## 1 Introduction  
The cerebral aneurysm is a potentially deadly degenerative vascular disease that can initially be asymptomatic. The physical processes that cause its formation are still being investigated, but many studies in this field have highlighted the importance of the local hemodynamics on the development of the aneurysm once it is present in a cerebral vessel [1; 2; 3; 4]. Due to local blood recirculation, the aneurysmic sac may rupture, leading to high rates of morbidity and mortality [5]. It is estimated that 8% of all strokes are caused by ruptured aneurysms [6]. So-called flow diverting stents (FDSs) can be placed inside a cerebral artery for the prevention of aneurysm rupture using a minimally invasive procedure. They are currently considered a promising alternative to traditional methods, such as coiling and clipping [6]. The FDS induces a flow reduction within the aneurysm sac, promoting a progressive aneurysm thrombosis in most patients. Slow blood flow recirculating inside the sac may lead to aneurysm occlusion if the necessary changes to the hemodynamics occur. Knowledge of the hemodynamics is crucial to planing an effective treatment and improving the designs of FDSs [7; 8; 9; 10], but unfortunately, they are still not fully understood. For that reason, this topic has been attracting significant interest from the scientific community and it is generally accepted that the success of a FDS procedure depends mainly on the induced hemodynamics [11; 12]. In particular, the porosity of the device is recognized to have a significant role in the flow reduction and in the rigidity of the stent. A lower porosity promotes flow reduction within the aneurys sac but it also leads to a high rigidity of the stent structure. In contrast, a device with a higher porosity is less effective at limiting flow into the aneurysm, but it is more flexible and can better adapt to the tortuous cerebral vessels [13]. In general, the evolution of new FDS technology has largely extended the usefulness of endovascular therapy. Innovative devices with better combinations of porosity and flexibility have been introduced with variable degrees of success [14; 15]. Moreover, the role of the design of a FDS in determining the effectiveness of the endovascular therapy is now well established [16]. Computational fluid dynamics (CFD) have been used to study the reduction of flow within the aneurys sac, analyze a number of hemodynamics indices as functions of different stent designs and determine the influences of the strut shape and size [2; 13; 17; 18]. The design of the stent has been analyzed within patient-specific cerebral vessel models by several authors [11; 19; 20; 21; 22; 23; 24; 25]. On the contrary, idealized models have also been frequently used for assessing the influences of the vessel geometry, aneurysm size and location on the effectiveness of the treatment [6; 13; 26; 27; 28; 29]. Notwithstanding the extensive work that has been performed over previous decades, [19; 25; 30] the real relationship between the indices local hemodynamics and clinical outcomes is still unclear. Treatment with FDS has been reported to be capable of promoting complete aneurysm occlusion in the majority of cases, but long term patency and delayed ruptures have also been reported as consequences [31]. The results summarized by [PERSON] and coworkers [16] from about 20 flow studies on idealized, in vivo, and patient-specific geometries suggest a spectrum of flow diversion responses. Flow diversion reduces the intraneurysmal flow activity in idealized/simplified sidewall-type geometries by about 75-95%, by about 20-40% in bifurcation-type geometries and by around 60% in in vivo aneurysms. Reduction in the intraneurysmal flow activity in clinical cases is 20-30% in bifurcation geometries and around 80-90% in sidewall geometries.