The expression for the hydraulic characteristic model of the blood pump is as follows:

$H={\beta }_{0}q+{\beta }_1\frac{\text{d}q}{\text{d}t}+{\beta }_2{\omega }^2$ (1)

where $H$ represents the pressure difference across the blood pump (unit: mmHg), $q$ denotes the flow rate generated by the pump (unit: mL/s), and $\omega$ indicates the rotational speed of the pump (unit: r/min). The terms ${\beta }_0$ to ${\beta }_2$ are constant coefficients of the hydraulic characteristic model.

As reference [17] mentioned, the blood pump was operated in a sine wave, square wave and triangle wave mode, during which data on the pressure difference across the pump and the flow rate were recorded over time. The relevant constant coefficients in Equation (1) were determined through numerical fitting using the least squares method, thus establishing the hydraulic characteristic model of the blood pump. The resulting constant coefficients of the model are presented below [17] :

${\beta }_0=-0.582,{\beta }_1=-0.06,{\beta }_2=2.841\times {10}^{-6}$

In this study, the left ventricular component of the model is developed based on reference [17] [18] . The schematic diagram of the lumped-parameter cardiovascular model with the blood pump is shown in Figure 1 .

The state equations of the blood pump‑cardiovascular coupling model are derived in accordance with Kirchhoff’s laws, and are expressed as:

$\stackrel{\dot{}}{x}=A\left(t\right)x+K\left(t\right)p\left(x\right)+c{\omega }^2$, where $\omega$ represents the rotational speed of the blood pump in rpm. The expressions for $A\left(t\right)$, $K\left(t\right)$, and $p\left(x\right)$ are:

$A\left(t\right)=\left[\begin{array}{cccccc}-\frac{\stackrel{\dot{}}{C}\left(t\right)}{C\left(t\right)}& 0& 0& 0& 0& -\frac{1}{C\left(t\right)}\\ 0& -\frac{1}{R_s{C}_R}& \frac{1}{R_s{C}_R}& 0& 0& 0\\ 0& -\frac{1}{R_s{C}_s}& \frac{1}{R_s{C}_s}& 0& \frac{1}{C_s}& 0\\ 0& 0& 0& 0& -\frac{1}{C_A}& \frac{1}{C_A}\\ 0& 0& -\frac{1}{L_S}& \frac{1}{L_S}& -\frac{R_C}{L_S}& 0\\ \frac{1}{L}& 0& 0& -\frac{1}{L}& 0& -\frac{R}{L}\end{array}\right]$

$K\left(t\right)=\left[\begin{array}{cc}\frac{1}{C\left(t\right)}& -\frac{1}{C\left(t\right)}\\ -\frac{1}{C_R}& 0\\ 0& 0\\ 0& \frac{1}{C_A}\\ 0& 0\\ 0& 0\end{array}\right]$

$p\left(x\right)=\left[\begin{array}{c}\frac{r\left(x_2-x_1\right)}{R_M}\\ \frac{r\left(x_1-x_4\right)}{R_A}\end{array}\right],r\left(\epsilon \right)=\{\begin{array}{l}\epsilon ,\epsilon >0\\ 0,\epsilon \le 0\end{array}$

$c=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ -\frac{{\beta }_2}{L}\end{array}\right]$

where $R=R_i+R_o+{\beta }_0$, $L=L_i+L_o+{\beta }_1$, the relevant parameters of the model are listed in Table 1 below:

In PSO, each potential solution to the optimization problem is represented as a “particle” within a D-dimensional search space. The entire population, through communication and collaboration among individual particles, follows the current best-performing particles to explore and exploit the solution space, ultimately guiding the swarm toward the global optimum or a satisfactory solution.

Consider a swarm consisting of N particles collaboratively searching for the optimal solution in a D-dimensional search space. The state of the i-th particle at iteration t is defined by two vectors:

Position Vector $X_{id}\left(t\right)=\left(x_{i1},x_{i2},\cdots ,x_{iD}\right)$: This represents the particle’s location in the search space and corresponds to a candidate solution for the optimization problem.

Velocity Vector $V_{id}\left(t\right)=\left(v_{i1},v_{i2},\cdots ,v_{iD}\right)$: This indicates the direction and step size of the particle’s movement in the next iteration.

Additionally, each particle retains the best position it has personally encountered, referred to as the personal best position $P_{id,pbest}\left(t\right)=\left(p_{i1},p_{i2},\cdots ,p_{iD}\right)$. The best position discovered by any particle in the entire swarm is known as the global best position $P_{d,gbest}\left(t\right)=\left(p_{1,gbest},p_{2,gbest},\cdots ,p_{D,gbest}\right)$.

During each iteration, particles update their velocity and position by tracking these two best solutions according to the following equations:

Velocity update equation:

$v_{id}\left(t+1\right)=\omega \cdot v_{id}\left(t\right)+c_1{r}_1\cdot \left(p_{id}-x_{id}\left(t\right)\right)+c_2{r}_2\cdot \left(g_d-x_{id}\left(t\right)\right)$ (2)

Position update equation:

$x_{id}\left(t+1\right)=x_{id}\left(t\right)+v_{id}\left(t+1\right)$ (3)

where:

$d=1,2,\cdots ,D$ denotes the dimension.

$w$ is the inertia weight, which balances the global exploration and local exploitation capabilities of the algorithm. A larger $w$ promotes global exploration, while a smaller $w$ facilitates local refinement.

$c_1$ and $c_2$ are acceleration coefficients (learning factors), typically set as positive constants. Specifically, $c_1$ is the “cognitive” coefficient, reflecting the particle’s reliance on its own experience, and $c_2$ is the “social” coefficient, indicating the particle’s tendency to trust collective information.

$r_1$ and $r_2$ are random numbers uniformly distributed in [0, 1], introducing stochasticity into the search process and helping to maintain population diversity.

The objective of this study is to minimize energy consumption during the pulsatile speed control of the blood pump. Therefore, the optimization target is defined as the pump’s energy consumption, quantified by its average shaft power over a specified operational duration from $t_1$ to $t_2$, with the unit in watts (W).

$P=\frac{{\displaystyle {\int }_{t_1}^{t_2}{T}_e\ast \omega \text{d}t}}{t_2-t_1}$ (4)

where $T_e$ is the electromagnetic torque of the blood pump motor in N·m, $\omega$ is the rotational speed of the blood pump in rad/s. The electromagnetic torque can be derived from the rotor dynamics model of the blood pump as follows：

$T_e-T_L-B\ast \omega =J\ast \alpha$ (5)

where $T_L$ is the load torque (N·m) during pulsation. $B$ is the rotor damping coefficient (N·m·s/rad). In this study, its effect is neglected and its value is set to 0. $\alpha$ is the motor angular acceleration (rad/s). $J$ is the rotor inertia (kg·m²)，and its value is 10⁻⁶ kg·m².

The load torque $T_L$ was characterized via experimental fitting. The specific load torque model utilized in the present paper is:

$T_L=a\ast {\omega }^2+b\ast \omega +c$ (6)

where $a=1.825{\text{e}}^{-5}$, $b=-0.002277$, $c=0.3291$.

In prior research on blood pump speed modulation, the rotational speed—being the primary control variable—directly governs the acceleration magnitude via the shape of its profile. Consequently, defining the rotational speed as the optimization variable enables the pursuit of an optimal speed profile that balances energy efficiency with pulsatile performance.

The number of control points, which defines the search space dimension D, is a critical parameter. In this study, it is set to 5. Setting D = 5 was determined through a trade-off analysis. A lower number would overly constrain the profile shape, potentially failing to capture essential hemodynamic features. Conversely, a higher number would exponentially increase the computational cost of the Particle Swarm Optimization (PSO) without guaranteeing a commensurate improvement in physiological performance. Therefore, D = 5 was selected as it provides a flexible yet computationally tractable parameterization for constructing effective speed profiles.

1) Rotational Speed Constraints

It is imperative to impose constraints on the rotational speed of the blood pump to ensure its normal operation and prevent adverse flow phenomena such as suction and regurgitation. Hemodynamic analysis identified that at 4000 rpm, regurgitation occurred, manifested as negative flow through the pump. At 10,000 rpm, suction was induced by suboptimal left ventricular pressure. Accordingly, the permissible operating speed for the blood pump is defined within the bounds of 4000 rpm and 10,000 rpm.

2) Acceleration Constraints

To minimize the energy consumption of the blood pump during operation, the acceleration was constrained within the range of −10,000 rpm/s to 10,000 rpm/s. This range is defined with reference to a 1 Hz sine wave, which has a baseline speed of 7000 rpm, a pulsatility amplitude of 2500 rpm, and a corresponding mean absolute acceleration of 10,000 rpm/s.

The purpose of imposing hemodynamic constraints is to ensure that the pulsatile modulation, driven by the PSO-optimized speed profile, restores key physiological parameters to within healthy ranges. The recovery of physiological health is evaluated based on aortic pressure and aortic flow, which must be maintained within specified physiological ranges.

1) Aortic Pressure Constraints

Hemodynamic performance was assessed against established normal ranges. The mean aortic pressure is defined as 85 - 100 mmHg, while the aortic pulse pressure (the difference between systolic and diastolic pressure) falls within 20 - 60 mmHg.

2) Aortic Flow Constraints

According to widely accepted medical standards, the mean aortic flow rate in healthy individuals is 80 - 100 ml/s.

The Particle Swarm Optimization (PSO) algorithm is a metaheuristic that mimics the collective behavior of bird flocks or fish schools. It operates by having a population of particles explore the solution space through iterative updates based on both individual and collective best-known positions, thereby efficiently converging to an optimal solution for minimizing the energy consumption of pulsatile blood pumps.

The workflow for optimizing the pulsatile energy consumption of the blood pump based on the Particle Swarm Optimization algorithm is illustrated in Figure 2 . The process begins with the initialization of the PSO parameters, including the number of optimization variables, inertia weight ( $w$), learning factors ( $c_1$ and $c_2$), and population size ( $m$). Subsequently, the position and velocity of each particle are randomly initialized. The position values of the particles are then assigned to the Simulink model for simulation. Following this, the position and velocity of each particle are updated. The fitness value for each particle is calculated, leading to the update of the personal best fitness value and position for the current iteration, as well as the global best fitness value and position. If the termination criteria are met, the global best fitness value and position are output. Otherwise, the Simulink model is executed again to proceed with the next iteration.