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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.