Needle Steering Force Model and Trajectory Planning

Reporter : Pan Li

Supervisors: Prof. Zhiyong Yang, A.P. Shan Jiang

Content

1 Needle Insertion Force Model

2 Trajectory Planning for insertion needle

Future Works3

Experimental designPVA Phantom Ultrasonic

motorSurgical needle 6-axis capacity ATI

force/torque sensor

Fig.1 Visual image acquisition hardware and software systems

Fig.2 The devices for force measuring experiments

High speed camera

Force sensor

Deformation

Insertion

Extraction

A

B

C

D

Experimental procedure

1) Deformation phase (from A to B): Tissue deformation occurs.2) Insertion phase (from B to C): The needle penetrates into the soft tissue. 3) Extraction phase (from C to D): The needle is exacted from the soft tissue.

The force acting on a needle is written as: ( ) ( ) ( ) ( )s f cF x f x f x f x F x

)()( xfxF s)()()( xfxfxF cf

)()( xfxF f

Stiffness Force

1

Friction and Cutting Force

2Friction Force

3

Material Relaxation and Friction

Fig.3 Insertion force-time profile Fig.4 Insertion force-deep profile

Stiffness ForceThe stiffness force corresponds to viscoelastic interaction and can be described as prepuncture force.

Needle Insertion Force Model

Surface of soft tissue

Soft tissue

Needle base

Needle shaft

Needle tip

Deformation h

r

y

o

a

)(rfy

2)tan(2 hrEf stiffness

bebFzf

axaxazfdxa

stiffness

stiffness

)(

012

2

0)()(

)(

指数函数模型：

型：模式项多polynomial model:

exponential function model:

Fig.5 Five experiments of force curve profile Fig.7 Comparison between stiffness model and experimental dataFig.6 Error comparison between polynomial and exponential model

1

0)()( dttaEzf rstiffness dx

x

xfh

1

0 21

)( dxx

xfxaEf rstiffness

1

0 2

2

1

)(2

Fig.8 Stiffness model schematic diagram

cot)( axxf

Stiffness Force2)tan(2 hrEf stiffness

Stiffness force：

Equivalent modulus:2

22

1

21 111

EEEr

Parameter acquisition:

Fig.9 Parameter acquisition experiment diagram of insertion needle

7.0od mm 4.0id mm

4244 10052.1)(64

mmddI io

Moment of inertia:

Deflection equation: )3(6 1

2

max xlIE

Fxw

Strain energy density expression:

Five experiment result: 1E =14.1±1.6GPa

)3()3( 2211 ICICW

Deformation tenser invariant: 23

22

211 I

213

232

2212 )()()( I

Mooney-Rivlin model experiment:

21

1

21

1

1 1

)1(2CCt

Optimization toolbox by ANSYS:

2E = 116KPa

Geometry of needle :

(a) Impale the soft tissue

Friction ForceThe friction force occurs along the length of the needle inside the tissue and is due to Coulomb friction, tissue adhesion and damping.

(b) Withdrawing of the needle (c) Two consecutive insertions

Measurement of friction force:

Fig.10 Measurement of friction force

Friction Force

Soft tissue

Foundation

Needle base

Needle shaft

z

k

h

Surface of soft tissue

nF

frictionf

Foundation modulus: 12

1

42

22

2

165.0

IEbEE

k

Friction force model: hIEDEEDf friction 12

1

42

22

22 )(

1

65.02

Normal force: kAFnContact area: DhA

nfriction Ff Coulomb friction

Fig.11 Modified Winker foundation model sketch diagram

Fig.12 Comparison between force model and experimental data

Cutting Force

Cfcutting

The cutting force ranges from 0.04N to 0.1N periodically, and the fluctuation can disappear when the insertion speed is equal to 3mm/s and the insertion forces of two consecutive insertions are parallel to each other.

Constant

(1) :

Complete force model

DCfriction

CBcuttingfriction

BArstiffness

needle

zzzzIEDEEDf

zzzCzIEDEEDff

zzzzEf

zf

12

1

42

22

22

12

1

42

22

22

2

)(1

65.02

)(1

65.02

)tan(2

)(

The overall shape of the model is coherent to the experimental results, although the fluctuations in the insertion phase make a perfect match impossible.

Comparison: )()( )(mod)( xfxf elsmeasures

？(2) :

？(3) :

)(and)( )(mod)( xfxf elfmeasuref

)(and)( )(mod)( xfxf elcmeasurec

Trajectory Planning for insertion needle

Key technology ：Avoiding vital tissues;Implanting into tumor accurately;Shorten trajectory path;Reducing implant errors.

Anterior

Posterior

LeftRight

Bladder

Rectum UrethraTumor

Pelvis

Prostate

Superior

Inferior

Posterior Anterior

Artificial potential field？Local minimum in concave curved surface model

G

O

P

3D trajectory planning result of the spiral pipe model

3D trajectory planning result of the concave curved surface model

3D trajectory planning result of local minimum

minp

)()( attatt pp UF

Tattattatt

att ,,)(

zyxp

UUUU

)(22

1)( goalattatt pkp U

goalattatt )( ppkp F

0)(

0)(0)(

repr

0

11

2

1

)(

p

ppep

if

ifkpU

0)(

0)()(

obstacle

)(2

0)(rep

rep

0

111

)(

p

pppp

if

ifpp

kp

F

Relationship of force and potential value: Attractive potential value and force: Repulsive potential value and force:

Improved conjugate gradient method

Escaping from Local minimum

)1( kS

kk gS )(

)2( kS

2 kg

)()( kk S1 kg

)1( kp)(kp

)2( kp

e 3l

2l

1l

s

ig

minp

Optimization of trajectory planning

Select as the initial point

k=0

Calculate the new search point

No

Yes

Calculate the gradient of the new

search point

Select the first search point and preset the deflection angle of needle tip

Start

171 ~ gg

minp

Calculate the gradient of the first point kg

Select the negative direction of the gradient )(ks

cos2

))1(( kpU

YesCompute the optimal

step size )( k

)1( kp

Escape from the local minimum?

1kg

Compute the value of parameter

Confirm the search direction of next

step )1( ks

k=k+1 Select the deflecting

direction Angle of needle tip

As next search Direction angle

k=k+1

Input the first insertion point s

Define four cubic elements around s and mark 17 nodes with letter and subscript number 171 ~gg

Calculate composition gradient field of each node

Search the node that gradient decent most quickly

ig

Connect initial point and

Reach local minimum?

Yes

No

Select as the initial point

s

No

Reach the target?

YesEnd

No

ig

s ig

APF

ICGA

(a) , is selected as search direction angle

(b) , is selected as search direction angle

kkk gpUS )( )()(

)(1

)()1()1( )( kk

kkk SgSpUS

2

2111

][][

k

k

kT

k

kT

k

g

g

gggg

？Judgment cos)(

2)1( kpU

)()()( )(min kkk SpU

)()()()1( kkkk Spp

Modeling interactive forces during needle insertion and experiment validation.

Modeling and simulation of tissue deformation.

Modeling needle deflection during insertion.

Trajectory planning considering tissue deformation and needle deflection.

Needle guidance and steering with experiment validation.

Future works

Fig.1 Accurate robotic needle steering requires a model that predicts needle motion within the tissue. Both a stochastic motion planner (used pre- and/or intra-operatively) and an image-guided model-based feedback controller can use the mechanics-based model

Insertion model

Mechanics model and insertion

model

Modeling interactive forces

Trajectory planning

Needle guidance and steering

Tissue deformation and needle deflection

Stiffness force:

Modeling interactive forces during needle insertion

(a) Symmetric needle tip

(b) Asymmetric (bevel) needle tip

Fig.2 Asymmetry of the bevel tip produces a resultant transverse load which causes a flexible needle to naturally bend

Surface shape function :

xxf )( cota

cot)( axxf

Stiffness force:

1

0 2

2

1

)(2 dxx

xfxaEf rstiffness Simplified into:

2)(tan2 hEf rstiffness

Whereas:

cot)( axxf

To asymmetric needle tip:

Surface shape function:

To symmetric needle tip:

1

0 2

2

1

)(2 dxx

xfxaEf rstiffness Deduction: ?

Friction force: Coulomb friction

Fig.3 Overall slip model

Friction force：

NoneMacroslip

nKN

f {Fig.4 Partial slip model

Elastic deformation phase

Partial slip phase

Entire slip phase

sl

s slll

sll

dsr

dsrdsrrK

dsrrK

f

slip Entire)(

slip Partial)()()(

n deformatio Elastic)()(

1 2

Friction force：

Evolutionary:

Friction force:

others)(sgn

&00)(

es

seef

FF

FFvFvvF

F

veNvvF

i

svv

ccf

)()(sgn)( 0

Stribeck friction model:

: the damping coefficient associated with

LuGre friction model:

0

( )

0

( , )( )

( ) ( , )L t

f

vdz t v zdt g v

F t dF t

( ) ( ) vc p cg v e

0 1 2( , ) ( , ) ( , ) ( , )nzdF t z t t v dF tt

Where v is the contact velocity of each differential element and L is the incision length; and

( , ) ( ) ( )n ndF t dF f d

0 :stiffness coefficient of the microscopic deformations

1 z2 : the viscous relative dampingc and are the normalized coulomb and stictions

( )

0 1 20( ) ( , ) ( , ) ( )

L t

f nzF t z t t v f dt

Karnopp friction model:

2/)(

2/0),min(02/),max(

2/)(

),(

vvvbvgc

vvfdvvfd

vvvbvgc

fvf

rrprp

rsp

rsn

rrnrn

srfriction

frictioninertiameasured fff

))()(( nnnnppppmeasured vbvgcvbvgclmaf

))()(( nnnnppppfriction vbvgcvbvgclf

Fig.5 Stribeck model Fig.6 LuGre friction model

Fig.7 Karnopp model

Modeling and simulation of needle insertion into soft tissue

Modeling needle deflection during insertion

Fig.8 Distributed compressive load acting on a needle shaft as it interacts with an elastic medium.

Rayleigh-Ritz: energy

workinput

input

energy

1 )(11

WSN EE

work

Q

energy

2 )()(22 RPEE WWWSN

load axialbending needle pureABE UUN Sum of energy:

Needle-tissue interaction energy:

ninteractioncompressio

TCE UUS

ilPW inputinput Insertion force work:

Transverse tip load: ))()(( 1Q iiiii lvlvQW

Axial tip load: ))0()((P iiii luPW Rupture the elastic medium:

iclaGW R

Fig.9 Free-body diagram of the forces acting on the needle tip as it interacts with the elastic medium.

Trajectory planning considering tissue deformation and needle deflection

Trajectory planning considering tissue deformation and needle deflection

Fig.10 Artificial potential field

Fig.11 Numerical analysis method Fig.12 Inverse Kinematics solution

Considering needle deflection？Neglecting tissue deformation

(c) needle deflection and local changes in needle orientation.

(a)Tissue deformation (b) trajectory changes

mathematic(al) model？physiological anatomical model

(a) Basic concept

(b)Subdivided sampling time for higher update rate of the MPC-based feedback controller.Fig.13. Concept diagram of a model predictive controller.

Needle guidance and steering with experiment validation

Needle guidance and steering with experiment validation

Input constraints:

6.1

2.16.1

8.0maxmin

State constraints:

4040- maxmin

Model predictive control:

Discrete control inputs

TTp

TT NkukukukU )1(...)1()()(

Fig. 14. The reference flexible probe, the measured real flexible probe, and their local coordinates.

Trrrrr yxq ][

Tyxq ][

Linearized tracking error model

THANK YOU