Resultant Radius of Curvature of Stylet-and-Tube
Steerable Needles Based on the Mechanical
Properties of the Soft Tissue, and the Needle
Fan Yang1, Mahdieh Babaiasl2, Yao Chen3, Jow-Lian Ding4, John P. Swensen5
Abstract— Steerable needles have been widely researched in
recent years, and they have multiple potential roles in the medical
area. The ﬂexibility and capability of avoiding obstacles allow the
steerable needles to be applied in the biopsy, drug delivery and
other medical applications that require a high degree of freedom
and control accuracy. Radius of Curvature (ROC) of the needle
while inserting in the soft tissue is an important parameter for
evaluation of the efﬁcacy, and steerability of these ﬂexible needles.
For our Fracture-directed Stylet-and-Tube Steerable Needles, it
is important to ﬁnd a relationship among the resultant insertion
ROC, pre-set wire shape and the Young’s Modulus of soft tissue
to characterize this class of steerable needles. In this paper, an
approach is provided for obtaining resultant ROC using stylet
and tissue’s mechanical properties. A ﬁnite element analysis is
also conducted to support the reliability of the model. This work
sets the foundation for other researchers to predict the insertion
ROC based on the mechanical properties of the needle, and the
soft tissue that is being inserted.
Index Terms— Steerable Needles, Medical Robots, SEBS Tis-
sue, Bending Stiffness, Curvature
Different factors affect the radius of curvature of steerable
needles while being inserted into the soft tissue. These fac-
tors include the mechanical properties of the soft tissue, tip
geometry, and mechanical properties of the needle , ,
. Several research groups have developed mechanics-based
as well as non-physics based models to predict the insertion
radius of traditional steerable needles based on the properties
of the needle, and soft tissue , , . An analysis of
interaction forces between the needle and soft tissue was
developed by Misra et al. for bevel-tip steerable needles. A
ﬁnite element analysis was conducted to evaluate the forces on
the needle tip. A method to calculate the stored kinetic energy
and the relationship between multiple tubes’ curvatures and
1Fan Yang is a Graduate Research Assistant with the School of Mechanical
and Materials Engineering, Washington State University, Pullman, Washing-
ton, USA. firstname.lastname@example.org
2Mahdieh Babaiasl is a Graduate Research Assistant with the School of
Mechanical and Materials Engineering, Washington State University, Pullman,
Washington, USA. email@example.com
3Yao Chen is a Graduate Research Assistant with the School of Mechanical
and Materials Engineering, Washington State University, Pullman, Washing-
ton, USA. firstname.lastname@example.org
5Jow-Lian Ding is a Professor with the School of Mechanical and Mate-
rials Engineering, Washington State University, Pullman, Washington, USA.
4John P. Swensen is an Assistant Professor with the School of Mechanical
and Materials Engineering, Washington State University, Pullman, Washing-
ton, USA. email@example.com
Fig. 1. Stylet-and-Tube Steerable Needles, in which the direction of the tissue
fracture is controlled with the inner wire, and then the outer tube follows .
The combined radius of curvature is depend on step length, preset radius of the
wire, tissue’s mechanical properties, and tube & wire mechanical properties.
their bending stiffnesses for concentric tube robots have been
developed by Rucker et al. .
We have proposed a new class of steerable needles that
we call fracture-directed stylet-and-tube steerable needles in
which the direction of the tissue fracture is controlled by
an inner stylet and then the ﬂexible outer tube follows. This
method has shown to achieve a radius of curvature as low
as 6 mm in soft tissue phantoms which is unachievable by
traditional steerable needles . In order to fully characterize
these steerable needles, we need to ﬁnd a model that can
predict the ROC of the steerable needle inside the soft tissue
based on the mechanical properties of the needle, and the
soft medium. Our proposed fracture-directed stylet-and-tube
steerable needles have similarities to concentric tube robots.
Thereby, we attempt to ﬁt the wire-tube-tissue channel model
into the concentric tube model to simplify the mechanical
analysis of the insertion and ﬁnd out the relationship between
insertion ROC and the parameters of selected pre-curved wire.
In other words, we propose to suppose the tissue channel as
an outer tube and treat the needle and the tissue channel as
In this paper, ROC of the stylet-and-tube steerable needle
while insertion into soft tissue is predicted which is a function
of bending stiffnesses of the tube, stylet, and soft tissue as well
as the step length. Fig. 1 shows the different parameters that
affect the insertion radius of stylet-and-tube steerable needles.
Note that ROC is also dependent on the step length of the
inner stylet that is followed by the outer tube; however, in
this paper, step length is considered to be constant and the
full model involving the step length will be presented in the
The experiments presented in this paper are based on
soft tissue phantoms with the Young’s Moduli in the range
from 112.89 KPa to 229.57 KPa. This range coincides with
the Young’s Moduli of human tendon and stays within one
magnitude difference with the Young’s modulus of human
muscle. . The tissue phantoms with lower polymer ratios
can be used for mimicking softer tissues in the human body
for the same type of experiments.
II. MATER IAL S AND ME THODS
In this section, the materials and methods used in the
experiments are described. In the experiments, there are three
different stiffnesses of tissue simulants, and three combinations
of Nitinol tubes and wires. The Young’s Moduli of tissue
simulants, Nitinol tubes, and Nitinol wires are chosen across
a certain range. The Nitinol wires involved in the experiments
are heat-treated to generate the pre-curved shape, and the
curvature of all pre-curved wires are constant. Nitinol tubes
are polished and remain straight.
A. Nitinol Tubes and Wires
Most of Nitinol wires and tubes used in the experiments
are manufactured by Conﬂuent Medical Technologies (Scotts-
dale, AZ, USA), except the smallest Nitinol tube, which has
0.33mm outer diameter and 0.26 inner diameter, which is
produced by Goodfellow Corporation (Coraopolis, PA, USA).
B. Heat Treatment of Nitinol Stylet and Its Recoverable Strain
The stylets used in the experiments are made of superelastic
shape memory alloy, Nickel Titanium. As such, before heat
treatment, it is necessary to calculate the maximum recover-
able strain when determining the minimum pre-curved radius
of stylet curvature for experiments. To ensure the stylet can be
fully straightened without any plastic deformation, we decided
to use the common conservative superelastic Nitinol strain
limit of ε=8% . The relationship between recoverable
strain limit (ε), and needle tip pre-curvature (κ) is:
Then the minimum radius of needle curvature, r, can easily
be calculated by inverting κ:
The characteristics of the needles (consisting of the tube,
and wire) used in our experiments are described in Table I.
For these needles, the minimum precurved radii of 0.19mm,
0.29mm, and 0.47mm diameter stylets without plastic defor-
mation are 1.29mm, 1.97mm, and 3.17mm, respectively. Three
different preset radii of Nickel-Titanium stylets are 30mm,
60mm, and 90mm.
To fabricate the 30mm, 60mm, and 90mm constant radii
of curvature, straight Nitinol wires are pressed in a steel
mold (refer to Fig. 2), heated to 500◦Cfor 30 minutes, then
quenched in water. Generally, for the heat treatment of Nitinol
stylets with different diameters and radii of curvatures, heat
treatment time needs to be raised up if Nitinol pieces have
smaller diameters or preset radii. In this paper, 30 minutes heat
treatment time was chosen to assure that the 0.19mm stylet
can hold 30 mm radius of curvature under room temperature.
Fig. 2 shows the three different radii after heat treatment that
are used during experimentation.
Fig. 2. Steel heat treatment mold for Nitinol wires with three different radii.
The Nitinol wires are put in the grooves and then put in the oven at 500oC
for about 30 minutes and then quenched to get the constant pre-curvatures.
C. Tissue Simulants
The material we used to simulate the biological tissues in
this paper is Poly (styrene-b-ethylene-co-butylene-b-styrene)
triblock copolymer (SEBS), produced by Kraton Polymers
LLC (G1650, Houston, TX, USA). Soft tissue simulants made
of SEBS are environmentally stable substitutes for water-based
hydrogels , , , . The tissues made of SEBS
in mineral oil are optically clear which makes the imaging
of the inside of the tissues easier. The Young’s Moduli of
SEBS tissue simulants are calculated from the compression
tests. The density of this material is ρSEBS =910 kg
oil is used as the solvent for this material. The mineral oil
involved in the experiments is white mineral oil with density
of ρoil =0.85 g
mL . SEBS material and white mineral oil are
mixed by weight, in which the fractions of polymer used
in the experiments are 15%, and 25%. The mixture is then
put into the oven at 120oCfor 2 to 8 hours (the more
SEBS the mixture contains, the more time will be needed for
melting), and is mixed occasionally to produce a homogeneous
solution without any visible undissolved powder. The solution
is then put in the vacuum chamber to release any air bubbles
trapped into the mixture. It is then poured into the molds
of rectangular shape with dimensions of 100 ×100 ×30 mm.
Pre-curved Nitinol wires were placed in the mold before the
solution was poured to create channel with constant curvatures
PARAMETERS OF THE TUBE,AND WIRE WITH DIFFE RENT BENDING ST IFFNE SSES : LOW, MEDIUM,AND HIGH
Low Bending Stiffness Medium Bending Stiffness High Bending Stiffness
Parameters Tube Wire Tube Wire Tube Wire
Outer Diameter (mm) 0.33 0.19 0.57 0.29 0.85 0.47
Inner Diameter (mm) 0.26 - 0.32 - 0.52 -
Bending Stiffness (N/m) 2.68e-05 4.85e-06 3.55e-04 2.68e-05 1.65e-03 1.80e-04
Length (mm) 150 180 150 180 150 180
Young’s Modulus (GPa) 75 75 75 75 75 75
Poisson’s ratio 0.33 0.33 0.33 0.33 0.33 0.33
(Refer to Fig. 3). The tissue simulants are then let to cool
down at room temperature before removing from the molds.
Universal mold release was used in order to peel the tissue
simulants off easily.
Fig. 3. Holes are drilled on walls of the acrylic container to setup Nitinol
wires. These wires are used to create constant pre-cruvatures in soft tissue
phantoms (The SEBS in mineral oil solution will be poured on them).
D. Compression Tests
In order to develop a suitable model for ﬁnding bending
stiffnesses of tissues, Young’s moduli of tissue phantoms need
to be calculated ﬁrst. Load versus displacement curves are
obtained from compression tests. The static compression tests
are performed with Instron 600DX machine controlled by
Bluehill 3 software. Samples with the dimensions of 30mm
diameter and 10mm thickness for each of the three types of
SEBS tissue simulants (different stiffnesses) are manufactured.
The strain rates of all the tests are ﬁxed at 0.001 1
insertion into the soft tissue is done at a low strain rate).
The change of the gauge length is measured with a built-in
sensor and the compressive load (F) was obtained from a 25
lb (approximately 111 N) S-type load cell connected to the
machine. The data of displacement and Force was taken each
0.1(s). The strain of the samples is calculated by the following
where L0is the initial thickness of the samples, and the stress
of the tests can be calculated by the following equation:
where D is the diameter of the samples.
Three tests are performed for each tissue stiffness. The
average of the stress-strain data is calculated for each material
using the three sets of data. The Young’s Moduli of the tissue
phantoms are calculated by ﬁnding the slope of the linear
portion of the stress-strain curve.
E. Finite Element Analysis of the Resultant Curvature of the
Needle, and Tissue Channel
Finite element analysis has been used for ﬁnding the behav-
ior and consequent deﬂection in both active and passive steer-
able needle researches by Khashei Varnamkhasti et al.,
Oldﬁeld et al. and Jushiddi et al..
In fracture-directed stylet-and-tube needle steering ap-
proach, a channel is created ﬁrst by an inner Nitinol stylet, and
then is followed by a tube. Therefore, knowing the equivalent
bending stiffness of the tissue channel after insertion of the
needle is essential for predicting the resultant ROC and path
planning. Thus, A ﬁnite element analysis has been developed
for predicting equivalent bending stiffness of the tissue channel
after needle insertion.
All SEBS tissue phantoms, Nitinol tubes, and wires involved
in ﬁnite element analysis are exactly the same as those in
experiments. Table II shows the mechanical properties of
each tissue phantom, and Table I represents the sizes and
mechanical properties of each combination of Nitinol tubes
and wires. The complete ﬁnite element analysis contains two
different parts, both are performed in static structural solver.
In the ﬁrst part, a serial of points are selected along a straight
Nitinol stylet, and then the moment reactions on these points
are computed based on desired stylet curvature. In the second
part, reversed moments are applied on these points along a
curved stylet to simulate a pre-loaded straight stylet being
released. In the end, resultant radii of curvature are calculated
based on displacement results from the FEA. Fig. 4 depicts
The FEA modeling to ﬁnd the resultant ROC of the needle
which is a function of bending stiffness of the wire, tube, and
soft tissue being inserted.
Fig. 4. FEA model of the needle insertion into soft tissue. (A) A straight Nitinol tube is loaded with moments to form the same curvature as the pre-curved
tissue channel. (B) Loads are released, and the tissue channel is deformed by Nitinol tube. (C) Overlap of parts A, and B, a clear curvature change can be
observed. The centerline of loaded Nitinol tube was marked as white and the centerline of unloaded Nitinol tube was marked as black.
F. Equivalent Bending Stiffness of the Tissue Channel, treat
tissue channel as an outer tube
The equations related to the bending stiffness and curvature
for calculating multiple overlapped curved tubes has been
developed by Webster et. al.:
where κCis the combined curvature where the tube and wire
are fully overlapped, and κt,κb,κwand Kt,Kb,Kware the
curvature and bending stiffness of the tissue channel, tube, and
wire, respectively. Iiis the cross-sectional moment of inertia
and Eis the Young’s Modulus.
The product of the Young’s Modulus (Modulus of Elasticity)
and cross-sectional moment of inertia is bending stiffness.
Here is where we make the assumption and treat the tissue
channel as an outer tube. Because of the relatively small
strains of the tissue using the relative stiffness heuristic, this
simplifying assumption is valid.
By switching Ktto the left side, we can obtain equation
below from equation(5):
Since the Nitinol tubes we used in the experiment are
straight and the tissue channel holds the same curvature as
the wire, κb=0, κt=κw.
Thus, equation(7) can be written as:
G. Experimental Setup
Figure 5-A shows the overall system with the insertion
device, overhead camera, a lightbox for transparent tissue
simulants, and the associated electronics. Figure 5-B depicts
the assembly of the insertion system, with each essential
component labeled. Each collet, and bearing was mounted on
the opposing linear slides, such that the bores of the collets
are collinear. The tube collet is located distally, closest to
the insertion point, and the wire collet is located proximally
such that the wire can be pushed out of the tube. Three limit
switches are located at the most distal limit of travel, the most
proximal limit of travel, and between the wire and tube stages.
To develop a predictive model for the tissue channel bending
stiffness, the experiments follow the procedures described
1) Overlap straight tube and pre-curved wire then ﬁx the
ends of them into tube and wire collets, respectively.
2) The wire platform moves forward to insert pre-curved
wire into existing channel, which has the same curvature
as the wire.
3) Then tube platform follows the wire platform until
Nitinol tube and wire are fully overlapped inside the
4) Overhead camera takes photos for curve ﬁtting program
written in MATLAB.
5) Retract tube and wire platform and replace tissue, wire
and tube with next combination then repeat step 1, 2, 3,
Since the bending stiffness of the tube in each tube-wire
combination is at least one magnitude higher than the bending
stiffness of the wire, the tube almost remains straight while
they are overlapped. The curvature of the overlapped tube and
wire is assumed to be equal to 0.
III. RES ULTS
This section describes the compression test results of SEBS
tissue phantoms, and the results obtained from multiple sets of
experiments with different wire radii, different tube and wire
combinations, and different tissue phantoms. Five experiments
for each wire radius were conducted, the data presented in
ﬁgures are the mean value of the experiments.
The approach we used for path tracking is based on image
analysis. By ﬁtting curve to the captured image via MATLAB
(The MathWorks, Inc., Natick, Massachusetts, United States),
the insertion curvature of each tube can be obtained.
Fig. 5. (A) The whole needle insertion system setup including the insertion device, two micro step drives that drive linear slides, a light box, a camera, a
microcontroller, and a power supply. (B) Insertion device setup, including three limit switches, simultaneous rotation mechanism 3D printed by ABS, tube
and wire collets and chucks, two linear slides and two bearings.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Engineering Stress (kPa)
Fig. 6. Stress vs. strain curve of SEBS soft tissue simulants. The Young’s
modulus of each tissue is obtained by ﬁnding the slope of the linear section
of the curves. Tissue 1, 2, and 3 are 15%, 20%, and 25% G1650 SEBS in
mineral oil soft tissue simulants, respectively.
A. Young’s Moduli of SEBS tissues
Figure 6 depicts the stress-strain curve of the soft tissue
simulants. The Young’s modulus is obtained from ﬁnding the
slope of the linear section of the curve, and the Young’s
Moduli of Tissue 1, 2, and 3 are 112.89, 159.80, and 229.57
kPa, respectively. The properties of the soft tissue simulants
used in the experiments are provided in Table II.
MEC HANI CAL PRO PERTI ES OF SEBS TISSUE PHANTOMS. TI SSUE S 1, 2,
AN D 3ARE 15%, 20%, AND 25% G1650 SEBS IN MINERAL OIL SOFT
TISSUE PHANTOMS,RES PECT IVELY.
Tissue Phantom Density ( g
cm3) Young’s Modulus (kPa) Poisson’s ratio
Tissue 1 0.85 112.9 0.49
Tissue 2 0.862 159.8 0.49
Tissue 3 0.865 229.57 0.49
B. Equivalent Bending stiffness of the Tissue Channel, and the
Resultant Radius of Curvature
Surface plots of equivalent bending stiffnesses generated
from both FEA data and experimental data are provided in
Fig. 7. Since equivalent bending stiffness is directly derived
from the resultant radius of curvature, these two groups of
plots hold high similarity and the equivalent bending stiffness
computed from experimental data is marginally higher than
FEA in the same type of tissue phantom.
As it is shown in Figure 7, the equivalent bending stiffness
of tissue is not only dependent on the bending stiffness of
the wire and tube being inserted, but also affected by pre-
set channel radius of curvature. In other words, the equivalent
bending stiffness of the tissue channel is dependent on the
Young’s Modulus and cross-sectional moment of inertia of the
wire and tube as well as the pre-set wire radius of curvature
(during insertion, the tissue channel is always created by the
pre-curved wire, so their radii of curvature remain the same).
Thus, tissue channel equivalent bending stiffness is a variable
that reacts to the inserted stylet, it will increase as the bending
stiffness of the stylet increases.
Fig. 7. The equivalent bending stiffness surfaces in experiments and FEA with different tissue phantoms and varying channel radii. (A) The equivalent
bending stiffness of tissue channel created by 0.1905mm wire and 0.26mm ID 0.33mm OD tube in experiments. (B) The equivalent bending stiffness of tissue
channel created by 0.2921mm wire and 0.3175mm ID 0.5715mm OD tube in experiments. (C) The equivalent bending stiffness of tissue channel created
by 0.47mm wire and 0.52mm ID 0.85mm OD tube in experiments. (D) The equivalent bending stiffness of tissue channel created by 0.1905mm wire and
0.26mm ID 0.33mm OD tube in FEA. (E) The equivalent bending stiffness of tissue channel created by 0.2921mm wire and 0.3175mm ID 0.5715mm OD
tube in FEA. (F) The equivalent bending stiffness of tissue channel created by 0.47mm wire and 0.52mm ID 0.85mm OD tube in FEA.
The resultant radius of curvature describes the combined
radii of curvature of a pre-curved wire, a straight tube and
a curved channel in the tissue phantom. The resultant radius
of curvature depends on the bending stiffnesses of the wire,
tube and tissue channel. Three dimensional surfaces were ﬁt
on both FEA data and experimental data.
In experiments, the insertion of 0.85mm diameter tube
exceeded the tolerance of the tissue channel in Tissue 1
for 30mm radius channel. It couldn’t follow the inner wire
properly and tend to break through the boundary of the channel
after the ﬁrst 10-12mm during insertion because of the tight
curvature of the channel and its high bending stiffness. The
FEA provided a reasonable prediction for 30mm channel in
Tissue 1 when high bending stiffness tube being inserted, the
reason is that the tube were pre-loaded and pre-bend then put
into the channel, so the pressure exerted on the channel wall
was smooth and uniform. In reality, it’s arduous to duplicate
the setup we used in FEA, so we used a different approach
to obtain the resultant radius of curvature. Since the FEA and
experimental data have high similarity, the FEA data will be
used to replace this speciﬁc data point when discussing tissue
behavior in future sections.
By comparing the FEA data with experimental data, it’s
obvious that the resultant radius of curvature in experiments
are slightly lower than the resultant radius of curvature in FEA
except for the data point where the high bending stiffness tube
inserted in Tissue 1 at 30mm channel. A hypothesis is that
some of the mineral oil was escaped from the phantoms during
storage since the experiments were conducted in around two
days. The ratio rise of SEBS polymer will result in a stiffer
tissue phantom which caused a decrease in resultant curvature.
In general, if tube and wire are selected, the stiffer the tissue
phantom is, the smaller the resultant radius of curvature is.
On the other hand, if tissue phantom has a speciﬁc Young’s
Modulus, the larger the bending stiffness of the tube, the larger
the resultant radius of curvature is.
In this paper, series of experiments are conducted to derive
the resultant radius of curvature from the bending stiffness
of the tube and wire, the pre-set curvature of the tissue
channel and the Young’s Modulus of the tissue. A ﬁnite
element analysis is also developed to provide support for the
reliability of the resultant radius model. To achieve a quality
insertion with scarce tissue damage, the bending stiffness of
the wire should be approximately one magnitude lower than
the bending stiffness of the tube. If the bending stiffness of
the wire is too close to the bending stiffness of the tube, the
combined curvature of the tube and wire will be relatively high
which increases the difﬁculty at the beginning of the insertion.
If the bending stiffness of the wire is too low from the bending
stiffness of the tube, it’s harder for the tube to follow the wire
in relatively soft tissue. Since the equivalent bending stiffness
of tissue channel is a variable that reacts to the inserted stylet
in this model, further research which also takes step length
into consideration is needed to provide a complete model of
resultant insertion radius of curvature. The similarity of the
models obtained from both FEA and experiments reveals that
the insertion radius of curvature is predictable by given the
mechanical properties of the combination of tissue phantom,
tube, and wire at a selected step length.
V. CONCLUSION AND FUTURE WORK
We have presented a method of treating tissue channel as an
outer tube and an approach to predict the resultant radius of
Fig. 8. Resultant radii of curvature surfaces in experiments and FEA with different tissue phantoms and varying channel radii. (A) 0.33mm OD, 0.26mm
ID tube, and 0.19mm wire in experiments. (B) 0.57mm OD, 0.32mm ID tube, and 0.29mm wire in experiments. (C) 0.85mm OD, 0.52mm ID tube, and
0.47mm wire in experiments. (D) 0.33mm OD, 0.26mm ID tube, and 0.19mm wire in FEA. (E) 0.57mm OD, 0.32mm ID tube, and 0.29mm wire in FEA.
(F) 0.85mm OD, 0.52mm ID tube and 0.47mm wire in FEA.
curvature for fracture-directed stylet-and-tube needle steering
technique. The experiments conducted for the resultant radius
of curvature serve as a part of fracture directed steerable needle
research. The model can be expanded across a wider range of
tissues’ Young’s Moduli or pre-set wire curvatures by simply
conducting experiments on the target data ﬁeld. By combining
with the step length model, a complete model for a speciﬁc
insertion can be described as
where κris the insertion curvature, `is step length, κt,κb,
κwand Kt,Kb,Kware the curvature and bending stiffness of
the tissue channel, tube, and wire, respectively.
The prompt future work is to accomplish the establishment
of the complete predictive model and develop a path planning
algorithm for vision-based closed-loop control. This vision-
based closed-loop control system will provide the capability of
control to any planned path within achievable insertion radius.
Fracture directed steerable needles furnished a new class of
needles for future research which can be potentially used for
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