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Chronic renal failure is a disease that affects a considerable population percentage that requires the hemodialysis procedure which is a blood filtration. This process is extremely stressful in many cases for the patient, because there is a continuous degradation of vessels and fistulae susceptible to this process due to the alterations of the stress at the blood vessel walls and flow patterns, leading to diseases such as intimal hyperplasia and consequent stenosis. Experimental
in vivo researches in this area are very difficult to perform. In this sense computational models become interesting non invasive options to understand what happens to the blood in non viscometric geometries. In this work, we analyse blood flow, through computational modeling, in arteriovenous fistulae used in hemodialysis, using geometries with dimensions close to real ones. Discretization of the governing equations was made through a finite volume technique with the PISO—Pressure Implicit with Splitting of Operators—algorithm, using as a basis the OpenFOAM software platform. Large vessel Newtonian fluid model was used to analyze six possible anastomotic angles (20
^{。}, 25
^{。}, 30
^{。}, 35
^{。}, 40
^{。}, 45
^{。}). To analyze the possibility of stenosis formation caused by hyperplasia, results for wall shear stress, oscillatory shear index (OSI), velocity and local circulation fields were obtained, showing that higher angles presented more secondary flows and larger extensions of stagnation regions near the critical areas of the junctions. Moreover, a range around 25
^{。}was identified to be the most suitable choice for clinical applications, minimizing possibility of diseases.

Chronic renal failure is a disease in which partial loss of the kidney function occur in a slow, progressive and irreversible way. As a leading alternative treatment for this deficiency, the blood filtering process called hemodialysis is used. This is a process which consists in filtering blood and removing impurities from this fluid, similar to the process executed by the kidney. The blood is first transported through a venous access connected to an artificial arterial fistula and subsequently the blood passes through the dialysis process being filtered [

The problems related to fistulae are believed to be caused due to friction generated by the shear forces of blood on the vessel walls, causing diseases like intimal hyperplasia and thrombosis [

Intimal hyperplasia is a process in which cells proliferate on the vessel walls in response to injury. This can reduce the diameter of the vessels, obstructing the passage of blood, and can lead to blood clotting [

Endothelial cells are sensitive to wall shear stress and in oscillatory flows these cells and those within the flowing blood are subjected to a wide range of varying shear stress. Both high and low shear influence the ways these cells respond to local hemodynamic conditions. There is a propensity to develop stenoses at specific sites, suggesting that the geometry of the fistula and the resulting local hemodynamic conditions have their role in the development of that disease [

In vascular vessels, shear stress acts mainly on the inner surface of the vessels wall (endothelium) and has an indirect function to regulate vessel diameter. Following [

According to several researches, low shear stresses are accompanied by unstable fluid conditions like turbulence, stagnation regions, recirculation and are related to several pathologies [

[

Aiming the suitability of minimally invasive techniques, computational models capable of simulating blood behavior are being widely required. Considering computational fluid dynamics, [

In this work, aiming to contribute to the investigation of better anatomical configurations that could generate more favorable hemodynamic conditions, to minimize the problems related with fistulae failures, computational analyses are performed considering six medium to small anastomotic angles: 20˚, 25˚, 30˚, 35˚, 40˚ and 45˚, before any stenosis or other pathologies have started. In this case, the vessel diameters are big enough (>0.1 mm) to consider large vessel Newtonian blood flow model [

3-D arteriouvenous radial-cephalic side-to-end fistulae with antegrade blood flow are considered here. Straight arteries of length 15.0 cm and 3.1 mm diameter; vein of length 10.0 cm and 4.01 mm of diameter, with anastomotic angles of 20˚, 25˚, 30˚, 35˚, 40˚ and 45˚ will be studied. In

At the artery inlet section, a pulsating velocity condition will be prescribed and at the outlets a developed flow condition will be considered.

Blood is very well known to be a suspension of several components, large and small ones, in almost Newtonian liquid called plasma, and is, in essence, a non-Newtonian fluid in the sense that it is able to alter its structure and adapt itself behaving differently depending on where it is flowing in. These differences come out depending on several factors, being very evident if blood is flowing in large or in small vessels. In normal large vessels (without stenosis, etc.), (diameters > 0.1 mm) Reynolds numbers are high enough to preclude aggregation of red cells (RBC) and Womersley numbers (Wo) are high, characterizing the pulsating effects to be considerable, [

Taking into account the geometries of the arterial-cephalic fistulae studied here, with dimensions greater than 1 mm, blood is being considered as a Newtonian fluid.

Consider a 3-D domain,

subject to the boundary conditions

where ^{3}),

Once determined the velocity field,

This is an important variable indicating the friction level at the wall. However, to analyze the WSS at each instant of time is unfeasible and it does not directly supply a measure of the oscillations of the shear stress field for the pulse period. Then, it will be used here the idea, first introduced by [

Equation (9) quantify the degree of deviation of WSS from its medium direction during the whole cardiac period,

To solve the equation system of the flow (Section 2.2) in the arteriovenuos geometric models for the six angles, a finite volume method, [^{−5} s in all cases.

Simulations have been performed for side-to-end radial-cephalic arteriovenous fistulae with pulsating antegrade flow, testing six anastomotic angles: 20˚, 25˚, 30˚, 35˚, 40˚ and 45˚. Results are presented for velocity contour fill, stream lines, WSS for the peak systole and the minimum diastole instants and OSI for the during time period

Figures 6(a)-6(f) present the results for WSS in the systolic peak. They are very non-homogenous, with the higher values occurring along the proximal artery branches and smaller ones at the veins. Long extents of higher values than the normal ones (2.0 - 7.0 Pa) are present in this peak for all configurations, but the angles of 35˚ to

45˚ present these values even at the ceiling of the distal artery branch. The most homogeneous distribution of WSS is presented by the angle of 25˚, considering the normal range (light blue to orange colors in the graphic scale). The same can be observed when looking the minimum diastolic results of Figures 7(a)-7(f). But in these critical cases, at the regions of the junctions, the most favorable (higher values of WSS in longer extents and bounded by the normal values) are the angles of 25˚, 20˚, and 30˚, in this order. Angles of 35˚ to 45˚ present unfavorable very low WSS values along the veins and at the junction.

Systolic peak and minimum diastolic are only two instants of the whole pulse period. Although that wall shear stress levels are very important to cause intimal pathologies, their variations and oscillations are reported as being of major importance for stenosis formation, especially at the junction region. Then, results for the oscillation shear index have been obtained here by Equation (9), previously defined, which takes into account a mean variation of the shear stress along the whole period at each point of the walls. Figures 8(a)-8(f) illustrate these results at the junctions. Dark blue color contours indicate very low or null oscillation levels, OSI < 0.01, while OSI > 0.01 indicate higher oscillation degrees. Two views for each angle case are presented to better locate the regions with higher OSI. It can be observed that the areas with higher oscillations are placed near the junctions, with the angles of 35˚ to 45˚ presenting greater extents of high OSI. The angle of 25˚ presented smaller areas of high OSI followed by the angles of 20˚ and 30˚.

In this work, 3-D arteriovenous radial-cephalic side-to-end fistulae with antegrade blood flow have been computationally modelled for anastomotic angles of 20˚, 25˚, 30˚, 35˚, 40˚ and 45˚, to investigate anatomical configurations that could generate more favorable hemodynamic conditions, to minimize the problems related with fistulae failures. Large vessel blood flow model has been considered and a Finite Volume computational method was used to solve the equations for the pulsating flow problems in the OpenFoam free software platform. To analyze the possibility of stenosis formation, results were obtained here for wall shear stress, oscillatory index, velocity contour fill and local circulation fields. From the results, it is possible to conclude:

・ From the velocity fields and, especially, from the stream lines, higher angles presented more secondary flows and larger extensions of stagnation regions near the critical areas of the junctions;

・ From the WSS results, higher stresses have been detected along the artery walls than along the veins, and smaller angles being more favorable since they presented higher shear stress levels along the areas close to the junctions as well as a more homogeneous distribution;

・ From the OSI, it was possible to better analyze the effects of oscillating shear at the walls, from which the higher it is, the higher is the possibility of pathologies to come up. The 25˚ angle presented less extension areas of high OSI, followed by 20˚ and 30˚;

・ As a consequence it is suggested here that the range 25˚ ± 5˚ is the most favorable for clinical applications.

・ The results here predict larger areas of oscillation shear at the fistulae as in [

・ There is an evident correspondence between the sites of stagnation areas and those of higher OSI.

J. A. Silva acknowledges the financial support of CAPES, J. Karam-Filho acknowledges CNPq/FAPERJ and C. C. H. Borges is thankful to CNPq, FAPEMIG and CAPES.