Radiofrequency ablation (RFA) is an increasingly used, minimally invasive, cancer treatment modality for patients who are unwilling or unable to undergo a major resective surgery. There is a need for RFA electrodes that generate thermal ablation zones that closely match the geometry of typical tumors, especially for endoscopic ultrasound-guided (EUS) RFA. In this paper, the procedure for optimization of an RFA electrode is presented. First, a novel compliant electrode design is proposed. Next, a thermal ablation model is developed to predict the ablation zone produced by an RFA electrode in biological tissue. Then, a multi-objective genetic algorithm is used to optimize two cases of the electrode geometry to match the region of destructed tissue to a spherical tumor of a specified diameter. This optimization procedure is then applied to EUS-RFA ablation of pancreatic tissue. For a target 2.5 cm spherical tumor, the optimal design parameters of the compliant electrode design are found for two cases. Cases 1 and 2 optimal solutions filled 70.9% and 87.0% of the target volume as compared to only 25.1% for a standard straight electrode. The results of the optimization demonstrate how computational models combined with optimization can be used for systematic design of ablation electrodes. The optimization procedure may be applied to RFA of various tissue types for systematic design of electrodes for a specific target shape.

## Introduction

Pancreatic cancer is the fourth leading cause of cancer death among men and women in the U.S. with an average 5 year survival rate of 7% [1]. Patients are typically asymptomatic until an advanced stage of the disease, leading to late diagnosis and limited treatment options. Surgical resection is the only treatment option that has been shown to increase the average 5 year survival rate; however, only 20% of patients are eligible for surgical resection. This is, in part, because these patients are typically older and cannot or will not tolerate this major surgery which carries a 2–5% risk of mortality and a 20–40% risk of major complication [2]. For those ineligible for surgery, less invasive methods need to be explored and improved to increase effective treatment options.

Radiofrequency ablation (RFA) is a technique that began in the early 1990s and has been applied to cancer treatment throughout the body including the lungs, liver, kidneys, and pancreas [3,4]. The treatment is typically performed percutaneously and more recently endoscopically. During the procedure, an electrode is inserted into the tumor and large grounding pads are placed on the skin of the patient. When a radiofrequency current is applied, a highly concentrated electric field forms around the electrode and disperses quickly out to the grounding pads. The large grounding pads placed on the skin provide a large surface area to reduce the concentration of the electric field and avoid burns on the skin. The tissue dipole molecules in the concentrated regions of the electric field rotate as they attempt to align their charge with the oscillating current. Frictional heating occurs due to the motion of the molecules and then spreads through conduction to the surrounding tissue [4,5]. The temperature and exposure time determine the extent of the damage to the tissue.

The target temperature for ablation is between 50 and 100 °C [5]. Outside of this range, tissue cells will not reliably reach the point of cell death. For temperatures below 50 °C, the cells require a long exposure time to be destroyed and at temperatures greater than 100 °C, the cells will char, increasing their resistance, and limit the RF current flow. In the range of 50–60 °C, reported values for the exposure time required to cause cell death vary greatly in the literature [3–7]. However, there is agreement in that for cells heated to temperatures between 60 and 100 °C, cell death is nearly instantaneous [3,6,8–10]. The region of cell death surrounding the electrode is referred to as the ablation zone.

One common challenge associated with RFA is incomplete destruction of the tumor cells at the periphery. In order to prevent reoccurrence, the ablation zone should fully envelope the tumor as well as a small margin around the periphery. In the case of RFA, it is difficult to match the geometry of the ablation zone with the geometry of the tumor and required margin. For example, if the ablation zone is ellipsoidal and the tumor is spherical, then the mismatch in the geometries results in inadequate destruction of the tumor, or unnecessary destruction of healthy tissue in order to destroy a margin of tissue around the tumor.

The shape of the ablation zone is determined primarily by the electrode shape and tissue properties. The electrode shape determines the initial heating profile in the tissue, and the thermal tissue properties play a critical role in the expansion of the temperature profile, because the primary mode of heating is conduction. The variability of tissue properties patient-to-patient are difficult to predict, but it is envisioned that the electrode shape could be optimized to meet the patient-specific tumor geometry.

A few different configurations of electrodes for percutaneous or laparoscopic procedures are available in attempt to match the ablation zone with the geometry of the tumor. Hong and Georgiades [5] gave a review of current single electrodes and electrode systems. The most basic shape of RFA electrodes is cylindrical which typically generates an ellipsoidal ablation zone. The exposed length of the electrode can be adjusted or multiple cylindrical electrodes can be inserted in an attempt to alter the geometry of the ablation zone. Hong and Georgiades also describe more complex expanding electrodes that broaden the ablation zone through multiple tines that deploy or expand into tissue as the electrode is inserted. In his review of RFA for pancreatic adenocarcinoma, D'Onofrio et al. describes how straight electrodes are used for more cylindrical tumors and deployable electrodes are used for more spherical tumors [11]. However, these unique designs have limited effectiveness, because the treatment for each patient's tumor is unique.

A second challenge of RFA is the ability to treat tumors in certain regions of the upper abdomen including the pancreas. Endoscopic approaches are commonly used to access these locations in the upper abdomen and as such, several animal and clinical trials were recently conducted for endoscopic ultrasound-guided (EUS) RFA treatment of pancreatic cancer [12–14]. Each found the procedure to be safe with minimal risk for complications.

While EUS-RFA has been shown to be safe and increases accessibility to some areas of the body, there are only two EUS-RFA electrodes which have been tested in clinical trials, the Habib EUS-RFA catheter (Emcision Ltd., London) [13–15] and the cryotherm probe (ERBE Elektromedizin GmbH, Tübingen, Germany) [16,17]. Each electrode is a straight cylinder, which generates an elliptical ablation zone. There is a need for specialized electrode designs that can reduce the geometrical mismatch that often occurs in treatment of pancreatic cancer with RFA.

In order to understand RFA in tissue, numerous computational models have been created. RFA models have been applied in thyroid tissue based on reconstructed models from magnetic resonance imaging and also in liver-tissue using a hyperbolic bioheat equation and a new-voltage calibration method [18,19]. Various ablation models have been used to study the effects of tissue properties such as electrical conductivity and blood perfusion rate [20–22]. Others models focus on specific commercially available electrodes [23–26]. Chang [20] focused on the effects of the blood perfusion rate and temperature-dependent electrical conductivity. Later, Chang and Nguyen [23] studied how isotherms can be used to predict the ablation zone. Tungjitkusolmun et al. [27] presented a 2D and 3D model of ablation using the Rita Medical Systems Electrode and explored the effects of ablation near blood vessels. In addition, Altrogge et al. [25] studied the optimization of the placement of a commercial electrode in order to increase the efficiency of RFA. Another model by Lim et al. [28] focused on improving efficiency by considering alternative input waveform patterns. While many models have been used to understand RFA, the use of these models within a formal design optimization of probe tip geometry to create a specific ablation zone has not been considered.

In this paper, a procedure for combining a computational RFA model and an evolutionary optimization algorithm for the design of RFA electrodes that generate spherical ablation zones is outlined. The procedure is applied to the design of a specially shaped EUS-RFA ablation electrode for pancreatic cancer or other abdominal tumors accessible by EUS. First, a compliant electrode design is proposed. Next, a thermal ablation model is developed to predict the ablation zone surrounding the electrode. Finally, the ablation model is coupled with a genetic algorithm to optimize the geometry of the electrode and match the ablation zone with a spherical tumor.

## Methods

### Compliant Endoscopic Ultrasound-Guided Radiofrequency Ablation Electrode Design.

The compliant EUS-RFA electrode design was inspired by the tail feathers of a peacock, which begin as a straight bundle and then expand out into a large circular display. An earlier version of the design was proposed by Hanks et al. [29]. The peacock-inspired electrode is designed to be introduced into the tumor endoscopically, through a hollow needle. As shown by Fig. 1, the endoscopic needle, or sheath for the electrode, will be advanced through an endoscope up to the periphery of the tumor. Then, the electrode will be delivered through the needle and deployed into the tumor. The design of the electrode is such that it may be retracted into the endoscopic needle and, if necessary, redeployed in another location.

Two variations of a generic or base design for the optimization, shown in Fig. 2, are based on a circular array of outer tines and a single central tine which are connected at the exposed electrode base. It is shown in both the stowed (Figs. 2(a) and 2(d)) and deployed (Figs. 2(b) and 2(e)) configurations. The principle of the design is that the natural state of the device is open. The outer tines are compressed around the central tine and may be stowed within a rigid sheath, such as an endoscopic needle. When the electrode is extended out of the rigid sheath, the electrode tines will deploy and cut paths radially outward into the tumor. The expansion of the electrode will broaden the ablation zone as compared to a traditional straight needle electrode. The electrode design is parameterized in terms of the outer diameter in the stowed configuration ($de$), length of the center tine ($lc$), number of outer tines ($nt$), length of each outer tine ($loi,i=1,\u2026,nt$), radius of curvature of each outer tine ($roi$, $i=1,\u2026,nt$), and length of the exposed electrode base ($lb$) which connects the outer tines to the center tine. Figure 2(c) shows the electrode design parameters $de$, $lc$,$loi$, $roi$, and $lb$ for the $i$th outer tine of an eight outer tine example ($nt=8$). Figure 2(f) shows the design parameters associated with a 12 outer tine variation ($nt=12$) of the electrode which will be used as a second case for optimization. The outer tines are designed to have constant curvature because studies of flexible needles show that during insertion, the curvature of the needle path is essentially constant [30–32].

### Thermal Ablation Model.

The electric currents and bioheat transfer modules of comsolmultiphysicsmodeling software (version 5.2) are used to develop the thermal ablation model. These modules couple the thermal and electrical components of the simulation of RFA in biological tissue through the multiphysics module. The finite element analysis is transient and simulates an ablation procedure to obtain the geometry of the ablation zone generated by an electrode. During the simulation, the electrode is held fixed, as defined by design variables shown in Fig. 2. The model uses similar underlying assumptions and boundary conditions as the models by Chang [20] and Tungjitkusolmun et al. [27].

A cylindrical block with a radius of $rt$ and height $ht$ is used to represent the tissue and should be large enough to avoid boundary interactions with the elevated temperatures generated by the ablation. An insulated shaft, that would deliver the power to the electrode in the actual procedure, is represented by a cylinder of radius $rb$ and height $hb$. It is placed concentrically inside the cylinder extending from the bottom of the tissue up to the bottom of the exposed electrode base. The compliant electrode design is generated in comsol and fixed to the insulated shaft, centered within the tissue block. In order to reduce the computational size of the model, a simplified design without the beveled tips was generated for the ablation modeling. RFA tissue heating is caused by the concentrated electric field near the electrode, and thus, the heat transfer in the tissue should remain unaffected by the simplified model.

In Eq. (1), $\sigma $ (S/m) is the electrical conductivity, $V$ (V) is the voltage potential, $Je$ (A/m^{2}) is an externally generated current density, and $Qj$ (A/m^{3}) is a current source. In Eq. (2), $\rho $ (kg/m^{3}) is the tissue density, $C$ (J/(kg K)) is the tissue specific heat, *T* (K) is the temperature, $u$ (m/s) is the velocity vector, $k$ (W/(m K)) is the tissue thermal conductivity, $\rho b$ (kg/m^{3}) is the blood density, $Cb$ (J/(kg K)) is the blood specific heat, $\omega b$ (1/s) is the blood perfusion rate, $Tb$ (K) is the arterial blood temperature, $Qmet$ (W/m^{3}) is the heat source from metabolism, and $Qext$ (W/m^{3}) is the heat source from the spatial heating.

Electric boundary conditions of the model are an applied voltage to the active portion of the electrode design ($V=V0$) and grounded external surfaces of the tissue block ($V=0$). The applied voltage which generates a direct current as opposed to the radiofrequency current used in the actual procedure is a simplification, which is also used in the thermal model by Chang and Nguyen [23]. The grounded external surfaces of the cylindrical tissue block are representative of the large grounding pads used in the procedure to disperse the current.

The thermal boundary condition is specified as a constant temperature on the external surfaces of the tissue block ($T=T\u221e$). Heat conduction is permitted through the active portion of the electrode and exposed electrode base. Tetrahedral elements are chosen to mesh the tissue and electrode. Element size limitations are imposed in three sections in order to accommodate the small features of the model without drastically increasing the computational size.

In order to simplify the model, quarter symmetry was used to reduce the computational time, see Fig. 3. On the symmetry planes used to divide the model, an insulation boundary condition was specified. For the electrical currents module of comsol, the insulation boundary requires that no current flows through the boundary. In other words, the electric potential is assumed to be symmetric about the boundary. For the bioheat transfer module, an insulation boundary condition requires that the heat flux through the boundary be zero, or that the temperature is symmetric about the boundary. For a summary of the initial and boundary conditions, see Table 1.

Initial condition | Boundary condition | ||
---|---|---|---|

Tissue | Electrical | $V=0$ | — |

Thermal | T = T_{∞} | — | |

Electrode | Electrical | $V=V0$ | $V=V0$ |

Thermal | T = T_{∞} | — | |

Grounding pad | Electrical | $V=0$ | $V=0$ |

Thermal | T = T_{∞} | T = T_{∞} | |

Symmetry plane | Electrical | $V=0$ | $n\xb7J=0$ |

Thermal | T = T_{∞} | $\u2212n\xb7q=0$ |

Initial condition | Boundary condition | ||
---|---|---|---|

Tissue | Electrical | $V=0$ | — |

Thermal | T = T_{∞} | — | |

Electrode | Electrical | $V=V0$ | $V=V0$ |

Thermal | T = T_{∞} | — | |

Grounding pad | Electrical | $V=0$ | $V=0$ |

Thermal | T = T_{∞} | T = T_{∞} | |

Symmetry plane | Electrical | $V=0$ | $n\xb7J=0$ |

Thermal | T = T_{∞} | $\u2212n\xb7q=0$ |

The ablation zone is defined by the 60 °C isothermal surface surrounding the electrode. Time- and temperature-dependent models for cell death vary in the literature; however, at 60 °C, literature agrees that cell death is essentially instantaneous [3,6,8–10]. Chang and Nguyen [23] found that the 60 °C isothermal surface was a good representation of the ablation zone by comparing their model with experimental data.

### Optimization.

The optimization of the electrode was performed using a multi-objective genetic algorithm in matlab and is an extension of previous work by Hanks et al. [33]. A genetic algorithm was chosen because of the increased likelihood of finding a global minimum compared to gradient-based approaches, and its ability to converge to an optimal solution without the need to calculate numerical or analytical gradients. The multi-objective genetic algorithm is part of the global optimization toolbox in matlab and is a variant of the nondominated sorting genetic algorithm (NSGA-II) algorithm [34]. The purpose of the algorithm is to find the Pareto-optimal set of designs, or Pareto front. The Pareto front is the set of designs which are nondominated by any other design. A nondominated design is a design for which no other design is better in all objective functions.

The NSGA-II algorithm variant in matlab works by evaluating a series of design sets using the thermal ablation model. Each design set, or generation, is created based on the results of previous generations, gradually converging to the Pareto-optimal set of designs. matlab creates the initial generation randomly distributed throughout the design space. The design space is defined by the user-defined constraints and the upper and lower bounds of each design variable. Next, the electrode geometry, defined by the design variables, is sent to comsol for simulation and the data from each simulation are tabulated. Following each comsol simulation, matlab collects the data and evaluates the objective functions for the design. After all the designs in the generation have been simulated, they are ranked based on the objective functions evaluations. The nondominated designs are given a rank of one, designs dominated by one other design are rank two, and designs dominated by two designs are given a rank of three, etc. After all the designs of the generation have been ranked, a portion of the best-ranked designs are carried over to the next generation and other designs are generated by creating crossovers and mutations according to matlab default settings. The crossovers and mutations allow the genetic algorithm to sample a large design space and help avoid local minima. The process is repeated for subsequent generations, gradually improving with each generation.

Multi-objective optimization algorithms present a unique advantage, because all designs along the final Pareto front are optimal. On the contrary, weighting factors or ratios are required when multiple objectives are combined into a single objective. In order to test different weights or ratios, the optimization problem has to be run multiple times. A multi-objective optimization enables the designer to view a set of Pareto optimal designs and select a solution based on the relative importance or limits of each objective function. In the case of treating tumors, the multi-objective optimization allows the designer to view the set of optimal solutions, consider the importance of each objective function, and select the best design while considering other patient specific factors such as surrounding anatomy.

As the optimization progresses, the metric used to determine convergence is the change in the distribution of the Pareto front. Once the Pareto front is essentially stagnant, the algorithm has determined a set of optimal designs which represent the best tradeoffs of multiple objective functions. The change in the distribution of the Pareto front is computed as the average relative change in the spread of the optimal designs. Where the spread is a distance measurement of the distribution of designs along the Pareto front. The algorithm is considered converged when the average relative spread change for the previous $nstall$ generations is less than $\eta $, the stall tolerance, and the current spread is less than the average of the previous five generations. In other words, the distribution of the optimal designs has had minimal changes for $nstall$ generations, and the spread of designs is decreasing.

The purpose of the optimization is to match the size and shape of the ablation zone for a particular electrode with the size and shape of target ablation zone. After all the variables and objective functions are described, a formal description of the optimization is given by Eqs. (5)–(9). The design variables included in the optimization are $lc$, $nt$, $loi$, $r0i$ as defined previously. The upper and lower bounds for each design variable, represented by Eq. (5), define the design space.

*θ*,

*φ*location; see Eq. (7). The resulting $fi$ values are grouped by whether they are greater or less than the target distance and then averaged into two objective functions, $fover$ and $funder$, see Eqs. (8) and (9). The objective function $fover$ focuses on minimizing the amount of tissue destroyed outside the target zone (healthy tissue beyond the target zone). The objective function $funder$ focuses on maximizing the amount of tissue destroyed within the target zone. While these two objectives functions are not always in direct conflict, they result in a set of Pareto-optimal designs from which the designer can make a more informed decision when compared with a single objective optimization. The use of these objective functions allows for consideration of optimal electrode designs ranging from no ablation outside the target region ($fover=0$) to ablating the entire target region ($funder=0$) and a variety in between. The constraint in Eq. (6) allows the designer to control the length of the center tine relative to the length of the outer tine through $Cli$

This metric is unitless and represents the measure of how close the ablation zone matches the target shape.

## Results

For the purpose of the EUS-RFA electrode for pancreatic cancer, the materials used in the thermal ablation model are assumed to be pancreatic tissue, polyurethane for the insulated shaft, and nickel–titanium (Nitinol) for the exposed electrode base, outer tines, and center tine. Nitinol was chosen for its superelasticity and biocompatibility. For Nitinol, the superelastic property means there is minimal increase in stress for a large increase in strain during the phase transition from austenite to martensite. The material properties used for each of the above listed materials are specified in Table 2 [35–40]. The target ablation zone was assumed to be a 25 mm sphere, based on typical tumors that would be treated with such a procedure. Each simulation was for an ablation procedure time of 6 min. A voltage $V0=20$ V was applied to the electrode. An ambient temperature $T\u221e=37$ °C was chosen based on standard body temperature. The size of the pancreatic tissue used during the simulation was a cylinder of radius $rt$ = 6 cm with a height $ht$ = 12 cm, for which boundary effects from the proximity of grounding pads are negligible. The convergence of the finite element model was performed for a set of electrode design parameters by continuously refining the mesh until the ablation zone volume remained essentially unchanged (within 0.5%). The insulated shaft was modeled as a cylinder of radius $rb$ = 0.48 mm with a height $hb$ = 5 cm, centering the electrode in the pancreatic tissue. The length of the exposed electrode base which connects the outer tines to the central tine was held fixed at 1 mm.

Material | Density $(kg/m3)$ | Thermal conductivity $(W/mK)$ | Electrical conductivity $(S/m)$ | Heat capacity $(J/kgK)$ | Ambient temperature $(K)$ | Perfusion coefficient (1/s) |
---|---|---|---|---|---|---|

Pancreatic tissue [35,36] | 1087 | 0.51 | 0.566 | 3164 | 37 | — |

Blood [35,37] | 1050 | — | — | 3617 | 37 | 0.0064 |

Polyurethane [38,39] | 70 | 0.026 | 1 × 10^{−8} | 1045 | — | — |

Nitinol [40] | 6450 | 18 | 1.25 × 10^{6} | 836.8 | — | — |

Material | Density $(kg/m3)$ | Thermal conductivity $(W/mK)$ | Electrical conductivity $(S/m)$ | Heat capacity $(J/kgK)$ | Ambient temperature $(K)$ | Perfusion coefficient (1/s) |
---|---|---|---|---|---|---|

Pancreatic tissue [35,36] | 1087 | 0.51 | 0.566 | 3164 | 37 | — |

Blood [35,37] | 1050 | — | — | 3617 | 37 | 0.0064 |

Polyurethane [38,39] | 70 | 0.026 | 1 × 10^{−8} | 1045 | — | — |

Nitinol [40] | 6450 | 18 | 1.25 × 10^{6} | 836.8 | — | — |

Two cases will be presented to illustrate how the optimization technique can be useful in systematic design of EUS-RFA electrodes. Each case is summarized in Table 3. Case 1, shown in Figs. 2(a)–2(c), presents an electrode design in which there are eight outer tines and each tine has the same length and radius of curvature. In keeping each of the outer tines the same length and curvature, the number of design variables is reduced to three: $lc$, $lo$, and $ro$. Case 2, shown in Figs. 2(d)–2(f), allows more freedom to the optimization by allowing for more complex electrode designs. The number of tines is increased to 12 and there are two configurations of outer tines, such that every other tine has the same configuration. All odd number tines have the same length ($loodd$) and radius of curvature ($roodd$) while all even number tines have the same length ($loeven$) and radius of curvature ($roeven$). The upper and lower bounds for each design variable are shown in Table 3. They are chosen such that a wide range of electrode designs could be simulated, from a long and thin profile to a very short and round profile.

Case 1 | Case 2 | |
---|---|---|

Design parameters | 3 | 5 |

Stowed diameter ($de$) | 0.96 mm | 0.96 mm |

Center length ($lc$) | $lc$ | $lc$ |

Number of tines ($nt$) | 8 | 12 |

Outer tine length ($loi$) | $loi=lo,i=1,\u2026,8$ | $loi=loodd,i=1,3,\u2026,11loeven,i=2,4,\u2026,12$ |

Outer tine radius ($roi$) | $roi=ro,i=1,\u2026,8$ | $roi=roodd,i=1,3,\u2026,11roeven,i=2,4,\u2026,12$ |

Exposed electrode base ($lb$) | 1.0 mm | 1.0 mm |

Constraint parameter ($Cli$) | 0.8 | 0.8 |

Stall generations ($nstall$) | 5 | 5 |

Pareto change tolerance ($\eta $) | 0.001 | 0.001 |

Grid points ($ngp$) | 23 | 23 |

Target radial distance ($ti$) | 12.5 mm | 12.5 mm |

Optimization bounds and constraints | $5mm5mm5mm\u2264lcloro\u226425mm25mm200mm\u2212lc+0.8l0i\u22640$ | $5mm5mm5mm5mm5mm\u2264lclooddloevenrooddroeven\u226430mm30mm30mm200mm200mm\u2212lc+0.8l0even\u22640\u2212lc+0.8l0odd\u22640$ |

Case 1 | Case 2 | |
---|---|---|

Design parameters | 3 | 5 |

Stowed diameter ($de$) | 0.96 mm | 0.96 mm |

Center length ($lc$) | $lc$ | $lc$ |

Number of tines ($nt$) | 8 | 12 |

Outer tine length ($loi$) | $loi=lo,i=1,\u2026,8$ | $loi=loodd,i=1,3,\u2026,11loeven,i=2,4,\u2026,12$ |

Outer tine radius ($roi$) | $roi=ro,i=1,\u2026,8$ | $roi=roodd,i=1,3,\u2026,11roeven,i=2,4,\u2026,12$ |

Exposed electrode base ($lb$) | 1.0 mm | 1.0 mm |

Constraint parameter ($Cli$) | 0.8 | 0.8 |

Stall generations ($nstall$) | 5 | 5 |

Pareto change tolerance ($\eta $) | 0.001 | 0.001 |

Grid points ($ngp$) | 23 | 23 |

Target radial distance ($ti$) | 12.5 mm | 12.5 mm |

Optimization bounds and constraints | $5mm5mm5mm\u2264lcloro\u226425mm25mm200mm\u2212lc+0.8l0i\u22640$ | $5mm5mm5mm5mm5mm\u2264lclooddloevenrooddroeven\u226430mm30mm30mm200mm200mm\u2212lc+0.8l0even\u22640\u2212lc+0.8l0odd\u22640$ |

The objective functions $fover$ and $funder$ are based on grid points around the ablation zone surface, see Fig. 4. These points correspond to positions at *θ* = 0 deg, 45 deg, and 90 deg. At each *θ* location (0 deg, 45 deg, and 90 deg), eight positions are calculated for corresponding to *φ* = 0 deg, 22.5 deg, 45 deg, 67.5 deg, 90 deg, 112.5 deg, 135 deg, 157.5 deg, and 180 deg. Repeated grid points at *φ* = 0 deg and 180 deg, for *θ* = 0 deg, 45 deg, and 90 deg, are removed, resulting in a total of 23 grid points to determine the objective functions. The target radial distance ($ti$) was equivalent for each point around the surface ($ti$ = 12.5 mm). The optimization algorithm seeks to minimize both objective functions, finding a range of optimal designs. Convergence is determined by an average relative spread change of less than 0.1% over the final five generations ($nstall$ = 5, $\eta $ = 0.001).

### Case 1.

The optimization was performed using a population of 75 designs per generation and converged after 138 generations. Figure 5 shows the entire design space, i.e., a comparison of the objective function values for each design evaluated. Each point on the plot represents a feasible design, the plot shows the tradeoff between objective functions which exist for some designs. From the objective functions, one can quickly understand the relative size and shape of the ablation zone. For example, for designs along the horizontal axis ($fover=0$), all 23 grid points are within the treatment zone. Other designs along the Pareto front decrease $funder$, or fill more of the target volume, but also treat regions of tissue beyond the target zone. This range of optimal designs will allow for discretion of whether it is preferable to only treat tissue within the target region or to increase volume of treated tissue within the target region at the expense of destroying other tissue outside the target region. In this study, a single optimal electrode design was selected for further consideration based on the design located closest to the utopia point.

For case 1, the parameters for the optimal electrode design selected for comparison against case 2 are $lc$, $lo$, and $ro$ are 18.33 mm, 15.57 mm, and 10.42 mm, respectively, with $fover$ and $funder$ equal to 0.0 and 0.1027, respectively. The distance from the utopia point to the optimal design is 0.1027. The temperature profile for the optimal design from case 1 is shown in Fig. 6 (top).

### Case 2.

The optimization was performed using a population of 75 designs per generation and converged after 91 generations. The entire design space is shown in Fig. 7. As with case 1, a wide range of optimal designs lie along the Pareto front providing a variety of options for optimal designs. From the final Pareto front of designs, the parameters for the optimal electrode design selected for comparison against case 1 are $lc$, $loodd$, $roodd$, $loeven$, and $roeven$ are 22.42 mm, 19.49 mm, 20.31 mm, 15.39 mm, and 8.49 mm, respectively, with $fover$ and $funder$ equal to 0.0434 and 0.0555, respectively. The distance from the utopia point to the optimal design is 0.0705. The temperature profile for the optimal design for case 2 is shown in Fig. 6 (bottom). The two optimal electrode designs from case 1 and case 2 are compared visually by their ablation zones in Fig. 8 and their design parameters are summarized in Table 4.

Center length | Outer length | Outer radius | ||||
---|---|---|---|---|---|---|

Case 1 | $lc$ (mm) | $lo$ (mm) | $ro$ (mm) | |||

Optimal | 18.33 | 15.57 | 10.42 | |||

Case 2 | $lc$ (mm) | $loodd$ (mm) | $loeven$ (mm) | $roodd$ (mm) | $roeven$ (mm) | |

Optimal | 22.42 | 19.49 | 15.39 | 20.31 | 8.49 |

Center length | Outer length | Outer radius | ||||
---|---|---|---|---|---|---|

Case 1 | $lc$ (mm) | $lo$ (mm) | $ro$ (mm) | |||

Optimal | 18.33 | 15.57 | 10.42 | |||

Case 2 | $lc$ (mm) | $loodd$ (mm) | $loeven$ (mm) | $roodd$ (mm) | $roeven$ (mm) | |

Optimal | 22.42 | 19.49 | 15.39 | 20.31 | 8.49 |

## Discussion

A systematic design optimization approach is useful in comparing electrode designs by reducing the amount of trial and error and intuition-based experimental work. A comparison of the two electrode designs in cases 1 and 2 can be used to show the benefit of such a systematic approach.

The purpose of cases 1 and 2 was to compare two electrode designs, where the second is a more complex design, which increases freedom of the optimization to find an electrode which generates an ablation zone that more closely matches the target, a 25 mm sphere. For case 1, all the outer tines have the same length and radius of curvature, limiting spherical ablation zones which can be generated. In particular, as the size of the target shape increases, it becomes more difficult for the electrode to fill the target shape while all outer tines have the same configuration. This can be seen in Fig. 6 (top), where the ablation zone could be improved by broadening the lower and upper portions. With a single outer tine configuration, the electrode is unable to fill these lower and upper portions. Increasing the number of tines and allowing multiple configurations should allow for better shape matching.

For case 2, Fig. 6 (bottom) shows that increasing the number of outer tines and allowing for two outer tine configurations resulted in significant improvement. The optimization improved the ablation zone by using the second set of tine configurations. As evident from the objective function values, the optimization was able to find a design with a decreased distance to the utopia point in case 2. Thus, the optimization approach was able to show that the case 2 design has the potential for improved shape matching. Depending on the increase in complexity for manufacturing case 2 designs, the improved objective function values may be desirable. For more complex, nonsymmetrical, target shapes, it is expected that electrode designs which allow more freedom will be necessary. The objective functions utilized for this optimization are useful for matching a sphere and can easily be expanded to more complex or arbitrary shapes.

In order to compare against currently available endoscopic electrodes, the ablation zone for a straight needle is also shown in Fig. 8 (left). The length of the straight electrode is typically adjustable up to 20 mm. In this case, the full electrode length (20 mm) was used for simulation and comparison. By comparing the ablation zone volumes with the target shape, one can obtain a quantitative sense of the improvements seen with the optimal designs. The percentage of the target volume filled by the ablation zone is computed by dividing the ablation zone volume inside the target zone by the total target volume. The percentage of the ablation zone outside the target volume is computed by dividing the ablation zone outside the target volume by the total ablation zone volume. The current straight electrode ablated only 25.1% of the target volume. The optimal solutions show increased treatment volumes of 70.9% and 87.0% for case 1 and case 2, respectively. The results for the straight electrode and optimal solutions for case 1 and case 2 are summarized in Table 5.

Single | Case 1 | Case 2 | |
---|---|---|---|

Percentage of target zone filled (%) | 25.1 | 70.9 | 87.0 |

Percentage of ablation volume outside target zone (%) | 0.5 | 0.4 | 3.2 |

Single | Case 1 | Case 2 | |
---|---|---|---|

Percentage of target zone filled (%) | 25.1 | 70.9 | 87.0 |

Percentage of ablation volume outside target zone (%) | 0.5 | 0.4 | 3.2 |

While this optimization approach has shown promise as a useful tool in simulating the ablation zone and designing RFA electrodes, limitations exist in the numerous variables of the human body. Experimental validation of the thermal ablation model has not yet been performed; although, when compared with other similar models reported in literature the average error between temperature profiles across the electrode was found to be 1.8% [20]. In addition, the pancreatic tissue in this model does not account for the inhomogeneity which exists in all biological tissue; however, the random and constant dividing of cancer cells will produce more homogeneous tissue as compared to typical healthy tissue. The tissue properties were also chosen based on an approximation of tumors, though, in reality, a variety of values are reported in literature for various pancreatic tissue properties.

One challenge with selecting tissue properties is that the reported values in literature vary and are often temperature or frequency dependent. For this optimization, the electrical conductivity and blood perfusion are assumed to be constant according to reported values for pancreatic tissue. The electrical conductivity is temperature dependent as discussed by Chang [23,35] and has been explored in many other studies [22,36,41]. The blood perfusion rate has also been shown to decrease with increased metabolic activity in tumors making it difficult to determine an appropriate value [37]. Although the overall size and temperature of the ablation zone was found to be sensitive to these parameters, the shape of ablation zone was not heavily affected. Thus, the current ablation model can be used to understand the general shape of the ablation zone and then tissue properties can be used to tune the model to match experimental ex vivo and in vivo testing for validation. It is also important to recognize that RFA may have limited effectiveness near large blood vessels because they act as a significant heat sink, limiting the temperature to which the tissue can be heated. While this model does not include the heterogeneities present in living tissue (blood vessels, surrounding organs, or other structures), the same optimization approach could be utilized with models of heterogeneous tissue to generate an electrode design for patient specific treatment planning.

While this approach for design optimization of RFA ablation electrodes shows great promise, this model only accounts for an electrode shape after it has been inserted in the tissue. The deployment of the electrode, and therefore the ablation zone produced, are likely to be affected by variations in tissue stiffness and other tissue properties. In order to consider the robustness of the current electrode designs, the outer tine radius of curvature was varied by $\xb1$10% for the case 1 solution, and the ablation zone volume was calculated. For an outer tine radius of curvature of 9.38 mm, the ablation zone volume increased by 3.3%. For an outer tine radius of curvature of 11.46 mm, the ablation zone volume decreased by 3.7%. Therefore, for slight variations in outer tine radius of curvature, the change in total ablation zone volume was small. In order to account for these variations in deployment, it will be necessary to develop modeling which predicts the deformation and mechanical response of the electrode as it deploys while being inserted into the tumor. This type of deployment model will be useful in relating the manufactured electrode shape with the final electrode shape deployed in tumors.

Ongoing work on the ablation electrode design includes studying the deployment of the tines through soft tissue both computationally and experimentally, ensuring manufacturability of the design, and ex vivo experimentation to validate the thermal ablation model.

## Conclusions

The purpose of this study was to optimize the shape of an RFA electrode to match the ablation zone for a target tumor geometry in order to improve the efficacy of treatment of pancreatic cancer. Using a computational model based on coupling the electric and bioheat transfer equations in comsolmultiphysics software, a thermal ablation model was used to predict the ablation zone for unique electrodes. The resulting electrode shape was optimized to generate a 2.5 cm spherical ablation zone using a multi-objective genetic algorithm. While this model and electrode are specifically focused on EUS-RFA of pancreatic cancer, RFA is a common procedure for tumors throughout the abdomen. This approach of optimizing the electrode shape for a target ablation zone has the potential to personalize treatment for many cancer patients and may be applied to a variety of tissues where RFA is practical.

Optimization of the electrode geometry with a thermal ablation model can be a valuable tool when designing specially shaped ablation probes. The potential to develop unique ablation probes, based on a specific target geometry, or a specific tumor, can help to customize patient care. In this case, the target shape of the ablation zone was spherical, however, the desired ablation zone may correlate to a patient specific tumor shape. In addition, more complex models may be developed which incorporate patient specific blood vessel structures to optimize the electrode while considering heat sink effects. This method of systematic design through optimization can be used to design specially shaped electrodes for patient specific cases. These optimization results provide evidence toward the benefits of computational modeling and its potential impact on personalized medicine.

## Funding Data

Division of Emerging Frontiers in Research and Innovation (1240459).

## Nomenclature

- $C$ =
specific heat of pancreatic tissue

- $Cb$ =
specific heat of blood

- $Cli$ =
constraint value for relative center and outer tine length

- $de$ =
stowed electrode outer diameter

- $du$ =
distance to the utopia point

- $fi$ =
percentage greater or smaller than $ti$ for a grid point

- $fover$ =
objective function representing ablation zone outside the target region

- $funder$ =
objective function representing ablation zone inside the target region

- $hb$ =
height of insulated shaft connected to the electrode

- $ht$ =
height of biological tissue during simulation

**J**=current density

- $Je$ =
externally generated current density

- $k$ =
thermal conductivity of pancreatic tissue

- $lb$ =
exposed electrode base length

- $lc$ =
electrode center tine length

- $lo$ =
length of outer tines

- $lclower$ =
lower bound of center tine length

- $lcupper$ =
upper bound of center tine length

- $loeven$ =
length of even numbered electrode tines for case 2

- $loi$ =
electrode outer tine length, $i=1,\u2026,nt$

- $lolower$ =
lower bound of outer tine length

- $loodd$ =
length of odd numbered electrode tines for case 2

- $loupper$ =
upper bound of center tine length

**n**=surface normal vector

- $ngp$ =
number of grid points for computing the objective functions

- $nover$ =
number of grid point locations outside the target region

- $nstall$ =
number of stall generations for determining convergence

- $nt$ =
number of electrode outer tines

- $nunder$ =
number of grid point locations inside the target region

**q**=heat flux

- $Qext$ =
heat source from spatial heating

- $Qj$ =
current source

- $Qmet$ =
metabolism heat source

- $rb$ =
radius of insulated shaft connected to the electrode

- $ri$ =
radial distance from the center of the ablation zone to a grid point

- $ro$ =
radius of curvature of outer tines

- $rt$ =
radius of biological tissue during simulation

- $roi$ =
electrode outer tine radius of curvature, $i=1,\u2026,nt$

- $roeven$ =
radius of curvature for even numbered electrode tines for case 2

- $rolower$ =
lower bound of outer tine radius of curvature

- $roodd$ =
radius of curvature for odd numbered electrode tines for case 2

- $roupper$ =
upper bound of outer tine radius of curvature

- $T$ =
temperature

- $ti$ =
target radial distance from the center of the ablation zone to a grid point

- $Tb$ =
blood temperature

- $T\u221e$ =
ambient tissue temperature

- $u$ =
velocity vector

- $V$ =
voltage potential

- $V0$ =
voltage applied to the electrode

- $\eta $ =
tolerance for Pareto front distribution change during $nstall$ generations

- $\rho $ =
density of pancreatic tissue

- $\rho b$ =
density of blood

- $\sigma $ =
electrical conductivity of pancreatic tissue

- $\omega b$ =
perfusion rate of blood