A quarter of a century biomechanical rupture risk assessment of abdominal aortic aneurysms. Achievements, clinical relevance, and ongoing developments

Abdominal aortic aneurysm (AAA) disease, the local enlargement of the infrarenal aorta, is a serious condition that causes many deaths, especially in men exceeding 65 years of age. Over the past quarter of a century, computational biomechanical models have been developed towards the assessment of AAA risk of rupture, technology that is now on the verge of being integrated within the clinical decision‐making process. The modeling of AAA requires a holistic understanding of the clinical problem, in order to set appropriate modeling assumptions and to draw sound conclusions from the simulation results. In this article we summarize and critically discuss the proposed modeling approaches and report the outcome of clinical validation studies for a number of biomechanics‐based rupture risk indices. Whilst most of the aspects concerning computational mechanics have already been settled, it is the exploration of the failure properties of the AAA wall and the acquisition of robust input data for simulations that has the greatest potential for the further improvement of this technology.


| WHAT IS AAA
Whilst an aneurysm, a permanent local enlargement of an artery by more than 50% of its diameter, may affect all arteries, the infrarenal aorta is most vulnerable to this disease and often leads to the development of an abdominal aortic aneurysm (AAA). The formation of AAAs is promoted by factors such as old age, male gender, smoking, and high mean arterial pressure (MAP). Furthermore, it is over-represented in persons with a family history of AAA and patients with coronary heart disease and Chronic Obstructive Pulmonary Disease. Aortic aneurysms can also present in patients with rare genetic diseases, such as Ehlers-Danlos syndrome type IV, Marfan syndrome, Loeys-Dietz syndrome, and fibromuscular dysplasia. Most AAAs remain small and therefore do not necessitate clinical intervention. However, an aneurysm may grow to a size that could result in aortic rupture, a life-threatening event. In urban areas only 50% of patients are given emergency treatment following rupture, out of which 50% survive the intervention, 1 resulting in an overall death rate of 75% from AAA rupture.

| Current guidelines and their limitations
Current guidelines suggest clinical intervention if the maximal transversal diameter exceeds a specified threshold value (55 mm in men and 50 mm in women), if the maximal transversal diameter grows by more than 5 mm in a year, or if the AAA becomes symptomatic. 2 Such intervention criteria are based on the outcome of randomized clinical trials, 3,4 where it has been shown that the risk of rupture for AAA below 55 mm is lower than 1%, a risk that is well below the 30-day-mortality for both open surgery and EVAR operation. 5 Intervention in cases below 55 mm is therefore not recommended. No strong evidence is reported in relation to the aforementioned growth-rate repair indication, and especially in females, the decision making is debated.
Whilst the sensitivity of the diameter-based repair indication is 99%, its specificity is suboptimal and the majority of AAAs that are operated on are in fact stable, 6,7 meaning that the intervention could have safely been postponed. The false positive rate, cases that are unnecessarily treated at a given time point (i.e., they are stable without the immediate threat of rupture), for AAAs that are smaller than 65 mm, is in excess of 90%. 5,6 A more specific repair indication criterion is therefore needed to reduce the number of surgical interventions that are performed unnecessarily. However, the sensitivity should be the same or greater as compared to current treatment approaches, towards the improvement of AAA patient care. Given the much higher mortality from AAA rupture as compared to the interventional risks (1%-5% 5,7 ), a small decrease in sensitivity would have immediately severe negative implications for the patient.

| Normal vessel wall structure
The histology of the normal aorta is well reported, and as with any conduit vessel, its wall is made up of intimal, medial and adventitial layers, see Figure 1. The intima consists of the endothelium, a monolayer of endothelial cells, that sits on a scaffold of collagen fibers, which themselves are dominantly aligned along the axial vessel direction. 8 The endothelium separates blood from the thrombogenic material of the vessel wall. The internal elastic lamina separates the intima from the media; a thick layer that is rich in smooth muscle cells (SMCs). Approximately 65 medial lamellar units (MLUs) 9 form the media of the human thoracic aorta, a lamellar structure that vanishes along the vascular tree and that is hardly visible F I G U R E 1 Artistic sketch relating to the structure of the healthy aorta. Intima, media and adventitia form the vessel wall. Structural integrity is provided by elastin and collagen, whilst endothelial cells, smooth muscle cells and fibroblast maintain the metabolism of the vessel wall. The vasa vasorum perfuses the adventitia and the outer media, whist convection across the vessel wall supports the intima, and the inner media in muscular arteries. Each MLU consists of elastin laminae, SMCs, and medial collagen. The external elastic lamina separates the media from the adventitia, a collagen fiber-rich layer with fibroblasts. Whilst the intima and the media are highly responsive to environmental cues, the adventitia acts to protect these layers from mechanical overload.
The orientation of collagen is inhomogeneous across the aortic wall. Whilst collagen is preferentially aligned along the circumferential direction in the inner media, the orientation disperses towards the outer media. 10 Conversely, the orientation of collagen in the adventitia is characteristically more isotropic, 11 see Figure 2A. The collagen in the media appears almost straight and displays a waviness (ratio between straight and curved length of a fiber) of no more than 0.9 (or 10%). 13 In contrast, the collagen in the adventitia is highly undulated, 11 see Figure 2B.
Endothelial cells, SMCs and fibroblasts are the most important cells in the vessel wall. Endothelial cells and SMCs of the contractile phenotype, equip the vessel wall with vasoreactivity, the ability to increase (vasodilation) and decrease (vasoconstriction) its diameter in response to environmental factors. SMCs can change to a synthetic phenotype where their functionality then resembles fibroblasts and they produce extracellular matrix components that are integrated into the vessel wall. Synthesis is counteracted by the degradation of extra cellular matrix (ECM) constituents (mostly through Matrix MetalloProteinases [MMP]), ultimately resulting in tissue turnover.
The mechanical response of the normal aortic wall is characterized by four distinct phases, see Figure 3. In the Phase I, the gradual straightening (mechanical engagement) of medial collagen results in the increase of stress with increasing stretch. In Phase II, between circumferential stretches of approximately 1.1 and 1.25, the first Piola-Kirchhoff F I G U R E 2 Orientation and waviness of collagen fibers across the wall thickness of the aorta. The graph shows the (not normalized) probability density function as a function of the collagen fibers' azimuthal orientation (left) and the waviness (right) across the vessel wall. An orientation of 90 corresponds to the circumferential vessel direction, and the interpolation of experimental data 8,10,12 determines the left plot. The waviness represents the ratio between straight and curved length of a fiber (i.e., the waviness of 1.0 denotes a straight fiber) and reported data 8,11,13,14 led to the right plot F I G U R E 3 Biaxial mechanical behavior of the healthy porcine aortic wall. Curves represent the first Piola-Kirchhoff stress versus stretch properties recordings form planar biaxial tissue testing. 10 Red, blue, and green curves represent have been acquired upon the prescription of the displacement ratios 1:5, 1:1, 5:1 along the circumferential and axial vessel wall direction, respectively stress versus stretch response is essentially linear. 15 The Phase III corresponds to the in-vivo deformation of the vessel wall, where adventitial collagen mechanically engages and brings about the progressive increase of stress with stretch. Phase IV, the final phase, relates to supraphysiological deformations. It is described by damage-related softening of the vessel wall, with failure occurring at circumferential stretches of approximately 1.7. Given that collagen is known to yield between 20% and 40% deformation, [16][17][18] the macroscopic stretch, that is, the stretch at the tissue level, may not be directly transferred to the microscopic stretch of a collagen fibril. The shear deformation of the interfibrillar matrix therefore may account for a significant portion of the total deformation.

| AAA disease
An aneurysm is the end-result of the irreversible pathological remodeling of the vessel wall. 19 It is characterized by the significant degradation of medial elastin, that is then replaced by collagen produced by SMCs. 20 More specifically, larger aneurysms show distinct pathological features, 21-24 such as • degradation and fragmentation of elastin fibers, • apoptosis of vascular SMC, • increased collagen content and collagen synthesis, • excessive inflammatory response, • increased oxidative stress Whilst the loss of elastin (and possibly also SMCs) triggers the initial dilatation, it is the collagen turnover that determines the enlargement and local weakening of the wall, which eventually leads to rupture. Given its high clinical relevance, the aneurysmatic infrarenal aorta has been studied extensively, with a number of pathophysiological mechanisms having been proposed. Figure 4 provides a summary of said mechanisms, which ultimately results in the destruction of the well-defined structural organization of the normal vessel wall. The AAA wall exhibits a degraded media with few SMCs, fragmented elastin structures, and an inflammatory and/or fibrotic adventitia that can be thicker than normal. Mast cells in the adventitia trigger degranulation and release different vasoactive factors linked to neovascularization. 25 The "perforation" by a large number of vasa vasorum vessels could then explain the diminished strength of the aneurysmatic vessel wall, 26 data that has been contradicted by a more recent study. 27 It is often challenging to clearly distinguish between medial and adventitial layers in large AAAs, as the entire wall resembles a fibrous collagenous tissue very similar to the adventitia in the normal aorta. 28 Almost all AAAs of a clinically relevant size contain an intraluminal thrombus (ILT), 29 a tissue that develops from the gradual deposition of blood-borne factors. It has solid-like mechanical properties 30,31 and its formation may be promoted by disturbed blood flow. 32 F I G U R E 4 The intra-luminal thrombus (ILT) and vessel wall of an abdominal aortic aneurysm (AAA). The ILT promotes proteolytic and oxidative activities and facilitates the breakdown of the extra cellular matrix, apoptosis of vascular smooth muscle cells (SMCs) and the activation of immune responses. This process also activates MMPs, such as MMP 8 and MMP 9, which together with other substances are transported into the vessel wall. The AAA wall is characterized by the depletion of SMCs, the fragmentation of elastic fibers, numerous inflammatory responses, and the presence of mast cells in the adventitia that promote neovascularization The ILT is composed of a fibrin mesh, traversed by a continuous network of interconnected canaliculi and contains blood particles, such as erythrocytes, neutrophils, aggregated platelets, blood proteins, and cellular debris, see Figure 4. In addition to vessel wall hypoxia, 26 the ILT creates an environment for increased proteolytic and oxidative activity, 33,34 possibly linked to the weakening 26 and thinning 21 of the vessel wall. The reaction products are advected with the fluid flow towards and then into the wall, where they contribute to ECM degradation and the adventitial immune response. Proteolysis that is catalyzed by proteases can degrade fibrillar ECM and intermediate adhesive proteins. It then provokes SMC detachment and apoptosis, 19 such that neither ECs nor SMCs can spread and proliferate in close contact with the ILT. It may perhaps explain the absence of an endothelium in the ILT-covered AAA wall.
As a consequence of the pathological alteration to vessel wall composition, the biomechanical properties of the AAA wall are very different from the normal aorta. 35,36 Elastin degradation results in a more non-linear response, with collagen engagement at lower stretches, that is, the "knee" region of the stress strain curve appears at lower stretches 35 as compared to normal aortic wall. Figure 5A illustrates this effect by summarizing an experimental study reported in the literature. 16 The less wavy organization of collagen fibers in the aneurysmatic wall 39 as compared to the normal vessel wall, 11 may explain this mechanical property. It leads to an almost instantaneous engagement of collagen and the respective sharp transition between soft and stiff vessel wall properties. Whilst the AAA vessel wall shows highly nonlinear stress-strain properties, the ILT is effectively linear, 30,31 see Figure 5B. Another factor concerns the directional dependence of AAA mechanical properties. Aneurysmatic disease leads to a collagen orientation that is characteristically less anisotropic than the normal aorta, 35,37 a factor that may be related to the more bulge-like geometry of an AAA, as compared with the cylindrical shape of the normal aorta.

| AAA BIOMECHANICAL MODELING
The importance of biomechanical modeling in AAA rupture risk assessment is steadily increasing. As with any mechanical model, an AAA rupture risk model represents the real aneurysm or rupture process up to the required degree of complexity. Model complexity is determined by the intended model application (IMA), a factor that guides model development and testing. The modeler should always try to keep a model as simple as possible. Is the model correct? To a great extent this is an irrelevant question, instead one should ask: Does the model fulfill its intended task? Is it fit for purpose?
A biomechanical AAA rupture risk assessment, such as the one shown in Figure 6, requires several assumptions to be made in the estimation of a real-world problem. This section summarizes the most common approximations that are made in AAA biomechanics.

| Geometry acquisition
An accurate description of AAA geometry is one of the most important factors towards the robust prediction of stress in the vessel wall. 40 A high resolution of the imaging modality is therefore crucial in keeping the variability of simulation F I G U R E 5 Elastic properties of infrarenal aortic tissue exposed to equi-biaxial tension. (A) Comparison of aged normal abdominal aortic tissue with the abdominal aortic aneurysm (AAA) wall. Properties were determined from in-vitro experimental tissue characterization. 35,37 (B) Intra-luminal thrombus (ILT) of the AAA. Data has been acquired from in-vitro experimental tissue characterization of the luminal and abluminal ILT sections 31,38 results low. It may explain why the majority AAA models have been generated from high-resolution computed tomography-angiography (CT-A) images. CT-A is also an established clinical imaging modality in aneurysm patient treatment and most protocols carry out image acquisition with an in-plane resolution of below 1 mm and an out-ofplane resolution of below 1.5 mm. Whilst magnetic resonance (MR) imaging at such high resolution is not clinically feasible, it allows for the discrimination between vessel the wall and ILT. CT-A cannot robustly provide this information in individual patients and the modeler is then forced to approximate the interface between the wall and ILT based on mean population data. The assumption of a constant wall thickness of 2 mm has frequently been applied, whilst the analysis of a large number of studies point towards a AAA wall thickness of 1.58 mm (SD 0.64 mm). 41 The wall thickness is, however inhomogeneous 21,42-44 and is influenced by several factors. Some models have considered a thinner wall behind a thick ILT, 45 a factor that increased the predictability of AAA rupture in a retrospective cohort. 45 In the clinical environment, non-gated CT-A is usually applied, it is then unknown as to which phase in the cardiac cycle the recorded images belong. Whilst this is potentially problematic with respect to the thoracic aorta whose diameter pulsates by approximately 10%, it presents as less of an issue for the infrarenal aorta, and especially the AAA, which is comparatively much stiffer, showing cyclic circumferential strain of below 2%.
The continued development of the medical image analysis field has equipped us with a variety of methods and software's to segment AAA anatomies from the remaining information in medical images. Whilst threshold-based approaches can robustly segment the contrast-enhanced lumen in CT-A, difficulties often present when segmenting the AAA's exterior boundary, as the AAA is almost always (partly) surrounded by tissue of very similar appearance in the images. In contrast, active segmentation models 46 exhibit much higher success rates and require only minimal manual interaction to accurately segment an AAA from CT-A data. In addition, the application of artificial intelligence methods may also contribute to a more accurate segmentation in the future.

| Experimental tissue characterization and its relation to AAA rupture
The biomechanical analysis of AAA requires the constitutive description of both the vessel wall and the ILT. The development, calibration and validation of such models has almost exclusively been based on in-vitro experimental tissue characterization. Tissue samples that are harvested either during autopsy or open surgery are mechanically tested towards the extraction of elastic and/or failure tissue properties. Most frequently, uniaxial tensile tests have been F I G U R E 6 Reconstruction of an AAA from CT-A images (left) towards the computation of wall stress (right). Finite deformation models, calibrated to mean population data, determine the biomechanical properties of vessel wall and the intra-luminal thrombus. Analysis performed with A4clinicsRE (Vascops GmbH) and took approximately 10 min on a standard Laptop computer AAA, abdominal aortic aneurysm; CT-A, computed tomography-angiography performed with samples either aligned along the circumferential, 27,47-49 axial, 44,50-57 or both 51,58,59 directions. The individual layers of the aneurysmal wall have rarely been characterized. 60 Given the large variability in strength data, no statistically significant difference was observed in the wall strength of AAA samples aligned in the axial or circumferential directions. 51,59 The mean vessel wall strength, however, appeared consistently higher in the circumferential direction and all the aforementioned studies did not adjust for inter-patient or intra-patient variability. In contrast, a recent study adjusted for inter-patient variability and demonstrated that the mean strength in the circumferential direction is approximately 400 kPa higher when compared to the axial strength of the aneurysmatic aorta. 58 The majority of AAA ruptures appear perpendicular to the circumferential direction. 44,51 The rupture defect then materializes longitudinally, as a consequence of the circumferential stress in the wall. It indicates the importance of robust estimates for AAA circumferential wall strength with respect to the prediction of rupture.
Most experimental tissue characterization stems from the anterior segment of AAA, the part accessible for tissue harvesting during open surgery. Strength data from other AAA segments therefore remains rather sparse, 44,51,61 as such we still have an incomplete picture concerning the wall strength distribution over the entire aneurysmatic aorta.
Uniaxial tensile tissue testing is a simple and robust protocol that allows for the exposure of tissue to stresses well above physiological levels. The testing of soft biological tissues requires preconditioning in order to achieve reproducible mechanical properties. If the desired outcome is the acquisition of elastic tissue properties, preconditioning should cover the same deformation (stress) domain within which the elastic properties are then characterized. The deformation (stress) level covered by preconditioning of a failure experiment is however unclear, and preconditioning between 0 and 200 kPa before stretching up to failure, 27,48,62 is recommended for the cross-comparison of the acquired data.
Uniaxial tensile testing exposes the tissue sample to a very different mechanical environment than the vessel wall experiences in vivo. Given that circumferential and axial stresses are the two dominating principal stress of the vessel wall, planar biaxial tensile testing 35,36,63,64 has been proposed in the mechanical characterization of AAA tissue, permitting a wide range of circumferential and axial stress combinations to be explored. However, the mounting of the specimen in the testing system limits the analysis to stress levels below approximately 100 kPa, a range that cannot cover peak systolic conditions. In comparison to uniaxial tensile testing, in biaxial tensile testing the AAA wall appears much softer at low strain levels (see Figure 7), an inconsistency that is difficult to explain with phenomenological constitutive models. 65 Bulge inflation is another biaxial loading protocol that allows for approximately equi-biaxial loading up to failure of the vessel wall. It has been used in the exploration of the thoracic aorta 66 but not yet applied to the study of the infrarenal aorta. In addition to the very different loading protocols, no standards exist for the post processing of the experimental raw data. The cross comparison of the experimental data is then not directly possible, and the development of test standards in soft biological tissue testing aims at overcoming the present short comings in AAA wall characterization. 67 F I G U R E 7 Predicted stress in the AAA wall at equi-biaxial tension. Blue and green curves represent predictions of models that have been calibrated to experimental data from uniaxial and respectively planar biaxial tissue characterization. Data represents the mean response according to the reported sets of parameters. Isotropic AAA wall models are represented by single curves. Anisotropic models are illustrated by their circumferential and axial stress responses, with area shaded in between. AAA, abdominal aortic aneurysm

| Constitutive modeling
Vascular tissue displays complex mechanical properties, such as finite deformation, stress-strain nonlinearity, anisotropy, and rate-dependency. 68 Not all of these factors are of major importance with respect to the modeling of AAAs, this section therefore presents the most common methods and underlying assumptions that have been proposed to describe AAA tissue.

| Wall
A linear elastic description of the vessel wall may only be used for an AAA with a low curvature and that presents without an ILT, conditions that rarely apply in practice. 64 As such, hyperelasticity is a frequently applied framework in the modeling of the AAA wall, due fundamentally to its ability to describe non-linear stress-strain tissue properties. The 2nd order Yeoh 69 strain energy density function (SEDF) is still the most widely used constitutive model, where I 1 denotes the first invariant of the right Cauchy-Green strain, and c 10 ,c 20 are stress-like material parameters identified from experimental data. The model was first proposed to describe the AAA wall in 2000, 40 with uniaxial tensile testing used to fit model parameters. A further study 70 from the same group, calibrated the Yeoh SEDF to largely unpreconditioned uniaxial failure curves, 59 an exercise that resulted in mean values of c 10 ¼ 174 kPa and c 20 ¼ 1881 kPa, respectively. Fifteen years later another study 56 from a different group identified the parameters c 10 ¼ 210 kPa and c 20 ¼ 4820 kPa, resulting in very similar stress-strain properties of the AAA wall. Conversely, another study recently used preconditioned uniaxial tensile data and observed a more pronounced nonlinearity in the mechanical response of the AAA wall, having identified the parameters c 10 ¼ 56 kPa and c 20 ¼ 4329 kPa: 27 This is likely a consequence of the preconditioning performed during the acquisition of experimental data. Biaxial tissue testing results in a highly non-linear stress strain response of the AAA wall, data that is difficult to capture with the 2nd order Yeoh SEDF. The 5th order Yeoh SEDF has therefore been proposed to characterize AAA tissue. 71 The calibration to planar biaxial experimental data resulted in the constants c 10 ¼ 5 kPa and c 20 ¼ c 30 ¼ 0, c 40 ¼ 3700 kPa and c 50 ¼ 13, 740 kPa. As compared to the aforementioned sets of parameters, a much more non-linear stress strain response is seen. Whilst aneurysmal disease damages the structural organization of the vessel wall, and thus the root cause of anisotropy, the AAA wall still shows mild anisotropy. The anisotropic SEDF. 72 proposed by Choi and Vito has therefore been used to describe the AAA wall, 35,37 as a function of the circumferential E 11 and axial E 22 Green-Lagrange strain components. The parameter 37 have been identified by two different groups and yield approximately the same biomechanical description of the AAA wall. As the SEDF is only a function of the normal strain components, the model predicts no shear stress in the reference configuration. Whilst shear strain components could easily be included as arguments of the SEDF, the related parameters need to be identified and the poly-convexity of the SEDF needs to be ensured. The material parameters in (3) cannot be independent from each other, see the analysis of the Fung SEDF elsewhere. 73 Whilst a number of other anisotropic SEDF's have been proposed for the vessel wall, 68 the relevance of anisotropic wall models in AAA biomechanics is limited. An anisotropic model requires the specification of the circumferential and axial vessel direction, an exercise that is often difficult given the complex AAA shape.
In addition to hyperelastic continuum models, the general theory of fiber-reinforced materials has also been used in AAA biomechanics. It allows for a structure-based description of biological tissues 74 through the introduction of a PDF that describes the fiber orientation in the undeformed configuration, that is, as a function of the referential direction vector M with jMj = 1. 68 The integration over the hemisphere À π 2 ≤ ϕ ≤ π 2 ; À π 2 ≤ θ ≤ π 2 È É then results in the vessel wall's Cauchy stress where, m ¼ FM denotes the push forward of M, and dev Á ð Þ ¼ Á ð Þ À I : Á ð ÞI=3 is the spatial deviator operator of the threedimensional problem. Here, σ λ ð Þ expresses the Cauchy stress of a fiber as a function of the fiber stretch λ and therefore prescribes the constitutive properties of the fibers. Such a 1D constitutive model eases the description of the detailed load-carrying mechanisms at the fiber level. The hydrostatic pressure p acts on the identity I and serves as a Lagrange parameter to enforce incompressibility.

| Collagen
Given the low quantity and disrupted nature of the elastin present in AAA, collagen remains as the principal load carrying protein. The load transmission of collagen in the vessel wall is therefore of importance in the assessment of AAA rupture and has consequently gained significant interest. Collagen fibrils (and additionally the fibers which they constitute) are undulated in the unloaded vessel wall; upon loading they gradually straighten and begin to bear load. It is this physiological feature that determines the strong nonlinear stiffening of the AAA wall at increasing macroscopic strain. Studies commonly assume that collagen fibrils are slender structures that can only carry tensile stress. Following its engagement, an individual collagen fibril may obey a linear Cauchy stress versus strain law according to where, λ f is the stretch of the collagen fibril. It is kinematically defined as λ f ¼ λ=λ st , where, λ st is the straightening stretch of a fibril, that is, the fiber stretch at which the collagen fibril becomes engaged. The total collagen fiber stress is then determined as the mean stress of all resident collagen fibrils. If the straightening stretch, λ st , is distributed according to a PDF, f λ st λ st ð Þ, then the fiber stress is expressed as Multiple variations of this approach are known, with the Cauchy-Lorentz, log-logistic, beta, and triangular distributions 68 having been used to characterize collagen engagement. In vivo, vascular collagen is continually synthesized and degraded by fibroblasts and SMCs, with a half-life on the order of 2 months in the normal vessel wall. It is thought that a deterioration or malfunction in this mechanism could contribute towards the aneurysmatic expansion of the vessel wall. Others have further extended the aforementioned framework to incorporate the mechanical homeostasis of collagen; the physiological tendency for the collagenous ECM to adapt towards a steady and stable internal environment. From a mechanical perspective, the effective characterization of collagenous remodeling for such a microstructural approach requires that two features be accounted for, namely the change in the relative density of collagen ρ and the adaption of its referential configuration, that is, the As a homeostatic measure, the physiological stretch λ ph is defined as the fiber stretch at which a specific percentage of fibrils are engaged according to f λ st λ st ð Þ. In the normal vessel wall for instance, approximately 6%-7% of collagen fibrils are engaged under physiological deformation. 75,76 The mechanical stimuli 41,77 : then informs how a given collagen fiber should remodel. In the case of ζ > 1, the fiber is stretched beyond its preferential homeostatic state and thus experiences a supraphysiological deformation. In the case of ζ < 1, the fiber contrarily experiences a subphysiological deformation. With respect to the change of the collagen density, a set of equations dictate the respective rates of synthesis, degradation and net change in collagen, where, _ ρ þ max is the maximum allowable synthesis. It is representative of an upper threshold for collagen production by fibroblasts and SMCs. 77 At supraphysiological deformations, collagen synthesis outweighs degradation, leading to a net increase in collagen fiber density. Conversely at subphysiological deformations degradation exceeds synthesis, causing an overall reduction in collagen fiber density.
Concerning the adaption of a fiber's referential configuration, weighted decomposition theory is used to adjust the straightening stretch distribution f λ st λ st ð Þ according to the mechanical stimuli ζ: 41 At supraphysiological deformations, the bounds of the distribution therefore move to higher stretches, meaning that in the undeformed state, collagen fibrils will appear more undulated. Once again, the opposite is evidently true of subphysiological deformations.
Application of this framework to the patient-specific AAA showcased its ability to avoid non-physiological stress gradients 41 and furthermore predictions of the growth of small AAA displayed typical features of AAA expansion reported from clinical studies. 77,78 Beyond such a collagen-related microstructural approach, the constrained mixture theory [79][80][81] and other concepts have been used to model aortic arteriogenesis. [82][83][84][85][86] Collagen deformation alone cannot fully explain the large deformations of the aorta at the tissue level, and a considerable amount of deformation has to be linked to the matrix shearing and/or the deformation of the interface between collagen and non-collagenous ECM. Microstructural models [87][88][89] have been developed with the aim of understanding such effects in the transmission of stress in soft biological tissues, and as such have incorporated a number of inelastic properties observed in tissue characterization experiments.

| ILT
The ILT itself is able to carry mechanical stress, as such it can have a protective effect, causing the unloading of the vessel wall situated directly beneath it. 90,91 Conversely and perhaps more importantly it leads to hypoxia, which acts to weaken the wall, most likely through upregulated neovascularization. The vessel wall is then perforated due to excessive vasa vasorum and eventually ruptures at lower stress levels. 26 Which of these factors that then dominates is dependent upon the AAA wall and ILT anatomy in question, an aspect that has prompted the in-vitro characterization of the ILT.
Whilst ILT tissue has been found to be isotropic, its properties are a function of the distance from the lumen, with different constitutive descriptions having been proposed. The Mooney-Rivlin SEDF has been used to describe ILT tissue, 92  In addition, the luminal layers of ILT tissue were characterized through preconditioned planar biaxial tensile testing and then calibrated to the Yeoh SEDF (1). 30 The study identified c 10 ¼ 7:98 kPa and c 20 ¼ 8:71 kPa, a much more compliant response as compared to the aforementioned uniaxial tissue characterization. These ILT properties have been corroborated by another study 31 based on preconditioned uniaxial tensile testing. The experimental data was calibrated to an Ogden-type SEDF where, λ i are the principal stretches, and the material parameter c was identified as 2.62, 1.98, and 1.73 kPa for luminal, medial, and abluminal ILT layers, respectively. As with these studies, another investigation of preconditioned biaxial tissue testing 38 confirmed the planar isotropy of ILT tissue, despite no attempt being made to fit a constitutive model to the experimental data. Regardless of the strong evidence for the ILT being an isotropic material, one study 36 suggested an anisotropic SEDF With the constitutive parameters describing the tissue identified as μ ¼ 9:7 kPa; 7:1 kPa; 5:1 kPa, k 1 ¼ 15:9 kPa; 6 kPa; 2:9 kPa, k 2 ¼ 2:7; 0:07; 0:03, ρ ¼ 0:33;0:05;0:05 and φ ¼ 84:1 ;86:7 ;89:1 . These parameters result, as one would expect given the aforementioned evidence, in an almost isotropic tissue description.

| Constitutive model-independent wall stress description
A membrane stress state has also been used in the description of AAA wall stress. 93,94 The Laplace equation then describes the problem, where, σ θ and σ z denote the circumferential and axial Cauchy stress, whilst r θ and r z are the respective radii. Here, the blood pressure is denoted by p, and t expresses the AAA wall thickness. It represents a statically determined problem, and the stress is then independent from material properties of the vessel wall. The application of the Laplace equation in the AAA wall stress analysis is possible for an AAA with a low wall curvature and without an ILT, conditions that rarely apply. 64 In most AAA the wall curvature is, at least in some areas, comparable to wall thickness. 44,49 Membrane theory is then not applicable, such that the shear stresses across the vessel wall contributes to the mechanical problem and so must not be neglected. Moreover, a thick ILT layer often covers the wall, and again prevents from the application of the thin wall assumption. 95

| Blood pressure
The aorta is exposed to pulsatile blood pressure, and the corresponding Neuman boundary condition strongly influences the stress state of vascular tissue. Whilst most models apply the pressure at the luminal boundary to the structure, both experimental 96 and numerical 97,98 studies have shown that the pressure propagates through the ILT and acts, with practically the same intensity, directly upon the aortic wall. It is the high porosity of the ILT that allows the fluid pressure to pass almost without resistance towards the wall. However, the ILT has also solid-mechanical properties and develops stress upon deformation, which then unloads the vessel wall underneath it. Although the ILT is much softer than the wall, it may be several times thicker than the wall, and carry a significant portion of the structural load. Whilst hypertension is known to be an independent risk factor for AAA rupture, 99 it is unclear whether the mean pressure, systolic pressure or pulse pressure causes most harm to the vessel wall. Furthermore, the failure of vascular tissue has not yet been explored in significant detail; the contribution of mean stress, peak stress and stress amplitude to the failure of the aortic wall is generally not understood. It is therefore unclear which blood pressure should be prescribed in an AAA rupture risk study. The first of such studies used the systolic blood pressure 100 or even the highest systolic blood pressure occurring over a year, 101 an approach that has since been generally supplanted by the prescription of MAP. Given the lack of, and the high uncertainty of, MAP measurements, a number of studies have used mean population MAP, see Table 1. The relation between wall stress and blood pressure in AAA biomechanics is non-linear, a factor that further complicates the relation between blood pressure and AAA rupture risk.
Based on the proposition that AAA wall rupture risk is linked to the peak in blood pressure, a pressure of 1.5 times the MAP was used to assess AAA rupture risk. 71 Whilst this figure lacked evidence, a recent campaign has aimed at acquiring more robust pressure data 117 , to be considered by the estimated annual risk of rupture index, see Section 4.2.

| Perivascular support
The adventitia anchors the aorta to surrounding tissue, as such the outer vessel boundary conditions are important to consider with respect to fluid structure interaction (FSI) studies of the aorta. 118 Regarding quasistatic simulations, perivascular support may not play an important role and most such studies prescribe a traction free outer vessel boundary.

| Stress-free reference configuration
At the time of medical image acquisition, a blood pressure is present, and thus the recorded aortic geometry is experiencing a mechanical stress. The majority of finite deformation analysis methods require the definition of a stressfree reference configuration, a configuration that in this case differs from the geometry obtained via medical imaging. Various approaches have been proposed to back-calculate an aneurysms reference configuration from medical image data, such as the backward incremental method 119 and the solution of the inverse problem. 50 With the backward incremental method, the image-based geometry serves as the initial reference configuration. The forward problem is solved, and the predicted displacements are then extracted from the reference configuration. The update is successively applied until the pressurized configuration approximates the image-based geometry with sufficient accuracy. Modifications of this approach have also been proposed 120,121 to improve its efficiency and robustness. It is however generally more convenient to solve the inverse problem, 122 which directly provides the wall stress. Whilst any commercial FE program supports the backward incremental method, solving of the (less common) inverse problem may not be directly be supported.
Both of the aforementioned approaches assume the segmented geometry to be in equilibrium. Given the AAA model will always approximate the real problem, equilibrium of the model is not equal to equilibrium of the real problem. Factors such as forces from the contact of the aneurysm with the spine and other surrounding organs are commonly not considered in the AAA model. It is therefore not surprising that both of the aforementioned approaches commonly predict local highly unrealistic stress in the vessel wall, artifacts that then need to be excluded in the data interpretation.

| Residual stresses
Vessel wall tissue is constantly synthesized and degraded at its in-vivo configuration, a process that allows the vessel to optimize its mechanical properties. 123 The continuous turnover of tissue constituents (arteriogenesis) results in residual stresses at the vessel's load-free configuration, a stress state that has attracted much scientific interest. 124 Residual stresses in the aorta appear in the circumferential, 89,91,125 and axial 126 directions. It has been hypothesized that residual stresses result in a homogenized stress across the vessel wall, or across an individual vessel wall layer, 89,91 such that the structure is then optimized for load-carrying.
Given the fundamentally different nature of growth-based and elastic deformations, the overall deformation is multiplicatively decomposed 127,128 subject to where, F g and F e are the growth-related and the elastic deformation gradient, respectively. For isotropic growth, F g ¼ 1 þ c ð ÞI expresses the residual stress state in the AAA wall, where c denotes the amount of local growth, 129 whilst F e is determined through the linear momentum. The problem is then identical to a thermal-elastic coupled system, where the (isotropic) thermal expansion would relate to tissue growth. The growth parameter c may be determined through the iterative minimization 126,127 of the stress across the wall, and thus the implementation of the homogeneous stress hypothesis. 89,91

| Spatial discretization and FE descriptions
At in-vivo conditions, the deformation of the vascular wall is incompressible, 130 and an adequate numerical discretization approach is therefore needed for efficient stress computation. Simple tetrahedral meshes 120,131 are not optimal and hexahedral 45,71,95,115,132,133 , or hexahedral-dominated meshes 107,134 , are known to perform much better. Furthermore, mixed finite element formulations, such as the Q1P0 elements 135 are most effective in the approximation of incompressible deformations. Interestingly, AAA biomechanics studies are not particularly sensitive to the discretization of the ILT. 120,136 Whilst Delauny-based algorithms directly provide tetrahedral meshes, more sophisticated algorithms, 46 that often include manual interactions, 71,137 are required in order to generate hexahedral meshes. A number of factors determine an effective FE mesh for AAA biomechanics studies. The non-linearity of the material model, and whether or not the stress gradient across the wall is important to be captured, strongly influence the required mesh density. 136 For clinical applications, a numerical error of approximately 5% is commonly accepted. Whilst a vascular biomechanics study should always include a mesh sensitivity analysis to verify mesh convergence, standard AAA rupture risk studies require roughly 10k to 50k Q1P0 finite elements, which compares to 200k to 1m for linear tetrahedral finite elements. For quadratic elements, approximately half of these numbers are required. 129,131

| Parameter sensitivity
There is always a degree of uncertainty associated with the input information of vascular biomechanical studies, a factor that needs to be carefully investigated to assess the robustness of the conclusions that can be drawn. Whilst the aortic geometry is one of the most important considerations, many other parameters influence the results of a vascular biomechanical study, some of which are discussed below.

| Effect of material model
The influence of the material model in AAA biomechanics studies needs to be discussed at a problem-specific level. For a statically determined problem, the stress in the aorta is independent from the material model 94 and the constitutive model only influences the strain in the aorta. Many vascular studies are however not statically determined, and the description of a material model therefore influences the results at the stress level.
Given the clinical relevance of AAA rupture, a number of studies have investigated the variability of PWS with respect to the constitutive description of the wall 40,64,95,134,138 and the ILT. [139][140][141] From a clinical perspective it is one of the most important parameters, however its use as a comparative measure disregards the spatial variation in stress throughout the vessel. The classification of stress levels, 64 allows for a more comprehensive comparison of stress predictions based on different modeling assumptions. The most comprehensive study to date 95 compared wall stress predictions from five different constitutive AAA wall models to preconditioned planar biaxial tensile tissue data. 35 It concluded that the non-linearity of the vessel wall had a pronounced influence on stress predictions, thus highlighting the importance of effectively characterizing the transition from soft to stiff properties (i.e., the "knee" region of the stress-strain curve) of the vessel wall. In the AAA wall, this region appears at strain levels of approximately 4%-11% 35 and is closely associated with the mechanical activation of collagen in the vessel wall. In contrast to these observations, the actual magnitude of the stiffness did not play an important role in the prediction of wall stress.
Whilst a linear elastic wall description should not be used in AAA studies, the wall stress insensitivity from the actual stiffness of the vessel wall allows modelers to use mean population data; it does not result in systematic errors as compared to the use of patient-specific data.
The definition/measurement of the material principal axes is challenging for complex vessel geometries and therefor only a small number of AAA biomechanical studies have used an anisotropic vessel wall description. A four-fold increase in PWS with an anisotropic model as compared to an isotropic wall description has been reported. 138 Whilst both models have been calibrated to the same experimental data, the "more flexible" anisotropic model resulted in a more pronounced non-linearity. A highly non-linear constitutive model then leads to a highly inhomogeneous wall stress, a factor that then could explain the observed difference in PWS.

| Effect of the intraluminal thrombus
Whilst the stiffness of the ILT is approximately 10-fold smaller than that of the vessel wall, it often covers a large volume of the aorta and as such is a structural component that should not be neglected in AAA biomechanics. Computational [141][142][143][144] and experimental 145 studies have consistently shown that the presence of an ILT acts to decrease wall deformation and stress. This observation does not conflict 97,98,146 with the observation that the blood pressure propagates almost undiminished towards the vessel 96 through the porous ILT.
Whilst the stress-buffering effect of the ILT likely protects the AAA from rupture, ILT-related wall tissue degradation could diminish the strength of the wall and increase the risk of rupture. Earlier biomechanical studies 27 support this hypothesis, but a more recent study did not confirm the presence of a lower wall strength behind a thick ILT. The effect of the ILT on wall strength therefore remains unclear. However, even if the presence of the ILT has no influence on wall strength, the wall is thinner behind the ILT 27,62 and will then rupture at a lower traction, that is, stress times wall thickness. 111

| Effect of the load-free configuration
The forward analysis of wall stress computations requires that the stress-free reference configuration be predicted from the geometry segmented from medical images. Given the complex AAA geometry, the stress-free configuration is not compatible 68 and therefore usually replaced by the load-free configuration of the AAA. The influence of the load-free configuration on wall stress predictions has been explored repeatedly 120,134,147 but remains somewhat linked to the aforementioned influence of other modeling assumptions, especially the constitutive description. Studies report an increase in PWS of 42% 147 and 21%, 120 and have also concluded that the impact is highly patient-specific. As discussed in Section 3.5, the predicted stress may show a rather high (unrealistic) bending component, a factor that could explain this elevation in PWS.
Given a statically determined problem, AAA wall stress would be independent from tissue properties. An overly stiff wall could then be used in FE models and the load-free geometry would be identical to the geometry seen in the image data. 94,148 However, this argument does not hold in general as AAA wall stress computation is only, in a small minority of cases (those presenting with no ILT and a low wall curvature 95 ), a statically determined problem.

| Effect of residual stress in the load-free configuration
The neglection of residual stresses in the load-free configuration leads to stress gradients across the wall at in-vivo loading of the vessel. The magnitude of the gradient depends on the material model and the wall thickness. Given that the AAA wall is relatively thin, the stress difference between the inside and outside of the wall remains below 10% for the most widely-used AAA wall model. 40 A characteristically more non-linear material model results in a greater stress difference. 64,129,136 It is believed that a membrane stress state determines the stress in the vessel wall, 89,91 as such a stress gradient across the vessel wall, or an individual vessel wall layer, is likely not physiological. The incorporation of residual stresses can provide such a membrane stress state. Membrane vessel wall models may therefore be a pragmatic way to avoid difficulties in the prescription of residual stresses. Alternatively, a single linear finite element across the wall, such as the Q1P0 finite element, results in a virtually homogeneous stress through the wall thickness.

| Effect of calcification in the vessel wall
With regard to the in-vitro characterization of the AAA wall, one is often forced to use homogenous vessel wall specimens, and patches that include large calcifications have to be excluded. The experimentally acquired data therefore represents calcification-free tissue, and a number of theoretical biomechanical studies 90,149,150 have explored the influence of large calcification pockets on wall stress. Conclusions remain, however, controversial and indicate that PWS may increase 90,149 or decrease 150 with the presence of calcifications. Moreover, studies suffer from unrealistic material properties for calcifications 90 and a very low number of considered cases. 149,150 Currently, all studies have considered calcification pockets to be a homogeneous material with an abrupt change in material properties relative to non-calcified tissue. This is however unrealistic, and more refined assumptions could lead to different conclusions concerning the role of calcifications.

| Effect of blood pressure
Whilst the blood pressure, within a reasonable range, is approximately linearly linked to wall stress, a normal aorta at higher MAP also has a thicker wall. 151,152 The influence of blood pressure on wall stress is therefore not trivial and may explain contradictory findings in the literature. Given the lack of knowledge concerning the failure mechanisms of the vascular wall, it remains unknown which blood pressure should be prescribed in biomechanical AAA rupture risk studies. Whilst a number of studies succeeded in the discrimination of ruptured and intact AAA by using MAP, 45,108,109 others 100,101,104,106 required the prescription of peak systolic blood pressure or an elevated blood pressure. 71 Both would be in accordance with the clinical observation that hypertension is a risk factor for aortic rupture. 99 The definition of stochastic-based risk indices 62,71 is another approach (see Section 4.2) and leads to much more sensitivity with respect to blood pressure.

| Effect of wall thickness
The thickness of the aorta is spatially inhomogeneous [42][43][44] and the wall thickness of a AAA negatively correlates with the thickness of the underlying ILT layer, 21,27 a factor that is important for successful discrimination of intact and ruptured AAAs. 45 Whilst the wall stress is inversely proportional to the wall thickness, a thinner wall in intact AAAs has a higher strength, 27,58,153 factors that compete with one another. Modelers that consider an inhomogeneous wall thickness should at the same time also prescribe an inhomogeneous wall strength in order to adequately reflect rupture risk.

| Effect of AAA growth
Whilst fast growth is commonly used as an AAA repair indication and biomechanical stress may influence AAA growth, 41,[154][155][156][157][158] all reported biomechanical rupture risk studies neglect the dynamics of the growing vessel wall. More clinical evidence and in-vitro tissue characterization would be needed to properly and effectively incorporate this effect within the biomechanical rupture risk assessment and some preliminary ideas have been reported elsewhere. 41

| BIOMECHANICAL RUPTURE RISK ESTIMATORS
In a biomechanical rupture risk assessment, AAA rupture appears in the event that the wall stress exceeds the strength of the vessel wall. Our understanding of failure in biological tissues stems mostly from the exploration of bone, [159][160][161][162] and the exploration of local failure mechanisms in soft tissue is limited to a few studies in skin. 163,164 It is therefore remains unclear which stress component, or combination of components, should be used in the design of a biomechanical risk assessment, that is, which failure hypothesis is to be used in (diseased) vascular tissue. The uncertainty surrounding the mechanical properties of vascular tissue is another inherent complexity associated with AAA rupture risk assessment. In spite of this incomplete information, a number of biomechanical rupture risk indices have been proposed to assess the individual risk of AAA rupture.

| Deterministic models
Wall stress naturally represents the loading of the vessel wall, and its largest value, the peak wall stress (PWS), has therefore been introduced as an AAA rupture risk index. 100 The stress state in the vessel wall is biaxial, and most studies have expressed the failure of the wall using the von Mises stress. 45,62,94 It is independent from the hydrostatic pressure and describes a plastic-like failure. In contrast, use of the largest principal stress assumes a more brittle-like failure of the wall. 71,101 The inhomogeneity in pathohistological findings 39 concerning the AAA wall, are indicative of a correspondingly inhomogeneous AAA wall strength, the local risk of rupture is then a function of stress and strength. The rupture risk potential (RRP) therefore relates the local stress to an estimated local strength 47 of the AAA wall. The largest value of the RRP has also been used as another, more refined biomechanical rupture risk index. 103 The peak wall rupture risk (PWRR) 45 is based on the same concept, and in addition, considers a strength reduction resulting from the fatigue of the vessel wall.
The finite element analysis rupture index (FEARI) 165 also concerns the relation of stress to strength, and uses constant AAA wall strength values assigned to eight different AAA segments.
The inverse relation between AAA wall thickness and strength 27,55,58 suggests that the traction, that is, stress times wall thickness, is a more suitable alternative to the stress itself. The peak wall tension (PWT) could consequently present as a more sensitive biomechanical risk index. 111 A similar result is achieved for PWS computed upon a AAA biomechanical model with constant, instead of the inhomogeneous wall thickness.
With the easiness of clinical interpretation in mind, biomechanical rupture risk indices may be explored with respect to the average AAA patient. 132 Such a relative measure results in an equivalent diameter, the diameter of a hypothetical AAA that experiences the same biomechanical risk as the actual case under consideration. It also allows one to link the biomechanical rupture risk assessment with the large diameter-based clinical AAA risk studies. 4

| Stochastic models
AAA biomechanical properties are uncertain, and recently a class of stochastic-based approaches have been proposed in the design biomechanical AAA risk indices. The AAA wall strength is one such uncertain parameter as there is significant intra-patient and inter-patient variability, 27,62 see Figure 7. Whilst the AAA wall shows a median strength of 825 kPa, 5% of wall samples would rupture already below 396 kPa.
For a given PDF, ρ σ ult , of AAA wall strength and the calculated PWS of an individual AAA, the probabilistic rupture risk index (PRRI) 71 PRRI ¼ expresses the risk of rupture. It outputs a value between zero and one, that represents the area beneath the PDF curve of Figure 8, between the wall strengths of zero and the respective PWS. This concept may additionally include any other uncertainty, such as the uncertainty in the wall thickness. 71 PWS then itself becomes a probabilistic variable with its own PDF, ρ PWS , and expresses the probabilistic risk of rupture. The review of experimental data showed the dependence of wall strength on PWS, a factor that has been used to refine the design of the PRRI. 62 Wall strength and wall thickness are inversely related, 27,55,58 and so it is of benefit that the joint probability distribution P PWS, σ ult ð Þbe used in the calculation of a probabilistic rupture risk index. Additional approaches towards a more refined computation of the probabilistic rupture risk have included the variability of the blood pressure. 166 Recently, the information that an AAA is intact at the time of examination has also been incorporated within the formulation of the probabilistic rupture risk. It permits the exclusion of certain combinations of wall thickness and wall strength, which resulted in a significant reduction in the uncertainty of the rupture risk 166 Whilst probabilistic risk indices support a direct interpretation of the results, deterministic risk indices require further calibration to clinical data. Which would then allow for the determination of the annual risk of rupture for a given risk index. 77

| VALIDATION
Validation is the most important, and often also the most challenging exercise of new bioengineering technology. Studies need to be well-designed to provide a clean test of causal connections between dependent and independent variables and to eliminate the influence of extraneous (or lurking) variables. 68 Randomization (using chance methods to assign study units) and replication (assignment of a large number of study units) are the most powerful tools to limit the influence of extraneous variables and ensure the study is not confounded. The validation of biomechanical risk indices is very seldomly performed blinded, 115 and the majority of investigators have been aware of the clinical outcome of the biomechanical study at the time it was being been performed.
Validation exercises must correlate with the IMA and are to be implemented at different levels. Biomechanical simulation pipelines are very sensitive to the individual operator who performs the study and specifies all modeling assumptions. The operator decides upon factors such as geometrical accuracy and constitutive descriptions of vascular tissue. Only very few simulation pipelines have been tested against this uncertainty and report intra-operator and interoperator variability of biomechanical indices. 167,168 Given the many constraints in the design of AAA rupture risk studies, none of the reported approaches in the literature are ideal and would be able to provide level 1 evidence in favor of the biomechanical AAA rupture risk assessment.

| What to compare?
An important aspect relating to validation is the assessment of the robustness of a risk index in the context of the clinical application. In addition to testing the statistical significance among cohorts, the receiver operating characteristics (ROC) curves are applied to rank risk indices. 62,101,115 However, simply comparing the area under the respective ROC curves between different risk indices is not sufficient, and does not fully reflect the clinical value of a risk predictor. Given that AAA rupture is one of the most severe cardiovascular events, the risk index must exhibit a sensitivity of almost 100% in order to best possibly avoid a false negative result. In contrast, a false positive result, whilst leading to the unnecessary repair of an AAA, would present much less harm to the patient. F I G U R E 8 Experimental (histogram) and fitted probability density function of wall strength relating to the largest available data set in the literature 62 5.2 | Retrospective observational studies comparing electively operated AAAs to ruptured AAAs A large number of studies retrospectively compared intact and ruptured cases in diameter-matched 45,71,[100][101][102][103][104]106,107,110,112,116 and non-diameter-matched 61,100,101,103,105,107,108,[111][112][113] cohorts. The adjustment for diameter aims at correcting for the strong correlation of PWS and diameter 82,119 ; a larger AAA will have a higher PWS, given that the wall thickness remains unchanged. However, diameter matching results in small cohorts artificially renders the problem independent from the diameter, one of the most important risk factors of AAA rupture. It is therefore desirable to perform the validation in non-diameter-adjusted (and as such larger) cohorts, and include a separate analysis of diameter-matched sub cohorts. 100,101,103,107,112 A power calculation may also be considered to a-priori estimate the cohort size and minimize type II error, 116 that is, the false rejection of the Null hypothesis and the prediction of a false negative case.

| Quasi-prospective comparison between ruptured and non-ruptured AAAs
Medical image data concerning patients with an AAA that was intact at the time of recording, yet ruptured at a later point in time (still in the past relative to the time of data processing), is a highly valuable resource in the validation of a biomechanical rupture risk assessment. A few such studies have been reported, 110,115 all with a relatively small number of cases due to the scarcity of this type of data. Given it is still a retrospective study, it has certain elements of a prospective study and is not flawed by the changes of the geometry at the site of the rupture, a draw-back of any study in Section 5.2. An important aspect is the time that passes between medical image acquisition and the point of AAA rupture. It is naturally much more difficult for a model to predict AAA rupture, the larger the discrepancy.

| Prospective studies and clinical trials
The strongest evidence is provided by prospective randomized trials. They are a priori double blinded and so far only one study of this kind has been published. 114 It provided evidence that counters the ability of biomechanical rupture risk indices to adequately predict AAA rupture. The biomechanical model used in this study was however not ideal and suffered from an overly simplified constitutive wall description and potentially large operator sensitivity. One other study, in favor of biomechanical rupture risk indices is about to be published.

| Summary of current validation exercises
A summary of AAA rupture risk studies is presented in Table 1, showing investigations with at least 10 patients. The rupture risk indices discussed in Section 4 have been tested, and only four studies 101,114-116 provided a threshold that would separate AAAs at low and high risk of rupture. Such a threshold criteria would be desired by clinicians in decision making. A threshold is always a tradeoff between sensitivity and specificity, and ROC curves as reported in studies, 62,101,115 then provide an ideal instrument to derive such thresholds.

| CONCLUSIONS AND FURTHER DIRECTION
The biomechanical AAA rupture risk assessment allows for the integration of many patient-specific risk factors and has been refined significantly over the past quarter of a century. AAA rupture is a localized event in the aneurysm wall, and global parameters, such as the maximum diameter, are limited in their ability to effectively indicate risk and therefore frequently fail to predict AAA rupture on an individual basis. In contrast, a biomechanical model allows for the estimation of the wall's local mechanical loading, that is, its risk of local rupture. Many biomechanical models are not sufficiently validated and therefore have not garnered significant clinician attention. A simulation model represents the real objective or process to the desired degree of complexity, and should be guided by clinical needs rather than by integrating all available (biomechanical) information to the given problem. 68 Whilst an isotropic elastic model that reflects the non-linearity of the AAA wall provides sufficiently accurate wall stress predictions, it is the strength of the aneurysmatic wall that requires much more attention. All reported data stems from uniaxial rupture tests, which does not accurately represent the in-vivo loading of the vessel wall. Moreover, research teams execute very different tissue characterization protocols, not supporting the cross-comparison of their results. Bulge inflation tests present as a more suitable alternative experiment, but so far this has only been applied to the thoracic aneurysm wall 66 as it supports the preparation of tissue samples that are of ample size. In addition, little is known concerning the actual failure mechanisms of the normal and diseased vessel wall. Fundamental aspects of fracture have not been explored and the most fundamental parameters in fracture mechanics, such as the fracture energy of the AAA wall are still unknown. The role of minor fractures and dissections may play a much more integral role in aortic aneurysms than originally thought, 169,170 but almost all research has been focused on the final wall fracture that determines AAA rupture.
Beyond an understanding of the passive interaction of structural components in the vascular wall, it is also of great importance to understand how the tissue responds and adapts to avoid rupture, that is, the growth and remodeling of the AAA wall. The biomechanical AAA rupture risk assessment is a generic approach that fully supports the integration of biochemical tissue activity, an aspect that has not attracted much attention in the literature.
The use of more accurate blood pressure loading in AAA biomechanical models could be another means of reinforcing the robustness of rupture predictions. The data in Table 1 indicates that only three out of nine studies (13%) reached statistical significance when mean population blood pressure was prescribed, whilst statistical significance was achieved in 13 out of 15 studies (87%) when prescribing patient specific blood pressure. The use of blood pressure from a single measurement point is not ideal and data recorded over the duration of a day or longer would be far more informative with respect to biomechanical models and actively improve their predictive capabilities.
In addition to the vessel wall, the ILT influences the progression and rupture of AAAs. Some AAA biomechanical models consider their radially changing properties, but the ILT shows much more complex heterogeneity, 36,38 a factor that could have a significant impact on AAA rupture risk. 140 In addition the biochemical activity of the ILT could contribute to AAA rupture risk, 19 and quantitative models in that respect are very much lacking. Whilst the blood flow through the aneurysmatic aorta may have a major influence on how the ILT forms, 32,171 blood flow inertia contributes minimally to the AAA wall stress. FSI studies showed therefore no significant improvement over wall stress predictions of quasi-static structural simulations. 134 However, blood flow influences the transport of particles from the blood stream into the wall or the thrombus, a factor that potentially influences the (local) growth of the aortic wall and then influences AAA rupture.
AAA rupture is influenced by many biomechanical, biochemical and clinical factors and their integration within biomechanically-dominated simulation models should present as a common research goal. Modern data processing approaches such a machine learning could play an important role in this respect. [172][173][174] DATA AVAILABILITY STATEMENT Data sharing not applicable to this article as no datasets were generated or analysed during the current study.