Computational Modeling and Simulation Examples in Bioengineering - Группа авторов

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n 1 Baseline ModifyingAbove bold upper F With ampersand c period circ semicolon Baseline Superscript left-parenthesis i minus 1 right-parenthesis Baseline minus Superscript n plus 1 Baseline bold upper F Superscript int left-parenthesis i minus 1 right-parenthesis"/>

      Here, ΔU(i) are the nodal displacement increments for the iteration “i,” and the system matrix Superscript n plus 1 Baseline ModifyingAbove bold upper K With ampersand c period circ semicolon Subscript tissue Superscript left-parenthesis i minus 1 right-parenthesis, the force vector Superscript n 1 Baseline ModifyingAbove bold upper F With ampersand c period circ semicolon Baseline Superscript left-parenthesis i minus 1 right-parenthesis, and the vector of internal forces n + 1Fint(i − 1) correspond to the previous iteration.

      We described the material nonlinearity of blood vessels which is used in further applications. The geometrically linear part of the stiffness matrix, left-parenthesis Superscript n plus 1 Baseline bold upper K Subscript normal upper L Baseline right-parenthesis Subscript tissue Superscript left-parenthesis i minus 1 right-parenthesis, and nodal force vector, n + 1Fint(i − 1), are defined:

      (1.10)left-parenthesis Superscript n plus 1 Baseline bold upper K Subscript normal upper L Baseline right-parenthesis Subscript tissue Superscript left-parenthesis i minus 1 right-parenthesis Baseline equals integral Underscript upper V Endscripts bold upper B Subscript normal upper L Superscript upper T Baseline Superscript n plus 1 Baseline bold upper C Subscript tissue Superscript left-parenthesis i minus 1 right-parenthesis Baseline bold upper B Subscript normal upper L Baseline normal d upper V comma left-parenthesis Superscript n plus 1 Baseline bold upper F Superscript i n t Baseline right-parenthesis Superscript left-parenthesis i minus 1 right-parenthesis Baseline equals integral Underscript upper V Endscripts bold upper B Subscript normal upper L Superscript upper T Baseline Superscript n plus 1 Baseline bold sigma Superscript left-parenthesis i minus 1 right-parenthesis Baseline normal d upper V

      Here, the consistent tangent constitutive matrix Superscript n plus 1 Baseline bold upper C Subscript tissue Superscript left-parenthesis i minus 1 right-parenthesis of tissue and the stresses at the end of time step n + 1σ(i − 1) depend on the material model used.

      1.7.4 FSI Interaction

      In many models of cardiovascular examples where deformation of blood vessel walls was taken into account, we can implement the loose coupling approach for the FSI [113–116]. The overall algorithm consists of the following steps:

      1 For the current geometry of the blood vessel, determine blood flow (with Arbitrary Lagrangian–Eulerian (ALE) formulation). The boundary conditions for the fluid are wall velocities at the common blood–blood vessel surface.

      2 Calculate the loads, which act on the walls from fluid domain (blood).

      3 Determine deformation of the walls taking the current loads from the fluid domain (blood).

      4 Check for the overall convergence which includes fluid and solid domain. If convergence is reached, go to the next time step. Otherwise go to step (1).

      5 Update blood domain geometry and velocities at the common solid–fluid boundary for the new calculation of the fluid domain. In case of large wall displacements, update the finite element mesh for the fluid domain. Go to step (1).

 Schematic illustration of (a) Shear stress distribution. (b) Drag force distribution.

      Together with CFD simulation, there are numerous statistics‐based machine learning methods that can be used to give more accurate and faster conclusions for clinicians [117].

      Filipovic et al. [121] combined DM techniques and CFD for the estimation of the wall shear stresses in AAA under prescribed geometry changes. They performed large‐scale CFD runs for creating machine learning data on the Grid infrastructure and their results showed that DM models provide good prediction of the shear stress at the AAA in comparison with full CFD model results on real patient data.

      The abovementioned studies have been limited by the use of geometric parameters and, in particular, the maximum diameter of lumen alone as factors contributing to the rupture of an AAA. But other parameters such as patient history and comorbidities and presence of stents or other geometric parameters such as the aneurysm neck angles, tortuosity, and genetics factors [122–126] should be included. Despite the fact that state‐of‐the‐art FEM approaches represent powerful tool for estimation of AAA, their application in clinical practice remains limited due to several reasons. Firstly, every patient has specific and complex anatomy and it is not possible to create a general or parametric human model. Moreover, accuracy of simulations depends on the considered level of details, meaning that increasing required computation time and power will be necessary for obtaining precise results. As a consequence, performing patient‐specific simulation may take a few hours (if patient scans are available in the first place). This makes current FEM approaches inadequate for urgent situations such as alerting patient in case of AAA rupture.

      In order to avoid described limitations, the patient‐specific DSS could be proposed. The main idea is to perform patient‐specific forward simulations in advance. AAA of different types (sizes and positions) will be simulated and calculated stress analysis will be used for training of intelligent model. But, in order to perform forward FEM simulations patient geometry is required. For this reason, during the registration into our system, in local workstation (user hospital), patient or his/her medical institution will be asked to provide us patients' medical scans (CT or MRI for example), if there are any. However, it is assumed

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