A lumped-parameter model of cardiovascular system coupled with LVAD is constructed to test the performance of the proposed PID control algorithm. The cardiovascular system model consists of left heart, right heart, systemic circulation, and pulmonary circulation, as shown in Figure 1 [10] . In each section, hemodynamic factors such as resistance, compliance and valve are represented by resistor, capacitor and diode. The relationship between these hemodynamic factors is characterized by the Kirchhoff’s law. In this work, a cardiovascular system model mimics a HF patient is considered, and all the parameter values used in this model are presented in Table 1 .

The LVAD bridges the left ventricle and the aorta, as shown in Figure 1 . A parameter-based pump model is used to describe the hydrodynamic characteristics of the RBP:

$H_{LVAD}=a_0{\omega }^2+a_2\omega Q_{LVAD}+a_3\omega +a_4{Q}_{LVAD}{}^2$

where $\omega$ is the pump speed in rpm, $Q_{LVAD}$ is the pump flow rate, and $H_{LVAD}$ is the pump pressure difference, respectively. $a_0$, $a_1$, $a_2$ and $a_3$ are constants fitted by experimental results of two commercial LVADs HeartMate 3 and Corheart 6 [11] [12] , and their values are shown in Table 2 .

The PID control system is shown in Figure 2 . In order to maintain adequate cardiac output for HF patients, the mean aortic flow rate ${\bar{Q}}_{ao}$ and the mean aortic pressure ${\bar{P}}_{ao}$ are chosen as control objects. These values are obtained after each cardiac cycle, and then transferred to the PID controller. After comparing with pre-settled values, the desired pump speed is calculated by following equation:

$\omega \left(\tau \right)=K_p\left(V-V_0\right)+K_i\underset{0}{\overset{{\rm T}}{{\displaystyle \int }}}\left(V-V_0\right)\text{d}\tau +K_d\frac{\text{d}}{\text{d}\tau }\left(V-V_0\right)$

where $V$ and $V_0$ indicate the control object and the pre-settled value, and $K_p$, $K_i$, and $K_d$ are control parameters. The PID gain values ( $K_p$, $K_i$, $K_d$) were selected through an empirical tuning process. This tuning process was based on simulation tests, where the controller gains were adjusted through trial and error to achieve the desired dynamic response. The goal was to ensure system stability and rapid convergence under various operating conditions, which is crucial for effective control of the LVAD pump speed. It should be noted that the equation above is not defined over time $t$, but over heartbeat $\tau$, where ${\rm T}$ is the total heartbeat. In each cardiac cycle, the pump speed remains unchanged, and its value is updated once per heartbeat. Hence the integral and the differential terms are calculated in a discrete manner. Once the pump speed is updated, the cardiac output in the next cycle can be simulated by the cardiovascular system model.

The efficiency and robustness of the proposed PID controller is evaluated numerically for a HF patient represented by the cardiovascular system-LVAD coupling model. The simulated heart rate is 75 beat per minute, corresponding to a cardiac cycle of 800 ms. The initial pump speed is set as 0 rpm. Then it is adjusted by the PID controller until a pre-defined mean aortic flow rate or mean aortic pressure is reached, satisfying the demand of a healthy human. It should be noted that the mean aortic flow and pressure are assumed to be directly measured in this simulation study. This assumption simplifies the modeling process; however, in real clinical applications, measuring total aortic inflow is challenging and would likely require additional sensors or a virtual-sensing strategy based on pressure and electrical variables. The aortic flow rate and the aortic pressure are assumed to be directly measured for PID control. The parameters of the PID controller remain unchanged for all tests reported in this work. All simulations are stopped once the pump speed reach a steady state. Physiological parameters such as LV volume, LV pressure, aortic flow rate, aortic pressure, etc. are obtained for data analysis.

All flow-target experiments follow the same protocol: heart rate 75 bpm (≈800 ms per cycle), initial speed = 0 rpm, aortic flow/pressure assumed directly measurable, and unchanged PID gains across all tests. The controller raises speed automatically until the mean aortic flow reaches the predefined setpoint; simulations stop when speed settles. Figures 3-5 summarize PV loops, steady waveforms, and beat-averaged trends.

At the 6.0 L/min setpoint, the beat-averaged flow exhibits a single under-damped peak ≈ 7.5 - 8.0 L/min within the first 10 - 20 beats, then decays exponentially to target; the steady-state speed ≈ 5.68 kRPM. At 5.4 L/min, the peak is smaller, settling is faster, and steady-state speed ≈ 4.68 kRPM. The speed trajectory follows the mean-flow envelope in phase—consistent with a beat-scale “second-order + integral” behavior shaped by peripheral resistance/compliance plus the integrator.

With the other pump, the same flow setpoints are reached at ≈2.80 and ≈3.18 kRPM, implying a higher effective static gain of its pump map around these operating points. Despite static-gain differences, damping and settling (in beats) remain comparable under identical PID gains.

We use mean-value filtering and take the derivative on the mean, which suppresses intrabeat ripples so the controller reacts to the beat-to-beat envelope; speed limits 2 - 9 kRPM with back-calculation anti-windup prevent integrator accumulation during saturation. Baseline gains: $K_p$ = 30 (kRPM)/(L/s), $K_i$ = 8 s⁻¹, $K_d$ = 0.3 kRPM∙s⁻¹∙L⁻¹.

As targets increase, the pump’s share rises and the aortic-valve opening fraction drops. On PV loops this appears as a left-downward shift with a slimmer loop area—evidence of stronger LV unloading. Lower flow targets place the operating point farther from frequent valve-opening regimes, attenuating nonlinearities and reducing transient peaks; conversely, higher targets achieve faster approach but require higher steady-state speeds. Under tested gains, no suction was observed, consistent with the chosen limits and mean-filtering.

When MAP = 85/95 mmHg is targeted, beat-to-beat responses remain lightly under-damped with exponential convergence. One device settles at ≈3.28/3.73 kRPM; for HeartMate 3 the corresponding speeds are ≈5.88/6.68 kRPM, illustrating the monotonic mapping “higher MAP → higher speed” and device-specific pump-map gains. Figures 6-8 summarize PV loops, steady waveforms, and beat-averaged trends.

Steady waveforms and beat-averaged MAP/speed trajectories indicate similar damping to flow-target control yet greater sensitivity to peripheral resistance; overshoot and settling remain acceptable with the tested gains.

All pressure-target experiments share the same PID gains, mean-value filtering, derivative on mean, and speed limits—ensuring fair side-by-side comparison across devices/targets. At MAP targets, increased afterload demands higher steady-state speed; PV loops still demonstrate unloading while preserving physiologic valve timing under the tested range—consistent with safe operation within the 2 - 9 kRPM limits.

To ensure a fair device-to-device comparison, both HeartMate 3 and Corheart 6 were tested under the same controller and protocol: HR = 75 bpm, initial speed = 0 rpm, identical PID gains, and the same measurement chain (mean-value filtering with derivative on mean, plus speed limits 2 - 9 kRPM with back-calculation anti-windup). All trials ended after the speed converged to steady state. This alignment means any cross-device differences mainly reflect the pump maps rather than tuning or protocol artifacts.

Under mean-flow targets, HeartMate 3 reaches 6.0 L/min with a steady-state speed of about 5.68 kRPM and 5.4 L/min at about 4.68 kRPM, with a lightly under-damped transient (single peak ≈ 7.5 - 8.0 L/min within the first 10 - 20 beats). Corheart 6 attains the same two flow setpoints at ≈2.80 kRPM and ≈3.18 kRPM (at the two setpoints), indicating a higher effective static gain around these operating points. Despite this static-gain difference, the damping and settling (in beats) are broadly similar between devices when gains are kept identical.

When targeting MAP = 85/95 mmHg, Corheart 6 settles at approximately 3.28/3.73 kRPM, while HeartMate 3 requires approximately 5.88/6.68 kRPM to achieve the same pressures. These results reaffirm the monotonic mapping “higher MAP → higher steady-state speed,” and again emphasize that the principal cross-device difference is the speed required by the respective pump maps, not the transient shape itself, which remains lightly under-damped with exponential convergence (Beat-averaged MAP/speed trajectories are summarized in Figure 9 ).

Across both targets, the transient envelopes of the two devices look alike at the same gains—rise, single modest overshoot, and settling within several beats—while the steady operating point differs due to pump-map gain. Physiologically, as either the target or the effective device gain increases, the LVAD contributes a larger share of total aortic inflow and the fraction of beats with aortic-valve opening declines; in PV space this appears as a left-downward shift and a slimmer loop, consistent with stronger LV unloading. These trends recur for both HeartMate 3 and Corheart 6 within the tested range.

For the same aortic-flow target, CoHeart 6 settles at a substantially lower speed (2.8kRPM) than HeartMate 3 (4.68 kRPM). This implies a higher static hydraulic gain for CoHeart 6 (more flow per unit rpm at comparable head), i.e., less rpm is needed to deliver the same perfusion.

HeartMate 3 shows a smaller first-peak flow overshoot but a longer settling envelope with slow decay of the speed ripple from 6.2 kRPM to 4.7 kRPM. Coheart 6 exhibits a larger initial overshoot yet settles faster toward 3.0 kRPM with minimal residual oscillation. Under identical PI gains and filtering, these contrasts are consistent with the pumps’ characteristic curves: higher plant gain (Coheart 6) → faster closed-loop and lower steady-state speed; lower gain (HeartMate 3) → slower decay and higher command.

In both cases the beat-wise average aortic flow converges to 5.4 L/min while the controller reallocates contributions between native aortic-valve ejection and LVAD flow through the speed command. The lower steady-state rpm of Coheart 6 suggests greater unloading per rpm (potentially lower shear for a given flow), whereas HeartMate 3 provides a slightly more tempered transient—useful trade-offs for pump selection and gain tuning.

1) Supervised variable focused on perfusion. Using total aortic inflow as the primary controlled variable directly constrains systemic perfusion. Compared with MAP-only or pump-head feedback, this target naturally captures pump-valve sharing and better reflects the clinical goal of delivering flow to the systemic circuit.

2) Robust measurement chain. Beat-to-beat averaging and taking the derivative on the mean attenuate intrabeat pulsatility and sensor noise, limiting differential amplification and reducing the risk of controller-induced oscillations.

3) Realistic actuator modeling. The controller explicitly uses the pump map with instantaneous head ΔP, avoiding the oversimplification “fixed speed ≈ fixed flow” and improving predictability across loading conditions.

4) Deployment simplicity. A positional PID with speed limiting and back-calculation anti-windup is lightweight, has few parameters, and is well suited to embedded real-time implementation.

5) Physiologic consistency. The 0D cardiovascular model includes AV delay and diode-valve logic; results reproduce expected LV unloading with left-downward PV-loop shifts and plausible changes in valve timing. These findings are consistent with prior research [11] [12] , which has demonstrated similar PV-loop shifts as a hallmark of effective LV unloading, further supporting the physiological validity of the proposed control method.

6) Compared to other commonly used control targets in the literature, such as estimated left atrial pressure or pulsatility index, the mean aortic flow and pressure were selected as control targets in this study. These targets are more directly indicative of overall blood flow and heart unloading, making them well-suited to meet the dynamic physiological demands of different patients.

The framework has practical and modeling constraints. First, performance depends on empirical pump maps, which can drift with viscosity, temperature, inter-patient variance, and device wear, biasing local gain and head estimates. Second, total aortic inflow is not routinely measurable; deploying this target will likely require additional sensors or a virtual-sensor strategy based on pressure and electrical variables, with added complexity and error sources. Third, the 0D model omits vascular inertance and valve opening dynamics and treats peripheral R/C as beat-wise constants, limiting representation of fast transients and reflex control. Fourth, beat-mean filtering introduces perceptual delay, potentially slowing response to acute load changes (posture, hemorrhage, vasoactive drugs). Fifth, no explicit suction detector is implemented; current safety relies on speed limits and averaging, leaving risk windows under extreme unloading. Finally, pathophysiologic heterogeneity (severe pulmonary hypertension, significant TR, arrhythmias) and real-time implementation issues (sampling, transport delays, quantization, packet loss) can degrade closed-loop quality and complicate patient-specific tuning.

1) Online identification & gain scheduling. Use recursive estimation/Kalman filtering to track effective static gain and head sensitivity and schedule by operating region; perform slow pump-map self-calibration with projection/limits and fall back to a conservative parameter set when uncertainty grows.

2) Virtual sensing & sensor fusion. Combine motor current/torque with head estimates and aortic-pressure/valve priors in EKF/UKF to reconstruct total inflow; if differential-pressure or ultrasound flow is available, fuse signals with confidence weighting and failure handling.

3) Minimal-state model refinement. Add key R-L-C inertance and simple orifice-valve dynamics in critical branches while constraining state growth to preserve real-time execution; pair with bounded estimators and noise shaping to capture fast transients without over-fitting.

4) Suction protection. Introduce explicit LV collapse indices (e.g., dP/dt patterns, suction proxies), a fault-management state machine with safe degradation, and hard constraints on speed, MAP, and pulse pressure.

The system has broad clinical applications, both for initial device setup and for continuous autonomous adjustment. It can assist clinicians in setting the initial LVAD pump speed according to the patient’s individual needs. Moreover, during long-term support, the system could autonomously adjust the pump speed in response to physiological changes, reducing the need for constant human intervention and improving the overall efficiency and effectiveness of patient care.