Adaptive Estimation of Pennes’ Bio-Heat Equation: Observer Design and PINNs-Based Implementation

Abstract

This article presents a multiple-model (MM) adaptive framework for state estimation of uncertain parabolic reaction-diffusion partial differential equations, with specific application to superficial oncological hyperthermia (HT). The effectiveness of such thermal processes critically depends on real-time temperature assessment—a significant difficulty when relying on intrinsically sparse thermometry. This challenge is further compounded by uncertainties in patient-specific properties, particularly blood perfusion rate, during tissue heating. The problem is formulated as a state estimation of the governing Pennes’ bio-heat equation (PBHE), developing an MM adaptive estimation framework capable of dealing with uncertain parameters in the process dynamics and using only noninvasive boundary measurements. The validation process revealed limitations in traditional numerical simulation methods, prompting a novel implementation leveraging physics-informed learning—an approach well-suited for scenarios with known governing equations but limited training data. The experimental validation using a muscle-equivalent phantom with controlled perfusion simulation provides evidence of the method’s effectiveness in realistic conditions. This is the first demonstration of adaptive estimation of parabolic partial differential equations (PDEs) based on physiscs informed learning methods. The proposed methodology offers three key advantages: elimination of domain discretization requirements, thereby enabling continuous spatiotemporal monitoring; real-time estimation capabilities for PDE solutions; scalability to higher dimensional input spaces for an increased adaptability.