Data-driven and physics-informed adaptive temperature estimation for oncological hyperthermia

Data-driven and physics-informed adaptive temperature estimation for oncological hyperthermia Guglielmo Cappellini∗, Emanuele De Santis∗, Samuele Leonardi†, Marilena Vendittelli∗ ∗DIAG “A. Ruberti”, Sapienza Universit` a di Roma, Rome, Italy Email: {cappellini, edesantis, vendittelli}@diag.uniroma1.it †Sapienza Universit` a di Roma, Rome, Italy Email: [email protected] Abstract—This paper integrates physics-informed neural networks with gated recurrent units to leverage both physical modeling and data-driven learning to achieve accurate and generalizable solutions in sparse-data regimes. The approach targets robot-assisted superficial oncological hyperthermia treatment, retrieving temperature distributions along a monodimensional domain from boundary measurements with uncertain domain properties and sparse internal measurements. Validation encompasses both simulations and experimental scenarios. Index Terms—Adaptive estimation, Physics-informed Neural Networks, Gate Recurrent Unit, Oncological Hyperthermia I. INTRODUCTION Heat transfer in biological tissues plays a critical role in various medical applications, especially in thermal therapies such as robot-assisted oncological Hyperthermia (HT) [1], where a fast and accurate assessment of the temperature distribution is crucial for more effective and safer treatments. The robotic framework is described in [2] and features impedance-based interaction control for accurate and safe delivery of the treatment, a digital twin of the operative setup, and communication modules with a pre-operative planning software. Heat transport in biological tissues is typically modeled by Pennes’ Bio-Heat Equation (PBHE) [3], a standard heat equation plus a linear reaction term that accounts for blood perfusion. Machine Learning (ML) approaches have recently emerged as powerful tools for approximating solutions to Partial Differential Equations (PDEs), especially when enriched with physical knowledge, such as the principle that lies behind Physics-Informed Neural Networks (PINNs) [4]. PINNs have been proven effective for providing fast solution to the inverse problem, however, physical systems in the real world may detach form ideal descriptions. Inspired by [5], this paper investigates a hybrid modeling approach that combines Gate Recurrent Unit (GRU), which are well suited to capture temporal dependencies, with PINNs. Using the integrated framework Physics-Informed Hybrid Neural This work is partially supported by Rome Technopole ECS00000024 funded by the European Union - Next GenerationEU, PNRR Mission 4 Component 2 Investment 1.5, the SoBigData research network HORIZON EUROPE G.A. 101079043, Rome Technopole Flagship Project 4, CUP B83C22002820006, through the Sapienza University of Rome project “Phygital Twin Technologies for innovative Surgical Training and Planning”, and by the Sapienza project ”SmartCare 75+. Designing Smart Wearables and Health Monitoring Systems for Home-Based Elderly Care” (project number RM1241910978DAC7). GRU Network PDE problem PINN DF block Fig. 1. Structure of the PIHNN model. Network (PIHNN), we aim to obtain good generalization throughout the spatial-temporal domain while maintaining physical consistency. After the validation of PIHNN in a benchmark heat equation problem, the setup is tested in an experimental scenario. II. PHYSICS-INFORMED HYBRID GRU MODELS The structure of PIHNN is reported in Fig. 1: inputs are time tand boundary measurements yi(t), which are processed by both GRU and PINN models before being combined in the data fusion block. The GRU leverages past knowledge of the system (e.g., previous internal measurements) to capture real-world behaviour, while PINNs encode fundamental mathematical laws governing the system. This combination allows PINNs to provide physically consistent predictions, while GRU contributes to context-aware adaptability. III. SIMULATION Given the characteristic time t∗and length L0, we introduce the scaled 1D Heat Equation: ∂τθ=αt∗ L2 0 ∂XX θ, (1) defined for (X, τ)∈[0,1] ×[0,1], where θ(X, τ)denotes the temperature field at spatial coordinate Xand time τ, and αrepresents the thermal diffusivity. The problem is subject to homogeneous Dirichlet Boundary Conditions (BCs) and a sinusoidal Initial Condition (IC): θ(0, τ) = θ(1, τ) = 0, τ ∈[0,1], θ(X, 0) = sin (πX), X ∈[0,1].(2) This configuration admits an analytical solution of the form: θ1D(X, τ) = e −π2αt∗ L2 0 τsin (πX). 2025 I-RIM Conference October 17-19, Rome, Italy ISBN: 9788894580570 10.5281/zenodo.17629726 101 Fig. 2. L2error of PIHNN vs Eq. (3) with different {¯α, (H, W )}and mixing strategies. The hybrid approach demonstrates superior robustness. Real-world systems deviate from the ideal 1D case in two ways. First, transverse heat diffusion is modeled through spatial dampening dependent on heating device dimensions (H, W ). Second, actual material diffusivity ¯αdiffers from theoretical values due to heterogeneities, environmental conditions, and manufacturing tolerances. We introduce a dampening multiplier Dto the 1D solution: θ(X, H, W, ¯α, τ) = D(H, W, ¯α, τ)θ1D(X, τ),(3) with D(H, W, ¯α, τ) = exph−¯ατζH−2+W−2i,where ζ is a fitting parameter. Eqs. (1)-(2) are used to train the PINN, while the GRU is trained on a dataset generated from Eq. (3) with (H, W )value based on the 3H CFMA specifications [6], and ¯αsampled inside the variation range of fat diffusivity from ITIS database. The data fusion block is governed by: θPIHNN =λ·θGRU + (1 −λ)·θPINN.(4) Results of PIHNN test on five simulations are given in Fig. 2, employing the relative L2error as metric. IV. EXPERIMENT The PIHNN is tested on cooling experiments from a muscleequivalent phantom with internal catheter thermocouples at two depths (gt1=0.571, gt2=0.143) and water-cooled catheters emulating perfusion. Since perfusion parameters are unknown, we adaptively mix a family of PINNs with different perfusion estimates based on superficial error [7], [8], while GRU is trained on previous experimental datasets. Performance analysis reveals complementary strengths (Fig. 3): PINNs excel at shallow depths due to valid 1D assumptions and boundary condition adaptivity but degrade with depth as 3D effects and heterogeneities emerge. GRU depends on training similarity—variable superficially but consistent at depth. This motivates spatially-adaptive mixing. Introducing the linear interpolation coefficient F=X−gt2 gt1−gt2, and with reference to Eq. (4), the weight is now: λ(X) =      0for X≤gt2 Ffor gt2< X < gt1 1for X≥gt1. Fig. 3. Absolute L2error of GRU and PINNs vs. measurements. PINNs are more accurate near the surface, GRU outperforms at bigger depths. Moreover, when X= (gt2,gt1)no training data for GRU is available, and a similar linear interpolation is adopted: θGRU(X) = θGRU(gt2) + F(θGRU(gt1)−θGRU(gt2)). V. RESULTS AND CONCLUSIONS This paper explores combining data-driven and physicsinformed AI for temperature estimation in robot-assisted hyperthermia, where full temperature profiles are needed but measurements are limited and physical models imperfect. Results show enhanced accuracy versus individual models, but optimal mixing weight λselection remains a fundamental limitation as it requires ground truth knowledge. A possible solution is heuristic λselection based on parameter distance from training conditions. REFERENCES [1] N. Datta, S. G. Ord´ o˜ nez, U. Gaipl, M. Paulides, H. Crezee, J. Gellermann, D. Marder, E. Puric, and S. Bodis, “Local hyperthermia combined with radiotherapy and-/or chemotherapy: Recent advances and promises for the future,” Cancer Treatment Reviews, vol. 41, no. 9, pp. 742–753, 2015. [2] M. Ferro, P. 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