Feedback Control for Steering Needles Through 3D Deformable Tissue Using Helical Paths

Kris Hauser, Ron Alterovitz, Nuttapon Chentanez, Allison Okamura, Ken Goldberg.

Yajia Zhang

Background

• Needles are used in medicine for a wide range of diagnostic and therapy delivery procedures.

• Needle tip must be positioned accurately at the target in the tissue. But the process requires skills. Errors may occur even under image guidance.

• Feedback controller that steers the needle and places the needle tip at the target even under the perturbation of the tissue and deflection of the needle trajectory.

Background

• Bevel-tip steerable needle• Steering the needle:

1. Force along the z axis. Cause the needle tip rotate about the x axis.

2. Rotate about the z axis. • Constant-Insertion-Speed and Constant-Twist-Rate

Helix Trajectory of needle tip• Needle tip position determined by inserted distance

d and twist rate φ.

Goal

• Feedback controller to steer the bevel-tip needle with:

1.Constant insertion speed2.Different twist rate

to reach the target in the tissue.

Why real time planner• Deformation of the tissue. Cause position of the target change.

• Deflection of the planned trajectory.

Controller Framework• For every iteration:

I. Propose: Generate a set P of proposal trajectories. Different φ maps to different trajectory.

II. Select:Find the trajectory with control φ(d) in the set P that achieves the minimal distance to the target.

III. Execute:Insert according to φ(d) and constant velocity for time Δt.

I. ProposeGenerate Proposal Trajectories

• When inserting the needle into the tissue, we build the coordinate frame according to the position of the needle tip.

Constant-Twist-Rate Helical Paths• After moving along the helix trajectory, we need to

know what the coordinate of the need tip according to the initial frame.

• :Helix with radius a, slope θ and oriented along the z axis.

)(dh

Constant-Twist-Rate Helical Paths

• : coordinate of the needle tip followed under a constant twist rate φ and insertion d.

• Rigid Transformation:

)(dt

)()( dtdAh

2sin)sec)/sin((cos

)sec)/cos((cossin)sec)/((sin

)(

drda

ardadrda

dt

2cosra rtan

Constant-Twist-Rate Reachable Set• The trajectory with infinity twist rate will almost

along the z axis.

}0),,(|)({)( ddtR

Alternating-Twist Maneuver

• For finite maximum twist rate, alternating-twist maneuver can reduce the gap along the z axis.

}0],,[|)({)( maxmaxmax ddtR )(R

Alternating-Twist Maneuver

• To fill in the gap, we consider the maneuver that makes a full turn of the helix with twist rate , and another with twist rate

II. SelectChoose the trajectory with minimum distance to target

• Minimize • + proposal trajectories in

the gap• Auxiliary function is used to calculate a tight

lower bound of given a region R.

||target)(||),( dtdf

}0,|||),{( max ddS

)(RfL),( df

Branch-and-Bound• A search tree recursively split the space into

subregions. We maintain the helix * and insertion distance d* which give the minimum value of f. If of subregion R gives value larger than f, we can safely prune the region. We continue the process until f achieves an ε tolerance.

)(RfL

III. Execute

• Insert according to the select trajectory for time Δt or some distance Δd.

Simulation

Simulation Result

• Accuracy: The final distance from the needle tip to the target when the controller terminates.

• Reference controller: A refresh occurs every 2%r of insertion distance, maximum twist rate = 10π rad/rmax

Simulation Result• Accuracy Without Perturbations:

Simulation Result

• Accuracy Under Perturbations and Modeling Errors:

• Gaussian Noise

Possible Improvement

• Avoid the obstaclesSet intermediate Target’. When reaching Target’, we may just assume the Target moved.

Target

Target’

Issue About Real Time Planning

• How often should we re-plan?• The reference controller refreshes after inserting

length s= 2%r to achieve high accuracy.• What if the deformation and deflection do not

happen (or some tiny changes)? • Complex system may require intense computation in

re-planning.