This paper deals with the probabilistic study and analysis of stresses produced due to the blood flow in a stenosed and a non stenosed artery using FEM. The probabilistic analysis was done using the random variables and their mean values (varying +/- 10%) to compute the sensitivity factors. Normal distribution was assumed for all the random variables. Similar procedure was employed for the stenosed artery and the solutions are produced in the form of easily understandable graphs and diagrams which can be used for diagnostic purposes

• Uncertainty quantification

• Probabilistic analysis

• Fluid/solid interaction

• Arteries

Gorla RSR (2024) Probabilistic Study of Fluid / Solid Interaction in Arteries. Ann Vasc Med Res 11(2): 1179.

Atherosclerosis is a dangerous disease caused due to the abnormal growth in the lumen of the arterial wall. Though the exact mechanism of the formation of stenosis (abnormal growth) is clearly not known, the factors that accelerate the growth of the disease are believed to be the deposition of various substances like cholesterol on the endothelium of the arterial wall and proliferation of the connective tissues. This disease is commonly located in large and medium sized arteries.

The difficulties in making realistic experimental and numerical investigations of the blood flow in stenotic arteries are the Non-Newtonian rheology of the blood, the compliant nature of the arterial wall, the pulsatile inlet flow, the mass transfer, the geometry of stenosis, and the transition to turbulent flow.

It has been observed that blood, being predominantly a suspension of erythrocytes in plasma, behave as a Non-Newtonian fluid at low shear rates in microvessels (Ex: Non-parabolic velocity profile and the existence of a peripheral layer). Various researchers studied the power law fluid model and assumed blood behaves like a Power Law fluid. Whereas few researchers: Forrester & Young, Lee & Fung, Morgan & Young, Shukla et al., Chaturni & Samy etc., assumed blood as a Newtonian fluid and ignored the suspension nature of the blood.

Alvaro and Martin [1], simulated unsteady Non-Newtonian blood flow and mass transfer in symmetric and non-symmetric stenotic arteries considering the fluid-structure interaction (FSI) using the code ADINA. They observed that the FSI affects significantly the hemodynamics on stenotic artery models and the stenotic severity have important influence on recirculation length.

Chakravarty, Datta and Mandal [2], analyzed the non-linear blood flow in a stenosed flexible artery, treating the aterial wall as an anisotropic, linear, viscoelastic, incompressible, circular cylindrical membrane shell. A pulsatile pressure gradient is applied and the axial and radial velocity profiles were obtained.

Misra and Shit [3], developed a mathematical model for studying the non-Newtonian flow of blood through a stenosed arterial segment. Herschel-Bulkley equation is used to represent the Non-Newtonian character of blood and the problem is investigated by combined use of analytical and numerical techniques. They finally concluded that the resistance of flow and skin-friction increase as the stenosis height increases.

Mandal [4], analyzed the non-Newtonian blood flow through tapered arteries with a stenosis and Dash, Jayaraman and Mehta [5], analyzed the flow in a catheterized artery with stenosis.

As the blood flows through the flexible artery it exhibits both normal and tangential stresses and due to the compliant nature of arteries they cause the wall to move in an unsteady manner. Hence this becomes a Moving Boundary Problem (MBP), where the interaction between the fluid motion and the surrounding solid cannot be neglected. In most published reports of the blood flow, the vessel is treated as a rigid wall, only a few researchers considering the moving wall. Moayeri and Zendehbudi [6], studied the effects of elastic property of the wall on flow characteristics through arterial stenosis. They assumed blood as a Newtonian fluid and the pulsatile of flow is modeled by using measured values of flowrate and pressure. A non-uniform distribution of the shear modulus is considered in axial direction at the stenosis location to model the high stiffness of the wall. The governing full Navier Equations are solved in the computational domain using the SIMPLER algorithm to study the effects of wall deformability, displacements and stress distributions.

Philip and Peeyush Chandra [7], analysed the blood flow through a stenosed artery considering the localized effects of the stenosis. The blood vessel is modeled as a rigid tube and the particulate nature of the blood is accounted through an Eringen Fluid model. It is observed that the resistance to the flow as well as shear stress increase as the height of the stenosis increases.

Zhao, Penrose, Zhao, Karayiannis and Collins [8], developed a 3 Dimensional fluid-solid transient coupling model of the blood flow in arteries to study the influence of fluid-solid interaction. They assumed blood to be incompressible, Newtonian fluid and axisymmetric, laminar flow in the simple geometry of the arteries. They ran a couple of analyses – one using CFX 4.4 and ANSYS and other using Bloodsim – CFX 5.5 – ANSYS and the results are compared with the available theoretical results.

In the present study, we assume blood as a Newtonian fluid and analyze its flow through a rigid stenosed and non-stenosed artery. The physical properties of blood and artery are obtained from Zhao, Penrose, Zhao, Karayiannis and Collins [8]. The formulation (Geometry) of the stenosed artery is obtained from Young.

Probabilistic Analysis

The ability to quantify the uncertainty of complex engineered systems subject to inherent randomness in loading, environment, material properties, and geometric parameters is becoming increasingly important in design and certification efforts. Traditional design approaches typically use worst case assumptions and safety factors to certify a design. This approach is overly conservative, does not quantify the reliability, nor does it identify critical parameters or failure modes affecting the system performance.

A probabilistic analysis approach characterizes input variabilities using probability density functions and then propagates these density functions through the performance model to yield uncertain model outputs, which can be related to failure metrics such as fatigue life, rupture, dose, or stress intensity. The approach quantifies the reliability, can reduce over-conservatism, and identifies critical parameters and failure modes driving the reliability of the system.

The programmers and researchers try to achieve the following in the development of the analysys algorithm:

• Identifying sources of errors and uncertainties
• Developing probability distributions for input variables
• Determining spatial and temporal variations
• Developing probabilistic load modeling
• Tailoring failure models for modeling uncertainty and obtaining appropriate system performance measures
• Creating system models (multiple failure mode and components)

Stress analyses play a critical role in understanding the structural performance and the mechanics of injury of biological systems. For more than two decades, researchers have endeavored to model the structural/dynamic behavior of the human body using increasingly complex and sophisticated computational modeling approaches such as finite element analysis techniques. The advantages of using finite element analysis are clear: complex geometry and boundary conditions can be modeled, heterogeneous and non-linear materials can be simulated, parametric studies isolating the effect of one or more variables can be performed, and delineating the stress distributions within the various components of the biological structure can be accomplished. This information may serve as a basis to evaluate the response of the biological structure to impact and long term exposure to high-g accelerations.

NESTEM enables designers to achieve reliable and optimum designs subjected to a life constraint with a probabilistic treatment of key uncertainties. The NESTEM code has been under development at the NASA Glenn Research Center.

uses deterministic analyses together with probabilistic methods to quantify the probability of failure of structural components which are subjected to complex mechanical and thermal loading.

The NESTEM code was developed to perform probabilistic analyses of structures subjected to either steady state or random thermal and mechanical loads. Probabilistic methods are becoming more and more useful due to the salient features of consistency, reliability and economy.

Problem Statement

A probabilistic analysis (using normal distribution) for the Von-Mises stress on the non-stenosed and stenosed artery models was performed. SolidWorks was used for the modeling the geometry of the artery. We assume a linear/elastic artery with the following physical properties: Young’s Modulus, E = 0.5 N/mm2 ; Poisson’s Ratio, v = 0.45.

The analysis was done using CosmosWorks, the SolidWorks analysis tool. It is a Finite Element Analysis (FEA) package embedded within the SolidWorks software which can perform design analysis, simulation and optimization directly from the SolidWorks interface.

A 3D half-symmetrical models of the stenosed and the non- stenosed artery is shown in the Figures 1 and 2.

Figure 1: Electrospun nanofibers membrane of poly-ε-caprolactone visualization after 21 days of human Osteoblasts culture (Cells visualization in blue (nucleus /DAPI) and PLLFITC labelled nanofibers in green): colonization and proliferation of osteoblasts into the nanofibers membrane.

Figure 2: Stenosed Artery Model

We consider a simple axisymmetric artery model with the inner radius 5mm, outer radius 6mm and length 50mm (From X.Zhao and J.M.T.Penrose). Based on the transient character of the blood flow, steady flow condition is assumed as constant static pressure 13320 Pa at inlet and 13300 Pa at outlet. Restricting the inlet of the artery, the analysis is ran for maximum Von - Mises stress.

The random variables and their mean values for the two cases considered are tabulated in following Table 1 and Table 2.

Table 1: Random Variables for Non Stenosed Artery

Random Variable	Mean Value
Young’s Modulus, E	0.5 N/mm2
Poisson’s Ratio, ν	0.45
Length of the artery, L	50 mm
Diameter of the artery, D	10 mm
Thickness of the artery, t	1 mm
Pressure Inlet, Pin	0.01332 N/mm2
Pressure Outlet, Pout	0.01330 N/mm2

Table 2: Random Variables for Stenosed Artery

Random Variable	Mean Value
Young’s Modulus, E	0.5 N/mm2
Poisson’s Ratio, ν	0.45
Length of the artery, L	50 mm
Diameter of the artery, D	10 mm
Thickness of the artery, t	1 mm
Length of the stenosis, Lst	10 mm
Depth of the stenosis, dst	5 mm
Pressure Inlet, Pin	0.01332 N/mm2
Pressure Outlet, Pout	0.01330 N/mm2

Stenosis Profiles

Similarly, using the same radii and thickness, a stenosed artery was modeled using the equation 

          1

where xst is the x coordinate of the center of the stenosis, x0 is the stenosis half-length, Lst is the total length of the vessel with stenosis and δ is the dimensionless thickness of the luminal reduction in the radial direction. The above equation is obtained from L. Ai and K.Vafai.

We consider a 5-Point Stenosis exactly half-way through the length of the artery and assuming the stenotic depth to be half the diameter of the artery, the dimensionless thickness of the luminal reduction in the radial direction, δ = 1/2.

Length of the stenosis, Lst=10mm

Therefore, for the artery with mean variables, the radial profile in the stenotic region takes the following cosine profile according to the equation.

for 20 < x < 30

Considering 5 points along the length of the artery from x = 20 to 30,

For x = 20

For x = 21,

=4.7612

For x = 23,

=3.441

For x = 27,

=3.441

For x = 29,

=4.7612

For x = 30,

=5

The values obtained are tabulated as followed (Table 3),

Table 3: Mean Profile

x	r(x)
20	5
21	4.7612
23	3.441
25	2.5
27	3.441
29	4.7612
30	5

As we can see the values of r(x) decrease initially and increase as we further increase the values of x, hence forming a Cosine curve. Similarly, the values of r(x) are determined for all the cases which affect the shape of the curve (Cosine Equation) i.e. Lst±5%, dst±5%, L±5% and D±5%.

The artery was modeled according to the dimensions obtained above for both Unstenosed and Stenosed arteries. As we are dealing with a constant pressure difference between the inlet and outlet of the artery, the load on the inner wall of the artery becomes a trapezoidal load. The inlet pressure is maintained at 13,320 Pa and the outlet pressure is maintained at 13,300 Pa. Therefore a 20 Pa pressure difference is assumed between the 50 mm length of the artery in both the cases hence maintaining the blood flow. The trapezoidal load is as shown in the Figure 3.

Figure 3: Load acting on the wall

Analysis Procedure

A CAD / Solid Modeling software, SolidWorks in this case was used to model the artery with the dimensions discussed previously. An axis created through the center of the artery and the inner wall of the artery was used to create a cylindrical coordinate system.

A fine mesh model was generated using the CosmosWorks tool. A new material model (Artery) is created using the material properties obtained from Zhao, Penrose, Zhao, Karayiannis and Collins [8].

In the CosmosWorks Editor, Static Stress with Linear Material Mode was chosen for our case. The inlet fixed boundary condition was applied and the trapezoidal load was applied to the inner surface of the artery. A mean variable analysis was made and the node at which the Maximum Von-Mises Stress obtained was noted.

One variable was changed by +/- 10% keeping the other random variables unchanged and the analysis was made and the new Von-Mises stress was recorded. The process was repeated till all the variables have been changed to +/- 10%.

The results were entered into a statistical analysis program, NESTEM, where probabilistic analysis is completed and the sensitivity factors for each variable were determined. A similar procedure was involved for the case of a Stenosed artery which involves the cosine profile.

The problem presented was solved iteratively by using a scattered set of values obtained by varying the mean variables by +/- 10% in the case of a Non – Stenosed Artery and by +/- 5% in the case of a Stenosed Artery. As shown in the figures, a 3-D model was created and was meshed using the Default SolidWorks Algorithm. The meshed results are as shown in the figure. The boundary conditions of restricting the Inlet of the artery to move in any direction (Fixed) and a pressure difference of 20 Pa between its ends was applied. All the random variables were assumed to be independent and a normal distribution was assumed for all random variables.

The maximum stress location was determined in the mean run of both the cases and this location was used to evaluate the cumulative distribution functions and the stresses produced in the arteries. A typical Von Mises stress distribution is shown. It was observed that the maximum stress location is at the stenosis and minimum stress location was observed at the inlet of the artery (Figures 4 and 5).

Figure 4: Non Stenosed Stress Results

Figure 5: Stenosed Stress Results

FEA Results

Figures 6-9 show the sensitivity factors for each of the variables.

Figure 6: Sensitivity Factors Vs Random Variables

Figure 7: Cumulative Probability of Stress. Results for the Stenosed Artery Model

Figure 8: Sensitivity Factors Vs Random Variables

Figure 9: Cumulative Probability of Stress

The sensitivity factors obtained from NESTEM are plotted for each probability value from 0.001 to 0.999 of the Von Mises stress. The input data files for the NESTEM for both the cases are given in the Appendix. The raw data obtained from the NESTEM analysis is also shown for the two cases. The maximum Von Mises stress after each variable is changed was noted.

Reliability Analysis Results

Results for the Non Stenosed Artery Model 

A probabilistic analysis was carried out from the results obtained from the Von Mises Stress Finite Element Analysis using CosmosWorks. Non – Stenosed and Stenosed (The abnormal growth in the arteries due to fatty deposits) Arteries are modeled using the 3D modeling SolidWorks software, the geometries taken from different research papers. Multiple analyses were made for different initial and boundary conditions by varying different random variables considered and the stress data was obtained.

Research can be carried out further by varying geometries and with different initial and boundary conditions. Fluid flow can be modeled and Non – Newtonian flow can be considered. Also, pulsatile blood flow can be considered to be more realistic. The equation of the stenosis can be varied for different depths and different radii.

All the variables have at least some effect on the Von Mises Stress whereas some variables have high impact. In the NonStenosed case we find that the Inner Diameter, Thickness and Inlet Pressure have the major impact on the stresses inside the artery with Inner Diameter playing a major role. Inner Diameter is 8 times more sensitive than the other two concerned factors.

Similarly in the case of the Stenosed artery, the Inner Diameter, Length of the stenosis, depth of the stenosis, Inlet Pressure and Poisson’s Ratio have a huge impact on the Von Mises Stresses with Poisson’s Ratio half as sensitive as the Inner Diameter in mean probabilities. Small changes in the above variables may lead to extreme stresses and possibly the rupture of the artery.

These variables may be considered while diagnosing patients with Atherosclerosis.

1. Alvaro Valencia, Martin Villanueva. Unsteady Flow and Mass Transfer in Models of Stenotic Arteries Considering Fluid-Structure Interaction. Int Communications Heat Mass Transfer. 2006; 33: 966-975.
2. Chakravarthy S, Datta A, Mandal PK. Analysis of Non Linear Blood Flow in a Stenosed Flexible Artery. Int J Eng Sci. 1995; 33: 1821-1837.
3. Misra JC, Shit GC. Blood Flow through Arteries in a pathological State: A Theoretical Study. Int J Engineering Sci. 2006; 44: 662-671.
4. Prashanta Kumar Mandal. An Unsteady Analysis of non-Newtonian Blood Flow through Tapered Arteries with a Stenosis. Int J Non- Linear Mechanics. 2005; 40: 151-164.
5. Dash RK, Jayaraman G, Mehta KN. Flow in a Catheterized Curved Artery with Stenosis. J Biomech. 1999; 32: 49-61.
6. Moayeri MS, Zendehbudi GR. Effects of Elastic Property of the Wall on Flow Characteristics through Arterial Stenoses. J Biomech. 2003; 36: 525-535.
7. Philip D, Peeyush Chandra. Flow of Eringen Fluid (Simple Microfluid) Through an Artery with Mild Stenosis. Int J Eng Sci. 1996; 34: 87-99.
8. Zhao X, Penrose JMT, Zhao SZ, Karayiannis TG, Collins MW. Numerical Study of Fluid Solid Interaction (FSI) in Arteries. 2004.
9. Ai L, Vafai K. A Coupling Model for Macromolecule Transport in a Stenosed Arterial Wall. Int J Heat and Mass Transfer. 2006; 49: 1568- 1591.
10. Gorla RSR, Pai SS, Rusick JJ. Probabilistic Study of Fluid Structure Interaction. Int J Eng Science. 2003; 41: 271-282.
11.   Tsun-Zee Mai and Anthony MJ Davis. Laminar Pulsatile Two-Phase Non-Newtonian Flow through a Pipe. Computers and Fluids. 1996; 25: 77-93.
12. Zendehbudi GR, Moayeri MS. Comparison of Physiological and Simple Pulsatile Flows through Stenosed Arteries. J Biomech. 1999; 32: 959- 965.
13. Narasimham MNL. Thermomicropolar Flow of a Nematic Liquid Crystal in a Flexible Stenosed Tube. Int J Eng Sci. 2000; 38: 1059- 1073.
14. Layek G, Gorla RSR. Unsteady Viscous Flow with Variable Viscosity in a Vascular Tube with an Overlapping Constriction. 2009; 47: 649- 659.