Stochastic Modelling and Dynamic Analysis of Cardiovascular System with Rotary Left Ventricular Assist Devices

The left ventricular assist device (LVAD) has been used for end-stage heart failure patients as a therapeutic option. The aortic valve plays a critical role in heart failure and its treatment with LVAD. The cardiovascular-LVAD model is often used to investigate the physiological demands required by patients and predict the hemodynamic of the native heart supported with a LVAD. As a bridge to recovery treatment, it is important to maintain appropriate and active dynamics of the aortic valve and the cardiac output of the native heart, which requires that the LVAD pump must be adjusted so that a proper balance between the blood contributed through the aortic valve and the pump is maintained. In this paper, our objective is to identify a critical value of the pump power to ensure that the LVAD pump does not take over the pumping function in the cardiovascular-pump system and share the ejected blood with left ventricle to help the heart to recover. In addition, hemodynamic often involves variability due to patients heterogeneity and the stochastic nature of cardiovascular system. The variability poses significant challenges to understand dynamic behaviors of the aortic valve and cardiac output. A generalized polynomial chaos (gPC) expansion is used in this work to develop a stochastic cardiovascular-pump model for efficient uncertainty propagation, from which it is possible to rapidly calculate the variance in the aortic valve opening duration and the cardiac output in the presence of variability. The simulation results show that the gPC based cardiovascular-pump model is a reliable platform that can provide useful information to understand the effect of LVAD pump on the hemodynamic of the heart.

💡 Research Summary

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This paper investigates how the electric power supplied to a rotary left ventricular assist device (LVAD) influences the dynamics of the aortic valve and the cardiac output of a failing heart, while explicitly accounting for patient‑to‑patient variability. A six‑state deterministic cardiovascular‑LVAD model is first constructed. The model includes left‑ventricular pressure, left‑atrial pressure, arterial pressure, aortic pressure, total systemic flow, and pump flow. The mitral and aortic valves are represented by ideal diodes that open or close depending on pressure differences, and the left‑ventricular compliance is modeled as a time‑varying elastance function E(t). Different values of the maximal elastance (Emax) simulate a healthy heart (Emax = 2 mmHg/ml), mild heart failure (Emax = 1 mmHg/ml), and severe heart failure (Emax = 0.5 mmHg/ml).

The LVAD is driven by an electric motor whose power P_E(t) is the control input u(t). Pump head (pressure gain) and flow are linked through a nonlinear resistance‑suction model that captures suction phenomena at high speeds. The key clinical question is to identify a “break‑point” in pump power: above this value the LVAD unloads the left ventricle so much that left‑ventricular pressure never exceeds aortic pressure, causing permanent closure of the aortic valve. Permanent valve closure is undesirable in a “bridge‑to‑recovery” scenario because it can lead to thrombosis, commissural fusion, and loss of native myocardial work.

Because physiological parameters such as systemic vascular resistance (SVR) vary across patients and within a patient (e.g., during rest, light exercise, or intense activity), the authors treat SVR as a random variable. To propagate this uncertainty efficiently, they employ a generalized Polynomial Chaos (gPC) expansion. SVR is modeled with a normal distribution; the gPC basis (Hermite polynomials) is truncated at third order, yielding a small set of deterministic coefficients that capture the full probability distribution of the model outputs. This approach dramatically reduces computational cost compared with Monte‑Carlo (MC) simulations while preserving accuracy in the estimation of means and variances.

Simulation experiments combine three activity levels (low, moderate, high) – reflected by different mean SVR values – with three heart‑failure severities (Emax = 2, 1, 0.5). For each combination, pump power is swept from 0 W to 10 W in 0.5 W increments. The outputs of interest are the aortic‑valve opening duration per cardiac cycle and the cardiac output (CO). The results show:

1. Critical Power Identification – For each heart‑failure level, a distinct power threshold (≈ 4–5 W for mild failure, lower for severe failure) separates regimes where the valve opens each beat from regimes where it remains closed. Operating above this threshold risks valve closure and associated complications.

2. Effect of Activity (SVR) – Lower SVR (exercise) reduces the required pump power to achieve a given CO and lengthens valve opening time, whereas higher SVR (rest) demands higher power and shortens valve opening.

3. Heart‑Failure Influence – With reduced contractility (lower Emax), the same pump power yields a shorter valve‑opening window, confirming that weaker hearts are more susceptible to valve closure for a given LVAD setting.

4. Uncertainty Propagation – The gPC‑based variance of CO spikes near the critical power, indicating that small SVR fluctuations can cause large output variability precisely where the system transitions from “valve open” to “valve closed”. This highlights the need for safety margins in controller design.

5. Computational Efficiency – The third‑order gPC method reproduces MC‑derived means and variances with less than 2 % error while requiring roughly 1/50 of the simulation time, making it suitable for real‑time controller synthesis.

The authors discuss clinical implications: LVAD controllers should not merely target a fixed flow rate but must also ensure that the aortic valve remains intermittently open. Real‑time estimation of SVR (e.g., via arterial pressure waveforms) combined with a gPC‑based predictor could enable adaptive power modulation that respects the identified critical power envelope. Future work is suggested to integrate patient‑specific data for model calibration and to embed the stochastic predictor within a feedback control loop, moving toward autonomous LVAD operation that supports myocardial recovery without compromising valve function.

In conclusion, the study provides a rigorous stochastic framework for evaluating LVAD‑induced hemodynamics. By coupling a physiologically detailed deterministic model with generalized Polynomial Chaos, the authors efficiently quantify how pump power, heart‑failure severity, and vascular resistance variability jointly shape aortic‑valve dynamics and cardiac output. The identification of a power break‑point and the characterization of output uncertainty constitute valuable guidance for the design of safe, recovery‑oriented LVAD control strategies.