The baseline design was based on a maglev blood pump previously developed in our group (Wu et al., 2021). The exploded view of the pump head is shown in Figure 1A. The magnetic rotor was combined with the rotor into an integral structure, and accommodated in a ring cavity. A secondary flow path was formed by the cavity and the lower portion of housing (cf. Figure 2A). The total blade number was eight, with four splitter blades. The baseline pump was designed using the speed coefficient method combined with the one-dimensional design theory (Wu et al., 2021). The design point was: 3,500 r/min, pressure head of 360 mmHg, and flow rate at 5 L/min. Figure 1B shows the portable blood pump. The portable blood pump shared the same impeller, annular rotor, and lower portion of housing with the baseline pump. The main difference is the upper portion, where the inlet is perpendicular to the pump axis and parallel to the outlet, a ring cavity was introduced to convert the circumferential inlet velocity to axial velocity. The main design parameters of the portable pump are shown in Figure 2 and Table 1.

Computational fluid dynamics (CFD) were employed to evaluate the hydraulic and hemolytic performance of both pumps. The grids were tetrahedral with five prism layers, generated using Ansys meshing (Ansys, Inc., Canonsburg, PA, United States). A cylindrical surface was placed downstream of the blade trailing edge, which acts as an interface between the rotating regions of the impeller and secondary flow path, and the region of volute and outlet. Similarly, another rotor/stator grid interface was place at the diaphragm, which separates the ring cavity and the region of impeller (as shown in Figure 3). A grid sensitivity analysis was also performed for the portable blood pump, with three grids of 4.58, 10.88, and 24.38 million, respectively (as shown in Table 2). The mean y⁺ was kept below 1 for the “middle” and “fine” grids. y⁺ may exceed 1 at some locations. To deal with this, the SST k-ω model was employed in this study, together with the wall function of “Enhanced wall treatment (EWT).” The EWT provides a unified empirical relationship for the buffer layer and log layer, and is y⁺ insensitive. The CFD simulations were performed using the commercial software Ansys Fluent. The flow rate of 5 L/min was applied at the inlet. The blood was regarded as a Newtonian fluid, and the viscosity was taken as 3.5 mPa s. A SIMPLE method was employed to solve the incompressible N-S equations. Turbulence was modeled using the RNG k-ε model. The steady “frame motion” approach was used to couple the rotating and stationary regions. Convergence criteria was set that the residuals of all equations drop below 10^–6. The computational setup of the baseline blood pump is the same as those of the portable pump. The same computational setup has been employed in our previous study and verified through hydraulic experiments (Wu et al., 2021). For grid sensitivity analysis, the rotational speed was set as 3,500 rpm, and inlet flow rate was set as 5 L/min for the portable pump. The H-Q curves of both pump models were computed, with flow rates ranging from 1 to 9 L/min and pressure head ranging from 100 to 900 mmHg. A detailed analysis and comparison of hydrodynamic and hemolytic performance was conducted at the design point (5 L/min, 360 mmHg) for both pump models. For each pump model, the rotational speed of the rotor was adjusted through iterative computations to meet the targeted pump head of 360 mmHg.

The energy-dissipation-based (EDB) formulation of pow-law models were employed in this study to estimate hemolysis level of the blood pumps. The EDB models relate hemolysis to energy dissipation ε and exposure time t through a power-law relationship (Giersiepen et al., 1990; Wu et al., 2019)

where HI(%) is the hemolysis index in percentage, ε is energy dissipation rate, Hb is the total hemoglobin concentration, hb represents the increase in plasma free hemoglobin; C, α, and β are empirical constants. Energy dissipation rate ε is calculated by adding up viscous dissipation rate ε_vis and turbulent dissipation rate ε_turb. Please refer to Wu et al. (2019) for the definition of ε_vis and ε_turb.

It has been shown that Reynolds stress is an inaccurate indicator of turbulent effects on hemolysis, and the stress-based formulation of power-law hemolysis models tend to over-predicted hemolysis (Jones, 1995; Hund et al., 2010; Wu et al., 2019, 2020). In contrast, the EDB formulation can improve the prediction of hemolysis for a wide range of flow conditions, especially turbulent flows (Wu et al., 2019, 2020). Three widely used sets of empirical constants were employed in this study. Table 3 lists the empirical constants of the three power-law models employed in this study. One can note that the exposure time in Eq. 1 is non-linear. It would be incorrect to consider that the hemolysis at the domain outlet is the sum of the local hemolysis, because of the non-linear dependency in time. This problem was avoided by introducing hb′ as a scalar variable equal to hb^1/β. Then Eq. 1 can be reorganized into a Eulerian scalar transport equation (Garon and Farinas, 2004).

where C is the source term defined as

The HI(%) was then calculated from the mass-weighted average of hb at the outlet of the device divided by Hb. Hemolysis predictions started after flow simulations had converged, with all flow variables frozen.

The results of grid sensitivity analysis are shown in Table 4. Pressure head and hemolysis index were compared to the results predicted with the fine grid. The HO model was used for hemolysis prediction. For the “Middle” grid, the error of pressure head was within 2%, and the error of hemolysis was within 1%. Therefore, a grid of around 10 million is the most appropriate, with results sufficiently resolved and relatively low computational cost compared with the finer mesh. Therefore, this grid was chosen for all the simulations of portable blood pump. The grid of baseline pump was generate using the same setting as the “Middle” grid of portable pump, also with size of around 10 million.

The H-Q curves of both pump models predicted using CFD were shown in Figure 4. The H-Q curves of the baseline pump were very flat, and were almost constant as flow rate increases. This was probably caused by the cylindrical volute. At low flowrates, secondary flows and turbulence intensities were very high, bringing high flow loss. The pressure head of the portable blood pump decreases with the increase of Q. Lower pressure head at high flow rates for portable pump means more flow losses, which might be caused by the ring cavity. This will be further analyzed in the following sections.

We focus on the pressure and flow field at the design point of 5 L/min, 360 mmHg. Table 5 shows the pressure head in different portions of the two pump models (as shown in Figure 3). Since steady simulations were conducted, total pressure was mass-weighted averaged at the inlet, outlet and interfaces of different portions. Pressure head of different portions were then calculated by subtracting the total pressure at the downstream interface/outlet from the upstream interface/inlet. The corresponding rotational speeds of the baseline and portable pumps were 3,160 and 3,350 rpm, respectively. The hydraulic efficiency of the baseline pump is around 3% higher than the portable pump. To achieve the same pressure head, the rotational speed of the portable pump was higher than the baseline pump. The pressure loss in the inlet (including ring cavity) portion of the portable pump is considerable, while the pressure loss is negligible in the inlet portion of the baseline pump. Figure 5 shows the streamlines at two cross-sections in the ring cavity, in which large vortices were clearly visible. The likely reasons for the formation of vortices in the ring cavity are the expansion of the flow path and the Coriolis force.

Figure 6 shows the flow patterns at the meridional plane, where surface streamlines and pressure contours are shown. Low-pressure zone exists at the center of ring cavity. The contours of positive axial velocity at the ring cavity/impeller interface are also shown in Figure 6. Positive axial velocity means backflow, happening at the outer area of the interface (as shown by the red arrows). The backflow is most likely caused by the low-pressure zone at the center of ring cavity.

Figures 7A,B show negative axial velocity contours at the meridional plane for both pump models. Regions of backflow can be observed at the entrance of the impeller. Figures 7C,D show contours of Reynolds stress. The Reynolds stress in the ring cavity of the portable pump is fairly high, reflecting the influence of vortices. On the other hand, the regions of backflow are also regions where Reynolds stress is low, showing that backflow reduced turbulence at the impeller. This explains that the pressure head of the impeller for the portable pump is higher than the baseline pump.

As aforementioned, three sets of empirical constants were employed to estimate hemolysis. These empirical constants were widely used and each has its own advantages in accuracy in different situations. In order to minimize the uncertainty brought by hemolysis model, a new variable, HI_diff, was introduced to represent the overall change of hemolysis over the baseline pump considering the results of all the three sets of empirical constants, defined as:

where the denominators represent the predicted hemolysis indices of the baseline pump, while primes represent the hemolysis indices of the portable blood pump. Subscripts “GW,” “HO,” and “TZ” refer to the hemolysis calculated using the three sets of empirical constants as shown in Table 3. At the design point (5 L/min, 360 mmHg), the value of HI_diff is 29%, i.e., the hemolysis level of the portable blood pump is 29% higher compared to the baseline pump.