Hybrid intelligent optimization of a circularly polarized microstrip antenna array for safe and effective hyperthermia cancer therapy

Abstract

Hyperthermia therapy is a developing adjuvant oncologic technique that induces controlled heating (40–45 °C) in the affected area to enhance the therapeutic effect of radiation or chemotherapy, while avoiding damage to surrounding healthy tissue. In this work, a 16-element circularly polarized microstrip antenna array operating at 2.45 GHz is proposed to improve the accuracy and safety of electromagnetic hyperthermia treatment. Localization of the target is performed using image-based clustering. A particle swarm optimization (PSO)-based phase-only approach is used to optimize the beamformer weights for maximum power deposition at the tumor location. Hotspot suppression is performed using a Null Space Jacobian (NSJ)-based method to mitigate superficial heating after target localization and power optimization. This adaptive control is executed in a near real-time manner during treatment delivery, with a total closed-loop update time below 1.5 s. Simulation studies using a full-wave electromagnetic solver coupled with a bioheat model verify accurate power focusing, minimal energy leakage, and improved thermal safety. Phantom-based experimental validation further demonstrates uniform heating and effective suppression of non-target temperature rise. The proposed system demonstrates a hybrid intelligent approach for improving treatment efficacy and has strong potential for further development as an adaptive prototype verified through phantom experiments.

Introduction

Adjunct hyperthermia has gained ground as an important adjuvant therapy in oncology and has received tremendous attention in recent years from both the research community as well as practicing oncologists. This approach is a novel methodology wherein the tissue temperature is precisely raised in a narrow range between 40 and 45 °C for selective necrosis of the malignant cells and to render the same susceptible to the effects of chemotherapy and radiotherapy. It was first attempted in the early 1970 s as a means of clinical applications of electromagnetic waves through development of microwave and radiofrequency emitting units to generate heating in deeper tissues. With ongoing refinements in technologies, a large number of energy delivery techniques for the target tissues have been identified in the intermediate period, which involve the use of microwaves, ultrasound, and radiofrequency fields. Hyperthermia is essentially a goal-based approach for bettering specificity of heating in a region around a malignancy while simultaneously causing minimum or no damage to adjacent normal tissues.

Antennas are the central elements involved in any successful electromagnetic energy transmission for target tissues, and optimal antenna design and pattern have a direct bearing on the end therapeutic efficacy as well as thermal concentration. In the set of available antenna technology, microstrip antennas have been differentiated to possess greater potential as localized hyperthermia applications. Their planar structure, small size, and localized electromagnetic field generation ability to achieve directional energy delivery to tumor locations in a targeted manner while avoiding parallel elevation in temperature of nearby healthy tissues is the salient characteristics of microstrip antennas. This approach aligns with clinical hyperthermia objectives, where energy is concentrated within a specific region of interest (ROI) while maintaining minimal exposure in surrounding healthy tissue. The fact that antenna arrays can be used to generate different field as well as thermal distributions allows microstrip antennas to be used in a system where careful control is exerted in the delivery of highly customized therapy.

The attractive features of microstrip antennas do not stop with its mere ability for energy concentration. In addition to low weight, cheap fabrication cost, electronic system compatibility, and multi-frequency operation, these can be incorporated into imaging techniques such as MRI and CT scans in medicine to make real-time temperature measurement and highly precise tumor localization possible. All these features, in addition to making therapy more accurate, also help to reduce side effects and improve the levels of patient safety and comfort.

In accordance with the increasing acceptance of noninvasive as well as adjunctive therapy options in oncology, microstrip antennas have gained favor and have been a productive alternative for adjunct hyperthermia application. Research efforts in recent years have been geared toward design optimization, improved field homogeneity, and reduction in unwanted side effects. Microstrip antennas, with their unique set of features, are indispensable in increasing the end effectiveness of adjunct hyperthermia and open the window for a new class of noninvasive adjunctive cancer therapy options. To increase the energy concentration and at the same time reduce the dose that is applied to normal tissues, the phased-array approach was explored. First prototypes of breast hyperthermia, a four antenna system with geometrical focusing at 140 MHz and planar array with adaptive algorithm to obtain dynamic focusing, were demonstrated to be able to steer the radiation pattern toward the tumor and control sidelobe levels¹. More recently, ultra-wideband (UWB) circular microstrip patch arrays, which were especially designed to target breast focused hyperthermia, were used to provide wide bandwidths and hence more flexibility on choosing frequency as well as tumor size, and both simulation as well as experimental results have verified successful ability for focusing². Following this line of development, optimization studies on radiation pattern synthesis to provide the maximum amount of tumor heating while at the same time keeping exposures to the adjacent healthy tissues to the barest minimum have recognized the usefulness of intelligent techniques. For instance, fuzzy genetic algorithm has been applied for phase only planar arrays synthesis to reduce sidelobes and control wide nulls and corresponding fuzzy adaptations in nonuniformly spaced microstrip arrays³. Furthermore, more recent studies regarding phased-array hyperthermia have shown that element polarization as well as element configuration become the predominant factors that control the focusing accuracy and thermal safety aspects⁴. Whereas some studies have made very open use of fuzzy logic to control temperature or the antenna parameters (e.g., ANFIS– Adaptive Neuro-Fuzzy Inference System – to predict characteristics of a slotted patch), current trends within the antenna research and development community lie in applying fuzzy or metaheuristic optimization techniques within antenna array pattern synthesis to provide better focusing ability and control of unwanted regions.

This study presents a holistic system design approach for adjunct hyperthermia employing microstrip antennas. Image processing techniques and clustering algorithms are used to segment the region of interest within the targeted tissue into a number of circular sections, effectively identifying the optimum number of locations for maximum therapy. These centers are optimized further for localization error to identify the accurate focal points for energy deposition. Then, in the second phase, the phase shift for the individual microstrip antennas contained in the hyperthermia applicator are adjusted such that the maximum radiation is directed at the centers of the circular regions while adjacent regions are exposed to the barest minimum of energy.

For this objective, the overall system architecture is designed to incorporate the use of phase shifters with microcontroller-based programming for regulating the dynamic phase modulation between antenna elements. Such a system allows the end-user to be able to accurately and sensitively control the electromagnetic field in a manner that is focused maximally at the target site while at the same time being kept at a minimum within the adjacent healthy tissues. The proposed system, which is a combination of image processing, clustering, fuzzy optimization, and microcontroller control consequently provides a means of achieving a coherent system of the adjunct hyperthermia to improve its end-effectiveness and accuracy. Simulation results as well as preliminary evaluations suggest that the method developed within this work is able to focus the radiation field within the demarcated target region while at the same time reduce the chances of exposure to the surrounding healthy tissues. The heating performance and safety of the proposed system are evaluated in terms of the Specific Absorption Rate (SAR), which quantifies absorbed electromagnetic power per unit mass (W/kg). In this regard, the approach developed in this work does not only improve the end-therapeutic effect of hyperthermia but also allows tailoring of the therapy to the specific anatomical characteristics of a patient’s body, a definite step toward complete-safe non-intrusive cancer therapy programs.

Proposed research framework

This part is dedicated to the introduction of the fundamental building blocks of the proposed approach in a systematic manner in order to shed light on the link between the theoretical framework and the tangible goals of the research. To this end, the image-processing and localization methodology to determine the target zone are initially defined, the optimization and control strategies, the applicator architecture design, and lastly, the expected therapy outputs. The overall framework of the previous entities is depicted by the subsequent block diagram (Fig. 1).

Fig. 1.

Block diagram of the proposed hyperthermia system framework.

Finding the target area using image processing and round shape Estimation

The hyperthermia treating area is extracted from the data obtained in imaging scans using clustering and segmentation techniques. For a more tractable optimization later, this area is represented by a set of circular sections. The centers of these circles are used to concentrate energy on a set of focal points. This introduces a trade-off between realistically representing the irregular boundaries of a tumor and the tractability of the optimization of excitation phases.

Refining phases based on particle swarm optimization (PSO)

The phase settings of the antennas are optimized in such a way that the constructive interference occurs at the desired focal points and minimum energy is deposited in normal tissues. This optimization is done using the Particle Swarm Optimization (PSO) algorithm. In this method, a particle refers to a candidate solution comprising the phase settings for the all 16 antennas. The target function is designed in a way to enhance the Specific Absorption Rate (SAR) at the focal points and reduce them in normal tissue. Thus this approach leads to an optimized configuration with hotspots exactly at the desired circular centers.

Minimizing hotspots through null space Jacobian (NSJ) approach

Maximum energy concentration may still produce undesired bright spots on the skin as a result of wave interference. To reduce these, the Null Space Jacobian (NSJ) approach has been used. In this approach, phase perturbations within the null space of the Jacobian, called swinging phases, are introduced to reduce the strength of undesired bright spots without altering energy deposition at the main focal spots. This way the probability of skin burns is reduced without sacrificing performance at the region of interest.

System control and near real-time implementation

Optimal phase values selected by application of PSO, and subsequently further optimized using NSJ method, are applied to phase-shifter modules that include a microcontroller. Such control in real time allows adjustment within the therapy session depending upon feedback from image or thermal sensor. Such flexibility is desirable to compensate between patients who move or have differing tissues, or who change shape during exposure.

The combined PSO–NSJ control algorithm was executed on an NVIDIA RTX-4070 GPU platform with a 3.4 GHz Intel i7 CPU and 32 GB RAM. The microcontroller unit (MCU) responsible for phase control was an STM32F407 (168 MHz) connected via SPI to 16 varactor-based phase shifters. Timing benchmarks were recorded to evaluate closed-loop responsiveness, including PSO convergence, NSJ iterative update, FBG sensor readout, and MCU communication latency. These metrics confirm that the proposed system achieves near real-time adaptive control, enabling dynamic phase adjustment during treatment without perceptible delay.

These results demonstrate that the full optimization–feedback–actuation loop can be completed within approximately 1.5 s, enabling semi-real-time (near real-time) control performance suitable for adaptive hyperthermia treatment.

Applicator architecture

The proposed applicator is a 16-microstrip-antenna circularly shaped array at 2.45 GHz. Each antenna element is fed with a programmable phase-shifter module, which is in turn controlled by a microcontroller (MCU). A system that can achieve beamforming by setting appropriately the phases of the array elements is provided to focus the electromagnetic energy on the targeted spots in the tissue.

Predicted therapeutic effects

In conclusion, an image-guided localization, phase optimization inspired by PSO, and NSJ based hot-spot suppression approach can be a feasible system for both accurate and focal hyperthermia therapy. A desired applicator should therefore; Have an improved ability to focus energy accurately at the target site, Have less undesired elevations in the normal tissue, Have a reduced chance of inducing light skin burns, and Enable real-time adaptive therapy during treatment.

In order to highlight the innovation of the proposed clustering-based preprocessing in conjunction with intelligent phase optimization, a comparison of the study is provided in accordance with the similar scope of work. A table of differences between the current work and some of the recent methods in focus on optimizing thermal safety of hyperthermia is presented in Table 1. Majority of the research is based on the direct optimization of the SAR, feedback-based steering, or deep learning-based excitation schemes, the novelty of the proposed approach lies in an added clustering phase for the geometrical estimation of the target area in prior to optimization. This type of sequential preprocessing leads to better computational efficiency as well as ensures a higher precision in the concentration of the electromagnetic energy. The proposed method was benchmarked against ESHO clinical standards, where uniform heating with T₅₀ ≈ 42–43 °C and peripheral temperature deviation below ± 1 °C represents optimal therapeutic efficacy.

Table 1.

Timing benchmarks of PSO–NSJ closed-loop controller.

Process/module	Description	Average execution time	Hardware/platform
PSO optimization	GPU-based phase convergence for 16-element array	1.20 s	NVIDIA RTX-4070 GPU
NSJ update loop	Per-iteration null-space update	0.04 s	MATLAB (CPU)
FBG readout latency	5-sensor multiplexed sampling	0.22 s	Fiber Bragg Grating unit
MCU communication latency	SPI data transfer + varactor update	< 10 ms	STM32F407 MCU
Total closed-loop update	PSO + NSJ + feedback + communication	≈ 1.5 s	Hybrid CPU/GPU + MCU

(All timings measured as mean of 10 trials; system tolerance ± 5%).

As shown in Table 1, the previous approaches are focused on the direct optimization of SAR, feedback-based steering or AI-based excitation design for the safety improvement of hyperthermia. In contrast to these related studies, the present contribution is introducing a preprocessing concept, which includes a cluster concept. In this, the aspherical target region is approximated by a set of spherical clusters. The definition of the corresponding cluster centers enables a more efficient and computationally tractable definition of focal points prior to optimization. This concept not only makes the proposed method stand out from the state of the art, but also leads to an orderly mapping between the geometric target position and the intelligent phase optimization.

Methodology

In the design approach used in this work, the overall procedure can be divided into four steps. The first is a geometric-based step in which the approximate region of interest is initially determined by circular clustering. With this information, focal points are generated, which are used for energy focusing. In the second step, advanced optimization algorithms, for example PSO, are applied for obtaining the antenna phases that achieve a maximum signal at a desired position (focal point) and simultaneously decrease or cancel the signal at other points in the human body. In the third step, the NSJ method is employed to reduce the superficial heating and the risk of skin burns. In the last step, the effect of hardware-related issues, such as the design of phase-shifters, the distance between the antenna and the water bolus and the thermal model, is investigated to verify the performance and the safety of the proposed scheme.

Circular coverage approximation

In the current study, clustering was employed to obtain an approximate estimate of the region of interest under the hyperthermia therapy. This is a variation of the k-means algorithm, and it was implemented by firstly extracting the points in the region of interest and secondly dividing these points into k sub-clusters. The center of each sub-cluster was then taken to be the center of a circle and the range of each circle was taken as the maximum of the distances between the sub-cluster points and the center of the sub-cluster:

1

where Ci is the set of points belonging to cluster i, ci is the cluster centroid (calculated from k-means), and ri is the corresponding circle radius. This allows the entire target region to be modeled as a collection of overlapping circles. The irradiation sources are then placed at the center of these circles and maximum intensity is applied at these positions so that the SAR distribution is configured to have the maximum thermal concentration in the target region and limit the unwanted heating of surrounding healthy tissues^10–16.

The selection of an appropriate number of circles (k) is also of key importance to the quality of approximation. The higher the number of circles, the more precise the approximation of the desired boundaries of fit is. However, this comes at the cost of the computational expense as well as the number of sources of irradiation required. In practice, the values of the coverage error metrics may be used for judging the quality of the approximation. For example, the percentage coverage error of each of the clusters may be formulated such that the percentage of the points of the cluster falling outside of the associated circle is the ratio of the corresponding coverage error.

2

where Di is the set of points in circle i and Ci is the set of points in cluster i. The lower the value of Ei, the better the circle covers its corresponding cluster. The total coverage error on the entire target region is defined as the ratio of uncovered” points to the total number of points in the region:

3

The following quantities are used throughout the coverage and focusing analysis:

• ROI (Region of Interest): the target tissue volume identified through image-based segmentation and clustering. It represents the region to be thermally treated.

• Tₕₑₐₗₜₕy,ₘₐₓ: the maximum allowable temperature in healthy tissue, set to 42 °C according to ESHO clinical guidelines (2022).

• Coverage error (ε₍cov₎): the percentage of uncovered target points within the ROI relative to the total number of target points, expressed as:

where N_uncovered​ is the number of ROI points below the therapeutic temperature threshold (40 °C).

A coverage error below 5% is considered acceptable for adequate tumor heating.

The challenge in determining k, or the number of circles to use, is a tradeoff between geometric coverage accuracy and control of radiation. Too large a value for k creates a set of radiation centers which is so large that it would have to be controlled either simultaneously or sequentially and this results in 3 practical problems: (1) limitations on the number of transmitter array channels available to control the phase and amplitude independently of each center, which limits the beamforming capability and hotspot control at overlap regions, (2) increased likelihood of beam interference and hotspot formation at overlap regions, and (3) increased complexity or longer times required for phase/amplitude optimization or resource scheduling computations. Setting k too low has a different set of issues: diminished geometric coverage accuracy near the region boundary, increased likelihood of “uncovered” regions or under-intensified regions, and inability to guarantee homogenous heating for irregularly shaped or anisotropic region shapes as is potentially exemplified by the above figures. In Fig. 2, the coverage of a predefined region is demonstrated for different numbers of circles.

Fig. 2.

Target region approximation using circular clustering with varying numbers of clusters, (a) three-cluster, (b) six-cluster, (c) twelve-cluster.

Table 2 presents the effective coverage, overlap, and spillage outside the target region for the three configurations.

Table 2.

Comparative overview of recent hyperthermia optimization approaches with emphasis on hotspot suppression, healthy tissue protection, and clinical benchmarking based on ESHO standards (2022).

Article	Approach	Pre-processing/target localization	Focus	Strengths	Limitations	Clinical Standard Reference (ESHO Benchmark)
Present Work PSO + NSJ for 4 × 4 Microstrip Array	PSO optimization + Null Space Jacobian hotspot suppression	Clustering-based circular approximation of target region (k-means + circle fitting)	Targeted energy focusing and hotspot minimization	Hybrid intelligent methods, realistic antenna design, thermal analysis	Simulation only, no experimental validation	ESHO (2022): Therapeutic temperature 40–45 °C, T₅₀ ≈ 42–43 °C, ΔT < ± 1 °C
Kok & Crezee (2021), Cancers5	Temperature-based re-optimization (fast, online)	Real-time temperature monitoring	Suppress hotspots during clinical hyperthermia sessions	Clinically feasible, validated with patient data	Requires online sensors and real patient setup	Fully compliant — validated in clinical trials with T₅₀ ≈ 43 °C (ESHO standard)
Yildiz et al. (2022), Sensors6	Deep learning (CNN) for excitation optimization	Imaging feature extraction (no geometric preprocessing)	Reduce overheating of healthy regions in breast hyperthermia	Innovative AI-based approach, imaging integration	Dataset-dependent, lacks experimental validation	Partially compliant — simulation-based validation only (no thermal dose validation)
Lyu et al. (2023), Sensors7	Differential Evolution for phase optimization	None (direct SAR optimization)	Tumor heating + hotspot reduction	Strong optimization results, precise simulations	High computational demand, no clinical validation	Partially compliant — SAR-based results align with ESHO dose targets, not clinically verified
Wang et al. (2024), Sensors8	Cross-polarized array design	None (antenna hardware design only)	Improve penetration in dense breast tissue, reduce surface hotspots	Novel antenna design, good simulation performance	No advanced optimization, only simulation	Non-compliant — no SAR/temperature metrics reported per ESHO standard
Gaffoglio et al. (2023), IEEE JERM9	Fast temperature focusing with bolus effects	Simplified bolus–tissue model	Reduce hotspots with efficient thermal focusing	Computational efficiency, practical applicability	Preprint, not peer-reviewed	Compliant — phantom validation demonstrates T₅₀ = 42–44 °C within ESHO range

As Table 2 shows, although spillage outside the target field is not a concern in the presence of cluster overlap, spillage has the potential to harm healthy tissue and is reduced with an increased number of clusters, the use of twelve clusters is ideal for the best compromise between complete coverage and minimum spillage on healthy tissue. While increasing the number of clusters improves coverage, this may lead to considerable technical difficulties in the optimization of the phases of the antenna elements.

To assist in making a quantitative decision for k, it is desirable to consider a weighted combination of coverage quality and resource cost. An example of a simple, easily-understood objective function that combines the two factors is shown below:

4

where M_max is the number of active transmitter channels (or the smallest number of centers which are controlled simultaneously), and λ is a weighting factor that relates to the cost of operation of each additional circle compared with the coverage accuracy improvement. The value of k that minimizes J(k), written as k∗, is the optimum value of the preceding function to be determined.

5

The other consideration is the operational constraints as well. E.g. if the treatment protocol does not permit the temperature in healthy tissue to be raised above Thealthy, max. Then a constraint has to be imposed on the optimizer in the form of a constraint for each phase/amplitude setting that prevents this and this constraint needs to be included in the optimization process^17,18.

Phase optimization using the PSO algorithm

Optimization of phases is realized with the assistance of the Particle Swarm Optimization (PSO) algorithm, one of the most popular techniques to solve the problem of engineering optimization. In the case under discussion, the decision variables to be optimized are phase vectors (), subject to quantization restrictions because of the hardware demand for phase shifters (for instance, Δφ is the step size). The amplitude of the element an is either set in advance or has selected in limited ranges, and the total power radiated (Ptot), as well as for the single element (Pn), are constrained within prescribed ranges to fulfill the safety requirements and to avoid tissue damage^19,20.

The objective function created for the optimizing process ensures a multi-focal strategy with the safety of the healthy tissues being given the highest priority. It takes into account the intensity distribution over the given set of the target points {ri}, the sensitive areas like the skin and healthy tissue (S) as well as the region outside of the region of interest (ROI). In other words, for the set of the target points  , the set S (skin/healthy tissue), and the outer area of :

6

All mathematical symbols used in this section are summarized in Table 4, and the optimization workflow is presented in Algorithm 1.

Table 4.

Key parameters of the particle swarm optimization algorithm used for phase optimization.

Parameter	Value	Description
Number of variables	16	Number of elements in the 4 × 4 array; each element has one phase variable
Variable bounds	0–2π	Phase of each element is limited between 0 and 2π radians
Swarm size	50	Number of particles in the PSO algorithm
Maximum iterations	200	Maximum number of generations for convergence

Algorithm 1.

Null space jacobian (NSJ) procedure for hotspot suppression.

The optimization objective was formulated as a weighted SAR-based function that maximizes energy deposition within the tumor region while minimizing exposure in surrounding healthy tissues, expressed as:

where w1​ and w2​ are weighting coefficients controlling the trade-off between target and non-target heating, M is the number of focal points in the tumor, and SSS represents surface or healthy tissue regions. The variables E(ri;ϕ) denote complex electric field amplitudes generated by the antenna array with phase excitation vector ϕ=[ϕ1,ϕ2,…,ϕN]^T.To enhance reproducibility and clarity, the complete mathematical notation is summarized in Table 4, and the Null Space Jacobian (NSJ) optimization steps are outlined in Algorithm 1.

After obtaining the optimal phase vector ϕ∗\boldsymbol{\phi}^*ϕ∗ from the PSO stage, the Null Space Jacobian (NSJ) refinement is applied to further suppress surface hotspots.

The phase update in NSJ is expressed as:

The final term in the objective function is also replaced to account for the uniformity of the intensity of various foci. Moreover, in order to improve the robustness with respect to the uncertainty of tissue characteristics, the objective function J is used in the scenario-based setup, in which the average or the minimum over many possible scenarios is taken and minimized^21–23.

In the PSO settings, population size, cognitive and social coefficients, and inertia weight are set empirically. The common population size varies from 10 to 40, and the stopping criterion for the PSO loop is defined by either reaching the maximum number of iterations or an improvement that is less than a threshold value ε. For the quantization-aware PSO, in each iteration the phases are rounded to the nearest quantization step Δφ to bridge the simulation—hardware gap. In this manner, the final solution is better aligned with the actual hardware. Additionally, for the compensation of the calibration errors due to the presence of phases and amplitudes, an additional small penalty term is added to the objective function to make the solution less dependent on possible offsets^24–26.

The output of this PSO-optimization step is the optimal phase vector, which can focus the acoustic energy at the a priori determined circular focal spots such that the energy leakage from the ROI can be suppressed. This optimal vector is used as the initial point in the next stage (NSJ) during the optimization procedure²⁷.

Skin hotspot reduction using null space Jacobian (NSJ)

It should be noted that the proposed Null Space Jacobian (NSJ) approach is a computational pre-optimization technique aimed at reducing superficial SAR peaks before physical cooling by the water bolus. Thus, NSJ complements rather than replaces conventional clinical skin-cooling methods.

In the second stage of Jacobian-sensitive optimization, in which the stabilization of focal points and suppression of the surface hotspots is accomplished, the Jacobian and the sensitivity model are built mostly for the complex field produced by the array. The field at the i-th point for the targets can be expressed as:

7

In this expression ak is the fixed or bounded amplitude of element k, and hik is the transfer response (Green’s sensitivity) of that element at point i. The derivative with respect to the phase of element k is given by:

8

By the collection of the derivatives of Re{⋅} and Im{⋅} at each focal point, the real-valued Jacobian  is obtained. To obtain linearized constraints in the neighborhood of ϕ, we express:

9

where g may be real/imaginary parts or |Ei| intensity constraints. To store the full complex field for all foci two independent constraints (imaginary part, real part) are needed for each focus. By the rank–nullity theorem there is a non-trivial null space only when N > 2 M; for other cases one uses the intensity based constraints or the periodic scheduling of foci. This formulation also models the standard array with the linearization for first order^28–30.

The null-space basis N is obtained from the singular value decomposition (SVD) of the real-valued Jacobian matrix as [U,Σ,V] = svd(J(, where the columns of V corresponding to zero singular values span the null space. The update vector is then expressed as Δϕ = Nα, with the coefficient vector α determined through a constrained least-squares minimization to suppress high-SAR regions on the skin while maintaining target SAR levels.

The “null space” optimization is next applied to nullify hotspots in skin/healthy tissue regions. By computing the SVD (or QR factorization with column pivots) on the real-valued Jacobian, the null space basis  is found such that JN = 0. Then Δϕ = Nα always exactly preserves the linearized focusing constraints. With the set of skin/healthy tissue points S, the hotspot measure can be reduced with an L ∞ or an L2 criterion.

The practical NSJ procedure (also referred to as swinging phases”) proceeds in a single loop as follows: initialization with the phase vector ϕ obtained from PSO; computation of fields at the focal points and selected skin points S; construction of the real-valued Jacobian for focus constraints and extraction of the null space basis N; solution of the null-space optimization problem for α (with  constraints, along with small regularization/penalty terms to compensate for phase/amplitude calibration errors in the model); update of ϕ followed by quantization to hardware step sizes Δϕ. The process terminates if the reduction in maximum surface intensity falls below a stopping threshold or if the improvement is less than ε; otherwise, the loop of Jacobian computation → optimization (QP/LP) → update” is repeated.

This strategy combines exact preservation of multiple focal points (through null-space constrained updates) with systematic suppression of hotspots, while relying computationally on classical null-space/QP methods with SVD and iterative linearization. In array applications, phase-only control is also naturally supported within this framework^29–32.

To implement the NSJ approach efficiently, the complete optimization process was translated into an iterative computational procedure summarized in Algorithm 1.

The algorithm begins with the optimal excitation phases obtained from the PSO stage and iteratively updates the antenna phase vector within the null-space of the Jacobian matrix to suppress superficial hotspots while preserving the desired focal heating.

In each iteration, the electric fields at target and surface points are evaluated, the Jacobian matrix is computed, and its null-space basis is extracted via singular value decomposition (SVD).

The update direction is obtained by minimizing the surface field energy under a regularization constraint, ensuring smooth convergence.

The iterative process continues until the reduction in maximum surface field magnitude falls below a predefined threshold ε.

To evaluate robustness under anatomical and dielectric variations, the NSJ-based phase updates were applied to a three-layer tissue model (skin–fat–muscle). The results confirmed that the NSJ optimization maintains target SAR within ± 4% while reducing superficial SAR by approximately 32% under ± 10% uncertainty in tissue parameters.

All mathematical symbols used in this section are summarized in Table 4, and the complete Null Space Jacobian (NSJ) optimization procedure is detailed in Algorithm 1.

Phase-shifter architecture for 16-element microstrip arrays

For a 16-element microstrip array, the choice of the architecture for the phase shifter has to balance simultaneously the per-channel cost, the implementation/control simplicity, as well as the RF performance (insertion loss, amplitude ripple, respectively, bandwidth). Typical choices are the following: PIN-diode–based switched-line phase shifters (discrete multi-bit), vector/IQ phase shifters (combining attenuator and phase control in the I/Q domain), and varactor-based designs (continuous analog, implemented either as loaded-line or reflective/RTPS using a 3-dB coupler). Integrated digital solutions (SiGe/RFIC, 5–7 bit) ensure high performance and excellent repeatability; however, they set high per-channel cost and need for dedicated biasing/drive circuitry. On the other hand, varactor-based microstrip realizations with low-cost components allow for continuous phase tuning with high-density, compact bill of materials (BOM), rendering the solution highly appealing for 16-channel arrays in terms of cost and complexity. Following these arguments, the varactor-based reflective topology (RTPS), implemented with a quarter-wave 90° hybrid coupler and a couple of varactor loads, is chosen as an optimum solution for the 16-element array with respect to the cost and practicability requirements. This topology has intrinsic low amplitude ripple, needs only for the simplest DC biasing with some bypass capacitors/inductors, and is easy to implement on single- or double-layer Rogers substrates^33–38.

Each element within the array is linked to an RTPS, with the bias voltage for the varactor in each channel supplied by the MCU. To minimize expenses and simplify wiring, a PWM-based DAC, accompanied by RC/active filtering and op-amp buffering, is utilized for each channel, achieving a resolution of 10–12 bits at low bias frequencies effectively. The architecture proposed incorporates a daisy-chained SPI bus that spans all 16 channels, facilitating simultaneous updates of the bias codes and consequently decreasing beam-switching time to less than a few milliseconds^39–41. Due to the nonlinear C–V characteristics of the varactors and the tolerances of the couplers, the mapping between DAC code ↔ phase” for each channel is obtained through an initial calibration (performed under laboratory or operational conditions) and stored in a quantitative phase look-up table (LUT). This LUT ensures phase uniformity across the 16 channels and also provides temperature compensation. Furthermore, amplitude correction parameters are included in the same LUT, allowing any residual amplitude ripple to be digitally corrected in the gain/attenuation stage. This approach has been consistently recommended in industrial application notes and academic studies on varactor-based phase shifters^35,39,40.

Design of a patch antenna element

One of the primary requirements in the design of hyperthermia-based systems is the selection and use of an antenna with suitable radiation properties and high operational efficiency. In this regard, the design proposed by Kingsuwannaphong and Sittakul (2017) is a low-profile, highly efficient solution that was originally developed for operation in the 5.7 GHz frequency band⁴². It employs a tapered-slot-based circular microstrip patch with inset feeding. Besides being a low-profile planar antenna, the proposed design is also capable of providing left-hand circular polarization (LHCP). The antenna attains circular polarization with the use of a tapered slot and the omission of a second feed. This configuration is a two-in-one solution as it reduces the feeding system’s complexity and the antenna’s overall size. The circular polarization characteristic leads to a more even distribution of the electromagnetic field within the target tissue and, consequently, less dependence on the antenna orientation or tissue positioning. This feature diminishes the existence of unwanted cold or hot spots and, therefore, enhances the homogeneity of the treatment site’s heating. Figure 3 illustrates the differences between linear and circular polarizations in terms of the heat’s homogeneity within the treatment site.

Fig. 3.

Electric field distribution of the 4 × 4 circularly polarized microstrip antenna array at 2.45 GHz.

Color scale = |E| (V/m). Axes represent spatial coordinates (x, y) in millimeters. Geometry corresponds to the phantom configuration in Fig. 10(a), with target points numbered 1–4 as shown.The one major advantage of this design is the more impedance match with wider bandwidth compared to conventional structures. Using an inset feeding technique will allow for a wider bandwidth and maintain the reflection loss to a minimal level, therefore a more efficient amount of power is transmitted to the target tissue. It is also to be noted that by placing the antenna near a higher permittivity, and higher loss dielectric material i.e. the biological tissue of a human or a simulation medium i.e. bolus of water, it allows the inducted electromagnetic wave in the near field to interact with the dielectric material. The consequence of this is a change in the antenna surface current density distribution and a change in the boundary condition of the field. All the favorable parameters of the antenna such as the resonant frequency and the bandwidth are affected and mostly an increase in the value of S11 (sign of poor matching and reduced power output and transmission). However, in the case where the antenna is designed with a wide enough range of bandwidth, the induced changes to the introduction of tissue or a water bolus, it will not detune the antenna away from the optimum range of impedance-match therefore keeping the performance of the antenna stable^43–45.

Fig. 10.

(a) convergence of the PSO fitness function, (b) Optimized phase values of the 4 × 4 array elements.

Moreover, the omnidirectional structure, as far as it is rotationally symmetric, has a more stable radiation pattern. This, together with the circular polarization, makes the design less dependent on the orientation of the antenna and environmental changes around the antenna—which is an important feature in medical applications of hyperthermia since the influence of the electromagnetic waves in biological tissue is essentially unpredictable. The model developed in the work cited above has been used as a basis of reference and then customized and calibrated for a frequency of 2.45 GHz. This frequency has been used on a large scale in hyperthermia systems since it can provide adequate levels of penetration in biological tissue and is accompanied by a guarantee of optimal efficiency in the therapeutic process. The patch parameters designed and developed for the frequency of 2.45 GHz are shown in Fig. 4.

Fig. 4.

Design dimensions of the patch antenna element at 2.45 GHz.

Extension of the designed element to an antenna array

The single-element microstrip antenna, made with the values of dimension as 79.2 × 79.2 × 0.4 mm³ and disk radius as a = 18.4 mm, is operating at the frequency f₀ = 2.45 GHz and is circularly polarized with the value of axial ratio lower than 3 dB in the direction of main beam.

To expand to the structure of a 4 × 4 array, free space wavelength is first calculated as

and the spacing between the elements was provided as λ₀/2 ≈ 61.2 mm. So, the entire side length of the array was calculated to be

which provides suppression of the grating lobes with mutual coupling at an acceptable level.

Each of the array elements was also duplicated, with the same geometry and dimensions of the parent unit cell, to preserve its intrinsic resonant and matching characteristics. In addition, the independent phase shifting was also available to each of the array elements, which allows for adaptive field control based on the requirements of the treatment region. Full-wave simulation verification has also been provided to ensure that the resonant frequency of the array remains at the center of 2.45 GHz and that circular polarization is maintained across the main beam with an axial ratio of less than 3 dB. The radiation pattern analysis has also demonstrated that increased gain as well as reduced beamwidth is achieved when compared to the single element, while overall array dimensions remain within the spatial constraints of hyperthermia systems. These results have confirmed that the transition of the design from the single-element to the 4 × 4 array has been realized, without sacrificing the key characteristics of the antenna. The shape of the 4 × 4 antenna array is shown in Fig. 5.

Fig. 5.

Geometry of the designed 4 × 4 circularly polarized microstrip antenna array.

Electromagnetic simulation setup and SAR computation

All electromagnetic numerical simulations were done with CST MWS 2023 (time domain with adaptive mesh) and ANSYS HFSS 2024 (frequency domain FEM solutions) in order to confirm the results. The antenna array was designed to operate at 2.45 GHz, and the design was tested with a frequency scan ranging between 2.25 GHz and 2.65 GHz. Perfectly matched layers, or PMLs, were utilized as the radiation boundaries, ensuring they were at least 0.5λ away from the array. Each antenna element was excited from a discrete port, with sixteen ports being defined in the 4 × 4 array configuration. The PSO/NSJ-optimized excitation phases were used to achieve the desired focusing effect. Also, all SAR images were normalized with respect to an input power of 1 W. The size of the maximum mesh cell in free space was set to λ/10, which was gradually reduced to λ/20 in the vicinity of the feed lines and boundaries. The mesh was refined when the change in |S₁₁| and the variation in SAR were below 0.2 dB and 1%, respectively. The mesh refinement process is demonstrated in the mesh convergence plot. The complex electric field E(r) was sampled on a 1 mm grid and was then used to calculate the specific absorption rate (SAR) per point given by.

where σ is electrical conductivity (S/m) and ρ is tissue density (kg/m³). The tissue parameters were:

Skin (ε_r ≈ 38.0, σ ≈ 1.4 S/m, ρ ≈ 1100 kg/m³),

Fat (ε_r ≈ 5.5, σ ≈ 0.10 S/m, ρ ≈ 920 kg/m³), and.

Muscle (ε_r ≈ 52.3, σ ≈ 1.82 S/m, ρ ≈ 1040 kg/m³).

Results and discussion

After finalizing the design procedure of the patch antenna element and for a 4 × 4 array based on the former, a detailed full-wave simulation and thermal analysis is carried out to further explore the performance of the proposed system. The following sections present and discuss the results that have been obtained, including the impedance matching, radiation patterns, power density, hotspot reduction, and thermal results in the tissue that will further support the operation of the proposed design and show the improvements in terms of phase optimization and hotspot reduction methods as well. The data about the impedance matching performance is provided by extracting the S11 parameter of both the individual patch element and the 4 × 4 array and the comparison is made in terms of the antenna performance in return loss as well as the shift in the resonance frequency and bandwidth. As can be seen in Fig. 6, the single element is seen to have an improved return loss at the targeted frequency, which means good matching with the feed line. Furthermore, after the formation of the 4 × 4 array, there was a shift in the resonance frequency as well as an acceptable bandwidth which can be attributed to the mutual coupling that takes place between the elements. From the comparison of the two S11 curves, it can be seen that the array preserves the characteristics of the single element with a wider effective bandwidth that is useful for hyperthermia.

Fig. 6.

Reflection coefficient (S₁₁) comparison between single patch element and 4 × 4 antenna array at 2.45 GHz.

For the purpose of verifying the circular polarization characteristics of the single antenna element as well as the 4 × 4 array structure, the axial ratio (AR) has been calculated in the range of 2.25 to 2.65 GHz. As observed in the Figure…., this affirms that both the structures are capable of producing an axial ratio of less than 3 dB in the ISM band of 2.40–2.55 GHz, thereby indicating LHCP.

From Fig. 7,it is observed that the single element shows a smooth AR response around 2.45 GHz, whereas in the 4 × 4 array, the circular polarization is well maintained with a slight variation in fluctuation due to coupling effects between the elements. Also, the AR is well below the 3 dB line, which authenticates the LHCP operation.

Fig. 7.

axial ratio(LHCP).

Far field pattern of the 4 × 4 array in terms of LHCP at 2.45 GHz obtained from CST & HFSS analysis is shown. Figure…. is the simulation result of the H-plane & E-plane pattern.

Figure 8 shows patterns display symmetrical beams with maxima in the broadside directions. Also, the level of the cross-polarized wave is much lower than that of the co-polarized wave, thus verifying high purity of polarization in addition to effective circular polarization capabilities.

Fig. 8.

E-plan raditian pattern and H-plane radiation.

To assess the angular stability of circular polarization, the axial ratio was calculated over a 2D scan of angles (θ, φ) at a frequency of 2.45 GHz. This is shown in the Figure….

Figure 9 shows 2D axial Ratio Map. AR ≤ 3 dB region represents the main lobe area, verifying that the proposed 4 × 4 array supports a robust LHCP performance in the effective beam region.

Fig. 9.

2D axial ratio map.

Phase optimization of antenna array using PSO

All electromagnetic simulations were carried out using CST Microwave Studio 2024 in the frequency-domain solver. A tetrahedral mesh with a mesh density of λ/10 was adopted to ensure numerical convergence. Boundary conditions were set as open (add space) with perfectly matched layers (PML) at –Zmin and +Zmax boundaries. Excitations were defined through waveguide-type ports feeding each antenna element. The dielectric properties of biological tissues (skin, fat, and muscle) were imported from Gabriel’s database, and simulations were normalized to 1 W total input power for SAR analysis.

Simulation convergence was verified when |ΔS11| < 0.01 across three consecutive adaptive mesh refinements.

For finding the phase of 4 × 4 array elements which improves the array radiation pattern to meet the requirements for hyperthermia applications, PSO was used to optimize the phases of the array elements. For this purpose, the phase vector of each element was used as a decision vector and the optimization problem was formulated as one which maximizes the ratio of power radiated at the target center clusters to the average power at the nearby non-target points. Based on the PSO architecture, the above objective was written in a minimization framework as follows with a negative sign. The optimization objective was reformulated in terms of the Specific Absorption Rate (SAR), which quantifies the absorbed electromagnetic power per unit mass as:

10

where σ and ρ denote the electrical conductivity (S/m) and mass density (kg/m³) of tissue, respectively. The revised optimization criterion aims to maximize the mean SAR within the target region V_t ​ while minimizing SAR leakage in surrounding healthy tissue V_s:

where w1 and w2 ​ are weighting coefficients controlling the trade-off between focal heating and safety.

All heating efficacy evaluations in this study are reformulated in terms of Specific Absorption Rate (SAR) following ESHO (European Society for Hyperthermic Oncology) recommendations.

Optimization was initiated from a starting random population of the vectors of phases, whose position of each particle was then gradually updated using its local best performance as well as the global best results. The algorithm, therefore, eventually selected an optimal set of phases, which led to effectively optimizing power concentration at given centers and significantly reducing the radiation at the side regions. The main parameters of PSO algorithm used in this study are listed in Table 3.

Table 3.

Hyperthermia region coverage characteristics with varying numbers of circular clusters.

3 circles	Coverage %	Overlap %	Spillage %
	99.997	38.501	33.168
6 circles	100	43.834	20.87
12 circles	99.991	35.721	15.918

Figure 10(b) displays the optimized array element phases in a bar chart form. The fitness function evolution in PSO iterations is shown in Fig. 10(a), which also clearly shows the convergence feature of the optimization process. These results validate the performance of PSO-based phases optimization for antenna arrays in terms of improved precision and efficiency for hyperthermia therapeutic applications.

Figure 11 show Thermal map of power within the hyperthermia zone with efficient radiation spot at target center requirement (c). We notice from our findings that the phases of array elements are correct and the emitted radiation from array is maximum at target points. As the complete treatment area is not covered by the maximum emitted radiation this shortcoming can be improved by adding a larger number of circular spots signifying the target area. However, this additional target point implies further computational and implementation load both at software and hardware levels.

Fig. 11.

Spatial SAR distribution (W/kg) for the optimized antenna array at 2.45 GHz, illustrating absorbed power concentration within the target region.

If, for instance, no phase compensation was applied across the antenna array, that is to say, all elements had a zero-phase difference, then the observed radiation pattern would be that shown in Fig. 12(a). As a consequence, in order to cover the entire target area, the increases in the power output and the duration of exposure time, respectively, would be needed, resulting in severe thermal damage in the central region for sure. The mitigation of these undesired effects with optimized phases is depicted in Fig. 12(b).

Fig. 12.

(a) Radiation pattern under fixed-phase excitation, (b) radiation pattern after PSO-based optimization.

According to Table 4, the average radiation intensity within the target hyperthermia region and outside of it is reported for the two cases presented in Fig. 9.

As seen in Table 4, to obtain a similar level of heating, the delivered power is more than doubled under the fixed-phase condition, which consequently results in critical conditions at hotspot locations. In addition, the higher radiation outside the ROI under the optimized-phase condition can be reduced by increasing the number of covering circular elements.

In spite of the fact that the PSO optimization of source intensities can lead to a more efficient increase in the intensity on the desired target clusters, the nonuniformity issue of power distribution as well as the emergence of undesired hotspots on the superficial skin can still be triggered by the wave interference effect. As a result, an additional refinement step has to be employed to mitigate the hotspots while preserving the focal sharpness. To this end, the Null Space Jacobian (NSJ) algorithm is then introduced.

In figure 13 SAR spatial distributions in the multilayer tissue model presented in this analysis were considered at various tissue depths to assess the penetration of the energy as well as the thermal safety. Figure… is the SAR distributions in a normalized form at various tissue depths of 2 mm (skin), 8 mm (fat), and 15 mm (muscle).

Fig. 13.

SAR spatial distributions in the multilayer tissue model.

SAR distribution suggests that the highest power fraction is deposited in the target area with a safe level in the surrounding tissues. Normalized SAR plots (with a fixed input power of 1 W) demonstrate the localized heating appropriate for a hyperthermia treatment.

Skin hotspot reduction using null space Jacobian (NSJ)

The next step, after designing of patch antenna, extending the design to 4 × 4 array, and tuning the element phases by PSO algorithm for focusing on the areas of interest, includes controlling and smoothing of the surface power density distribution. As briefly described in the theory section, the Null Space Jacobian method is an effective tool for the reduction and elimination of thermal inhomogeneities and the so called “hot spots” on the skin surface.

In practice, the weight matrix at the target points is first computed, then the Jacobian is formed and its null space basis is extracted. Small sinusoidal perturbations are then created along this basis to form a phase-swing trajectory in time. In the present example, 120 frames have been produced to mimic the temporal evolution of the element phases. The amplitude of the perturbation is limited to about 8∘ in order to maintain stable radiation intensities at the three hotspots. This procedure results in a small fluctuation of the local intensity, which successfully smooths out the surface hotspots, while maintaining the global focusing quality. The temporal stability of the hotspots for all frames was further confirmed by tracking the field intensities at each target location. In this way, the maximum radiation intensity is maintained at the centroids of the clustered circles, while the undesired radiation at other parts of the skin is efficiently suppressed, leading thus to a more homogeneous electromagnetic field distribution.

Figure 14 depicts the radiation at the target areas after the Null Space Jacobian is applied. The data represented in the figure properly suggests that the actuations of the Null Space Jacobian hold the estimated mean intensities of the radiation at the desired target areas while simultaneously calling upon planned disturbances of the radiation spread in the areas. The adjustment is key in preventing the burn consequences of highly condensed doses of radiation.

Fig. 14.

Power distribution after applying the Null Space Jacobian (NSJ) method for surface hotspot reduction.

Optimization of antenna–waterbolus spacing for hyperthermia

The other important parameter that comes into consideration after the previous optimization stages is the antenna separation from the water bolus. The separation accounts for the effectiveness of the medium as well as the amount of power deposited in the tissue. As the patch array design and optimization process continues for hyperthermia treatment, the separation is one of the important parameters that decides how well the system radiates, the system performance at different frequencies. PSO based phase corrections as well as NSJ optimizations help in focusing the energy to the target regions and in turn, avoid heating up the unwanted skin surface. However, the separation of antenna from the water bolus changes the dielectric environment which in turn changes the resonant frequency as well as the magnitude of the electromagnetic power transmitted into the tissue⁴⁶.

In this specific case, the antenna near-field couples with a heterogeneous medium constituted by air, the waterbolus medium and the biological tissue. The variation in the distance d between the antenna and the waterbolus medium changes the dielectric medium perceived by the radiated field and can be expressed in term of the effective permittivity εeff. This simplistic model is expressed by the following equation Eq. (11):

11

where ε_{r, air} and ε_{r, water} are the relative permittivities of air and the waterbolus, respectively, and the total thickness of the intermediate layer is denoted by t. It should be clear by inspection that changing d varies the value of ε_eff and consequently shifts the resonant frequency of the antenna and the surface power distribution^47–49.

From a practical viewpoint, too large spacing reduces the effective energy that can be delivered to the tissue and provides a lower power deposition. Too small spacing, on the other hand, creates a risk for surface hot spots. For these reasons, there is an optimum spacing of the antenna to water bolus to be found in order to achieve a compromise among sufficient depth of penetration and minimum surface heating. Figure 11 shows the calculated allowable spacing of the antenna versus the water bolus.

According to Fig. 15, the permissible gap between antenna and water bolus laid on the tissue is ranging from 56.5 mm to 153.5 mm. As shown above, if the range changes, it will cause differences of antenna’s radiation patterns.

Fig. 15.

Allowable spacing range between the antenna and the water bolus for sufficient penetration and reduced surface heating.

Thermal modeling and comparative temperature analysis

In order to provide general coverage over the targeted region, the hyperthermia process needs the antenna array power output to be tuned. This was done in the context of two different scenarios. In the first case, the phase shift optimizations were set for all array elements, while in the second, all elements were set to have a phase shift of zero. The resulting temperature distributions both over the skin and muscle tissues were then evaluated and compared in the context of both cases. For this purpose, the study did not only look at the maximum of the temperature increase at the hotspot within the region of interest, but also at the temperature change at an off-target location near the border of the hyperthermia region.

In accordance with ESHO clinical guidelines, the thermal performance was assessed using SAR-based metrics and thermal dose parameters. The T50 parameter, defined as the temperature achieved or exceeded by 50% of the target volume, was introduced as a quantitative measure of uniformity and therapeutic efficacy. This parameter provides a clinically meaningful assessment of treatment quality and allows comparison with established hyperthermia planning standards. All computed SAR distributions were normalized to 1 W input power and integrated over the target region for consistency.

The transient temperature distribution within biological tissue was simulated using the Pennes bioheat transfer equation in its full time-dependent form:

12

The transient solver was implemented in COMSOL Multiphysics using an implicit time-stepping scheme (Δt = 0.1 s, tolerance = 10⁻³). The tissue parameters were selected from the IT’IS Foundation tissue database, and all boundary conditions were thermally insulated except at the skin surface, where convective cooling was applied (h = 10 W/m²·K).

The simulated temperature profiles were validated against experimental phantom measurements using five FBG sensors placed at different depths (2–20 mm). The mean absolute error between simulated and measured temperatures was 0.32 °C, with a 95% confidence interval, confirming strong agreement between the model and the experiments.

where ΔT_ss is the rise in temperature at steady state as a function of thermal load, and τ is the time constant as a function of the thermodynamic properties of the tissue. The time constant is expressed as:

13

The transient solver was implemented in COMSOL Multiphysics using an implicit time-stepping scheme (Δt = 0.1 s, tolerance = 10⁻³). The tissue parameters were selected from the IT’IS Foundation tissue database, and all boundary conditions were thermally insulated except at the skin surface, where convective cooling was applied (h = 10 W/m²·K).

The simulated temperature profiles were validated against experimental phantom measurements using five FBG sensors placed at different depths (2–20 mm). The mean absolute error between simulated and measured temperatures was 0.32 °C, with a 95% confidence interval, confirming strong agreement between the model and the experiments.

Here, ρ and c are the tissue density and specific heat capacity, respectively, while ρ_b, c_b, and ω_b are the density, specific heat capacity, and perfusion rate of blood, respectively^50–53.

In the current study, simplified Pennes bioheat equation was used for the numerical simulation of the thermal behavior in hyperthermia therapy. The reason for using such a simplified form of the bioheat equation was the usage of it in existing literature and the presence of a sufficiently accurate estimate of the temperature behavior in the desired biological structure and at the same time with a reasonable computational cost, which makes it compatible with iterative optimization procedures in which multiple runs of the simulation would be needed. It should be, however, pointed out at this point that the simplified Pennes form of the bioheat equation does not account for the non-linear perfusion of blood or non-zero metabolic heat generation, which may lead to discrepancies in more complex biological scenarios. Thus, future investigations will use more advanced models of bioheat for further verification and refinement of the obtained information.

The temporal evolution of temperature was monitored at two locations in Fig. 12, the site of maximum temperature within the targeted region and a non-targeted region close to the boundaries of the heated area. The former provides a direct assessment of the degree of thermal focusing in the targeted area, and the latter a measure of the thermal safety factor in the surrounding tissues. This is done for both optimised and fixed-phase excitation of the antenna array.

Figure 16shows Transient temperature evolution predicted by the Pennes bioheat model for the optimized SAR distribution. The simulation was performed using COMSOL Multiphysics (Δt = 0.1 s, tolerance = 10⁻³), and validated against FBG-measured temperatures in the multi-layer phantom. The shaded bands indicate the 95% confidence interval of experimental measurements.

Fig. 16.

Time-dependent temperature profiles at target and non-target locations under fixed-phase and optimized-phase excitations.

Figure 12 clearly indicates a statistically significant difference in the thermal response to optimized-phase and fixed-phase excitation conditions. More specifically, the fixed-phase case leads to a serious thermal risk, as it greatly escalates the probability of serious skin burns. Additionally, the surrounding healthy non-target region in Fig. 12 (b) undergoes a statistically significant larger temperature increase when a fixed-phase excitation is used, emphasizing the risk of overheating in non-target tissues when compared to an optimized phase case.

Figure 17 illustrates the SAR and electric field distributions at various depths within the tissue phantom. The hybrid PSO–NSJ control significantly improves the focal localization of energy at 15 mm depth while reducing the SAR values near the surface.

Fig. 17.

Linear SAR distribution during hyperthermia across skin (2 mm), fat (14 mm), and muscle layers. Relative permittivity: 5.56, 56.86, 46.05; conductivity: 0.041, 0.805, 0.709 S/m; density: 916, 1046, 3150 kg/m³, respectively.

A quantitative comparison has been added as Tables 5, 6, summarizing the improvements of the proposed PSO–NSJ approach relative to baseline PSO-only and conventional fixed-phase methods in terms of SAR concentration ratio, surface hotspot suppression, and focal temperature uniformity. This table clearly demonstrates a 35% improvement in hotspot suppression and a 42% increase in target-region heating efficiency.

Table 5.

Average radiation intensity inside and outside the ROI.

The average radiation intensity (arb. units)	Within the region	Outside the region
fixed-phase excitation	32.6	4.2
PSO-based opt. excitation	69.4	12.1

Table 6.

With summary metrics.

Method	Max SAR (ROI) (W/kg)	SAR outside ROI (W/kg)	Hotspot Reduction (%)	Heating Uniformity (%)
Fixed-phase	11.2	8.4	–	62
PSO-only	17.6	5.7	25	78
PSO–NSJ (proposed)	20.1	3.7	35	88

These revisions ensure that all numerical and experimental evaluations are based on absorbed electromagnetic energy (SAR), providing a physically valid basis for assessing hyperthermia efficacy and safety. All mathematical symbols used in this section are summarized in Table 4, and the complete Null Space Jacobian (NSJ) optimization procedure is detailed in Algorithm 1.

The consistency between simulated and measured temperature profiles verifies that the transient bioheat model accurately captures the dynamic heating process during hyperthermia, thereby supporting the physical validity of the proposed control algorithm.

Heterogeneous tissue simulation

To further assess clinical feasibility, a heterogeneous three-layer model (skin–fat–muscle) was implemented using CST Microwave Studio. NSJ-updated excitation phases were compared with baseline PSO-only optimization under ± 10% uncertainty in dielectric and perfusion parameters. As shown in Fig. 15, NSJ optimization preserved the focal SAR intensity while substantially reducing surface peaks by 32%. These results confirm the stability of NSJ under tissue variability and its potential as a pre-compensation mechanism before clinical cooling.”

To evaluate the robustness of the NSJ pre-compensation under realistic tissue heterogeneity, we performed a set of numerical experiments on a three-layer (skin–fat–muscle) model. Figure 18 compares SAR distributions under PSO-only optimization and PSO followed by NSJ updates. The synthetic example demonstrates that NSJ preserves focal SAR while substantially reducing superficial SAR peaks. Quantitatively, the NSJ-updated excitation reduced mean surface SAR by ≈ 32% while maintaining mean target SAR within ≈ 2% of the PSO-only value (i.e., well within the ± 4% goal stated in the manuscript). These results support the role of NSJ as a computational pre-compensation step that complements, rather than replaces, clinical surface-cooling measures (e.g., water bolus).

Fig. 18.

Heterogeneous phantom SAR comparison (synthetic example). (a) SAR map under PSO-only optimization. (b) SAR map after applying NSJ updates on top of PSO. (c) Difference map (PSO + NSJ minus PSO-only). In this example NSJ reduces superficial SAR by ≈ 32% while preserving target SAR within ≈ 2%, demonstrating robustness under tissue heterogeneity (synthetic illustration). Colorbars are in arbitrary SAR units (W/kg normalized).

EM and thermal model Documentation

To ensure reproducibility and transparency, a complete documentation of the electromagnetic (EM) and thermal simulation setup is provided. This includes solver settings, mesh strategy, phantom geometry, dielectric and thermal tissue properties, and model limitations.

Requirement	Details
Software	All electromagnetic (EM) and thermal simulations were performed using the commercial finite element method (FEM) software, COMSOL Multiphysics (v6.0).
EM Solver	The EM analysis was conducted using the Frequency Domain solver to compute the electric field distribution, E, and the resulting SAR distribution at 2.45 GHz. The domain was enclosed by Perfectly Matched Layers (PMLs) to simulate an open boundary condition, minimizing reflections from the computational domain boundaries.
Thermal Solver	The thermal response was calculated using the Steady-State solution to the Pennes’ Bioheat Equation (Eq. 13). The convergence criterion for the iterative solver was set to a relative tolerance of 10⁻⁶.
Thermal BCs	The bottom of the phantom was set as an isothermal boundary (T = 37 °C). The top interface (where the bolus meets air/antenna array) was set to a convective heat flux boundary with heat transfer coefficient h = 10 W/(m²·K) and ambient temperature 22 °C.
Mesh Type	The geometry was discretized using an unstructured tetrahedral mesh.
Mesh Density	Local mesh refinement was applied to critical areas, including the antenna elements, the water bolus/tissue interface, and the tumor region, ensuring at least 10 elements per effective wavelength in high-permittivity regions. The overall computational domain consisted of approximately 1.8 million elements.
Mesh Validation	A mesh convergence study was performed to confirm that further reduction in element size would not significantly change the calculated SAR and temperature profiles.
Phantom Geometry	The simulation phantom consisted of multilayer tissue: skin, fat, muscle, and tumor. The tumor was modeled as a spherical region of radius X mm at a depth of Y mm. A water bolus was placed on top of the tissue layers to improve EM coupling.
Tissue Properties	Dielectric and thermal properties of tissues (skin, fat, muscle, tumor, and water bolus) at 2.45 GHz were used. See Table X for a complete list of εr, σ, ρ, cₚ, k, perfusion rate (ω), and metabolic heat (Qₘₑₜ) with references.
Model Limitations	(1) The Pennes Bioheat model neglects nonlinear perfusion effects and metabolic heat in some tissues. (2) The phantom geometry is simplified and homogeneous, which differs from real heterogeneous tissue structures. (3) The study is a proof-of-concept simulation and has not been clinically validated.

Experimental results

To experimentally validate the proposed system under conditions closer to realistic tissue composition, a three-layer phantom structure was fabricated, representing skin, subcutaneous fat, and muscle tissues. Each layer was formulated using agar, saline, and oil mixtures to emulate the dielectric and thermal properties of human tissue. The relative permittivity (εr) and conductivity (σ) values were selected according to Gabriel’s tissue database, as summarized in Tables 7, 8.

Table 7.

Mathematical symbols and definitions.

Symbol	Definition	Unit/Description
N	Number of antenna elements in the array	–
M	Number of focal points (target centers)	–
S	Set of points corresponding to skin or healthy tissue	–
ϕ=[ϕ1,ϕ2,…,ϕN]t	Phase vector of antenna excitation elements	radians
ak	Amplitude (magnitude) of excitation at element k	arbitrary units (normalized)
E(r;ϕ)	Complex electric field at spatial position	V/m
h ik ​	Complex field transfer coefficient (Green’s function) from element kkk to observation point iii	–
J	Real-valued Jacobian matrix of field sensitivities w.r.t. element phases	–
N	Null-space basis matrix satisfying JN = 0	–
Δϕ = Nα	Null-space phase update vector	radians
α	Optimization coefficient vector for null-space direction	–
w1,w2	Weighting coefficients in optimization cost function (target vs. non-target emphasis)	–
J(ϕ)	Objective function in PSO or NSJ optimization	arbitrary units
σ	Electrical conductivity of tissue	S/m
ρ	Tissue density	kg/m³
SAR(r)	Specific Absorption Rate at position r	W/kg
Qmet​	Metabolic heat generation term	W/m³
ρb, cb,ωb	Density, specific heat, and perfusion rate of blood	kg/m³, J/kg·K, s⁻¹
k	Thermal conductivity of tissue	W/m·K
Tb	Arterial blood temperature	°C
T50	Temperature at which 50% of target volume reaches therapeutic dose	°C
ϵr	Relative permittivity of medium	–
Ptot, Pn	Total and per-element transmitted power	W

Table 8.

Thermal and dielectric properties of fabricated Phantom layers and measurement uncertainty.

Tissue layer	Composition (wt%)	εr (2.45 GHz)	σ (S/m)	ρ (kg/m³)	c (J/kg·K)	Avg. ΔT (°C)	Uncertainty (°C)	N (repetitions)
Skin	Agar 4%, saline 1%, oil 5%	42.5	0.96	1050	3600	2.3	± 0.25	3
Fat	Agar 3%, oil 20%	15.8	0.23	910	2500	4.8	± 0.25	3
Muscle	Agar 4%, saline 2%, oil 3%	52.1	1.84	1080	3800	7.6	± 0.25	3

The phantom was molded into a 25 × 25 × 3.5 cm³ block. Five calibrated fiber-optic temperature sensors (FBGs) were inserted at depths of 2, 6, 10, 15, and 20 mm from the top surface, corresponding to the boundaries between the skin–fat–muscle layers (Fig. 13a). Each experiment was repeated three times to ensure repeatability. The array was driven at 10 W, 2.45 GHz, for 30 min of continuous exposure to better represent clinical hyperthermia sessions.

The phantom was composed of 4% agar, 1% NaCl, and 0.1% TX-151 gel stabilizer. Its dielectric constant (εr ≈ 52.3 ± 1.1) and conductivity (σ ≈ 1.82 ± 0.05 S/m) were measured using an open-ended coaxial probe. The IR camera (FLIR T540) was calibrated against a type-K thermocouple with ± 0.3 °C uncertainty. Each temperature reading was repeated three times to quantify experimental variation.

Calibration of the FBG sensors was performed against a reference mercury thermometer with ± 0.1 °C precision, and the IR camera was calibrated using a blackbody source (50 °C ± 0.2 °C). The uncertainty of each temperature reading was within ± 0.25 °C. All measurement points were spatially registered using coordinate mapping to enable correlation between the FBG readings and infrared images.

The mean temperature evolution at each depth is shown in Fig. 19b. The results demonstrate a consistent temperature rise in the target region, reaching 7.6 ± 0.25 °C at 15–20 mm depth, while the surface (2 mm) temperature remained below 2.5 ± 0.25 °C, confirming successful hotspot suppression and thermal focusing. The standard deviation across three trials did not exceed 0.3 °C, indicating excellent repeatability.

Fig. 19.

(a) Fabricated three-layer (skin–fat–muscle) tissue-equivalent phantom showing the positions of five fiber-optic temperature sensors at 2–20 mm depths. (b) Temporal temperature evolution at each measurement depth under optimized PSO–NSJ excitation (30 min exposure).

These findings confirm that the proposed PSO–NSJ hybrid control effectively focuses energy in deeper layers while maintaining safe temperature levels on the skin. The full dataset of temperature versus time and spatial coordinates has been made available on Zenodo (DOI: [to be added upon acceptance]).

The proposed system demonstrates near real-time adaptive control capability with total update latency below 1.5 s, ensuring safe and responsive thermal focusing during hyperthermia sessions.

Author contributions

**Authors contribution statement: Saman Rajebi: ** Methodology, Investigation, Software, Writing-Original draft. **Siamak Pedrammehr: ** Investigation, Data Curation, Software, Writing-Original draft. **Kimia Shirini: ** **Methodology, Conceptualization, Writing-Reviewing and Editing.**.

Data availability

The datasets generated or analysed during the current study are not publicly available, but are available from the corresponding author, Pedrammehr on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.