We performed10simulation runs with simulated perturbations for each target location in the environment starting from the same initial insertion location. Similar to the Sim-Test material, we performed both open-loop executions of feasible motion plans and closed-loop steering using our rapid replanning approach (Chapter. 2) for both optimization criteria. This is similar to the experiments conducted in the ex vivo porcine loin tissue sample (Sec. 2.7.3). For the shortest path criterion, we enlarged all obstacles by a safety buffer of 5 mm.
Fig. 3.6b shows the mean targeting error and standard deviations across multiple simulated insertions using the meansµtand covariancesMtestimated using our method (Eqns. (3.48) - (3.57)).
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Scene #1 Scene #2 Scene #3 Scene #4
Tar
geting Err
or
(mm
)
Targeting Error (Tissue Phantom)
Rapid Replanning Shortest Path Open-loop Shortest Path
Rapid Replanning Maximum Clearance Open-loop Maximum Clearance
(a) Physical experiments in Sim-Test material
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Scene #1 Scene #2 Scene #3 Scene #4
Tar
geting Err
or
(mm
)
Targeting Error (Tissue Phantom) - Simulation
Rapid Replanning Shortest Path Open-loop Shortest Path
Rapid Replanning Maximum Clearance Open-loop Maximum Clearance
(b) Simulations with estimated parameters (µtandMt)
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Scene #1 Scene #2 Scene #3 Scene #4
Tar
geting Err
or
(mm
)
Targeting Error (Tissue Phantom) - Simulation
Rapid Replanning Shortest Path Open-loop Shortest Path
Rapid Replanning Maximum Clearance Open-loop Maximum Clearance
(c) Simulations with single estimated meanµand covarianceM
Figure 3.5: We compare the targeting error using closed-loop, rapid replanning steering and open- loop execution for the two proposed optimization criteria in simulation in Sim-Test material ((b)-(c)). Error bars indicate one standard deviation of the targeting error over repeated trials. The errors, both for open-loop execution and closed-loop steering using our rapid replanning approach are comparable
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Scene #2 Scene #1
Targeting Error (mm)
Targeting Error (Porcine Loin)
Rapid Replanning Shortest Path Open-loop Shortest Path
Rapid Replanning Maximum Clearance Open-loop Maximum Clearance
(a) Physical experiments in ex vivo porcine loin tissue sample
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Scene #2 Scene #1
Targeting Error (mm)
Targeting Error (Porcine Loin) - Simulation Rapid Replanning Shortest Path Open-loop Shortest Path Rapid Replanning Maximum Clearance Open-loop Maximum Clearance
(b) Simulations with estimated parameters (µtandMt)
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Scene #2 Scene #1
Targeting Error (mm)
Targeting Error (Porcine Loin) - Simulation Rapid Replanning Shortest Path Open-loop Shortest Path Rapid Replanning Maximum Clearance Open-loop Maximum Clearance
(c) Single
Figure 3.6: We compare the targeting error using closed-loop, rapid replanning steering and open- loop execution for the two proposed optimization criteria in simulation in ex vivo porcine loin tissue. Error bars indicate one standard deviation of the targeting error over repeated trials. The errors, both for open-loop execution and closed-loop steering using our rapid replanning approach are comparable to the errors encountered in steerable needle insertions in actual ex vivo porcine tissue.
For comparison, we also include the mean targeting error and corresponding standard deviations for the actual experiments in ex vivo porcine loin tissue sample as described in Sec. 2.7.3 (Fig. 3.6a) and when we used a single estimated meanµand covarianceM as given by Eqn. (3.58) (Fig. 3.6c).
The mean targeting error for open-loop execution of motion plans for the maximum clearance criterion for the two test scenes (Scene #1 and #2) are in close agreement. However, the mean targeting error and standard deviation error for the shortest path criterion is overestimated in simulation. This can be attributed to the fact that we only have limited statistics for the physical experiments from 3runs for each target within the ex vivo porcine loin tissue sample. The mean targeting error for closed-loop execution of motion plans for both criteria are comparable to the errors encountered in practice (Fig. 3.6a).
3.8
Discussion
We presented a data-driven stochastic model of the motion of the needle tip. We use an expectation maximization (EM) algorithm for estimating the parameters of this stochastic model from data gathered from experiments and prior procedures. Since modeling all sources of error and uncertainty during steerable needle procedures is challenging, our data-driven method provides an alternate means of creating stochastic models that capture needle behavior to serve as a basis for algorithms for preoperative procedure optimization, motion planning, state estimation, and control of steerable needles. Since the objective is to capture the cumulative effects of all sources of error and other unmodeled effects, we assume that the available data is representative of the errors encountered during steerable needle procedures. The hypothesis is that the greater the amount of available data, the better our understanding will be of uncertainty in steerable needle procedures. Our method estimates different parametersµtandMtfor each time steptto better capture uncertainty due to local effects by explicitly considering the impact of insertion depth on uncertainty.
Our approach has a few limitations. We are restricted by our assumption of a discrete time kinematic motion model. Since the model parameters are estimated based on measurements that are obtained at discrete time intervals and the number of time intervals in the procedures, our stochastic model cannot be used to simulate needle procedures that might vary greatly in the time step or length
which is important from the perspective of state estimation, planning, and control. The dependence of our method on the curvature occurs through the computation of the physical control inputs that are applied to the needle during duty cycled rotation of the needle. However, our method is general enough to work for needles with similar curvature characteristics.
CHAPTER 4
Unified Framework for Planning and Control in
Deformable Environments
In this chapter, we present a unified framework for motion planning and feedback control for closed-loop steering of steerable needles in deformable tissue. We use a sampling-based motion planner based on a physically-based simulator of the deformable environment to generate a set of candidate plans based on expected deformations. We use the simulator and optimal control to numerically estimate time-dependent state distributions based on uncertain parameters (e.g. deformable material properties, actuation errors, and noisy sensing) and then select the plan with the highest estimated probability of successfully avoiding obstacles and reaching the target region. Using FEM-based simulation of deformable tissues, we demonstrate that our method can generate high quality plans for guiding steerable needles around obstacles to the desired target region under considerable deformations and uncertainty under 2D image guidance.