Data Driven Control of Vagus Nerve Stimulation for the Cardiovascular System: An in Silico Computational Study

FIGURE 1. LSTM predictions (shown in red) compared to the output from the in silico physiological model of the rat cardiac system (PM, shown in black) for 200 simulated cardiac cycles of (A) heart rate and (B) mean arterial blood pressure (MAP). For cycles 1–50 (light green), location 1 was active with a pulse width of 0.36 ms and pulse frequency of 31 Hz. For cycles 51–100 (light blue), location 2 was active with a pulse width 0.19 ms and a pulse frequency 17 Hz. For cycles 101–150 (yellow), location 3 was active with a pulse width of 0.42 ms and a pulse frequency of 47 Hz. The last 50 cycles did not have any locations active.

FIGURE 2. A Schematic showing our model-based predictive closed-loop optimal control framework for optimizing the vagus nerve stimulation (VNS) parameters to control multiple cardiac physiological biomarkers such as heart rate (HR) and mean arterial blood pressure (MAP). Here, a long short-term memory (LSTM) based recurrent neural network model has been used to make predictions of HR and MAP in response to VNS and optimize the VNS parameters within the model predictive control (MPC) framework.

FIGURE 3. Sparsity-promoted closed-loop control of the heart rate (HR) and the mean arterial blood pressure (MAP) of the in silico physiological model of the rat cardiac system. (A) and (B) show the closed-loop control response from the in silico physiological model (shown in red dots) and the target (or reference) for the controller (shown in black line) for HR and MAP, respectively. (C) and (D) show the pulse width and the stimulation frequency, respectively, designed by the controller at each VNS location.

FIGURE 4. Minimum energy-based closed-loop control of the heart rate (HR) and the mean arterial blood pressure (MAP) of the in silico physiological model of the rat cardiac system. (A) and (B) show the closed-loop control response from the in silico physiological model (shown in red dots) and the target (or reference) for the controller (shown in black line) for HR and MAP, respectively. (C) and (D) show the pulse width and the stimulation frequency, respectively, designed by the controller at each VNS location.

FIGURE 5. Minimum overshoot-based closed-loop control of the heart rate (HR) and the mean arterial blood pressure (MAP) of the in silico physiological model of the rat cardiac system. (A) and (B) show the closed-loop control response from the in silico physiological model (shown in red dots) and the target (or reference) for the controller (shown in black line) for HR and MAP, respectively. (C) and (D) show the pulse width and the stimulation frequency, respectively, designed by the controller at each VNS location.

FIGURE 6. Comparison between the physiological outputs predicted from the modified in silico physiological model (IP, shown in blue), the in silico physiological model (PM, shown in black) and LSTM model (shown in red) in response to the vagus nerve stimulation (VNS) for 200 simulated cardiac cycles. (A) shows the heart rate (HR, bpm) and (B) shows the mean arterial blood pressure (MAP, mmHg). For cycles 1–50 (light green), location 1 was active with a pulse width of 0.36 ms and pulse frequency of 31 Hz. For cycles 51–100 (light blue), location 2 was active with a pulse width 0.19 ms and a pulse frequency 17 Hz. For cycles 101–150 (yellow), location 3 was active with a pulse width of 0.42 ms and a pulse frequency of 47 Hz. The last 50 cycles did not have any locations active.

FIGURE 7. Sparsity-promoted closed-loop control of the heart rate (HR) and the mean arterial blood pressure (MAP) of the modified in silico physiological model of the rat cardiac system. (A) and (B) show the closed-loop control response from the modified in silico physiological model (shown in blue dots) and the target (or reference) for the controller (shown in black line) for HR and MAP, respectively. (C) and (D) show the pulse width and the stimulation frequency, respectively, designed by the controller at each VNS location.

FIGURE 8. Offset-free closed-loop control of the heart rate (HR) and the mean arterial blood pressure (MAP) of the modified in silico physiological model of the rat cardiac system in the presence of intra-patient variability in the VNS response to HR and MAP. (A) and (B) show the closed-loop control response from the modified in silico physiological model (shown in blue dots) and the target (or reference) for the controller (shown in black line) for HR and MAP, respectively. (C) and (D) show the pulse width and the stimulation frequency, respectively, designed by the controller at each VNS location.

FIGURE 9. Comparison between the physiological outputs predicted from the modified in silico physiological model of an overactive sympathetic system (Sympathetic, shown in purple), the original in silico physiological model (PM, shown in black) and LSTM model (shown in red) in response to the vagus nerve stimulation (VNS) for 200 simulated cardiac cycles. (A) shows the heart rate (HR, bpm) and (B) shows the mean arterial blood pressure (MAP, mmHg). For cycles 1–50 (light green), location 1 was active with a pulse width of 0.36 ms and pulse frequency of 31 Hz. For cycles 51–100 (light blue), location 2 was active with a pulse width 0.19 ms and a pulse frequency 17 Hz. For cycles 101–150 (yellow), location 3 was active with a pulse width of 0.42 ms and a pulse frequency of 47 Hz. The last 50 cycles did not have any locations active.

FIGURE 10. Offset-free closed-loop control of the heart rate (HR) and the mean arterial blood pressure (MAP) in an in silico overactive sympathetic system model of the rat cardiac system. (A) and (B) show the closed-loop control response from the modified in silico physiological model (shown in purple dots) and the target (or reference) for the controller (shown in black line) for HR and MAP, respectively. (C) and (D) show the pulse width and the stimulation frequency, respectively, designed by the controller at each VNS location.

FIGURE 11. Performance comparison of different trained neural network architectures based on the mean absolute error for the same test set. (A) shows a comparison of baseline performance to a traditional (or conventional) RNN, gated recurrent unit (GRU) and LSTM. (B) Shows the a comparison of the effect of the number of LSTM layers on the model’s predictive performance. (C) Shows a comparison of the effect of the number of inputs on the model’s predictive performance.