Evaluation In Vitro of a Treatment Planning Algorithm for an Epicardial Crawling Robot

Abstract

HeartLander is a small, mobile robot designed to assist surgical procedures on the surface of the heart. It crawls within the pericardial sac surrounding the heart. Numerous potential clinical uses for HeartLander involve injections or other interventions at multiple locations on the epicardial surface. To minimize treatment time, we have developed an algorithm that optimizes a plan for reaching a given set of treatment targets. Results from in vitro evaluation on a beating heart model show improvement over a greedy technique.

I. Introduction

HeartLander is a small, mobile robot that has been designed to assist surgical procedures on the surface of the heart (the epicardium). It is inserted through a subxiphoid incision (below the sternum), and then through a small incision in the pericardium (the sac surrounding the heart). This technique is less invasive than traditional laparoscopic techniques, which require the left lung to be collapsed in order to access the heart behind it. By attaching to the epicardium, HeartLander passively compensates for the movement of the heartbeat. These advantages allow the patient to breathe normally, and obviate cardiopulmonary bypass. By allowing regular breathing and heartbeat, general anesthesia is potentially obviated as well.

HeartLander has two body segments, or feet, which attach to the epicardium using suction (See Fig. 1). Using flexible push-wires connected to offboard stepper motors to modulate the distance between the feet, and alternating suction between them, HeartLander achieves an inchworm-like locomotion [1]. This allows HeartLander to move to various treatment targets on the heart, where it can administer treatments such as injection, pacemaker lead placement, and tissue ablation for cardiac resynchronization [2]. Such treatments often involve reaching multiple targets on the heart. For example, after a patient suffers a heart attack, a portion of her heart muscle may be damaged. Administering treatments to the area of damage may encourage the muscle to heal properly. Taking the least amount of time to administer these treatments results in faster post-operative recovery, and decreases operating room costs.

Fig. 1.

A view of the HeartLander crawling robot.

A magnetic tracker provides information about HeartLander's position and orientation with six degrees of freedom, allowing movement around the heart to be coordinated. A tube running from outside the body through the front foot provides a channel through which the treatments are deployed.

HeartLander can coordinate its motion in two ways: regular locomotion and fine positioning locomotion. In regular locomotion, the back foot attaches to the heart using suction, and the front foot is left free. The stepper motors push the front foot forward. The front foot applies suction, and the back foot is released. This allows the stepper motors to pull the back foot up to the front foot. This completes one step. In order to turn, one wire is extended farther than the other. Using this locomotion, HeartLander can travel to any position on the heart surface. In fine positioning locomotion, the back foot remains adhered to the epicardium, while the front foot reaches for a treatment target.

While on the heart surface, HeartLander experiences friction forces that can result in slippage while using regular locomotion. This friction is a result of the close proximity of the organs within the chest to each other. The forces increase during certain phases of the heartbeat and respiration. The slippage that these forces create affects the efficiency and accuracy of regular locomotion. In fine positioning locomotion, the adhered back foot adds greater stability, and allows HeartLander to reach nearby treatment targets with greater accuracy [1].

The slippage experienced with regular locomotion increases procedure times, which increases treatment cost and influences patient outcomes. Combining both types of locomotion, and maximizing the use of fine positioning can decrease procedure times. We have developed an algorithm that first finds the smallest set of points, referred to as base locations, on the heart surface that allows HeartLander to use fine positioning to reach all the required treatment targets. Secondly, it finds the shortest path from the apex of the heart (where HeartLander is inserted), to each target in the set, and back to the apex for removal. To address these issues, we look to the literature on the Facility Location Problem [3] and the Traveling Salesman Problem [4] respectively, using the linear programming solver lpsolve [5] to find the solutions. We describe how we used these problems to develop an algorithm that combines HeartLander's two types of locomotion, which we compare to a greedy algorithm. We use three patterns of treatment targets to evaluate our algorithm, and analyze our findings.

II. Methods

A. Selection of Base Locations

The Facility Location Problem was solved in order to determine the base locations from which HeartLander used fine positioning to reach each treatment target. The problem deals with the problem of designating optimal warehouse (or facility) locations for a set of given stores (or sites). There are many possible facility locations, each with an associated building cost and service cost to each individual site. The Facility Location Problem selects the set of facilities that serve each site exactly once, with the lowest combination of building and service costs.

The Facility Location Problem can be solved as a Mixed Integer Linear Program (MILP) [6]. Given a set of variables and a set of constraints, a MILP solver finds the optimal values for each of the variables. To define the Facility Location Problem as a MILP, we define a set of variables F = (f₁, f₂,…, f_n) to represent our possible facilities, and a set of values Y=(y₁, y₂,…, y_n) to be the associated building costs for each facility. The MILP solver will set f_x = 1 if facility x is to be created, and f_x = 0 if it is not. The cost for facility i to serve site j is defined as c_ij. The MILP solver will set x_ij = 1 if facility i is to serve site j.

The constraints are defined to find the cheapest solution such that each site is served by exactly one facility, and that facility is one that will be created. This is defined formally as follows.

$$\begin{array}{c}min\sum _{i\le m}{f}_i{y}_i+\sum _{i\le m}\sum _{j\le n}{c}_{ij}{x}_{ij}\\ \mathit{\text{s.t.}}\\ \sum _{i\le m}{x}_{ij}=1,\forall j\le n\\ x_{ij}\le y_i,\forall i\le m,\forall j\le n\\ x_{ij},y_i\in \{0,1\},\forall i\le m,\forall j\le n\end{array}$$

In the case of HeartLander, the sites are the treatment targets required for the procedure. The facility locations are the places on the heart surface that HeartLander can stop and use fine positioning. In reality, those locations are continuous, but for the purposes of the algorithm, the locations are discretized. The fine positioning motion is the method of serving a site. The cost of reaching site j from facility i is determined by an equation based on both the distance HeartLander much stretch to reach the site from the facility point (d_ij), and the angle with which it much reach (θ_ij). Based on empirical observations of HeartLander's fine positioning, we used the cost function c_ij = 7.5θ_ij + 2.5d_ij. The angle is weighted more heavily because it has a stronger influence on the speed and accuracy of the fine positioning than the distance. If either the angle or the distance is beyond HeartLander's physical limitations, c_ij = •, ensuring that facility i will never be chosen to serve site j. To reduce the amount of regular locomotion HeartLander must use, we want to minimize the total number of facilities created. To achieve this, we set the values in Y to be high enough that it is not preferable to create extra facilities over using a high service cost. Each facility is given the same cost.

B. Order of Base Locations

Once the base locations are established, the problem of determining the order in which HeartLander will reach them is formulated as a Traveling Salesman Problem. This problem looks to find the minimal tour from a salesman's home, to each city in which he has business, and back to his home. His tour must be one continuous loop, and he only wants to visit each city once. The cost of the tour can be travel time, distance, or price. It can also be solved as a MILP. We define the cost between city i and city j to be c_ij, for i, j < n, where n is the number of cities. The MILP solver sets the variable x_ij = 1 if we choose to travel from city i to city j. The variable u is used to ensure that our tour is continuous, rather than a number of small, separate loops. The other constraints are defined such that we find the least expensive solution, while ensuring that we visit each city once.

$$\begin{array}{c}min\sum _i\sum _j{c}_{ij}{x}_{ij}\\ \mathit{\text{s.t.}}\\ x_{ij}\ge 0,\forall i,j\\ x_{ij}\le 1,\forall i,j\\ \sum _{j\ne i}{x}_{ij}=1,\forall i\\ u_i-u_j+n{x}_{ij}\le n-1,\forall i\in \{0,1,\dots ,n-1\},j\in \{1,2,\dots ,n-1\}\end{array}$$

In the case of HeartLander, the cities are the base locations that were determined with the Facility Location Problem. The start and end location is the apex, the bottom of the heart where HeartLander is inserted. Because horizontal motion with HeartLander is more difficult and requires more time, tours involving much horizontal motion needed to have a higher cost than those that did not. Therefore, the cost function is defined as c_ij = d_ij + 10 h_ij, where d_ij is the distance between city i and city j, and h_ij is the horizontal component of that distance.

C. Other Considerations

By attaching to the surface of the heart, HeartLander moves along with the heart, passively compensating the beating motion. This is advantageous when administering treatments, because HeartLander does not need to actively predict the motion of the heart beat and move along with it. However, the signal from the magnetic tracker that is used to coordinate HeartLander's locomotion on the heart reflects the movement of the heartbeat. Because the treatment targets on the heart are defined statically, this artifact must be removed in order for HeartLander to reach targets accurately.

To remove the noise from the heartbeat artifact from the tracker signal, we applied a third-order Chebyshev Type II low-pass filter, with a stopband cutoff of 20 dB at a frequency of 1.0 Hz to the tracker signal. The filter attenuates the noise from the heartbeat to levels similar to the noise from the tracker itself, while introducing a minimal delay (See Fig. 2).

Fig. 2.

The Z signal from the magnetic tracker. HeartLander begins at rest of the heart, then takes a step, and is at rest again. The unfiltered signal is in red, and the filtered signal is in green.

Experiments were conducted on a Chamberlain Group no. 1008 heart model, which uses compressed air to create a simulated heartbeat. This model is fitted with a fabric cover to simulate the pericardium. Treatment points were defined on a 3D model generated using CT data from a scan of the heart model [7]. The points that define the heart surface in the 3D model were used as the set of potential base locations.

D. Experimental Setup

We evaluated our algorithm against a greedy algorithm, which was the approach used by HeartLander previously. This approach did not take advantage of fine positioning, nor weighted distances. The greedy algorithm chose the closest target to the previously chosen target, until all targets had been visited, starting and ending with the apex.

The treatment targets were defined in three patterns. The patterns start simply, and increase in complexity and relevance in a clinical environment. In the first pattern, a set of treatment targets were randomly defined on the surface of the heart. This pattern is not typical of a real surgical procedure. The second pattern simulated the treatment of the perimeter of an area of damaged tissue, while the third simulated the treatment of the entire surface of an area of damaged tissue.

An experiment was run for each of the three patterns, each with 5 paired trials. Each trial consisted of two parts, the greedy algorithm and our algorithm. Simulations of these experiments had been run previously, suggesting that procedure times would decrease with our algorithm [8].

The time for each experiment was measured in seconds from the beginning of locomotion at the apex until HeartLander's return to the apex after reaching each treatment target. Time to administer a treatment was not included.

III. Results

For each paired trial, a registration process was run to map the CT coordinate frame (where the treatment targets were defined), to the tracker coordinate frame (where the treatment targets were reached). This is accomplished by choosing points on the CT model, and choosing the corresponding points on the heart model. This results in a transformation matrix that allows us to translate between the two frames. Due to error in selecting the position of the points, and the changes of the heart shape while beating, this process is imprecise, and must be repeated regularly to maintain accuracy. The same transformation matrix was used in the two parts of each paired trial. The more precise the registration was, the easier it was for HeartLander to reach the treatment targets, and therefore the treatment times of both the greedy algorithm and our algorithm were reduced. This causes the two results from each trial to be paired.

Out of the 15 paired trials that were run, 4 resulted in the greedy algorithm obtaining the shorter procedure time, and 11 resulted in our algorithm having the shorter procedure time. We can apply the Two-Tailed Fisher's Exact Probability Test to test the Null Hypothesis that the greedy algorithm and the path planning algorithm are equally likely to have the shorter treatment time for any given trial. In doing so, we find that p < 0.05, and we can reject the null hypothesis. Therefore, there is a statistically significant association between using our algorithm and obtaining the shorter treatment time. At this time, more data is needed to statistically analyze the amount of time saved using our technique.

Fig. 3-5 show the paths that were generated by both our algorithm and the greedy algorithm for the random, perimeter, and area patterns respectively. The results in Table I show that the greatest improvement in procedure times occurred for the pattern that simulated treatment of the entire surface of an area of damaged tissue. This was expected, since in the other two patterns, most of the treatment targets were spread farther apart. HeartLander could only take advantage of fine positioning locomotion when treatment targets were within one step of each other. The improvement for the area pattern is beneficial, because it has the greatest clinical relevance. It is the pattern that is most likely to be used in a surgical procedure.

Fig. 3.

Path plans for treating a random pattern. The plan created by the greedy algorithm is on the left, and the plan created by our algorithm is on the right. Grey shows all potential base locations, regular locomotion is shown in cyan, and fine positioning is shown in magenta.

Fig. 5.

Path plans for treating a pattern simulating the treatment of the entire surface of an area of damaged tissue. The plan created by the greedy algorithm is on the left, and the plan created by our algorithm is on the right. Grey shows all potential base locations, regular locomotion is shown in cyan, and fine positioning is shown in magenta.

TABLE I. Planning Algorithm vs. Greedy ApproacH.

	Random	Perimeter	Area
Number of Treatment Sites	13	21	26
Number of Facilities	10	13	12
Error: Plan (mm)	2.5 ± 0.1	1.9 ± 0.2	2.1 ± 0.2
Error: Greedy (mm)	2.4 ± 0.3	2.1 ± 0.3	2.0 ± 0.3
Time: Plan (min)	17.7 ± 2.3	18.5 ± 2.8	18.8 ± 3.9
Time: Greedy (min)	19.9 ± 7.9	20.4 ± 1.1	25.9 ± 10.4
Mean Time Decrease	12.65%	9.41%	27.43%

IV. Discussion

Decreasing procedure times results in faster post-operative recovery, and decreased operating room costs. By optimizing the treatment plan using a combination of the Facility Location Problem and the Traveling Salesman Problem, the algorithm presented herein achieved shorter treatment times than those attained with a greedy algorithm, while maintaining roughly equal positioning accuracy. Future work will involve gathering more data, and moving towards experimental verification in vivo in a porcine model.

Fig. 4.

Path plans for treating a pattern simulating the treatment of the perimeter of an area of damaged tissue. The plan created by the greedy algorithm is on the left, and the plan created by our algorithm is on the right. Grey shows all potential base locations, regular locomotion is shown in cyan, and fine positioning is shown in magenta.