Simulation-informed kernel-based regression models for predicting temperature distribution in microwave hyperthermia

Abstract

This work analyzes heat transfer in microwave heating process using hyperthermic approach. Predictive performance of artificial intelligence (AI) models coupled with data of mechanistic model were evaluated. The resulting predictive models provide a surrogate representation of the simulated temperature field and may support temperature assessment in microwave hyperthermia applications. The model is developed in two stages, where in the first stage, electromagnetic heating is modeled via solution of bioheat equations to compute temperature field in the tissue. Then, the results are used for building several artificial intelligence-based models. The predictive stage of the study was carried out using three nonlinear kernel-driven learning approaches, namely Support Vector Machine (SVM), Kernel Ridge Regression (KRR), and Relevance Vector Machine (RVM), with the spatial coordinates serving as predictors of the temperature field. Model-specific hyperparameters were determined through a Cuckoo Search-based optimization procedure. Among the evaluated algorithms, SVM delivered the strongest overall predictive performance, yielding R² values of 0.9766 for the training data and 0.9768 for the test data. Its mean R² obtained from five-fold cross-validation was likewise 0.9768, with a very small standard deviation of 0.00063, while it also produced the minimum MSE and MAE values. In contrast, RVM exhibited the smallest worst-case deviation, with a maximum absolute error of 9.3946 K. Although KRR and RVM achieved satisfactory test-set R² values of 0.9200 and 0.9560, respectively, the collective evaluation of predictive accuracy, error magnitude, and validation stability identified SVM as the most suitable approach for dependable and precise estimation of the spatial temperature distribution. These results offer significant insights into the utilization of kernel-based regression models for predicting spatial temperatures, with implications for diverse domains including cancer therapy.

Introduction

There are different methods for cancer treatment among which chemotherapy is the most widely used approach. Despite the effectiveness of chemotherapy in cancer treatment, there are major side effects associated with this method for patients who suffer from cancer. So, other efficient methods can be explored to be used for cancer treatment alone or in combination with chemotherapy^1,2. One of the useful methods for cancer therapy is hyperthermic oncology by which cancer tumors are treated by applying localized heat. The method has been shown to be efficient in which the temperature could be increased up to 113 °F to kill cancer cells. Different techniques can be applied for heating among which microwave coagulation is usually used with the aid of microwave antenna to create localized heat in the cancer tissue³.

In oncological hyperthermia, precise regulation of the delivered thermal load is essential to minimize unintended heating of healthy tissue. Computational modeling offers a practical means of estimating the temperature field established around the microwave applicator and within the adjacent biological medium. This is typically achieved by formulating the governing electromagnetic and bioheat-transfer equations and solving them numerically to obtain the spatial temperature response of the tissue^4,5. Such physics-based simulations can also serve as the foundation for data-driven surrogate modeling. In the framework adopted here, the numerical model is first employed to generate the temperature distribution over the computational domain, after which the resulting simulation outputs are used as training data for machine-learning regression algorithms. Accordingly, the physics-based and learning-based stages operate in sequence, rather than as a simultaneously trained or tightly integrated physics-informed architecture. Similar simulation-to-learning strategies have been successfully applied across a range of thermal, physical, and chemical systems, where they have demonstrated strong predictive fitting capability^6–8.

Recent studies have quantitatively demonstrated the predictive capability of machine-learning models coupled with numerical or thermal simulations. For example, Wang et al⁸. developed a CFD–machine-learning framework for temperature prediction in a rotary kiln and reported that genetic-algorithm optimization of the random-forest model reduced the RMSE from 25.0879 to 17.5330 and the MAE from 6.5600 to 4.7661. Related hybrid regression studies have also emphasized quantitative evaluation using multiple performance indices. Kachalla and Ghiaus reported an R² of 0.9766, RMSE of 0.0223, and MAE of 0.0074 for an hourly SARIMA–ANN forecasting model, while another hybrid energy-forecasting study reported an R² of 0.988 and RMSE of 0.020 for the best-performing seasonal configuration^9,10. Although the predicted quantities and their numerical scales differ among these applications, these studies demonstrate the importance of evaluating simulation-informed and hybrid machine-learning models using both goodness-of-fit and error-based performance metrics.

In addition, recent studies have further demonstrated the value of systematic model comparison, hyperparameter optimization, and cross-validation across different machine-learning applications. Uluocak and Yavuz developed hybrid CNN–LSTM and CNN–GRU architectures for multi-class fault diagnosis in manufacturing systems and combined these models with Bayesian hyperparameter optimization and ten-fold cross-validation, reporting an overall classification accuracy of 0.98 and an average F1-score of 0.96. In another application, Uluocak comparatively evaluated LSTM, GRU, CNN–GRU, and CNN–LSTM models for atmospheric methane forecasting and showed that CNN–LSTM achieved the best predictive performance, with an MAE of 0.6567 and a Nash–Sutcliffe efficiency of 0.933^11,12. Although these studies address different physical applications, they reinforce the importance of comparative model assessment, optimization, and multi-metric validation when selecting data-driven predictive models.

Machine-learning methodologies have become widely adopted across scientific and engineering research because of their ability to uncover complex data structures and construct predictive relationships from large datasets, particularly in situations where direct application of mechanistic models is computationally demanding or impractical. Within this broader class of data-driven methods, kernel-based regression techniques are especially effective for representing intricate dependencies between explanatory variables and model outputs. Through kernel functions, the original predictors are implicitly projected into richer feature spaces, enabling the representation of relationships that may not be readily described in the original input domain^13,14. In the present study, three kernel-oriented regression algorithms, namely Kernel Ridge Regression (KRR), Support Vector Machine (SVM), and Relevance Vector Machine (RVM), are comparatively examined for reconstructing the spatial temperature field from the corresponding coordinate information. Closely related work has demonstrated the feasibility of combining electromagnetic–bioheat simulation with machine learning for microwave hyperthermia. Yang et al. employed simulation-derived spatial coordinates to predict tissue temperature using tree-based regression models, including DT, RF, and LGBM, and reported high predictive accuracy after hyperparameter optimization. This previous work supports the use of simulation-informed learning as a practical strategy for approximating spatial temperature fields in microwave-based thermal therapy¹⁵.

The three methods differ substantially in their mathematical formulation and learning strategy. KRR extends conventional ridge regression by incorporating kernel transformations, thereby allowing nonlinear input–output behavior to be represented while retaining explicit regularization. SVM provides a flexible framework applicable to both classification and regression; in its regression form, the model seeks a function that maintains prediction deviations within an acceptable tolerance while controlling model complexity. RVM, in contrast, adopts a probabilistic Bayesian formulation in which sparsity is promoted automatically, allowing the final predictive relationship to be determined using a limited number of relevance vectors. This sparse representation is particularly attractive when compact models are desired for complex, high-dimensional learning problems.

To obtain suitable configurations for each regression algorithm, their tunable parameters were determined using the Cuckoo Search optimization technique, a population-based metaheuristic inspired by the reproductive behavior of cuckoo species and designed for efficient exploration of complex search spaces. The principal objective of the present investigation is to establish a consistent comparison among KRR, SVM, and RVM for spatial temperature prediction. Model performance is therefore assessed using several complementary indicators, including the coefficient of determination R², mean squared error, mean absolute error, and maximum absolute error. Considering these measures collectively enables identification of the regression approach that provides the most accurate and robust approximation of the temperature distribution within the investigated domain.

The contributions of this paper lie in elucidating the strengths and weaknesses of various kernel-based regression models for spatial temperature prediction tasks. The machine learning models are integrated with numerical simulation of heat transfer in the cancer tissue to predict temperature distribution. The contribution of this study is the systematic evaluation of three kernel-based regression approaches, KRR, SVM, and RVM, as simulation-informed surrogate models for predicting spatial temperature distributions generated by coupled bioheat and electromagnetic simulations. Their hyperparameters are optimized using Cuckoo Search, and their predictive performance is comparatively assessed using goodness-of-fit, error-based, and cross-validation metrics. The study therefore focuses on evaluating the suitability and relative performance of these regression approaches for rapid approximation of the simulated temperature field in microwave hyperthermia [6–8]. The findings of this research can inform the selection of appropriate ML techniques for similar applications in fields such as environmental monitoring, energy systems optimization, and materials science, thereby advancing our understanding and capabilities in modeling complex physical systems.

Data set and pre-processing

The dataset contains 64,870 records with the following variables:

• r (m): The radial coordinate spans the interval from 0.000 to 0.030 m. Its average value is 0.0058 m, with a standard deviation of 0.0073 m. The first, second, and third quartiles are located at 0.0015, 0.0024, and 0.0058 m, respectively.

• z (m): Measurement in meters, ranging from 0.000 to 0.080. The mean value for this variable is of 0.0406 and its standard deviation is 0.0222. Quartiles are 0.0206 (25th), 0.0403 (50th), and 0.0599 (75th).

• T (K): The simulated temperature values extend from 310.236 to 354.790 K, with an overall mean of 321.886 K and a standard deviation of 12.864 K. The corresponding 25th, 50th, and 75th percentiles are 311.489, 315.515, and 330.354 K, respectively.

The inputs are r(m) and z(m), and the output is T(K). Indeed, temperature as a function of coordinates are obtained in cylindrical coordinate. The data for machine learning are derived from numerical simulation of heat transfer for microwave heating. Axial symmetry was assumed for the geometry which consisted of antenna and the tissue for treatment. Both electromagnetic equations and bioheat equations must be solved to find the temperature distribution. More details regarding the equations, numerical method and geometry can be found elsewhere¹⁶. The complete dataset of 64,870 records was extracted from a single steady-state coupled electromagnetic–bioheat COMSOL simulation. Each record corresponds to a distinct spatial location (r, z) within the computational domain and its associated simulated temperature T. Therefore, the records represent spatial samples from one simulated temperature field rather than outputs from multiple independent simulation runs. Accordingly, the machine-learning models developed here should be interpreted as surrogate models of this specific simulated temperature field. A closely related study employed a simulation-generated dataset of more than 64,000 spatial observations with the same input–output structure, using the radial and axial coordinates (r, z) to predict tissue temperature T in microwave hyperthermia¹⁵. This similarity provides useful methodological context for the present dataset, although the current analysis is conducted using the specific COMSOL-derived temperature field described above.

The bioheat equation is expressed as¹⁷:

1

Here, k represents the thermal conductivity of the biological tissue, while the associated blood-related parameters correspond to blood density, blood specific heat capacity, and the perfusion rate, respectively. The thermal contributions arising from metabolic activity and externally imposed heating are represented by the metabolic and external heat-source terms , respectively.

The externally supplied thermal energy originates from electromagnetic absorption within the tissue and can be formulated as follows¹⁷:

2

The computational domain was constructed as a two-dimensional axisymmetric r-z representation of liver tissue containing a slender coaxial-slot microwave applicator. The modeled tissue region extended to approximately 0.030 m in the radial direction and 0.080 m along the axial coordinate. The applicator geometry comprised a coaxial structure with a 0.29 mm diameter inner conductor and an outer conductor having inner and outer diameters of 0.94 and 1.19 mm, respectively. The assembly was enclosed within a PTFE catheter of 1.79 mm diameter. A circumferential slot with a width of 1 mm was positioned 5 mm from the short-circuited antenna tip. During operation, the antenna was excited at 2.45 GHz with an applied microwave power of 10 W. These dimensional and operating specifications are consistent with the microwave coagulation-therapy configuration adopted in the numerical formulation^16,17.

For the electromagnetic component of the model, the relative dielectric constants assigned to the coaxial dielectric, PTFE catheter, and liver tissue were 2.03, 2.60, and 43.03, respectively. The electrical conductivity of the liver was set to 1.69 S/m, while the relative magnetic permeability was assumed to be unity. Metallic portions of the antenna were modeled as perfect electric conductors, imposing a zero tangential electric-field condition at their surfaces. Electromagnetic energy was introduced through a coaxial-port excitation corresponding to 10 W input power. To minimize nonphysical reflections from the truncated computational boundaries, low-reflecting or scattering-type conditions were prescribed at the outer electromagnetic boundaries. Axisymmetric behavior was enforced along the r = 0 line.

For the bioheat calculation, the liver thermal conductivity was 0.56 W/(m·K), the blood density was 1000 kg/m³, the blood specific heat capacity was 3639 J/(kg·K), the blood perfusion rate was 3.6 × 10^− 3, and the blood temperature was 37 °C. Metabolic heat generation was neglected (Q_met =0), whereas the electromagnetic resistive-loss density was used as the external heat source Q_ext. Thermal insulation was imposed on the exterior boundaries of the truncated tissue domain.

A free triangular finite-element mesh was employed. To resolve the steep electromagnetic-field gradients around the coaxial antenna, the maximum element size was limited to 3 mm over the computational domain and further reduced to 0.15 mm in the coaxial dielectric region. Fourth-order Lagrange elements were used for the electromagnetic field formulation. This locally refined discretization reproduces the benchmark SAR distribution reported for the underlying microwave coagulation configuration¹⁷, providing a check on the adequacy of the spatial discretization. The stationary bioheat problem and the frequency-domain electromagnetic problem were solved using FEM, with UMFPACK employed as the direct solver. Because the thermal calculation was steady-state, no transient time-stepping scheme or initial-temperature condition was required. Moreover, as UMFPACK is a direct linear solver for the resulting discretized system, an iterative linear-solver convergence tolerance was not involved.

The temperature field obtained from the numerical simulation was subsequently used as the data source for developing the machine-learning surrogates. In this formulation, the radial and axial coordinates, r and z, serve as the explanatory variables, while the corresponding simulated temperature represents the prediction target. The resulting learning models are therefore intended to emulate the COMSOL-derived spatial temperature response rather than substitute the electromagnetic and bioheat mechanisms embedded in the original physics-based model. Their role is to learn an efficient mapping from spatial position to temperature once the numerical data have been generated.Figures 1 and 2 summarizes the linear relationships among r, z, and T through a correlation matrix. The radial and axial coordinates are essentially uncorrelated, with a coefficient of − 0.0098, indicating negligible linear dependence between the two spatial directions. Temperature exhibits a more pronounced inverse association with the axial coordinate, with a correlation coefficient of − 0.64, whereas its relationship with the radial coordinate is weaker but remains negative, with a coefficient of − 0.39. The heatmap presents these pairwise associations graphically using a color-coded scale, thereby providing an immediate visual indication of both the direction and relative strength of the correlations among the variables.

Fig. 1.

Meshed geometry of the system for numerical simulations.

Fig. 2.

Correlation heatmap of the dataset.

Figure 3 provides a distributional view of the three variables r, z, and T through violin plots. The radial coordinate r is strongly concentrated near the lower end of its range and exhibits a pronounced right-skewed pattern, with a relatively long tail extending to its maximum value of 0.030 m. In contrast, the axial coordinate z is distributed more evenly across its interval, although modest density variations are visible around the quartile regions. The temperature variable T also shows an asymmetric distribution, with most observations clustered around the central portion of the data and a tail extending toward higher temperature values. Collectively, these plots provide a compact representation of the location, dispersion, and shape of each variable distribution and help characterize the structure of the dataset prior to machine-learning analysis.

Fig. 3.

Violin Plots of Variables.

The descriptive statistics and visual distributions were also considered when preparing the data for machine-learning analysis. The comparatively broad and skewed distributions of r and T, together with the different numerical ranges of the input coordinates, motivated the use of feature scaling before model training. The correlation analysis further indicates that the two spatial coordinates provide complementary information: z shows a stronger association with temperature than r, while the near-zero correlation between r and z indicates that the two input variables do not provide strongly redundant linear information. These statistical and graphical assessments therefore support the use of both spatial coordinates as model inputs.

The dataset contained no missing values; therefore, no imputation procedure was required. Potential anomalous observations were examined using the Local Outlier Factor (LOF) method, which evaluates each observation according to its local density relative to neighboring samples¹⁸. Before model training, the input variables were normalized using min-max scaling to place the spatial coordinates on a comparable numerical scale and to prevent differences in feature magnitude from disproportionately affecting the kernel-based models.

Following normalization, the dataset (64,870 records) was randomly shuffled and partitioned into training and testing subsets using an 80:20 split ratio, yielding 51,896 records for training and 12,974 records for testing. A fixed random seed (random_state = 42) was used for the shuffling and splitting operation to ensure reproducibility. Since the r and z coordinates originate from a structured numerical simulation grid rather than spatially clustered sampling locations, random (non-spatial) shuffling was adopted for the split; this avoids introducing spatial autocorrelation bias while preserving the overall distribution of r, z, and T across the domain. The 5-fold cross-validation reported in the Results section was performed on the training subset only (also with random_state = 42), and the held-out 20% test set was used exclusively for final, independent performance evaluation, with no overlap between the data used for hyperparameter tuning/cross-validation and the data used for testing.

The present surrogate formulation intentionally uses only the spatial coordinates r and z as input variables because the objective is to approximate the temperature field for a fixed numerical configuration. Additional inputs may be useful when extending the framework to varying treatment or patient-specific conditions. Such variables could include tissue thermal and dielectric properties, blood perfusion, antenna position, microwave power, and other operating parameters. Although neighboring mesh temperatures could potentially improve local prediction accuracy, their use would require careful treatment because they are derived from the same temperature field as the prediction target and may introduce information leakage. Accordingly, expansion of the input feature set is considered an important direction for future work.

Kernel-based regression models

Kernel-based regression models estimate relationships between variables using kernel functions, which map input features into a higher-dimensional space to capture non-linear patterns in data. This work applies three such models: Relevance Vector Machine (RVM), Support Vector Machine (SVM), and Kernel Ridge Regression (KRR). RVM yields a sparse, probabilistic solution; SVM maximizes the margin between predicted values; and KRR incorporates explicit regularization. Each is described in the following subsections¹⁹.

Kernel Ridge Regression (KRR)

Kernel Ridge Regression (KRR) is a resilient ML algorithm that efficiently solves regression and classification problems using ridge regression and the kernel trick. This method works well when the input variables and target variables have complex, non-linear relationships. KRR is a kernel method, which is widely used to capture complex data patterns.

KRR employs a kernel function to map the feature space into a higher-dimensional space. This transformation enables KRR to conduct linear regression in the mapped space, leading to a linear amalgamation of transformed features that optimally captures the data.

In its most basic form, the objective function for this model can be expressed as follows²⁰:

3

Here, n signifies the total number of instances, denotes the main value for the i-th target, signifies the dual variables (Lagrange multipliers) optimized by Kernel Ridge Regression (KRR). Additionally, the feature vectors for the i-th and j-th data points are indicated by and , denotes the kernel function that calculates the similarity between and in the transformed space, and the regularization parameter serves an important function in balancing mitigating overfitting and the fitting of the training subset.

The flexibility and potency of KRR are derived from the selection of the kernel function. Some frequently employed kernel functions are²¹:

1. Linear Kernel:

4

2. Polynomial Kernel:

5

3. Radial Basis Function (RBF) Kernel:

6

4. Sigmoid Kernel:

7

The regularization parameter (λ) is crucial for the objective function of KRR. It balances accurate training data fitting with overfitting risk. While lowering λ helps the model match training data, it also increases the risk of overfitting. Increased λ values result in stricter regularization, reducing overfitting risk but potentially causing underfitting.

After the KRR model has been trained, for unseen data points predictions can be calculated using the following Eq. 2⁰:

8

Where, stands for the predicted target value for the new (unseen) data point with feature vector x. By choosing kernel functions, KRR can model complex non-linear relationships between input parameters and target output, making it predictive. It achieves a harmonious equilibrium between the adaptability of the model and the application of regularization techniques, rendering it a highly valuable instrument in ML for a broad range of tasks, including regression and classification.

Support vector machine (SVM)

Support Vector Machine (SVM) represents a widely used family of supervised learning techniques that can be formulated for both classification and regression tasks. In regression applications, the objective is to determine a prediction function that captures the relationship between the input variables and the continuous response while maintaining an appropriate balance between model complexity and fitting accuracy. Rather than attempting to reproduce every training observation exactly, support vector regression seeks a function for which deviations from the observed responses remain within a prescribed tolerance whenever possible²². For a linear regression formulation, the predictive function can be written as²³:

9

where x denotes the input vector, w represents the model-weight vector, and b is the intercept or bias term. Model training consists of identifying suitable values of w and b while simultaneously controlling the magnitude of prediction errors. This requirement leads to the corresponding constrained optimization problem²⁴:

10

subject to the following conditions²⁵:

11

12

13

In these expressions, denotes the observed response associated with the i-th training sample, whereas ε specifies the width of the error-insensitive region around the regression function. The variables and account for deviations that exceed this tolerance on either side of the fitted function. The regularization coefficient C governs the trade-off between maintaining a relatively simple regression function and penalizing prediction errors outside the ε-insensitive region. Increasing C places greater emphasis on minimizing such deviations, whereas smaller values allow stronger regularization and greater tolerance for training errors.

The slack variables ξ_i, ξ_i* allow flexibility around the ε-margin, while C balances margin width against training error. Because the decision function depends only on the support vectors, SVM regression is robust to outliers, and the kernel trick extends it to non-linear decision boundaries.

Relevance vector machine (RVM)

Relevance Vector Machines (RVMs) are a prominent subset of ML algorithms extensively applied for various predictive tasks. In regression, RVM aims to find a predictive function that optimizes the model’s performance while achieving a sparse representation, thereby enhancing generalization and computational efficiency^26,27. Unlike SVM, which constructs a hyperplane to maximize the margin, RVM focuses on determining a probabilistic model that selectively uses a subset of data points for prediction.

The Relevance Vector Machine (RVM) formulates regression within a probabilistic Bayesian setting, where the response variable is represented through a weighted combination of kernel-transformed input features. The corresponding predictive relationship can be expressed as²⁶:

14

In this formulation, x denotes the input vector, while the kernel-based feature representation is used to transform the original predictors into a suitable feature space. The vector w contains the model coefficients, and b represents the intercept term. Rather than estimating these parameters solely by minimizing a deterministic fitting error, RVM infers them probabilistically by combining the likelihood of the observed data with sparsity-inducing prior distributions. This mechanism encourages many model weights to approach zero, leaving only a limited number of influential basis functions in the final regression model.

Model estimation is consequently based on the posterior distribution of the unknown weights conditioned on the available observations. Determining the most probable parameter configuration under this Bayesian formulation can be represented through an equivalent minimization problem, written as follows²⁸:

15

subject to²⁹:

16

where is the observed target for the i-th data point, are hyperparameters controlling the sparsity, is the noise variance, and denotes the Gaussian distribution. The hyperparameters are adjusted to promote sparsity in the weight vector w, effectively reducing the number of relevance vectors.

By prioritizing sparsity, RVM’s probabilistic formulation resists overfitting and produces a decision function based on a small subset of relevance vectors, analogous to the support vectors in SVM, while retaining the ability to model non-linear relationships via kernel functions.

Model optimization

Model hyperparameters were optimized using the Cuckoo Search (CS) algorithm (Fig. 4), a nature-inspired metaheuristic based on the brood-parasitism behavior of cuckoo birds, effective for locating optima in multi-dimensional search spaces³⁰. In hyperparameter tuning, CS maintains a population of candidate hyperparameter sets and iteratively updates them to identify the combination that yields the best model performance.

Fig. 4.

Flowchart for CS.

For each regression model, hyperparameter optimization was performed independently using the Cuckoo Search algorithm. Candidate solutions represented combinations of the hyperparameters associated with the corresponding model, and their quality was evaluated according to the predictive performance obtained during the optimization process. The objective was to minimize the mean squared error (MSE), while the corresponding R² value was monitored as an additional measure of predictive quality. The use of error-based criteria such as MSE, together with assessment of convergence behaviour, provides a standard quantitative basis for evaluating optimization and predictive algorithms³¹.

For KRR, the regularization parameter (λ) was searched within the range of 1.0 × 10⁻⁴ to 1.0 × 10², while the kernel type was selected from linear, polynomial, radial basis function (RBF), and sigmoid kernels. For nonlinear kernels, the kernel coefficient (γ) was explored between 1.0 × 10⁻⁴ and 1.0 × 10¹, and the polynomial degree, where applicable, was searched between 2 and 5.

For SVM regression, the penalty parameter (C) was optimized over the range of 1.0 × 10⁻¹ to 1.0 × 10³. The ε-insensitive loss parameter (ε) was searched between 1.0 × 10⁻³ and 1.0, while the kernel function was selected from linear, polynomial, RBF, and sigmoid kernels. The kernel coefficient (γ) for the nonlinear kernels was explored within the range of 1.0 × 10⁻⁴ to 1.0 × 10¹, and the polynomial degree was varied between 2 and 5 when the polynomial kernel was selected.

For RVM, the kernel type was optimized using the same candidate kernel functions, including linear, polynomial, RBF, and sigmoid kernels. The kernel coefficient (γ) for nonlinear kernels was explored between 1.0 × 10⁻⁴ and 1.0 × 10¹, while the polynomial degree was searched between 2 and 5. The sparsity-controlling hyperparameters (αⱼ, Eq. 15) were not included in the CS search space; consistent with standard RVM formulations, these are estimated automatically and internally through the model’s own type-II maximum-likelihood (evidence) update during training, rather than externally tuned. Only the kernel-related hyperparameters (kernel type, γ, polynomial degree) were optimized via Cuckoo Search, as listed in Table 1.

Table 1.

Hyperparameter search ranges and Cuckoo Search configuration adopted for optimization of the investigated regression models.

Model	Hyperparameter	Search strategy/range
KRR	Kernel	Linear, Polynomial, RBF, Sigmoid
	λ	1.0 × 10⁻⁴–1.0 × 10²
	γ	1.0 × 10⁻⁴–1.0 × 10¹
	Polynomial degree	2–5
SVM	Kernel	Linear, Polynomial, RBF, Sigmoid
	C	1.0 × 10⁻¹–1.0 × 10³
	ε	1.0 × 10⁻³–1.0
	γ	1.0 × 10⁻⁴–1.0 × 10¹
	Polynomial degree	2–5
RVM	Kernel	Linear, Polynomial, RBF, Sigmoid
	γ	1.0 × 10⁻⁴–1.0 × 10¹
	Polynomial degree	2–5
CS	Number of nests	25
	Maximum iterations	50
	Discovery probability, pₐ	0.25
Objective function	All models	Minimize MSE

The Cuckoo Search optimization was conducted using a population of 25 nests and a maximum of 50 iterations. The discovery probability was set to pₐ = 0.25. At each iteration, new candidate hyperparameter combinations were generated through Lévy flights and compared with the existing candidate solutions. A candidate solution was retained when it produced a lower objective function value. The optimization was terminated once the maximum of 50 iterations was reached, without an additional early-stopping/convergence tolerance; across the three models, the best objective value was observed to stabilize (change in MSE < 1 × 10⁻³ over the last 10 iterations) well before this limit, confirming convergence within the allotted budget. The optimization procedure was conducted separately for KRR, SVM, and RVM, and the best hyperparameter combination for each model was subsequently used for the final model training and evaluation.

The Cuckoo Search algorithm integrates a Levy flight mechanism, enabling it to efficiently manage the trade-off between exploitation and exploration. To accelerate convergence, the algorithm randomly explores hyperparameter space and prioritizes promising regions. The inherent randomness of this algorithm and diversity render it well-suited for hyperparameter optimization tasks aimed at discovering the optimal combination to enhance model performance³².

Results and discussion

The spatial temperature field was first obtained by numerically solving the coupled bioheat problem over the computational domain. As illustrated in Fig. 5, the highest temperatures occur in the region immediately surrounding the antenna, where electromagnetic energy deposition is strongest. This localized heating pattern reflects the intended behavior of the microwave applicator, with thermal intensity concentrated near the treatment region rather than distributed uniformly throughout the surrounding tissue. The resulting temperature field is inherently dependent on the operating conditions of the heating system, and variations in parameters such as the electromagnetic field strength can modify both the magnitude and spatial extent of the generated heat.

Fig. 5.

Temperature distribution in the domain (K) calculated using numerical simulation.

The simulation-derived temperature data were then used to develop and assess the machine-learning surrogate models. Kernel Ridge Regression (KRR), Support Vector Machine (SVM), and Relevance Vector Machine (RVM) were trained to estimate temperature T from the radial and axial coordinates (r, z). The tunable parameters of each model were optimized using the Cuckoo Search (CS) algorithm. Their predictive capabilities were subsequently evaluated against the numerical-simulation results using a common set of performance criteria, allowing the relative accuracy, robustness, and limitations of the three regression approaches to be compared systematically.

Model performance was evaluated using the coefficient of determination R², mean squared error (MSE), mean absolute error (MAE), and maximum absolute error. These metrics were calculated using their standard formulations¹⁸:

17

18

19

20

where and denote the numerical-simulation temperature and the corresponding model prediction, respectively, is the mean observed temperature, and n is the number of evaluated samples. Higher R and lower MSE, MAE, and maximum error indicate better predictive performance.

From a treatment perspective, the predicted temperature field should be interpreted in relation to clinically relevant thermal thresholds. Conventional hyperthermia aims to elevate tumor temperature to approximately 45 °C (about 318 K) while limiting excessive heating of surrounding tissue³. The numerical temperature field used in the present study spans approximately 310–355 K, indicating that some locations, particularly near the antenna, reach temperatures substantially above the conventional hyperthermia range. These high-temperature regions should therefore be interpreted as potential localized overheating or ablative zones rather than as the desired temperature throughout the treated tissue. In this context, the surrogate model can assist in rapidly identifying the spatial distribution of temperature and regions approaching or exceeding the intended thermal range. However, the present model should not be interpreted as a clinical control system, since patient-specific safety margins and prospective clinical validation were not investigated in this study. The localized temperature maximum observed near the microwave antenna is consistent with the thermal pattern reported by Yang et al., who similarly observed the highest tissue temperatures adjacent to the applicator followed by a progressive decrease with increasing distance¹⁵. This agreement supports the physical consistency of the spatial temperature pattern obtained in the present simulation.

The cross-validation standard deviations provide an additional indication of the stability of the evaluated models across folds. In particular, the relatively small standard deviations observed for KRR, SVM, and RVM indicate limited variation in predictive performance across the five validation subsets. Among the evaluated models, SVM exhibited the smallest cross-validation standard deviation (0.0006289), supporting the consistency of its predictive performance. These cross-validation statistics were interpreted together with the independent test-set MSE, MAE, and maximum absolute error to avoid relying on a single performance measure.

The model selection was based on a comprehensive comparison of predictive accuracy and generalization performance. The primary criterion was the coefficient of determination (R²) obtained on the independent test dataset, with a higher R² indicating better predictive capability. This criterion was complemented by the error-based metrics, including mean squared error (MSE), mean absolute error (MAE), and maximum absolute error, for which lower values were preferred. To assess the stability and generalization consistency of the models, the mean and standard deviation of the R² values obtained from 5-fold cross-validation were also considered. Accordingly, the final model was selected by jointly considering high test R², low prediction errors, high cross-validation mean R², and low cross-validation variability rather than relying on a single performance indicator.

The results of ML modeling and their comparisons are listed in Tables 2 and 3 for the three models employed in this study. Based on the predefined model selection criteria, the models were compared according to their independent test R², prediction error metrics, and 5-fold cross-validation performance. The results demonstrate the superior overall performance of SVM. It exhibits the highest R² scores for both the training and testing datasets, as well as the highest cross-validation mean. Additionally, SVM achieves the lowest MSE and MAE among all models, indicating superior overall predictive accuracy. Although RVM exhibits a lower maximum error than SVM, the selection was based on the collective assessment of all predefined criteria rather than on maximum error alone. Considering its higher test R², lower overall prediction errors, and strong and stable cross-validation performance, SVM was selected as the preferred model for predicting temperature based on spatial coordinates.

Table 2.

R² Scores of Trainings, Test and Validation.

Model	Train R2 Score	Test R2 Score	5-Fold CV Mean	5-Fold CV Standard Deviation
KRR	0.9218	0.9200	0.9320	0.0012683
SVM	0.9766	0.9768	0.9768	0.0006289
RVM	0.9574	0.9560	0.9612	0.0008780

Table 3.

Error Rates (Test data).

Model	MSE (K²)	MAE (K)	Max Error (K)
KRR	1.1490E + 01	1.7946E + 00	1.3253E + 01
SVM	3.6647E + 00	8.9252E-01	1.2233E + 01
RVM	6.6324E + 00	1.3257E + 00	9.3946E + 00

A direct wall-clock comparison between the COMSOL simulation and the machine-learning models was not recorded during the original computational runs. Because computational time depends strongly on the hardware configuration, solver settings, software implementation, and model-specific parameters, retrospective numerical estimates would not provide a reliable or reproducible benchmark. In the present framework, the COMSOL finite-element simulation represents the physics-based computational stage used to generate the temperature field, whereas training of the surrogate models is performed offline using the resulting simulation data. Once trained, the surrogate models evaluate the temperature response directly from the spatial coordinates without repeatedly solving the coupled bioheat and electromagnetic equations. Therefore, their principal computational advantage is expected during repeated prediction, although no quantitative speed-up factor is claimed in the present study. A controlled runtime comparison performed on identical hardware is identified as an important direction for future work.

While KRR and RVM also exhibit competitive performance, SVM provides the best overall performance based on its higher R², lower MSE and MAE, and stable cross-validation results. RVM, however, achieves the lowest maximum absolute error among the evaluated models.

Although the SVM achieved a relatively low overall MAE of 0.893 K, its maximum absolute error reached 12.23 K. This distinction is clinically important because a low average error does not exclude larger local deviations in individual regions. Consequently, the present surrogate should be considered a temperature-prediction tool requiring further validation before application to treatment control or safety-critical decision making.

The present comparison is restricted to the three investigated kernel-based surrogate models. Therefore, the reported superiority of SVM should be interpreted only relative to KRR and RVM within the present dataset and simulation setting. Future work should extend the benchmarking to simpler statistical baselines and other machine-learning families, such as linear regression, tree-based regressors, and neural-network models.

Based on the comprehensive evaluation of R² scores and error rates, the Support Vector Machine emerges as the optimal model for predicting temperatures in the given system. Distribution of residuals for this model are shown in Fig. 6. This model achieved the highest R² scores (training: 0.9766, testing: 0.9768), signifying a robust fit to the data, along with the lowest error rates, comprising MSE (3.6647E + 00) and MAE (8.9252E-01), showcasing accurate predictions. RVM achieved the smallest maximum error (9.3946 K), whereas SVM showed the best overall performance based on its higher R² and lower MSE and MAE. The residual distribution shown in Fig. 6 provides an additional diagnostic view of the selected SVM model beyond the tabulated error metrics. Together with the reported MSE, MAE, and maximum absolute error, this residual analysis helps characterize the spread of prediction errors on the independent test set. More detailed visualization tools, such as parity plots and spatial error maps, may further enhance performance assessment and are therefore identified as useful extensions for future work.

Fig. 6.

Distribution of errors in test dataset for SVM model.

Feature importance analysis for the dataset is indicated in Figs. 7, 8 and 9, where it is observed that the axial coordinate (z) has a greater influence on temperature than the radial coordinate (r). While this ranking is consistent with the correlation structure reported in Fig. 2 (|r|: z–T = 0.64 vs. r–T = 0.39), it is corroborated here through the trained SVM surrogate itself rather than the raw data alone, since Fig. 7 reflects the sensitivity of the fitted model’s predictions rather than a simple correlation coefficient. To further probe model interpretability beyond this global ranking, Figs. 10 and 11 present one-dimensional partial dependence plots, showing the predicted temperature as a function of r and z respectively, with the complementary coordinate held at representative fixed values. These partial dependence curves reveal the functional form of each variable’s effect — including local nonlinearities and the location of maximum sensitivity along each axis — that a single importance score cannot capture. In addition, the local gradients of predicted temperature with respect to r and z (∂T̂/∂r, ∂T̂/∂z), estimated numerically from the trained SVM model across the domain, confirm that the largest sensitivity to z occurs near the antenna tip region, consistent with the location of peak predicted temperature shown in Figs. 8 and 9. Moreover, 3D and 2D illustrations of temperature distribution are shown in Figs. 8, 9, 10 and 11. These results have been constructed using ML models and show great agreement with the numerical simulations. In both 3D and 2D illustrations, the region with the maximum T can be observed accurately. The spatial pattern reconstructed by the surrogate model is also consistent with previous microwave-hyperthermia modelling, where peak temperatures were concentrated near the antenna and decreased toward more distant tissue regions¹⁵.

Fig. 7.

Feature importance of input coordinates.

Fig. 8.

Three-dimensional visualization of the predicted temperature field as a function of the radial and axial coordinates, r and z.

Fig. 9.

Contour map illustrating the spatial distribution of the predicted temperature across the computational domain.

Fig. 10.

Isolated influence of the radial coordinate, r, on the predicted temperature response.

Fig. 11.

Isolated influence of the axial coordinate, z, on the predicted temperature response.

Conclusion

A simulation-informed surrogate-modelling framework was investigated for predicting spatial temperature distributions in microwave hyperthermia. Temperature data generated from the coupled bioheat and electromagnetic numerical simulation were used to train and compare three kernel-based regression models, namely KRR, SVM, and RVM, with their hyperparameters optimized using the Cuckoo Search algorithm. Among the evaluated models, SVM provided the best overall predictive performance, achieving training and test R² values of 0.9766 and 0.9768, respectively, together with a 5-fold cross-validation mean R² of 0.9768 and a standard deviation of 0.00063. On the independent test data, SVM also produced the lowest MSE of 3.6647 K² and MAE of 0.8925 K. RVM, however, yielded the lowest maximum absolute error of 9.3946 K, compared with 12.233 K for SVM, indicating that model selection should consider both average and worst-case prediction errors.

These results demonstrate that kernel-based surrogate models can reproduce the principal spatial variation of the COMSOL-generated temperature field with high overall predictive accuracy within the investigated simulation domain. Such models may provide a useful computational representation for assessing temperature distributions and identifying regions approaching or exceeding intended thermal levels without repeatedly solving the underlying numerical equations. Nevertheless, the models are trained exclusively on simulation-generated data and should not be interpreted as clinically validated treatment-control systems. Their application to patient-specific treatment planning would require further validation using independent experimental or clinical data, together with controlled assessment of computational speed and safety-related prediction errors.

The generalizability of the present surrogate model is limited to the numerical configuration used for its development. The current model was trained using a fixed liver-tissue geometry, material properties, antenna configuration, operating frequency, and input power, with spatial coordinates r and z as the model inputs. Therefore, direct application of the trained model to other tissue types, skin layers, or patient-specific anatomies cannot be assumed. Extension to such applications would require retraining and validation using data that account for differences in tissue geometry, dielectric and thermal properties, blood perfusion, antenna placement, and treatment conditions. Nevertheless, the proposed simulation-informed surrogate-modelling framework itself can be adapted to other tissues or patient-specific configurations when appropriate training data are available.

Author contributions

M.A.A.: writing manuscript, visualization of results, investigation, validation, method development. N.H.A.: writing manuscript, visualization of results, investigation, validation, supervision. K.H.A.: writing manuscript, software, investigation, validation, method development. All authors reviewed the manuscript.

Funding

Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, under grant no. (IPP: 134-135-2026 ).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics approval

was not required for this study because it was based exclusively on numerical simulation-generated data and did not involve human participants, human biological materials, patient data, or animal experiments.