Figure 2.

Figure 2: Velocity field after 2.4 s.

The pressure at both ends of the stenosis over the last cycle is shown in Figure 3

Figure 3: Pressure over the last cycle at proximal and distal ends of the stenosis.

Notes About the COMSOL Implementation

The modeling process begins by opening an mph file that includes a mesh part. This mesh part imports the artery’s mesh, which is then adjusted to include the stenosis area and refined to ensure good resolution for fluid flow calculations.

The computation starts with a stationary step to get an initial velocity distribution for the subsequent time-dependent step over three blood flow cycles.

References

1. F. He, X. Wang, L. Hua, and T. Guo, “Non-Newtonian Effects of Blood Flow on Hemodynamics in Pulmonary Stenosis: Numerical Simulation,” Appl. Bionics Biomech., 1434832, 7 pages, 2023; doi.org/10.1155/2023/1434832.

2. P. Owasit and S. Sriyab, “Mathematical modeling of non-Newtonian fluid in arterial blood flow through various stenoses,” Adv. Differ. Equ., 340, 2021; doi.org/10.1186/s13662-021-03492-9.

Polymer_Flow_Module/Tutorials/pulmonary_artery_stenosis

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Component 1 (comp1)

Start by importing an mphbin file of the artery.

Mesh-Based Geometry 1

In the toolbar, click and choose .

Import 1

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file pulmonary_artery_stenosis_mesh.mphbin.

Click

Clear the checkbox.

Intersect with Plane 1

The stenosed area is introduced by modifying the imported mesh.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 8 and 9 only.

Locate the section. From the list, choose .

Select Boundary 7 only.

In the text field, type 3.8[cm].

Select the checkbox.

Click

Create Domains 1

In the toolbar, click

and choose .

In the window for , click

Free and Porous Media Flow, Brinkman (fp)

Now, continue with the setup of the physics.

Fluid Properties 1

In the window, under > click .

In the window for , locate the section.

Find the subsection. From the list, choose .

From the list, choose .

Porous Medium 1

In the toolbar, click

and choose .

Select Domain 5 only.

Fluid 1

In the window, click .

In the window for , locate the section.

Find the subsection. From the list, choose .

From the list, choose .

Find the subsection. From the list, choose .

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	1120	kg/m³	Basic
Zero shear rate viscosity	mu0	0.0560	Pa·s	Carreau model
Infinite shear rate viscosity	mu_inf	0.00345	Pa·s	Carreau model
Relaxation time	lam_car	3.313	s	Carreau model
Power index	n_car	0.3568	1	Carreau model
Porosity	epsilon	0.5	1	Basic
Permeability	kappa_iso ; kappaii = kappa_iso, kappaij = 0	1e-7	m²	Basic

Free and Porous Media Flow, Brinkman (fp)

Porosity and permeability are highly influenced by the degree of stenosis. The values used here are estimates (see Table 1).

Inlet 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Global Definitions

The inlet velocity is a time-dependent function that models the blood flow cycle and will be defined in the following steps.

Interpolation 1 (int1)

In the toolbar, click

and choose > .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file pulmonary_artery_stenosis_interpolation.txt.

Locate the section. From the list, choose .

Click

To make the interpolation function periodic, create an analytic function that references the interpolation function and enforces periodicity.

Analytic 1 (an1)

In the toolbar, click

and choose > .

In the window for , type v_in in the text field.

Locate the section. In the text field, type int1(t).

In the text field, type t.

Click to expand the section. Select the checkbox.

In the text field, type 0.8.

Locate the section. In the text field, type m/s.

In the table, enter the following settings:


Argument	Unit
t	s

Locate the section. In the table, enter the following settings:


Plot	Argument	Lower limit	Upper limit	Fixed value	Unit
√	t	0	2.4	0	s

Click

Next, call the function for the inlet velocity, which is time dependent and should be invoked with the argument t using the expression v_in(t). Since the computation first calculates a stationary solution to establish an initial velocity distribution for the subsequent time-dependent study step, the variable t for time is not defined at that stage. This would cause the stationary study to produce an error with the v_in(t) expression. To prevent this error, use the try_catch operator in the form v_in(try_catch(t,0)). This means that the expression will first attempt to evaluate v_in for t, and if t is undefined, it will default to evaluating v_in for 0.

Free and Porous Media Flow, Brinkman (fp)

Inlet 1

In the window, under > click .

In the window for , locate the section.

In the U0 text field, type v_in(try_catch(t,0)).

Outlet 1

In the toolbar, click

and choose .

Select Boundaries 4, 7, 10, and 19 only.

Now, build the physics-controlled mesh.

Mesh 1

In the window, under click .

In the window for , locate the section.

In the table, clear the checkbox for The geometric details might not be accurately captured this way, but they could just be artifacts from the imported mesh rather than real details. So there is no strong justification for including them in the mesh, and excluding them reduces computational effort.

Click