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**Computational vascular fluid dynamics: problems, models and methods.**
*(English)*
Zbl 1096.76042

From the text: Three different issues will be mainly addressed in this note.

1. Definition of suitable mathematical models: due to the complexity of the cardiovascular system, a preliminary analysis aiming at introducing suitable simplifying assumptions in the mathematical modelling process is mandatory. Obviously, different kinds of simplifications are suitable for different vascular districts. These issues will be discussed in Sect. 3 for blood, and in Sect. 4 for the vascular walls, with a particular emphasis on large- and medium-sized vessels, where atherosclerosis usually develops.

2. Preprocessing of clinical data: the suitable treatment of clinical data is crucial for the definition of a real (i.e., taken from a patient) geometrical model, which is of utmost importance for the meaningfulness of numerical results. This aspect demands geometrical reconstruction algorithms in order to achieve simulation in real vascular morphologies. We briefly consider this issue in Sect. 3.4.

3. Development of appropriate numerical techniques: the geometrical complexity of the vascular districts suggests the use of unstructured grids (in particular for finite element method), while the strongly unsteady nature of the problem demands effective time-advancing methods. In particular, for the numerical simulation of the flow equations in a vascular district, fractional step methods that separate the computation of velocity from that of pressure field seem to be adequately accurate and computationally effective (see Sect. 5.1).

When the compliance of the vascular walls is taken into account, specific techniques such as the arbitrary Lagrangian Eulerian (ALE) method for the numerical solution of the fluid equations in moving domains have to be used and will be considered in Sect. 5.2 and Sect. 5.3. Finally, in Sect. 5.4 we consider the coupling of the equation for the fluid in ALE formulation and those for the vessel walls, in view of the numerical simulation of fluid-structure interaction problems.

The virtual hemodynamic environment based on mathematical models, preprocessing of clinical data and numerical devices, as described in this note, provides the physicians approaching this complex bio-medical field with different options, among which we mention: the understanding of the intimate correlation between any modification of vessel morphology and the associated alteration of flow patterns, with its impact on long-term results; the basis for a sophisticated planning, before performing real procedures on the patient; the possibility of altering vascular geometries in order to study or foresee end-results of any vascular procedure. The numerical results addressed at the end of this work demonstrate the potentiality of these interdisciplinary studies.

1. Definition of suitable mathematical models: due to the complexity of the cardiovascular system, a preliminary analysis aiming at introducing suitable simplifying assumptions in the mathematical modelling process is mandatory. Obviously, different kinds of simplifications are suitable for different vascular districts. These issues will be discussed in Sect. 3 for blood, and in Sect. 4 for the vascular walls, with a particular emphasis on large- and medium-sized vessels, where atherosclerosis usually develops.

2. Preprocessing of clinical data: the suitable treatment of clinical data is crucial for the definition of a real (i.e., taken from a patient) geometrical model, which is of utmost importance for the meaningfulness of numerical results. This aspect demands geometrical reconstruction algorithms in order to achieve simulation in real vascular morphologies. We briefly consider this issue in Sect. 3.4.

3. Development of appropriate numerical techniques: the geometrical complexity of the vascular districts suggests the use of unstructured grids (in particular for finite element method), while the strongly unsteady nature of the problem demands effective time-advancing methods. In particular, for the numerical simulation of the flow equations in a vascular district, fractional step methods that separate the computation of velocity from that of pressure field seem to be adequately accurate and computationally effective (see Sect. 5.1).

When the compliance of the vascular walls is taken into account, specific techniques such as the arbitrary Lagrangian Eulerian (ALE) method for the numerical solution of the fluid equations in moving domains have to be used and will be considered in Sect. 5.2 and Sect. 5.3. Finally, in Sect. 5.4 we consider the coupling of the equation for the fluid in ALE formulation and those for the vessel walls, in view of the numerical simulation of fluid-structure interaction problems.

The virtual hemodynamic environment based on mathematical models, preprocessing of clinical data and numerical devices, as described in this note, provides the physicians approaching this complex bio-medical field with different options, among which we mention: the understanding of the intimate correlation between any modification of vessel morphology and the associated alteration of flow patterns, with its impact on long-term results; the basis for a sophisticated planning, before performing real procedures on the patient; the possibility of altering vascular geometries in order to study or foresee end-results of any vascular procedure. The numerical results addressed at the end of this work demonstrate the potentiality of these interdisciplinary studies.

### MSC:

76M27 | Visualization algorithms applied to problems in fluid mechanics |

76Z05 | Physiological flows |

92C10 | Biomechanics |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |