Optimization of Maturation of Radio-Cephalic Arteriovenous Fistula Using a Model Relating Energy Loss Rate and Vascular Geometric Parameters

Yang Yang^{1},
Nellie Della Schiava^{2},
Pascale Kulisa^{3},
Mahmoud El Hajem^{4},
Benyebka Bou-Saïd^{1},
Serge Simoëns^{3},
Patrick Lermusiaux^{2}

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1. INTRODUCTION

Patients with end-stage renal disease (ESRD) need to undergo hemodialysis (HD) to survive. The arteriovenous fistula is the gold standard for vascular access for hemodialysis, and it is the access recommended by the NFKDOQI [1]. Among all possible AVFs, distal RCAVF is recommended as the first choice of native AVFs. However, due to its small flow capability and thin vessels, the venous outflow after the operation cannot reach the expected blood flow. It will lead to the non-maturation of RCAVF and causes a higher early failure rate. Therefore, the choice of radial artery (RA) and cephalic vein (CV) and the design of the anastomosis play an essential role in the postoperative maturity rate of RCAVF.

We proposed in this study a model to optimize the design of the RCAVF. This new complete model using the interrelation between the different GPs brought scientific probes about the hemodynamic of RCAVF. Our research was inspired by Murray’s law [2]. The geometric structure of human blood vessels has evolved according to natural biological selection. The structure of blood vessels follows the principle of minimum energy loss. AVF is an artificial connection of blood vessels. When the energy loss generated by the designed vascular circuit is small, the blood flow is easier to establish so that venous blood flow can reach the target value more easily and quickly, which facilitates the maturation of the AVF.

The research review [3] summarized previous studies and recommended surgeons to choose *D _{ra}* and

2. METHODS

2.1. Theoretical Analysis

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

*E _{avf}* in the RCAVF included energy lost due to vessel bending (

(a) (b)

Figure 1. Radio-cephalic arteriovenous fistula (RCAVF) schematic diagram and geometric parameters (GPs).

• Blood is an incompressible Newtonian fluid (*ρ* = 1060 kg/m^{3}, *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.

2.1.1. Energy Loss Rate in Bending Cephalic Vein

The energy loss rate along a bending CV segment (Figure 2) consists of *E _{b}* and

${E}_{b}={\zeta}_{M}\cdot \frac{\rho {Q}_{cv}^{3}}{2{S}_{cv}^{2}}$ , (1)

where *z** _{M}* is the minor loss coefficient of the bending vessel, it can be defined as:

${\zeta}_{M}={A}_{1}{B}_{1}{C}_{1}$ . (2)

*A*_{1}, *B*_{1}, and *C*_{1} are defined by:

$\{\begin{array}{l}{A}_{1}=0.9\mathrm{sin}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\theta \le 90\u02da\\ {B}_{1}=0.21{\left[\frac{{D}_{cv}}{{R}_{0}}\right]}^{i},{R}_{0}=\frac{h-{D}_{ra}/2}{1-\mathrm{cos}\theta}\\ {C}_{1}=1\end{array}$ (3)

*R*_{0} is the radius of curvature of the bending segment. When the ratio of *D _{cv}* to

${E}_{b}=0.189\mathrm{sin}\theta {\left[\frac{{D}_{cv}\left(1-\mathrm{cos}\theta \right)}{h-{D}_{ra}/2}\right]}^{i}\frac{\rho {Q}_{cv}^{3}}{2{S}_{cv}^{2}}$ . (4)

When blood transports in vessels, the frictional forces will cause pressure loss, thereby consuming energy. The energy loss rate due to friction [2] in this segment could be expressed as:

Figure 2. Blood flow in bending cephalic vein.

${E}_{f}=\frac{128\mu \left(h-{D}_{ra}/2\right)\theta {Q}_{cv}^{2}}{\pi {D}_{cv}^{4}\left(1-\mathrm{cos}\theta \right)}$ . (5)

2.1.2. Energy Loss rate at Anastomosis

Placement of the fistula allows the blood flow in RA to bypass the distal resistance generated by the capillary in the fingers and return to the heart through the anastomosis. For that reason, *Q _{t}* flow rate direction of the distal artery changes and flows into the vein together with

${E}_{a}={E}_{t}+{E}_{h}=\left({K}_{t}+{K}_{h}\right)\frac{\rho {U}_{con}^{2}}{2}{Q}_{con}=\left({K}_{t}+{K}_{h}\right)\frac{\rho {Q}_{cv}^{3}}{2{S}_{con}^{2}}$ . (6)

The minor loss coefficients at toe and heel are *K _{t}* and

${K}_{t}=2{\left(1-{q}_{t}\right)}^{2}\frac{1}{{\phi}_{t}}\mathrm{cos}\left(\frac{\pi +3\theta}{4}\right)-2{q}_{t}^{2}\frac{1}{{\phi}_{t}}\mathrm{cos}\left(\frac{3\theta}{4}\right)+{q}_{t}^{2}\frac{1}{{\phi}_{t}}+1$ (7)

${K}_{h}=2{q}_{h}^{2}\frac{1}{{\phi}_{h}}\mathrm{cos}\left(\frac{\pi +3\theta}{4}\right)-2{\left(1-{q}_{h}\right)}^{2}\frac{1}{{\phi}_{h}}\mathrm{cos}\left(\frac{3\theta}{4}\right)+{q}_{h}^{2}\frac{1}{{\phi}_{h}}+1$ , (8)

where *q _{t}* and

$\{\begin{array}{l}{q}_{t}=\frac{{Q}_{t}}{{Q}_{con}}\\ {q}_{h}=\frac{{Q}_{h}}{{Q}_{con}}\end{array}$ , $\{\begin{array}{l}{\phi}_{t}=\frac{{S}_{t}}{{S}_{con}}\\ {\phi}_{h}=\frac{{S}_{h}}{{S}_{con}}\end{array}$ . (9)

Next, we introduced the calculation of the cross-sectional area of the total blood flow at the confluence (*S _{con}*), as shown in Figure 4. In the anastomosis operation, surgeons cut CV obliquely to obtain an elliptical cross-section

${A}_{a\text{\_}cv}=\frac{{D}_{cv}}{\mathrm{sin}\theta},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{a\text{\_}cv}={D}_{cv}$ . (10)

Then RA is cut along the axial direction and opened along the radial direction of the blood vessel to obtain an elliptical anastomosis *S _{a}* with a major axis

Figure 3. Blood flow at anastomosis.

Figure 4. 3D simplified diagram of the anastomosis operation. Top is the original CV artery before being stretched (middle) to be linked to the RA (bottom).

Figure 4 with the major axis *A _{con}* and the minor axis

$\{\begin{array}{l}{A}_{con}={D}_{a}\mathrm{sin}\theta -\frac{\left({D}_{a}\mathrm{sin}\theta -{D}_{cv}\right){D}_{a}\mathrm{cos}\theta}{2h+{D}_{a}\mathrm{sin}\theta \left(1-\mathrm{cos}\theta \right)}\\ {B}_{con}=\frac{{D}_{ra}}{2}+\frac{\left(2{D}_{cv}-{D}_{ra}\right){D}_{a}\mathrm{cos}\theta \left(1-\mathrm{cos}\theta \right)}{4h\theta}\end{array}$ (11)

Therefore, the cross-section of the total blood flow at the *S _{con}* confluence could be obtained from:

${S}_{con}=\frac{\pi {A}_{con}{B}_{con}}{4}$ . (12)

2.1.3. Total Energy Loss Rate in RCAVF

From Equations (4), (5), and (5), the total energy loss rate in RCAVF (*E _{avf}*) was given by the following equation:

$\begin{array}{c}{E}_{avf}={E}_{b}+{E}_{f}+{E}_{a}\\ =0.189\mathrm{sin}\theta \sqrt{\frac{{D}_{cv}\left(1-\mathrm{cos}\theta \right)}{h-{D}_{ra}/2}}\frac{\rho {Q}_{cv}^{3}}{2{S}_{cv}^{2}}+\frac{128\mu \left(h-{D}_{ra}/2\right)\theta {Q}_{cv}^{2}}{\pi {D}_{cv}^{4}\left(1-\mathrm{cos}\theta \right)}+\left({K}_{t}+{K}_{h}\right)\frac{\rho {Q}_{cv}^{3}}{2{S}_{con}^{2}}\end{array}$ (13)

where *K _{t}*,

Figure 5. Top view, 3D view, and front view of cephalic vein (CV) after stretching. (a) Top view; (b) 3D view; (c) Front view.

2.2. Model Validation

In order to validate the model, we compared *E _{avf}* calculated by Equation (13) and

In Equation (13), there was a total of 5 GPs (*D _{ra}*,

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}*,

$\frac{\rho}{2}\frac{{Q}_{t}^{3}}{{S}_{t}^{2}}+{P}_{t}{Q}_{t}+\frac{\rho}{2}\frac{{Q}_{h}^{3}}{{S}_{h}^{2}}+{P}_{h}{Q}_{h}=\frac{\rho}{2}\frac{{Q}_{cv}^{3}}{{S}_{cv}^{2}}+{P}_{cv}{Q}_{cv}+{E}_{avf\text{\_}num}$ . (14)

*Q _{t}* and

3. RESULTS AND DISCUSSIONS

3.1. Comparison between Numerical and Analytical Results

We calculated *E _{avf}* for 25 models using Equation (13) and compared them with

Figure 6. RCAVF simulation model.

Figure 7. Comparison of numerical results and analytical results in 25 models.

Table 1. The research ranges of 5 GPs.

Table 2. Parameters of simulation models.

Table 3. Relative difference between numerical results and analytical results.

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

3.2. Relationship between *E _{AVF}* and GPs

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

Thus, Figure 8 shows the relationship between *E _{avf}* and

In Figure 8(b), we set *D _{cv}*.

Figure 8(c) portrayed the relationship curve between *E _{avf}* and

Figure 8(d) gives the evolution of *E _{avf}* as a function of

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

Figure 8. The relationship between energy loss (*E _{avf}*) and radial artery diameter (

Figure 9. The relationship between energy loss (*E _{avf}*) and cephalic vein diameter (

Figure 10. The relationship between energy loss (*E _{avf}*) and vessel distance (

Figure 11. The relationship between energy loss (*E _{avf}*) and anastomotic diameter (

Figure 12. The relationship between energy loss (*E _{avf}*) and anastomotic angle (

2) Compared to *D _{ra}*,

Figure 8 and Figure 9 show that when *D _{ra}* changed, the range of variation of

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

4) When *h* was large, it was recommended to choose a larger *D _{cv}* or decrease the distance between the vessels to reduce

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

3.3. Critical Energy Loss Rate (CEL)

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

3.3.1. CEL Definition

The vascular resistance generated at the anastomosis makes it difficult to increase the blood flow in RA, which will lead the blood flow in CV difficult to reach the dialysis standard, thus making the non-mature fistula. Critical energy loss rate referred to the maximum allowable energy loss rate in the RCAVF. When the energy loss rate in RCAVF was less than this threshold, the maturity of the fistula could be guaranteed.

To define the CEL, *h *was set according to the distribution of blood vessels on the forearm of each patient. From the result (3.2.c), since *D _{a}* had a relatively lesser impact on energy loss compared to

This value of CEL was calculated for the critical values of *D _{ra}* and

3.3.2. CEL to Help for RCAVF Anastomoses Design

In reality, *D _{ra}*,

Table 4. Definition of CEL.

In this analysis, we maintained *D _{a}* at a constant value of 10 mm, which corresponded to the value practiced by our hospital partners.

We divided the anastomosis design into three possible levels, as shown in Table 5. When *θ* was greater than or equal to *θ _{a}*,

Not all the blood vessels could get *θ _{a}* and

Figure 13. The relationship between *E _{avf}* and

Table 5. Recommended level for anastomosis design.

Type I: *θ _{a}* and

Type II: Only *θ _{b}* could be calculated,

Type III: Neither of *θ _{a}* or

We recommended that surgeons choose the blood vessel Type I with an A-level anastomosis design to ensure RCVAF maturity and primary patency. For the blood vessel of Type I and Type II with B-level anastomosis design, the surgeons needed to evaluate the patient’s vascular performance in advance. For RCAVF with C-level anastomosis, all three types of blood vessels were not recommended.

4. CONCLUSIONS

In most cases, native RCAVF is the preferred VA strongly recommended by clinical practice guidelines [1 , 20]. It has the advantages of longer survival patency, fewer complications, and lower risk of infection, making it have lower morbidity and mortality and could reduce the patient’s financial burden. Nevertheless, its high early failure rate (46%) [21] is still a big problem now. A recent study [22] proposed that the traditional vein-to-artery configuration could be changed to the artery-to-vein, thereby increasing the maturity rate and primary patency of the native fistula. Our study also concentrated on fistula configuration optimization. It was the first study that proposed to help surgeons design the fistula shapes from the perspective of energy loss, thereby increasing the RCAVF maturity rate.

Through this study, we analyzed the relationships between the energy loss rate in RCAVF and its five associated GPs, and then verified their accuracies with numerical results. It was found that *D _{a}* and

To help surgeons create a RCAVF plan for patients, we defined two energy loss rate thresholds: *CEL _{a}* and

This new complete model studied the different GPs. The anastomotic angle seemed to be an important parameter, more than the anastomotic diameter. This result may raise some questions about surgical theories regarding the treatment of AVF complications, such as high flow correction. Depending on the model, reducing *D _{a}* to reduce high-flow AVF may not have real meaning. To verify this result, other studies must be conducted on mature AVF using this model.

This model can also be used to design other native AVFs and AVG (arteriovenous graft) shapes. In use, the inlet flow rate and GPs need to be reset, and the minor loss coefficients *K _{t}* and

ACKNOWLEDGEMENTS

Thanks to the Chinese government for funding this research and to the Hospices Civils de Lyon to provide the clinical data for this study and participate in this work.

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