Computational Modeling and Simulation Examples in Bioengineering. Группа авторов

that ILT reduced AAA wall stress, progressive AAA growth was not related to the diameter but AAA volume and relative ILT volume, and that higher wall stress was related to AAA growth only when ILT was not included in the simulations.

Kontopodis et al. [90] analyzed a single uncommon case of a 75‐year‐old Caucasian male diagnosed with fast‐growing AAA without family history but with other medical conditions such as smoking, hypertension, diabetes, and COPD. The aneurism which was initially small (45 mm in diameter) presented a growth of 1 cm in only six months. The focus of research was on thrombus and lumen volumes, thrombus maximum thickness, maximum centerline curvature together with biomechanical aspects such as PWS and areas with high stress and stress redistribution. Two 3D AAA models (initial and follow up) were reconstructed from 2D CT angiography images with manual and automatic segmentation process performed with open source software ITK‐SNAP [91]. The results indicated that the total volume increased from 85 to 120 ml for six months while lumen volume remained constant (72 ml for initial and 71 for follow up examination). Moreover, ILT increased from 14 to 50 ml and maximum thickness increased from 0.3 to 1.6 cm in the anterior part with maximum curvature of the centerline increased from 0.4 to 0.5 cm⁻¹ suggesting an anterior bulging. Additionally, area under high wall stress remained almost constant while there was a marked redistribution of wall stress from anterior site and AAA neck to posterior wall. Total posterior wall area exposed to stress was 0 cm² at the initial stage and 9.7 cm² at the follow up. It was interesting to note that PWS did not increase as was the case with previous studies (e.g. [89]). Authors suggested that although large‐scale studies are needed, information gathered by the analysis could be of use in determining AAA natural history and patient‐specific rupture risk estimation. Anton et al. [92] provided an insight into a comparative study, numerical and experimental, to analyze the pressure field in AAA models and validate numerical model's ability to predict experimental pressure field. They used patient‐specific AAA geometry with and without ILT with iliac arteries included. Iliac arteries were taken into account because numerical observations suggest that their inclusion is critical for accurate prediction of pressure, flow pattern, and wall stress [93]. Four CFD mathematical models were employed for ILT models and three CFD models were used for model without ILT because Reynolds Stress Model did not converge satisfactorily. CFD models predicted pressure fields substantially well with average difference of 1.1% for the model without ILT and 15.4% for ILT model. Moreover, they aimed to compare the spatial pressure drop before and after the formation of ILT although only few studies before performed spatial pressure drop measurements in AAA (e.g. [94, 95]). The special pressure drop in the ILT model was 5000 Pa while in the model without ILT, it was around 1500 Pa which is explained by the fact that with smaller lumen, flow velocities are higher. The study of Polzer et al. [96] broadened finite element single‐phase AAA models (e.g. [69,97–100]) with poroelastic description of ILT. The model was loaded by a pressure step and a cyclic pressure wave. The numerical results of the studied idealized axisymmetric two‐phase AAA models proved that the entire blood pressure was transmitted to the AAA wall in spite of the existence of the ILT, which was also previously confirmed by [101]. The study found that the stress in the AAA wall, at steady state, did not depend on the permeability of ILT and did not differ from single‐phase conventional models. Consequently, the results of the study suggested that single‐phase description could be used reliably to predict the stress in AAA wall instead of biphasic one which is less computationally efficient. The study also showed that ILT reduced wall tensile stress by a value between 46 and 62%. Additionally, the analysis helped the explanation of differences between in‐vivo and in‐vitro measurements by demonstrating that pressure under ILT depends on the local geometry of the AAA. On the other hand, poroelastic description cannot be replaced when investigating transport phenomena through AAA tissue. Baek et al. [102] used a set of 39 patients' CT images data from 9 different patients. The images were divided into two groups – low and high ILT group, so that the effect of ILT on the AAA expansion could be established. The results indicate that the relationship between AAA expansion rate and maximum diameter can be changed by inhomogeneous distribution of ILT thickness. Although generally ILT presence is associated with aneurysm expansion rate, a slowdown of expansion was established in areas of thick ILT.

      By using an inverse optimization method, Zeinali and Baek [103] created a computational framework toward patient‐specific AAA modeling. Namely, using a 3D geometry from medical images, they identified initial material parameters for healthy aorta to satisfy homeostatic condition and then created different computational shapes and considered multiple spatiotemporal forms of elastin degradation and stress‐mediated collagen turnover. The results exhibited the importance of the role of elastin damage extent, geometric complexity of an enlarged AAA, and sensitivity of stress‐mediated collagen turnover on the wall stress distribution and the rate of expansion. Also, the study showed that the distributions of stress and local expansion initially correspond to the extent of elastin damage, but change because of stress‐mediated tissue growth and remodeling dependent on the aneurysm shape. The specificity of their study lies in the fact that the authors did not use AAA patient‐specific model, but medical images of a healthy subject. On the other hand, they suggest that in spite of the model used for the present study, their computational framework could be used in a patient‐specific modeling to predict AAA shape and mechanical properties if improved in the domain of boundary conditions, description of aortic tissue, growth and remodeling, and the development of inverse scheme using AAA patients' longitudinal images.

The studies continuously prove that ILT has the potential to influence AAA both biochemically and biomechanically. Namely, ILT induces localized hypoxia, possibly leading to increased neovascularization, inflammation, and local wall weakening [104]. On the other hand, ILT development is strongly influenced by agonists and antagonists of platelets activation, aggregation, adhesion, and the proteins involved in the coagulation cascade which are not thoroughly discussed in the present literature. The study of Biasetti et al. [105] analyzed the evolution of chemical species involved in the coagulation cascade, their relation to coherent vertical structures, and the possible effect on ILT development. The authors developed a fluid–chemical model that simulates the coagulation cascade through a series of convection–diffusion–reaction equations. They followed the coagulation cascade model ([106]) which consisted of 18 species and involved plasma‐phase and surface‐bound enzymes and zymogens, with both plasma‐phase and membrane‐phase reactions. Blood was modeled as a non‐Newtonian incompressible fluid. The coupling was achieved by a set of convection–diffusion–reaction equations that were added to the computed blood flow field in order to predict the distribution of chemicals. Since the model was able to couple the fluid and chemical domains, it represents an integrated mechanochemical potential for further understanding of ILT formation and development mechanisms.

      Meaningful limitation of the study is mirrored in the fact that authors assumed a rigid wall which may alter the results since ILT tissue can deform during a cardiac cycle and consequently influence the fluid dynamics and the distribution of chemicals.

1.7 Finite Element Procedure and Fluid–Structure Interaction

      Blood flow in aorta has a time‐dependent 3D flow, so the time‐dependent and full three‐dimensional Navier–Stokes equations were solved. The laminar flow condition appropriate for this type of analysis [107] was used. The finite element code was validated using the analytical solution for shear stress and velocities through the curved tube [108]. A penalty formulation will be used [109]. The incremental–iterative form of the equations for time step and equilibrium iteration “i” is:

      (1.1)

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