Want to save time when ordering extrusions for your next catheter project? This article will help you calculate the diameter of your next coiled catheter.

## Background

Many neuro and peripheral catheters are made by a process of reflowing or laminating polymers and reinforcement materials over a mandrel. The reinforcement material we'll consider in this article is a coil. A coil winder is used to wind a wire over a catheter mandrel. Then a polymer jacket is sized to slide over the coil during assembly. Heat shrink tubing is then slid over the jacket. During the catheter reflow process, the jacket melts and is compressed by the heat-shrink. This causes the jacket to reduce in diameter and increase in wall thickness.

## Conservation of Area

For starters, let's think of the cross section of a reflowed catheter without a coil. There's a mandrel, liner, and polymer jacket. During the catheter reflow process the jacket melts and is compressed by the heat shrink tubing. This reduces inner and outer diameter of the jacket. Since no material is lost during the catheter reflowing process, the cross-sectional area of the jacket must be the same before and after reflowing. We can use this to calculate the relationship between the pre and post reflowed jacket diameters.

Keep in mind that the final ID of the jacket would be the liner OD, or the diameter of the mandrel plus twice the liner thickness.

## Enter the coil

Now let's add a coil. The conservation of

area method is invalidated because the coil is not a uniform tube. In fact, the jacket is no longer a uniform tube.

The reflowed jacket has a helical void in it because of the space the coil takes up inside of it. Instead of thinking of a conservation of area, we can use a conservation of volume to equate the pre and post lamination conditions. Again, we assume that we're not losing any material during catheter reflow. Let's cut a section of coiled catheter into a unit length and work out the volumes of each component. We'll use 1 inch for the length unit and use inches for all the diameter and volume calculations too.

Finding the volume of the jacket is straightforward. We just multiply the cross-sectional area of the jacket (first equation) by the length. Since the length is 1, it's unchanged. For the coil volume it's the same plan. We'll multiply the area of the coil wire by the length of the coil wire in a unit length of catheter. Visualize un-wrapping the coil wire along the coiled length. The result forms a right triangle.

The height of the triangle (a) is 1 unit (the same unit as the rest of your calculations). For the base of the triangle (b), think of how many times the cylinder would turn if you unwrapped the coil. The number of turns can be represented by the length of the cylinder (1 unit) divided by the pitch of the coil. We multiply the number of turns by the coil mean diameter and Pi to find this leg of the triangle.

Great, we have the length of each leg, now we just apply the Pythagorean theorem to get the wire length. Note that this number expresses how much coil wire is required per length of catheter. So, if you wanted to know how much total wire a catheter would take, you just multiply this number by the length of the catheter.

To find the volume of coil wire in a unit length, we now multiply the cross-sectional area of the wire by the length of the wire. If using a round wire you'd use the equation for the area of a circle. If you're using flat wire, you'd multiply the wire width by the thickness. Add the volume of the coil to the volume of the jacket and that gets us the total volume before and after reflowing! Let's remember the equation for the area of a pipe and assume that the area is equal to the volume since the length is 1. Solving for OD we get:

## Conclusion

Using this method, you can calculate the OD and wall thickness of polymer jackets for reflow. It should save you a few prototype iterations when designing a new catheter. Additionally, you can fine-tune the OD of the device by varying the coil pitch. If reflowing a braided shaft, the volume of the braid can be calculated similarly based on the braid angle, number of wires, and cross section of the wire. Of course, these calculations should only be used as a starting point. Fine-tune your catheter OD with prototypes and testing. Good luck!