Computational Modeling and Simulation Examples in Bioengineering. Группа авторов

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СКАЧАТЬ These differential equations are integrated with a selected time step size Δt. The displacements ^{n + 1}U at end of time step are finally obtained according to equation:

      (1.8)

where the tissue stiffness matrix and vector are expressed in terms of the matrices and vector in (1.7). Above equation is obtained under the assumption that the problem is linear: the viscous resistance is constant, displacements are small, and the material is linear elastic.

      In many real examples as in case of aneurism or heart ventricle motion, the wall displacements can be large, hence the problem becomes geometrically nonlinear. Also, the tissue of blood vessels has nonlinear constitutive law which has to be expressed with materially nonlinear finite element formulation. Therefore, the linear formulation of the equation may not be appropriate. For a nonlinear problem, there is incremental–iterative equation

      (1.9)

      Here, ΔU⁽ⁱ⁾ are the nodal displacement increments for the iteration “i,” and the system matrix , the force vector , and the vector of internal forces ^{n + 1}F^{int(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, , and nodal force vector, ^{n + 1}F^{int(i − 1)}, are defined:

      (1.10)

      Here, the consistent tangent constitutive matrix of tissue and the stresses at the end of time step ^{n + 1}σ^{(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).

The shear stress and drag force distribution have been presented in Figures 1.2 and 1.3 for two different patients for proximal and distal AAA.

Figure 1.2 (a) Shear stress distribution. (b) Drag force distribution.

Figure 1.3 (a) Shear stress distribution. (b) Drag force distribution.

1.8 Data Mining and Future Clinical Decision Support System

      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].

Kolachalama et al. [118] proposed a DM technique that accounted for the geometric variability in patients for predicting cardiovascular flows. A Bayesian network‐based algorithm was used to understand the influence of key parameters through a sensitivity analysis. Martufi et al. [119] investigated a geometrical characterization of the wall thickness distribution in AAA. They were able to train a model to differentiate the wall thicknesses in ruptured and un‐ruptured AAA. Shum et al. [120] developed a model from 66 ruptured data sets and 10 non‐ruptured data sets and their geometric indices and wall thickness variations. The results of this study СКАЧАТЬ