Advanced computational approaches for bioheat transfer in minimally invasive cancer thermal therapies

1 Introduction

Beyond its classical thermodynamic formulation, contemporary bioheat modeling increasingly resides within the broader domain of computational biomedical engineering, integrating image-based anatomical reconstruction, multiphysics energy transport, and advanced numerical simulation frameworks. Modern treatment planning workflows routinely incorporate MRI- and CT-derived geometries, electromagnetic or acoustic field solvers, temperature-dependent material models, and coupled thermal–fluid representations to capture vascular heat transport and device–tissue interaction (Berjano, 2006; Ji and Brace, 2011; Chiang et al., 2013). These developments position bioheat modeling not merely as a standalone thermal equation, but as a multiphysics computational transport problem embedded within image-guided oncology and high-performance numerical simulation frameworks (Quarteroni et al., 2017; Kok et al., 2020). Foundational developments in bioheat and thermal therapy modeling, including classical perfusion-based formulations and early focused ultrasound frameworks, have established the basis for modern computational approaches (Pennes, 1948; Hynynen and McDannold, 2004). More recently, hybrid physics-informed and data-driven approaches have further expanded this landscape, linking classical partial differential equation–based modeling with machine learning–assisted surrogate prediction and uncertainty quantification (Suzuki et al., 2023; Fu et al., 2024).

Thermal ablation techniques have become integral to the management of primary and metastatic tumors in the liver, lung, brain, kidney, and prostate. By elevating tissue temperatures beyond cytotoxic thresholds, these therapies induce coagulative necrosis while minimizing systemic toxicity (Jia and Feng, 2025; Kok et al., 2020). Therapeutic success is highly sensitive to spatiotemporal temperature distributions governed by tissue composition, vascular perfusion, applicator geometry, and energy delivery protocols (Ahmed et al., 2014; Kok et al., 2020).

The maturation of image-guided tumor ablation as a clinical discipline has been accompanied by parallel efforts to standardize terminology, procedural reporting, and outcome assessment. A landmark contribution was the consensus document issued by the International Working Group on Image-Guided Tumor Ablation, first published in 2003 and subsequently updated through international collaboration with major professional societies (Jia and Feng, 2025; Ahmed et al., 2014). This framework established unified definitions for ablation modalities, applicators, procedural parameters, imaging assessment, complications, and clinical end points, enabling reproducible reporting and meaningful comparison across technologies and institutions (Jia and Feng, 2025; Ahmed et al., 2014). Importantly, the consensus explicitly acknowledged the growing complexity of tumor ablation, including the influence of organ-specific physiology, vascular perfusion, and evolving energy-based modalities (Jia and Feng, 2025; Ahmed et al., 2014; Kok et al., 2020).

These standardized definitions and reporting requirements provide a necessary foundation for computational bioheat modeling, as they define the physical, biological, and procedural parameters that must be represented within numerical simulations. Consequently, the consensus framework underpins not only clinical trial design and regulatory evaluation, but also the development, validation, and clinical translation of computational bioheat models aimed at improving treatment planning, intra-procedural guidance, and outcome prediction (Ahmed et al., 2014; Kok et al., 2020).

Computational modeling of bioheat transfer has therefore emerged as a cornerstone of treatment planning, device design, and intra-procedural guidance. While early models provided essential conceptual insight, contemporary clinical demands increasingly require computational frameworks capable of representing tissue heterogeneity, nonlinear thermophysical behavior, and patient-specific anatomy (Pennes, 1948; Rossmann and Haemmerich, 2014). This review critically evaluates advances that extend beyond classical bioheat formulations and assesses their implications for clinical translation.

In parallel with developments in biomedical thermal modeling, advances in computational transport physics—including studies of nonlinear heat transfer, magnetohydrodynamic transport, and bioconvective nanofluid dynamics—have expanded the theoretical understanding of coupled thermal–fluid processes in complex media. For example, numerical analyses of entropy generation and heat transport irreversibility in magnetized porous channels have demonstrated how magnetic forces, viscous dissipation, and porous media interactions influence thermal transport behavior in heterogeneous systems (Marzougui et al., 2021). Similarly, modeling studies of laser–tissue thermal interactions incorporating local thermal non-equilibrium frameworks have highlighted how beam profiles, tissue porosity, and exposure parameters shape spatial temperature distributions and thermal damage patterns in biological materials (Hassan et al., 2025). Research on bioconvective nanofluid transport further illustrates the role of microorganisms, electromagnetic forces, and slip effects in stabilizing nanoparticle suspensions and modulating thermal transport in multiphase fluids (Dharmaiah et al., 2023). Related investigations of Casson and Williamson nanofluid flows under magnetic fields have demonstrated how coupled thermal, chemical, and convective transport mechanisms govern heat and mass transfer behavior in complex non-Newtonian fluids (Iranian et al., 2026; Sankari et al., 2025). Although these studies originate primarily from broader thermal-fluid engineering contexts, they provide conceptual insights into nonlinear heat transfer, multiphysics coupling, and porous-media transport phenomena that are also relevant to biological heat transfer systems. Incorporating perspectives from these broader transport frameworks can therefore enrich bioheat modeling strategies for heterogeneous and perfusion-dominated tissues encountered in tumor thermal therapies.

2 Novelty and distinct contribution of this review

Several authoritative reviews have examined thermal oncology technologies (Kok et al., 2020), temperature-dependent tissue properties (Rossmann and Haemmerich, 2014), and modality-specific computational modeling approaches (Berjano, 2006; Ji and Brace, 2011). More recently, systematic analyses have highlighted heterogeneity in computational modeling strategies for liver ablation (van Erp et al., 2023) and advances in extended bioheat formulations (Yuan et al., 2025). However, most existing reviews primarily emphasize model formulation, physical mechanisms, or modality comparison in isolation, rather than providing an integrative translational framework linking physiological realism, computational scalability, validation maturity, and clinical deployment.

The present review differs in five key respects. First, we introduce a three-axis evaluative taxonomy that positions bioheat modeling strategies along the dimensions of physiological fidelity, computational tractability, and clinical validation maturity. Physiological fidelity encompasses vascular resolution, multiphysics coupling, and temperature-dependent thermophysical properties (Rossmann and Haemmerich, 2014; Tucci et al., 2022). Computational tractability addresses solver stiffness, runtime scalability, reduced-order modeling feasibility, and suitability for real-time deployment (Mattingly et al., 1998; Quarteroni et al., 2017). Clinical validation maturity incorporates evidence from phantom studies, animal models, MRI thermometry comparison, and regulatory credibility frameworks (Odéen et al., 2016a; van Erp et al., 2023; Morrison et al., 2019). This multidimensional framework enables structured comparison across classical Pennes formulations (Pennes, 1948), non-Fourier and dual-phase-lag extensions (Tzou, 1995; Sharma and Kumar, 2020), vascular-resolved and porous-media approaches (Hassanpour and Saboonchi, 2016), multiphysics electromagnetic–thermal coupling (Ji and Brace, 2011; Chiang et al., 2013), and physics-informed surrogate modeling strategies (Suzuki et al., 2023; Fu et al., 2024). The proposed three-axis taxonomy complements existing bioheat modeling classifications, which traditionally categorize models according to their governing physical formulation (e.g., Pennes-type, non-Fourier, vascular-resolved, or multiphysics frameworks) (Berjano, 2006; Rossmann and Haemmerich, 2014). Rather than replacing these formulation-based categorizations, the present framework introduces additional evaluative dimensions that capture practical considerations critical for clinical translation. By jointly considering physiological fidelity, computational tractability, and validation maturity, the taxonomy provides a structured means of comparing modeling approaches that may differ substantially in mathematical formulation but ultimately aim to solve similar predictive tasks in thermal therapy planning. In this sense, the framework can serve as a systematic evaluation tool for emerging computational strategies—including hybrid AI–physics models—by situating them within a multidimensional landscape that reflects both modeling realism and clinical feasibility. Importantly, this perspective also highlights how progress toward patient-specific predictive modeling requires simultaneous advances in physiological representation, computational efficiency, and experimental validation, thereby supporting the development of real-time, uncertainty-aware digital twin frameworks for image-guided tumor thermal therapy.

Second, rather than descriptively cataloguing modeling techniques, this review emphasizes regime-dependent applicability. We explicitly examine the physical and anatomical conditions under which simplified perfusion models remain adequate and contrast them with scenarios—such as perfusion-dominated liver ablation near major vessels—where vascular-resolved or hybrid multiphysics frameworks become necessary (Hassanpour and Saboonchi, 2016; Yuan et al., 2025). Third, we integrate recent developments in physics-informed machine learning and hybrid AI–PDE solvers directly within the context of thermal oncology, aligning surrogate modeling advances (Suzuki et al., 2023; Du et al., 2024) with MRI thermometry validation strategies (Odéen et al., 2016b) and emerging digital twin paradigms in cancer therapy (Kok et al., 2020). Fourth, we incorporate regulatory credibility considerations, including risk-based verification and validation principles (Morrison et al., 2019; U.S. Food and Drug Administration, 2023), thereby bridging computational mechanics, clinical implementation, and medical device approval pathways.

Finally, this review is organized around a central translational question: how can computational bioheat models achieve real-time, uncertainty-aware, patient-specific predictive capability suitable for adaptive intraoperative thermal therapy control? By integrating modeling hierarchy, uncertainty quantification, validation maturity, and practical translational constraints, the review provides a structured perspective for advancing computational bioheat modeling beyond descriptive comparisons toward clinically actionable predictive frameworks (Figure 1). Furthermore, the review situates bioheat modeling within the broader landscape of computational transport science by highlighting conceptual links with nonlinear thermal–fluid transport studies developed in magnetohydrodynamic, porous-media, and bioconvective flow systems. Although these studies arise primarily from engineering transport physics, they offer methodological insights into multiphysics coupling, nonlinear heat transfer, and heterogeneous transport processes that may inform future multiscale and multiphysics modeling strategies in thermal oncology (Marzougui et al., 2021; Dharmaiah et al., 2023; Sankari et al., 2025; Iranian et al., 2026; Hassan et al., 2025).

3 Classical and extended bioheat models

3.1 Pennes bioheat equation

The classical Pennes bioheat equation is written as:where:

ρ c ∂ T ∂ t = ∇ · k ∇ T + ρ b c b ω T a − T + Q e x t

• ρ– tissue density (kg·m^-3)
• c– specific heat capacity of tissue (J·kg^-1·K⁻¹)
• T– local tissue temperature (K or °C)
• t– time (s)
• k– thermal conductivity of tissue (W·m^-1·K⁻¹)
• ρ b– blood density (kg·m^-3)
• c b– specific heat capacity of blood (J·kg^-1·K⁻¹)
• ω– volumetric blood perfusion rate (s^-1)
• T a– arterial blood temperature (K or °C)
• Q e x t– externally applied heat source term (W·m^-3), representing energy deposition from radiofrequency, microwave, laser, or ultrasound modalities

The left-hand side represents transient heat storage within tissue. The first term on the right-hand side describes conductive heat transfer according to Fourier’s law. The second term represents perfusion-mediated heat exchange, modeled as proportional to the temperature difference between arterial blood and local tissue. The final term accounts for externally delivered energy.

The foundational bioheat framework introduced by Pennes (1948)—derived from forearm temperature measurements and formalized through a perfusion-based heat exchange term proportional to blood flow and the arterial–tissue temperature difference—established a simplified yet practical model that continues to underpin bioheat analysis in hyperthermia and tumor thermal therapy. Despite its simplifying assumptions, including spatially averaged perfusion and isotropic conduction, the Pennes equation remains the reference formulation against which extended and multiphysics bioheat models are evaluated.

Despite its enduring utility, the Pennes formulation is limited in its ability to represent vascular geometry and spatially heterogeneous perfusion (Pennes, 1948; Rossmann and Haemmerich, 2014). In particular, the assumption of spatially averaged perfusion and isotropic heat conduction restricts predictive accuracy in highly vascularized tumors (Rossmann and Haemmerich, 2014; Hassanpour and Saboonchi, 2016). Comprehensive reviews of temperature-dependent thermal, dielectric, and perfusion properties have demonstrated substantial nonlinear variation of tissue parameters between 40 °C and ablation temperatures, underscoring the necessity of incorporating temperature-dependent coefficients in predictive bioheat models (Rossmann and Haemmerich, 2014).

The clinical diversity of thermal therapies—including hyperthermia and thermal ablation delivered via electromagnetic, ultrasonic, perfusional, and conductive modalities—has been comprehensively reviewed by Kok et al. (2020). That work highlights the central importance of precise thermal dosimetry, image guidance, and treatment planning, while emphasizing how patient-specific anatomy, perfusion dynamics, and device-dependent energy deposition generate highly heterogeneous temperature distributions in vivo (Kok et al., 2020). These clinical and technological constraints directly motivate the development of physics-based and computational bioheat models capable of predicting spatiotemporal temperature fields under modality- and tissue-specific conditions. Consequently, subsequent modeling efforts have focused on refining bioheat formulations through temperature-dependent tissue properties, vascular-aware representations, and numerical frameworks that better capture spatial heterogeneity and dynamic thermal responses, as discussed in subsequent section (Trujillo and Berjano, 2013; Hassanpour and Saboonchi, 2016). However, beyond refinement of thermophysical parameters, fundamental structural limitations of the Pennes formulation emerge under specific physiological and thermal regimes, as discussed below.

3.1.1 Regime-dependent limitations of the Pennes bioheat equation

Although the Pennes bioheat equation remains the foundational framework for thermal therapy modeling (Pennes, 1948; Wissler, 1998), its simplifying assumptions introduce regime-dependent limitations that become clinically significant under specific anatomical and energetic conditions.

The principal limitation arises from the treatment of blood perfusion as a spatially homogeneous, isotropic volumetric sink term proportional to the local arterial–tissue temperature difference. This formulation neglects discrete vascular geometry and directional convective heat transport. In tissues containing large vessels—particularly hepatic tumors adjacent to portal or hepatic veins—convective heat removal along vessel axes produces asymmetric temperature fields and distorted lesion geometries that cannot be captured by spatially averaged perfusion terms (Berjano, 2006; Hassanpour and Saboonchi, 2016; Yuan et al., 2025). Under such conditions, discrepancies between Pennes-based predictions and vascular-resolved simulations may exceed 15%–25% in predicted ablation margins (Hassanpour and Saboonchi, 2016; Yuan et al., 2025), potentially compromising oncologic safety margins.

A second limitation emerges under rapid heating regimes characteristic of high-power microwave and laser ablation. The classical Pennes model assumes instantaneous heat flux response governed by Fourier conduction, implying infinite thermal propagation speed. While acceptable at macroscopic time scales, this assumption may introduce error during steep thermal gradients and short-duration energy deposition, particularly at microscale vascular interfaces or near applicator tips (Tzou, 1995; Sharma and Kumar, 2020). In these high-gradient regimes, delayed thermal response and local thermal non-equilibrium effects can influence transient temperature evolution.

Third, the Pennes formulation does not inherently account for temperature-dependent perfusion shutdown or vascular collapse occurring at cytotoxic temperatures. Experimental and modeling studies demonstrate that perfusion may initially increase with moderate heating but subsequently decrease sharply beyond approximately 60 °C due to coagulative damage (Rossmann and Haemmerich, 2014). Because perfusion in the Pennes model is typically treated as constant or piecewise-defined, failure to dynamically couple perfusion to tissue injury may lead to overestimation of convective cooling in late-stage ablation.

Fourth, phase change phenomena—particularly water vaporization near 100 °C—introduce latent heat effects, abrupt changes in thermal conductivity and electrical properties, and potential impedance rises in radiofrequency ablation (Trujillo and Berjano, 2013; Sebek et al., 2016). The classical Pennes equation does not explicitly incorporate vaporization enthalpy or multiphase transitions, limiting predictive accuracy in high-temperature ablation regimes.

Finally, uncertainty propagation within the Pennes framework remains substantial. Spatial variability in perfusion, conductivity, and boundary conditions—often inferred rather than directly measured in vivo—can produce significant variability in predicted necrosis margins (Odéen et al., 2016a; van Erp et al., 2023). Because the perfusion term aggregates microvascular effects into a single scalar parameter, model calibration may mask underlying physiological heterogeneity.

Collectively, these limitations do not invalidate the Pennes equation; rather, they define the physiological and thermal regimes in which it remains appropriate. In moderately perfused tissues without large vessel proximity and under controlled heating rates, Pennes-based models provide computationally efficient and clinically useful approximations. However, in perfusion-dominated, high-gradient, or phase-transition regimes—such as liver tumors near major vessels or high-power microwave ablation—vascular-resolved, multiphysics, or hybrid frameworks become necessary to preserve predictive fidelity.

3.2 Extended and non-fourier bioheat models in tumor thermal therapies

In solid tumor ablation, classical Pennes-based bioheat models have been widely applied to predict temperature distributions during radiofrequency, microwave, laser, and focused ultrasound therapies, particularly in relatively homogeneous and moderately perfused tissues such as liver and kidney (Pennes, 1948; Berjano, 2006; Ji and Brace, 2011). However, under rapid heating conditions—such as those encountered in high-power microwave or laser ablation—the assumption of instantaneous heat flux response embedded in Fourier’s law may become less accurate.

To address this limitation, non-Fourier heat conduction models have been introduced. One commonly used formulation is the dual-phase-lag (DPL) model, expressed as:where:

τ q ∂ q ∂ t + q = − k ∇ T + τ T ∂ ∇ T ∂ t

• q– heat flux vector (W·m^-2)
• k– thermal conductivity (W·m^-1·K⁻¹)
• τ q– phase lag of the heat flux (s)
• τ T– phase lag of the temperature gradient (s)
• ∇ T– spatial temperature gradient

The parameter τ q represents the relaxation time required for the heat flux to respond to an imposed temperature gradient, while τ T accounts for delayed development of the temperature gradient itself. When both lag parameters approach zero, the equation reduces to classical Fourier heat conduction.

In tumor thermal therapies characterized by steep thermal gradients and rapid energy deposition, DPL formulations better capture finite thermal propagation speed and transient heat transport effects, particularly at short time scales or in microscale vascular environments (Tzou, 1995; Hassanpour and Saboonchi, 2016). The thermophysical parameters governing both Fourier and non-Fourier formulations—including thermal conductivity, specific heat, density, and dielectric properties—are tissue-dependent and vary across organ systems (Duck, 2013). Moreover, these properties exhibit strong temperature dependence during ablation, further complicating predictive modeling.

Trujillo and Berjano (2013) systematically evaluated piecewise temperature-dependent formulations for thermal and electrical conductivities in hepatic radiofrequency ablation models, demonstrating that, despite substantial differences in functional form—including a 1.5%–2% °C⁻¹ increase in electrical conductivity up to 100 °C followed by a sharp vaporization-induced drop—predicted lesion dimensions and impedance metrics remained largely consistent. These findings support the use of simplified yet physiologically informed conductivity models in contemporary in silico RFA studies (Trujillo and Berjano, 2013). Expanded electromagnetic–thermal models incorporating sigmoidal temperature-dependent dielectric properties have demonstrated substantial improvements in predictive accuracy during microwave ablation, reducing mean temperature prediction error from approximately 27%–32% (static/linear models) to ∼4% when validated against experimental data (Ji and Brace, 2011).

Nevertheless, the use of spatially averaged perfusion and isotropic conduction assumptions constrains predictive fidelity, especially in highly vascularized tumors characterized by spatially heterogeneous, temperature-dependent, and directionally structured blood flow along complex vascular architectures (Rossmann and Haemmerich, 2014; Hassanpour and Saboonchi, 2016). These limitations have motivated the development of advanced computational frameworks incorporating temperature-dependent perfusion (Kazemi Alamouti et al., 2021), volumetric heat capacity and vaporization modelling (Sebek et al., 2016), non-Fourier heat transport, and porous-media or vascular-resolved representations—including co- and counter-current microvascular networks—to better capture perfusion-mediated heat transport (Tzou, 1995; Hassanpour and Saboonchi, 2016; Heltai and Caiazzo, 2019; Yuan et al., 2025). Variable-porosity porous media formulations based on local thermal non-equilibrium (LTNE) theory have further improved in vivo microwave ablation predictions in liver tumors by accounting for spatially varying vascular density and experimentally derived perfusion heterogeneity (Tucci et al., 2022).

Recent computational investigations in laser-based cardiac ablation further demonstrate the limitations of classical Fourier heat conduction models at fast time scales. Bianconi et al. (2025) compared first-order Fourier formulations with higher-order heat transfer models incorporating optical–thermal coupling and anisotropic tissue properties, showing that second-order formulations more accurately predicted transient temperature dynamics and lesion morphology. Their inclusion of a three-state cellular death model additionally enabled differentiation between reversible and irreversible thermal injury, underscoring the importance of coupling advanced heat transfer theory with biologically grounded damage kinetics. The key assumptions, governing features, and primary limitations of classical and extended bioheat formulations are summarized in Table 1.

TABLE 1. Tumor thermal therapy modalities and dominant bioheat modeling challenges.

Summary of principal energy delivery mechanisms in common tumor thermal therapies and the dominant bioheat modeling challenges associated with each modality. These challenges motivate the development of extended, multiphysics, and patient-specific bioheat models to accurately predict temperature distributions and treatment margins in heterogeneous tumor environments.

3.3 Comparative performance of bioheat modeling paradigms

While classical and extended bioheat models differ in physiological fidelity, their relative performance varies depending on tumor location, vascularity, and energy modality. Comparative studies indicate that in moderately perfused tissues, lesion diameter predictions obtained using the Pennes equation typically deviate by less than 5%–10% from vascular-resolved formulations. However, in highly vascularized liver tumors adjacent to major vessels, discrepancies can exceed 15%–25%, primarily due to unmodeled heat-sink effects (Hassanpour and Saboonchi, 2016; Yuan et al., 2025).

Multiphysics models incorporating temperature-dependent conductivity and dielectric property variation improve impedance prediction during radiofrequency ablation by approximately 10%–20% relative to constant-property models (Trujillo and Berjano, 2013). Nevertheless, these improvements often come at increased computational cost, with vascular-resolved simulations requiring hours to days of computation compared with minutes for classical finite-element Pennes-based models.

Machine learning–assisted surrogate models reduce simulation time from hours to seconds (Suzuki et al., 2023), but their predictive reliability depends heavily on training data coverage and may degrade under out-of-distribution thermal conditions such as vaporization or tissue charring.

Collectively, these observations suggest a trade-off between physiological realism, computational efficiency, and validation maturity. Classical models remain valuable for rapid planning, whereas vascular-resolved and multiphysics frameworks offer superior predictive fidelity in perfusion-dominated regimes. Hybrid and surrogate approaches provide real-time capability but require rigorous validation and uncertainty quantification before routine clinical adoption. A structured comparison of major bioheat modeling paradigms across physiological fidelity, computational requirements, validation maturity, and reported predictive error ranges is provided in Table 2.

TABLE 2. Comparative analysis of bioheat modeling approaches in tumor thermal therapy.

Overview of commonly used computational frameworks, highlighting differences in physiological realism, computational cost, predictive accuracy, validation maturity, and typical clinical application. Representative lesion prediction error ranges are derived from published comparative modeling studies and vary depending on tissue type, energy modality, and validation context. * Values represent approximate ranges reported in literature and may vary depending on tissue type, modality, and validation conditions.

3.4 Thermal damage kinetics and coupling to bioheat models

Prediction of temperature alone is insufficient for clinical decision-making; thermal therapy outcomes are ultimately governed by cumulative cellular injury. Most computational frameworks couple bioheat equations with damage kinetics models, most commonly the Arrhenius injury integral:where A is the frequency factor, E a the activation energy, R the universal gas constant, and T τ the transient temperature field obtained from bioheat modeling. A value of Ω = 1 corresponds to approximately 63% cell death, often used as a threshold for coagulative necrosis.

Ω t = ∫ 0 t A ⁡ exp ⁡ − E a R T τ d τ

An alternative empirical metric, cumulative equivalent minutes at 43 °C (CEM43), is frequently applied in hyperthermia contexts to quantify sub-lethal thermal exposure. While Arrhenius-based models provide mechanistic grounding, parameter uncertainty and tissue-specific variability remain significant challenges.

Importantly, emerging evidence indicates that thermal therapies may induce immunogenic modulation rather than purely necrotic destruction (Yang et al., 2018; Haen et al., 2011). Sub-lethal heating can alter tumor microenvironmental signaling, vascular permeability, and immune cell recruitment, suggesting that future computational frameworks may need to integrate bioheat transfer with systems-level biological response models. Higher-order thermal models combined with multi-state cellular death dynamics provide improved prediction of reversible versus irreversible injury compared with classical Arrhenius-only approaches (Bianconi et al., 2025).

4 Multiphysics and multiscale modeling frameworks

Modern computational frameworks for tumor thermal therapies increasingly adopt multiphysics formulations that explicitly couple bioheat transfer with the dominant energy transport mechanisms of each modality. In radiofrequency and microwave ablation, electromagnetic field propagation is solved concurrently with bioheat equations to account for frequency-dependent tissue dielectric properties, applicator geometry, and dynamic changes in electrical conductivity during heating, all of which strongly influence local energy deposition and lesion morphology (Berjano, 2006; Ji and Brace, 2011). In focused ultrasound therapies, acoustic wave propagation models are coupled with thermal solvers to capture nonlinear propagation effects, focal intensity distributions, and acoustic absorption–induced heating within tumor tissue (Hynynen and McDannold, 2004; Pernot et al., 2003). Similarly, laser-based thermal therapies couple photon transport models—such as Monte Carlo or diffusion-based formulations—with bioheat equations to resolve wavelength-dependent optical absorption, scattering, and spatial energy deposition within heterogeneous tissues (Welch and van Gemert, 2011; Hawasli et al., 2013). Fully coupled electromagnetic–thermal models further account for dynamic changes in dielectric and thermal properties during heating, improving prediction of lesion morphology and impedance evolution in radiofrequency and microwave ablation (Chiang et al., 2013; Sebek et al., 2016; Rossmann and Haemmerich, 2014). In focused ultrasound applications, acoustic–thermal coupling frameworks explicitly model nonlinear wave propagation, absorption, and focal energy concentration to capture heat generation in perfusion-rich and structurally heterogeneous tumors (Quadri et al., 2018).

From a methodological perspective, developments in computational transport modeling outside the biomedical domain provide additional conceptual frameworks relevant to advanced bioheat simulations. Studies of magnetohydrodynamic heat transport, porous-media thermal flow, and bioconvective nanofluid dynamics demonstrate how coupled momentum, thermal, and mass transport processes can generate complex temperature distributions in heterogeneous systems (Marzougui et al., 2021; Dharmaiah et al., 2023; Sankari et al., 2025). Similarly, investigations of magnetically influenced non-Newtonian fluids and bioconvective transport illustrate how electromagnetic forces, microorganism-induced convection, and nonlinear rheological effects modify heat transfer behavior in multiphase media (Iranian et al., 2026). Although these models originate primarily from engineering transport physics, their treatment of nonlinear heat transfer, multiphysics coupling, and porous-media flow provides methodological insights that may inform the development of more physiologically realistic bioheat models for tumor thermal therapy. Such interdisciplinary perspectives reinforce the importance of integrating thermal transport theory with biological and clinical constraints when designing predictive computational frameworks for thermal oncology (Hassan et al., 2025).

At the multiscale level, explicit representation of tumor vasculature has emerged as a critical advancement, as blood vessels act as dominant heat sinks that can substantially distort thermal lesion geometry. Vascular-resolved and porous media models enable simulation of convective heat transfer at the microvascular scale while remaining computationally tractable at the organ level (Hassanpour and Saboonchi, 2016). By linking cellular-scale thermal damage kinetics with tissue- and organ-scale heat transport, these multiscale frameworks provide more mechanistically grounded predictions of tumor response, offering improved guidance for treatment planning and interpretation of post-therapy outcomes.

Such multiphysics and multiscale approaches are increasingly regarded as essential for capturing the spatial heterogeneity and dynamic thermal response that characterize solid tumors, particularly in perfusion-rich organs (Bosque et al., 2021; Kok et al., 2020). Moreover, the inherently multiscale nature of heat transfer in tumors provides a direct conceptual bridge between cellular damage mechanisms and organ-scale temperature distributions, reinforcing the clinical relevance of integrated modeling strategies. Recent advances in physics-informed machine learning further extend this paradigm by embedding governing physical laws directly into data-driven models, thereby improving interpretability, generalization, and data efficiency in computational medical imaging, particularly for MRI, CT, and ultrasound applications (Ahmadi et al., 2025).

5 Numerical and computational techniques

To simulate heat propagation during tumor thermal therapy, the affected organ is discretized into numerous small elements, allowing local temperature changes to be computed and subsequently assembled into a global thermal map (Mattingly et al., 1998; Berjano, 2006). Finite element and finite volume methods are the most widely adopted numerical frameworks for this purpose, as they accurately accommodate irregular tumor geometries, heterogeneous tissue properties, neighbouring organs, and complex applicator placements (Zienkiewicz et al., 2005; Berjano, 2006). Adaptive mesh refinement dynamically increases spatial resolution in regions exhibiting steep thermal gradients—such as near ablation probes or tumor margins—while maintaining a coarser mesh elsewhere, thereby resolving sharp temperature transitions without excessive computational cost (Collins et al., 2020). In parallel, GPU-accelerated solvers and ultrasound acoustic power computations leverage massive parallelism to reduce simulation times from hours to seconds, enabling near–real-time thermal prediction during image-guided interventions (Zhang et al., 2019; Gu et al., 2011; Suzuki et al., 2023). Reduced-order modeling strategies further enhance computational efficiency by approximating high-fidelity simulations with compact surrogate representations that preserve essential thermal behavior, enabling rapid exploration of multiple treatment scenarios (Quarteroni et al., 2017; Collins et al., 2020). Early foundational work demonstrated that balanced-realization–based reduced-order modeling could significantly decrease computational complexity in hyperthermia simulations while retaining acceptable accuracy in temperature prediction (Mattingly et al., 1998). Such approaches laid the groundwork for contemporary surrogate and hybrid modeling strategies used in real-time treatment planning.

Collectively, these numerical and computational techniques underpin modern intra-procedural decision-support systems, where fast, reliable, and patient-adaptive thermal predictions are critical for optimizing treatment safety, efficacy, and personalization (Ahmed et al., 2014; Kok et al., 2020).

Recent work has demonstrated analytically derived and hybrid bioheat formulations capable of voxel-level temperature reconstruction across microwave, laser, and radiofrequency ablation, with validation against MRI thermometry (De Melo Antunes et al., 2025). In ex vivo experiments, 83%–98% of voxels exhibited root-mean-square error ≤1 °C, with total simulation and image-alignment times below 0.3 s per repetition, highlighting the feasibility of near–real-time intraoperative temperature estimation. Such approaches illustrate the growing potential of hybrid analytical–numerical frameworks to bridge computational efficiency and experimental validation.

6 Patient-specific and image-based modeling

Magnetic resonance imaging and computed tomography are commonly used to reconstruct patient-specific tumor geometry, surrounding organs, and applicator positioning, allowing thermal predictions to reflect the true clinical scenario (McDannold, 2005; Hawasli et al., 2013).

Previous studies have demonstrated simulated tumor lesions generated using a range of methodologies, including analytical shape models, patient-derived image segmentations, and biologically informed growth models, enabling controlled assessment of imaging and therapy performance under known ground truth conditions (Xu, 2014; Abadi et al., 2020; Barufaldi et al., 2021).

For example, in liver tumor ablation, CT-derived anatomy can be used to explicitly account for proximity to major vessels, which are known to induce heat-sink effects and incomplete ablation margins (Berjano, 2006; Ahmed et al., 2014). The incorporation of routine clinical imaging into bioheat modeling has enabled simulations tailored to individual patients rather than idealized anatomy. Beyond anatomy, advanced imaging techniques provide functional information that further personalizes treatment. Imaging-based perfusion estimates allow models to incorporate patient-specific blood flow, while MR thermometry enables real-time temperature mapping during therapy, offering a mechanism for intra-procedural validation and model correction (Odéen et al., 2016a; Odéen et al., 2016b). Despite these advances, substantial uncertainties persist. Tissue thermal properties, perfusion changes during heating, and boundary conditions at organ interfaces are often inferred rather than directly measured, limiting predictive reliability (Bosque et al., 2021; Hassanpour and Saboonchi, 2016). Addressing these uncertainties remains a prerequisite for the widespread clinical adoption of patient-specific bioheat modeling as a robust decision-support tool. Thermal transport behavior varies substantially across organ systems, reflecting differences in vascular density, perfusion magnitude, tissue anisotropy, and safety constraints. As summarized in Table 3, liver tumors require perfusion-adaptive or vascular-resolved modeling to address heat-sink effects, whereas brain applications demand MRI-guided, uncertainty-bounded frameworks due to tight safety margins. Lung, prostate, pancreatic, and osseous tumors present distinct dielectric, conductive, and anatomical constraints that alter dominant heat-transfer mechanisms. These organ-specific factors reinforce that no single bioheat formulation is universally optimal; instead, modeling selection must be context-driven, balancing physiological fidelity with computational feasibility and validation maturity.

TABLE 3. Organ-specific sources of uncertainty and corresponding bioheat modeling strategies in tumor thermal therapy.

Organ-dependent physiological, anatomical, and procedural uncertainties that influence heat transfer during tumor thermal therapies, along with preferred computational strategies to address these challenges. The table underscores the necessity of tumor- and organ-specific modeling frameworks to improve predictive accuracy and clinical safety.

7 Data-driven and hybrid modeling approaches

Data-driven methods, particularly machine learning, are increasingly used to complement physics-based bioheat models in tumor thermal therapies. A major application is surrogate modeling, in which learning algorithms are trained on large datasets generated from numerical simulations or experimental measurements to rapidly predict temperature distributions and ablation zones (Suzuki et al., 2023; Collins et al., 2020). For example, neural-network-based surrogates have been developed for radiofrequency and microwave ablation to estimate three-dimensional temperature fields within seconds, enabling interactive exploration of probe positioning and power settings during procedures (Suzuki et al., 2023). Deep learning prediction for Real-Time Ablation Zone Measurement in Ultrasound Imaging have been explained by Zimmermann et al. (2024). Image data–driven predictive modeling frameworks integrating intra-procedural imaging with biomechanical and thermal simulations have further demonstrated the feasibility of patient-adaptive guidance during thermal ablation (Collins et al., 2020).

Machine learning has also been applied to treatment parameter optimization, identifying energy-delivery strategies that maximize tumor coverage while minimizing damage to surrounding critical structures (Collins et al., 2020). In liver and kidney tumor ablation, such approaches have demonstrated improved predicted margins compared with empirically selected parameters (Ahmed et al., 2014; Suzuki et al., 2023). Adaptive control strategies are further emerging, in which real-time imaging—most notably MR thermometry—is integrated with learning algorithms to dynamically adjust power delivery and compensate for perfusion changes or unexpected heat-sink effects (McDannold, 2005; Odéen et al., 2016a).

Hybrid modeling frameworks that combine mechanistic bioheat equations with physics-informed neural networks or constrained learning architectures are attracting growing interest. Sparse physics-informed surrogate modeling approaches capable of reconstructing nonlinear heat transfer dynamics from limited time-series data further demonstrate the potential for interpretable and extrapolatable thermal prediction with minimal training requirements (Suzuki et al., 2023). By embedding physical laws directly into the learning process, these approaches improve robustness and interpretability while reducing dependence on large training datasets (Fu et al., 2024; Collins et al., 2020). Such hybrid strategies form the computational backbone of emerging patient-specific digital twin frameworks in thermal oncology, enabling real-time prediction and adaptation while remaining grounded in biophysical principles (Kok et al., 2020). These developments parallel broader advances in physics-informed learning, where governing equations, device physics, and sparse clinical data are jointly leveraged for real-time prediction, uncertainty quantification, and treatment optimization (Ahmadi et al., 2025). Recent advances in physics-constrained robust learning of partial differential equations from limited and noisy datasets further enhance the stability and interpretability of surrogate models, supporting more reliable reconstruction of governing heat transfer dynamics under sparse clinical data conditions (Du et al., 2024).

The relative strengths and limitations of commonly used bioheat modeling approaches in tumor thermal therapy are compared in Table 2.

Despite promising performance, ML-assisted models exhibit important limitations. Training datasets in oncology are often organ-specific, protocol-dependent, and relatively small, limiting generalizability across institutions and imaging platforms. Domain shift—arising from differences in applicator type, power delivery protocols, or perfusion variability—can lead to significant degradation in predictive performance. Even physics-informed surrogate approaches, while more interpretable and data-efficient, remain sensitive to sparse or biased training data and model misspecification (Suzuki et al., 2023; Fu et al., 2024). Furthermore, surrogate models may extrapolate poorly under extreme thermal conditions such as vaporization or tissue carbonization, where physical behavior deviates from the training distribution. Recent work on physics-constrained learning of partial differential equations highlights both the potential and the challenges of reconstructing nonlinear heat transfer dynamics from limited and noisy datasets, particularly in high-gradient regimes (Du et al., 2024). These considerations underscore the necessity of hybrid frameworks that embed physical constraints and incorporate rigorous uncertainty quantification to mitigate failure modes and improve clinical robustness.

8 Validation, uncertainty, and clinical translation

Analogous to virtual imaging trials, physics-based bioheat simulations can be viewed as modality-specific virtual clinical trials, in which patient anatomy, device physics, and treatment interpretation are explicitly modeled to assess therapeutic performance under controlled and reproducible conditions (Abadi et al., 2020).

Robust validation is a prerequisite for the clinical adoption of computational bioheat models in tumor thermal therapies, yet remains particularly challenging in oncology due to biological heterogeneity and organ-specific physiology. Validation strategies typically progress from tissue-mimicking phantoms, which allow controlled assessment of energy deposition and thermal spread, to animal tumor models that capture vascular perfusion and heat-sink effects, and ultimately to clinical datasets incorporating imaging-based temperature measurements and oncologic outcomes (Bosque et al., 2021; Hassanpour and Saboonchi, 2016; Sharma and Kumar 2020). Differences in tissue composition, spatial scale, and perfusion characteristics across these platforms often limit direct comparability, underscoring the need for standardized, tumor-specific validation frameworks (Kok et al., 2020). A recent systematic review of computational modeling of liver ablation zones highlighted substantial heterogeneity in modeling assumptions, validation strategies, and reported predictive accuracy, underscoring the need for standardized benchmarking and prospective validation frameworks (van Erp et al., 2023). The review further emphasized that limited external validation and variability in perfusion modeling remain major barriers to clinical translation.

Organ-dependent factors further complicate validation. In liver tumors, proximity to large vessels such as the portal vein or hepatic veins can produce pronounced heat-sink effects, rendering accurate prediction of ablation margins highly sensitive to perfusion modeling (Berjano, 2006; Yuan et al., 2025). In lung tumors, low tissue density, respiratory motion, and variable aeration substantially alter heat conduction and energy deposition, challenging both experimental validation and computational accuracy (Ji and Brace, 2011). In brain tumors, the need to preserve surrounding eloquent tissue places stringent demands on spatial precision, and validation efforts increasingly rely on MRI-based thermometry and post-treatment imaging rather than histologic confirmation (McDannold, 2005; Hawasli et al., 2013). These considerations emphasize the importance of context-aware validation strategies rather than universal performance metrics.

Uncertainty quantification has therefore emerged as a central requirement for tumor-specific bioheat modeling. Variability in tissue thermal properties, perfusion dynamics, applicator positioning, and boundary conditions can substantially influence predicted temperature fields and treatment margins. Explicit quantification and propagation of these uncertainties enable prediction intervals to be reported alongside nominal temperature estimates, supporting more informed clinical decision-making (Odéen et al., 2016a; Collins et al., 2020). Such probabilistic outputs are increasingly recognized as essential for regulatory evaluation and for building clinician trust in model-assisted therapy planning (U.S. Food and Drug Administration, 2023; European Medicines Agency, 2022). Key sources of organ-specific uncertainty and corresponding bioheat modeling strategies are summarized in Table 3.

From a regulatory and translational perspective, integration of computational bioheat models into routine oncologic workflows must align with evolving medical device approval frameworks. Regulatory agencies such as the U.S. Food and Drug Administration and European notified bodies increasingly emphasize model credibility, transparency, and validation when computational tools are used to support treatment planning or device performance claims (U.S. Food and Drug Administration, 2023). Demonstrating compliance with established principles of verification, validation, and uncertainty quantification is therefore critical for achieving regulatory clearance or CE marking of software-assisted thermal therapy platforms.

Despite substantial methodological advances, significant translational barriers continue to limit routine clinical deployment of advanced bioheat models. Time constraints during image-guided interventions, incomplete access to patient-specific physiological parameters such as perfusion and temperature-dependent tissue properties, and the need for intuitive clinician-facing interfaces often restrict implementation of computationally intensive frameworks. Addressing these constraints will require streamlined numerical architectures, prospective tumor-specific validation studies, and sustained interdisciplinary collaboration among engineers, imaging scientists, oncologists, and regulatory experts. Importantly, clinical translation depends not only on computational accuracy, but on demonstrable improvements in treatment planning precision, safety margins, and long-term oncologic outcomes across diverse tumor types.

From a regulatory perspective, evaluation of computational medical models is increasingly guided by structured credibility frameworks, including the FDA-recognized ASME V&V 40 standard, which defines model credibility in relation to context-of-use, verification rigor, validation evidence, and uncertainty assessment (Morrison et al., 2019; U.S. Food and Drug Administration, 2023). Risk-informed credibility assessment is particularly critical for bioheat models employed in treatment planning, where prediction error may directly influence ablative safety margins and adjacent organ preservation. In this context, physics-informed machine learning approaches—by embedding governing equations as physical priors—support interpretable uncertainty quantification and enhanced model robustness, thereby strengthening regulatory confidence (Ahmadi et al., 2025). Similarly, hybrid analytical–numerical models validated against MRI thermometry exemplify emerging model–measurement convergence, reinforcing the evidentiary foundation required for regulatory acceptance and broader clinical integration (De Melo Antunes et al., 2025).

9 Conclusion and future directions

Computational bioheat modeling in minimally invasive cancer thermal therapies has evolved from the foundational Pennes formulation to an increasingly sophisticated spectrum of extended, multiphysics, vascular-resolved, and hybrid data-driven frameworks. This evolution reflects growing recognition that temperature prediction in solid tumors is governed not only by conduction, but by heterogeneous perfusion, nonlinear thermophysical properties, phase transitions, and organ-specific anatomical constraints (Pennes, 1948; Rossmann and Haemmerich, 2014; Kok et al., 2020; Yuan et al., 2025). Classical Pennes-based models remain computationally efficient and clinically useful for moderately perfused tissues and rapid pre-procedural planning (Wissler, 1998; Berjano, 2006). However, in perfusion-dominated regimes—particularly hepatic tumors adjacent to major vessels—vascular-resolved and porous-media approaches more accurately capture heat-sink effects and asymmetric lesion morphology (Hassanpour and Saboonchi, 2016; Tucci et al., 2022; van Erp et al., 2023). Non-Fourier and higher-order formulations further address limitations associated with rapid heating and steep thermal gradients, though their practical relevance remains modality- and scale-dependent (Tzou, 1995; Sharma and Kumar, 2020; Bianconi et al., 2025).

The emergence of machine learning–assisted surrogate models and physics-informed neural networks introduces transformative potential for real-time thermal prediction and adaptive intraoperative control (Suzuki et al., 2023; Fu et al., 2024; Du et al., 2024). Yet the clinical reliability of such approaches depends critically on training data coverage, robustness to domain shift, explicit uncertainty quantification, and alignment with structured regulatory credibility frameworks (Collins et al., 2020; Morrison et al., 2019; U.S. Food and Drug Administration, 2023). Hybrid computational architectures that embed governing physical constraints within accelerated surrogate models appear particularly promising, offering a pathway toward interpretable, uncertainty-aware patient-specific digital twins (Kok et al., 2020; De Melo Antunes et al., 2025).

Looking forward, future advances in tumor thermal therapy are likely to be driven by integration of closed-loop treatment systems, patient-adaptive digital twins, and multiscale biological response modeling. Closed-loop platforms combining real-time imaging—most notably MR thermometry—with predictive bioheat models and adaptive control algorithms enable dynamic adjustment of energy delivery during treatment, compensating for perfusion changes, tissue heterogeneity, and unanticipated heat-sink effects (McDannold, 2005; Odéen et al., 2016b; Collins et al., 2020). Such systems hold particular promise in anatomically complex and perfusion-rich organs such as the liver and brain.

Beyond temperature prediction alone, next-generation modeling frameworks increasingly seek to couple bioheat transport with immunological and molecular response pathways, reflecting evidence that thermal therapies modulate tumor immunity and microenvironmental signaling rather than inducing purely necrotic destruction (Haen et al., 2011; Keisari and Kelson, 2023). Parallel developments in multimodal machine learning—integrating imaging, structured clinical data, and procedural parameters—suggest complementary opportunities for data fusion within image-guided thermal therapy workflows (Ul Haq et al., 2025).

Despite these advances, three persistent translational barriers must be overcome: uncertainty in patient-specific parameters such as perfusion and vaporization thresholds (Rossmann and Haemmerich, 2014; van Erp et al., 2023); the need for sub-second computational performance during intervention (Mattingly et al., 1998; Suzuki et al., 2023); and rigorous verification, validation, and risk-informed uncertainty assessment consistent with regulatory expectations (Morrison et al., 2019; U.S. Food and Drug Administration, 2023). Sustained convergence of computational mechanics, advanced imaging, clinical oncology, and regulatory science will therefore be essential.

As these elements mature, computational bioheat modeling is poised to transition from a retrospective planning instrument to an active, adaptive component of precision thermal oncology. Achieving real-time, uncertainty-bounded, patient-specific predictive capability will define the next stage of image-guided cancer therapy—transforming bioheat modeling from a simulation tool into a clinically embedded decision-support system for adaptive, biologically informed tumor ablation.

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Footnotes

Edited by: Yatao Ren, Harbin Institute of Technology, China

Reviewed by: Fateh Mebarek-Oudina, University of Skikda, Algeria

Reza Rahpeima, National Taiwan University, Taiwan

Footnotes

ASME, American Society of Mechanical Engineers; CEM43, Cumulative Equivalent Minutes at 43 °C; CT, Computed Tomography; DPL, Dual-Phase-Lag; EM, Electromagnetic; EMA, European Medicines Agency; FDA, U.S. Food and Drug Administration; GPU, Graphics Processing Unit; HIFU, High-Intensity Focused Ultrasound; LITT, Laser Interstitial Thermal Therapy; LTNE, Local Thermal Non-Equilibrium; MRI, Magnetic Resonance Imaging; MWA, Microwave Ablation; PDE, Partial Differential Equation; PINN, Physics-Informed Neural Network; RFA, Radiofrequency Ablation; V&V, Verification and Validation.