A Model to Predict Deflection of an Active Tendon-Driven Notched Needle Inside Soft Tissue

Abstract

The last decade has witnessed major progress in the field of minimally invasive and robotic-assisted surgeries. Needle insertion, a minimally invasive technique, has proven its efficacy in procedures such as brachytherapy, ablation, drug delivery, and biopsy. Manual needle steering inside tissue is a challenging task due to complex needle-tissue interactions, needle and tissue movement, lack of actuation and control, as well as poor sensing and visualization. Recently, active tendon-driven notched needles, and robotic manipulation systems have been proposed to assist surgeons to guide the needles in desired trajectories toward target positions. This work introduces a new deflection model for the active tendon-driven notched needle steering inside soft tissue for intention to use in model-based robotic control. The model is developed to predict needle deflection in a single-layer tissue. To validate the proposed deflection model, five sets of needle insertion experiments with a bevel-tipped active needle into single-layer phantom tissues were performed. A real-time robot-assisted ultrasound tracking method was used to track the needle tip during needle insertion. It was shown that the model predicts needle deflection with an average error of 0.58 ± 0.14 mm for the bevel-tipped active needle insertion into a single-layer phantom tissue.

1 Introduction

Robotic minimally invasive procedures have received traction in recent years for their improved targeting precision inside the human body while leaving minimal damage to tissue, resulting in improvements in procedure's outcome. Dexterity and visualization of robotic instruments are key factors in the procedure's success. Extensive research has been conducted to improve flexibility of the instruments as well as sensing and visualization of the instruments inside the body. Examples of robotic tools with improved flexibility include tendon-driven notched needles [1,2], steerable guidewires [3], continuum robotic tools [4,5], concentric tubes [6,7], and three-dimensional printed active flexible needles [8,9]. Several sensing and visualization techniques have been developed to provide feedback to the surgeon or use them in robotic closed-loop control systems. Recent progress in sensing and visualization of the robotic tools include ultrasound (US) tip tracking [10–12], shape prediction [13,14], Fiber Bragg Gratings based force estimation and shape sensing [15,16], and shape memory alloy self-shape and self-force sensing [17,18]. Modeling and control of continuum robotic tools have been previously studied [19–22]. Additionally, multiple mechanics- and dynamic-based models have been developed to estimate the needle deflection for passive needles [23–25], or with an external lateral force to decrease the amount of required axial rotations at the needle base [26].

An example of a minimally invasive procedure that benefits from active flexible needles is prostate brachytherapy (BT). Prostate cancer is the second most common cancer among men in the U.S. [27]. High Dose Rate (HDR) BT is an internal and temporary radiotherapy method to remove cancerous tumors. The conventional HDR BT uses straight, passive needles to place the radiation sources at the desired locations. Although this conventional method has been promising, studies have reported side-effects such as edema in tissue, incontinence, and impotence. The side effects are a result of excessive radiation and needle penetration into sensitive organs such as the urethra, bladder, rectum, penile bulb, cavernous veins, and neuro-vascular bundles. Also, the use of HDR BT is limited in patients whose pubic arch obstructs the transperineal path to the prostate, thereby interfering with needle placement [28]. A study showed that the procedure was practical for only 24 out of the 40 patients studied due to pelvic bone arch interference [29]. Substantial pubic arch interference, which is more likely in patients with a large prostate, makes it difficult to achieve adequate source placement in the anterior and lateral portions of the prostate [30]. Even a narrow pubic arch may prevent proper implantation in a small prostate gland [30]. Known strategies to overcome this problem such as oblique catheter insertion and pelvic rotation [31,32] are not optimal. Other methods suggested the insertion of skew-line [33] or oblique needles [32] to pass the pubic arc. Active tendon-driven needle steering is a promising method, which alleviates these concerns via precise needle steering in the desired trajectory. This work introduces a new mechanics-based model according to Euler–Bernoulli beam theory to predict active tendon-driven notched needle deflection inside tissue. Needle insertion experiments have been conducted in single-layer phantom tissues to validate the model.

2 Materials and Methods

Section 2.1 explains the mechanics-based model and method to predict needle deflection. Section 2.2 introduces the robotic system to insert and manipulate the active tendon-driven needle inside tissue. Section 2.3 presents the experimental validation of the model. Experimental results and discussion are presented in Sec. 3. Finally, conclusion and future work are discussed in Sec. 4.

2.1 Needle-Tissue Interactions and Deflection Estimation.

The mechanics-based model (Fig. 1) is developed based on the Euler–Bernoulli beam theory [34] to predict beam deflection inside tissue. The theory is valid for deflections that are smaller than 10% of the total length of the beam. This is the case for most needle-based interventions including prostate HDR BT.

Fig. 1.

Mechanics of an active needle deflection in a two-layer tissue

The input parameters to the model are the needle tip force $F_t$, and the tendon-pulling force $F_p$. The following equation shows the overall potential energy of the system, consisting of the energy stored in the needle and tissue and the external work applied to the needle during a needle insertion task

$$\mathrm{\Pi }\left(u\right)=U\left(u\right)+V=U_s\left(u\right)+U_d\left(u\right)+V_t+V_p$$ (1)

where $\mathrm{\Pi }\left(u\right)$ is the total stored energy in the system, $U\left(u\right)$ is the energy stored in the system due to the needle displacement, V is the work applied to the system, $U_s$ is the strain energy due to the needle bending, $U_d$ is the energy stored in the system due to displaced tissue, $V_t$ is the work applied at the tip due to the needle-tissue interactions, $V_p$ is the work applied by the internal tendons of the needle, and $u$ is the needle deflection.

The strain energy stored in the needle during the insertion, $U_s$, is obtained by

$$U_s\left(u\right)={\int }_0^l\frac{\text{EI}}{2}{\left(\frac{{\partial }^{2}u\left(d,z\right)}{{\partial z}^2}\right)}^{2}dz$$ (2)

where $z$ is the horizontal coordinate, $E$ is the Young's modulus, $I$ is the needle's second moment of inertia, $l$ is the needle length, $d$ is the insertion depth, and $u(d,\hspace{0.17em}z)$ is the needle deflection shape at the insertion depth of d and z.

The equation below shows $U_d\left(u\right)$, the energy stored in the system due to the tissue displacement

$$U_d\left(u\right)=\frac{K_1}{2}{\int }_0^{l-d_k}{\left(u\left(d,z\right)-u_t\left(d,z\right)\right)}^2\hspace{0.17em}dz+\frac{K_2}{2}{\int }_{l-d_k}^l{\left(u\left(d,z\right)-u_t\left(d,z\right)\right)}^2\hspace{0.17em}dz$$ (3)

where $u_t\left(d,l\right)$ is the measured path at the tip by the US probe during the insertion task, $K_1$ and $K_2$ are the tissue stiffnesses expressed as force per unit area in two-layer tissue (for a single layer tissue, $K_2$ is zero), and $d_k$ is the depth at which the tissue layer and the tissue stiffness changes (zero for the single-layer tissue). The work applied at the tip due to the needle-tissue interactions is

$$V_t=F_t\hspace{0.17em}u\left(d,l\right)$$ (4)

where $F_t$ is the needle tip force.

The work applied by the internal tendons of the needle is

$$V_p=F_p\hspace{0.17em}\text{sin}\theta \hspace{0.17em}u(d,l)$$ (5)

in which $F_p$ is the tendon-pulling force, and $\theta$ is the bending angle of the needle.

Replacing Eqs. (2)–(5) into Eq. (1)

$$\mathrm{\Pi }\left(u\right)={\int }_0^l\frac{\mathrm{EI}}{2}{\left(\frac{{\partial }^{2}u\left(d,z\right)}{{\partial z}^2}\right)}^{2}dz+\frac{K_1}{2}{\int }_0^{l-d_k}{\left(u\left(d,z\right)-u_t\left(d,z\right)\right)}^2\hspace{0.17em}dz+\frac{K_2}{2}{\int }_{l-d_k}^l{\left(u\left(d,z\right)-u_t\left(d,z\right)\right)}^2\hspace{0.17em}dz-F_{t}u(d,l)-F_p\mathrm{s}\mathrm{in}\theta \hspace{0.17em}u(d,l)$$ (6)

The Rayleigh–Ritz method [26] can be used to solve this equation to predict the deflected shape of the needle. The following finite series represent the weighted shape functions

$$u_n\left(d,z\right)=\sum _{i=1}^n{q}_i\left(z\right)g_i\left(d\right)$$ (7)

where $u_n\left(d,z\right)$ is the weighted shape function, $q_i\left(z\right)$ is the ith shape function, and $g_i\left(d\right)$ is the corresponding weighting coefficient. $q_i\left(z\right)$ the ith vibration mode and can be found as

$$q_i\left(z\right)=\frac{1}{k_i}\left(\text{sin}\left({\beta }_i\frac{z}{l}\right)-\sinh \left({\beta }_i\frac{z}{l}\right)-{\gamma }_i\left[\text{cos}\left({\beta }_i\frac{z}{l}\right)-\cosh \left({\beta }_i\frac{z}{l}\right)\right]\right)$$ (8)

where ${\gamma }_i$ and $k_i$ are

$${\gamma }_i=\frac{\text{sin}{\beta }_i+\text{sin}h{\beta }_i}{\text{cos}{\beta }_i+\text{cos}h{\beta }_i}$$ (9)

$$k_i=\text{sin}{\beta }_i-\text{sin}h{\beta }_i-{\gamma }_i(\text{cos}{\beta }_i-\text{cos}h{\beta }_i)$$ (10)

in which ${\beta }_i$ is a constant. For a cantilever beam (clamped-free), ${\beta }_i=\pi (i-\frac{1}{2})$, and thereby ${\beta }_1$– ${\beta }_4$ are 1.857, 4.695, 7.855, and 10.996, respectively. By inserting the Eq. (7) into Eq. (6), the following equation is obtained

$$\mathrm{\Pi }\left(u_n\right)=\frac{\text{EI}}{2}{\int }_0^l{\left(\sum _{i=1}^n{q}_i^{\prime \prime }\left(z\right)g_i\left(d\right)\right)}^{2}dz+\frac{K_1}{2}{\int }_0^{l-d_k}{\left(\sum _{i=1}^n{q}_i\left(z\right)g_i\left(d\right)-u_t(d,z)\right)}^{2}dz+\frac{K_2}{2}{\int }_{l-d_k}^l{\left(\sum _{i=1}^n{q}_i\left(z\right)g_i\left(d\right)-u_t(d,z)\right)}^{2}dz-F_t\sum _{i=1}^n{q}_i\left(l\right)g_i\left(d\right)-F_p\mathrm{s}\mathrm{in}\theta \sum _{i=1}^n{q}_i\left(l\right)g_i\left(d\right)$$ (11)

where $q_i^{″}\left(z\right)$ denotes the second derivative of $q\left(z\right)$ with respect to $z$. The condition for minimizing the potential $\mathrm{\Pi }\left(u_n\right)$ is $\frac{\partial \mathrm{\Pi }}{\partial g_j}=0$ for $j=1,\hspace{0.17em}\dots ,\hspace{0.17em}n$

$$\frac{\partial \mathrm{\Pi }\left(u_n\right)}{\partial g_j\left(d\right)}=\text{EI}{\int }_0^l\left(\sum _{i=1}^n{q}_i^{\prime \prime }\left(z\right)g_i\left(d\right)\right)q_j^{\prime \prime }\left(z\right)dz+K_1{\int }_0^{l-d_k}\left(\sum _{i=1}^n{q}_i\left(z\right)g_i\left(d\right)-u_t(d,z)\right)q_j\left(z\right)dz+K_2{\int }_{l-d_k}^l\left(\sum _{i=1}^n{q}_i\left(z\right)g_i\left(d\right)-u_t(d,z)\right)q_j\left(z\right)\hspace{0.17em}dz-F_t-F_p\mathrm{s}\mathrm{in}\theta =0$$ (12)

Equation (12) can be simplified into

$$\sum _{i=1}^n{\phi }_{\mathrm{ji}}{g}_i\left(d\right)-{\omega }_j-F_t-F_p\mathrm{s}\mathrm{in}\theta =0$$ (13)

in which

$${\phi }_{\mathrm{ji}}=\text{EI}{\int }_0^l{q}_i^{\prime \prime }\left(z\right)q_j^{\prime \prime }\left(z\right)dz+K_1{\int }_0^{l-d_k}{q}_i\left(z\right)q_j\left(z\right)dz+K_2{\int }_{l-d_k}^l{q}_i\left(z\right)q_j\left(z\right)dz$$ (14)

$${\omega }_j\left(z\right)=K_1{\int }_0^{l-d_k}{u}_t\left(d,z\right)q_j\left(z\right)dz+K_2{\int }_{l-d_k}^l{u}_t\left(d,z\right)q_j\left(z\right)dz$$ (15)

The simplified equation can be written in matrix form as

$$\mathrm{\Phi }\left(\mathrm{z}\right)g\left(\mathrm{d}\right)=F_t{1}_{n\times 1}+F_p\mathrm{s}\mathrm{in}\theta 1_{n\times 1}+\mathrm{\Omega }\left(z\right)$$ (16)

where the matrices are

$$\begin{array}{ccc}\mathrm{\Phi }=\left[\begin{array}{ccc}{\phi }_{11}& \cdots & {\phi }_{1n}\\ ⋮& \ddots & ⋮\\ {\phi }_{n1}& \cdots & {\phi }_{\mathrm{nn}}\end{array}\right];& \mathrm{\Omega }=\left[\begin{array}{c}{\omega }_1\\ ⋮\\ {\omega }_n\end{array}\right];& g=\left[\begin{array}{c}{g}_1\\ ⋮\\ g_n\end{array}\right]\end{array}$$ (17)

$g$ can be obtained as

$$g={\mathrm{\Phi }}^{-1}(F_t{1}_{n\times 1}+F_p\mathrm{s}\mathrm{in}\theta 1_{n\times 1}+\mathrm{\Omega })$$ (18)

Finally, the needle deflected shape can be obtained by inserting Eqs. (8) and (18) into Eq. (7).

2.2 Robotic System for Active Tendon-Driven Needle Insertion and Tracking.

The robotic needle insertion system, shown in Fig. 2(a), was used to insert and actuate the active needle inside a phantom tissue. The robotic system consists of (i) a needle manipulation system (Fig. 2(b)), which consists of a Maxon motor that is programed to pull the tendons and actuate (bend) the active needle, (ii) a US machine (Chison, ECO 5) and an Arducam USB camera (Fig. 2(c)) to track the needle tip in real-time, and (iii) a linear motorized stage (Velmex, Inc., Bloom- field, NY) and a guide template (Fig. 2(d)) for axial movement (insertion) of the needle inside the tissue.

Fig. 2.

(a) Robotic needle insertion system consisting of (b) a needle manipulation (actuation) system to pull the tendon and bend the active needle, (c) an ultrasound probe and camera mounted on a linear stage for R-AUST of the needle tip in single-layer phantom tissue, and (d) a linear stage and grid template for needle insertion

A robot-assisted ultrasound tracking (R-AUST) method was used in this work to track the needle tip in real-time. Using two-dimensional transverse US images, the R-AUST method visualizes the cross section of the needle using a series of imaging techniques to identify the coordinates of the needle tip. A Python code was programed to capture the image streaming from a USB port connected to a frame grabber (Epiphan Av.io HD, Epiphan Systems, Ottawa, Canada), and perform image analyses. It is important for the US probe to move with the needle tip. Therefore, a motorized linear stage was assembled and programed (actuated by a Maxon motor) to carry the US probe and follow the needle tip during the needle insertion process. The Maxon motor was programed in Python to manipulate its velocity with respect to the needle insertion velocity to ensure visibility of the needle tip in US images. For further details please see our previous publications [10,13].

Figure 3 shows the active needle, fabricated on a superelastic nitinol tube (1.80 and 1.50 mm, outer and inner diameters, respectively, with other important parameters shown in Table 1), for model validation experiments. Figure 3(a) shows the needle's flexure section with a series of six slits 1.40 mm deep and 0.30 mm wide for improved flexibility. The flexure section was insulated (Fig. 3(b)) using an ultrathin heat shrink plastic to prevent tissue from penetrating inside the needle tube. Figure 3(c) shows the design parameters of each notch carved on the needle tube. The notches were made in the lab using typical machining tools such as a Dremel, and ultrathin cut-off disks with a thickness of 0.30 mm and 0.127 mm (Gesswein & Co., Inc., Bridgeport, CT). Based on our calculations (details are outlined in our previous work [2]), the estimated maximum bending angle of this needle is 60 deg. A 30 deg bevel-tip was attached to the active needle, as shown in Fig. 3(d) to facilitate the initial puncture of the needle in tissue. Listed in Table 1, $r_o$ is the outer radius of the needle, $r_i$ is the inner radius of the needle, $t$ is the width of the cutouts, $d$ is the depth of the cutouts, $d_n$ is the distance between two cutouts, and $\overline{y}$ is the position of the neutral axis.

Fig. 3.

Active tendon-driven notched needle: (a) flexure section, (b) insulated flexure section in bent position, (c)design parameters, and (d) bevel-tip active needle

Table 1.

Design parameters of the active needle shown in Fig. 3

$r_o\hspace{0.17em}\hspace{0.17em}\left(\mathrm{mm}\right)$	$r_i\hspace{0.17em}\hspace{0.17em}\left(\mathrm{mm}\right)$	$t\hspace{0.17em}\hspace{0.17em}\left(\mathrm{mm}\right)$	$d\hspace{0.17em}\hspace{0.17em}\left(\mathrm{mm}\right)$	$d_n\hspace{0.17em}\hspace{0.17em}\left(\mathrm{mm}\right)$	$\overline{y}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{mm}\right)$
0.90	0.75	0.30	1.40	0.64	0.71

2.3 Experimental Model Validation.

For model validation, the needle insertion experiment was repeated five times. We performed active needle insertion tests with a bevel-tip in a soft single-layer phantom tissue of 20 kPa stiffness [35,36] to a depth of 60 mm. During the needle insertions, the internal tendon of the active needle was gradually pulled using the Maxon motor to realize bending. The Maxon motor's current and the tendon displacement were collected to estimate the tendon pulling force and the needle's tip path was measured in real-time using the R-AUST. The inputs to the model are the tip path, the force at the tip, and the tendon pulling force. The needle tip force was estimated as 0.29 N for a 30 deg bevel tip angle in a single-layer tissue [26]. To calculate the tendon pulling force, equation below for the lead screw was used [37]

$$T=\frac{F_p{d}_m}{2}\left(\frac{l+\pi f{d}_m\sec \alpha }{\pi d_m-fl\sec \alpha }\right)$$ (19)

where $T$ is the applied torque by the Maxon motor to pull the tendons, $d_m$ is the mean diameter of a single thread lead screw, which is 8.00 mm here, $F_p$ is the tendon pulling force to be found, $l$ is the tendon displacement, $f$ is the coefficient of friction for threaded pairs of a steel and dry screw, which is 0.25 here, and $\alpha$ is the thread angle, which is zero for a square thread [37].

The applied torque (T) was estimated using Maxon motor's current, nominal current, and torque constant using Eqs. (20) and (21) [38]. For the Maxon motor (0.5 W Maxon DC motor RE8 Ø8.00 mm, Precious Metal Brushes), the nominal current and torque constant are 86.50 mA and 7.11mNm/A, respectively

$$\mathrm{Torque}\hspace{0.17em}\mathrm{actual}\hspace{0.17em}\mathrm{value}\hspace{0.17em}\left(\mathrm{per}\hspace{0.17em}\mathrm{mil}\right)=\frac{\mathrm{Current}\hspace{0.17em}\left(\mathrm{mA}\right)}{\mathrm{Nominal}\hspace{0.17em}\mathrm{current}\hspace{0.17em}\left(\mathrm{mA}\right)}*1000$$ (20)

$$T=\left(\frac{\mathrm{Torque}\hspace{0.17em}\mathrm{actual}\hspace{0.17em}\mathrm{value}\hspace{0.17em}\left(\mathrm{per}\hspace{0.17em}\mathrm{mil}\right)}{1000}\right)\mathrm{*}\left(\frac{\mathrm{Nominal}\hspace{0.17em}\mathrm{current}}{1000}\right)\mathrm{*}\mathrm{Torque}\hspace{0.17em}\mathrm{constant}$$ (21)

To find the bending angle $\theta$ in each step of insertion, the following equation could be use

$$\theta ={\text{tan}}^{-1}\frac{u_t\left(i\right)-u_t\left(0\right)}{z\left(i\right)-z\left(0\right)}$$ (22)

where $u_t\left(i\right)$ and $u_t\left(0\right)$ are the needle tip paths at moment $i$ and at the initial moment, respectively, and $z\left(i\right)$ and $z\left(0\right)$ are the insertion depths at moment $i$ and at the initial moment, respectively. The average maximum bending angle and average mean bending angle for the bevel-tipped active needle insertions in the single-layer tissue were 4.92 deg and 2.66 deg, respectively. The average mean angle was used here in the equations.

3 Results and Discussion

This section evaluates the model prediction by directly comparing the estimated deflection of the needle (predicted by the model) with the measured deflection (provided by the R-AUST method) and estimating the maximum and average error in shape prediction (difference between the predicted and measured needle deflection). The experiments were performed for the bevel-tip active needle insertion in a single-layer phantom tissue.

Figure 4(a) compares the estimated deflection of a bevel-tipped active needle in a single-layer phantom tissue, predicted by the model, with the deflection of the needle measured (recorded) by the R-AUST method.

Fig. 4.

Bevel-tip active needle insertion in a single-layer phantom tissue to a depth of 60 mm: (a) estimated (model predicted) needle deflection versus measured needle deflection via R-AUST and (b) absolute error in needle deflection prediction

The maximum measured and estimated (model-predicted) deflections were 5.23 mm, and 6.72 mm, respectively at 60 mm of insertion depth. Figure 4(b) shows the absolute error of needle deflection model prediction. The absolute error was found by estimating the difference between the estimated (model-predicted) and measured needle deflections. The maximum error was 1.69 mm, with an average error of 0.59 mm. A slightly larger error was observed at higher insertion depths. The tendon pulling (needle actuation and the consequent bending) initiated at a depth of 13 mm and ends at 37 mm of depth.

Table 2 lists the maximum deflection, maximum and average errors for five bevel-tipped active needle insertion trials in a single-layer phantom tissue to a depth of 60 mm. The maximum and average errors of model prediction for five needle insertion trials were 1.55 ± 0.11 mm, and 0.58 ± 0.14 mm, respectively.

Table 2.

Summary of the active needle insertion experiments in phantom tissues to a depth of 60 mm

Trial	Max deflection model prediction/measured (mm)	Max error (mm)	Average error (mm)
1 (Fig. 4)	6.31/4.90	1.65	0.48
2	5.91/5.11	1.51	0.82
3	5.34/4.35	1.43	0.53
4	6.13/4.68	1.49	0.51
5	6.72/5.23	1.69	0.59

4 Conclusion

This work presents a mechanics-based model to estimate the needle deflection of active tendon-driven notched needle insertions in single-layer phantom tissues for the first time. The energy-based model provides estimation and prediction of needle deflection during insertion considering the pulling force of the internal tendons force and needle-tissue interactive forces. The model is appropriate for model-based needle deflection control. The model is experimentally validated and shown to estimate the needle deflection with reasonable accuracy for single-layer phantom tissues. This work takes a first step toward control methods for active tendon-driven notched needle deflection the pulling force of internal tendons.

Funding Data

• The National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health (Award No. K25EB030562; Funder ID: 10.13039/100000002).

Nomenclature

BT =

brachytherapy

d =

depth of the needle cutouts

d =

insertion depth

$d_k$ =

depth at which the tissue layer and tissue stiffness changes

$d_m$ =

mean diameter of a single thread lead screw

$d_n$ =

distance between two needle cutouts

E =

Young's modulus

f =

coefficient of friction for the threaded pairs of a steel and dry screw

$F_p$ =

tendon-pulling force

$F_t$ =

needle tip force

FBG =

fiber Bragg gratings

$g$ =

weighted coefficient

HDR =

high dose rate

I =

needle's second moment of inertia

k =

constants based on trigonometric functions

K =

tissue stiffness

l =

needle's length

$l$ =

tendon displacement

PAI =

pubic arch interference

$q$ =

vibration mode

R-AUST =

robot-assisted ultrasound tracking

$r_i$ =

inner radius of the needle

$r_o$ =

outer radius of the needle

t =

width of the needle cutouts

T =

applied torque by the Maxon motors to pull the tendons

$u$ =

needle deflection

$U$ =

energy stored in the system due to needle displacement

$U_d$ =

energy stored in the system due to the displaced tissue

$u_n$ =

weighted shape function

$U_s$ =

strain energy due to the needle bending

US =

ultrasound

$u_t$ =

measured path at the tip from the US probe

V =

work applied to the system

$V_p$ =

work applied by the internal tendons of the needle

$V_t$ =

work applied at the tip due to the needle-tissue interactions

$\overline{y}$ =

position of the needle's neutral axis

$z$ =

axial coordinate of the needle

$\alpha$ =

thread angle

$\beta$ =

constants

$\gamma$ =

constants based on trigonometric functions

$\theta$ =

bending angle of the needle

$\mathrm{\Pi }$ =

total stored energy in the system

$\phi$ =

function based on equation simplification

$\mathrm{\Phi }$ =

matrix containing $\phi$

$\omega$ =

function based on equation simplification

$\mathrm{\Omega }$ =

matrix containing $\omega$