Deep Learning-Based Koopman Analysis For The Prediction Of Multi-Beat Cardiovascular Dynamics

Cardiovascular diseases are the leading cause of death worldwide, highlighting the critical need for accurate and efficient models in cardiology. Numerical models play a vital role in therapy planning, control of cardiacassist devices, and population studies. However, achieving accurate multi-heartbeat predictions in real-time for nonlinear systems, such as the cardiovascular system, remains a significant challenge. Unlike linear dynamical systems, nonlinear systems pose difficulties that extend beyond the medical field. Lusch et al. recently combined Deep Learning (DL) with Koopman analysis to linearize multiple nonlinear dynamical systems, providing a promising approach to address this challenge. This thesis aims to extend their work to the prediction and linearization of hemodynamic signals over multiple heartbeats. This study applied the approach of Lusch et al. to predict and linearize 18 hemodynamic signals from a lumped cardiovascular system model. Two datasets were used: one containing time-series data for 10 heartbeats and another for 30 heartbeats, with the latter featuring a significant preload step to trigger the Frank-Starling mechanism. The system starts by transforming initial conditions into an optimized representation using a deep autoencoder. A second neural network constructs the Koopman operator for linearized predictions in the encoded space, which are then transformed back to the original space by the autoencoder. Hyperparameters were optimized using Particle Swarm Optimization (PSO) and grid search was conducted to determine the sliding window overlap in training data. The optimized training process for wide and shallow networks resulted in a maximal overlap of sliding windows. The Koopman operator was constructed using 11 eigenvalues, reducing the dimensionality of the intermediate linear space. Predictive steps were calculated in less than 1:5 10􀀀4 seconds, achieving an 18-fold speed increase compared to the numerical model. The system demonstrated a z-normalized Root Mean Squared Error (RMSE) of 0.020 for the first dataset and 0.067 for the second. The low variance in Koopman operator eigenvalues (less than 1 10􀀀8) indicated successful linearization of the nonlinear dynamics. Notably, the model accurately simulated the sudden preload increase, showcasing its potential for practical applications. The extended framework presented in this thesis successfully linearizes the mid- to long-term dynamics of the nonlinear cardiovascular system, including the hemodynamic response to changes in preload. The substantial speed increase and high precision in simulating hemodynamic signals suggest the system’s suitability for real-time applications in cardiovascular modeling. These findings provide a robust foundation for further research and development, potentially enhancing patient outcomes through more efficient and accurate predictive models. In conclusion, this study demonstrates that the application of a DL approach combined with Koopman analysis effectively linearizes and predicts the dynamics of a nonlinear cardiovascular system. The significant improvements in speed and accuracy, particularly in simulating hemodynamic responses to preload changes, underscore the potential of this system for real-time applications. Future research can build on this foundation to further refine and expand the capabilities of cardiovascular modeling, ultimately contributing to improved clinical outcomes.