IDEALIZED ABDOMINAL AORTIC ANEURYSM (AAA) GEOMETRY AS PREDICTOR OF HEMODYNAMICS STRESSES

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1 European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September 10-14, 2012 IDEALIZED ABDOMINAL AORTIC ANEURYSM (AAA) GEOMETRY AS PREDICTOR OF HEMODYNAMICS STRESSES Soudah E 1*, Vilalta G 2, Bordone M 1, Vilalta JA 3, Nieto F 2, Pérez MA 4, Vaquero C 5 1 International Center for Numerical Methods in Engineering (CIMNE) Technical University of Catalonia, Barcelona, 08034, Spain. {esoudah, 2 CARTIF Centro tecnológico, Mechanical Engineering Division. Boecillo, 47151, Spain {guivil, felnie}@cartif.es 3 Polytechnical University of Havana, Industrial Engineering Department Havana, 19340, Cuba jvilalta@ind.cujae.edu.cu 4 ITAP Institute. University of Valladolid, 47011, Spain marrue@cartif.es 5 University and Clinic Hospital of Valladolid, Valladolid, 47011, Spain. cvaquero@med.uva.es Keywords: AAA, Predictors, Correlations, Hemodynamics stresses, Pearson s correlation Abstract. The objective of this study is the correlation between peak wall stress (PWS), abdominal pressure and abdominal aorta aneurysm (AAA) geometric parameters using computational fluid dynamics. Idealized AAA models were created by three-dimensional (3D) reconstruction based on real AAA acquired computed tomography (CT) images, considering the main aneurysm geometric parameters (maximum diameter D, the length L and the asymmetry β). Factorial experiment using these variables and three levels (3 3 factorial) was carried out, in order to optimize the number of the models. Based on 3D surface meshes of the idealized AAA, a high quality volume mesh was created using an optimal tetrahedral aspect ratio for whole domain. In order to quantify the PWS and the recirculation inside the AAA, a 3D CFD approach was used. The CFD computation was performed assuming that the arterial wall is rigid and the blood is considered an homogeneous Newtonian fluid with density 1050 kg/m 3 and a kinematic viscosity of Pa.s. A linear relation between PWS and geometric parameters and geometric biodeterminants were observed using of Pearson s rank correlation coefficient. Parallelization procedures were used in order to increase the performance of the CFD calculations.

2 1 INTRODUCTION In its most accepted definition abdominal aortic aneurysm (AAA) is a localized, progressive and permanent dilation of the aortic wall. Usually are thought to be the end results of irreversible pathological remodelling of the arterial connective tissue, which causes changes, over time, in AAA geometry, in the constitutive formulation an in the failure criterion. The statistics associated to the AAA are of great concern because of its catastrophic consequences in term of rupture. Evaluating rupture risk is very important in avoiding unnecessary surgical treatment in patients whose aneurysm has diameter higher than upper threshold value (50-55 mm) or predicting the rupture in patients whose aneurysm diameter is less than that this reference value. Nowadays, it is recognized that current clinical criteria for assessment of the abdominal aortic aneurysm rupture risk can be considered insufficient because they have not a physically sound theoretical basis [1], despite they are based on a wide empirical evidence. Hence, in last years, researchers and physicians have had the challenge to identify a more reliable criterion associated with the actual rupture risk of the patient-specific aneurysm. The literature begins to reflect the existence of a consensus that, rather than empirical criteria, the biomechanical approach could facilitate a better method to assess the AAA rupture risk. The basic premise of the biomechanical approach to estimate the AAA rupture risk, is that this phenomenon follows the principle of material failure, that is, an aneurysm ruptures when the stresses acting on the arterial wall exceeding its failure strength, reflecting the interaction between the arterial wall structural remodelling and the forces generated by blood flow within the AAA. Simple geometric parameters, those obtained directly from the computed tomography CT, can define the AAA from a geometrical point of view. It is well documented that aneurysm shape, defined by means of appropriate relations between simple parameters which are known as Geometric BioDeterminants (GBDs) [2], has strong influence on flow patterns and consequently on wall stress distribution (peak values and locations). Therefore, it has been recognized that the AAA morphology has a significant influence in its rupture potential. Aim of this study was to characterize the relation of the main geometric parameters of the AAA (simple and GBDs) in the hemodynamic stresses (Wall Shear Stress-WSS and Hemodynamic Pressure-HP) acting on intimal layer. 2 METHODS 2.1 AAA Geometry Twenty eight virtual aneurysm models which represent the true vessel lumen surface, were generated in CATIA V.5R19 (Dessault Systèmes, Paris) using the mathematical equation presented in [3], which has been modified to allow the parametrization the 28 aneurysm models. The virtual aneurysm model is characterized by circular cross sections perpendicular to the z-axis of the geometry, which coincides with its centerline and is divided in three regions. The inlet or proximal region is 6 cm in length with constant diameter, d = 2 cm. The distal or out-

3 let segment is also 2 cm constant in diameter and its length is about 5d, in order to guaranty that the applied numerical boundary conditions do not affect the global flow characteristics within the aneurysmal sac. A dilated segment of the geometry represents the aneurysmal abdominal aorta whose characteristic dimensions are the length L and a variable diameter between d and the maximum diameter of the aneurysm in the middle section, D. The asymmetry of the model is governed by is governed by the eccentricity (ε) of the aneurysmatic sac as shown in the inset of Table 1. The anterior length is measured from point of intersection between the maximum diameter planes with aneurysm centerline to anterior wall. Thus, β = 1.0 yields an axisymmetric aneurysm. The factor which assesses the length (L) and the diameter (D) of the AAA sac is known as Saccular index (γ)(see table 1). Clinical observations indicate that the smaller the saccular index the higher is the possibility of AAA rupture. The deformation rate (χ) characterizes the abdominal the non-deformed abdominal aorta diameter (d) and the maximum diameter of the aneurysm sac, D (see table 1). The value that defines a low rupture risk is taken as the lower deformation condition of the artery (lower values D and higher d), and for the most critical condition, as the higher deformation (higher values D and lower d). Table 1: Asymmetry index (β), Saccular index (γ) and Deformation rate (χ). The parametrized model differs in maximum diameter aneurysm sac (D), the length aneurysm sac (L) and aneurysm asymmetry (, while considering the value of the renal artery diameter, d=20 mm, constant (typical healthy human abdominal aorta). The criteria used to select the range of values for different geometric parameters are related to physiological and pathological conditions. The maximum diameter was varied between 40 mm (considered as small AAA) and 80 mm (a value upper than threshold value for elective repair and unusual for the clinical practice). Considering the clinical statistics, the length of the segments that represent the aneurysmatic abdominal aorta, was assumed between 90 mm and 130 mm. The degrees of asymmetry are taken between 0.2 (high asymmetry degree) and 1 (axi-symmetric). A non-aneurysmal aorta was also included in the work as control geometry with a constant diameter (D = d). The combination of these parameters allows represent the geometric models for different stages of aneurysm development, which is appropriated to characterize its rupture risk. In order to optimize the number of the models that were used, a factorial experiment using these variables and three levels (3 3 factorial) was carried out, and the matrix of the geometric factors that represent the aneurysm models was determined. 28 virtual AAA models based of the combination of these parameters have been created according to the combination of GDB parameters (D, L, β). Next table shows the geometric parameters used for the virtual AAA models. D (mm) L (mm) (-) Table 2. Factorial experiment resulting matrix to define the geometric parameters of the aneurysm models.

4 Figure 1 shows a schematic representation of the virtual models that have been parametrized: a) Diameter(D), length(l) and eccentricity (ε) of the abdominal aneurysm sac has been modified. Figure 1. Schematic representation of the AAA geometry model used: a) varying D, L and ε. 2.2 Meshing, governing equation and boundary conditions The computational grid mesh was generated using the graphical pre and post processor for computer simulation and analysis, GiD [4]. All meshes generated contain 1 million (± 10%) of elements. Based on the boundary mesh obtained from the parametric models, an Advancing Front [5] method to fill the interior with tetrahedron was used. The Advancing Front [5] is an unstructured grid generation method. Grids are generated by marching from boundaries (front) towards the interior. Tetrahedral elements are generated based on the initial front. As tetrahedral elements are generated, the initial front is updated until the entire domain is covered with tetrahedral elements, and the front is emptied. For these models, this procedure is ideal to generate a volume mesh with quality warranty of the elements, this mean angle and volume bounds are guaranteed. Using this Advancing front method we have obtained a smooth element and an aspect radio for whole of the meshes upper than 0.85 (ideal ratio=1 for an equilateral triangle). For defining the governing equations of the blood flow, we must consider that the blood is a suspension of red and white cells, platelets, proteins and other elements in plasma. In vitro studies on blood flow through narrow tubes have revealed complex rheological behavior of blood. Blood behaves as a nonnewtonian fluid in arteries with internal diameter < 500 μm and newtonian with internal diameter > 500 μm. The viscosity of blood depends on the arteries diameter and this behavior is known as the Fahraeus-Lindqvist effect, this effect is characterized by a decrease in the apparent blood viscosity as the arteries diameter decreases below 500 μm. The minimum apparent viscosity is reached when the tube diameter is 8 μm. Upon further decreases in tube diameter, the apparent viscosity increases very rapidly. The physical

5 reason behind the Fahraeus-Lindqvist effect is the formation of a cells-free layer near the wall of the tube [6]. The layer is devoid of Red Blood Clots(RBCs) and has a reduced local viscosity. The core of the tube, on the contrary, is rich with RBCs and has a higher local viscosity. The extent of the cell-free layer, which depends on the vessel size and hematocrit, is a major factor that determinate the apparent viscosity of the blood. However, in large arteries with internal diameter > 500 μm, the blood may be considered a homogeneous fluid, with standard behavior (Newtonian fluid)[6] and constant density of 1050 kg/m3 and a kinematic viscosity of 4x10-3 Pa.s. The Newtonian approximation is acceptable in large arteries. In this work, blood is assumed as three-dimensional, steady, incompressible, homogeneous, Newtonian fluid with no external forces applied on it. In our case, the idealized AAA can be considered large arteries. Mathematically an incompressible flow in the absence of body force these equations are expressed in compact form as follows: (1) (2) where V is the velocity vector, is the flow rotation tensor and p is the pressure. The boundary conditions for the velocity V are given by the equations set (3). These are: no-slip at the walls (3a), fully developed parabolic profile at the inlet (3b), and time dependent normal traction due to luminal pressure at the outlet (3c). These conditions are expressed, in mathematical form, as follows: (3a) (3b) where d r is the inner radius of the abdominal aorta, u r is the Cartesian components of the velocity vector in the Y direction, u(t) and p(t) are the time-dependent velocity and pressure waveforms, τ nn is the normal traction, designate of the respective boundary, and I is the standard identity matrix. Two waveforms taken from previous studies have been selected to reproduce in vivo pulsatile velocity (prescribed at the abdominal aorta inlet [7]) and pressure (prescribed at the abdominal aorta outlet [3]). The main characteristics are the cycle period is 1 s, with peak flow occurring at 0.4 s, the time-average Reynolds number R e =355 is acceptable for the conditions of the simulation. Figure (2) shows the pulsatile waveforms used as boundary conditions. Womersley number which characterizes the flow frequency, the geometry and the viscous fluid properties is 16.1, what is a typical value for the aortic segment. The pressure wave is also a triphasic pulse that it is appropriate for normal hemodynamics conditions in the infrarenal segment of the human aorta, with peak pressure occurring at 0.5 s. It is important to note, that boundary conditions (3c)

6 with similar characteristics have been used in previous works and its outcomes reflect stable CFD simulations of blood flow. 0, , Velocity (m/s) 0,300 0,200 0,100 Pressure (Pa) ,000-0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Time (s) Time (s) Figure 2. Boundary conditions for the CFD simulations. Left: velocity wave at the inlet. Right: pressure wave at the outlet. For the flow field computations, the arterial wall is comprised from non-elastic and impermeable material due to the movement provoked by the impact of the flow over the wall might be negligible, as indicated [8]. A framework for building multi-disciplinary finite elements program Kratos[9] was used for solving the Computational fluid problem. 3 RESULTS The wall shear stresses and the internal pressure were correlated with both simple geometric parameters and geometric biodeterminants. The relation between hemodynamics stresses and simple geometric parameters and geometric biodeterminants was assessed by using of Pearson s rank correlation coefficients (Minitab for Windows, release 15.0, standard version). To assess the reference variables on hemodynamic stresses it was carried out a multiple regression analysis with a stepwise strategy. In all models, peak WSS varied from to Pa (1.118 ± 0.18 Pa) and peak pressure varied from to Pa ( ± 12.7 Pa). 3.1 Wall Shear Stress vs Aneurysm Geometric Parameters Figure 3 shows the relation between peak WSS and simple geometric parameters (amaximum diameter, b-length). In figure (1a) (r=-0.275, p=0.164) and (2b) (r=-0.36, p=0.065), it is possible to observe the existence of a weak linear and negative relation which is significant at 10% for both parameters for all the 28 cases studied Figure 3. Relation between peak WSS and Simple Geometric Parameters: a) maximum diameter, b) length.

7 Figure 4 shows the relation between peak WSS and GBDs. Fig.2a WSS against the asymmetry factor (r=-0.298, p=0.131), (fig.2b) WSS against the saccular index (r=-0.104, p=0.607), (fig.2c) WSS against the deformation rate (r= , p=0.164)). Saccular index is no significant and the asymmetry and deformation rate are weak with significance less than 15%. Figure 4. Relation between peak WSS and Geometric biodeterminants: a) asymmetry, b) saccular index, c) deformation rate. 3.2 Pressure vs Aneurysm Geometric Parameters Figure 5 shows the relation between peak hemodynamic pressure and simple geometric parameters (a-maximum diameter, b-length). In figure (3a) it is observed a strong linear and negative relation between pressure and aneurysm diameter (r=-0.696, p=0.000). In (fig.3b), this trend continues although with less intensity (r=0.373, p=0.06). Figure 5. Relation between peak hemodynamic pressure and Simple Geometric Parameters: a) maximum diameter, b) length.

8 In figure 6 is shown the relation between peak hemodynamic pressure and GBDs. Pressure correlate better with the saccular index (strong and negative, r=-0.75, p=0.000) and deformation rate (strong and negative, r=-0.7, p=0.000). With the asymmetry, the correlation is nosignificant (r=-0.25, p=0.21). Figure 6. Relation between peak hemodynamic pressure and Geometric biodeterminants: a) asymmetry, b) saccular index, c) deformation rate. 4 DISCUSSION The prevailing aetiology hypothesis is that AAA results from the coupling between structural changes in inner layers of the arterial wall and disturbed patterns of hemodynamics stresses acting on the vessel wall. Any structural wall change may influence the flow in the abdominal aorta and downstream of it. Similarly, changes in the blood flow will result in altered pressure and wall shear stress and may lead to different destructive processes: calcification, wall inflammation etc [10]. As presented in previous works [8, 11], it has been shown that the so-called disturbed flow conditions, that develop within the AAA, such as rapid decrease in the velocity and regions of very high (or low) hemodynamic stresses gradients, may all contribute in various ways to the vascular disease, primarily via their effects on the endothelium. According to velocity and pressure waveform prescribed as boundary conditions, at peak systolic pressure the velocity profile is in its late systolic phase and thus the blood at the inlet has begun to decelerate. Consequently, recirculating regions develop establishing local and temporal hemodynamic stresses distribution. General behavior of these variables is summarized. The pressure load distribution is strongly influenced by the behavior of the flow, which in this case is characterized by periodic temporal accelerations and decelerations. At any given phase of the cycle, the pressure distribution remains almost constant along aneurysm wall. The results show that the high surface complexity at aneurysmatic bulge, characterized by a diverging-converging shape, seems to have a little effect on the pressure distribution, except for those stages when a change of phase of the flow waveform occurs. According with pulsatile nature of the blood flow, the change in wall pressure distribution is also time dependent with similar patterns in different points of the

9 wall and with a distribution that tends to follow the pulsatile waveform. The largest favorable wall pressure gradients are obtained at systolic acceleration. Due to this, the flow remains laminar and fully attached to aneurysmatic sac wall and the largest adverse wall pressure gradients are obtained during late systolic deceleration, nearly at the beginning of the flow reversal period. An interesting behavior take place nearly the distal neck where a characteristic rise of the wall pressure value is seen in decelerating phase of the flow alternating with sharp pressure drop in acceleration phase of the pulsatile flow. The results also show that the aneurysmatic sac wall is exposed to low intensity of spatial WSS during the cardiac cycle except from the proximal and distal end sections which were exposed to higher shearing force mainly during early systolic acceleration. These findings are in accordance with the pattern flow in these regions. The change in WSS distribution is also time dependent. In general, this behavior may cause degenerative lesions of aneurysmal wall, altering the wall thickness and could eventually cause rupture. Taking into consideration the growing interest by the AAA morphology and morphometry, other geometric parameters related with 3D lumen centerlines have recently been proposed and assessed in the literature. According to [12], the resulting centerline is a piecewise linear line defined on the Voronoi diagram, whose vertices lie on Voronoi polygon boundaries [13]. Values of Voronoi sphere radius R(x) are therefore defined on centerlines, so that centerline points are associated with maximal inscribed spheres. Since centerlines were constructed to lie on local maxima of distance from the boundary, there is a tight connection between maximal sphere radius and minimum projection diameter used in clinical evaluation. In fact, classic angiographic vessel diameter evaluation is performed considering the minimum diameter obtained by measurements on different projections. The availability of a robust method for centerline computation and diameter measurement allows characterizing blood vessel geometry in a synthetic way, therefore giving the opportunity of performing a study on a population of models. Since it has been shown that planarity, tortuosity and branching angles have a major influence on complex blood flow patterns, such a study may reveal if particular vessel configurations are involved in vascular pathology. Three MBDs have been defined using this approach: tortuosity, curvature and torsion centerline. Algorithms have been developed to 3D reconstruction of the lumen centerline geometry. Figure 5 shows the visual representation of these determinants. Tortuosity, an absolute number, expresses the fractional increase in length of a tortuous vessel in relation to the imaginary straight line and has been described in [14]. Torsion is measured in 1/cm 2 and curvature is measured in 1/cm. Figure 5.Schematic visualization of tortuosity, curvature and torsion[7].

10 Among these parameters, the tortuosity in the presence of ILT could serve as potential predictor of rupture risk and may become a useful addition to maximum diameter in the decisionmaking process of AAA treatment [14]. 5 CONCLUSIONS This work represents a numerical investigation of 28 virtual abdominal aortic aneurysm models for the rupture risk prediction based on the geometrical parameters related to the wall shear stress and the luminal pressure. The effects of GDBs (asymmetric, diameter and length) of the aneurysm sac are studied in detail with respect to the wall stress, while maintaining the aorta diameter constant at 2 cm. This study demonstrates that the luminal pressure is the primary mechanical load on the aneurismal wall; while the WSS shows a weak significance respect to the variation of the saccular index, asymmetry and deformation rate. Therefore the peak hemodynamics pressure related to the GDBs can be used as predictors capable of identifying aneurysms that are at high risk of rupture, with higher accuracy than only using the maximum diameter criterion. 6 REFERENCES [1] D.A Vorp: Biomechanics of abdominal aortic aneurysm. J. Biomech., 40(2007), [2] G. Vilalta, F. Nieto, C. Vaquero, et al.: Patient-specific clinical assessment of abdominal aortic aneurysm rupture risk based on its geometric parameters. In Proceedings of the 8th International Conference on Biomedical Engineering, Innsbruck-Austria, February 16-18, [3] C.M Scotti, A.D. Shkolnik, S.C. Muluk, E.A Finol: Fluid-structure interaction in abdominal aortic aneurysms: Effects of asymmetry and wall thicknes. Biomedical Engineering OnLine. Accessed: 2012 Jan 21. [4] GiD. The personal pre and postprocessor, CIMNE (2011). [5] R. Löhner - A Parallel Advancing Front Grid Generation Scheme; AIAA (2000) [6] Robin Fahraeus: evolution of his concepts in cardiovascular physiology Am J Physiol Heart Circ Physiol September 1, :(3) H1005-H1015. [7] Y. Papaharilaou, J.A Ekaterinaris, E. Manousaki, et al: A decoupled fluid structure approach for estimating wall stress in abdominal aortic aneurysm. J. Biomech. 40 (2007), [8] Biasetti J, Hussain F, Gasser TC: Blood flow and coherent vortices in the normal and aneurysmatic aortas: a fluid dynamical approach to intra-luminal thrombus formation. J.R. Soc. Interface, Doi: /rsif [9] Dadvand, P.; Rossi, R.; Oñate, E.An object-oriented environment for developing finite element codes for multi-disciplinary applications. Archives of computational methods in engineering.17-3, pp /201. ISSN

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