Course of Fenomeni di Trasporto nei Sistemi BiologiciLab. of Simulazione Computazionale

Prof. A. Dubini – Eng. E. Bianchi

Giulio De Riccardis 837403

A******** B******** ******

Fluid-dynamics study on an axisymmetric model of a drugeluting stent

• Abstract

The Arteriosclerosis disease is the thickening and hardening of a medium-large caliberartery wall which is established because of risk factors like: age, sex, tobacco, diet,diabetes mellitus, obesity, hypertension, etc. It results as a fibrous accumulation and alumen constriction, until complete obstruction of the vessel. Arteriosclerosisrepresents the first cause of death, with the 35-38% of the total deaths (USA andEurope, 2005).

If preventively recognized, modern surgical approaches concern in drug treatment,angioplasty, vessel substitution, its bypass or stent implant.

Using a catheter and a balloon, the stent is placed around the plaque. The balloon isinflated, expanding the stent until complete adhesion to the internal wall of the artery,the plaque is crushed and the balloon removed.

The major development in stent design evolution is represented by DES, drug elutingstent, with metal core and polymeric coating filled of anti-thrombotic and anti-proliferative drug.

Aim of this project is therefore to realize, through the use of the ANSYS Fluentsoftware, a CFD analysis on a stent inserted in the cardiovascular system and thetransport mechanism which regulates the release of drug from the same.

The stent is modeled as a series of n.5 struts with circular section and distance of 0.03m from the center of the struts to the interior wall of the vessel, while the arterysection under consideration as straight circular tube with wall of porosity andpermeability known.

The adopted axisymmetric model let us study the fluid-dynamics in just one section ofthe vessel and extend results and considerations to the 3D structure, significantlyreducing the computational cost of the analysis and the time necessary to design thegeometry.

The input fluid has an already developed velocity profile, described by a parabolicfunction which good approximates the velocity profile of the blood in situ. Thecomputation of the flow rate and the flux on the external wall provides us moreinformations about transport modalities.

• Geometry

Name Value x 10 - 3 Unit of MeasureH1 9.6 [m]H2 3 [m]

H3, H4, H5, H6 0.9 [m]V1 1.5 [m]V2 0.5 [m]V3 1.47 [m]

D1, D2, D3, D4, D5 0.2 [m]

• Mesh

31'181 total elements, with minimum and maximum dimensions of 4.8877E-6 m and 9.7755E-4 m respectively.

• Fluid Properties

1) Blood

Name Value Units of MeasureDensity 1057 [Kg/m3]Viscosity 0.0035 [Pa*s]

Cp 1006.43 [j/(kg*K)]Thermal Conductivity 0.0242 [W/(m*K)]

2) Drug

Name Value Units of MeasureDensity 1030 [Kg/m3]Viscosity 0.0035 [Pa*s]

Cp 1006.43 [j/(kg*K)]Thermal Conductivity 0.6 [W/(m*K)]

Molecular Weight 28.966 [Kg/kmol]

3) Mixture

Name Value Units of MeasureDensity Mixing Law* [Kg/m3]Viscosity 0.0035 [Pa*s]

Cp Mixing Law** [j/(kg*K)]Thermal Conductivity 0.0454 [W/(m*K)]Diffusion Coefficient 1E-6 [m2/s]

* ρ =Ff* ρ f + (1- Ff)* ρ s

where: ρf = Drug density ρs = Blood density Ff = Drug volumetric fraction

** Cp= Ff*Cpf + (1- Ff)*Cps

where: Cpf = Drug specific heat Cps = Blood specific heat Ff = Drug mass fraction

4) Vessel wall

Name Value Units of MeasurePorosity 0.61 -Permeability 2E-10 [m2]

• Boundary Conditions

Name Type Expression Units of MeasureWall_in Wall Zero Diffusive Flux [Kg/(m2*s)]

Wall_out Wall Zero Diffusive Flux [Kg/(m2*s)]

External_wall pressure outlet P = 0ωf =0 ( Hp: zero retroflux)

[Pa][-]

Inlet velocity inletv=

vmax

r vaso2 ∗y2+vmax

ωf =0

[m/s]

[-]

Outlet pressure outlet P = 0ωf =0 ( Hp: zero retro-flux)

[Pa][-]

Axis axis Symmetric axis [-]Strut wall ωf =1 [-]

• Regimen, Solver and Settings

Regimen: StationarySolver: Pressure-Based

Convergence residuals:

Variable Value

Continuity 1E-6

Velocity in x dir. 1E-6

Velocity in y dir. 1E-6

Drug mass fraction 1E-9

• Solutions and Results

1) Plot of convergence residuals:

Convergence condition for velocity and continuity residuals is E-6; while for drug mass fraction is equal to E-9. The residuals converge quickly, indication of a well built model.

2) Plot of test variables stabilization:

The monitored variables are: the maximum of output velocity and the mass fraction of drug on the wall. Both tend to a horizontal asymptote which confirms, associated to the residuals convergence, the validity of the found solution.

3) Check of the mass flow rate conservation law:

The difference between the input and output mass flow rate is of E-11 order, can thenbe approximated to zero.

4) Input and output velocity profile:

Due to the small size of the model, an already developed input velocity profile isimposed, then parabolic, with a maximum value equal to 0.14 m/s. The outputmaximum value results lower (0.13 m/s), since the flux in the radial direction throughthe vessel wall, considered porous, is not zero.

5) Compute of the Reynolds number:

Re= ρ∗v̄∗D

μ = 84.56

The motion regimen is laminar, in fact Re results two orders of magnitude lower thanthe limit (Re=2300). The computed Re refers to the only blood motion and itsproperties, since the velocity profile has been solved decoupled from the drugtransport. Moreover it has been used the average value of the input velocity, whichrepresents the worst case.

6) Colored maps of velocity:

Magnitude:

Axial Component:

Radial Component:

The colored maps show that preferred direction is the axial one, while the radialvelocity maintains low values. It is noticed for the latter peaks of E-3 order (positivesand negatives) in proximity of the struts, where the fluid motion in axial direction isimpeded.

7) Colored map of the drug concentration:

The drug diffuses preferentially in the radial direction and part along the wall. The presence of drug within the lumen is approximated to zero.

8) Compute of the local (plot) and total (value) mass flow rate:

Total mass flow rate = 1.624E-5 Kg/s

9) Compute of local (plot) and average (value) flux:

Average flux= 0.135 Kg/(m2*s)

The flow rate and the flux plots along the External_wall (“parete” in these screenshots)of the vessel reflect the trend of the radial velocity and the drug distribution. In factthey present local maximum in proximity of the struts.

10) Evaluation of blood recirculation / zero flux areas:

The map does not show blood recirculation or zero flux areas in proximity of the firstthree struts, while downstream of the fourth and the fifth one can be seenaccumulation zones due to the reunification of the two fluxes.

Closed to the Wall_out and in proximity of the separation surface among wall andlumen, due to the discontinuity of the flux between Outlet and Wall_out, anaccumulation zone is also present.

• Conclusions

The results of the analysis confirm the validity of the model and the assumptionsmade to study the behavior in situ of a drug eluting stent and its local interactionswith blood flux.

It results of particular interest the colored map of the drug concentration, showing howthe latter is only transported in the radial direction and along the wall, performing itsjob without affecting zones of the body system different from that under analysis. Infact, role of the drug is to prevent coagulation of blood, thrombogenesis andneointimal hyperplasia in the neighborhood of the stent struts, where wall lesions andpossible stagnation / recirculation of blood are present.

The areas which present blood stagnations ( downstream of the fourth and the fifthstrut) are due to the reunification of the two fluxes and to the circular section of thestrut, which with the quadrilateral section today represents the state of art.Consequently the oxygen and nutrients supply results reduced in the interested zones.The presence of recirculations, not highlighted in this study, would subject endothelialcells to not physiological shear stress and would push the same to proliferate,resulting in re-narrowing of the lumen. Instead, the stagnation area closed to theWall_out and in proximity of the separation surface between wall and lumen is only theresult of the boundary conditions imposed, which create discontinuity among the twosurfaces.

The team is satisfied of the results obtained in the short time available and grated tohave learned, under the guide of the Eng. Bianchi, the use of ANSYS Fluent as apowerful instrument for fluid-dynamics analysis.