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82 5. IMPACT OF THE FLUID-STRUCTURE INTERACTION MODELING ON THE HUMAN VESSEL HEMODYNAMICS

(A) (B)
FIG. 5.2 Grids of the considered models used for the computations: aorta (A) and carotid artery (B).

elements for the aorta and the carotid artery, respectively. These sizes have been selected after a mesh-independence
study realized by means of CSM analyses. These are based on the computational displacements and strains because the
maximum principal stress is difficult to be accurately resolved in analogy with the WSS for the fluid computation. Also,
in this case, the number of elements composing the final solid grids has been chosen as a compromise between the
necessary requirements (error percentage on the strain computations) and the computational costs. With the aforemen-
tioned cluster, the final FSI computations required about 336 h to be completed with the computational requirements
previously detailed.

5.2.3 Boundary Conditions Dilemma
For FSI problems, it is known that mixed conditions are required for correctly computing the intravascular flow and
pressure. In particular, pressure information has to be given to the model for adequately computing the arterial stresses
and strains. Theoretically, patient-specific measured blood flow and pressure are the perfect candidates in this sense.
Unfortunately, while measured flow can be obtained in a noninvasive way by means of the laser Doppler technique,
pressure measures are normally very invasive and required the use of a probe that may negatively influence the
measure. In addition, flow measurements are clinically standardized while pressure measures are nonstandard
and performed only in specific cases.
The impedance-based method may help to overcome this problem, providing a powerful computational tool that,
starting from flow measurements, allows the computation of pressure waveforms. In the following sections, this attrac-
tive method originally proposed by Olufsen [24] will be explained. This method will be used for both the computations
of the aorta and the carotid artery.

5.2.3.1 Aortic and Carotid Inflow
Patient-specific data of blood flow and blood pressure were not available for this study. Therefore, intravascular
Doppler ultrasonic measured flows for the two considered arteries were taken from the literature [24, 25]. Murray’s
law was then used to impose the flow rate in each outlet branch. Using this law, the flow rate in each outlet is inversely
proportional to the third power of the diameter of each branch:

Q root Q p1 Q p2
+ , (5.1)
3 ¼ 3 3
D root D p1 D p2
where Q is the flow rate and D the diameter of each arterial branch [26]. In other words, for the child branches, the flow
coming from the mother branch (root) splits following the cubic power of the parent diameters D p1 and D p2 . Each
branch outflow was then calculated starting from the root flow waveform using Murray’s law and later used for
the computation of the impedance-based pressure waveforms. For the carotid artery, we used the averaged flow given
in Lee et al. [5] for the CCA. Then, this flow was divided following Murray’s law at the ICA and ECA.
5.2.3.2 The Impedance-Based Method
Following the recursive approach developed by Olufsen et al. [24, 25] and extended by Steele et al. [27] for calcu-
lating the impedance of the human vascular tree, we first modeled the circulatory system as a structured fractal

I. BIOMECHANICS