Before the RCAVF creation, surgeons measured and analyzed the quality of the patient vessels through duplex ultrasound and selected the a priori most appropriate AVF according to the D_ra and D_cv based on their clinical experiences. During the operation, surgeons release the vein over a certain distance to bring the vein to the artery, then cut the vein (CV) obliquely following an angle θ and cut along the axial direction of the artery (RA) a length D_a as shown in Figure 1(a). Then RA and CV will be connected to form the RCAVF vascular access, as depicted in Figure 1(b). The blood flow in the proximal artery (PA) enters the fistula through the heel of the anastomosis with a flow rate Q_h equivalent to that of the normal radial artery (22 mL/min). The blood flow in the distal artery (DA) enters the fistula through the toe of the anastomosis at a flow rate Q_t, with the same order of magnitude as that of the wrist (5 mL/min). Q_h and Q_t were calculated by multiplying the average flow velocity in the blood vessel by the cross-section of the vessel [12 - 14].

E_avf in the RCAVF included energy lost due to vessel bending (E_b) and friction (E_f) in CV, as well as the dissipation of energy due to change in the direction of blood flow (E_d) at the anastomosis. In this research, the blood was laminar flow, and the following assumptions were used:

• Blood is an incompressible Newtonian fluid (ρ = 1060 kg/m³, µ = 0.0035 Pa·s) flowing in a rigid horizontal blood vessel;

• The potential energy and the heat transfer are negligible;

• The energy loss due to metabolism is negligible.

In order to validate the model, we compared E_avf calculated by Equation (13) and E_{avf_num} obtained with Foam-extend 4.0 numerical simulations. To ensure the relevance of the model, the comparison was made for 25 RCAVF 3D shapes created with Catia V5.

In Equation (13), there was a total of 5 GPs (D_ra, D_cv, h, D_a, and θ). We created a standard model (as a reference of fixed values of GPs as shown in Table 1), then only changed one GP each time. For a given GP, we selected five values within its range (Table 1) and kept the remaining GPs consistent with the standard model. Therefore, we created a total of 25 models. The details of the models are shown in Table 2.

In foam-extend 4.0, we used pisoFoam [18] solver to simulate the flow in these geometries with the same initial conditions (flow rate and pressure). After numerical convergence, the pressure at anastomosis inlets (S_t, S_h) and outlet (S_cv) (Figure 6) could be extracted. They denoted respectively P_t, P_h, and P_cv. According to the law of energy conservation, E_{avf_num} of each model can be calculated by:

ρ 2 Q t 3 S t 2 + P t Q t + ρ 2 Q h 3 S h 2 + P h Q h = ρ 2 Q c v 3 S c v 2 + P c v Q c v + E a v f _ n u m . (14)

Q_t and Q_h are the flow rates at arterial inlets (Table 2). According to mass conservation, Q_cv is equal to the sum of Q_t and Q_h.

We calculated E_avf for 25 models using Equation (13) and compared them with E_{avf_num} obtained by numerical simulations. The comparisons were shown in Figure 7. The blue dots in Figure 7 represent the analytical results (E_avf), and the red dots represent the numerical results (E_{avf_num}). We could observe that, for each geometric parameter, E_{avf_num} and E_avf had the same profile. The relative difference between the two results defined as (|E_avf − E_{avf_num}|/E_avf) was shown in Table 3 for each case. The relative difference between the two approaches was less important when analyzing the impact of the variation of D_cv or h. It varied between 1% and 7% for D_cv. For h, it oscillated between 1% and 9%. In the case of D_ra and D_a, the relative difference was greater and reached 12%. In all cases, the difference between the two approaches was

minimal for the standard value of the geometric variables. The difference became greater when the geometric variables moved away from the reference values. Whereas for the θ, the difference between the numerical approach and the model tended to increase from 1% to 11% when the angle increased.

These differences could be explained by the simplifications of the geometry of the RCAVF made for this study. Another possible explanation was linked to the assumptions made initially in [16] on which we relied to establish the expression of E_a given by Equation (6). Nevertheless, we could consider that these differences between E_avf and E_{num_avf} remain within an acceptable range. Therefore, Equation (13) established here could reasonably be exploited to predict the energy loss rate in RCAVF.

We used Matlab2017 to analyze deeper the relationship between E_avf and the different GPs. The curves illustrating this relationship are represented in Figures 8-12. For each figure, the evolution of E_avf as a function of a GP was illustrated. For each GP, the evolution of the E_avf was parametrized by one of the remaining GPs (subfigures a to d). The three remaining GPs were kept constant and equaled to their respective value corresponding to the standard model given in Table 2.

Thus, Figure 8 shows the relationship between E_avf and D_ra. In Figure 8(a), we set h, D_a, and θ to the standard constant values and changed D_cv. From this figure, we observed that for all the values of D_cv, the E_avf decreased when the D_ra increased. For a fixed value of D_ra, the energy loss rate was lower when the D_cv increased. The decrease in E_avf as a function of D_ra was more pronounced for the smaller D_cv values. This general tendency to have lower losses as the diameter of the artery or vein increases could be explained by lower blood velocities at constant flow and larger diameters. As a result, the friction decreased.

In Figure 8(b), we set D_cv. D_a, and θ to the constant values and changed h. The behavior of E_avf versus D_ra was the same as in the previous case. The energy loss rate decreased when D_ra increased. Here, on the other hand, for a fixed value of D_ra, the energy loss was more significant when the distance h increased. This result seemed consistent since the friction loss rate was proportional to the distance traveled by the blood. A greater distance h implied a greater length of the bending vein, so a more significant energy loss rate appeared.

Figure 8(c) portrayed the relationship curve between E_avf and D_ra when D_cv, h, and θ were unchanged, and D_a was changed. Here the behavior was the same as for Figure 8(a). A larger diameter was the source of a more limited energy loss rate.

Figure 8(d) gives the evolution of E_avf as a function of D_ra and θ when D_cv, h, and D_a are unchanged. The decrease in the angle accentuated the decrease in E_avf as a function of the diameter.

Without detailing the analysis of all Figures 8-12, it was possible to conclude that we had the following general tendencies:

1) E_avf was inversely proportional to D_ra, D_cv, D_a, and θ, and proportional to h. In other words, if surgeons want to improve the maturity rate of RCAVF, we suggested them choose RA and CV with larger diameters and closer distance and make an anastomotic plan with a larger anastomotic diameter and anastomosis angle.

2) Compared to D_ra, D_cv had a more significant impact on E_avf. Large-diameter CV (D_cv) could effectively reduce the energy loss rate in the fistula and increase the maturity rate of RCAVF than large-diameter RA (D_ra).

Figure 8 and Figure 9 show that when D_ra changed, the range of variation of E_avf was much smaller than the change caused by changing D_cv. We could also see that the curves in Figure 9 were steeper than Figure 8. Therefore, we could prove that the maturity rate of RCAVF had a significant correlation with D_cv, and the correlation with D_ra was relatively weak. This result agreed with [19], which summarized a clinical survey carried out on 96 patients who used RCAVF for dialysis. The author concluded that fistula maturation correlated with D_cv, but there was no correlation with D_ra.

3) To increase the maturity rate of RCAVF, the choice of θ was more important than D_a.

By comparing Figure 11 with Figure 12, we could observe that variation of E_avf with θ was more significant than D_a. When θ increased, the energy loss rate could be effectively reduced in RCAVF. It could be seen from Figure 9(c) and Figure 11(b) that when D_cv changed, the effect of D_a on E_avf was almost negligible. It could be seen from Figure 11(d) and Figure 12(d) that only when θ was very small, the D_a had some effect on the energy loss. This phenomenon could be explained. The energy loss at the RCAVF anastomosis is mainly caused by the friction E_f of the venous bending segment. The loss due to friction is called the major loss because it is much larger than the minor loss (E_a and E_b) due to local shape changes. The change of θ will cause the length of the venous bending segment to change and thus change the major loss, while the change of D_a will only cause a slight change in E_a. Therefore, compared with θ, the influence of D_a on E_avf is almost negligible.

4) When h was large, it was recommended to choose a larger D_cv or decrease the distance between the vessels to reduce E_avf.

It could be observed from Figure 10(b) and Figure 10(d) that the greater the h, the greater the influence of D_cv on E_avf. Therefore, for some patients with RA and CV on different sides of the arm, the selection of D_cv was particularly important. If the patient’s CV (D_cv) was too small or the anastomosis design could not be achieved due to the limitation of the muscles or other tissues in the arm, in order to guarantee the maturity at this point, we could decrease the h (for example, releasing part of the CV in the forearm) to reduce the energy loss rate or choose other types of AVF.

To ensure the maturity and primary patency of RCAVF, the Kidney disease outcomes quality initiative (KDOQI) [1] and the European Society for Vascular Surgery (ESVS) [20] pointed out in the 2019 vascular access clinical guidelines that it was recommended to use an artery and vein with a diameter greater or equal to 2 mm. From a systematic review [3], the author summarized 12 articles (1200 patients in total) and proposed to use CV and RA with a diameter greater than or equal to 2 mm, and not recommended to use CV and RA with a diameter less than 1.5 mm. In [4 - 6], the authors also had different proposals for the recommended D_cv and D_ra ranges. All these suggested critical D_cv and D_ra values could ensure a high RCAVF maturity rate range from 1.5 mm - 2 mm. However, no research has highlighted a precise critical value of D_cv and D_ra that can guarantee the maturity of RCAVF. Our analysis found that the reason for this phenomenon was that they ignored the impacts of θ. It could be observed from Figure 12(a) that when D_cv = 3 mm, D_a = 6 mm, and h = 12 mm, the E_avf generated when D_ra = 1 mm and θ = 70˚ is less than and the case when D_ra = 2 mm and θ = 10˚ mm. Consequently, we could explain why there were still mature cases when D_cv or D_ra was under 1.5 mm. Therefore, to ensure the RCAVF maturity, the choice of θ was also essential while considering the vessel diameters. We proposed a Critical Energy Loss rate (CEL) concept to help surgeons choose the range of θ reasonably while considering patient’s specific D_cv and D_ra