Inverse elastostatic stress analysis in pre-deformed biological structures: Demonstration using abdominal aortic aneurysms

In stress analysis of membrane-like biological structures, the geometry constructed from in vivo image, which often corresponds to a deformed state, is routinely taken as the initial stress-free geometry. In this paper, we show that this limitation can be completely removed using an inverse elastostatic approach, namely, a method for finding the initial geometry of an elastic body from a given deformed state. We demonstrate the utility of the inverse approach using a patient-specific abdominal aortic aneurysm model, and identify the scope of error in stress estimation in the conventional approach within a realistic range of material parameter variations.     

Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability

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.     

Use of the photoelastic method and finite element analysis in the assessment of wall strain in abdominal aortic aneurysm models

Abdominal aortic aneurysm (AAA) is a significant health problem. Current clinical rupture-risk relies primarily on the maximum diameter of the AAA and also growth rate. However, AAAs are a patient-specific problem and recently, numerical tools have been employed to estimate rupture-potential. Alternatively, experimental assessment of AAA biomechanics receives less attention, yet, rigorous validation of numerical tools is required prior to clinical acceptance. This paper examines the use of the photoelastic method to assess wall strain and its validation using finite element analysis (FEA) in a small number of patient-specific AAA models. Experimental models were manufactured in-house using the injection-moulding procedure together with a commercially available photoelastic material. The material was mechanically characterised prior to testing, with models examined under three loading regimes (80, 120 and 160mmHg). Each experimental model was imaged using computed tomography (CT) and reconstructed in three dimensions (3D) for numerical analyses. Experimental wall strain was measured and numerical wall strain calculated with finite element analysis (FEA). Results were qualitatively and quantitatively compared. There was good qualitative agreement between the experimental and numerical methods, with similar trends apparent throughout all models at all pressures. Overall, acceptable percentage errors between the techniques were observed for all models. Median errors of -6.5%, -0.4% and 3.9% for the models at 80, 120 and 160mmHg pressures, respectively, were determined. The photoelastic method is a very useful experimental tool that provides instant, easy to interpret, information regarding wall strain. The technique is useful for validation of numerical AAA studies.     

Patient-specific stress analyses in the ascending thoracic aorta using a finite-element implementation of the constrained mixture theory

It is now a rather common approach to perform patient-specific stress analyses of arterial walls using finite-element models reconstructed from gated medical images. However, this requires to compute for every Gauss point the deformation gradient between the current configuration and a stress-free reference configuration. It is technically difficult to define such a reference configuration, and there is actually no guarantee that a stress-free configuration is physically attainable due to the presence of internal stresses in unloaded soft tissues. An alternative framework was proposed by Bellini et al. (Ann Biomed Eng 42(3):488–502, 2014). It consists of computing the deformation gradients between the current configuration and a prestressed reference configuration. We present here the first finite-element results based on this concept using the Abaqus software. The reference configuration is set arbitrarily to the in vivo average geometry of the artery, which is obtained from gated medical images and is assumed to be mechanobiologically homeostatic. For every Gauss point, the stress is split additively into the contributions of each individual load-bearing constituent of the tissue, namely elastin, collagen, smooth muscle cells. Each constituent is assigned an independent prestretch in the reference configuration, named the deposition stretch. The outstanding advantage of the present approach is that it simultaneously computes the in situ stresses existing in the reference configuration and predicts the residual stresses that occur after removing the different loadings applied onto the artery (pressure and axial load). As a proof of concept, we applied it on an ideal thick-wall cylinder and showed that the obtained results were consistent with corresponding experimental and analytical results of the well-known literature. In addition, we developed a patient-specific model of a human ascending thoracic aneurysmal aorta and demonstrated the utility in predicting the wall stress distribution in vivo under the effects of physiological pressure. Finally, we simulated the whole process preceding traditional in vitro uniaxial tensile testing of arteries, including excision from the body, radial cutting, flattening and subsequent tensile loading, showing how this process may impact the final mechanical properties derived from these in vitro tests.     

Characterizing heterogeneous properties of cerebral aneurysms with unknown stress-free geometry: a precursor to in vivo identification

Knowledge of elastic properties of cerebral aneurysms is crucial for understanding the biomechanical behavior of the lesion. However, characterizing tissue properties using in vivo motion data presents a tremendous challenge. Aside from the limitation of data accuracy, a pressing issue is that the in vivo motion does not expose the stress-free geometry. This is compounded by the nonlinearity, anisotropy, and heterogeneity of the tissue behavior. This article introduces a method for identifying the heterogeneous properties of aneurysm wall tissue under unknown stress-free configuration. In the proposed approach, an accessible configuration is taken as the reference; the unknown stress-free configuration is represented locally by a metric tensor describing the prestrain from the stress-free configuration to the reference configuration. Material parameters are identified together with the metric tensor pointwisely. The paradigm is tested numerically using a forward-inverse analysis loop. An image-derived sac is considered. The aneurysm tissue is modeled as an eightply laminate whose constitutive behavior is described by an anisotropic hyperelastic strain-energy function containing four material parameters. The parameters are assumed to vary continuously in two assigned patterns to represent two types of material heterogeneity. Nine configurations between the diastolic and systolic pressures are generated by forward quasi-static finite element analyses. These configurations are fed to the inverse analysis to delineate the material parameters and the metric tensor. The recovered and the assigned distributions are in good agreement. A forward verification is conducted by comparing the displacement solutions obtained from the recovered and the assigned material parameters at a different pressure. The nodal displacements are found in excellent agreement.     

Identification of in vivo material and geometric parameters of a human aorta: toward patient-specific modeling of abdominal aortic aneurysm

Recent advances in computational modeling of vascular adaptations and the need for their extension to patient-specific modeling have introduced new challenges to the path toward abdominal aortic aneurysm modeling. First, the fundamental assumption in adaptation models, namely the existence of vascular homeostasis in normal vessels, is not easy to implement in a vessel model built from medical images. Second, subjecting the vessel wall model to the normal pressure often makes the configuration deviate from the original geometry obtained from medical images. To address those technical challenges, in this work, we propose a two-step optimization approach; first, we estimate constitutive parameters of a healthy human aorta intrinsic to the material by using biaxial test data and a weighted nonlinear least-squares parameter estimation method; second, we estimate the distributions of wall thickness and anisotropy using a 2-D parameterization of the vessel wall surface and a global approximation scheme integrated within an optimization routine. A direct search method is implemented to solve the optimization problem. The numerical optimization method results in a considerable improvement in both satisfying homeostatic condition and minimizing the deviation of geometry from the original shape based on in vivo images. Finally, the utility of the proposed technique for patient-specific modeling is demonstrated in a simulation of an abdominal aortic aneurysm enlargement.     

Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis

Rupture risk estimation of abdominal aortic aneurysms (AAA) is currently based on the maximum diameter of the AAA. A more critical approach is based on AAA wall stress analysis. For that, in most cases, the AAA geometry is obtained from CT-data and treated as a stress free geometry. However, during CT imaging, the AAA is subjected to a time-averaged blood pressure and is therefore not stress free. The aim of this study is to evaluate the effect of neglecting these initial stresses (IS) on the patient-specific AAA wall stress as computed by finite element analysis. Additionally, the contribution of the nonlinear material behavior of the AAA wall is evaluated.Thirty patients with maximum AAA diameters below the current surgery criterion were scanned with contrast-enhanced CT and the AAA's were segmented from the image data. The mean arterial blood pressure (MAP) was measured immediately after the CT-scan and used to compute the IS corresponding with the CT geometry and MAP. Comparisons were made between wall stress obtained with and without IS and with linear and nonlinear material properties.On average, AAA wall stresses as computed with IS were higher than without IS. This was also the case for the stresses computed with the nonlinear material model compared to the linear material model. However, omitting initial stress and material nonlinearity in AAA wall stress computations leads to different effects in the resulting wall stress for each AAA. Therefore, provided that other assumptions made are not predominant, IS cannot be discarded and a nonlinear material model should be used in future patient-specific AAA wall stress analyses.     

Including surrounding tissue improves ultrasound-based 3D mechanical characterization of abdominal aortic aneurysms

ObjectivesIn this study the influence of surrounding tissues including the presence of the spine on wall stress analysis and mechanical characterization of abdominal aortic aneurysms using ultrasound imaging has been investigated.MethodsGeometries of 7 AAA patients and 11 healthy volunteers were acquired using 3-D ultrasound and converted to finite element based models. Model complexity of externally unsupported (aorta-only) models was complemented with inclusion of both soft tissue around the aorta and a spine support dorsal to the aorta. Computed 3-D motion of the aortic wall was verified by means of ultrasound speckle tracking. Resulting stress, strain, and estimated shear moduli were analyzed to quantify the effect of adding surrounding material supports.ResultsAn improved agreement was shown between the ultrasound measurements and the finite element tissue and spine models compared to the aorta-only models. Peak and 99-percentile Von Mises stress showed an overall decrease of 23–30%, while estimated shear modulus decreased with 12–20% after addition of the soft tissue. Shear strains in the aortic wall were higher in areas close to the spine compared to the anterior region.ConclusionsImproving model complexity with surrounding tissue and spine showed a homogenization of wall stresses, reduction in homogeneity of shear strain at the posterior side of the AAA, and a decrease in estimated aortic wall shear modulus. Future research will focus on the importance of a patient-specific spine geometry and location.     

A simple method of estimating the stress acting on a bilaterally symmetric abdominal aortic aneurysm

We extended a method of estimating the stress acting on an axisymmetric abdominal aortic aneurysm (AAA) under a load in vivo (Elger, D. F., Blackketter, D. M., Budwig, R. S., Johansen, K. H. (1996) The influence of shape on the stresses in model abdominal aortic aneurysms, Journal of Biomechanical Engineering, 118, pp. 326-32.) to bilaterally-symmetric AAAs, which are symmetric about the sagittal plane. Stresses were calculated along the anterior and posterior median lines of the AAA wall. Of the two force equilibrium equations, the Laplace equation held in this study. The longitudinal equilibrium was extended to hold by approximating the meridional tension and the directional cosine of the wall surface as constants along the circumference. The estimated stresses were compared with the results of a finite element analysis. Comparisons showed that the maximal principal stress, usually the circumferential stress or sometimes the meridional stress depending on location, sufficiently represented the wall stress. The proposed method provides a reasonable index for evaluating the rupture risk using the peak value of the maximal principal stress and its location without using the stress-free geometry and constitutive equation.     

A volumetric model for growth of arterial walls with arbitrary geometry and loads

Stress and deformation in arterial wall tissue are factors which may influence significantly its response and evolution. In this work we develop models based on nonlinear elasticity and finite element numerical solutions for the mechanical behaviour and the remodelling of the soft tissue of arteries, including anisotropy induced by the presence of collagen fibres. Remodelling and growth in particular constitute important features in order to interpret stenosis and atherosclerosis. The main object of this work is to model accurately volumetric growth, induced by fluid shear stress in the intima and local wall stress in arteries with patient-specific geometry and loads. The model is implemented in a nonlinear finite element setting which may be applied to realistic 3D geometries obtained from in vivo measurements. The capabilities of this method are demonstrated in several examples. Firstly a stenotic process on an idealised geometry induced by a non-uniform shear stress distribution is considered. Following the growth of a right coronary artery from an in vivo reconstructed geometry is presented. Finally, experimental measurements for growth under hypertension for rat carotid arteries are modelled.     

An augmented iterative method for identifying a stress-free reference configuration in image-based biomechanical modeling

Continued advances in computational power and methods have enabled image-based biomechanical modeling to become an important tool in basic science, diagnostic and therapeutic medicine, and medical device design. One of the many challenges of this approach, however, is identification of a stress-free reference configuration based on in vivo images of loaded and often prestrained or residually stressed soft tissues and organs. Fortunately, iterative methods have been proposed to solve this inverse problem, among them Sellier’s method. This method is particularly appealing because it is easy to implement, convergences reasonably fast, and can be coupled to nearly any finite element package. By means of several practical examples, however, we demonstrate that in its original formulation Sellier’s method is not optimally fast and may not converge for problems with large deformations. Fortunately, we can also show that a simple, inexpensive augmentation of Sellier’s method based on Aitken’s delta-squared process can not only ensure convergence but also significantly accelerate the method.     

In vitro validation of finite-element model of AAA hemodynamics incorporating realistic outlet boundary conditions

The purpose of this study is to validate numerical simulations of flow and pressure in an abdominal aortic aneurysm (AAA) using phase-contrast magnetic resonance imaging (PCMRI) and an in vitro phantom under physiological flow and pressure conditions. We constructed a two-outlet physical flow phantom based on patient imaging data of an AAA and developed a physical Windkessel model to use as outlet boundary conditions. We then acquired PCMRI data in the phantom while it operated under conditions mimicking a resting and a light exercise physiological state. Next, we performed in silico numerical simulations and compared experimentally measured velocities, flows, and pressures in the in vitro phantom to those computed in the in silico simulations. There was a high degree of agreement in all of the pressure and flow waveform shapes and magnitudes between the experimental measurements and simulated results. The average pressures and flow split difference between experiment and simulation were all within 2%. Velocity patterns showed good agreement between experimental measurements and simulated results, especially in the case of whole-cycle averaged comparisons. We demonstrated methods to perform in vitro phantom experiments with physiological flows and pressures, showing good agreement between numerically simulated and experimentally measured velocity fields and pressure waveforms in a complex patient-specific AAA geometry.     

A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments

The prediction of patient-specific proximal femur mechanical response to various load conditions is of major clinical importance in orthopaedics. This paper presents a novel, empirically validated high-order finite element method (FEM) for simulating the bone response to loads. A model of the bone geometry was constructed from a quantitative computerized tomography (QCT) scan using smooth surfaces for both the cortical and trabecular regions. Inhomogeneous isotropic elastic properties were assigned to the finite element model using distinct continuous spatial fields for each region. The Young's modulus was represented as a continuous function computed by a least mean squares method. p-FEMs were used to bound the simulation numerical error and to quantify the modeling assumptions. We validated the FE results with in-vitro experiments on a fresh-frozen femur loaded by a quasi-static force of up to 1500 N at four different angles. We measured the vertical displacement and strains at various locations and investigated the sensitivity of the simulation. Good agreement was found for the displacements, and a fair agreement found in the measured strain in some of the locations. The presented study is a first step toward a reliable p-FEM simulation of human femurs based on QCT data for clinical computer aided decision making.     

Numerical simulation of blood flow in the left ventricle and aortic sinus using magnetic resonance imaging and computational fluid dynamics

Understanding cardiac blood flow patterns has many applications in analysing haemodynamics and for the clinical assessment of heart function. In this study, numerical simulations of blood flow in a patient-specific anatomical model of the left ventricle (LV) and the aortic sinus are presented. The realistic 3D geometry of both LV and aortic sinus is extracted from the processing of magnetic resonance imaging (MRI). Furthermore, motion of inner walls of LV and aortic sinus is obtained from cine-MR image analysis and is used as a constraint to a numerical computational fluid dynamics (CFD) model based on the moving boundary approach. Arbitrary Lagrangian–Eulerian finite element method formulation is used for the numerical solution of the transient dynamic equations of the fluid domain. Simulation results include detailed flow characteristics such as velocity, pressure and wall shear stress for the whole domain. The aortic outflow is compared with data obtained by phase-contrast MRI. Good agreement was found between simulation results and these measurements.     

Impact of calcifications on patient-specific wall stress analysis of abdominal aortic aneurysms

As a degenerative and inflammatory desease of elderly patients, about 80% of abdominal aortic aneurysms (AAA) show considerable wall calcification. Effect of calcifications on computational wall stress analyses of AAAs has been rarely treated in literature so far. Calcifications are heterogeneously distributed, non-fibrous, stiff plaques which are most commonly found near the luminal surface in between the intima and the media layer of the vessel wall. In this study, we therefore investigate the influence of calcifications as separate AAA constituents on finite element simulation results. Thus, three AAAs are reconstructed with regard to intraluminal thrombus (ILT), calcifications and vessel wall. Each patient-specific AAA is simulated twice, once including all three AAA constituents and once neglecting calcifications as it is still common in literature. Parameters for constitutive modeling of calcifications are thereby taken from experiments performed by the authors, showing that calcifications exhibit an almost linear stress–strain behavior with a Young’s modulus E ≥ 40 MPa. Simulation results show that calcifications exhibit significant load-bearing effects and reduce stress in adjacent vessel wall. Average stress within the vessel wall is reduced by 9.7 to 59.2%. For two out of three AAAs, peak wall stress decreases when taking calcifications into consideration (8.9 and 28.9%). For one AAA, simulated peak wall stress increases by 5.5% due to stress peaks near calcification borders. However, such stress singularities due to sudden stiffness jumps are physiologically doubtful. It can further be observed that large calcifications are mostly situated in concavely shaped regions of the AAA wall. We deduce that AAA shape is influenced by existent calcifications, thus crucial errors occur if they are neglected in computational wall stress analyses. A general increase in rupture risk for calcified AAAs is doubted.     

Finite element and photoelastic modelling of an abdominal aortic aneurysm: a comparative study

Rupture prediction of abdominal aortic aneurysms (AAAs) remains a clinical challenge. Finite element analysis (FEA) may allow for improved identification for intervention timing, but the method needs further substantiation. In this study, experimental photoelastic method and finite element techniques were compared using an idealised AAA geometry. There was good agreement between the numerical and experimental results. At the proximal and distal end of the AAA model, the maximum differences in principle strain for an internal pressure of 120?mmHg had differences ranging from 0.03 to 10.01%. The maximum difference in principle strain for the photoelastic and the finite element model at a pressure of 120?mmHg was 0.167 and 0.158, respectively. The current research strengthens the case for using FEA as an adjunct to the current clinical practice of utilising diameter measurement for intervention timing.     

Finite element and photoelastic modelling of an abdominal aortic aneurysm: a comparative study

Rupture prediction of abdominal aortic aneurysms (AAAs) remains a clinical challenge. Finite element analysis (FEA) may allow for improved identification for intervention timing, but the method needs further substantiation. In this study, experimental photoelastic method and finite element techniques were compared using an idealised AAA geometry. There was good agreement between the numerical and experimental results. At the proximal and distal end of the AAA model, the maximum differences in principle strain for an internal pressure of 120 mmHg had differences ranging from 0.03 to 10.01%. The maximum difference in principle strain for the photoelastic and the finite element model at a pressure of 120 mmHg was 0.167 and 0.158, respectively. The current research strengthens the case for using FEA as an adjunct to the current clinical practice of utilising diameter measurement for intervention timing.