Multi-Objective Optimization Design and Numerical Study of Water-Cooled Microwave Ablation Antennas

Abstract

Microwave ablation, as a minimally invasive technique used for the treatment of tumors, is highly dependent on the performance of ablation antennas for its therapeutic effect. Clinically, antennas are required to form roughly spherical ablation zones with sufficient volume within a limited time. To meet this requirement, this paper establishes finite element models and conducts multi-objective optimization on fully water-cooled dipole antenna and partially water-cooled choke dipole antenna based on different water-cooled structures. On the premise of minimizing reflection coefficient and maximizing ablation volume, a three-dimensional objective space is constructed by introducing the minimization of roundness error, and the set of Pareto solutions is solved. The CRITIC-TOPSIS method is used to balance multi-objective conflicts and select the unique optimal solution from the Pareto set. By analyzing the optimal solution, simulation results show that the optimized antennas can effectively form near-spherical ablation shapes while minimizing the reflection coefficient and maximizing the ablation volume. Among these, the partially water-cooled antenna exhibits superior electromagnetic characteristics and ablation profile, whereas the fully water-cooled antenna demonstrates better temperature field behavior.

1. Introduction

As a core component of the global tumor burden [1], liver tumors have always been a focus of clinical treatment. Surgical resection remains the primary clinical approach for liver tumors. However, this procedure, which involves removing tumor-invaded tissue, creates large wounds, inevitably sacrifices healthy tissue, and may lead to a series of complications [2,3,4]. Against this background, minimally invasive ablation techniques have gradually gained clinical attention. This technology features high safety and small wounds, making it increasingly preferred by patients. Common minimally invasive ablation techniques include ultrasound, radiofrequency, microwave, laser, and irreversible electroporation ablation [5,6,7,8]. Among these, microwave ablation has become the first-choice minimally invasive ablation surgery in clinical practice due to its excellent penetration ability, efficient ablation speed and low incidence of complications [9,10,11,12,13].

The therapeutic effect of microwave ablation mainly depends on the performance of the ablation antennas. Currently, ablation antennas are categorized into monopole, dipole, and slot antennas based on their basic structural characteristics [14,15,16]. Most of them are designed to minimize the reflection coefficient and obtain sufficient ablation area by increasing the input power or prolonging ablation time. However, there is a serious problem of reverse heating caused by reflected currents that mainly originate from reflected microwaves induced by impedance mismatch between the coaxial cable and biological tissue. This reverse heating leads to the formation of long strip-shaped or comet-shaped ablation zones, namely tailing [17]. Such tailing-induced irregular ablation zones make it difficult for clinicians to delineate an ablation range that minimizes unnecessary damage to healthy tissue, thereby severely impairing patients’ postoperative recovery. Therefore, in clinical practice, spherical ablation zones are preferred [18], as they not only facilitate the standardization of surgical procedures but also offer predictable ablation margins. These features enable clinicians to accurately define the ablation range, directly reducing the risk of unnecessary damage to healthy tissue. In addition, while high input power and adequate ablation time are regarded as effective approaches to ensuring the ablation range completely covers the tumors, this strategy is largely constrained by patients’ tolerance and elevates the risk of complications. Therefore, microwave ablation antennas that can form a sufficient ablation area in a short period of time while minimizing damage to surrounding healthy tissue are still the main focus of current research.

The improvement of coaxial cable structure and the introduction of water-cooling systems are currently the main research directions for designing antennas that meet this requirement. In terms of the design of feed lines, Cavagnaro [19] proposed a choke dipole antenna, which adds a thin metal block inside the antenna to block the backward propagation of microwaves and concentrate the microwave energy at the needle tip. Compared with dipole antennas, it effectively suppresses the tailing in the ablation area. Brace [20] optimized the coaxial dual slot ablation antenna with the axial ratio of heating area and reflection coefficient as the optimization objectives, obtaining a reflection coefficient of −20.9 dB and an axial ratio of 0.7, although there was a small tailing in the axial direction of the ablation area. However, there is a lack of thermal physical properties of the feed line during the simulation process. In practical situations, metal feed line with good thermal conductivity further transfers heat, causing unnecessary damage to healthy tissues in the axial direction. In contrast, unlike improving the structure of the feed line to suppress reverse heating, the water-cooling systems directly absorb this excess heat through cooling water to suppress the formation of tailing [21]. Current studies often simplify the antenna model as a single feed line, focusing solely on optimizing the electromagnetic characteristics of the coaxial cable, without integrating a water-cooling system into the model or conducting collaborative optimization of water-cooling system and antenna structural parameters within a unified framework. This oversight means that when the water-cooling system is subsequently integrated into the already optimized antenna, the high relative permittivity of water (${\epsilon }_r$ = 78) and adjustment of the cooling range will cause changes in the effective dielectric environment, which may further disrupt the previously optimized impedance matching and energy distribution. In addition, when there is a clear conflict between optimization objectives, current optimization methods tend to rely on subjective weight allocation or expert experience, which makes it challenging to objectively quantify the weights of various design parameters and balance the trade-offs between optimization objectives.

To bridge these gaps, the finite element method is used to simulate and analyze the performance of microwave ablation antennas in liver tissue. This paper establishes models that incorporate the water-cooling system and perform co-optimization of the water-cooling system and the antenna structure. Based on the differences in water-cooling structures, the partially water-cooled antenna and fully water-cooled antenna are selected as optimization objects to solve for the Pareto optimal solutions. For these two antennas, the fully water-cooled antenna achieves complete water-cooling coverage of the entire feed line, while the partially water-cooled antenna lacks cooling at the tip due to the additional structure for suppressing reflected currents occupying its axial space. In terms of optimization objectives, while ensuring the minimization of reflection coefficient and maximization of ablation volume, the roundness error is minimized as much as possible, with a focus on the balance between the tailing and the under-temperature zones. For Pareto solution sets that exhibit obvious conflicts, the CRITIC-TOPSIS method is used to screen the unique optimal solution. Then the water-cooled antennas under the optimal solution are characterized and analyzed. The final results indicate that both antennas exhibit excellent ablation performance and can form a nearly spherical ablation area with sufficient volume. The difference in water-cooled structures is mainly manifested in the temperature field. Compared with the partially water-cooled antenna, the fully water-cooled antenna can effectively reduce the temperature of the needle tip and minimize the risk of carbonization.

2. Materials and Methods

2.1. Finite Element Model

This paper uses COMSOL MULTIPHYSICS 6.3 for numerical analysis and simulates the ablation performance in liver tissue by establishing models of microwave ablation antennas. In the modeling of antennas and liver, partially water-cooled choke dipole antenna and fully water-cooled dipole antenna are selected as optimization objects based on the differences in water-cooling structures. The cooling range of choke dipole antenna extends from the needle handle to the choke, while the range of dipole antenna extends to the dipole tip. To simplify numerical simulations, this paper treats the liver as a uniform biological tissue. Combined with the microwave ablation antenna’s axisymmetric structure, the ablation zone exhibits approximate symmetry about the axis, so a two-dimensional axisymmetric model is constructed for performance simulation. The rectangular region of the simulated liver tissue is set to a width of 80 mm and a height of 110 mm, as well as the initial temperature of the liver tissue is 37 °C. The shapes and dimensions of the models constructed are shown in Figure 1. Both antennas exhibit differences solely in the axial dimensions of the coaxial cables and the extension length of the water-cooled tubes, with the dimensions of all other components remaining identical.

In terms of parameter settings for antenna models, the contact surface between the dielectric of the coaxial cable and the outside is designated as a microwave input port with a power of 50 W. The microwave frequency is 2.45 GHz and the ablation duration is 6 min. The edge of the liver is defined as a thermal insulation boundary to assume that the heat transfer process only occurs between the liver tissue and the antennas [22,23], while the scattering condition is defined as an electromagnetic scattering boundary, which means that the boundary does not disturb the electromagnetic field distribution [23].

Regarding the material selection of the antennas, the inner and outer conductors of the coaxial cable are made of copper, and the dielectric is filled with PTFE. The puncture cap of ablation antenna is made of zirconia ceramic, which has both resistance of high temperatures and good puncture performance. The outer needle tube is made of stainless steel, its high strength ensures that the antenna does not deform during insertion into tissue. About the design of heat dissipation, water-cooled tube made of PTFE serves as pathway for heat exchange to suppress tailing, and the initial temperature of the cooling water is 20 °C. The partially water-cooled antenna is additionally equipped with a choke made of copper. The thermal and physical properties of liver tissue and microwave ablation antennas are shown in

Table 1. The thermal and physical properties of materials [24,25].

Material	Density (kg/m3)	Specific Heat (J/kg·K)	Thermal Conductivity (W/m·K)
Liver	1079	3570	0.52
Zirconia ceramic	6050	400	2.2
PTFE	800	1000	0.25
Stainless steel	8055	500	16.2
Copper	8960	386	401

, and the electromagnetic characteristics at 2.45 GHz are shown in

Table 2. The electromagnetic properties of materials at microwave frequency of 2.45 GHz [19,23].

Material	Relative Permittivity	Conductivity (S/m)	Relative Permeability
Liver	43.03	1.69	1
Zirconia ceramic	25	0	1
PTFE	2.03	0	1
Stainless steel	1	1.1 × 106	1
Copper	1	5.8 × 107	1

.

Due to the fact that liver tissue is a lossy medium, the propagation of electromagnetic waves in liver tissue follows Helmholtz Equation (1) [26]:

$$\nabla \times {{\mu }_r}^{-1}(\nabla \times \stackrel{⃑}{E})-{k_0}^2({\epsilon }_r-{\displaystyle \frac{j\sigma }{w{\epsilon }_0}})\stackrel{⃑}{E}=0$$

(1)

where ${\mu }_r$ is the relative magnetic permeability of the liver, $\stackrel{⃑}{E}$ is the electric field intensity vector, $k_0$ is the propagation constant of electromagnetic wave, ${\epsilon }_r$ is the relative permittivity of the liver, $\sigma$ is the electrical conductivity of the liver, $w$ is the angular frequency of electromagnetic wave, and ${\epsilon }_0$ is the vacuum permittivity.

When electromagnetic waves penetrate liver tissue, their energy is absorbed and converted into thermal energy. The pennes biological heat transfer Equation (2) [27] is used to describe the conduction of heat within liver tissue.

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}=\nabla \cdot (k\nabla T)+w_b{\rho }_b{c}_b(T_b-T)+Q_{met}+Q_{ext}$$

(2)

where $\rho$ is density, $\mathrm{c}$ is specific heat capacity, $k$ is thermal conductivity, $w_b$ is blood perfusion rate (3.6 × 10⁻³ s⁻¹) which mediates heat exchange between liver tissue and circulating blood in the simulation to realize tissue heat transfer and dissipation. ${\rho }_b$ is blood density (1040 kg/m³) [27], $c_b$ is blood specific heat capacity (3639 J/kg·K), $Q_{\mathit{ext}}$ is resistance heating, and $Q_{\mathit{met}}$ is heat generated by metabolism. Among them, the heat source $Q_{\mathit{met}}$ is usually much smaller than the electromagnetic heat source, so it is ignored.

In addition, the energy deposition of microwaves in liver tissue is characterized by the specific absorption rate (SAR) (3) [28]:

$$SAR={\displaystyle \frac{\sigma |E{|}^2}{2\rho }}$$

(3)

With the heating of microwaves, the temperature of liver tissue continues to rise, and irreversible thermal damage occurs after accumulating a certain amount of heat. It should be noted that irreversible thermal damage of tissue is not solely determined by a single threshold of temperature. While reaching the critical temperature can directly induce the necrosis of tissue, it is essential to quantify the cumulative time effect using the damage fraction $F_d$ when the temperature remains below this threshold. In other words, if the sub-threshold temperature is sustained for a sufficient duration, the accumulated heat can still lead to necrosis. Therefore, arrhenius Equation (4) [29] is used to describe the effect of temperature on thermal damage to liver tissue, and the effective boundary of the damaged tissue is determined by the damage fraction (5).

$$\Omega \left(t\right)=\mathit{ln}\left({\displaystyle \frac{C\left(0\right)}{C\left(t\right)}}\right)=A{\int }_0^t{e}^{{\displaystyle \frac{-Ea}{R\cdot T\left(t\right)}}}dt$$

(4)

$$F_d=1-e^{-\Omega \left(t\right)}$$

(5)

where Ω(t) is the thermal injury coefficient, A is the frequency factor of liver tissue (7.39 × 10³⁹ s⁻¹ [30]), Ea is the activation energy of liver tissue (257,700 J/mol [30]), R is gas constant (8.3143 J/mol·K), T(t) is the temperature change over time. When the damage fraction reaches 0.632, it is considered that irreversible damage has occurred, and a value of 1 represents the maximum degree of thermal damage, indicating that the cells have died irreversibly.

2.2. Numerical Evaluation of Antennas

2.2.1. Coupling Power Between Antenna and Biological Tissue

During microwave ablation, the energy coupling efficiency between the antenna and biological tissue directly affects the morphology of the ablation area, and the reflection coefficient S₁₁ is the core parameter for quantifying the quality of this coupling [31]. It is defined as the proportional relationship between the reflected power Pr and the incident power Pi at the antenna’s feed port (6), measured in dB, which directly reflects the degree of microwave energy transmission from the antenna to biological tissue.

$$S_{11}=10\times {\mathit{log}}_{10}\left({\displaystyle \frac{\mathit{Pr}}{\mathit{Pi}}}\right)$$

(6)

A smaller S₁₁ indicates good impedance matching between the antenna and tissue, facilitating the efficient conversion of microwave energy into thermal energy to form a stable ablation zone. Conversely, a larger one means significant energy reflection, which may shift the ablation area and damage healthy tissue even the generator.

2.2.2. Ablation Volume

The actual antenna design requires sufficient ablation area within a limited time to fully cover the tumor boundary. This paper calculates the ablation volume by rotating the ablation surface of the two-dimensional axisymmetric model to generate a three-dimensional damage area. In assessment of damaged tissue, a damage fraction of 1 is defined as the threshold for determining the boundary of necrotic damaged area. This paper presents a statistical analysis of the necrotic ablation volume V.

2.2.3. Ablation Roundness

The quality of ablation roundness directly determines the coverage of the ablation area and the control of damage to surrounding healthy tissue. However, in the actual process of ablation, there are often two typical defects in the ablation contour, namely tailing and under-temperature zone [32], which together destroy the ablation roundness.

From the perspective of defect causes, tailing originates from the thermal damage caused by reverse heating of the antennas, and the thermal conductivity of the feed line and metal outer needle tube further exacerbates this problem. The under-temperature zone is directly related to the thermal regulation of the water-cooling system, manifested as a local depression [33] along the antenna axis at the ablation boundary, although this area generates thermal damage due to microwave radiation, the intensity of energy deposition is already weak due to its location at the edge of the ablation zone. Combined with the absorption of local heat by the cooling water, the degree of thermal damage is greatly reduced. Both types of defects will widen the deviation between the actual contour and the ideal circle.

Notably, as the under-temperature zone is inherently associated with water-cooling regulation, its extent and potential interaction with tailing may change with the flow rate of cooling water. To explore this variation and the interplay between the two defects, we studied the thermal damage distribution of liver tissue under different flow rates, plotted a curve with a damage threshold of 1, and marked the extreme points of the tailing and under-temperature zone, as shown in Figure 2.

Results indicate that there exists a regulatory contradiction between the under-temperature zone and tailing. Increasing the flow rate can further suppress tailing but enhances heat absorption at the axial edge, resulting in a deeper depression in the under-temperature zone. Conversely, reducing the flow rate alleviates the under-temperature zone yet exacerbates tailing. Notably, as the depression deepens, the influence of flow rate diminishes gradually. Currently, most water-cooled antennas adopt a fixed flow rate across multiple power levels, this makes it common in actual ablation to have significant under-temperature zones when the input power is low. Medical staff typically conduct a secondary ablation on the under-temperature zones during antenna withdrawal to avoid the risk of incomplete ablation. But this will inevitably cause unnecessary damage.

To better capture the features of the contour, this paper employs the minimum zone method [34] to quantify the deviation between the actual ablation contour and an ideal circle. Sampling was conducted along the boundary of the damaged area, with the damage fraction threshold set to 1. Based on the discrete points $P_i(x_i,y_i)$ on the boundary, $i$ = 1, 2, …, n, the center $O\left(x_o,y_o\right)$ and radius R of the reference circle were fitted. The coordinates of the reference circle’s center are determined by minimizing the sum of squared residuals from all discrete points to the fitted circle (7):

$$J=\sum _{i=1}^n\left[\sqrt{(x_i-x_o{)}^2+(y_i-y_o{)}^2}-R\right]$$

(7)

Subsequently, inner and outer circles were constructed with the reference circle’s center as their common center, and all sampling points fall completely between the two circles. Specifically, the inner circle radius $R_{\mathit{inner}}$ is the minimum of all sampling points’ distances to the reference circle center, ensuring it just encloses the closest sampling point. The outer circle radius $R_{\mathit{outer}}$ is the maximum of these distances, ensuring it just encloses the farthest sampling point and covers the entire actual ablation contour. The difference in radius between the inner and outer circles is defined as the ablation roundness error E (8). A smaller error indicates better ablation roundness.

$$\mathit{E}\mathsf{=}{\mathit{R}}_{\mathit{outer}}-{\mathit{R}}_{\mathit{inner}}$$

(8)

2.3. Multi-Objective Optimization

With the core of multi-objective optimization defined as minimizing S₁₁ and roundness error while maximizing the ablation volume, the optimization objectives are identified as the axial dimensions of the feed lines and the flow rates of cooling water. Furthermore, due to structural disparities, the partially water-cooled antenna additionally features an adjustable water-cooling range compared to the other type. The specific design variables to be optimized and their respective ranges are shown in

Table 3. The range of values for the parameters to be optimized.

	Design Variables	Range of Values
Partially water-cooled antenna	Length of dipole tip, l1	1–20 mm
	Width of feed gap, w1	0.1–10 mm
	Length of choke, l2	1–10 mm
	Location of choke, l3	0.1–15 mm
	Flow rate of cooling water, Q1	10–40 mL/min
Fully water-cooled antenna	Length of dipole tip, l4	1–20 mm
	Width of feed gap, w2	0.1–10 mm
	Flow rate of cooling water, Q2	10–40 mL/min

.

The multi-objective gray wolf optimizer (MOGWO) [35], a group optimization algorithm, is employed for the multi-objective optimization of both water-cooled antennas. The MOGWO algorithm is an enhanced and more efficient multi-objective optimization algorithm derived from GWO. It incorporates an external archiving mechanism to store non-dominated solutions and adopts a leader selection strategy based on the roulette wheel method to balance the convergence and diversity of solutions. The algorithm parameters used in MOGWO algorithm are shown in

Table 4. Algorithm parameters used in MOGWO algorithm in the study.

Parameters	Values
Population size	120
Number of generations	50
Size of external archive	120
Repository Member Selection Pressure	2
Leader Selection Pressure	4
Grid Inflation Parameter	0.1
Number of Grids per each Dimension	10

.

Specifically, the joint optimization of COMSOL simulation and optimization algorithm was implemented in COMSOL MULTIPHYSICS 6.3 with MATLAB. The ablation volume is transformed into a minimization problem by taking a negative value, and together with the minimization of S₁₁ and roundness error, it forms a three-dimensional target space. The objective functions of water-cooled antennas are given in Equation (9):

$$\min \{{\mathrm{S}}_{11},-\mathrm{V},\mathrm{E}\}$$

(9)

2.4. Multi-Objective Decision-Making

The Pareto solution sets obtained by multi-objective optimization algorithms consist of a series of non-dominated solutions. By analyzing the Pareto solutions of the two antennas, the partially water-cooled antenna shows a significant trade-off between the S₁₁ and the roundness error E. In contrast, the fully water-cooled antenna has a fixed cooling range, which maintains the stability of the equivalent dielectric environment around the antenna. Therefore, its S₁₁ is mainly determined by inherent structural parameters. Its relatively simple structure allows the solutions to converge to consistent numerical values. No obvious trade-off between objectives exists in this converged set. Thus, the optimal solution for the fully water-cooled antenna only adopts these fixed values, while the multi-objective decision-making method is applied to the Pareto solution set of the partially water-cooled antenna to screen the optimal solution.

The Technique for Order Preference by Similarity to Ideal Solution with Criteria Importance through Intercriteria Correlation (CRITIC-TOPSIS) [36] is adopted to select the optimal solution from the Pareto set. Prior to weight assignment, the data characteristics of the Pareto set were analyzed, where S₁₁ has a large degree of dispersion. Since S₁₁ is expressed on a logarithmic scale to characterize the ratio of reflected power to incident power, small fluctuations are significantly amplified when reflected power is extremely low. This leads to substantial variations in S₁₁ and abnormally increased data dispersion. From a practical engineering standpoint, S₁₁ of −30 dB corresponds to only 0.1% reflected power which fully meets the core energy utilization requirements of microwave ablation. Further reductions in S₁₁ offer virtually no practical enhancement in key ablation indicators such as ablation volume and sphericity. Nevertheless, these negligible reflected power fluctuations greatly increase the data dispersion, this interferes with the CRITIC method’s objective weighting accuracy which relies on data variability. Thus, all S₁₁ values below −30 dB were uniformly set to −30 dB, and the Pareto solutions were reselected to eliminate extreme discrete data’s interference on weight assignment while preserving the antenna’s core ablation performance.

After weight allocation, the weights of S₁₁ and E are 0.4742 and 0.3483, respectively. The ablation volume has a weight of only 0.1774 due to its small data fluctuations. Based on the weight proportions, TOPSIS method is used to quantify the degree of deviation between different schemes and ideal point. The smaller the deviation, the better the scheme performs. The Pareto curve obtained for the parameter design of partially water-cooled antenna is shown in Figure 3 and the optimal parameters generated by the MOGWO algorithm are shown in

Table 5. The optimal design parameters were generated from the MOGWO algorithm.

	Design Variables	Values
Partially water-cooled antenna	Length of dipole tip, l1	7.13 mm
	Width of feed gap, w1	1.08 mm
	Length of choke, l2	5.51 mm
	Location of choke, l3	3.02 mm
	Flow rate of cooling water, Q1	18 mL/min
Fully water-cooled antenna	Length of dipole tip, l4	8.77 mm
	Width of feed gap, w2	0.68 mm
	Flow rate of cooling water, Q2	22 mL/min

.

3. Results

3.1. Analysis of Ablation Performance

Performance analysis of the multi-objective optimized antennas was conducted via finite element models, and comparisons were made against the initial models of single-objective optimization. Among them, the partially water-cooled antenna was optimized only for S₁₁, while the fully water-cooled antenna aimed for roundness error. Figure 4 shows the final thermal damage distribution of water-cooled antennas under single-objective and multi-objective optimization. The damage boundary is the black contour corresponding to a damage fraction threshold of 1, and a yellow dashed line representing a perfect circle is used to fit it. Corresponding key objective parameters are compared in

Table 6. Performance of antennas with single and multi-objective optimization.

		${\mathbf{S}}_{11}$ (dB)	$\mathit{V}$ (cm3)	E (mm)
Partially water-cooled antenna	Single objective	−27.798	25.753	9.54
	Multi-objective	−24.781	26.489	6.80
Fully water-cooled antenna	Single objective	−8.889	22.013	5.79
	Multi-objective	−17.528	26.065	5.48

.

The original single objective optimization for water-cooled antennas only targeted one performance indicator, leading to one-sided performance improvement. Although single objective optimization has shown good performance in the S₁₁ of the partially water-cooled antenna, it has poor values in the parameter of roundness error. The tailing is suppressed by adjusting the flow rate. However, relying solely on the adjustment of flow rate of cooling water, the ablation profile of the antenna cannot be represented as a good circle, the overall shape still tends to be elliptical. For the fully water-cooled antenna, optimization is limited to optimizing roundness error E, resulting in excessive reflection of microwave energy and a smaller ablation volume.

Compared with single objective optimization, although sacrificing some performance in terms of S₁₁, the partially water-cooled antenna after multi-objective optimization reduces the difference in longitudinal and transverse size by 6.43 mm, showing good ablation roundness, while the fully water-cooled antenna increases the ablation diameter by 3.28 mm and significantly improves the S₁₁. This confirms that multi-objective optimization can form a well-balanced solution and achieve an improvement in the overall performance of water-cooled microwave ablation antennas.

For the water-cooled antennas optimized by the multi-objective algorithm, both an-tennas can generate a nearly spherical necrotic volume of 26 cm³ under an input power of 50 W and within a duration of 6 min. Regarding the S₁₁, the partially water-cooled antenna exhibits superior performance due to the integrated choke structure. As presented in Figure 5, the S₁₁ of partially water-cooled antenna reaches −24.781 dB, while the S₁₁ of fully water-cooled antenna is −17.528 dB. The S₁₁ values of the two antennas at 2.45 GHz remain sufficiently low relative to the entire frequency band to meet clinical impedance matching requirements. This result fully demonstrates the effectiveness of multi-objective optimization.

The characteristics of SAR are shown in Figure 6. By observing power deposition 5 mm away from the antenna axis, it was found that both antennas’ heat deposition concentrates around the slot. Compared to fully water-cooled antenna, the partially water-cooled antenna focuses more energy in the slot’s near-field. The SAR peak of partially water-cooled antenna is significantly higher than that of fully water-cooled antenna, with the former reaching a peak of 6504 W/kg, while the latter peaks at only 5778 W/kg. In addition, the suppression of reflected currents by the choke of the partially water-cooled antenna effectively reduces SAR tailing compared to fully water-cooled antenna.

3.2. Distribution of Temperature Field

In microwave ablation therapy, the temperature field is the key link between the deposition of microwave energy and thermal damage to tissue. Its spatial distribution characteristics not only directly determine the effectiveness of the ablation area, but also affect the safety of treatment. The temperature field distribution of the water-cooled antennas after ablation is shown in Figure 7.

According to [37], when the temperature exceeds 130 °C, the intracellular water will undergo severe vaporization, which can lead to carbonization of tissue in severe cases. During the process of removing the ablation antenna, carbonized tissue is prone to adhering to the antenna surface and even causing tissue tearing, significantly increasing the risk of postoperative infection. Based on this temperature standard, the temperature field difference between the two antennas can be analyzed through the correlation between water-cooled structural characteristics.

The cooling range of partially water-cooled antenna only extends from the needle handle to the choke, and there is no active cooling structure at the needle tip. The heat generated by microwave radiation continues to accumulate around the tip and cannot be timely exported through the water-cooled channel. This structural limitation results in the tip being directly exposed to the concentrated area of microwave energy, where the temperature continues to rise above 130 °C. Therefore, there is a significant risk of carbonization and it is easy to cause adhesion between the antenna and the tissue.

In contrast, the fully water-cooled antenna extends the water-cooling range to the dipole tip. The cooling water can directly flow through the needle tip, taking away the heat generated by deposition of microwave energy in real time and suppressing the rise in temperature. The temperature of needle tip is always controlled at a low level and does not directly contact area with temperature above 130 °C. The volume of this area is significantly less than that of the partially water-cooled antenna, effectively reducing the risk of tissue adhesion and tearing caused by carbonization. However, this does not mean that partially water-cooled antenna has lost the value of clinical application. In response to the problem of carbonization and tissue adhesion at the needle tip, adding a PTFE coating on the surface of antenna for physical isolation can effectively avoid tissue adhesion. At the same time, periodic intermittent pulsed microwave output can be used to reduce the continuous deposition of local energy [37], which can effectively minimize the formation of carbonized tissue and alleviate the above clinical hazards.

Moreover, both antennas only lead to a slight indentation feature in the temperature distribution curve at the axial edge of the ablation zone, which is consistent with the ablation results defined by damage fraction, further verifying the correlation between temperature field distribution and tissue thermal damage.

3.3. Verification of Robustness

To verify the robustness of the optimized antennas, the performance of the microwave ablation antennas was analyzed by changing the insertion depth and the relative permittivity of liver tissue. By analyzing the SAR distribution of the two antennas under different insertion depths, the results show that the SAR peak values exhibit no significant fluctuation with the change in insertion depth. Specifically, the maximum variation in the SAR peak for the partially water-cooled antenna is no more than 33 W/kg, while that for the fully water-cooled antenna is within 15 W/kg. Both antennas maintain good power dissipation near the feed gap, as shown in Figure 8. Similarly, considering individual differences, assuming that the dielectric properties of the tissue vary within ±10% of the average characteristics, the performance of S₁₁ at 2.45 GHz under different relative permittivities is validated, as shown in

Table 7. S₁₁ of water-cooled antennas under different relative permittivities of liver tissue.

	Relative Permittivity	S11 (dB)
Partially water-cooled antenna	47.33	−22.644
	43.03	−24.781
	38.73	−26.204
Fully water-cooled antenna	47.33	−18.023
	43.03	−17.528
	38.73	−17.045

. Both antennas also show good stability in terms of S₁₁.

On the basis of maintaining the parameters of the antenna structure determined at the initial power of 50 W, the ablation performance of the antenna under different input powers was further explored by adjusting the flow rate Q. Since the antenna structure is fixed, S₁₁ hardly changes, and the core mode of the energy distribution is determined by the fixed structure. The flow rate Q only serves as an auxiliary adjustment to adapt to power variations, and while ensuring ablation roundness, it has minimal impact on the ablation volume. Therefore, the roundness error E was used as the sole optimization target. Using a finite element model, the ablation effects at power gradients of 70 W and 90 W were analyzed, with the ablation duration kept at 6 min. The optimization results are shown in Figure 9.

Under multi-power conditions, the roundness error of the two antennas was well maintained. But as the ablation volume increases, the defects at some contour boundaries are magnified. Under a heating duration of 6 min, the intersection of the maximum transverse dimension and the longitudinal dimension of the ablation zone is defined as the heating center. Analysis of the ablation morphology evolution of the two antennas reveals significant differences. For the fully water-cooled antenna, due to the lack of suppression of reflected currents, there is an uneven distribution of heat at both ends of the ablation center, resulting in the axial extension length on the side away from the needle tip being significantly greater than that on the other side. At 70 W, the difference in axial extension lengths between the two ends reaches 5.62 mm, which directly causes the ablation shape to deviate from the regular sphere and form a sharp contour at the axial edge. When the power increases to 90 W, this difference further widens to 6.59 mm.

In contrast, the partially water-cooled antenna, with its integrated choke structure, can effectively block reverse microwave propagation and balance energy distribution. Across multiple power levels, the extension difference between the two edges of the center remains extremely small. This minimal extension discrepancy ensures the ablation shape always approximates a regular sphere, fully demonstrating the antenna’s superior adaptability in maintaining morphological regularity under multi-power conditions.

4. Discussion

This study designed and optimized microwave ablation antennas with distinct water-cooling structures, aiming to generate sufficiently large near-spherical ablation zones within a limited time. With the core objectives of minimizing S₁₁ and roundness error while maximizing ablation volume, the study conducted collaborative optimization of the feed line structure and water-cooling design under a multi-objective framework, yielding two optimized antennas with distinct performance characteristics.

Structural differences in water-cooling coverage and electromagnetic regulation lead to distinct performance trade-offs for the two antennas. The partially water-cooled antenna features a choke structure that blocks reflected currents to concentrate SAR distribution while alleviating tailing and constraining the electromagnetic field. Therefore, the antenna achieves superior electromagnetic matching, regular ablation contours and stable performance under multi-power conditions. It is suited for scenarios requiring high ablation regularity, but the choke restricts water-cooling extension to the tip, resulting in local heat accumulation and increased risk of carbonization or adhesion.

In contrast, the fully water-cooled antenna extends cooling range to the tip. This design dissipates tip heat in real time and significantly reduces risk of carbonization, rendering it ideal for carbonization-sensitive applications. Since it lacks a choke to regulate reflected currents, uneven heat distribution emerges under multi-power conditions and leads to sharpened ablation contours as well as shifted centers, which compromise conformality for irregular tumors.

Although the above results reflect the performance trade-offs between these two antenna designs, this study has inherent limitations. The model assumes that the liver is a homogeneous tissue and does not consider the temperature dependence of the material parameters of liver tissue. In clinical settings, the liver exhibits inherent heterogeneity, including differences in parameter characteristics between tumor and normal tissue as well as the heat sink effect induced by blood vessels. These factors can alter the actual paths of electromagnetic energy deposition and heat diffusion, potentially leading to smaller ablation volumes and larger roundness errors in practical applications compared to the simulation results. Furthermore, temperature-dependent dielectric properties may cause a slight overestimation of energy deposition in high-temperature regions. However, this does not alter the relative performance advantages of the optimized antennas nor does it affect the effectiveness of the collaborative optimization method.

5. Conclusions

This paper focuses on the multi-objective optimization of microwave ablation antennas and integrates the synergistic optimization of their water-cooling systems and feed line structures. Additionally, the performance differences between partially and fully water-cooled antennas with a dipole feed line structure are systematically analyzed. Both antennas optimized through multi-objective design can form near-spherical ablation zones with sufficient volume, which meet basic clinical requirements. The partially water-cooled antenna equipped with a choke structure can sustain more stable sphericity of the ablation zone across different power levels but carries a notable risk of needle tip carbonization. In contrast, while the fully water-cooled antenna lacks strong adaptability across power settings, it offers distinct advantage in temperature field performance. This paper provides a practical reference for the structural optimization of microwave ablation antennas and offers insights for antenna selection under different application scenarios. Future work will focus on optimizing antenna performance in complex environments, through physical experiments to verify the effects of liver heterogeneity and the vascular heat sink effect on ablation.