Reduced-Order Modeling for Thermal Dose Forecasting in Wearable Devices

The "always-on" functionalities of modern wearables encourage users to maintain direct contact with their powered devices for long durations each day. These now-ubiquitous features of smart watches and fitness trackers include step counting, GPS tracking, heart-rate monitoring, and sleep quality assessment. Microprocessors, batteries, and other active components of wearable devices continuously dissipate heat while in use, some of which is transferred to the user’s skin tissues. The thermal dose received from the device by the user at any given instant is typically low; however, irreversible tissue damage can result from cumulative low dose exposure [1]. Therefore, it would be advantageous for a wearable device to be able to monitor accumulated thermal dose and predict future thermal dosing accurately as it is being used.The heat transfer between a device and its wearer is governed by many parameters, including the device materials and geometry, use conditions, ambient conditions, and the thermal properties of the wearer’s skin. The heat flow can be simulated by solving partial differential equations (PDEs), but such models become cumbersome to solve over long time horizons or for complex geometries and are therefore impractical for real-time forecasting of thermal dose. In this paper, we establish a methodology for building tractable models that can characterize and forecast the heat transfer between a device and a user and the potential for thermal injury. Specifically, we developed reduced-order models (ROMs) based on proper orthogonal decomposition (POD) for modeling the transient heat transfer between a powered wearable device and a user. These reduced-order models approximate solutions to Pennes’ Bioheat Equation [2] for perfuse tissue and the classical heat equation for other materials. The time-temperature forecasts from these models can then be used to calculate the skin tissues’ equivalent duration exposure at 43 °C (formally known as Cumulative Equivalent Minutes at 43 °C or CEM43 °C), which has been widely used to characterize the onset of necrosis at the basal layer of the epidermis [3].The development of the ROMs in this work has two distinct phases: an "offline" step in which the structure of the ROM is built or inferred from training data and an "online" step in which the model can be solved rapidly to predict the thermal response of the skin to time-varying heat sources or external conditions. The offline step can either be performed using classical methods such as Galerkin’s Method of Weighted Residuals or modern machine learning techniques [4]. The online step then only requires the solution of a low-dimensional ordinary differential equation system in place of the original partial differential equations and therefore has substantially lower CPU and memory requirements.When the ROMs are constructed using representative training data, they can provide fast, accurate temperature forecasts for arbitrarily complex time-varying boundary conditions and heat sources. Very little error is observed between the solution of the ROM and the full PDE model, and the solution of these reduced-order models is orders of magnitude faster than a classical finite-volume solution of the governing equations for the same scenario. The output of these ROMs could therefore be incorporated into e.g., device control algorithms to determine the need for power throttling or device shutdown to prevent thermal injury to the wearer.