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Evolution of (a) elastin stretch λ, (b) collagen fibre stretch λ C β and (c) collagen fibre concentration m C of a spherical membrane with time t for four growth functions GF; see (46) and (56).
Source publication
Experimental and theoretical guidance is needed to understand how the collagen fabric evolves during the development of aneurysms. In this paper, we model the development of an aneurysm as a cylindrical/spherical membrane subject to 1D enlargement; these conceptual models reflect the development of fusiform and saccular cerebral aneurysms. The mech...
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... that GF 4 prevents the initial rapid increase in fibre stretch which is observed for GF 2 and thus a lower value of m C is required to stabilize the geometry (see Fig. 4c). Figure 5 illustrates the evolution of (a) the elastin stretch λ, (b) the collagen fibre stretch λ C and (c) the collagen fibre concentration m C , with respect to time t for the spherical membrane. As can be seen from Fig. 5(a), GF 2 and GF 4 stabilize the growth of the spherical membrane, while for GF 1 and GF 3, it continues to enlarge. ...
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... in fibre stretch which is observed for GF 2 and thus a lower value of m C is required to stabilize the geometry (see Fig. 4c). Figure 5 illustrates the evolution of (a) the elastin stretch λ, (b) the collagen fibre stretch λ C and (c) the collagen fibre concentration m C , with respect to time t for the spherical membrane. As can be seen from Fig. 5(a), GF 2 and GF 4 stabilize the growth of the spherical membrane, while for GF 1 and GF 3, it continues to enlarge. The collagen fibre stretch λ C continues to increase for GF 1, while for GF 2, it undergoes a rapid increase initially but then decreases towards the attachment stretch λ C AT . Interestingly, GF 4, which is additionally ...
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... for GF 1, while for GF 2, it undergoes a rapid increase initially but then decreases towards the attachment stretch λ C AT . Interestingly, GF 4, which is additionally sensitive to the rate of degradation of elastin, prevents the rapid increase in fibre stretch which is observed for GF 2. Of course, GF 3 alone fails to stabilize the membrane (see Fig. 5a,b). On inspection of Fig. 5(c), it can be seen that m C stabilizes for GF 2 and GF 4 as the membrane stabilizes in size and λ C → λ C AT . For GF 1, the concentration continues to increase as the increase in collagen fibre concentration is not sufficient to stabilize the growth of the membrane. As for the cylindrical membrane, given that ...
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... undergoes a rapid increase initially but then decreases towards the attachment stretch λ C AT . Interestingly, GF 4, which is additionally sensitive to the rate of degradation of elastin, prevents the rapid increase in fibre stretch which is observed for GF 2. Of course, GF 3 alone fails to stabilize the membrane (see Fig. 5a,b). On inspection of Fig. 5(c), it can be seen that m C stabilizes for GF 2 and GF 4 as the membrane stabilizes in size and λ C → λ C AT . For GF 1, the concentration continues to increase as the increase in collagen fibre concentration is not sufficient to stabilize the growth of the membrane. As for the cylindrical membrane, given that GF 3 is proportional to the ...
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... collagen fibre concentration is not sufficient to stabilize the growth of the membrane. As for the cylindrical membrane, given that GF 3 is proportional to the magnitude of the rate of loss of elastin, the increase in collagen fibre concentration decreases as a function of time and, therefore, the collagen concentration is constant for t > 5 (see Fig. 5c), and thus the membrane increases non-linearly in size (see Fig. ...
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... of the membrane. As for the cylindrical membrane, given that GF 3 is proportional to the magnitude of the rate of loss of elastin, the increase in collagen fibre concentration decreases as a function of time and, therefore, the collagen concentration is constant for t > 5 (see Fig. 5c), and thus the membrane increases non-linearly in size (see Fig. ...
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... we compare the qualitative behaviour of the remodelling for the cylinder and the sphere. Comparing Figs 4(a) and 5(a), it can be seen that when using GF 1, the evolution of the stretch of the sphere has greater non-linearity than for the cylinder; similarly, the fibre stretch increases more rapidly for the sphere (compare Figs 4b and 5b). It is of interest to note how much more rapidly m C is increasing for the sphere (compare Figs 5c and 4c) under GF 1; however, these increases are insufficient to stabilize the geometry and achieve material equilibrium, i.e. the fibre stretch continues to increase to maintain mechanical equilibrium, and thus it continues to enlarge. ...
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... we compare the qualitative behaviour of the remodelling for the cylinder and the sphere. Comparing Figs 4(a) and 5(a), it can be seen that when using GF 1, the evolution of the stretch of the sphere has greater non-linearity than for the cylinder; similarly, the fibre stretch increases more rapidly for the sphere (compare Figs 4b and 5b). It is of interest to note how much more rapidly m C is increasing for the sphere (compare Figs 5c and 4c) under GF 1; however, these increases are insufficient to stabilize the geometry and achieve material equilibrium, i.e. the fibre stretch continues to increase to maintain mechanical equilibrium, and thus it continues to enlarge. ...
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... Figs 4(a) and 5(a), it can be seen that when using GF 1, the evolution of the stretch of the sphere has greater non-linearity than for the cylinder; similarly, the fibre stretch increases more rapidly for the sphere (compare Figs 4b and 5b). It is of interest to note how much more rapidly m C is increasing for the sphere (compare Figs 5c and 4c) under GF 1; however, these increases are insufficient to stabilize the geometry and achieve material equilibrium, i.e. the fibre stretch continues to increase to maintain mechanical equilibrium, and thus it continues to enlarge. Note that GF 4 prevented the rapid increase in the fibre stretch that was observed with GF 2 for both the cylinder and the sphere (compare Figs 4b and 5b) and thus the membranes are stabilizing with smaller geometries than using GF 2 for each case. ...
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... the table. It can be seen that for the growth functions GF 2 and GF 4, the evolved values of m C are close to (but below) m C UB for both the cylindrical and the spherical membranes. As t → ∞, the solutions are such that m C → m C UB . Note that for the spherical membrane, the simulation using GF 1 has a larger m C at t = 30 compared to GF 4 (see Fig. 5c), however, for GF 1, the geometry continues to enlarge whereas GF 4 yields stabilization (see Fig. 5a). This is due to the fact that the increases of m C for GF 1 are not sufficient to achieve material equilibrium given the large change in dimensions, i.e. the remodelled value of m C is well below the required value for m C UB for ...
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... to (but below) m C UB for both the cylindrical and the spherical membranes. As t → ∞, the solutions are such that m C → m C UB . Note that for the spherical membrane, the simulation using GF 1 has a larger m C at t = 30 compared to GF 4 (see Fig. 5c), however, for GF 1, the geometry continues to enlarge whereas GF 4 yields stabilization (see Fig. 5a). This is due to the fact that the increases of m C for GF 1 are not sufficient to achieve material equilibrium given the large change in dimensions, i.e. the remodelled value of m C is well below the required value for m C UB for material equilibrium at a given stretch: consequently, the membrane continues to enlarge due to the ...
Citations
... They found that at the beginning, the aneurysm has a reduction in its maximum Von Mises stress, but then it increases along with the aneurysm size, meaning a higher rupture risk. Other works have used cylindrical/spherical membranes subjected to 1D enlargement to model the development of an aneurysm where the mechanical response is attributed to collagen and elastin [22]. ...
Cerebral aneurysms come in a wide range of shapes and sizes, they can evolve over time and present significant changes. It is generally accepted that large aneurysms are more prone to rupture, but the rupture of small aneurysms has also been observed, indicating the presence of additional risks factors. The aim of this study is to assess the effects of the aneurysm size and wall thickness on its rupture risk by using fluid-structure interaction simulations. Six patient-specific geometries were chosen, four were used for studying the size effect and two for studying the thickness effect. Additional cases where the aneurysm was removed were included. It was found that thinner walls suffer from significantly greater stresses, whereas an increment in size means, in general, lower wall shear stress and greater equivalent stress. By removing the aneurysm, it could be noted that at the rupture point, the reduction in the time-averaged wall shear stress was 75%. Although the size of an aneurysm has a great impact in its rupture risk, its wall thickness needs to be considered, since even maintaining its size, the aneurysm could suffer from a thin-ning of its walls that can lead to structural failure.
... However, some publications show impressive technical achievements in modeling the biomechanics of aneurysms and blood vessels, for instance, considering incompressible nonlinear material models such as Mooney-Rivlin solid multi-layered materials or incorporating fiber orientations [15,16]. Unfortunately, these works consider either simple theoretical geometries (dome = sphere, vessel = cylinder) or a low number of cases (1)(2). ...
Rupture of intracranial aneurysms is the most common cause of spontaneous subarachnoid bleeding, related to high morbidity and mortality rates. However, intracranial aneurysms have a higher prevalence than that due to their spontaneous rupture rate, exacerbated by the risks associated with occlusion intervention, which motivates the development of technological tools to support clinical diagnosis and endovascular occlusion intervention planning. In particular, the aneurysm dome is sensitive to applied loads in the contiguous surroundings to the aneurysm neck. Indeed, this region shows high complexity due to the arterial wall nature of the pathology. This work presents preliminary statistical analysis results of a thin shell model, with varying material and geometrical parameters, under a localized load emulating the effect of a microcatheter pressing the neck area. In a selection of 34 cases, we show that dimensionality reduction techniques such as Isomap can help determine non-trivial regions of interest under concentrated loads, leading to more general machine learning classification models for sensitive area identification.KeywordsEndovascular planningComputational biomechanicsForward jumpDimensionality reduction
... The collagen is configured to be non load bearing at the onset of voiding. Hence the parameter k nc can be analytically determined from the force-balance equation given the stress function σ nc for the non-collagenous constituents where for the deformation considered σ nc = 4k nc λ 2 (1 − 1/λ 6 ) (Watton et al., 2009a). ...
... Membrane model of bladder-The governing equation for quasi-static inflation of a spherical membrane composite is(Watton et al., 2009a): ...
We present a constrained mixture-micturition-growth (CMMG) model for the bladder. It simulates bladder mechanics, voiding function (micturition) and tissue adaptations in response to altered biomechanical conditions. The CMMG model is calibrated with both in vivo and in vitro data from healthy male rat urinary bladders (cystometry, bioimaging of wall structure, mechanical testing) and applied to simulate the growth and remodelling (G&R) response to partial bladder outlet obstruction (BOO). The bladder wall is represented as a multi-layered, anisotropic, nonlinear constrained mixture. A short time scale micturition component of the CMMG model accounts for the active and passive mechanics of voiding. Over a second, longer time scale, G&R algorithms for the evolution of both cellular and extracellular constituents act to maintain/restore bladder (homeostatic) functionality. The CMMG model is applied to a spherical membrane model of the BOO bladder utilizing temporal data from an experimental male rodent model to parameterize and then verify the model. Consistent with the experimental studies of BOO, the model predicts: an initial loss of voiding capacity followed by hypertrophy of SMC to restore voiding function; bladder enlargement; collagen remodelling to maintain its role as a protective sheath; and increased voiding duration with lower average flow rate.
This CMMG model enables a mechanistic approach for investigating the bladder’s structure-function relationship and its adaption in pathological conditions. While the approach is illustrated with a conceptual spherical bladder model, it provides the basis for application of the CMMG model to anatomical geometries. Such a mechanistic approach has promise as an in silico tool for the rational development of new surgical and pharmacological treatments for bladder diseases such as BOO.
... The main challenge in such models is introducing or adopting proper degradation and remodeling functions for the vessel wall. Growth studies may just deal with tissue modeling, called growth and remodeling (G&R) studies [32,33,34,35,36,37,38,39,40] or contain both tissue and blood flow field [41,15,42,43,44,45,46,47,16,48]. The second group may benefit from introducing flow-dependent degradation functions, which are discussed later. ...
... and Hill [66] proposed a time-dependent function for disruption of elastin content of the wall for abdominal aortic aneurysm (AAA) formation that was later used by cerebral studies too [33,38]: ...
... Watton and Hill [66] proposed the following remodeling functions for AAA formation that were used widely in cerebral studies later [33,38,16,47,36,37,67]. They defined the time rate of collagen recruitment stretch , and collagen content , as functions of deviation of collagen stretch , from the attachment stretch : ...
Predicting the future behavior of cerebral aneurysms was the target of several studies in recent years. When an unruptured cerebral aneurysm is diagnosed, the physician has to decide about the treatment method. Often more giant aneurysms are diagnosed at higher risk of rupture and are candidates for intervention. However, several clinical and morphological parameters are introduced as risk factors. Therefore, some small size aneurysms with a higher growth rate and rupture risk may be missed.
Nowadays, computational models and artificial intelligence can help physicians make more precise decisions, not only according to the aneurysm size. Therefore, the target can be developing a tool that receives the patient history and medical images as input and gives the aneurysm growth rate and rupture risk as output. Achieving this target can be possible by developing a proper computational growth model and using artificial intelligence. This requires knowledge of the vascular microstructure and the procedure of disease development, including degradation and remodeling mechanisms. Moreover, geometrical and clinical risk factors should also be recognized and considered.
The present article is a step-by-step indication of this concept. In this paper, first, a review of different computational growth models is presented. Then, the morphological and clinical risk factors are described, and at last, the methods of combining the computational growth models with machine learning are discussed. This review can help the researchers learn the fundamentals and take the proper future steps.
... It is possible to find impressive technical improvements in modeling the biomechanics of blood vessels. For example, by considering incompressible nonlinear material models such as Mooney-Rivlin solid, multi-layered materials, or even incorporating fiber orientations [15,16]. Unfortunately, these works consider either simple theoretical geometries (dome = sphere, vessel = cylinder) or a few number of cases. ...
The mechanism of aneurysm rupture is still not fully understood. The rupture risk of the intervention may increase during endovascular occlusions of cerebral aneurysms due to a localized load in the parent vessel close to the neck, a common day-today situation. As a first attempt on the road towards developing a plausible analysis capable of dealing with many cases in a statistical sense, we describe the deformation kinematics using a geometrically nonlinear thin shell model under Kirchhoff-Love's assumptions in conjunction with a simplistic Kirchhoff-St. Venant's hyperelastic material model. Though it cannot assess the artery's complexity, this more straightforward yet not trivial approach enable us to statistically study the application of a concentrated load in many locations, which mimics the action of an instrument during the endovascular treatment. We performed numerical simulations on 34 cases from the AneuriskWeb Database. We present preliminary results considering a smoothly varying thickness between the parent vessel and the aneurysm dome, focusing in the mesh construction process and loading.
... However, some publications show impressive technical achievements in modeling the biomechanics of aneurysms and blood vessels, for instance, considering incompressible nonlinear material models such as Mooney-Rivlin solid multi-layered materials or incorporating fiber orientations [20,21]. Unfortunately, these works consider either simple theoretical geometries (dome = sphere, vessel = cylinder) or a low number of cases (1)(2). ...
Rupture of intracranial aneurysms is the most common cause of spontaneous subarachnoid bleeding, related to high morbidity and mortality rates. However, intracranial aneurysms have a higher prevalence than that due to their spontaneous rupture rate, exacerbated by the risks associated with occlusion intervention, which motivates the development of technological tools to support clinical diagnosis and endovascular occlusion intervention planning. In particular, the aneurysm dome is sensitive to applied loads in the contiguous surroundings to the aneurysm neck. Indeed, this region shows high complexity due to the arterial wall nature of the pathology. This work presents preliminary statistical analysis results of a thin shell model, with varying material and geometrical parameters, under a localized load emulating the effect of a microcatheter pressing the neck area. In a selection of 34 cases, we show that dimensionality reduction techniques such as Isomap can help determine non-trivial regions of interest under concentrated loads, leading to more general machine learning classification models for sensitive area identification.
... Biomechanical G&R theories help understand and quantify the evolution of the biological tissues. In the specific case of arteries, several theoretical frameworks generally focusing on the question of aneurysm development (Baek et al. 2006;DiCarlo et al. 2009;Watton et al. 2009) adopt a macroscopic point of view describing the evolution of tissue growth as an irreversible (as opposed to elastic) component of the deformation gradient (Kuhl 2014). ...
... Each has different natural configurations as revealed by early experiments using elastase, collagenase and chondroitinase to selectively degrade individual constituents; each also has different material properties and rates of turnover in the form of synthesis and degradation. In parallel, therefore, different constrained mixture models arose to study arterial growth and remodelling, including predictions of aneurysmal enlargement [67,162] ( figure 9). Indeed, such models helped explain the effects of different levels of elastolytic insults versus different rates of collagen deposition on overall rates of lesion enlargement, the latter of which predicted a subsequent experimental finding that suggests the therapeutic potential of antagonizing certain microRNAs to control rates of matrix production. ...
One of the most remarkable differences between classical engineering materials and living matter is the ability of the latter to grow and remodel in response to diverse stimuli. The mechanical behaviour of living matter is governed not only by an elastic or viscoelastic response to loading on short time scales up to several minutes, but also by often crucial growth and remodelling responses on time scales from hours to months. Phenomena of growth and remodelling play important roles, for example during morphogenesis in early life as well as in homeostasis and pathogenesis in adult tissues, which often adapt to changes in their chemo-mechanical environment as a result of ageing, diseases, injury or surgical intervention. Mechano-regulated growth and remodelling are observed in various soft tissues, ranging from tendons and arteries to the eye and brain, but also in bone, lower organisms and plants. Understanding and predicting growth and remodelling of living systems is one of the most important challenges in biomechanics and mechanobiology. This article reviews the current state of growth and remodelling as it applies primarily to soft tissues, and provides a perspective on critical challenges and future directions.
... The outermost layer, the tunica adventitia, forms the transition to the surrounding tissue. The arrangement of its fibers counteracts the longitudinal expansion of the vessel [400]. In contrast to systemic arteries, intracranial arteries have significantly less elastin in the media, which is assumed to make them more susceptible to aneurysm formation. ...
Cerebral aneurysms are weak vessel areas that can bulge out and balloon, caused by a pathologically altered structure of the vascular wall. They bear the risk of rupture, leading to internal bleeding causing high risks of mortality. Although most aneurysms will never rupture, the potential risk of bleeding makes the detection and risk-assessment of aneurysms a critical issue. Imaging methods are used for the detection and localization of aneurysms. The decision as to whether or not aneurysms should be treated must be carefully considered, as there is a risk of fatal outcome during surgery. Their initiation and progression depend strongly on the interplay of vascular morphology and hemodynamics. Unfortunately, the processes causing aneurysm growth and rupture are not well understood. Blood flow simulations can obtain information about the patient-specific hemodynamics.
It is also the basis for the development for new, low-risk treatment options since treatment success depends on blood flow characteristics. In clinical routine, risk assessment and treatment planning are just based on morphological characteristics
of an aneurysm and its surrounding vasculature. However, this information allows no reliable evaluation of the aneurysm state. To improve decision-making, medical and biomedical researchers analyze simulated flow data, which are multi-attribute data with high spatial and temporal complexity. The data exploration is performed quantitatively and qualitatively, where the former focuses on the evaluation of specific scalar values such as pressure or wall thickness and the latter focuses
on the analysis of flow patterns such as vortices. Correlations between qualitative and quantitative characteristics can be revealed and formed into hypotheses that can lead to a better understanding of the internal aneurysm procedures. However, the visual exploration of flow data is a time-consuming process, which is affected by visual clutter and occlusions. The goal of our work is to develop computer-aided methods that support the quantitative and qualitative visual exploration of morphological and hemodynamic characteristics in cerebral aneurysm data sets. Since this is an interdisciplinary process involving both physicians and fluid mechanics experts, redundancy-free management of aneurysm data sets is required to enable efficient analysis of the information. We developed a consistent structure to document aneurysm data sets, where users can search for specific cohorts, and individual cases can be analyzed more detailed to assess the aneurysm state as well as to weigh different treatment scenarios. The prerequisite for the visual exploration is the extraction of the ostium, which is a curved surface that separates the parent vessel from an aneurysm. We provide an automatic determination of the ostium. Based on this several other morphological descriptors are computed automatically. Besides an analysis of morphological aspects,
the aneurysm data exploration comprises four more parts: a simultaneous evaluation of wall- and flow-related characteristics, a simultaneous analysis of multiple scalar information on the aneurysm wall, the analysis of mechanical wall processes as well as a qualitative characterization of the internal flow behavior. We provide methods for each of these parts: occlusion-free depictions of the vessel morphology and internal blood flow, interactive 2D and 3D visualizations to explore multi-attribute
correlations, comparative glyph-based visualizations to explore mechanical wall forces and automatic classification of qualitative flow patterns. Our methods were designed and evaluated in collaboration with domain experts who confirmed their usefulness and clinical necessity.
... We utilize existing mathematical approaches which model the arterial tissue as a constrained mixture [9,10] and attribute its load bearing to the distinct contributions from elastin [11], distributions of collagen fibers [12][13][14] and VSMCs [15]. Here we overview the key features that the mathematical model captures; further details are provided in Appendix A for the interested reader. ...
... At the onset of vasospasm, the VSMCs contract and as a result the diameter at systolic pressure is decreased (Fig. 1b); for purposes of illustration, we simulate a 50% reduction in systolic diameter so that the arterial diameter at systolic pressure is reduced to 1.46 mm. Following this initial constriction, the following changes are conjectured to occur over a time scale on the order of days VSMCs remodel to restore their stretch towards their preferred value (VSMC attachment stretch) on the vasospastic vessel (via remodeling [9,10]: this is achieved through reconfiguration of their internal cytoskeleton and/or of their attachments to the ECM [17,18]). As VSMCs remodel their stretch, to maintain force balance, VSMCs' cell stiffness increases, captured by increasing material parameters of VSMCp and VSMCa [18,19]. ...
... The arterial wall is modelled as a nonlinear elastic cylindrical membrane [10] with thickness H and radius R in its unloaded configuration. In vivo, it is subject to an internal pressure p and axial stretch Z . ...
Background
Cerebral vasospasm (CVS) following subarachnoid hemorrhage occurs in up to 70% of patients. Recently, stents have been used to successfully treat CVS. This implies that the force required to expand spastic vessels and resolve vasospasm is lower than previously thought.
Objective
We develop a mechanistic model of the spastic arterial wall to provide insight into CVS and predict the forces required to treat it.
Material and Methods
The arterial wall is modelled as a cylindrical membrane using a constrained mixture theory that accounts for the mechanical roles of elastin, collagen and vascular smooth muscle cells (VSMC). We model the pressure diameter curve prior to CVS and predict how it changes following CVS. We propose a stretch-based damage criterion for VSMC and evaluate if several commercially available stents are able to resolve vasospasm.
Results
The model predicts that dilatation of VSMCs beyond a threshold of mechanical failure is sufficient to resolve CVS without damage to the underlying extracellular matrix. Consistent with recent clinical observations, our model predicts that existing stents have the potential to provide sufficient outward force to successfully treat CVS and that success will be dependent on an appropriate match between stent and vessel.
Conclusion
Mathematical models of CVS can provide insights into biological mechanisms and explore treatment approaches. Improved understanding of the underlying mechanistic processes governing CVS and its mechanical treatment may assist in the development of dedicated stents.
