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A patient-specific computational model of fluid-structure interaction in abdominal aortic aneurysms
Abstract
It is generally believed that knowledge of the wall stress distribution could help to find better rupture risk predictors of abdominal aortic aneurysms (AAAs). Although AAA wall stress results from combined action between blood, wall and intraluminal thrombus, previously published models for patient-specific assessment of the wall stress predominantly did not include fluid-dynamic effects. In order to facilitate the incorporation of fluid-structure interaction in the assessment of AAA wall stress, in this paper, a method for generating patient-specific hexahedral finite element meshes of the AAA lumen and wall is presented. The applicability of the meshes is illustrated by simulations of the wall stress, blood velocity distribution and wall shear stress in a characteristic AAA. The presented method yields a flexible, semi-automated approach for generating patient-specific hexahedral meshes of the AAA lumen and wall with predefined element distributions. The combined fluid/solid mesh allows for simulations of AAA blood dynamics and AAA wall mechanics and the interaction between the two. The mechanical quantities computed in these simulations need to be validated in a clinical setting, after which they could be included in clinical trials in search of risk factors for AAA rupture.
... In this article, the authors have focused exclusively on the innovation linked to the customization of the stent/graft in terms of the curvature and dimensions of the local diameter of the rings. As far as the design of the metallic stent is concerned, the next step will be to optimize it for a potential increase in fatigue life related to the number of loops and the central bending angle [5,6]. Furthermore, given that such a thoughtful stent/graft requires a new production technology, in particular additive manufacturing, the design will have to be optimized also for this reason. ...
... The results obtained and presented in this work are only partially comparable with the literature on studies of aneurysms or endoprostheses [1][2][3]6,8,10,11], since the cases are dealt with separately or through the CFD study of the reconstructed aorta or the study of the stent/graft, which is always simplified at the level of geometry. The simplification on the geometry considering a perfectly tubular vessel with a straight axis and the stent/graft also allows faster analyses with less computational effort. ...
... In line with various studies in the literature [6,8,56,57] , it is noted that the wall shear stress at the site of the aneurysm is zero. This indicates that the blood flow does not exert tangential stresses on the wall due to the presence of low-velocity areas in the site, as can be seen in Figure 12. ...
Background: The treatment of abdominal aortic aneurysm (AAA) is today commonly treated by inserting a stent-graft by the endovascular route, without resorting to open surgery. However, some clinical cases do not allow this less invasive approach, meaning that the stent-graft cannot be inserted and open surgery is used. Methods: In the study, we propose a fluid–structure interaction (FSI) analysis of an aneurysmatic aorta that could not be treated with Endovascular Aneurysm Repair (EVAR). The vessel is reconstructed through segmentation from CT scans and subsequently modeled on CAD software to create the surface and thickness of the vessel itself. Subsequently, we proceeded to carry out Computational Fluid Dynamics (CFD) and FSI simulation. We propose a computational study on a vessel geometry that is faithful to reality and customized. Results: Hemodynamic variable results of the carried out simulations indicate that low velocity and consequently very low WSS areas located in aneurysmal site are no longer found when conventional or patient-specific grafts are inserted. The wall stress distribution of aorta FEM analysis enabled the identification of the area at risk of failure, that is, in the posterior part of the aneurysm (∼107 Pa), while FSI analysis of the patient-specific graft led to a uniform von Mises stresses distribution (∼105 Pa), except for the junctions where peak stress occurred. Conclusion: The importance of this study is to highlight the benefits of the personalized stent/graft. As the authors expected, the study shows the numerous benefits of the customized stent/graft in terms of blood flow trend and wall stress compared to a traditional stent/graft by supporting the tendency to want to shift the target towards customized stents/grafts, also in the vascular surgery sector.
... Lagrangian description is typically used to track the deformation of the solid domain ( Donea et al., 1982;Bathe and Zhang, 2004). Neglecting the gravitational forces (f B s ) is a commonly used approach due to their insignificant effect on wall stresses ( Wolters et al., 2005). ...
... For the solid domain, principal wall stresses (σ 1 , σ 2 , σ 3 ), Von Mises stress and wall displacements are critical parameters for AAA rupture risk assessment. The peak wall stresses are better indicators when compared to the maximum AAA diameter, since rupture is the mechanical failure of the wall where the strength of wall is not sufficient to withstand the peak wall stress ( Fillinger et al., 2003;Wolters et al., 2005). Von Mises stress is a measure used for failure prediction based on three principal stresses as given in Equation (3). ...
... WSS is a measure of flow-driven tangential forces per unit area on the AAA wall. The magnitude of WSS at the fluid-structure interface can be determined by multiplying the viscosity (µ) with the local shear rate ( ˙ γ ) ( Wolters et al., 2005). In large arteries, WSS amplitudes typically range from 1 to 5 Pa, therefore WSS values smaller than 1 Pa are evaluated as low WSS (Ene-Iordache and Remuzzi, 2012;Qiu et al., 2018). ...
Abdominal aortic aneurysm (AAA) is the dilatation of the aorta beyond 50% of the normal vessel diameter. It is reported that 4–8% of men and 0.5–1% of women above 50 years of age bear an AAA and it accounts for ~15,000 deaths per year in the United States alone. If left untreated, AAA might gradually expand until rupture; the most catastrophic complication of the aneurysmal disease that is accompanied by a striking overall mortality of 80%. The precise mechanisms leading to AAA rupture remains unclear. Therefore, characterization of disturbed hemodynamics within AAAs will help to understand the mechanobiological development of the condition which will contribute to novel therapies for the condition. Due to geometrical complexities, it is challenging to directly quantify disturbed flows for AAAs clinically. Two other approaches for this investigation are computational modeling and experimental flow measurement. In computational modeling, the problem is first defined mathematically, and the solution is approximated with numerical techniques to get characteristics of flow. In experimental flow measurement, once the setup providing physiological flow pattern in a phantom geometry is constructed, velocity measurement system such as particle image velocimetry (PIV) enables characterization of the flow. We witness increasing number of applications of these complimentary approaches for AAA investigations in recent years. In this paper, we outline the details of computational modeling procedures and experimental settings and summarize important findings from recent studies, which will help researchers for AAA investigations and rupture mechanics.
... Decades of studies on the AAAs have shown that in addition to biological factors (Choke et al., 2005), several mechanical factors affect the wall stress Isotropic Uniform -- (Fillinger et al., 2002) Isotropic Uniform -Stress better than diameter (Wolters et al., 2005) Isotropic Uniform -Early FSI (Lu et al., 2007) Isotropic Uniform -- (Speelman et al., 2006) Isotropic Uniform Nonlinear, Isotropic - (Scotti et al., 2008) Isotropic Variable -FSI, idealized geometry (Rodríguez et al., 2008) Anisotropic Uniform -Idealized geometry (Rissland et al., 2008) Anisotropic Uniform Linear FSI (Dorfmann et al., 2010) Isotropic Uniform -- (Maier et al., 2010) Isotropic Uniform Nonlinear, Isotropic - (Gasser et al., 2010) Isotropic Variable Variable stiffness - (Raut et al., 2013) Isotropic Variable -- (Xenos et al., 2015) Anisotropic Uniform Linear FSI including flow regime (Dua and Dalman, 2010;Scotti and Finol, 2007), vessel geometry (Di Martino and Vorp, 2003;Kleinstreuer and Li, 2006;Li and Kleinstreuer, 2006;Rodríguez et al., 2008;Sacks et al., 1999;Scotti et al., 2005;Venkatasubramaniam et al., 2004;Vorp et al., 1998), and mechanical properties of the arterial wall (Di Martino and Vorp, 2003;Polzer et al., 2013;Rodríguez et al., 2008;Vorp, 2007). ...
... It is now well accepted that the wall stress is an important index in the assessment of AAA rupture. Furthermore, in large arteries, the deformability of the vessel wall greatly affects blood hemodynamics, thus for a more accurate simulation, fluid-structure interaction (FSI) analysis must be taken into account (Rissland et al., 2008;Scotti and Finol, 2007;Vorp, 2007;Wolters et al., 2005). ...
... One popular approach for stress analysis of AAA is to use patient-specific models that are constructed from clinical images (Di Martino et al., 2001;Dorfmann et al., 2010;Doyle et al., 2010;Raut et al., 2013;Rissland et al., 2008;Speelman et al., 2006;Wolters et al., 2005;Xenos et al., 2010). Clearly, the shape of an AAA and the thickness of the arterial wall are the major factor in determination of the arterial wall stress distribution (Raut et al., 2013;Scotti et al., 2008). ...
... Decades of studies on the AAAs have shown that in addition to biological factors (Choke et al., 2005), several mechanical factors affect the wall stress Isotropic Uniform -- (Fillinger et al., 2002) Isotropic Uniform -Stress better than diameter (Wolters et al., 2005) Isotropic Uniform -Early FSI (Lu et al., 2007) Isotropic Uniform -- (Speelman et al., 2006) Isotropic Uniform Nonlinear, Isotropic - (Scotti et al., 2008) Isotropic Variable -FSI, idealized geometry (Rodríguez et al., 2008) Anisotropic Uniform -Idealized geometry (Rissland et al., 2008) Anisotropic Uniform Linear FSI (Dorfmann et al., 2010) Isotropic Uniform -- (Maier et al., 2010) Isotropic Uniform Nonlinear, Isotropic - (Gasser et al., 2010) Isotropic Variable Variable stiffness - (Raut et al., 2013) Isotropic Variable -- (Xenos et al., 2015) Anisotropic Uniform Linear FSI including flow regime (Dua and Dalman, 2010;Scotti and Finol, 2007), vessel geometry (Di Martino and Vorp, 2003;Kleinstreuer and Li, 2006;Li and Kleinstreuer, 2006;Rodríguez et al., 2008;Sacks et al., 1999;Scotti et al., 2005;Venkatasubramaniam et al., 2004;Vorp et al., 1998), and mechanical properties of the arterial wall (Di Martino and Vorp, 2003;Polzer et al., 2013;Rodríguez et al., 2008;Vorp, 2007). ...
... It is now well accepted that the wall stress is an important index in the assessment of AAA rupture. Furthermore, in large arteries, the deformability of the vessel wall greatly affects blood hemodynamics, thus for a more accurate simulation, fluid-structure interaction (FSI) analysis must be taken into account (Rissland et al., 2008;Scotti and Finol, 2007;Vorp, 2007;Wolters et al., 2005). ...
... One popular approach for stress analysis of AAA is to use patient-specific models that are constructed from clinical images (Di Martino et al., 2001;Dorfmann et al., 2010;Doyle et al., 2010;Raut et al., 2013;Rissland et al., 2008;Speelman et al., 2006;Wolters et al., 2005;Xenos et al., 2010). Clearly, the shape of an AAA and the thickness of the arterial wall are the major factor in determination of the arterial wall stress distribution (Raut et al., 2013;Scotti et al., 2008). ...
Abdominal Aortic Aneurysm (AAA) is one of the main cardiovascular diseases, which threats human's health while it appears, develops and in crucial cases ruptures and leads to hemorrhage. In the current work, we aim to investigate numerically the transient blood flow in a patient-specific AAA model, while effects of wall compliance is considered by employing the fluid-structure interaction method. The AAA model is reconstructed from acquired CT angiographic data of a patient diagnosed with AAA and an intraluminal thrombus (ILT). For the comparison purposes two different material models, i.e. isotropic and anisotropic are considered. Additionally, to have a better estimation, wall thickness variability iscompared with simpler uniform wall thickness model. In this study Navier-Stokes equations along with elastodynamics equation are coupled through Arbitrary Lagrangian-Eulerian formulation method and solved numerically. Findings demonstrate that the isotropic material model with uniform wall thickness significantly underestimates wall stresses as compared to the anisotropic material model with variable wall thickness. Indeed, results emphasize that considering vessel wall as an anisotropic, heterogeneous (variable thickness) structure estimates much higher wall stresses comparing with isotropic, uniform thickness model. Therefore, given realistic vessel wall structure and the fact that the anisotropic, variable wall thickness model predicts higher wall stresses, it could be a more reliable model to give an accurate estimation to physicians to diagnose the stage of a disease and choosing an appropriate therapeutic procedure.
... Of the recent FSI studies of patient-specific AAAs [9,10,33,34], none modeled the supraceliac aorta, as would be crucial when assuming an axisymmetric inlet velocity profile [35], such as the parabolic and plug profiles assumed in these studies. Using the open-source software package SimVascular [36,37], we modeled a patient-specific AAA from the computed tomography angiogram of a 75-year-old male, including a total of 11 outlets between the supraceliac aorta and common iliac arteries. ...
... While Lin et al. [45] noted that the rigid wall assumption overestimates the wall shear stress (WSS) at the aneurysm neck and underestimates WSS in the aneurysm sac, we instead observed similar OSI (0.163 -0.180) and TAWSS (3.15 -3.39 dyn/cm 2 ) values averaged over the aneurysm sac across all three cases. These results are in agreement with the similar velocity profiles and WSS gradients observed across rigid wall and FSI models in [33]. ...
We recently demonstrated the reduction of the unified continuum and variational multiscale formulation to a computationally efficient fluid-structure interaction (FSI) formulation via three modeling assumptions pertaining to the vascular wall. Similar to the coupled momentum method introduced by Figueroa et al., the resulting semi-discrete formulation yields a monolithically coupled FSI system posed in an Eulerian frame of reference with only a minor modification of the fluid boundary integral. To achieve uniform second-order temporal accuracy and user-controlled high-frequency algorithmic damping, we adopt the generalized-α method for uniform temporal discretization of the entire coupled system. In conjunction with a fully consistent, segregated predictor multi-corrector algorithm preserving the block structure of the incompressible Navier–Stokes equations in the implicit solver’s associated linear system, a three-level nested block preconditioner is adopted for improved representation of the Schur complement. In this work, we apply our reduced unified continuum formulation to an appropriately prestressed patient-specific abdominal aortic aneurysm and investigate the effects of varying spatial distributions of wall properties on hemodynamic and vascular wall quantities of interest.
... Of the recent FSI studies of patient-specific AAAs [10,29,9,30], none modeled the supraceliac aorta, as would be crucial when assuming an axisymmetric inlet velocity profile [31], such as the parabolic and plug profiles assumed in these studies. Using the open-source software package SimVascular [32,33], we modeled a patient-specific AAA from the computed tomography angiogram of a 75-year-old male, including a total of 11 outlets between the supraceliac aorta and common iliac arteries. ...
... While Lin et al. [41] noted that the rigid wall assumption overestimates the wall shear stress (WSS) at the aneurysm neck and underestimates WSS in the aneurysm sac, we instead observed similar OSI (0.163 -0.180) and TAWSS (3.15 -3.39 dyn/cm 2 ) values averaged over the aneurysm sac across all three cases. These results are in agreement with the similar velocity profiles and WSS gradients observed across rigid wall and FSI models in [29]. ...
We recently demonstrated the reduction of the unified continuum and variational multiscale formulation to a computationally efficient fluid-structure interaction (FSI) formulation via three sound modeling assumptions pertaining to the vascular wall. Similar to the coupled momentum method introduced by Figueroa et al., the resulting semi-discrete formulation yields a monolithically coupled FSI system posed in an Eulerian frame of reference with only a minor modification of the fluid boundary integral. To achieve uniform second-order temporal accuracy and user-controlled high-frequency algorithmic damping, we adopt the generalized-$\alpha$ method for uniform temporal discretization of the entire coupled system. In conjunction with a fully consistent, segregated predictor multi-corrector algorithm preserving the block structure of the incompressible Navier-Stokes equations in the implicit solver's associated linear system, a three-level nested block preconditioner is adopted for improved representation of the Schur complement. In this work, we apply our reduced unified continuum formulation to an appropriately prestressed patient-specific abdominal aortic aneurysm and investigate the effects of varying spatial distributions of wall properties on hemodynamic and vascular wall quantities of interest.
... We did not have the measured blood flow for this specific patient, so we took the average values. Also, we did not consider wall properties and fluidstructure interaction [5], [9], [11] which have important role in mechanisms of aneurysmal growth and rupture, as well as in the stent graft migration and durability. ...
... Prescribed blood flow rate[8],[9]: (a) enddiastole (b) peak-systole; (c) late-systole; (d) end-systole; ...
The aim of this study was to demonstrate how fluid dynamic parameters are affected by aortic geometry and flow condition in two cases. Case A included blood flow analysis in aortic system with abdominal aortic aneurysm and left common iliac aneurysm before stent graft placement, while in case B was included stent graft geometry, at the site of the aneurysms. An individual patient-specific geometry and a 3D finite element meshes were reconstructed, based on Computed tomography (CT) scan images. The analysis was performed using the possibilities of computational fluid dynamics. It uses numeric methods and algorithms for the simulation of blood flow by solving the Navier-Stokes equations on computational meshes. The computational simulations of cardiac cycles were performed for average blood properties and blood flow rate. The velocity field, pressure and shear stress, as main fluid dynamics parameters, were visualized and compared for cases A and B.
... Our medium-term goal is to expand the capabilities of FEBIO by including robust fluid-structure interactions (FSIs). Such interac- tions occur commonly in biomechanics and biophysics, most nota- bly in cardiovascular mechanics where blood flows through the deforming heart and vasculature [12][13][14], diarthrodial joint lubri- cation where pressurized synovial fluid flows between deforming articular layers [15], cerebrospinal mechanics where fluid flow through ventricular cavities may cause significant deformation of Manuscript [16][17][18], vocal fold and upper airway mechanics [19,20], viscous flow over endothelial cells [21,22], canalicular and lacunar flow around osteocytes resulting from bone deformation [23][24][25], and many applications in biomedical device design [26][27][28]. Some FSI capabilities already exist in sev- eral of the open-source CFD codes referenced previously; these implementations were developed as addendums to sophisticated fluid analyses, often employing a limited range of solid material responses, such as small strain analyses of linear isotropic elastic solids. ...
... 12 Results reported later employ the Intel Math Kernel Library implementation of PARDISO. 13 ...
The mechanics of biological fluids is an important topic in biomechanics, often requiring the use computational tools to analyze problems with realistic geometries and material properties. This study describes the formulation and implementation of a finite element framework for computational fluid dynamics (CFD) in FEBio, a free software designed to meet the computational needs of the biomechanics and biophysics communities (febio.org). This formulation models nearly incompressible flow with a compressible isothermal formulation that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, while fluid density varies with dilatation as prescribed by the axiom of mass balance. Using this approach, equal order interpolations may be used for both essential variables over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian viscous responses as well as inviscid fluids. The efficiency of numerical solutions is enhanced using Broyden's quasi-Newton method. The results of finite element simulations were verified using well-documented benchmark problems as well as comparisons with other free and commercial codes. These analyses demonstrated that the novel formulation introduced in FEBio could successfully reproduce the results of other codes. The analogy between this CFD formulation and standard finite element formulations for solid mechanics makes it suitable for future extension to fluid-structure interactions.
... Several mathematical models based on idealised and realistic geometries are provided in literature, which show the effects of the aortic bulging on the redistribution of the pressure and velocity of the blood and the subsequent stresses in the aortic wall, compared to a healthy artery [11][12][13][14][15][16] ; the effects of stent-graft EVAR on the interaction between the aorta and the blood flow have also been investigated in idealised and real case studies [17][18][19][20][21][22][23][24][25][26] . ...
... One of the most challenging features of the modelling proposed in this work, which differentiates it from the current literature on computational vascular biomechanics [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] , is represented by the interaction between the components of the EVAS system. In particular, the so-called contact interaction (namely, the analysis of the behaviour of solids touching each other at one or more points or zones) was implemented, taking into account the possibility of slipping (with and without friction) and separation of the EVAS components, thus allowing for the detection of possible relative displacement and/or the separation between the endobags as well as between the endobags and the thrombus/aorta. ...
Endovascular sealing is a new technique for the repair of abdominal aortic aneurysms. Commercially available in Europe since 2013, it takes a revolutionary approach to aneurysm repair through minimally invasive techniques. Although aneurysm sealing may be thought as more stable than conventional endovascular stent graft repairs, post-implantation movement of the endoprosthesis has been described, potentially leading to late complications. The paper presents for the first time a model, which explains the nature of forces, in static and dynamic regimes, acting on sealed abdominal aortic aneurysms, with references to real case studies. It is shown that elastic deformation of the aorta and of the endoprosthesis induced by static forces and vibrations during daily activities can potentially promote undesired movements of the endovascular sealing structure.
... Several mathematical models based on idealised and realistic geometries are provided in literature, which show the effects of the aortic bulging on the redistribution of the pressure and velocity of the blood and the subsequent stresses in the aortic wall, compared to a healthy artery [11][12][13][14][15][16]; the effects of stent-graft EVAR on the interaction between the aorta and the blood flow have also been investigated in idealised and real case studies [17][18][19][20][21][22][23][24][25][26]. ...
... One of the most challenging features of the modelling proposed in this work, which differentiates it from the current literature on computational vascular biomechanics [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], is represented by the interaction between the components of the EVAS system. In particular, the so-called contact interaction (namely, the analysis of the behaviour of solids touching each other at one or more points or zones) was implemented, taking into account the possibility of slipping (with and without friction) and separation of the EVAS components, thus allowing for the detection of possible relative displacement and/or the separation between the endobags as well as between the endobags and the thrombus/aorta. ...
Endovascular sealing is a new technique for the repair of abdominal aortic aneurysms. Commercially available in Europe since~2013, it takes a revolutionary approach to aneurysm repair through minimally invasive techniques. Although aneurysm sealing may be thought as more stable than conventional endovascular stent graft repairs, post-implantation movement of the endoprosthesis has been described, potentially leading to late complications. The paper presents for the first time a model, which explains the nature of forces, in static and dynamic regimes, acting on sealed abdominal aortic aneurysms, with references to real case studies. It is shown that elastic deformation of the aorta and of the endoprosthesis induced by static forces and vibrations during daily activities can potentially promote undesired movements of the endovascular sealing structure.
... Moreover, the FEM has contributed to the optimisation of cardiovascular stent design, enhancing their mechanical performance and biocompatibility while reducing tissue injury risks (Gundert et al. 2012;Baradaran et al. 2021;Russ et al. 2021). Additionally, researchers have used the FEM to create patient-specific models of aortic aneurysms, providing valuable insights into mechanical properties and rupture risks (Wolters et al. 2005;Martin et al. 2015;Wang et al. 2018), ultimately aiding in the decision-making process for treatment options. ...
This study presents novel applications for conducting left ventricular (LV) shape analysis within a finite element (FE) framework. The 3D endocardial surface data of both healthy and dilated LVs are obtained using TomTec 4D LV-Analysis software through echocardiography and then imported into the FE framework for LV geometry. By applying FE analysis to 3D echocardiography data, we propose novel methods for evaluating LV shape and remodelling, including the nodal strain approximation along myocardial fibre directions, the assessment of LV sphericity and eccentricity in multiple planes, a new method for LV apical conicity, and the analysis of regional LV rotations. The feasibility of these novel formulations and the developed FE based framework are demonstrated by evaluating and comparing 3D echocardiogra-phy data from two objects, one with a healthy LV and the other with dilated cardiomyopathy. ARTICLE HISTORY
... In addition, the surrounding organs were imposed, not causing a load at the outer wall of blood vessels, which means the residual stresses in the AAAs wall assumed to be zero [45]. The local variances of the blood vessel thickness cannot be captured by the CT scan; hence, it was presumed to be 2 mm [38,[48][49][50] and uniformly distributed at all regions of the AAA wall. Hypertension contributes to the formation of AAA and a key factor causing the rupture [51]. ...
... In several studies, the inclusion of ILT provided improved solution accuracy in terms of the mechanical stresses on the wall [29,30]. Mechanical stress can be considered as the most important factor in determining the risk of rupture because the rupture occurs when the mechanical stresses that arise due to the flow-induced hemodynamic forces are lower than the ultimate stress of AAA wall [27,31]. Therefore, determining the peak mechanical stresses has utmost importance to predict an impending mechanical failure on AAA wall. ...
Abdominal aortic aneurysm (AAA) is a critical health disorder, where the abdominal aorta dilates more than 50% of its normal diameter. Enlargement in abdominal aorta alters the hemodynamics and flow-induced forces on the AAA wall. Depending on the flow conditions, the hemodynamic forces on the wall may result in excessive mechanical stresses that lead to AAA rupture. The risk of rupture can be predicted using advanced computational techniques such as computational fluid dynamics (CFD) and fluid–structure interaction (FSI). For a reliable rupture risk assessment, formation of intraluminal thrombus (ILT) and uncertainty in arterial material properties should be taken into account, mainly due to the patient-specific differences and unknowns in AAAs. In this study, AAA models are computationally investigated by performing CFD simulations combined with FSI analysis. Various levels of ILT burdens are artificially generated in a realistic AAA geometry, and the peak effective stresses are evaluated to elucidate the effect of material models and ILT formation. The results indicate that increasing the ILT burden leads to lowered effective stresses on the AAA wall. The material properties of the artery and ILT are also effective on the stresses; however, these effects are limited compared to the effect of ILT volume in the AAA sac.
... However, reliable, profound and risk-based recommendations for clinical support are as of now only available for a few selected problems, but are currently very active areas of research. Today, many impressive patient-specific case studies focusing on various different cardiovascular diseases, taking for example aortic dissection (see, e.g., [266,[385][386][387][388] among others) or abdominal aortic aneurysma (see, for example [389][390][391][392]) can be found in literature, while non-invasive parameter identification and propagation of uncertainties through the computationally expensive models is becoming more and more feasible given robust and fast solution algorithms for the individual subproblems and FSI [393][394][395][396][397][398]. ...
Countless applications in science and engineering such as blood flow, dam or ship construction, insect flight, car design or wind interacting with buildings and bridges besides many other problems involve incompressible flows and their interaction with elastic bodies. Whenever the flow field triggers corresponding forces acting on wetted structures large enough to cause finite deformations, this can, in turn, significantly alter the flow domain and thereby the velocity and pressure present in the fluid. As a consequence, any reliable model of such physical processes must capture the interaction of fluids and solids.
As a prime example from the biomedical context, complex clinical applications in the cardiovascular system continue to challenge numerical methods and well-established algorithms. Here, incorporating standard mathematical methods can impair the solution scheme's performance drastically. However, there is vast potential for computational science and engineering to assist in clinic, e.g., in training medical personnel or evaluating treatment options in-silico on digital twins. Parameter studies on virtual cohorts or patient-specific modelling of cardiovascular diseases to foster understanding of the underlying principles at work additionally continue to motivate research efforts to develop tailored numerical methods.
Within this work, we first devise novel schemes to tackle generalised Newtonian fluid flow, which incorporate shear-thinning effects contributing to the complex behaviour of blood. Such fluids show a dependence of the viscosity on the local shear rate, such that subtle differences in the flow field's gradients might impact the shear resistance and hence the overall problem drastically. In this context, we extend well-established concepts towards generalised Newtonian fluid flow, devising a stabilised coupled velocity-pressure formulation and an iteration-free split-step scheme decoupling velocity and pressure spaces. Both approaches allow for equal-order interpolation with standard continuous finite elements, which might be considered advantageous in practical biomedical applications as spatial discretisations derived from segmented medical image data are often lower-order accurate.
The stabilised and split-step flow solvers build upon a pressure Poisson equation as a fundamental ingredient, which is derived from the fluid's momentum balance equation, implicitly enforcing mass balance. Together with fully consistent boundary conditions, this yields increased accuracy compared to standard inf-sup stabilisation schemes for the coupled approach or even allows recovering the pressure given velocity. Furthermore, the design of efficient solution schemes decoupling the involved fields and linearising the governing equations based on higher-order accurate extrapolation combined with adaptive timestepping results in a substantial overall speed-up. In this way, the proposed novel techniques facilitate practical applications in science, industry and medicine, since the time to solution can be reduced significantly.
Targeting the haemodynamic regime, we couple the split-step flow solver to a nearly incompressible, hyperelastic, fibre-reinforced continuum modelling the arterial tissue. In this regard, major contributions of this work are (i) combining added-mass stable, semi-implicit coupling of fluid and structure, Robin interface conditions and acceleration schemes such as Aitken's relaxation or the Interface Quasi-Newton Inverse Least-Squares method and (ii) incorporating modelling aspects and numerical techniques necessary for clinical application. The latter aspect involves lumped parameter models to account for the neglected downstream vasculature, construction of suitable inflow profiles on general inflow sections given volumetric flow rates only, stabilising re-entrant flow and dominant convection, incorporating stress states present in the structure at the time of image acquisition and automatic construction of proper material orientation for complex geometries as encountered, e.g., in aortic dissection.
The final fluid-structure interaction scheme thus merely couples the fluid's pressure and the structural solver iteratively, while the remaining subproblems are solved only once per time step, all of which consist of standard problems frequently encountered in science and engineering such as Poisson, mass-matrix or advection-diffusion-reaction equations. Thus, we heavily profit from standard algebraic multigrid preconditioners and linear solvers, which enable us to simulate realistic scenarios using physiological problem parameters.
We carefully investigate the proposed flow and fluid-structure interaction schemes with respect to their accuracy, robustness and reliability via several numerical tests ranging from academic setups over benchmark scenarios all the way to patient-specific simulations, were we demonstrate favourable mathematical properties such as expected convergence rates and required robustness. Thus, the numerical examples presented herein aim to bridge the gap to real-world problems, considering state-of-the-art modelling aspects, physiological parameters and showcasing the framework's versatility and thereby substantially contribute to the field of computational (bio-)mechanics.
... This also leads to the induced damage of endothelium due to the endothelial cells sensitivity and elevated level of WSS and Time Average Wall Shear Stress (TAWSS) changes [71][72][73]. Moreover, the aortic rupture has a high potential to be appeared on the ascending aorta wall due to the high WSS effects acting on the aorta tissue [13,74,75]. ...
... This poses a problem, since the use of contrast agents and radiation hampers frequent use. 21 Since previous FSI studies used either idealized AAA geometries 18,22,23 or a small set of CT-derived geometries, 10,24,34,36 no longitudinal studies employing FSI simulations are executed yet. For such a study, timeresolved 3-dimensional ultrasound (3D+t US) is the preferred image modality to extract the patient-specific geometry, since it is safe, fast and affordable. ...
To improve abdominal aortic aneurysm (AAA) rupture risk assessment, a large, longitudinal study on AAA hemodynamics and biomechanics is necessary, using personalized fluid-structure interaction (FSI) modeling. 3-dimensional, time-resolved ultrasound (3D+t US) is the preferred image modality to obtain the patient-specific AAA geometry for such a study, since it is safe, affordable and provides temporal information. However, the 3D+t US field-of-view (FOV) is limited and therefore often fails to capture the inlet and aorto-iliac bifurcation geometry. In this study, a framework was developed to add parametric inlet and bifurcation geometries to the abdominal aortic aneurysm geometry by employing dataset statistics and parameters of the AAA geometry. The impact of replacing the patient-specific inlet and bifurcation geometries, acquired using computed tomography (CT) scans, by parametric geometries was evaluated by examining the differences in hemodynamics (systolic and time-averaged wall shear stress and oscillatory shear index) in the aneurysm region. The results show that the inlet geometry has a larger effect on the AAA hemodynamics (median differences of 7.5 to 18.8%) than the bifurcation geometry (median differences all below 1%). Therefore, it is not feasible to replace the patient-specific inlet geometry by a generic one. Future studies should investigate the possibilities of extending the proximal FOV of 3D+t US. However, this study did show the feasibility of adding a parametric bifurcation geometry to the aneurysm geometry. After extending the proximal FOV, the obtained framework can be used to extract AAA geometries from 3D+t US for FSI simulations, despite the absence of the bifurcation geometry.
... For accurate patient-specific finite element (FE) analysis, robust geometry segmentation and meshing is important, 20,21 however, using commercially available finite element solvers to hexahedral mesh complex 3D arterial geometries is very difficult. To-date, studies have created structured hexahedral meshes of carotid bifurcations, 17,18,20,21,24,[26][27][28] but in many cases simplifying assumptions are used. These assumptions range from assigning a uniform wall thickness by extruding the lumen, to assuming the geometry is cylindrical or using a one-layer vessel model. ...
The carotid bifurcation experiences a complex loading environment due to its anatomical structure. Previous in‐vivo material parameter estimation methods often use simplified model geometries, isotropic hyperelastic constitutive equations or neglect key aspects of the vessel, such as the zero‐pressure configuration or residual stress, all of which have independently been shown to alter the stress environment of the vessel wall. Characterising the location of high stress in the vessel wall has often been proposed as a potential indicator of structural weakness. However, excluding the afore‐mentioned zero‐pressure configuration, residual stress and patient specific material parameters can lead to an incorrect estimation of the true stress values observed, meaning that stress alone as a risk indicator of rupture is insufficient. In this study, we investigate how the estimated material parameters and overall stress distributions in geometries of carotid bifurcations, extracted from in‐vivo MR images, alter with the inclusion of the zero‐pressure configuration and residual stress. This approach consists of the following steps: (1) geometry segmentation and hexahedral meshing from in‐vivo magnetic resonance images (MRI) at two known phases; (2) computation of the zero‐pressure configuration and the associated residual stresses; (3) minimisation of an objective function built on the difference between the stress states of an “ almost true” stress field at two known phases and a “deformed” stress field by altering the input material parameters to determine patient specific material properties; and (4) comparison of the stress distributions throughout these carotid bifurcations for all cases with estimated material parameters. This numerical approach provides insights into the need for estimation of both the zero‐pressure configuration and residual stress for accurate material property estimation and stress analysis for the carotid bifurcation, establishing the reliability of stress as a rupture risk metric. This article is protected by copyright. All rights reserved.
... (B) Structured hexahedral solid mesh using idealized AAA model. (C) Unstructured tetrahedral fluid mesh using patient-specific AAA model[1].Figure2:Boundary conditions FSI analysis[2]. ...
Background: About 18 million people die each year from cardiovascular disorders, accounting for 31% of all deaths worldwide. Abdominal Aortic Aneurysm (AAA) is a serious clinical condition manifested as dilation of the aorta beyond 50% of the normal vessel diameter. Current clinical practice is to surgically repair large AAAs with a diameter > 5.5 cm. However, the practice is questionable based on small AAA rupture and large AAA no rupture cases. Currently, there is no accepted technique to quantify the risk of rupture for individual AAAs. It is believed that rupture locations are where peak wall stresses act. Hemodynamic forces by the flowing blood such as shear stress are also thought to contribute to the formation of aneurysms leading to rupture. Aim: Our aim is to perform precise computational analysis for the assessment of rupture risk for AAA patients. Methods: In this IRCC funded project, we will develop a patient-specific computational modeling methodology to assess wall stresses acting on the diseased AAA, for reliable rupture risk assessment of the conditions. In the computational simulations, we will adapt the fluid-structure interaction approach to account for both tissue displacements and hemodynamic forces, for enhanced accuracy. We have recruited 20 AAA patients at HMC and collected CT scans and ultrasound images for these patients. Using these medical data, we are developing accurate 3D model geometries. Doppler ultrasound measurements are used as velocity boundary conditions in the simulations. Expected Results: Findings from this project will contribute significantly to understanding the biomechanics and mechanobiology of AAA rupture and will help to establish a computational modeling approach for rupture risk assessment of AAAs.
... At the external boundary, a pressure load acting on the outer surface of the wall is sometimes included to account for the surrounding tissue and organs that affect aortic pulsatility. However, due to the limited published data on the exact magnitude of the abdominal pressure [40], we assume that the aortic wall is exposed to zero external ambient pressure, as often done in the literature [35,53,56]. ...
We introduce a poroelastic model for intraluminal thrombus (ILT) that captures both the flow within ILT and its deformation. The model for ILT is coupled with blood flow and arterial wall deformation and used to study the biomechanics in image-based abdominal aortic aneurysm (AAA) models. Three different patient-specific geometries were considered in this study. Using finite element analysis, numerical simulations were performed to investigate the role of ILT on the risk of AAA rupture as assessed by Peak Wall Stress (PWS). Our results indicate that the presence of ILT may reduce wall stress in AAA. However, our results indicate that ILT permeability has little effect on AAA PWS, and that similar values of PWS are obtained with both porous and nonporous ILT models.
... Previous studies employing FSI simulations either used idealized AAA geometries (Scotti et al., 2005(Scotti et al., , 2008Lin et al., 2017) or a small set of patient-specific geometries acquired using computed tomography (CT) (Di Martino et al., 2001;Wolters et al., 2005;Scotti and Finol, 2007;Xenos and Bluestein, 2011). ...
Currently, the prediction of rupture risk in abdominal aortic aneurysms (AAAs) solely relies on maximum diameter. However, wall mechanics and hemodynamics have shown to provide better risk indicators. Patient-specific fluid-structure interaction (FSI) simulations based on a non-invasive image modality are required to establish a patient-specific risk indicator. In this study, a robust framework to execute FSI simulations based on time-resolved three-dimensional ultrasound (3D+t US) data was obtained and employed on a data set of 30 AAA patients. Furthermore, the effect of including a pre-stress estimation (PSE) to obtain the stresses present in the measured geometry was evaluated. The established workflow uses the patient-specific 3D+t US-based segmentation and brachial blood pressure as input to generate meshes and boundary conditions for the FSI simulations. The 3D+t US-based FSI framework was successfully employed on an extensive set of AAA patient data. Omitting the pre-stress results in increased displacements, decreased wall stresses, and deviating time-averaged wall shear stress and oscillatory shear index patterns. These results underline the importance of incorporating pre-stress in FSI simulations. After validation, the presented framework provides an important tool for personalized modeling and longitudinal studies on AAA growth and rupture risk.
... While individual fluid particles experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect. The third term of the equation (4) represents convective acceleration, and it is a nonlinear quantity [6,[7][8][9][10][11][12][13][14][15]. ...
The main purpose of this study was to reconstruct 3D aorta models based
on a series of 2D CT images of two patients suffering from an aneurysm.
After the 3D models had been made, a numerical solution of the problem
of the abdominal aortic aneurysm was obtained. The numerical solution
incorporated mathematical models of biomechanical systems. Various
parameters such as shear stress, pressure, and velocity of fluid through
blood vessel could be calculated using these methods. The first part of the
paper introduces the abdominal aorta and aneurysm. The second part of
this paper describes the process of getting 3D model geometries of the
aortic aneurysms. Finally, the results obtained from the models are
discussed with the aim of predicting a rupture of an abdominal aortic
aneurysm and selecting the right treatment for this disease.
... On the cerebral aneurysm wall, WSS is a measure of tangential flow-driven forces per unit area. It is possible to determine the magnitude of WSS at the interface between the fluid structures by multiplication of the viscosity (l) with the local shear rate (c_) [23]. Time-averaged wall shear stress (TAWSS) can be calculated as given in Equation (2) where T is the period of integration [24]. ...
This study investigates the effect of stent thickness and stent porosity which are important factors determining the post-treatment intra-aneurysmal hemodynamics. The study uses computational fluid dynamics (CFD) to estimate the hemodynamic behaviour: flow velocity, pressure distributions, time-averaged wall shear stress (TAWSS), oscillatory shear index (OSI), besides relative residence time (RRT) blood flow distribution in a proposed stent and three other commercially available stents. The hemodynamic behaviour is compared between four different cases. In each case, each stent has the specific thickness and porosity values. The results show that the velocity magnitude inside the sac declined in thinner stents and lower porosity stents, TAWSS on the aneurysmal wall declined linearly in lower porosity stents, OSI and RRT increased obviously in thicker stents and higher porosity stents. Finally, the results conclude that the stent with the lowest thickness and porosity has the best performance that leads to post-stent thrombus formation and healing. However, the proposed stent design, a more porous construct, has a higher RRT compared to the used commercially available stents, which helps promote the thrombus growth inside the aneurysm sac.
... Momentum conservation is the governing equation in solid domain as given in Equation (5), where τ s defines solid stress tensor, f s defines body forces on the solid, ρ s defines the solid mass density, and a s is acceleration vector of the solid particles. Similar to the fluid domain, gravitational acceleration is the main body force and it is neglected in many studies due to indiscernible effect on solid deformations [44]: ...
The heart is the first functional organ in a developing embryo. Cardiac development continues throughout developmental stages while the heart goes through a serious of drastic morphological changes. Previous animal experiments as well as clinical observations showed that disturbed hemodynamics interfere with the development of the heart and leads to the formation of a variety of defects in heart valves, heart chambers, and blood vessels, suggesting that hemodynamics is a governing factor for cardiogenesis, and disturbed hemodynamics is an important source of congenital heart defects. Therefore, there is an interest to image and quantify the flowing blood through a developing heart. Flow measurement in embryonic fetal heart can be performed using advanced techniques such as magnetic resonance imaging (MRI) or echocardiography. Computational fluid dynamics (CFD) modeling is another approach especially useful when the other imaging modalities are not available and in-depth flow assessment is needed. The approach is based on numerically solving relevant physical equations to approximate the flow hemodynamics and tissue behavior. This approach is becoming widely adapted to simulate cardiac flows during the embryonic development. While there are few studies for human fetal cardiac flows, many groups used zebrafish and chicken embryos as useful models for elucidating normal and diseased cardiogenesis. In this paper, we explain the major steps to generate CFD models for simulating cardiac hemodynamics in vivo and summarize the latest findings on chicken and zebrafish embryos as well as human fetal hearts.
... The Navier-Stokes equations with arbitrary Lagrangian-Eulerian formulation are used as the governing equations in the frame of artery zone [16]. ...
Background
The effects of arterial wall compliance on blood flow have been revealed using fluid-structure interaction in last decades. However, microcirculation is not considered in previous researches. In fact, microcirculation plays a key role in regulating blood flow. Therefore, it is very necessary to involve microcirculation in arterial hemodynamics.
Objective
The main purpose of the present study is to investigate how wall compliance affects the flow characteristics and to establish the comparisons of these flow variables with rigid wall when microcirculation is considered.
Methods
We present numerical modeling in arterial hemodynamics incorporating fluid-structure interaction and microcirculation. A novel outlet boundary condition is employed to prescribe microcirculation in an idealised model.
Results
The novel finding in this work is that wall compliance under the consideration of microcirculation leads to the increase of wall shear stress in contrast to rigid wall, contrary to the traditional result that wall compliance makes wall shear stress decrease when a constant or time dependent pressure is specified at an outlet.
Conclusions
This work provides the valuable study of hemodynamics under physiological and realistic boundary conditions and proves that wall compliance may have a positive impact on wall shear stress based on this model. This methodology in this paper could be used in real model simulations.
... In terms of structural modelling, if the aim of the simulation is to predict blood hemodynamics, even simple material models can be sufficient, for instance, based on isotropic linear elasticity [135][136][137][138]. On the other hand, to compute realistic transmural stress distributions resulting from the interior blood flow in an artery, nonlinear and anisotropic model of the wall structure should be accounted for [139,140]. ...
This review aims to highlight urgent priorities for the computational biomechanics community in the framework of mechano-chemo-biological models. Recent approaches, promising directions and open challenges on the computational modelling of arterial tissues in health and disease are introduced and investigated, together with in silico approaches for the analysis of drug-eluting stents that promote a pharmacological-induced healing. The paper addresses a number of chemo-biological phenomena that are generally neglected in biomechanical engineering models but are most likely instrumental for the onset and the progression of arterial diseases. An interdisciplinary effort is thus encouraged for providing the tools for an effective in silico insight into medical problems. An integrated mechano-chemo-biological perspective is believed to be a fundamental missing piece for crossing the bridge between computational engineering and life sciences, and for bringing computational biomechanics into medical research and clinical practice.
... Moreover, it is also observed that the smallest opening produced the highest pressure at the ascending wall of aorta due to the jet flow effects. If these circumstances continuously occurred, the aortic rupture has higher potential to appear at the ascending aortic wall due to the high WSS effects acting on the aorta tissue [19,65,66]. ...
This study investigated the impact of paravalvular leakage (PVL) in relation to the different valve openings of the transcatheter aortic valve implantation (TAVI) valve using the fluid structure interaction (FSI) approach. Limited studies were found on the subject of FSI with regards to TAVI-PVL condition, which involves both fluid and structural responses in coupling interaction. Hence, further FSI simulation with the two-way coupling method is implemented to investigate the effects of hemodynamics blood flow along the patient-specific aorta model subjected to the interrelationship between PVL and the different valve openings using the established FSI software ANSYS 16.1. A 3D patient-specific aorta model is constructed using MIMICS software. The TAVI valve identical to Edward SAPIEN XT 26 (Edwards Lifesciences, Irvine, California), at different Geometrical Orifice Areas (GOAs), is implanted into the patient’s aortic annulus. The leaflet opening of the TAVI valve is drawn according to severity of GOA opening represented in terms of 100%, 80%, 60%, and 40% opening, respectively. The result proved that the smallest percentage of GOA opening produced the highest possibility of PVL, increased the recirculatory flow proximally to the inner wall of the ascending aorta, and produced lower backflow velocity streamlines through the side area of PVL region. Overall, 40% GOA produced 89.17% increment of maximum velocity magnitude, 19.97% of pressure drop, 65.70% of maximum WSS magnitude, and a decrement of 33.62% total displacement magnitude with respect to the 100% GOA.
... The aortic wall was formulated as nearly incompressible, using linear elastic material with isotropic Young's Modulus as suggested in published literature (Gao et al., 2006;Lantz et al., 2011). The density of aortic wall was set to 1080 kg/m 3 while the Poisson's ratio was set to 0.499 consistent with the literature (Lantz et al., 2011;Wolters et al., 2005). The wall stiffness is always subjected to specific parameters and a universal value does not exist (Lantz et al., 2011). ...
Thoracic aortic aneurysm (TAA) is a severe cardiovascular disease with a high mortality rate, if left untreated. Clinical observations show that aneurysm growth can be linked to undesirable hemodynamic conditions of the aortic aneurysm. In order to gain more insight on TAA formation, we developed a computational framework in vitro to investigate and compare the flow patterns between pre-aneurismal and post-aneurismal aorta using a deformable wall model. This numerical framework was validated by an in vitro experiment accounting for the patient-specific geometrical features and the physiological conditions. The complex flow behaviors in the preaneurismal and post-aneurismal aorta were evaluated experimentally by particle image velocimetry (PIV). Our experimental results demonstrated flow behaviors similar to those observed in the fluid-structure interaction (FSI) numerical study. We observed a small vortex induced by the non-planarity of pre-aneurismal aorta near the aortic arch in pre-aneurysmal aorta may explain the aneurysm formation at the aortic arch. We found that high endothelial cell action potential (ECAP) correlates with the recirculation regions, which might indicate possible thrombus development. The promising image-based fluid-structure interaction model, accompanied with an in vitro experimental study, has the potential to be used for performing virtual implantation of newly developed stent graft for treatment of TAA.
... -27 We then simulated ...
Objective—
Isolated common iliac artery aneurysms (CIAA) are rare. Their prognosis and influence on aortoiliac blood flow and remodeling are unclear. We evaluated the hypotheses that morphology at and distal to the aortic bifurcation, together with the associated hemodynamic changes, influence both the natural history of CIAA and proximal aortic remodeling.
Approach and Results—
Twenty-five isolated CIAAs (15 intact, 10 ruptured), in 23 patients were reconstructed and analyzed with computational fluid dynamics: all showed abnormal flow. Then we studied a series of 24 hypothetical aortoiliac geometries in silico with varying abdominal aortic deflection and aortic bifurcation angles: key findings were assessed in an independent validation cohort of 162 patients. Wall shear stress in isolated unilateral CIAAs was lower than the contralateral common iliac artery, 0.38±0.33 Pa versus 0.61±0.24 Pa, inversely associated with CIAA diameter ( P <0.001) and morphology (high shear stress in variants distal to a sharp kink). Rupture usually occurred in regions of elevated low and oscillatory shear with a wide aortic bifurcation angle. Abdominal aortas deflected towards the CIAA for most unilateral isolated CIAAs (14/21). In silico, wider bifurcation angles created high focal regions of low and oscillatory shear in the common iliac artery. The associations of unilateral CIAA with aortic deflection and common iliac artery diameter with bifurcation angle were confirmed in the validation cohort.
Conclusions—
Decreasing wall shear stress is strongly associated with CIAA progression (larger aneurysms and rupture), whereas abnormal blood flow in the CIAA seems to promote proximal aortic remodeling, with adaptive lateral deflection of the abdominal aorta towards the aneurysmal side.
... Abdominal aortic aneurysms (AAA) occur when the maximum diameter of the abdominal aorta increases by 50% or the diameter is greater than 3.0 cm. Computational haemodynamic studies have suggested that the infrarenal aorta experiences reversed flow due to the bifurcation of the aorta into the iliac arteries and has been linked to the dilation of the aorta wall (Wolters et al. 2005). Around 75% of AAA contain an intraluminal thrombus (ILT) (Wang et al. 2002) although the contribution of thrombus to aneurysm rupture risk is debated. ...
Computational modelling of the cardiovascular system offers much promise, but represents a truly interdisciplinary challenge, requiring knowledge of physiology, mechanics of materials, fluid dynamics and biochemistry. This paper aims to provide a summary of the recent advances in cardiovascular structural modelling, including the numerical methods, main constitutive models and modelling procedures developed to represent cardiovascular structures and pathologies across a broad range of length and timescales; serving as an accessible point of reference to newcomers to the field. The class of so-called hyperelastic materials provides the theoretical foundation for the modelling of how these materials deform under load, and so an overview of these models is provided; comparing classical to application-specific phenomenological models. The physiology is split into components and pathologies of the cardiovascular system and linked back to constitutive modelling developments, identifying current state of the art in modelling procedures from both clinical and engineering sources. Models which have originally been derived for one application and scale are shown to be used for an increasing range and for similar applications. The trend for such approaches is discussed in the context of increasing availability of high performance computing resources, where in some cases computer hardware can impact the choice of modelling approach used.
... Computational modelling of the heart is a powerful technique for detailed investigations of cardiac behavior, and enables the study of mechanisms and processes that are not directly accessible by experimental methods. There is currently a drive towards adapting these computational models to individual patient data, to aid in the creation of individualized diagnosis, clinical decision support, and treatment planning [1][2][3][4][5][6][7]. However, this model adaptation presents a number of challenges related to the lack of available data and the fact that measurable data, needed for patient-specific model input parameters, are inherently subject to measurement uncertainties or intrinsic biological variability. ...
Computational cardiac modelling is a mature area of biomedical computing, and is currently evolving from a pure research tool to aiding in clinical decision making. Assessing the reliability of computational model predictions is a key factor for clinical use, and uncertainty quantification (UQ) and sensitivity analysis are important parts of such an assessment. In this study, we apply new methods for UQ in computational heart mechanics to study uncertainty both in material parameters characterizing global myocardial stiffness and in the local muscle fiber orientation that governs tissue anisotropy. The uncertainty analysis is performed using the polynomial chaos expansion (PCE) method, which is a non-intrusive meta-modeling technique that surrogates the original computational model with a series of orthonormal polynomials over the random input parameter space. In addition, in order to study variability in the muscle fiber architecture, we model the uncertainty in orientation of the fiber field as an approximated random field using a truncated Karhunen-Lo\'eve expansion. The results from the UQ and sensitivity analysis identify clear differences in the impact of various material parameters on global output quantities. Furthermore, our analysis of random field variations in the fiber architecture demonstrate a substantial impact of fiber angle variations on the selected outputs, highlighting the need for accurate assignment of fiber orientation in computational heart mechanics models.
... studied exploiting FSI techniques in recent years. Some of them include yacht sails [8,79,131], parachutes [120,149] and other fabric dynamics [11,105], rotor dynamics [21,47], and a vast number of biomedical applications including arterial blood flow [44,50,57], aortic heart valves [32,140], and aortic aneurysms [128,148]. ...
This work is inspired by natural flyers such bats and insects. They show outstanding aerodynamic performance due to their flexible membrane wings and their ability to control its stiffness to improve manoeuvrability. In this work the fluid-structure coupling as well as the physics and the control of electroactive membranes have been simulated in a multiphysics framework. This study has allowed not only to have an insight of the flow mechanisms which allow a membrane wing to enhace lift and delay stall at high angles of attack but also lays the basis of the understanding of how an active control of the membrane’s stiffness in response to the unsteadiness of the fluid-structure coupling can deliver a more stable flight. In particular, numerical simulations are conducted for an electroactive membrane wing in a laminar incompressible flow. The fluid-structure interaction problem is simulated for electroactive polymers whose shape and stiffness can be modified by applying an electric potential. The Maxwell stresses generated by the electric field across the membrane produce an in-plane relaxation. Results from this work show that a fixed voltage applied to a prestretched membrane results in an increased camber and therefore enhanced mean lift. Moreover, the effect of a partial activation is considered as well as an oscillating voltage across the membrane. The results presented in this work indicate that the lift is increased at angles of attack up to a = 12 when the back section of the membrane is activated. In addition, lift is increased at higher angles of attack when the voltage oscillates at frequencies close to resonance of the coupled fluid-structure system. Finally, an active control has been simulated exploiting the electromechanical characteristics of electroactive polymers and using the membrane itself as a sensor. This work shows that when the whole surface of the membrane is used as sensor and actuator, a proportional integral control is able to reduce the membrane’s oscillations at medium angles of attack, delivering a more stable flight and smoother response to a gust.
... Fluid-structure interaction models (FSI) are creeping into all disciplines in science [2]. Its applications can range from guiding the engineering design of airplanes, boats, and cars for transportation [3,4,5] to understanding the locomotion of aquatic organisms [6,7,8] and animal flight [9,10,11], or personalized medicine [12,13,14] and surgical planning and practice [15,16] to understanding the role of blood flow in heart development [17,18,19,20]. ...
The immersed boundary method (IB) is an elegant way to fully couple the motion of a fluid and deformations of an immersed elastic structure. In that vein, the IB2d software allows for expedited explorations of fluid-structure interaction for beginners and veterans to the field of computational fluid dynamics (CFD). While most open source CFD codes are written in low level programming environments, IB2d was specifically written in high- level programming environments to make its accessibility extend beyond scientists with vast programming experience. Although introduced previously by Battista et al. 2015, many improvements and additions have been made to the software to allow for even more robust models of material properties for the elastic structures, including a data analysis package for both the fluid and immersed structure data, an improved time-stepping scheme for higher accuracy solutions, and functionality for modeling slight fluid density variations as given by the Boussinesq approximation.
Computational methods have been used to simulate hemodynamics for several decades now. However, the field has experienced a remarkable advancement in the past 20 years, due to concurrent breakthroughs in medical imaging and computer hardware and software. It is now possible to create sophisticated patient‐specific models of hemodynamics to study cardiovascular disease, test the performance of medical devices, perform noninvasive diagnostics, and even virtually plan complex surgeries. In this chapter, we provide an overview of classic, well‐established methods for blood flow modeling and a summary of novel computational tools. We then review several salient clinical applications in which computational modeling has had a prominent role in the past few years and end the chapter with the discussion of current challenges and future opportunities.
Objective:
we propose a procedure for calibrating 4 parameters governing the mechanical boundary conditions (BCs) of a thoracic aorta (TA) model derived from one patient with ascending aortic aneurysm. The BCs reproduce the visco-elastic structural support provided by the soft tissue and the spine and allow for the inclusion of the heart motion effect.
Methods:
we first segment the TA from magnetic resonance imaging (MRI) angiography and derive the heart motion by tracking the aortic annulus from cine-MRI. A rigid-wall fluid-dynamic simulation is performed to derive the time-varying wall pressure field. We build the finite element model considering patient-specific material properties and imposing the derived pressure field and the motion at the annulus boundary. The calibration, which involves the zero-pressure state computation, is based on purely structural simulations. After obtaining the vessel boundaries from the cine-MRI sequences, an iterative procedure is performed to minimize the distance between them and the corresponding boundaries derived from the deformed structural model. A strongly-coupled fluid-structure interaction (FSI) analysis is finally performed with the tuned parameters and compared to the purely structural simulation.
Results and conclusion:
the calibration with structural simulations allows to reduce maximum and mean distances between image-derived and simulation-derived boundaries from 8.64 mm to 6.37 mm and from 2.24 mm to 1.83 mm, respectively. The maximum root mean square error between the deformed structural and FSI surface meshes is 0.19 mm. This procedure may prove crucial for increasing the model fidelity in replicating the real aortic root kinematics.
Large aortic aneurysm and acute and chronic aortic dissection are pathologies of the aorta requiring surgery. Recent advances in medical intervention have improved patient outcomes; however, a clear understanding of the mechanisms leading to aortic failure and, hence, a better understanding of failure risk, is still missing. Biomechanical analysis of the aorta could provide insights into the development and progression of aortic abnormalities, giving clinicians a powerful tool in risk stratification. The complexity of the aortic system presents significant challenges for a biomechanical study and requires various approaches to analyse the aorta. To address this, here we present a holistic review of the biomechanical studies of the aorta by categorising articles into four broad approaches, namely theoretical, in vivo, experimental and combined investigations. Experimental studies that focus on identifying mechanical properties of the aortic tissue are also included. By reviewing the literature and discussing drawbacks, limitations and future challenges in each area, we hope to present a more complete picture of the state-of-the-art of aortic biomechanics to stimulate research on critical topics. Combining experimental modalities and computational approaches could lead to more comprehensive results in risk prediction for the aortic system.
Abdominal aortic aneurysm (AAA) is an irreversible dilation of abdominal aorta, which may rupture if not surgically treated. To date, most aorta stent-graft used in clinical practice are batch manufactured devices with a uniform diameter. Custom abdominal aortic stent-grafts are able to overcome standard stents limitations. In this study, a customized aortic stent-graft (NiTi -Dacron) for the treatment of AAA has been proposed. Fluid dynamics analyses were performed to deepen the hemodynamic of aneurysm vessel and the proposed patient-specific graft. By means this study, the authors have shown the real benefits of the device for the patient and the possibility to apply this new stent-graft in the near future.
Cardiovascular disease (CVD) is the leading cause of mortality worldwide and its incidence is rising due to an aging population. The development and progression of CVD is directly linked to adverse vascular hemodynamics and biomechanics, whose
in-vivo
measurement remains challenging but can be simulated numerically and experimentally. The ability to evaluate these parameters in patient-specific CVD cases is crucial to better predict future disease progression, risk of adverse events, and treatment efficacy. While significant progress has been made toward patient-specific hemodynamic simulations, blood vessels are often assumed to be rigid, which does not consider the compliant mechanical properties of vessels whose malfunction is implicated in disease. In an effort to simulate the biomechanics of flexible vessels, fluid-structure interaction (FSI) simulations have emerged as promising tools for the characterization of hemodynamics within patient-specific cardiovascular anatomies. Since FSI simulations combine the blood's fluid domain with the arterial structural domain, they pose novel challenges for their experimental validation. This paper reviews the scientific work related to FSI simulations for patient-specific arterial geometries and the current standard of FSI model validation including the use of compliant arterial phantoms, which offer novel potential for the experimental validation of FSI results.
Objective
Intracranial aneurysms of patients with comorbidities like hypertension and atrial fibrillation have elevated rupture risk. Besides, the height and width of the aneurysm sacs are also essential features for pre-operative planning. However, clinicians still use only medical images and aspect ratios to identify which aneurysm has a higher risk of rupture. The rupture risk assessment often remains incomplete for lack of crucial insights. Therefore, the implication of a computational study on aneurysm diagnosis is analyzed in this study.
Method
Here, the two-way fluid-structure interaction method is used to reproduce these effects of clinical conditions and geometry on aneurysms. Wall shear stress, oscillatory shear index, total deformation, and other parameters are calculated.
Results
The wall shear stress (WSS) is the stress exerted by the wall on the fluid in a direction on the local tangent plane. The WSS inside a larger aneurysm decreases by around 50% compared to the aneurysms with a lower aspect ratio, indicating higher potential growth and rupture risk. However, the maximum pressure during systole for the aneurysm with the highest aspect ratio is 33% lower than the second-largest one during atrial fibrillation because of its unusual geometry. Although these hemodynamic parameters can be used to identify the effects of the size and shape of the aneurysms, they cannot completely depict the impact of the medical conditions. On the other hand, structural parameters like effective stress and total deformation successfully show the influence of size, hypertension, and atrial fibrillation on the aneurysm walls. The time-averaged wall deformation for the largest aneurysm is 1.3 and 2 times higher than the medium and smallest aneurysms. Moreover, compared to the normal condition, the total deformation at the aneurysm sac walls is 4–8% more for atrial fibrillation and 25–33% more for hypertension. A similar observation is made for the time-averaged value of effective stress exerted on the aneurysm walls, another parameter used to assess the rupture risk of aneurysms. Most importantly, these computational results align with previous clinical studies on patients with hypertension and atrial fibrillation.
Conclusion
Based on these observations, it is certain that to understand the impact of the comorbidities of the patients with intracranial aneurysms, results obtained from the structural analysis are required alongside hemodynamic parameters.
The carotid bifurcation experiences a complex loading environment due to its anatomical structure. Previous in-vivo material parameter estimation methods often use simplified model geometries, isotropic hyperelastic constitutive equations or neglect key aspects of the vessel, such as the zero-pressure configuration or residual stress. These factors have independently been shown to alter the stress environment of the vessel wall. Characterising the location of high stress in the vessel wall has often been proposed as a potential indicator of structural weakness. However, excluding the afore-mentioned zero-pressure configuration, residual stress and patient specific material parameters can lead to an incorrect estimation of the true stress values observed, meaning stress alone as a risk indicator of rupture is insufficient. In this study, we investigate how the estimated material parameters and overall stress distributions in geometries of carotid bifurcations, extracted from in-vivo MR images, alter with the inclusion of the zero-pressure configuration and residual stress.
This approach consists of the following steps: (1) geometry segmentation and hexahedral meshing from in-vivo MRI images at two known phases; (2) computation of the zero-pressure configuration and the associated residual stresses; (3) minimisation of an objective function built on the difference between the stress states of an “ almost true” stress field at two known phases and a “deformed” stress field by altering the input material parameters to determine patient specific material properties; and (4) comparison of the stress distributions throughout these carotid bifurcations for all cases with estimated material parameters. This numerical approach provides insights into the need for estimation of both the zero-pressure configuration and residual stress for accurate material property estimation and stress analysis for the carotid bifurcation, establishing the reliability of stress as a rupture risk metric.
Graphical Abstract
Aneurysms are bulges in arteries which reflect unhealthy state of conduit in which blood is flowing. In the aorta, they are typically found in the abdominal region as well as thoracic region. Understanding the rupture risk of these vessels is critical to preventing failure and fatalities. In the current clinical practice, treatment modalities are initiated, when the out‐pouching exceeds maximum diameter (Dmax). However this approach is very crude as it does not account for the fluid mechanical forces and the attendant stresses. Since it is medically and ethically not possible to follow the patients to study the rupture risk potential, fluid structure interaction (FSI) modelling would be an apt tool to develop adequate understanding on various hemodynamic parameters. On the other hand, performing patient‐specific studies would demand adequate lead time and they are computationally expensive as well. In the present study, the shape of the aneurysm and its interaction with the flowing fluid are accounted through the shape indices to study the FSI effects on the hemodynamic parameters. Numerical simulation of Newtonian flow through five axi‐symmetric geometries with different shape indices coupled with a linear elastic vessel wall model is considered. From these simulations, it was observed that (Dmax) to height ratio (DHr) is the most significant shape index which influences the variation of all hemodynamic parameters, which makes it a potential candidate for predicting rupture risk. Wall acceleration due to pulsatile flow was found to cause the onset of re‐circulation zones at the centre of the aneurysm during early systole and the temporal deceleration resulted in the generation of near wall eddying structures during late diastole. Investigation of turbulence carried out with k‐ω Shear Stress transport turbulence model, predicts a turbulence intensity of greater than 1.5% in the diseased segment as well as the distal end of the aneurysm.
An understanding of the haemodynamics of pulsatile blood flow and the response of the arterial wall to blood pressure in health and disease is vital for those managing vascular disease. Computational Fluid Dynamics (CFD), Finite Element Analysis (FEA) and Fluid-Solid Interaction (FSI) modelling are approaches which can be used to understand the behaviour of blood flow forces and resultant deformation of the arterial wall. CFD is a flow simulation technique which provides a powerful tool for the study of haemodynamic and image-based modelling of blood flow, using haemodynamic parameters, in the development, diagnosis, and also treatment of cardiovascular disease. The FSI method is helping to make links between blood flow shear stress on arterial wall (WSS) and the distribution of stress into the blood vessel to explain why atherosclerotic plaque develops at arterial junctions for example. These techniques allow engineers and clinicians to study vascular diseases such as atherosclerosis, aneurysm formation and dissection. These techniques are also able to analyse the effects of devices developed to treat vascular disease.
Objective:
The purpose of this study was to determine whether computed tomography (CT) angiography with machine learning (ML) can be used to predict the rapid growth of abdominal aortic aneurysm (AAA).
Materials and methods:
This retrospective study was approved by our institutional review board. Fifty consecutive patients (45 men, 5 women, 73.5 years) with small AAA (38.5 ± 6.2 mm) had undergone CT angiography. To be included, patients required at least 2 CT scans a minimum of 6 months apart. Abdominal aortic aneurysm growth, estimated by change per year, was compared between patients with baseline infrarenal aortic minor axis. For each axial image, major axis of AAA, minor axis of AAA, major axis of lumen without intraluminal thrombi (ILT), minor axis of lumen without ILT, AAA area, lumen area without ILT, ILT area, maximum ILT area, and maximum ILT thickness were measured. We developed a prediction model using an ML method (to predict expansion >4 mm/y) and calculated the area under the receiver operating characteristic curve of this model via 10-fold cross-validation.
Results:
The median aneurysm expansion was 3.0 mm/y. Major axis of AAA and AAA area correlated significantly with future AAA expansion (r = 0.472, 0.416 all P < 0.01). Machine learning and major axis of AAA were a strong predictor of significant AAA expansion (>4 mm/y) (area under the receiver operating characteristic curve were 0.86 and 0.78).
Conclusions:
Machine learning is an effective method for the prediction of expansion risk of AAA. Abdominal aortic aneurysm area and major axis of AAA are the important factors to reflect AAA expansion.
Background:
The analysis of the correlation between blood flow and aortic pathology through computational fluid dynamics (CFD) shows promise in predicting disease progression, the effect of operative intervention, and guiding patient treatment. However, to date, there has not been a comprehensive systematic review of the published literature describing CFD in aortic diseases and their treatment.
Methods:
This review includes 136 published articles which have investigated the application of CFD in all types of aortic disease (aneurysms, dissections, and coarctation). We took into account case studies of both, treated or untreated pathology, investigated with CFD. We also graded all studies using an author-defined Grading of Recommendations Assessment, Development and Evaluation (GRADE) approach based on the validation method used for the CFD results.
Results:
There are no randomized controlled trials assessing the efficacy of CFD as applied to aortic pathology, treated or untreated. Although a large number of observational studies are available, those using clinical imaging tools as independent validation of the calculated CFD results exist in far smaller numbers. Only 21% of all studies used clinical imaging as a tool to validate the CFD results and these were graded as high-quality studies.
Conclusions:
Contemporary evidence shows that CFD can provide additional hemodynamic parameters such as wall shear stress, vorticity, disturbed laminar flow, and recirculation regions in untreated and treated aortic disease. These have the potential to predict the progression of aortic disease, the effect of operative intervention, and ultimately help guide the choice and timing of treatment to the benefit of patients and clinicians alike.
Many physiological systems involve strong interactions between fluids and solids, posing a signicant challenge when modeling biomechanics. The objective of this study was to implement a fluid-structure interaction (FSI) solver in the free, open-source finite element code FEBio (febio.org), that combined the existing solid mechanics and rigid body dynamics solver with a recently-developed computational fluid dynamics (CFD) solver. A novel Galerkin-based finite element FSI formulation was introduced based on mixture theory, where the FSI domain was described as a mixture of fluid and solid constituents that have distinct motions. The mesh was defined on the solid domain, specialized to have zero mass, negligible stiffness and zero frictional interactions with the fluid, whereas the fluid was modeled as isothermal and compressible. The mixture framework provided the foundation for evaluating material time derivatives in a material frame for the solid and in a spatial frame for the fluid. Similar to our recently reported CFD solver, our FSI formulation did not require stabilization methods to achieve good convergence, producing a compact set of equations and code implementation. The code was successfully verified against benchmark problems and an analytical solution for squeeze-film lubrication. It was validated against experimental measurements of the flow rate in a peristaltic pump, and illustrated using non-Newtonian blood flow through a bifurcated carotid artery with a thick arterial wall. The successful formulation and implementation of this FSI solver enhances the multiphysics modeling capabilities in FEBio relevant to the biomechanics and biophysics communities.
This volume comprises the latest developments in both fundamental science and patient-specific applications, discussing topics such as: cellular mechanics, injury biomechanics, biomechanics of the heart and vascular system, algorithms of computational biomechanics for medical image analysis, and both patient-specific fluid dynamics and solid mechanics simulations. With contributions from researchers world-wide, Computational Biomechanics for Medicine: Measurments, Models, and Predictions provides an opportunity for specialists in the field to present their latest methodologies and advancements. © Springer International Publishing AG, part of Springer Nature 2019. All rights reserved.
The increasing use of Computational Fluid Dynamics (CFD) for simulating blood flow in clinics demands the identification of appropriate patient‐specific boundary conditions for the customization of the mathematical models. These conditions should ideally be retrieved from measurements. However, finite resolution of devices as well as other practical/ethical reasons prevent the construction of complete data sets necessary to make the mathematical problems well posed. Available data need to be completed by modeling assumptions, whose impact on the final solution has to be carefully addressed. Focusing on aortic vascular districts and related pathologies, we present here a method for efficiently and robustly prescribing PC‐MRI based patient‐specific data as boundary conditions at the domain of interest. In particular, for the outlets, the basic idea is to obtain pressure conditions from an appropriate elaboration of available flow rates based on a 3D/0D dimensionally heterogeneous modelling. The key point is that the parameters are obtained by a constrained optimization procedure. The rationale is that pressure conditions have a reduced impact on the numerical solution compared to velocity conditions, yielding a simulation framework less exposed to noise and inconsistency of the data, as well as to the arbitrariness of the underlying modelling assumptions. Numerical results confirm the reliability of the approach in comparison with other patient‐specific approaches adopted in the literature.
Aortic dissection is where the medial layer of the arterial wall is separated by a tear leading to intramural bleeding. The blood forms an alternate channel of flow known as the false lumen. Thrombosis of the false lumen (FL) is common in cases of dissection as flow conditions are typically more stagnant than in the true lumen (TL). Central to the process of thrombosis is the activation and aggregation of platelets in the blood. Therefore, the aim of this work is to simulate the transport of platelets in a case of Type-B aortic dissection in a clinically-relevant timeframe.
Blood flow in the large systemic arteries is modeled using one-dimensional equations derived from the axisymmetric Navier–Stokes equations for flow in compliant and tapering vessels. The arterial tree is truncated after the first few generations of large arteries with the remaining small arteries and arterioles providing outflow boundary conditions for the large arteries. By modeling the small arteries and arterioles as a structured tree, a semi-analytical approach based on a linearized version of the governing equations can be used to derive an expression for the root impedance of the structured tree in the frequency domain. In the time domain, this provides the proper outflow boundary condition. The structured tree is a binary asymmetric tree in which the radii of the daughter vessels are scaled linearly with the radius of the parent vessel. Blood flow and pressure in the large vessels are computed as functions of time and axial distance within each of the arteries. Comparison between the simulations and magnetic resonance measurements in the ascending aorta and nine peripheral locations in one individual shows excellent agreement between the two. 2000 Biomedical Engineering Society.
PAC00: 8719Uv
In this paper a combined arbitrary Lagrange-Euler fictitious domain (ALE-FD) method for fluid-structure interaction problems in cardiovascular biomechanics is derived in terms of a weighted residual finite-element formulation. For both fluid flow of blood and solid mechanics of vascular tissue, the performance of tetrahedral and hexahedral Crouzeix-Raviart elements are evaluated. Comparable convergence results are found, although for the test cases considered the hexahedral elements are more accurate. The possibilities that are offered by the ALE-FD method are illustrated by means of a simulation of valve dynamics in a simplified left ventricular flow model.
In this paper a family of one-dimensional nonlinear systems which model the blood pulse propagation in compliant arteries is presented and investigated. They are obtained by averaging the Navier-Stokes equation on each section of an arterial vessel and using simplified models for the vessel compliance. Different differential operators arise depending on the simplifications made on the structural model. Starting from the most basic assumption of pure elastic instantaneous equilibrium, which provides a well-known algebraic relation between intramural pressure and vessel section area, we analyse in turn the effects of terms accounting for inertia, longitudinal prestress and viscoelasticity. The problem of how to account for branching and possible discontinuous wall properties is addressed, the latter aspect being relevant in the presence of prosthesis and stents. To this purpose a domain decomposition approach is adopted and the conditions which ensure the stability of the coupling are provided. The numerical method here used in order to carry out several test cases for the assessment of the proposed models is based on a finite element Taylor-Galerkin scheme combined with operator splitting techniques
Laminar and turbulent numerical simulations of steady flow in an aneurysm model were carried out over Reynolds numbers ranging from 300 to 3600. The numerical simulations are validated with Digital particle Image Velocimetry (DPIV) measurements, and used to study the fluid dynamic mechanisms that characterize aneurysm deterioration, by correlating them to in vitro blood platelet deposition results. It is shown that the recirculation zone formed inside the aneurysm cavity creates conditions that promote thrombus formation and the viability of rupture. Wall shear stress values in the recirculation zone are around one order of magnitude less than in the entrance zone. The point of reattachment at the distal end of the aneurysm is characterized by a pronounced wall shear stress peak. As the Reynolds number increases in laminar flow, the center of the recirculation region migrates toward the distal end of the aneurysm, increasing the pressure at the reattachment point. Under fully turbulent flow conditions (Re = 3600) the recirculation zone inside the aneurysm shrinks considerably. The wall shear stress values are almost one order of magnitude larger than those for the laminar cases. The fluid dynamics mechanisms inferred from the numerical simulation were correlated with measurements of blood platelet deposition, offering useful explanations for the different morphologies of the platelet deposition curves.
A numerical and experimental investigation of unsteady entry flow in a 90 degrees curved tube is presented to study the impact of the non-Newtonian properties of blood on the velocity distribution. The time-dependent flow rate for the Newtonian and the non-Newtonian blood analog fluid were identical. For the numerical computation, a Carreau-Yasuda model was employed to accommodate the shear thinning behavior of the Xanthan gum solution. The viscoelastic properties were not taken into account. The experimental results indicate that significant differences between the Newtonian and non-Newtonian fluid are present. The numerical results for both the Newtonian and the non-Newtonian fluid agree well with the experimental results. Since viscoelasticity was not included in the numerical code, shear thinning behavior of the blood analog fluid seems to be the dominant non-Newtonian property, even under unsteady flow conditions. Finally, a comparison between the non-Newtonian fluid model and a Newtonian fluid at a rescaled Reynolds number is presented. The rescaled Reynolds number, based on a characteristic rather than the high-shear rate viscosity of the Xanthan gum solution, was about three times as low as the original Reynolds number. Comparison reveals that the character of flow of the non-Newtonian fluid is simulated quite well by using the appropriate Reynolds number.
There is a need to understand why and where the abdominal aortic aneurysm may rupture. Our goal therefore is to investigate whether the mechanical properties are different in different regions of the aneurysm. Aorta samples from five freshly excised whole aneurysms, > or = 5 cm in diameter, from five patients, average age 71 +/- 10 years, were subjected to uniaxial testing. We report the wall thickness, yield stress and strain, and parameters that describe nonlinear stress-strain curves for the anterior, lateral and posterior regions of the aneurysm. The posterior region was thicker than the anterior region (2.73 +/- 0.46 mm versus 2.09 +/- 0.51 mm). The stress-strain curves were described by sigma = a epsilon(b), where sigma is true stress and epsilon is engineering strain. In the circumferential direction, the wall stiffness increased from posterior to anterior to lateral. In the longitudinal direction, the lateral and anterior regions showed greater wall stiffness than the posterior region. The wall stiffness was greater in the circumferential than longitudinal direction. The anterior region was the weakest, especially in the longitudinal direction (yield stress sigmaY = 0.38 +/- 0.18 N mm(-2)). For a less complex model the aneurysmal wall could be considered orthotropic with sigma = 12.89epsilon(2.92) and 4.95epsilon(2.84) in the circumferential and longitudinal directions. For the isotropic model, sigma =7.89epsilon(2.88). In conclusion, different regions of the aneurysm have different yield stress, yield strains, and other mechanical properties, and this must be considered in understanding where the rupture might occur.
Abdominal aortic aneurysm (AAA) disease is a degenerating process whose ultimate event is the rupture of the vessel wall. Rupture occurs when the stresses acting on the wall rise above the strength of the AAA wall tissue. The complex mechanical interaction between blood flow and wall dynamics in a three dimensional custom model of a patient AAA was studied by means of computational coupled fluid-structure interaction analysis. Real 3D AAA geometry is obtained from CT scans image processing. The results provide a quantitative local evaluation of the stresses due to local structural and fluid dynamic conditions. The method accounts for the complex geometry of the aneurysm, the presence of a thrombus and the interaction between solid and fluid. A proven clinical efficacy may promote the method as a tool to determine factual aneurysm risk of rupture and aid the surgeon to refer elective surgery patients.
Whether elective surgical repair of small abdominal aortic aneurysms improves survival remains controversial.
We randomly assigned patients 50 to 79 years old with abdominal aortic aneurysms of 4.0 to 5.4 cm in diameter who did not have high surgical risk to undergo immediate open surgical repair of the aneurysm or to undergo surveillance by means of ultrasonography or computed tomography every six months with repair reserved for aneurysms that became symptomatic or enlarged to 5.5 cm. Follow-up ranged from 3.5 to 8.0 years (mean, 4.9).
A total of 569 patients were randomly assigned to immediate repair and 567 to surveillance. By the end of the study, aneurysm repair had been performed in 92.6 percent of the patients in the immediate-repair group and 61.6 percent of those in the surveillance group. The rate of death from any cause, the primary outcome, was not significantly different in the two groups (relative risk in the immediate-repair group as compared with the surveillance group, 1.21; 95 percent confidence interval, 0.95 to 1.54). Trends in survival did not favor immediate repair in any of the prespecified subgroups defined by age or diameter of aneurysm at entry. These findings were obtained despite a low total operative mortality of 2.7 percent in the immediate-repair group. There was also no reduction in the rate of death related to abdominal aortic aneurysm in the immediate-repair group (3.0 percent) as compared with the surveillance group (2.6 percent). Eleven patients in the surveillance group had rupture of abdominal aortic aneurysms (0.6 percent per year), resulting in seven deaths. The rate of hospitalization related to abdominal aortic aneurysm was 39 percent lower in the surveillance group.
Survival is not improved by elective repair of abdominal aortic aneurysms smaller than 5.5 cm, even when operative mortality is low.
Among patients with abdominal aortic aneurysm (AAA) who have high operative risk, repair is usually deferred until the AAA reaches a diameter at which rupture risk is thought to outweigh operative risk, but few data exist on rupture risk of large AAA.
To determine the incidence of rupture in patients with large AAA.
Prospective cohort study in 47 Veterans Affairs medical centers.
Veterans (n = 198) with AAA of at least 5.5 cm for whom elective AAA repair was not planned because of medical contraindication or patient refusal. Patients were enrolled between April 1995 and April 2000 and followed up through July 2000 (mean, 1.52 years).
Incidence of AAA rupture by strata of initial and attained diameter.
Outcome ascertainment was complete for all patients. There were 112 deaths (57%) and the autopsy rate was 46%. Forty-five patients had probable AAA rupture. The 1-year incidence of probable rupture by initial AAA diameter was 9.4% for AAA of 5.5 to 5.9 cm, 10.2% for AAA of 6.0 to 6.9 cm (19.1% for the subgroup of 6.5-6.9 cm), and 32.5% for AAA of 7.0 cm or more. Much of the increased risk of rupture associated with initial AAA diameters of 6.5-7.9 cm was related to the likelihood that the AAA diameter would reach 8.0 cm during follow-up, after which 25.7% ruptured within 6 months.
The rupture rate is substantial in high-operative-risk patients with AAA of at least 5.5 cm in diameter and increases with larger diameter.
Quantitative functional analysis of the left ventricle plays a very important role in the diagnosis of heart diseases. While in standard two-dimensional echocardiography this quantification is limited to rather crude volume estimation, three-dimensional (3-D) echocardiography not only significantly improves its accuracy but also makes it possible to derive valuable additional information, like various wall-motion measurements. In this paper, we present a new efficient method for the functional evaluation of the left ventricle from 3-D echographic sequences. It comprises a segmentation step that is based on the integration of 3-D deformable surfaces and a four-dimensional statistical heart motion model. The segmentation results in an accurate 3-D + time left ventricle discrete representation. Functional descriptors like local wall-motion indexes are automatically derived from this representation. The method has been successfully tested both on electrocardiography-gated and real-time 3-D data. It has proven to be fast, accurate, and robust.
In the abdominal segment of the human aorta under a patient's average resting conditions, pulsatile blood flow exhibits complex laminar patterns with secondary flows induced by adjacent branches and irregular vessel geometries. The flow dynamics becomes more complex when there is a pathological condition that causes changes in the normal structural composition of the vessel wall, for example, in the presence of an aneurysm. This work examines the hemodynamics of pulsatile blood flow in hypothetical three-dimensional models of abdominal aortic aneurysms (AAAs). Numerical predictions of blood flow patterns and hemodynamic stresses in AAAs are performed in single-aneurysm, asymmetric, rigid wall models using the finite element method. We characterize pulsatile flow dynamics in AAAs for average resting conditions by means of identifying regions of disturbed flow and quantifying the disturbance by evaluating flow-induced stresses at the aneurysm wall, specifically wall pressure and wall shear stress. Physiologically realistic abdominal aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50 < or = Rem < or = 300, corresponding to a range of peak Reynolds numbers 262.5 < or = Repeak < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is depicted by a sequence of four different flow phases in one period of the cardiac pulse. Peak wall shear stress and peak wall pressure are reported as a function of the time-average Reynolds number and aneurysm asymmetry. The effect of asymmetry in hypothetically shaped AAAs is to increase the maximum wall shear stress at peak flow and to induce the appearance of secondary flows in late diastole.
CT Angiography (CTA) is an established technique for the minimally invasive imaging of arteries. The technique of maximum intensity projection (MIP) is often used to get a comprehensive overview of the vascular anatomy. On a MIP, however, arterial wall calcifications may hinder the visualization of the arterial lumen. These calcifications are in direct contact with the contrast-enhanced blood, which makes removal difficult. We present a local subtraction method for the automatic removal of these calcifications. In our approach a second CT scan has to be made, prior to contrast injection. The calcifications in both scans are registered prior to subtraction to compensate for displacements in between the two scans. Local subtraction results are compared with results obtained by thresholding. The method was tested in a phantom and with data from four patients. The phantom represented an artery with different types of stenosis. Data were used from patients for which CTA of the renal arteries was performed. For two patients the electrocardiogram (ECG) was recorded during the CTA examination, making retrospective cardiac gated reconstructions possible. Both the phantom and the patient study showed that the local subtraction method is capable of removing calcifications and visualizing the residual lumen. In the patient study it appeared that some artifacts remained for higher pitch values. We conclude that the local subtraction method is less subjective and more accurate than thresholding. Best results are obtained by use of a small pitch, at the expense of the volume covered during a single breath hold.
The abdominal aortic aneurysm (AAA) is a degenerating disease for which the end stage is the rupture of the vessel wall. Accurate prediction of the stresses acting on the aneurysm tissue may be used to determine the actual risk of rupture of a specific aneurysm. To accomplish this, a correct constitutive model for the aneurysmal aortic wall and any intraluminal thrombus (ILT) present within it are needed. Our laboratory has previously reported the mechanical properties of ILT. The aim of this work is to investigate the reliability of using population-mean values of ILT constitutive parameters to estimate AAA wall stress distribution. For this, a three-dimensional asymmetric model of an aneurysm including ILT was generated and a parametric study was conducted varying ILT constitutive properties within a physiological range. Results show that the presence of any ILT reduces and redistributes the stresses in the aortic wall markedly. Maximum variation in the peak wall stresses for all the models analyzed was 5%. Adopting a nonhomogeneous ILT did not alter the stress distribution. On the basis of these results, we infer that population mean parameters for ILT material characteristics can be used to reasonably estimate the wall stresses in patient specific aneurysm models. © 2003 Biomedical Engineering Society.
PAC2003: 8719Rr, 8719Xx, 8710+e
We investigated the effects of sequential and prolonged exposure to high and low wall shear stress on arterial remodeling using a rabbit arteriovenous fistula (AVF) model. Blood flow was increased by approximately 17-fold to 20-fold when the AVF was open, and returned to normal when the AVF was occluded. Repeated opening and closing of the AVF resulted in sequential exposure of the artery to high and low wall shear stress. High flow and high wall shear stress induced arterial dilatation, elongation, and tortuosity, without intimal thickening. The common carotid artery was elongated 37% after 4 weeks of high flow, and was shortened 10% after 6 weeks of normal flow. Subsequent cycles of high flow induced less elongation, with less shortening after return to normal flow. Enlargement of the distal segment was more dramatic than in the proximal segment, despite exposure to the same volume of flow and the same initial high wall shear stress after creation of the AVF. The distal carotid segment enlarged more than did the proximal segment during each exposure to high flow. In segments of carotid artery exposed to low wall shear stress (<5 dynes/cm(2)) intimal thickening developed. These changes were maximal in the distal carotid segment, just before the AVF. Each cycle of low wall shear stress induced intimal thickening accompanied by medial hyperplasia. Intimal thickening was inhibited during periods of high flow when wall shear stress was high. Three cycles of flow alteration induced three layers of intimal thickening in the distal arterial segment, two layers of intimal thickening in the middle segment, and one layer of intimal thickening in the proximal segment. Long-term exposure to low wall shear stress induced severe intimal thickening and medial hyperplasia in different segments. Thus the response of the carotid artery afferent to an AVF varies along the length of the artery, with maximum enlargement, elongation, and tortuosity in the distal segment, just proximal to the AVF. Similarly, intimal thickening in response to low wall shear stress is maximal in the distal carotid artery. It appears that intimal thickening is related to local levels of low wall shear stress, and occurs when wall shear stress chronically falls to less than 5 dynes/cm(2).
In this paper we describe a new tool for interactive free-form fair surface design. By generalizing classical discrete Fourier analysis to two-dimensional discrete surface signals -- functions defined on polyhedral surfaces of arbitrary topology --, we reduce the problem of surface smoothing, or fairing, to low-pass filtering. We describe a very simple surface signal low-pass filter algorithm that applies to surfaces of arbitrary topology. As opposed to other existing optimization-based fairing methods, which are computationally more expensive, this is a linear time and space complexity algorithm. With this algorithm, fairing very large surfaces, such as those obtained from volumetric medical data, becomes affordable. By combining this algorithm with surface subdivision methods we obtain a very effective fair surface design technique. We then extend the analysis, and modify the algorithm accordingly, to accommodate different types of constraints. Some constraints can be imposed without any modification of the algorithm, while others require the solution of a small associated linear system of equations. In particular, vertex location constraints, vertex normal constraints, and surface normal discontinuities across curves embedded in the surface, can be imposed with this technique. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/image generation - display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - curve, surface, solid, and object representations;J.6[Com- puter Applications]: Computer-Aided Engineering - computeraided design General Terms: Algorithms, Graphics. 1
Background:
Early elective surgery may prevent rupture of abdominal aortic aneurysms, but mortality is 5-6%. The risk of rupture seems to be low for aneurysms smaller than 5 cm. We investigated whether prophylactic open surgery decreased long-term mortality risks for small aneurysms.
Methods:
We randomly assigned 1090 patients aged 60-76 years, with symptomless abdominal aortic aneurysms 4.0-5.5 cm in diameter to undergo early elective open surgery (n=563) or ultrasonographic surveillance (n=527). Patients were followed up for a mean of 4.6 years. If the diameter of aneurysms in the surveillance group exceeded 5.5 cm, surgical repair was recommended. The primary endpoint was death. Mortality analyses were done by intention to treat.
Findings:
The two groups had similar cardiovascular risk factors at baseline. 93% of patients adhered to the assigned treatment. 309 patients died during follow-up. The overall hazard ratio for all-cause mortality in the early-surgery group compared with the surveillance group was 0.94 (95% CI 0.75-1.17, p=0.56). The 30-day operative mortality in the early-surgery group was 5.8%, which led to a survival disadvantage for these patients early in the trial. Mortality did not differ significantly between groups at 2 years, 4 years, or 6 years. Age, sex, or initial aneurysm size did not modify the overall hazard ratio.
Interpretation:
Ultrasonographic surveillance for small abdominal aortic aneurysms is safe, and early surgery does not provide a long-term survival advantage. Our results do not support a policy of open surgical repair for abdominal aortic aneurysms of 4.0-5.5 cm in diameter.
Background Early elective surgery may prevent rupture of abdominal aortic aneurysms, but mortality is 5-6%. The risk of rupture seems to be low for aneurysms smaller than 5 cm. We investigated whether prophylactic open surgery decreased long-term mortality risks for small aneurysms. Methods We randomly assigned 1090 patients aged 60-76 years, with symptomless abdominal aortic aneurysms 4.0-5.5 cm in diameter to undergo early elective open surgery (n=563) or ultrasonographic surveillance (n=527). Patients were followed up for a mean of 4.6 years. If the diameter of aneurysms in the surveillance group exceeded 5 5 cm, surgical repair was recommended. The primary endpoint was death. Mortality analyses were done by intention to treat. Findings The two groups had similar cardiovascular risk factors at baseline. 93% of patients adhered to the assigned treatment. 309 patients died during follow-up. The overall hazard ratio for all-cause mortality in the early-surgery group compared with the surveillance group was 0.94 (95% CI 0.75-1.17, p=0.56). The 30-day operative mortality in the early-surgery group was 5.8%, which led to a survival disadvantage for these patients early in the trial. Mortality did not differ significantly between groups at 2 years, 4 years, or 6 years. Age, sex, or initial aneurysm size did not modify the overall hazard ratio. Interpretation Ultrasonographic surveillance for small abdominal aortic aneurysms is safe, and early surgery does not provide a long-term survival advantage. Our results do not support a policy of open surgical repair for abdominal aortic aneurysms of 4 0-5.5 cm in diameter.
Context Among patients with abdominal aortic aneurysm (AAA) who have high operative
risk, repair is usually deferred until the AAA reaches a diameter at which
rupture risk is thought to outweigh operative risk, but few data exist on
rupture risk of large AAA.Objective To determine the incidence of rupture in patients with large AAA.Design and Setting Prospective cohort study in 47 Veterans Affairs medical centers.Patients Veterans (n = 198) with AAA of at least 5.5 cm for whom elective AAA
repair was not planned because of medical contraindication or patient refusal.
Patients were enrolled between April 1995 and April 2000 and followed up through
July 2000 (mean, 1.52 years).Main Outcome Measure Incidence of AAA rupture by strata of initial and attained diameter.Results Outcome ascertainment was complete for all patients. There were 112
deaths (57%) and the autopsy rate was 46%. Forty-five patients had probable
AAA rupture. The 1-year incidence of probable rupture by initial AAA diameter
was 9.4% for AAA of 5.5 to 5.9 cm, 10.2% for AAA of 6.0 to 6.9 cm (19.1% for
the subgroup of 6.5-6.9 cm), and 32.5% for AAA of 7.0 cm or more. Much of
the increased risk of rupture associated with initial AAA diameters of 6.5-7.9
cm was related to the likelihood that the AAA diameter would reach 8.0 cm
during follow-up, after which 25.7% ruptured within 6 months.Conclusion The rupture rate is substantial in high-operative-risk patients with
AAA of at least 5.5 cm in diameter and increases with larger diameter.
This work is provided as a back up to two series of lectures at universities in the Netherlands and comes in four parts. The first one gives a general introduction to the finite element analysis of partial differential equations of elliptic type. Three approaches to the Navier-Stokes equations are studied, including time dependent terms. Thirdly, a more mathematical approach is given and in part four, an overview is provided of research up to the time of the book's compilation. (R.I.H.)
The sixth editions of these seminal books deliver the most up to date and comprehensive reference yet on the finite element method for all engineers and mathematicians. Renowned for their scope, range and authority, the new editions have been significantly developed in terms of both contents and scope. Each book is now complete in its own right and provides self-contained reference; used together they provide a formidable resource covering the theory and the application of the universally used FEM. Written by the leading professors in their fields, the three books cover the basis of the method, its application to solid mechanics and to fluid dynamics.* This is THE classic finite element method set, by two the subject's leading authors * FEM is a constantly developing subject, and any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in these books * Fully up-to-date; ideal for teaching and reference
Necessary and sufficient conditions are derived for a cubic to be monotone on an interval. These conditions are used to develop an algorithm that constructs a visually pleasing monotone piecewise cubic interpolant to monotone data. Several examples are given that compare this algorithm with other interpolation methods. 5 figures.
The well-known and popular Savitzky-Golay filter has several disadvantages. A very attractive alternative is a smoother based on penalized least squares, extending ideas presented by Whittaker 80 years ago. This smoother is extremely fast, gives continuous control over smoothness, interpolates automatically, and allows fast leave-one-out cross-validation. It can be programmed in a few lines of Matlab code. Theory, implementation, and applications are presented.
At present, the diameter of an abdominal aortic aneurysm (AAA) is considered to be the primary indicator of rupture risk. Surgical treatment is usually not considered before the diameter exceeds 5 cm, but rupture does occur for diameters less than 5 cm. We investigate if better rupture risk indicators can be obtained by patient-specific hemodynamic modeling. This paper discusses the steps involved in this modeling and describes our initial results for each of the steps.
The rheological properties of blood, nonNewtonian viscosity in steady flow, frequency dependence and shear rate dependence of viscoelasticity in oscillatory flow, and thixotropy, are brought together by a unifying concept. Rheological states are defined which separate nonequilibrium properties, such as thixotropy, from equilibrium properties, such as steady flow viscosity and sustained oscillatory flow viscoelasticity. It is considered that the aggregation of erythrocytes is the primary process governing the conditions of equilibrium. A generalized Maxwell model is developed to provide a basis for quantitative analysis of equilibrium properties. A shear rate dependent degradation function serves to adjust the model elements to the flow conditions. Characteristic relaxation times become significant rheological parameters for equilibrium viscosity and viscoelasticity while other characteristic times are important to thixotropy. Numerical data are evaluated for the several rheological properties by comparison with the theory using a computerized regression analysis. These determinations show that non-Newtonian viscosity and viscoelasticity can be calculated using the same numerical properties. Thus, the theory provides a rational framework into which several rheological tests of blood can be placed.
Finite element analysis (FEA), a computer-based method for solving complex structural problems, was used to determine the wall stress distribution in three cases of model infrarenal abdominal aortic aneurysms representing common problems in determining risk of aneurysm rupture. The point of maximum circumferential wall stress in a spherical model aneurysm was located near the junction of the aneurysm and the nondilated aorta, while maximum longitudinal wall stress was located at the point of maximum diameter of the aneurysm. FEA showed that cylindrically shaped constant thickness model aneurysms had a higher maximum circumferential stress (sigma c = 11.9 X 10(5) dyn/cm2) and a comparable maximum longitudinal wall stress (sigma L = 6.6 X 10(5) dyn/cm2) when compared with spherical model aneurysms of the same diameter (sigma c = 8.1 X 10(5) dyn/cm2 and sigma L = 6.2 X 10(5) dyn/cm2). Analysis of the aorta to aneurysm diameter ratio (A:a gradient) indicated that aortic size is important in determining aneurysm wall stress and that the relationship between aortic size and wall stress is dependent upon aneurysm wall thickness. We conclude that the ability of the aneurysm wall to withstand stress in the longitudinal as well as the circumferential direction is an important factor determining aneurysm rupture. Finally, this investigation showed that FEA is a versatile tool for use in studying the mechanics of vascular structures, making it potentially more useful than size alone in estimating the clinical significance of abdominal aortic aneurysms.
Previous statistical studies by the authors lent strong support to an aggressive policy of case selection in the surgical treatment of asymptomatic abdominal aortic aneurysms. To test further the value of narrow operative contraindications, the clinical course of 156 patients with such lesions, who between the years of 1952 and 1971 had been deemed unfit for surgical treatment, was surveyed. Among the 127 patients who never came to operation, 90 died during the 20-year followup period. Although organ-fixed atherosclerosis (mostly coronary) was the leading cause of death (55.0%), rupture of the aneurysm continued to be an important source of loss of life (27.8%). Upon the whole, the findings corroborated the validity of an aggressive surgical approach and emphasized the prognostic threat of aneurysmal size and arterial hypertension in the nonsurgically treated patient.
Hitherto the size of abdominal aortic aneurysms (AAA) has been considered the most important factor in determining the risk of rupture. For this reason most interest has been devoted to physical, echographic and tomographic analyses of the shape of AAA. However, it is known that rupture can also occur in small AAA. Other factors must be considered to have an important role in the natural history of aneurysms. The aim of this study was to characterise the mechanical stress in the wall of an AAA due to pressure in the presence of atherosclerosis, intraluminal thrombus and anatomical restraints. The Finite Elements Method (FEM) was used to determine wall stress distribution. Due to the simplicity of the AAA structure an axisymmetric model has been built. The results of the structural analysis confirms that maximum stress increases with diameter. These effects may be reduced by the presence of intraluminal thrombus, which in the models reduces maximum stress by up to 30%; however this is not the case for dissecting thrombus. On the other hand atherosclerotic plaques cause stress concentration and a significant increase in maximum wall stress. The risk of rupture can increase by about 200%. Finally the investigation shows the FEM is a versatile tool for studying the mechanics of vascular structures. It enables the influence of various parameters on wall stress to be quantified in diagnostic settings, and so could be useful for predicting the rupture of AAA, although at present such predictions are limited by data leakage and by the approximations used in the model.
Abdominal aortic aneurysms (AAAs) rupture when the wall stress exceeds the strength of the vascular tissue. Intraluminal thrombus may absorb tension and reduce AAA wall stress. This study was performed to test the hypothesis that intraluminal thrombus can significantly reduce AAA wall stress.
AAA wall stresses were determined by axisymmetric finite element analysis. Model AAAs had external diameters ranging from 2.0 to 4.0 cm. Model parameters included: AAA length, 6 cm; wall thickness, 1.5 mm; Poisson's ratio, 0.49; Young's modulus, 1.0 MPa; and luminal pressure, 1.6 x 10(5) dyne/cm2. Stresses were calculated for each model without thrombus, and then were recalculated with thrombus filling 10% of the AAA cavity. Calculations were repeated as thrombus size was increased in 10% increments and as thrombus elastic modulus increased from 0.01 MPa to 1.0 MPa. Maximum wall stresses were compared between models that had intraluminal thrombus and the unmodified models. Stress reduction greater than 25% was considered significant.
The maximum stress reduction of 51% occurred when thrombus with elastic modulus of 1.0 MPa filled the entire AAA cavity. Stresses were reduced by only 25% as modulus decreased to 0.2 MPa. Similarly, decreasing thrombus size by 70% resulted in stress reduction of only 28%. Large AAAs experienced greater stress reduction than small AAAs (48% vs 11%).
Intraluminal thrombus can significantly reduce AAA wall stress.
Risk for rupture of an abdominal aortic aneurysm is widely believed to be related to its maximum diameter. From a biomechanical standpoint, however, risk is probably more precisely related to mechanical wall stress. Many abdominal aortic aneurysms are asymmetric (for example because of anterior bulging with posterior expansion limited by the vertebral column). The purpose of this work was to investigate the effect of maximum diameter and asymmetric bulge on wall stress.
Three-dimensional computer models of abdominal aortic aneurysms were generated. In one protocol, maximum diameter was held constant while bulge shape factor was varied. The shape factor took into account the asymmetric shape of the bulge. In a second protocol, the shape of the aneurysmal wall was held constant while maximum diameter was varied. Wall stress was computed in each instance with a commercial software package and assumption of physiologic intraluminal pressure.
Both maximum diameter and the shape factor were found to have substantial influence on the distribution of wall stress within the aneurysm. In some instances the maximum stress occurred at the midsection, and in others it occurred elsewhere. The magnitude of peak stress acting on the aneurysm increased nonlinearly with increasing maximum diameter or increasing asymmetry.
Our computer models showed that the stress within the wall of an abdominal aortic aneurysm and possibly the potential for rupture are as dependent on aneurysm shape as they are on maximum diameter. This information may be important in determining severity of individual abdominal aortic aneurysms and in improving understanding of the natural history of the disease.
To evaluate the role played by biomechanical and geometrical parameters of endoluminal thrombus and of aortic wall on abdominal aortic aneurysm (AAA) behaviour.
Tensile tests on 21 AAA thrombus specimens from six patients undergoing AAA repair and numerical evaluation of aneurysmal aortic wall stress and strain distribution. Parameters of the analysis were lumen eccentricity, thrombus Young's Modulus and the aortic wall constitutive equation.
There was a linear stress/strain for all the thrombus specimens. The numerical analyses show the mechanical behaviour of AAA as a function of lumen eccentricity and biomechanical parameters.
Well organised thrombus reduces the effect of the pressure load on the aneurysmal aortic wall.
Abdominal aortic aneurysm (AAA) is a local, progressive dilation of the distal aorta that risks rupture until treated. Using the law of Laplace, in vivo assessment of AAA surface geometry could identify regions of high wall tensions as well as provide critical dimensional and shape data for customized endoluminal stent grafts. In this study, six patients with AAA underwent spiral computed tomography imaging and the inner wall of each AAA was identified, digitized, and reconstructed. A biquadric surface patch technique was used to compute the local principal curvatures, which required no assumptions regarding axisymmetry or other shape characteristics of the AAA surface. The spatial distribution of AAA principal curvatures demonstrated substantial axial asymmetry, and included adjacent elliptical and hyperbolic regions. To determine how much the curvature spatial distributions were dependent on tortuosity versus bulging, the effects of AAA tortuosity were removed from the three-dimensional (3D) reconstructions by aligning the centroids of each digitized contour to the z axis. The spatial distribution of principal curvatures of the modified 3D reconstructions were found to be largely axisymmetric, suggesting that much of the surface geometric asymmetry is due to AAA bending. On average, AAA surface area increased by 56% and abdominal aortic length increased by 27% over those for the normal aorta. Our results indicate that AAA surface geometry is highly complex and cannot be simulated by simple axisymmetric models, and suggests an equally complex wall stress distribution. © 1999 Biomedical Engineering Society.
PAC99: 8719Rr, 8759Fm, 8757Gg
Pulsatile flow in abdominal aortic aneurysm (AAA) models has been examined in order to understand the hemodynamics that may contribute to growth of an AAA. The model studies were conducted by experiments (flow visualization and laser Doppler velocimetry) and by numerical simulation using physiologically realistic resting and exercise flow conditions. We characterize the flow for two AAA model shapes and sizes emulating early AAA development through moderate AAA growth (mean and peak Reynolds numbers of 362 < Re(mean) < 1053 and 3308 < Re(peak) < 5696 with Womersley parameter 16.4 < alpha < 21.2). The results of our investigation indicate that AAA flow can be divided into three flow regimes: (i) Attached flow over the entire cycle in small AAAs at resting conditions, (ii) vortex formation and translation in moderate size AAAs at resting conditions, and (iii) vortex formation, translation and turbulence in moderate size AAAs under exercise conditions. The second two regimes are classified in the medical literature as disturbed flow conditions that have been correlated with atherogenesis as well as thrombogenesis. Thus, AAA disturbed hemodynamics may be a contributing factor to AAA growth by accelerating the degeneration of the arterial wall. Our investigation also concluded that vortex development is considerably weaker in an asymmetric AAA. Furthermore, turbulence was not observed in the asymmetric model. Finally, our investigation suggests a new mode of transition to turbulence: vortex ring instability and bursting to turbulence. The transition process depends on a combination of the pulsatile flow conditions and the tube cross-sectional area change.
Abdominal aortic aneurysm (AAA) rupture is believed to occur when the mechanical stress acting on the wall exceeds the strength of the wall tissue. Therefore, knowledge of the stress distribution in an intact AAA wall could be useful in assessing its risk of rupture. We developed a methodology to noninvasively estimate the in vivo wall stress distribution for actual AAAs on a patient-to-patient basis.
Six patients with AAAs and one control patient with a nonaneurysmal aorta were the study subjects. Data from spiral computed tomography scans were used as a means of three-dimensionally reconstructing the in situ geometry of the intact AAAs and the control aorta. We used a nonlinear biomechanical model developed specifically for AAA wall tissue. By means of the finite element method, the stress distribution on the aortic wall of all subjects under systolic blood pressure was determined and studied.
In all the AAA cases, the wall stress was complexly distributed, with distinct regions of high and low stress. Peak wall stress among AAA patients varied from 29 N/cm(2) to 45 N/cm(2) and was found on the posterior surface in all cases studied. The wall stress on the nonaneurysmal aorta in the control subject was relatively low and uniformly distributed, with a peak wall stress of 12 N/cm(2). AAA volume, rather than AAA diameter, was shown by means of statistical analysis to be a better indicator of high wall stresses and possibly rupture.
The approach taken to estimate AAA wall stress distribution is completely noninvasive and does not require any additional involvement or expense by the AAA patient. We believe that this methodology may allow for the evaluation of an individual AAA's rupture risk on a more biophysically sound basis than the widely used 5-cm AAA diameter criterion.
Knowledge of the wall stresses in an abdominal aortic aneurysm (AAA) may be helpful in evaluating the need for surgical intervention to avoid rupture. This must be preceded by the development of a more suitable finite strain constitutive model for AAA, as none currently exists. Additionally, reliable stress analysis of in vivo AAA for the purposes of clinical diagnostics requires patient-specific values of the material parameters, which are difficult to determine noninvasively. The purpose of this work, therefore, was three-fold: (1) to develop a finite strain constitutive model for AAA; (2) to estimate the variation of model parameters within a sample population; and (3) to evaluate the sensitivity of computed stress distribution in AAA due to this biologic variation. We propose here a two parameter, hyperelastic, isotropic, incompressible material model and utilize experimental data from 69 freshly excised AAA specimens to both develop the functional form of the model and estimate its material parameters. Parametric analyses were performed via repeated finite element computations to determine the effect of varying each of the two model parameters on the stress distribution in a three-dimensional AAA model. The agreement between experimental data and the proposed functional form of the constitutive law was very good (R2 > 0.9). Our finite element simulations showed that the computed AAA wall stresses changed by only 4% or less when both the parameters were varied within the 95% confidence intervals for the patient population studied. This observation indicates that in lieu of the patient-specific material parameters, which are difficult to determine the use of population mean values is sufficiently accurate for the model to be reasonably employed in a clinical setting. We believe that this is an important advancement toward the development of a computational tool for the estimation of rupture potential for individual AAA, for which there is great clinical need.
Factors influencing the development or rupture of abdominal aortic aneurysms (AAAs) have not yet been confirmed. This study delineated the risk factors for rupture of AAAs as evaluated by means of a combination of three-dimensional (3D) reconstruction and clinical data analysis.
The study population comprised Japanese patients in whom an atherosclerotic AAA had been diagnosed between January 1980 and December 1997. We obtained 3D-based data by means of computer-aided 3D reconstruction from computed tomography studies of AAAs. The data included the tortuosity of the aneurysm, maximum transverse diameter, length of the aneurysm, aneurysmal volume, aneurysmal surface area, largest aneurysmal cross-sectional area, ratio of transverse aneurysmal diameter to the length of the aneurysm (T/L), and amount of mural thrombus. Clinical data were collected from patient files. All data were assessed by means of multivariate analysis for their predictive value for expansion or rupture of AAA.
The most efficient predictor of annual expansion rate of maximum transverse diameter (EX-D) was a combination of largest aneurysmal cross-sectional area, tobacco use, and tortuosity. The most efficient predictor of annual expansion rate of aneurysmal volume (EX-V) was a combination of aneurysmal volume and blood urea nitrogen level. The most efficient predictors of aneurysmal rupture was a combination of EX-D, diastolic blood pressure, and T/L.
Three-dimensional-based data on aneurysmal morphology, including T/L, largest aneurysmal crosssectional area, and aneurysmal volume, had strong predictive value for expansion and rupture of AAAs.
To estimate when an abdominal aortic aneurysm (AAA) may rupture, it is necessary to understand the forces responsible for this event. We investigated the wall stresses in an AAA in a clinical model. Using CT scans of the AAA, the diameter and wall thickness were measured and the model of the aneurysm was created. The wall stresses were determined using a finite element analysis in which the aorta was considered isotropic with linear material properties and was loaded with a pressure of 120 mmHg. The AAA was eccentric with a length of 10.5 cm, a diameter of 2.5 to 5.9 cm, and a wall thickness of 1.0 to 2.0 mm. The aneurysm had specific areas of high stress. On the inner surface the highest stress was 0.4 N/mm2 and occurred along two circumferentially oriented belts--one at the bulb and the other just below. The stress was longitudinal at the anterior region of the bulb and circumferential elsewhere, suggesting that a rupture caused by this stress will result in a circumferential tear at the anterior portion of the bulb and a longitudinal tear elsewhere. In the mid-surface the highest stress was 0.37 N/mm2 and occurred at two locations: the posterior region of the bulb and anteriorly just below. The stress was circumferential, suggesting that the rupture caused by this stress will produce a longitudinal tear. The location and orientation of the maximum stress were influenced more by the tethering force than by the wall thickness, luminal pressure, or wall stiffness. In conclusion, the rupture of an AAA is most likely to occur on the inner surface at the bulb. Such analytical approaches could lead to a better understanding of the aneurysm rupture and may be instrumental in planning surgical interventions.
The treatment of patients with abdominal aortic aneurysms (AAAs) is typically based on the potential for rupture. Current rupture assessments are in turn based on statistics from aggregate populations and are incapable of providing precise risk estimates for individual AAAs. Significant benefit could be realized if rupture potential for individual AAAs could be reliably determined on the basis of simple geometric characteristics or the results of symmetric thin-shell analysis. This study seeks to determine whether it is possible to estimate wall stresses by use of these simple measures.
Linear finite element analysis was used to estimate the distribution of von Mises stresses in a series of homogeneous, isotropic, three-dimensional AAA models subject to static loading and assumed to have zero residual stresses. The magnitude of the peak stress was tabulated for each model along with the following characteristics: aneurysm volume; maximum diameter; maximum radius; maximal wall distention; aspect ratio (ratio of greatest anteroposterior diameter to transverse diameter); local radii of curvature (in both longitudinal and circumferential directions); and maximum symmetric thin-shell stress estimates (on the basis of the meridional contour). The relationship between peak stress and each of the characteristics was assessed by use of Spearman rank correlation coefficients, with values less than 0.95 interpreted as signifying unreliable associations.
Peak stresses in the individual models ranged from 1.79 x 10(6) dyne/cm2 to 15.1 x 10(6) dyne/cm2. The circumferential and longitudinal radii of curvature were frequently able to predict the locations of high stress, but were unreliable in predicting the magnitude of peak stress. The aspect ratio showed the strongest correlation with peak wall stress (r = 0.88, 95% CI, 0.68-0.96), whereas the other characteristics showed even less correlation. Symmetric thin shell analysis accurately predicted stresses in axially symmetric models, but it was incapable of predicting either the location or magnitude of peak stress in asymmetric models.
Simple geometric criteria and symmetric thin shell analyses are unreliable in predicting AAA stresses. Future attempts to estimate wall stress and assess risk of rupture for individual AAAs may require detailed three-dimensional modeling.
In the United States, cardiovascular disease, e.g., atherosclerosis and hypertension, that lead to heart failure and stroke, is the leading cause of mortality, accounting for over 40 percent of deaths in those aged 65 years and above. Over 80 percent of all cardio-vascular deaths occur in the same age group. Thus, age, per se, is the major risk factor for cardiovascular disease. Clinical manifestations and prognosis of these cardiovascular diseases likely become altered in older persons with advanced age because interactions occur between age-associated cardiovascular changes in health and specific pathophysiologic mechanisms that underlie a disease. A fundamental understanding of age-associated changes in cardiovascular structure and function ranging in scope from humans to molecules is required for effective and efficient prevention and treatment of cardiovascular disease in older persons. A sustained effort over the past two decades has been applied to characterize the multiple effects of aging in health on cardiovascular structure and function in a single study population, the Baltimore Longitudinal Study on Aging. In these studies, community dwelling, volunteer participants are rigorously screened to detect both clinical and occult cardiovascular disease and characterized with respect to lifestyle, e.g. exercise habits, in an attempt to deconvolute interactions among lifestyle, cardiovascular disease and the aging process in health. This review highlights some specific changes in resting cardiovascular structure and function and cardiovascular reserve capacity that occur with advancing age in healthy humans. Observations from relevant experiments in animal models have been integrated with those in humans to provide possible mechanistic insight.
A method is introduced to automatically find the coronary axis based on two or more user-defined points, even in the presence of a severe stenosis. The coronary axis is determined by finding a minimum cost path (MCP) in a feature image in which the tubular-like structures are enhanced. The results of the proposed method were compared with manually drawn central axes to estimate the accuracy. In 32 3D TFE-EPI acquisitions of patients and volunteers, 14 right coronary arteries (RCAs), 15 left anterior descending arteries (LADs), and eight left circumflex arteries (LCXs) were manually tracked twice by two operators to determine a reference axis and to assess the inter- and intra-user variability. On average, the maximum distance to the reference axis, based on only two user-defined points, is less than 1.5 mm; the average distance is around 0.65 mm, which is less than the average in-plane resolution. The results of the method are comparable to those of the manual operators.
The purpose of this study was to calculate abdominal aortic aneurysm (AAA) wall stresses in vivo for ruptured, symptomatic, and electively repaired AAAs with three-dimensional computer modeling techniques, computed tomographic scan data, and blood pressure and to compare wall stress with current clinical indices related to rupture risk.
CT scans were analyzed for 48 patients with AAAs: 18 AAAs that ruptured (n = 10) or were urgently repaired for symptoms (n = 8) and 30 AAAs large enough to merit elective repair within 12 weeks of the CT scan. Three-dimensional computer models of AAAs were reconstructed from CT scan data. The stress distribution on the AAA as a result of geometry and blood pressure was computationally determined with finite element analysis with a hyperelastic nonlinear model that depicted the mechanical behavior of the AAA wall.
Peak wall stress (maximal stress on the AAA surface) was significantly different between groups (ruptured, 47.7 +/- 6 N/cm(2); emergent symptomatic, 47.5 +/- 4 N/cm(2); elective repair, 36.9 +/- 2 N/cm(2); P =.03), with no significant difference in blood pressure (P =.2) or AAA diameter (P =.1). Because of trends toward differences in diameter, comparison was made only with diameter-matched subjects. Even with identical mean diameters, ruptured/symptomatic AAAs had a significantly higher peak wall stress (46.8 +/- 4.5 N/cm(2) versus 38.1 +/- 1.3 N/cm(2); P =.05). Maximal wall stress predicted risk of rupture better than the LaPlace equation (20.7 +/- 5.7 N/cm(2) versus 18.8 +/- 2.9 N/cm(2); P =.2) or other proposed indices of rupture risk. The smallest ruptured AAA was 4.8 cm, but this aneurysm had a stress equivalent to the average electively repaired 6.3-cm AAA.
Peak wall stresses calculated in vivo for AAAs near the time of rupture were significantly higher than peak stresses for electively repaired AAAs, even when matched for maximal diameter. Calculation of wall stress with computer modeling of three-dimensional AAA geometry appears to assess rupture risk more accurately than AAA diameter or other previously proposed clinical indices. Stress analysis is practical and feasible and may become an important clinical tool for evaluation of AAA rupture risk.
The role of intraluminal thrombus (ILT) on abdominal aortic aneurysm rupture is still not clear. Rupture of an aneurysm occurs when the wall stress exceeds the wall strength at any location on the wall. The purpose of this study was to address the hypothesis that the presence of ILT alters the wall stress distribution or wall stress magnitude in AAA.
Patient-specific 3D AAA geometries were reconstructed from computed tomographic images. Two geometric features, ILT surface ratio (ILT surface area divided by the total AAA surface area) and ILT volume ratio (ILT volume divided by the total AAA volume), were calculated for each AAA. Two models were created for each patient: one with ILT and one without ILT. Systolic pressure measured at the time of computed tomographic imaging was applied to the internal surface of each model. A nonlinear large deformation algorithm was used to compute wall stress distribution with the finite element method. The Wilcoxon matched pairs test was used to compare the peak wall stress between the two models of each patient.
Four patients were studied with ILT surface ratios that ranged from 0.29 to 0.72 and ILT volume ratios that ranged from 0.12 to 0.66. The peak wall stress was reduced (range, 6% to 38% reduction; P =.067) for all models with ILT included (range, 28 to 37 N/cm(2)) as compared with models with no ILT (range, 30 to 44 N/cm(2)). Visual inspection also revealed a marked effect of ILT on the wall stress distribution.
The presence of ILT alters the wall stress distribution and reduces the peak wall stress in AAA. For this reason, ILT should be included in all patient-specific models of AAA for evaluation of AAA wall stresses.
We previously showed that peak abdominal aortic aneurysm (AAA) wall stress calculated for aneurysms in vivo is higher at rupture than at elective repair. The purpose of this study was to analyze rupture risk over time in patients under observation.
Computed tomography (CT) scans were analyzed for patients with AAA when observation was planned for at least 6 months. AAA wall stress distribution was computationally determined in vivo with CT data, three-dimensional computer modeling, finite element analysis (nonlinear hyperelastic model depicting aneurysm wall behavior), and blood pressure during observation.
Analysis included 103 patients and 159 CT scans (mean follow-up, 14 +/- 2 months per CT). Forty-two patients were observed with no intervention for at least 1 year (mean follow-up, 28 +/- 3 months). Elective repair was performed within 1 year in 39 patients, and emergent repair was performed in 22 patients (mean, 6 +/- 1 month after CT) for rupture (n = 14) or acute severe pain. Significant differences were found for initial diameter (observation, 4.9 +/-.1 cm; elective repair, 5.9 +/-.1 cm; emergent repair, 6.1 +/-.2 cm; P <.0001) and initial peak wall stress (38 +/- 1 N/cm(2), 42 +/- 2 n/cm(2), 58 +/- 4 N/cm(2), respectively; P <.0001), but peak wall stress appeared to better differentiate patients who later required emergent repair (elective vs emergent repair: diameter, 3% difference, P =.5; stress, 38% difference, P <.0001). Receiver operating characteristic (ROC) curves for predicting rupture were better for peak wall stress (sensitivity, 94%; specificity,81%; accuracy, 85% [with 44 N/cm(2) threshold]) than for diameter (81%, 70%, 73%, respectively [with optimal 5.5 cm threshold). With proportional hazards analysis, peak wall stress (relative risk, 25x) and gender (relative risk, 3x) were the only significant independent predictors of rupture.
For AAAs under observation, peak AAA wall stress seems superior to diameter in differentiating patients who will experience catastrophic outcome. Elevated wall stress associated with rupture is not simply an acute event near the time of rupture.
The purpose of this study was to analyze anatomic characteristics of patients with ruptured abdominal aortic aneurysms (AAAs), with conventional two-dimensional computed tomography (CT), including comparison with control subjects matched for age, gender, and size.
Records were reviewed to identify all CT scans obtained at Dartmouth-Hitchcock Medical Center or referring hospitals before emergency AAA repair performed because of rupture or acute severe pain (RUP group). CT scans obtained before elective AAA repair (ELEC group) were reviewed for age and gender match with patients in the RUP group. More than 40 variables were measured on each CT scan. Aneurysm diameter matching was achieved by consecutively deleting the largest RUP scan and the smallest ELEC scan to prevent bias.
CT scans were analyzed for 259 patients with AAAs: 122 RUP and 137 ELEC. Patients were well matched for age, gender, and other demographic variables or risk factors. Maximum AAA diameter was significantly different in comparisons of all patients (RUP, 6.5 +/- 2 cm vs ELEC, 5.6 +/- 1 cm; P <.0001), and mean diameter of ruptured AAAs was 5 mm smaller in female patients (6.1 +/- 2 cm vs 6.6 +/- 2 cm; P =.007). Two hundred patients were matched for diameter, gender, and age (100 from each group; maximum AAA diameter, 6.0 +/- 1 cm vs 6.0 +/- 1 cm). Analysis of diameter-matched AAAs indicated that most variables were statistically similar in the two groups, including infrarenal neck length (17 +/- 1 mm vs 19 +/- 1 mm; P =.3), maximum thrombus thickness (25 +/- 1 mm vs 23 +/- 1 mm, P =.4), and indices of body habitus, such as [(maximum AAA diameter)/(normal suprarenal aorta diameter)] or [(maximum AAA diameter)/(L3 transverse diameter)]. Multivariate analysis controlling for gender indicated that the most significant variables for rupture were aortic tortuosity (odds ratio [OR] 3.3, indicating greater risk with no or mild tortuosity), diameter asymmetry (OR, 3.2 for a 1-cm difference in major-minor axis), and current smoking (OR, 2.7, with the greater risk in current smokers).
When matched for age, gender, and diameter, ruptured AAAs tend to be less tortuous, yet have greater cross-sectional diameter asymmetry. On conventional two-dimensional CT axial sections, it appears that when diameter asymmetry is associated with low aortic tortuosity, the larger diameter on axial sections more accurately reflects rupture risk, and when diameter asymmetry is associated with moderate or severe aortic tortuosity, the smaller diameter on axial sections more accurately reflects rupture risk. Current smoking is significantly associated with rupture, even when controlling for gender and AAA anatomy.
The decision to repair an asymptomatic abdominal aortic aneurysm (AAA) is currently based on diameter (> or =5.5 cm) alone. However, aneurysms less than 5.5 cm do rupture while some reach greater than 5.5 cm without rupturing. Hence the need to predict the risk of rupture on an individual patient basis is important. This study aims to calculate and compare wall stress in ruptured and non-ruptured AAA.
The 3D geometries of AAA were derived from CT scans of 27 patients (12 ruptured and 15 non-ruptured). AAA geometry, systolic blood pressure and literature derived material properties, were utilised to calculate wall stress for individual AAA using finite element analysis.
Peak wall stress was significantly higher in the ruptured AAA (mean 1.02 MPa) than the non-ruptured AAA (mean 0.62 MPa). In patients with an identifiable site of rupture on CT scan, the area of peak wall stress correlated with rupture site.
Peak wall stress can be calculated from routinely performed CT scans and may be a better predictor of risk of rupture than AAA diameter on an individual patient basis.
Prediction of rupture risk in abdominal aortic aneurysm during observation: wall stress versus diameter A comparative study of aortic wall stress using finite element analysis for ruptured and non-ruptured abdominal aortic aneurysms
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SEPRAN introduction, user's manual, programmer's guide and standard problems
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