

ORIGINAL ARTICLE 

Year : 2021  Volume
: 8
 Issue : 4  Page : 351354 

The mathematical basis of multisheath vascular access
Arvind Lee^{1}, Omar Aziz^{2}, Gurkirat Singh^{2}
^{1} Department of Vascular Surgery, Nepean Hospital, Penrith, NSW; Department of Surgery, Nepean Clinical School, University of Sydney, Sydney, Australia ^{2} Department of Vascular Surgery, Nepean Hospital, Penrith, NSW, Australia
Date of Submission  23Jun2021 
Date of Acceptance  02Jul2021 
Date of Web Publication  9Dec2021 
Correspondence Address: Arvind Lee Department of Vascular Surgery, Nepean Hospital, Penrith, NSW; Department of Surgery, Nepean Clinical School, University of Sydney, Sydney Australia
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/ijves.ijves_69_21
In complex endovascular interventions, there is often a need for multiple smaller sheaths to be placed parallelly inside a larger sheath to gain simultaneous access into different vessels. Here, we describe one such case of fenestrated repair of a juxtarenal aneurysm from our recent experience. This case required simultaneous cannulation with sheaths in both renal arteries and the superior mesenteric artery through the fenestration of a custommade fenestrated stent graft. This paper aims to discuss, in simple terms, the mathematical basis behind calculating the diameters of smaller sheaths inside a larger sheath. Three different configurations are discussed – two, three, and four sheaths within a larger sheath. For simplicity, the inside sheaths are all of the same outer diameter and the diameter of all the sheaths is assumed to remain uniform throughout their lengths.
Keywords: Access, endovascular, fenestrated aneurysm
How to cite this article: Lee A, Aziz O, Singh G. The mathematical basis of multisheath vascular access. Indian J Vasc Endovasc Surg 2021;8:3514 
Introduction   
Every sheath has an inner and outer diameter, with a circular profile. While the inner diameter is easy to estimate based on the French size of the sheath, the exact outer diameter is often only available with the manufacturer of the particular sheath. For example, a 45 cm 6 Fr Pinnacle Destination sheath (Terumo) has an outer diameter of 2.77 mm while the same sheath but 65 cm long has an outer diameter of 2.82 mm.^{[1]} [Figure 1] shows a case example of a 3 vessel fenestrated repair for a juxtarenal abdominal aortic aneurysm. The 3 fenestrations and visceral arteries were simultaneously cannulated with 6 Fr 45 cm Terumo sheaths placed within a 20Fr sheath in the contralateral groin.  Figure 1: (a) Preoperative image of a juxtarenal abdominal aortic aneurysm not amenable for standard endovascular repair. (b) Threevessel custommade fenestrated stent graft inserted from the right side and three parallel 6 Fr sheaths inserted through and 20 Fr sheath in the left groin. (c) Both renal and superior mesenteric arteries simultaneously cannulated with a sheath and covered stent. (d) Completion angiogram showing successful exclusion of aneurysm with preserved visceral branches
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In the three scenarios that are discussed here, this paper will work out the relationship between the inner diameter of the outside sheath (D) and the outer diameter of the inner sheaths (d) and their respective radii (R and r) using the considerations below.
When a circle is divided into portions which are enclosed by two radii and an arc of its circumference, the portion is called a sector. The area of any sector is determined by the angle between the 2 radii, θ [Figure 2].  Figure 2: A sector shown within a circle of radius R and angle between radii of θ
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The largest circle that one can place inside a given sector will be the one that contacts the arc and the 2 radii tangentially [Figure 3].
If one was to drop a line from the center of the circle inside the sector, to the tangent at each radius, the lines would meet at a right angle and we get two equal triangles [Figure 4]. The two triangles will divide the angle θ equally. As shown in [Figure 4], the side opposite each θ/2 angle would have a length equal to the radius of the smaller circle, r. The hypotenuse of each triangle will be the difference between the radius of the sector, R, and the radius of the smaller circle, r.  Figure 4: 2 equal right angle triangles with angle θ/2 opposite the side with length r and hypotenuse Rr
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We know that the sine rule for triangles states that sine θ = opposite/hypotenuse. Hence, we obtain the formula, Sine θ/2 = r/R − r.
Equal Sheaths Inside a Larger Sheath   
Two samesized sheaths inside a larger sheath will occupy their own equal semicircular sectors. Common sense would suggest that the two largest but equal sheaths that can be placed inside would each have an outer diameter equal to half the inner diameter of the larger sheath.
Proving this mathematically is quite simple. The angle θ in this scenario is 180 [Figure 5]. Using the above formula, Sin 180/2 = r/R − r
Now, we know, Sin 90 is 1.
Hence, it follows that
1 = r/R − r
R − r = r
R = 2r
As an example, if one needed to place two 6 Fr Destination (Terumo) 45 cm sheaths inside a larger sheath, the minimum sheath size to allow this would have a radius of 2.77 mm and a diameter of 5.54 mm. This in French terms would be 16.62 Fr. The same configuration using two 65 cm long sheaths would require a minimum sheath size of 16.92 Fr.
Equal Sheaths Inside a Larger Sheath   
Three samesized sheaths inside a larger sheath will divide the inside of the larger sheath into three equal sectors, with an angle θ of 120 [Figure 6]. Using the above formula,
Sin 120/2 = r/R − r
Now, we know Sin 60 is
Hence, it follows that
Using the numerical value for which is 0.866 leads to
r = 0.866R/1.866
As an example, if one needed to place three 6 Fr Destination 45 cm sheaths inside a larger sheath, then as the outer diameter of each sheath is 2.77 mm (radius 1.385 mm), the minimum radius of the larger sheath would have to be 2.984 mm and hence diameter 5.96 mm. This in French terms would be 17.9 Fr.
Equal Sheaths Inside a Larger Sheath   
Four samesized sheaths inside a larger sheath will divide the inside of the larger sheath into four equal sectors, with an angle θ of 90 [Figure 7]. Using the above formula,
Sin 90/2 = r/R − r
Now, we know Sin 45 is
Using the numerical value for which is 0.707
r = 0.707R/1.707
As an example, if one needed to place four 6 Fr Destination 45 cm sheaths, then the minimum radius of the larger sheath would have to be 3.344 mm and diameter 6.688 mm. This in French terms would be 20.06 Fr which in practice would be a 22 Fr outer sheath.
Addition of a Central Wire   
In the above scenarios, there is no adjustment made for the presence of a central luminal wire of the main sheath. If the central wire is maintained, the following calculation can be used to determine the respective diameters.
In both the threesheath and foursheath scenarios, the following formula applies [Figure 8] and [Figure 9].  Figure 8: The addition of a central wire with diameter 2x along with 3 equal sheaths with diameter 2r
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 Figure 9: The addition of a central wire with diameter 2x along with 4 equal sheaths with diameter 2r
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If x is the radius of the largest wire possible, then R = x + 2r and hence x = R − 2r.
In the above example of three 6 Fr 45 cm Destination sheaths, if the outer sheath was 18 Fr, then the maximum possible radius of the central luminal wire would be 3 mm (radius of the inner diameter of an 18 Fr sheath) minus 2.77 mm (outer diameter of the 6 Fr sheath) which is 0.23 mm. Hence, the maximum possible diameter of the central wire is 0.46 mm, which is an 0.018” wire and not an 0.035” wire. To allow for a 0.035” wire (0.889 mm), the outer sheath will have to have a diameter of 6.429 mm which will be a 20 Fr sheath.
[Table 1] summarizes the above findings and may be used as a handy reference prior to cases. The table provides a formula to calculate the diameter (D) of the outer sheath required to accommodate the three scenarios with and without a central wire (diameter X) using the known value of the outer diameter of the inner sheaths (d).  Table 1: Summary of formulae for calculating the diameter of larger sheath D with respect to multiple smaller sheaths of diameterd
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Discussion   
We believe that this is the first paper that addresses the mathematical principles that govern the fitting of multiple sheaths inside a larger sheath. Multisheath access is an essential part of complex endovascular procedures such as fenestrated,^{[2]} chimney,^{[3]} and sandwich^{[4]} repairs of aortic aneurysms. In a fenestrated repair for example, the contralateral groin requires placement of a large sheath to accommodate multiple smaller sheath access into the branches, as shown in the case example [Figure 1]. The required size of the larger sheath to achieve this depends on the number of branches that need cannulation, the choice of stents for the branches, and the size and length of the sheaths used to secure the cannulation.
There are, of course, variables other than the individual sheath size that will determine the freedom with which each individual smaller sheath moves within the larger sheath. This includes tortuosity, extrinsic compression by calcification, the presence of dried blood with increase in friction, and the number of smaller sheaths that are already in place. In a fourvessel fenestrated repair for example, there is a higher likelihood of competition for space during advancement of a fourth sheath. If there is significant resistance in advancing the final sheath in these cases, the final branch may be left cannulated with a wire alone. If any such variables exist in a case and are anticipated at planning, then a larger outer sheath than the minimum required may be chosen at the outset.
For multisheath access, the type of valve on the larger sheath must also be considered. In our experience, the crosscut valves of the Cook Medical sheaths are preferred over the Gore DrySeal valve. The crosscut valve allows each smaller sheath to occupy distinct quadrants of the valve ensuring there is no leakage of blood through the valve and also limiting competition between the sheaths when one is moved.
We hope that this paper helps practitioners understand, in simple terms, the mathematical basis for sheath size selection during cases that require multisheath access.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
References   
1.  
2.  Ricotta JJ 2 ^{nd}, Oderich GS. Fenestrated and branched stent grafts. Perspect Vasc Surg Endovasc Ther 2008;20:17487. 
3.  Kansagra K, Kang J, Taon MC, Ganguli S, Gandhi R, Vatakencherry G, et al. Advanced endografting techniques: Snorkels, chimneys, periscopes, fenestrations, and branched endografts. Cardiovasc Diagn Ther 2018;8:S17583. 
4.  Lobato AC, CamachoLobato L. A new technique to enhance endovascular thoracoabdominal aortic aneurysm therapyThe sandwich procedure. Semin Vasc Surg 2012;25:15360. 
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9]
[Table 1]
