A new method to compute the blood flow equations using the physics-informed neural operator

Abstract

Using the Navier-Stokes equation for real blood flow simulations in patients is a challenging task. Among many of the challenges, initial and boundary conditions are not directly measurable. Many parameters for the flow model are also not directly measurable and differ from patient to patient and over time. New machine learning techniques, like physics-informed neural networks (PINN), offer some new ways to handle these difficulties. In this work, we aim to learn the operator that maps some easily measurable physiological signals to the solution of the blood flow equation. We use our proposed model based on Navier-Stokes equation and PINN to fit real data on blood pressure. A Windkessel boundary condition is used to produce physically correct reflection waves. A time-periodic condition is used to capture the periodicity of blood flow and enables our model to simulate the blood flow without initial and boundary conditions. Furthermore, we allow the periods of each instance of solution to be different, which makes the training of neural operators computationally expensive, but more accuracy and physical correct towards real blood pressure data. Further more, we also propose an efficient implementation to incorporate the periodic condition into our model. Estimating the hyper-parameters in the Navier-Stokes equation is also difficult. We then introduce a hyper-parameter network to estimate these parameters during the training process as well. The blood flow data contains useful information for disease detection and diagnosis, but directly measuring the entire blood flow remains a significant challenge. We apply our proposed method to cuffless blood pressure estimation. More specifically, we aim to predict the blood pressure waveform (continuous blood pressure and velocity in both time and space) from Electrocardiogram (ECG) and photoplethysmogram (PPG) signals, which can be easily measured using wearable devices. Compared to other methods, our method is the first one that can predict blood flow continuously, both with location and time which are valuable for cardio-vascular medical treatments and diagnoses.