### Artikel

## Wall Shear Stress simulations in a CT based human abdominal aortic model

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### Autoren

Veröffentlicht: | 14. März 2007 |
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### Gliederung

### Abstract

Abnormal Wall Shear Stress (WSS) distributions may be a predictor for the risk of development or rupture of an aneurysm and a tool to control the efficiency of the therapy. In this work, we present a method for the computation of WSS based on Computational Fluid Dynamics (CFD) simulations. The approach is applied to a CT based abdominal aortic model and realistic boundary conditions are considered based on MR flow measurements. WSS simulations are computed using the CFD program Fluent. The simulation results show realistic mean WSS distributions and provide a quantitative description of the stress conditions in the aortic model. The mean range of the computed values is compatible with that obtained by other in vivo and in vitro methods [1], [2]. The method provides a numerical approach for the computation of WSS. However, further development and validation should be investigated in order to reach computations that can be implemented clinically.

### Introduction

Aortic pathologies such as aneurysms or dissections are associated with high mortality on the event of rupture. For both entities wall hypertension is the main predisposing factor. A dilatation of the fatigued vessel wall is known as aneurysm. The risk of rupture is positively correlated with the diameter of the vessel and hence the degree of dilatation. A diameter larger than 5 cm is associated with a high risk of rupture and requires surgical intervention. Aortic aneurysms occur with increasing incidence in patients above the age of 50. Only 20% of the patients survive an aortic rupture.

Due to the high mortality and in order to intervene duly, it is essential to identify reliable predictors for development and rupture of aortic aneurysms. The extent of dilatation is the current predictor; however this is not always reliable and sufficient enough.

A dilatation of the aorta is mainly due to a highly fatigued vessel wall. If left untreated, it will enlarge and may rupture. Ruptures occur when the mechanical stresses acting on the inner wall exceed the failure strength of the diseased aortic tissue. Wall Shear Stresses (WSS) are the forces acting on the vessel, which arise from the blood moving along the inner wall surface. Abnormal WSS distributions may therefore be a causative and hence predictive factor for the development of an aortic pathology. Spring et al. showed that WSS in the common carotid artery is reduced in patients with Abdominal Aortic Aneurysms (AAA) and in those exposed to vascular risk factors such as age, smoking and hypertension [3]. Jiang et al. found that in hypertensive patients WSS was significantly lower than in normotensive patients in the carotid aorta [4]. Thus WSS analysis may be a valuable tool in risk evaluation and therapy follow up, giving the possibility of controlling the efficiency of the intervention.

The aim of the present study is therefore to establish a numerical approach to determine non-invasively WSS distributions within patient-specific aortic models. The computations are based on Computational Fluid Dynamics (CFD) simulations and aim to further understand the underlying hemodynamics of aortic pathologies such as AAA.

### Methods

#### Mathematical background

Wall Shear Stresses (WSS) are the tangential forces acting on the vessel wall. They arise from the blood moving along the inner wall surface. WSS can be computed from the tangential traction vectors [5] as a function of the near-wall gradients of the velocity fields and the blood viscosity.

The total fluid stress given by the Cauchy stress tensor is defined by:

where *p* is the fluid pressure, the identity matrix, *µ* the blood viscosity, the rate of deformation tensor and the velocity gradient.

The total traction vector can be written as: t=σn*
_{s}
*, where n

*is the local unit normal vector to the wall surface at a given position.*

_{s}The tangential component t*
_{s}
* of the traction vector, which is the component in flow direction, can be derived by simply subtracting out the normal part, such that:

The WSS ז at the fluid-solid boundary is determined by the magnitude of this tangential traction vector. For a Newtonian fluid, ז can be estimated by a simple multiplication of the shear rate with the viscosity μ, assumed to be constant:

To compute the velocity gradients , the flow field along the luminal surface is derived by solving the equations governing the blood flow, represented by the Navier-Stokes and the continuity equations. For an incompressible and Newtonian fluid they are given by:

These include the mass, momentum and energy conservation equations. In the lumen region, they represent a mathematical relationship between the pressure *p*, the velocity *v* in the flow direction, the mass density *ρ* and the viscosity *µ* of the blood.

#### Computational Fluid Dynamics Simulations

For the computations of patient-specific WSS, Computational Fluid Dynamics (CFD) simulations have been carried out to compute the blood flow velocity gradients and to derive the WSS distributions. The CFD process is based on the Finite Volume Method (FVM) and consists of the segmentation of the patient tomographs (Figure 1 [Fig. 1]), the creation of a 3D geometrical model (Figure 2 [Fig. 2]), the generation of high-quality surface (Figure 3 [Fig. 3]) and volume (Figure 4 [Fig. 4]) mesh models, the setting of realistic boundary and initial conditions (Figure 5 [Fig. 5]), and on the generation of the physical model for the simulations.

a) *Segmentation*: An essential prerequisite for the CFD simulations is a precise description of the geometry of interest. The CT based patient-specific geometry was therefore accurately segmented, extracting the region of interest represented by the inner boundary of the aortic wall. A total of 210 slices with a thickness of 1mm were acquired to reconstruct the abdominal aortic model from 70 mm above the renal arteries to the aortic bifurcation. A first order approximation of the boundary surface was first generated from the 3D scan using the region growing approach. Then a manual segmentation of the 2D slices in axial direction was required to improve the 3D segmentation.

b) *Geometrical Model*: The processing of the segmented patient data was performed using MediFrame [6]. A 3D image model was created by combining all 2D segmented slices into one image file. Then, the boundary surface of the model was constructed based on the marching cubes algorithm and was idealized using Laplacian smoothing and surface cleaning. For blood simulations free of backflows, a clip filter was applied in order to adjust the boundary surfaces such that their normal is parallel to the flow direction. Finally, the surface was exported into an .stl file consisting of 3D triangles.

c) *Surface Mesh Model*: High-quality three dimensional surface meshes of the patient-specific aortic boundary have to be created to guarantee the stability of the CFD simulations. Using the mesh generator Gambit, the .stl created in the previous step was first imported and the old mesh was removed. A size function (cSF) accounting for the curvature at the wall surface was then created, based on which the wall mesh model was generated.

d) *Volume Mesh Model*: Blood simulations based on the Finite Volume Method require the generation of 3D volume meshes. The volume model represented by the closed surface of the fluid domain was therefore generated first. Then a Boundary Layers (BL) mesh control function was created to control the distribution of the fluid cell size near the wall. A meshed Size Function (mSF) was defined from the wall into the fluid domain, in order to control the propagation of the mesh from the surfaces into the volume. Finally, based on these control functions, the volume mesh model was generated.

e) *Boundary and Initial Conditions*: In order to solve the system of partial differential equations governing the blood flow, a set of boundary and initial conditions needs to be defined. For patient-specific simulations, an unsteady homogeneous velocity profile based on MR flow measurements was used and smoothed in a plane above the renal arteries to set the boundary conditions at the inlet of the aortic model. At the outlets, the boundary conditions were determined in terms of outflow rates and at the wall, the no-slip boundary condition was defined. For the initial conditions, the whole model was initialized with zero-velocity at t=0 s.

f) *Simulations (Physical Model and Solver Settings)*: The Wall Shear Stress distributions and the blood flow through the aortic model were simulated using the CFD program Fluent. The segregated (implicit) solver was used to discretize the Navier-Stokes equations. The field variables were interpolated to the faces of the control volumes using a second-order scheme. For the pressure-velocity coupling, the solver employed the Pressure Implicit Splitting of Operators (PISO) algorithm, useful for unsteady problems. As for the physical properties, blood was considered as a homogeneous Newtonian fluid with a constant dynamic viscosity of 0.003 Nsm^{-2}. With a Reynolds number of Re≈2000, the flow was assumed to be laminar, and incompressible with a constant density of 1050 kgm^{-3}. The blood vessel was modelled as rigid.

A detailed description of the CFD workflow can be found in [7].

### Results

The simulation results were obtained using the FVM program FLUENT 6.2. The CFD computations were carried out to describe the shear stress distributions along the wall at any instant of time. The WSS were computed from the resultant of the velocity gradients near the wall. A whole cardiac cycle of period T=0.85 s was modelled using 2000 equally spaced time steps.

Figure 6 [Fig. 6], 7 [Fig. 7], 8 [Fig. 8], 9 [Fig. 9] and 10 [Fig. 10] show the WSS distributions along the aortic model at the early systole (t=0.09 s), the peak systole (t=0.18 s), the late systole (t=0.36 s), during the diastole (t=0.50 s) and at the peak diastole (t=0.53 s) respectively. They nicely show how the WSS increase from the early systole to the peak systole, where they become maximal, then decrease to the late systole, until they reach a value near zero during the diastole, and increase again at the peak diastole. The figures on the left represent a colored scale of the magnitude of the WSS in Pascal (1 Pascal = 10 dynes/cm^{2}), and the figures on the right represent an XY plot of the WSS (y-axis in Pascal) along the z position in aortic flow direction (x-axis in mm).

Regions of relative low WSS were observed below the renal arteries level (blue on left-figures and z>70 mm on right-figures), while higher WSS values were found within the small arteries (red on left-figures and z<70 mm on right-figures). This can be estimated, according to the definition of the WSS ז, by the large diameter and the low velocity fields in regions of large diameters i.e. below the renal arteries, and by the small diameters and the high velocity fields within the small arteries. This explanation can be also approximated to the Hagen-Poiseuille formulation which assumes, for parabolic blood velocity profiles in near-circular lumens, the mean shear stresses to be proportional to the volume flow and inverse proportional to the radius *R*
^{3}.

The mean WSS were computed in terms of Area Weighted Averages (AWA) along the whole cardiac cycle. The maximum mean WSS was found at the peak systole with a relative high value of 68.5 dynes/cm^{2}. The high systolic WSS at t=0.18 s can be explained by the high peak velocity at that time i.e. where the velocity field exhibits its maximum. As for the lowest mean WSS, it was found to be 2.8 dynes/cm^{2} at t=0.50 s during the diastole, and is due to the nearly zero inlet velocity profile at that time.

Further, the mean WSS at early systole (t=0.09 s), late systole (t=0.36 s) and peak diastole (t=0.53 s) were 10.3 dynes/cm^{2}, 8.6 dynes/cm^{2} and 15.8 dynes/cm^{2} respectively. Note that these values are the averages along the whole model, including high values in the small arteries, and that the percentage p of cells at the wall surface with WSS values less than 5 dynes/cm^{2} is 91.1% at early systole, 88.4% at peak systole, 93.2% at late systole, 99.5% at diastole and 88.7% at peak diastole.

Indeed, the mean range of the computed values agrees with the range obtained by other in vivo and in vitro methods. [1] and [2] reported mean WSS values varying between 0 dynes/cm^{2} and 10.4 dynes/cm^{2} under resting conditions with an approximate blood flow of 3 litres/min. Their values however excluded the small arteries, thus only included the infrarenal and supraceliac regions. This corresponds to our results simulated at early (t=0.09 s) and late (t=0.36 s) systole, for z-positions larger than 70 mm. Figure 6 [Fig. 6] and Figure 8 [Fig. 8] show mean values of approximately 5 dynes/cm^{2} in these regions.

### Discussion and further development

The WSS computations show realistic distributions and agree with elsewhere published results under similar conditions, as reported in [1], [2] and the references therein. Nevertheless, further development and validation of the simulations still have to be investigated. An experimental setting as well as clinical trials will be developed to allow the validation. Indeed, a physical verification of the computations based on the generation of results that are mesh-independent, will also be performed.

In the described simulations, blood was considered as a Newtonian fluid. However, an important aspect in modelling the blood flow is to accuratetly describe the physical nature of the blood as a fluid. Thus, the blood model will be extended to a non-Newtonian fluid, considering the stresses within the blood as nonlinearly dependent on the deformation rate.

In order to show the WSS distributions in a normal subject, the simulations were applied to a model of a healthy abdominal aorta. An investigative study with a large dataset and complex aneurysm geometries would further provide a tool to better understand aortic aneurysms in terms of identification and prediction and to give an insight of the WSS distributions within deseased (aneurysmal) and repaired (stented) aortic models.

The vessel wall was considered as rigid and thus the elasto-mechanical deformation of the vessel wall was not taken into account. A structural model should therefore be developed to allow strain and stress analysis. And finally, a coupled system between the blood model and the structural model will provide a patient-specific Fluid-Structure Interaction in the aorta to achieve computations that can be implemented clinically.

### Conclusion

We established a numerical approach to compute the distribution of Wall Shear Stresses within patient specific aortic models. The method is evaluated using a CT based model of the abdominal aorta and provides a quantitative description of the stress conditions along the vessel wall. The simulations are based on our Computational Fluid Dynamics workflow developed in a previous study [7]. The CFD computations were run using the program Fluent, which uses the Finite Volume Method to solve the Navier-Stokes equations to compute the blood flow within the aortic model. The WSS distributions are then derived from the gradients of the velocity fields near the wall. WSS are important for predicting the risk of rupture associated with aortic pathologies such as aneurysms and dissections. Based on these predictions, the necessity of a surgical intervention can be evaluated. Further, it is then possible to plan optional and optimal therapies as well as to determine the efficiency of the treatment post-operationally.

### Notes

#### Acknowledgements

Grateful and deepest thanks to R. Kröger from Fluent Germany for the extensive contribution and the precious advices regarding the simulations. Thanks also to H. von Tengg-Kobligk, K. Ruf and J. Ziko for the help with the clinical data.

The present study was conducted within the setting of the “Research training group 1126: Intelligent Surgery - Development of new computer-based methods for the future workplace in surgery” founded by the German Research Foundation (DFG).

#### Conflicts of interest

None declared.

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