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Backflow mitigation in stepped catheters

Raghavan R.1, Brady M.L.1, Grabow B.3,

and Block W.2,3,4

1Therataxis LLC, Baltimore, MD 21218, USA;

Departments of 2Medical Physics, 3Bioengeneering , University of Wisconsin, Madison, WI

53715, USA; 4inseRT MRI, Madison, WI

CORRESPONDING AUTHOR:

Raghu Raghavan

Therataxis, LLC

1101 E. 33rd Street, Ste. B305, Baltimore, MD 21218

(443) 451-7154 (phone); (443) 451-7157 (fax)

raghu@therataxis.com

SHORT TITLE:

Stepped catheters

KEY WORDS:

Convection-enhanced delivery, backflow, intraparenchymal infusion, magnetic resonance

imaging, reflux-resistant, catheters, prestress.

2

ABSTRACT

Background: Backflow of infusate along the catheter shaft can lead to undesirable distributions

of drug during convection-enhanced delivery into the brain parenchyma. Stepped catheter

geometries have been suggested to limit backflow.

Objective: The principal objective of this study was to compare two different-sized step catheter

designs, SmartFlow™ 16- (SF16)and 14-gauge (SF14) catheters, for their backflow

characteristics at various flow rates.

Methods: The experiments were conducted in live porcine brains, with catheters were inserted

into the thalamus. In-vivo infusions, monitored continuously, of diluted gadodiamide at flow

rates increasing from 2.5 L/min to 10 L/min were conducted to observe the backflow in the SF

catheters which were placed at the same location in either thalamus.

Results: The smaller SF16 was far more likely to restrain backflow at the 3mm step than the

larger SF14, particularly at the lower flow rates. This led to lower average backflow (3.9±2.0

mm for the 16-gauge vs. 9.0±5.0 mm for the 14-gauge at 2.5 μl/min).

Conclusions: The SF16 stepped catheter proved more likely to restrict backflow at the step. A

simple theoretical estimate, based on our previous study of backflow, suggests that prestress in

the tissue is an additional effect on backflow –compared with the effects accounted for in the

usual theories of backflow -- that may account for the superior performance. However, the

SF16 has then two features which reduce backflow in comparison with the SF14: the smaller

diameter of the tip, and the larger width of the step. The experiments could not disentangle the

effects due to each, and unclear whether the smaller diameters of the step and shaft, or the

larger step size are primarily responsible for the difference. Further studies would be valuable

in clarifying the causes of backflow reduction.

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INTRODUCTION

MATERIALS AND METHODS

All experiments were performed at the University of Wisconsin, Madison.

Animal model: All infusions were targeted in the thalamus of adolescent swine, as it is a large

enough volume to contain multiple high-volume infusions during experiments at high flow rates.

The swine was chosen over NHP models for its similarity of head size to humans, appropriate

use of animal resources, lack of disease transmission risks, and substantial reduction in

required animal care personnel during the procedure. The swine has also been extensively

utilized elsewhere for investigations involving CED procedures in the brain [1-3].

Surgery and Targeting Method.

We utilized a platform for MR-guided neurological surgery that was first implemented at UW-

Madison [4-6] and is now being extended by a commercial entity (inseRT MRI, Madison, WI).

The platform accesses the scanner hardware through a commercial portal termed RTHawk,

(HeartVista, Palo Alto, CA) [4], that permits the design and execution of image-guided

procedures using high level imaging plug-ins to define a procedural workflow. The platform

allows the interventionalist to manipulate the brain ports used to orient catheters with real-time

feedback similar to a stereotactic OR suite, rather than the iterative “shoot and scoot”

manipulation provided commercially in MR image guidance today.

Through a bidirectional link with RTHawk, image data acquired in real-time was displayed by

Vurtigo (Visual Understanding of Real-Time Image Guided Operations, Sunnybrook Health

Sciences Centre; Toronto, Canada), an open-source visualization platform that allows

simultaneous display and interaction with multiple 3D and 2D datasets [7]. Changes to the

desired acquisition geometry made in Vurtigo were sent to RTHawk and then on to the scanner,

allowing visualization of dynamically acquired 2D planes overlaid on previously acquired 3D

volumes. This link was implemented using a “geometry server” software program that

communicates with RTHawk and Vurtigo.

Real-time scan control and visualization was conducted on a high-performance external

workstation with two quad-core Intel Xeon E5620 2.4 GHz CPUs, 12 GB of memory, an NVIDIA

GF100 Quadro 4000 graphics card, and gigabit Ethernet controller, running 64-bit Linux.

Scanner interface was via an internal Ethernet switch. Visualization display was available on a

screen in the control room, which was placed in the scanner room window so that an operator

could lean into the bore and reposition the MR-visible fluid-filled alignment guide to the optimal

trajectory angle

The device targeting capability is based on a real-time implementation of prospective stereotaxy

[8]. The NavigusTM brain port consists of a MR-visible fluid-filled alignment guide seated in a

ball-and-socket pivot joint with two degrees of rotational freedom. The visible tip of the guide

was positioned at the center of the pivot joint, and the body of the guide extended away from the

skull along the trajectory of device insertion. We identified the desired target point in the brain

and the location of the pivot point by first acquiring a high resolution 3D T1-weighted “roadmap”

volume with inversion recovery preparation. After identifying location markers of the alignment

guide pivot point and the Ce target point, the trajectory alignment software tool calculated an

“aiming point” outside the skull that was co-linear with the target and alignment guide pivot

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points. Real-time imaging of a plane perpendicular to and centered on the aiming point provided

dynamic feedback during the alignment step, allowing the operator to incrementally move the

alignment guide until the distal end of the alignment guide overlapped with the software-

displayed aiming point.

When the trajectory angle (antero-postrerior, medio-lateral direction) of the fluid-filled alignment

guide was confirmed to be on target, the alignment stem was locked into position. The fluid-filled

alignment guide was removed, the remote introducer was fastened to the stem, and the guiding

insert was placed in the guiding stem. The catheter for the infusion was threaded through the

remote introducer and the guiding insert, and was fastened to the remote introducer by a locking

mechanism.

Catheters

Two sizes of SmartFlow™ multi-stepped catheter (MRI Interventions, Inc., Memphis, TN, USA)

were compared in this study. The two catheters, shown in Fig. 1, differ primarily in the outer

diameters of their three sections. Both catheters have a 3mm length tip with open end port,

followed by at step to an 18mm long central shaft of larger diameter, which then tapers to a 16-

or 14- gauge main shaft. Table 1 lists the outer diameters of tip and central shaft, and the

resulting computed step width (half the difference between shaft and tip diameters).

Infusion.

Silica infusion lines were used to connect the catheter to 5cc. Hamilton gas-tight syringes. The

syringes were driven by an MRI-compatible syringe pump (PHD 2000, Harvard Apparatus, Inc.,

Holliston, MA, USA) placed at approximately the same height as the catheter tip. A solution of

gadodiamide (2 mmol/l; Omniscan™, GE Healthcare Inc., Princeton, N.J., USA) in phosphate-

buffered saline was infused. For each catheter placement, backflow measurements were

obtained at multiple flow rates. In three of the four animals, the initial flow rate was 2.5 μl/min,

while the fourth was started at 5.0 μl/min. The infusion was run at the initial flow rate for 5

minutes, and then stopped for 8-9 minutes while a high resolution 3D MRI image was acquired.

This process was repeated at flow rates increasing by 2.5 μl/min at each iteration. The flow rate

was incremented in this manner until either the backflow reached a CSF boundary (at which

point no additional backflow is expected) or a maximum flow rate of 10.0 μl/min was reached.

MRI Acquisition.

All procedures were monitored with a 3-Tesla GE SIGNA (MR 750) MRI scanner (GE

Healthcare, Waukesha, WI). During infusion, the infusion was monitored with real-time 2D

imaging. A single slice aligned with the catheter was acquired at 6 sec intervals (TR = 33 ms,

TE = 4.5 ms, flip angle = 50 o, FOV = 140 x 105 mm, matrix = 256x192, slice thickness = 2.5

mm). After each infusion, a 3D fast spoiled gradient echo (FSPGR) scan was acquired (TR =

9.1 ms, TE = 3.9 ms, in-plane FOV = 180 mm, matrix = 256x224, slice thickness = 0.8 mm, 248

contiguous slices spaced at 0.4 mm intervals.)

Backflow Measurement

Backflow was identified and monitored during each five-minute infusion using in real time MRI.

Backflow distance was measured in the 3D T1-weighted scan acquired just after the flow was

stopped. The distance was measured from the catheter tip to the furthest extent of visible tracer

along the catheter shaft using custom 3D image processing software. Measures were recorded

to 0.5mm precision, which was limited by the resolution of the MR imaging.

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Statistical Analysis

Backflow distances were compared using a two-sample t-test, assuming equal variances. Two-

tailed tests were used, with statistical significance defined as p < 0.05. Grouped data are

presented in the test as M ± SD.

RESULTS

Four pigs were used for a total of 16 infusion sites. Each pig therefore served for four infusion

sites which consisted of paired sites in each hemisphere, with two sites in the thalamus in each

hemisphere. Tables 2A and 2B summarize the set of backflow distances measured in the

study.

Most infusions started at 2.5 μl/min, and all reached at least 5.0 μl/min before the backflow

reached a CSF boundary. At both 2.5 and 5.0 μl/min, the average backflow of the 16-gauge

catheter was less than that of the 14-gauge, reaching statistical significance at 2.5 μl/min

(p=0.040) and just failing at 5.0 μl/min (p=0.051). However, it should be noted that most (6 out

of 8) of the 14-gauge infusions reached a CSF boundary at 5.0 μl/min, which limited their

backflow distance. The remaining two reached CSF at 7.5 μl/min. In contrast, none of the 16-

gauge infusions reached CSF at 5.0 μl/min and only two at 7.5 μl/min.

Statistical analysis of the backflow distances may not be appropriate in this study, due to the low

number of samples and high standard deviation of the results. Furthermore, the backflow

distance measure does not appear to be normally distributed. In every case, the backflow was

observed to reach at least to the catheter step, 3 mm from the tip. Furthermore, in the 16-gauge

catheter backflow distances were clustered around the 3mm value. Five out of six backflows at

2.5 μl/min and five of eight at 5.0 μl/min are in the range of 3-4mm. The remaining four

backflow that overran the step had widely distributed backflows, from 6 – 12 mm. By contrast,

only one of the 14-gauge catheter infusions, at 2.5 μl/min, had a backflow in the 3-4 mm range.

Images of the full set of eight infusions with each catheter at 5.0 μl/min are shown in Fig. 2. T1-

weighted 3D MR data sets acquired after completion of the five-minute 5.0 μl/min infusion were

rotated so that the catheter appears in-plane in each image. The infusate appears to be

concentrated around the tip in most of the 16-gauge infusions (upper row), while the 14-gauge

infusions are distributed around a greater length of the central shaft (lower row).

DISCUSSION

It appears from this study that the step in the 16-gauge catheter does tend to restrict backflow.

The barrier is not absolute. A fraction of the infusions did backflow past the step, and this

fraction increased with the flow rate. For those infusions that did overrun the step, the backflow

varied widely. Typically, a large increase in backflow was seen at the lowest flow rate to

overrun the step, and much smaller increases with flow rate thereafter. Furthermore, among

those infusions at a given flow rate that had overrun the step, there was little difference in

between the 16- and 14-gauge catheters; i.e., their difference lies mainly in the tendency to

overrun the step, not in the performance for backflow beyond the step.

It is not unexpected that a smaller catheter would display lower backflow. Theoretical models

and experiment have demonstrated a relation between catheter radius and backflow distance in

straight open end port catheters: the simplest such model is described in [9]. For more

numerical and recent approaches, see [12], [13]. However, the mechanism underlying the

backflow-restricting properties of stepped designs is not well understood. It is important to note

that the 16-gauge catheter is not a proportionally scaled down version of the 14-gauge. While

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the tip and shaft are both smaller than their counterparts in the 14-gauge design, the size of the

step is in fact larger by nearly a factor of two. It is not clear from the data itself whether the

larger step or the smaller radii play a greater role in the improved performance.

In order to better visualize the effect of the step, we undertook some experiments in an Agarose

gel, which we describe here.

Materials and Methods

All experiments were performed at the Engineering Resources Group in Hialeah, FL, USA. The

gel used was a 0.6% agarose gel prepared in a standard way [10]. A catheter with a step, with

the same dimensions as the SF16 was then used in the experiment.

Infusion.

The images shown were taken when the catheter was inserted at a relatively fast insertion (5

mm/sec). This is to ensure that it encounters sliding friction when inserted, and not stick-slip

friction which will obtain at slow insertion speeds. The latter would allow a punch to develop

elastically and then tear, resulting in reflux due to gel damage rather than backflow which is due

to response within the domain of elastic response and porous flow.

Results

It is seen that the flow has difficulty penetrating a small region in front of the step, which

therefore acts as a barrier. It is also interesting to compare this with the behavior reported in

[11] where the shaft of a cylindrical catheter with no step is shown to have accumulated tissue in

front of it at slow insertion speeds, and this has the opposite effect that the step does: namely it

seems to increase the reflux due to the tissue damage caused.

In the Appendix, we show a simple calculation that indicates that prestress in addition to the

conventional factors that affect backflow may be sufficient to explain the phenomena observed.

Further discussion of the issues that need to be examined is discussed there.

ACKNOWLEDGEMENTS

This research was supported by the Kinetics Foundation.

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REFERENCES

[1] Martin L. Brady, Raghu Raghavan, Deep Singh, P J Anand, Adam S. Fleisher, Jaime Mata,

William C. Broaddus, and William L. Olbricht. In-vivo performance of a micro-fabricated catheter

for intraparenchymal delivery. Journal of Neuroscience Methods. 229: 76-83, 2014.

[2] Martin Brady, Raghu Raghavan, Zhi jian Chen, and William C. Broaddus. Quantifying fluid

infusions and tissue expansion in brain. IEEE Transactions on Biomedical Engineering,

58:2228-2237, 2011.

[3] A. Jones, A. Bienemann, N. Barua, P. J. Murison, and S. Gill, "Anaesthetic complications

pigs undergoing MRI guided convection enhanced drug delivery to the brain: a case series,"

Veterinary anaesthesia and analgesia, vol. 39, pp. 647-52, 2012.

[4] Emborg ME, Joers V, Fisher R, Brunner K, Carter V, Ross C, et al. Intraoperative

intracerebral MRI-guided navigation for accurate targeting in nonhuman primates. Cell

transplantation. 2010;19(12):1587-97. doi: 10.3727/096368910X514323. PubMed PMID:

20587170; PubMed Central PMCID: PMC3278961.

[5] Brodsky EK, Block WF. Intraoperative device targeting using real-time MRI. Biomedical

Sciences …. 2011:6-9.

[6] Grabow B, Block W, Alexander AL, Hurley S, CD R, Sillay K, et al., editors. Extensible real-

time MRI platform for intraoperative targeting and monitoring. Society of Brain Mapping and

Therapeutics; 2012; Toronto.

[7] Radau, P. E. et al. VURTIGO : Visualization Platform for Real-Time , MRI-Guided Cardiac

Electroanatomic Mapping. 244-253 (2012).

[8] Truwit, C.L and Liu, H. Prospective stereotaxy: a novel method of trajectory alignment using

real-time guidance. Journal of Magnetic Resonance Imaging, vol 13, 452 – 457 (2001).

[9] Raghu Raghavan, Samuel Mikaelian, Martin Brady, and Zhi-Jian Chen. Fluid infusions from

catheters into elastic tissue I: azimuthally symmetric backflow in homogeneous media. Physics

in Medicine and Biology, 55:281-304, 2010.

[10] Chen ZJ1, Gillies GT, Broaddus WC, Prabhu SS, Fillmore H, Mitchell RM, Corwin

FD, Fatouros PP. A realistic brain tissue phantom for intraparenchymal infusion studies. J

Neurosurg. 2004 Aug;101(2):314-22.

[11] Casanova F, Carney PR and Sarntinoranont M. Influence of needle insertion speed on

backflow for convection-enhanced delivery J Biomech Eng. 2012 134(4):041006.

[12] G. A. Orozco, J. H. Smith, and and J. J. García. Backflow length predictions during flow-

controlled infusions using a nonlinear biphasic finite element model. Medical &

Biological Engineering & Computing, 52:841–849, 2014. doi:10.1007/s11517-014-1187-1.

[13] J. J. García, A. B. Molano, and J. H. Smith. Description and validation of a finite

element model of backflow during infusion into a brain tissue phantom. Journal of

Computational and Nonlinear Dynamics, 8:011017, 2013. doi:10.1115/1.4007311.

8

Figure 1: A side view photograph of the SFh (top) and SFa (bottom) catheters.

Catheter Tip diameter

(mm)

Shaft diameter

(mm)

Step width

(mm)

16

gauge 0.36 0.76 0.2

14

gauge 0.665 0.877 0.106

Table 1. The measured dimensions of the catheters

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Pig Infusion

Backflow (mm)

2.5

μl/min

5.0

μl/min

7.5

μl/min

10.0

μl/min

18 3 3.0 4.0 10.0

18 4 3.0 7.0 9.0

19 3 3.0 12.0 13.0

19 4 3.0 3.0 3.0 7.0

20 3 3.0 4.0 8.0 13.0

20 4 8.0 11.0 12.0

25 2 3.5 6.0 8.0 9.0

25 3 3.0 4.0 6.5 8.0

Mean ± SD 3.9 ± 2.0 5.8 ± 3.7 7.7 ± 3.5 9.3 ± 2.0

Table 2A. Backflow in the 16-gauge SmartFlow catheter.

Pig Infusion

Backflow (mm)

2.5

μl/min

5.0

μl/min

7.5

μl/min

10.0

μl/min

18 1 5.0

18 2 11.0

19 1 8.0 8.0

19 2 6.5 10.0

20 1 18.0 18.0

20 2 3.0 5.0 13.0

25 1 9.0 9.0 9.0

25 4 10.0 17.5

Mean ± SD 9.0 ± 5.0 10.4 ± 5.0

Table 2B. Backflow in the 14-gauge SmartFlow catheter.

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Figure 2: T1-weighted MR imaging of eight paired infusions of the SmartFlow 16- and 14-gauge

catheters showing the backflow at 5 L/min. The top row shows the backflow around the 16-

gauge catheter, while the bottom row shows the infusion from the 14-gauge catheter. The 16

gauge catheter has a smaller tip diameter as well as a larger step width than the 14-gauge one.

11

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 3: A series of “time lapse” images of a dye infusion within gel from a stepped catheter.

Appendix: pre-stress and ﬂow redirection in stepped catheters

Raghu Raghavan

A. The effect of prestress from a step

The step catheter was ﬁrst claimed by its inventors to reduce backﬂow by redirecting the

ﬂow laterally. However, we surmise that the eﬀect of prestress is more likely to be eﬀective

for this purpose. We use the simple theory of backﬂow developed earlier [9] to estimate

the reduction of backﬂow due to pre-stress from a step catheter. In a second subsection,

we consider the eﬀect of ﬂow redirection, which can be added to the design of the tests,

but the eﬀect is negligible for the designs being considered. The more important eﬀect

is due to the prestress. The calculations here are at best order of magnitude estimates:

more detailed work, necessarily numerical, may be called for when moreso studies are

done to investigate the phenomenon. We should also mention that, since this work was

completed, we have studied another catheter designed to have a so-called bullet nose that

has a maximum width considerably larger [1]: correspondingly it seems to limit backﬂow

even more strongly, giving further evidence for our pre-stress hypothesis.

We make the following assumptions to simplify the treatment here:

1. The microtip at the end of either the 14 gauge (OD: 0665 mm) or 16 gauge (OD:

036 mm) SmartFlow(TM) catheter tears thetissueapartsothatthereisnostress

onthetissueforthelength(3mm) of this microtip. These tips are the same

length (3mm) in both (though unfortunately not at all of the same diameter which

complicates understanding of the phenomenon as well as the analysis).

2. The wider cannula that has a step (radial extent 01mmfor the 14 gauge and 02mm

for the 16 gauge) that applies a stress or a deformation to the tissue and does not

tear it: we assume the deformation is entirely radial.

3. We will need two kinds of quantities to estimate a third, given the above data on

the catheters. Namely, we need to know the ﬂow rate ref that has a backﬂow just

1

Appendix: pre-stress and flow redirection in stepped catheters 2

equal to the tip length (3mm) in each catheter. Since the catheters are of diﬀerent

diameters, from the assumption made in 1 above, we will get diﬀerent numbers for

the two catheters. These ﬂow rates were not measured, so we will assume such

a distance for one catheter, and estimate it for the other from theory (see below).

The other quantity we need is the ﬂow rate 0(

ref ) which is just stopped by the

step in question. Then we can estimate the ﬂow rate that is just prevented by the

other catheter.

4. Theory does allow us to estimate these ﬂow rates directly, but at the cost of knowing

tissue parameters, which we shall avoid in this preliminary analysis.

5. We thus need two quantities which we have to estimate. We can take these to be

ref [14] and 0[14] which are the ﬂow rates for the 14 gauge catheter at which the

backﬂow has reached 3mm, and at which the step size of 01just prevents further

backﬂow. We shall then estimate the corresponding quantities for the 16 gauge

catheter, the quantity 0[16] being of the most interest.

Of course, had we had identical tip diameters for both catheters, and only a varying width

for the step, the comparison would involve fewer parameters to be ﬁtted, but unfortunately

this was not the case. As stated above, we shall make use of the formulas derived in [9].

The formulas are only approximate but they are simple, leading to simple rough criteria.

It is shown there that ﬂuid pressure in the annulus between catheter of outer radius and

soft tissue will shear away a tissue—free width

()'()

2()(1)

“Soft” here means that the shear modulus of tissue is much less than the compressional

modulus. In the formula above it is assumed there is a ﬂuid pressure ()at a point

adjacent to the catheter, along its length, and measured back from its tip. Then, if the

ﬂow rate is =0 , along with the corresponding to pressure rise at catheter tip to =0,

Appendix: pre-stress and flow redirection in stepped catheters 3

we know from the previous work that

=1

3µ5000×3

=0 ×4×4

1923¶1

5

;()==0 ³1−

´;()==0 µ1−

¶2

3

(2)

The relationship between the pressure =0 at the tip of the catheter and the ﬂow rate

=0 there is also known (see previous paper for the approximations involved: in par-

ticular =0 is not quite the pump ﬂow rate, but can be computed from it if a reﬁned

estimate is needed):

=0 =25

=0 µ1203

4¶1

5

=: 25

0−45(3)

where depends on ﬂuid and tissue parameters but not on the radius of the tip, nor on

the ﬂow rates. Now we have at the step of a catheter, which is up a distance 1,say,

from the tip, the ﬂow rate and pressure are givenbyequation(2),with=1.Ifwe

then increase the catheter radius at 1to a larger value +, then we assume that there

is an extra radial pre-stress at this point in accordance with (1):

=

2(4)

see [9]. If

(1)

(5)

we assert that there is no further backﬂow. In other words, we have the equation for the

critical pressure at the tip

=0 µ1−1

0¶2

3

=2

(6)

Letusassumethespecialconditionthatwehavecalibratedtheexperimentsothatatthe

ﬂow rate ref , the tip length is just the backﬂow length for the given tip radius. This

means

1

0

=µref

=0 ¶35

(7)

Appendix: pre-stress and flow redirection in stepped catheters 4

Then we ﬁnally get, using (3), (4), and (6),

=25

015Ã1−µref

0¶35!2

3

=15³35

=0 −35

ref ´2

3(8)

that the critical step size that just prevents backﬂow given an input ﬂow rate =0

is given by the right hand side of (8). Consider the two SmartFlow catheters, we note

that [16][14] is about 19from the measurements (see Table 1). Similarly, we ﬁnd

[16][14] ≈185. Given all the other errors we have we neglected the eﬀect of the

diﬀerent tip radii in the ratio of the step lengths that stop the corresponding backﬂow.

Unfortunately, we do not have good data on the other quantities involved, either but let

us take ref [14] ∼1Lmin (which means that this ﬂow rate causes a backﬂow of 3mm,

just up to the step in this catheter). The scaling of backﬂow lengths with radii would give

ref [16] ∼25Lmin for the ﬂow rates that reach the step for the smaller diameter

catheter. Using the values from Table 1 then, we have

³35

=0[16] −35

ref [16]´2

3

³35

=0[14] −35

ref [14]´2

3

=192 (9)

We now need one more number, say, =0[14],whichistheﬂow rate that is just stopped

by the 14 gauge catheter step. Let us assume this is 2Lmin, in rough accordance

with the discussion above. Then we ﬁnd

³35

0[16] −2535´2

3

¡235−1¢2

3

=192 (10)

which gives the ﬂow rate that is stopped by the 16 gauge catheter as 0[16] = 66Lmin.

It should be noted that a more reﬁned analysis (which is available) of backﬂow results

in a reduction of the quantity ref[16], given the two catheter diameters. However, we

postpone such reﬁnements and their consequences to a later date. The conclusion of this

back-of-the-envelope calculation is to indicate that prestress may be suﬃcient to explain

Appendix: pre-stress and flow redirection in stepped catheters 5

the behavior of the step in resisting backﬂow. Clearly more studies are needed: the

theory can easily be extended and reﬁned to support accurate measurements and test this

hypothesis more rigorously.

It should be noted that absent pre-stress — as is well known, and also evident from

equation (2) — larger diameter catheters are subject to greater backﬂow. Thus, once a

ﬂow makes it past a step, the larger diameter will allow greater backﬂow than a catheter

with the narrow tip diameter throughout. Is there then an optimal step size so that ﬂows

make it past do not result in catastrophic backﬂow? Wecanusetheaboveformulasto

analyze this problem as well, but it would be an academic exercise, since one should design

pre-stress inducing geometries to forbid backﬂow at the maximum ﬂow rates envisaged,

and not design them to fail to do so within the range of ﬂow rates we wish to use.

B. Flow redirection is ineffective in reducing backflow at a step

In the discussion above, we did not allow for the ﬂow rate entering the step to be greater

than the ﬂow rate attempting to leave it and cause further backﬂow. As stated, people

originally speculated that this might be the cause for backﬂow reduction. While it is

obvious that an inﬁnite sized step would stop backﬂow, it is also intuitive that the small

steps in question should not have a signiﬁcant eﬀect on the ﬂow rate. We conﬁrm this

with a calculation here. The reduction is essentially a ‘radial backﬂow’ phenomenon,

which we can develop in full analogy to the axial backﬂow of [1]. We recall that we

(i) assume fully developed Poiseuille ﬂow in the circular annulus between the step of the

catheter and a putatively sheared away tissue, and (ii) compute an approximation to the

Darcy ﬂow that results from ﬂuid leaving this annulus and entering the tissue due to the

reduction in ﬂuid ﬂux that results from the Poiseuille ﬂow. (The paper referred to may

be consulted for details.) First we compute radial Poiseuille ﬂow. Let the ﬂow enter the

space at =and let the annuli be be between =−and =.Inotherwords2

is the unknown width of the channel, and will be a function of , the radial distance. We

use cylindrical coordinates. Assuming the lossless, incompressible ﬂow is fully developed

Appendix: pre-stress and flow redirection in stepped catheters 6

for

,wehave

=0

1

()=0

Stokes ﬂow obeys

=µ

µ1

()¶+2

2¶

The no slip conditions at the walls are

=0 =±

The ﬂow rate into the annulus is

=Z

−

2 =: 4

which deﬁnes if is known. Let us assume that the zero of pressure occurs at a distance

, the same used in the NIH backﬂow model, loc.cit. Thus, now calling 2=: ,the

width of the annulus, we get

()=6 ()

3ln

for constant , which does depend on because it is 2in contrast to 1Poiseuille ﬂow

through a cylindrical annulus where the ﬂow does not depend on axial distance. Also,

=3

8

1

µ1−³

´2¶

This completes the Poiseuille ﬂow half of what we need. The second half is to compute the

pressure required to sustain Darcy ﬂow given the ﬂow 0()leaving the tissue—free circular

annulus. This problem is identical to that of the well known electrostatic potential due

to a charged annulus. However, we further simplify everywhere, we shall not use it for

thesamereasonthatweusethesimpliﬁed model of an “inﬁnite cylinder” in the usual

Appendix: pre-stress and flow redirection in stepped catheters 7

backﬂow model. So we will use the on-axis result, since that will allow us not to deal with

the polar coordinate and retain cylindrical symmetry. We can begin with the on-axis

potential of the ring between and +, see any electrosatics text or the web:

=

40p2+2

If the charge density is a constant, then we have

(= 0)=2 Z2

1

p2+2=

20µq2+2

2−q2+2

1¶

We ch e ck that as 1→0

2→∞,

=−

20

+const

giving the correct constant electric ﬁeld. In order to proceed with the backﬂow calculation,

we need to identify and 0.,asbefore,is

=0()

2

while we should identify 0with . So, on axis we have

( =)= 0()

4 µp2+2−q2+2

1¶

=: −0()

We ignore since it will be much smaller than even 1.So

≈

4

to a good approximation.

We have

()=6 ()

3ln

;()=−

4

0()

Appendix: pre-stress and flow redirection in stepped catheters 8

We also need the relation between or and . From dimensional analysis, we set

=

for now, where is the radius of the host catheter. So

4()=63()

3ln

(11)

So we have

−

4

0()

=µ63

3¶1

4µln

¶1

4

1

4()

Integration yields

−Z

1

−1

4 =4

3

3

4

0−4

33

4=µ63

3¶1

44

×"2

8×21

4

Γµ1

42ln

¶+1

22µln

¶1

4#

1

where Γis the incomplete gamma function. The right hand side evaluates to 2431 8 ×

10−6. To immediately see how little diﬀerence this makes, let us convert from cgs to

min so that we get

3

4

0−3

48×10−3(12)

with the ﬂow rates on the left being expressed in min.Thusthelossofis completely

negligible over the distance of (this) step for any clinical ﬂow rate (of the order of 1−20

min), since the right hand side is less than two orders of magnitude smaller than the

lowest ﬂow rates used. Only when the ﬂow rates approach 1100 min will this ﬂow

redirection be of any eﬀect: at those ﬂow rates of course there is no measureable backﬂow

no matter what catheter one uses. We conclude that ﬂow redirection is of no eﬀect in

reducing backﬂow.