The configuration of a centrifugal pump has been widely used as a ventricular assist device (VAD) for adults and recently for infants and small children with failing ventricles because of its compact size, flatter H-Q characteristics, and wider range of peak efficiency.1–3 The hydraulic design and optimization of a miniature pediatric centrifugal blood pump presents a great challenge because of its very small size, relatively greater hydrodynamic losses, and higher biocompatibility requirements. The traditional trial-and-error design approaches for a blood pump are usually based on the one- or two-dimensional theory and semiempirical equations. Furthermore, the design cycle to reach an anticipated objective is usually very long with the need to validate designs by extensive in vitro model testing.
Computational fluid dynamics (CFD) has been widely used for designing blood wetted artificial organs.4–6 CFD offers an inexpensive and rapid means to acquire detailed flow-field information for a blood pump prototype that cannot be achieved by in vitro testing. Particularly, when integrated with parameterized models of the geometry, automatic mesh generator, and mathematical models of blood damage, CFD-based design optimization can become a powerful tool to assess the performance of a design and to offer a more efficient means of optimization than the traditional trial-and-error approaches.
Our consortium has investigated the design and optimization of a miniature magnetically levitated centrifugal blood pump, the PediaFlow, intended to deliver 0.3–1.5 l/min against 100 mm Hg to neonates and infants weighing 3–15 kg. The back clearance gap between the housing and large volume of the rotor, where the suspension and motor bearings are located, forms a continuous leakage flow path. Within the gap, flow demonstrates a very complex three-dimensional structure: the fluid adjacent to the rotating disk tends to accelerate by centrifugal force to flow radially outwards toward the outlet of the impeller against an unfavorable pressure gradient, which in turn forces blood to return along the stationary housing surfaces. Consequently, one or multiple vortices may be generated in the gap to block blood flow and induce uneven pressure distribution at the gap outlet. This phenomenon can cause the formation of a retrograde and antegrade leakage flow pattern. Our previous flow visualization for an adult centrifugal pump, the scaled-up version of PediaFlow, also confirms this interesting effect.8 Such an undesirable flow pattern in the gap not only directly impacts pump performance and causes unbalanced axial thrust, but also potentially induces cellular trauma and thrombosis. For industrial pumps, many efforts have been made in adding slots and/or secondary blades (ribs) on the back of the rotor to balance the axial thrust.10,11 However, for blood pumps, it is our interest to eliminate the adverse “retrograde” leakage flow at the gap outlet and to eliminate or attenuate the strength of the vortices within the gap to avoid potential cellular trauma and thrombosis.
This study presents a CFD-based design optimization system, which integrates our internally developed three-dimensional inverse design methods, parameterized models of the geometry, automatic mesh generator, and mathematical models of blood damage with a commercial CFD software package, STAR-CD (CD-Adapco, Melville, NY). The system has the potential to reduce or perhaps replace physical experiments with numerical simulation testing by iteratively generating and modifying a design until a specified target function is satisfied, through an automated process or with human expert guidance.17,25 In this report, we apply this system to eliminate the unfavorable retrograde leakage flow of a pediatric centrifugal pump, while putatively enhancing pump performance and biocompatibility.
Materials and Methods
PediaFlow Computational Model
The computational model of PediaFlow was developed by our CFD-based design optimization system for a design point of 0.5 l/min, 100 mm Hg, and 12,000 rpm, as illustrated in Figure 1. Here the primary and secondary blades are designed and optimized by an internally developed three-dimensional inverse design tool in the system,14,17 and other components, such as the inlet and outlet volutes of the pump, are designed by in-house parameterized models of the geometry based on B-spline curves or surfaces and other mathematical models.12 The semiopen impeller without front shroud includes six primary blades with an impeller diameter of 12.5 mm. It is driven by a brushless DC motor with toroidally wound motor coils within the rotor, which is fully suspended by the magnetic bearing system. A symmetrical annular inlet volute, similar to the one used in MedQuest Heartquest pump (Salt Lake City, UT), is used in the PediaFlow to uniformly direct the fluid into the impeller eye. For computational simplification, the inlet volute is not included in the computational domain. Included in the analysis are the exit volute, which collects blood existing from the impeller and converts a portion of the kinetic energy into pressure energy, and the front blade tip clearance, which also has a significant effect on pump performance and efficiency as well as blood damage.
CFD-Based Design Optimization System
It is important to integrate a robust and flexible inverse design tool in a CFD-based design optimization system to allow automatic or expert-interfaced generation and modification of the design geometry. In the turbomachinery industry and the field of blood pumps, tremendous advances have been made through the direct approach applying Quasi-three-dimensional, Euler three-dimensional, and viscous three-dimensional Navier-Stokes methods to solve the flow fields for a given blade geometry.2,3,13 However, a robust and fully three-dimensional inverse design approach, by which the required flow distribution is specified and the corresponding blade geometry is computed, is still not commonly available.14 The fundamental equation commonly used for the inverse design of the blade camber-line can be written as:
Where θ is the wrap angle, and m denotes the meridional distance along the camber-line. In Equation 1, as an approximation, the meridional velocity along the meridional streamlines can be computed by one- or two-dimensional potential flow theories, whereas the distribution of the angular momentum (rVθ) or its derivative can be specified according to the pressure loading, as originally proposed by Jansen and Kirschner,15 and recently by Zangeneh16 and Goto et al.14 Therefore, a three-dimensional blade shape can be stacked together by integrating Equation 1 along each streamline, provided that the blade thickness distribution is given.
Equation 1 can be further simplified as:
According to Equation 2, if the blade angle β distribution along a streamline is known, similarly the three-dimensional blade geometry can be obtained. Compared with Equation 1, the number of unknown variables in Equation 2 for the blade is reduced to only one. According to previous work by Wu et al.,17 an inverse design method by Equation 2 is usually more flexible and efficient than by Equation 1.
An alternative method to that of Equation 2 is to directly and somewhat arbitrarily draw the camber-line in a conformal mapping plane (I-U), which is defined as:
Then, according to Equation 2, the local blade angles can be easily obtained by using:
Although the three basic inverse design methods as described above are theoretically derived from the same equation or its deformation, it practically involves quite different techniques to adapt each into a flexible and robust design tool. The first method requires statistical or empirical data or a designer's experience in prescribing the distribution of angular momentum (rVθ), or the distribution of its derivative. The second method requires the same in describing a blade angle distribution. The third method utilizes a designer's experience or intuition in drawing a camber-line in the conformal mapping plane. Each of these approaches, if integrated with the advanced three-dimensional viscous CFD codes, can become a very practical and powerful tool in the design and optimization of turbomachinery and blood pumps.14,17 All three inverse design methods have been adopted in our CFD-based design optimization system to allow a designer options for achieving a rapid optimization for the impeller blades and guide vanes of various types of centrifugal, mixed-flow, and axial-flow pumps.
Mesh generation is one of the most important steps in CFD-based design optimization. Although some commercial CFD packages, like STAR-CD, FLUENT (Fluent Inc., Lebanon, NH) and CFX (ANSYS Inc., Canonsburg, PA), have automatic mesh-generation capabilities, these packages usually require the boundary (surface) data in the form of output from a CAD system. Furthermore, these packages usually can only generate tetrahedral or hybrid cells and user manipulation of the grid is often needed. In the present study, an internally developed automatic mesh generator based on the elliptic method18 is used to generate multiblock structured grids with high-quality hexahedral cells and with boundary orthogonality18 for all of the wall surfaces (Figure 1). A clustering O-grid is generated around the primary and secondary blades (Figures 1c and 1d) to efficiently capture the steep velocity gradients near these surfaces. The meridional mesh of the model is shown in Figure 1b, where in k- direction, 6 nodes are used for primary blade tip clearance, 17 nodes for the primary blade region, 7 nodes for the secondary blade region, and 5 nodes for the secondary blade tip clearance. The automatic mesh-generation code inputs data or curves directly either from the inverse design tool or from the parameterized geometric models. Thus it is more efficient, more flexible, and less time-consuming than the commercial automatic mesh-generation packages.
The CFD solver used in the design optimization system, STAR-CD, incorporates an element-based finite volume method suitable for unstructured and structured meshes with various available turbulence models. It solves steady-state or transient flow problems with single or multiple frames of reference, including sliding mesh simulation for turbomachinery and rotary blood pumps.
A low Reynolds number κ−ε turbulence model20 is selected for the turbulent simulation in this study based on a Reynolds number calculated with the impeller tip speed and its outlet diameter (Re = ρU2D2/μ = 29,714). This selection particularly takes account of the possible relatively lower Reynolds number at the gap region. The steady-state flow is solved based on the multiple frames of reference with an implicit coupling method to account for the differing rotations,20 particularly for the flow in the front blade tip clearance and the back clearance. Approximately 1.5 million cells are used for the entire pump model with a grid independency check performed for all the cases in the optimization loop to ensure a reasonable CPU time and appropriate density of grid cells. The y+ values for the entire computational model are also monitored by postprocessing for each case and they are found to be < 1.6.
Although blood exhibits non-Newtonian behavior at very low shear rates, studies21,22 have shown that blood can be modeled as a Newtonian flow at shear rates larger than the threshold of 100 s−1. According to our CFD results, the maximum shear stress in the secondary blade tip clearance is about 800 Pa, and the mean shear stress is about 500 Pa. Also, according to previous work8 for PIV measurement of a scaled-up version of the PediaFlow VAD, the flow in the gap was proven to be turbulent. Accordingly, in this study, blood is modeled as a Newtonian fluid even in the gap regions with a constant viscosity of 0.0035 Pa-s and a density of 1,040 kg/m3.
The primary flow path of the original design, with six primary blades as shown in Figure 1, had been optimized by our CFD-based design system for the maximum effiency and minimum blood damage. However, as shown in Figure 2, there is inward and outward leakage flow at the outlet of the back gap. An objective for this study is to eliminate the unfavorable retrograde leakage flow. If the net antegrade leakage flow, QS_net, is taken as a target function, it has a relationship with the main geometric parameters of, for instance, the secondary blades and the back gap as follows:
where Ns is the number of secondary blades, βs1 and βs2 are the leading and trailing angle, respectively, θs is the wrap angle, hs is the height, ws is the width, ls is the length, and the δhs is the blade tip clearance.
However, QS_net also affects the pump performance, axial thrust, and the blood damage. This relationship can be expressed as:
where H is the pump head, η is the efficiency, Tf is the axial thrust, and NIH is the index of hemolysis.
From the above discussion, it is apparent that there is a multitude of geometric and performance variables affecting the target function QS_net. It would be extremely difficult and very time consuming if a traditional trial-and-error approach is used to realize such a challenging objective. This problem can also not be solved merely by using CFD analysis alone. A CFD-based design optimization system has the potential to address this challenge, although a number of iterative loops, in which modifications of the secondary blade geometry are made and their effects on the target function QS_net evaluated according to Equation 7. Three representative cases are summarized in Table 1: original design without secondary blades, an interim result with 12 secondary blades, and the optimized one with 18 secondary blades.
Results and Discussion
Figure 2 demonstrates the complex three-dimensional flow structure within the gap captured by CFD for case 1. Although a net unfavorable retrograde leakage flow from gap outlet to inlet without distinguishable vortex is observed at the back gap within one r-z meridional section as shown in Figure 2a, Figure 2b displays complex and interesting “inward” and “outward” flow patterns at the outlet of the gap, and Figure 2c illustrates a distinguished vortex at the lower part of the back gap at another meridional section. The strength of the vortex depends on the balance between the centrifugal force caused by the rotational disk and the unfavorable pressure gradient imposed by the fluid from the outlet of the primary blades. Such an interesting “inward” and “outward” leakage flow pattern was confirmed by our flow visualization testing for a scaled-up version of the PediaFlow pump as shown in Figure 3.
Slots have been used in industrial pumps and blood pumps to balance the axial thrust.10,11 Different combinations of numbers and dimensions of slots were investigated in this study to mitigate the “inward and outward” leakage flow patterns. Although blood flows radially outwards within the slots, at the side gap, blood does not flow outwards towards the outlet as expected. Although the volume rate of retrograde flow reduces slightly with an increase in the number of slots, their effectiveness in inducing net antegrade flow is still not satisfactory.
Secondary blades/radial ribs are usually more effective than slots to balance the axial thrust.7 Because the radial blades can generate a higher pressure rise than the primary blades, which in most cases use backward type vanes with blade angles < 90 degrees, they are optimized by our CFD-based design optimization system to generate the net antegrade leakage flow as expected. In the iterative optimization loops, an intermediately optimized result for 12 radial (90-degree) secondary blades with a height and width of 0.2 × 0.2 mm, located along the back surface of the rotor is reported here. Figure 4 displays the computed pressure distribution on the surfaces of the six primary blades and on the surfaces of 12 radial secondary blades and the rotor outer surfaces as well. For a better view, the back shroud of the primary impeller is not shown in this figure. Compared to the curved backward type primary blades, the pressure distribution on the blade-to-blade region and outlet of the secondary blades are quite nonuniform although the mean pressure near the outlet of the gap is higher than the mean pressure at the outlet of the primary blades. As illustrated in Figure 5, the higher mean pressure at the outlet of the gap leads to a net antegrade flow with only localized areas of retrograde flow. Figure 6 shows the velocity vectors at the midspan cut-plane of the 12 secondary blades. As commonly found in the blood pumps with 90-degree straight blades,23 a strong passage vortex with a rotational direction opposite to that of the impeller is predicted by CFD. The vortex is developed partially due to the large flow incidence angle at the blade leading edge, and primarily due to the rotational effects and the blade shape itself which forms a more diffusive flow passage than the normally used backward-curved blades. Although the vortex can be washed out by the flow, it may be a source for increased risk of mural thrombosis.
By increasing the number of the secondary blades to 18 and extending them along the outer axial surface of the rotor to a blade trailing-edge angle of 135 degrees through the loop of optimization, the pressure distribution within the blade-to-blade passage and outlet of the secondary blades becomes more uniform (Figure 7). The mean pressure near the outlet of the gap also becomes higher than that in case 2 due to the functioning of the axial part of the secondary blades. Consequently, a net antegrade flow with minimal zones of retrograde flow is achieved (Figure 8). Due to the increase in the number of the secondary blades and their extension to the axial surfaces of the rotor, the strength of the passage vortex becomes weaker and its center shifts towards the outer ring of the rotor (Figure 9).
Figure 10 shows the computed net leakage flow through the back clearance gap for the three cases listed in Table 1. For the original design without secondary blades, the negative net leakage flow implies a retrograde blood flow dominated in the back clearance gap. Retrograde blood flow is undesirable, because a portion of the blood flowing through the pump makes two or more passes through the primary blades before exiting the pump—not only potentially causing cellular trauma and/or thrombosis, but also impacting the pump head and efficiency. Although a small amount of positive net leakage flow, as in case 2, does not exclude the possibility of some volume of the retrograde flow in the back clearance gap, it is more favorable than the net negative flow, as in case 1, to increase the pump head. A net antegrade leakage flow with very minimal retrograde flow, as in case 3, is desirable for the pressure rise while not causing excessive disk friction losses.
For our miniature magnetically levitated centrifugal PediaFlow pump, the axial thrust is balanced by the suspension system. It is one of the objectives to optimize the pump design to minimize both the radial and axial hydro-dynamic forces. Figure 11 displays the computed axial thrusts for the three cases, where negative axial thrust means a direction to the impeller inlet and a positive value means a direction towards the back of the rotor. For the original design without secondary blades, the axial thrust is highest. When the number of the secondary blades increases to 12, the axial thrust is almost balanced. With the further increase of the number of secondary blades, the axial thrust increases.
Several mathematical models of blood damage have been integrated into our CFD-based design optimization system to estimate the mechanically induced blood damage.24,25 For this study, CFD analyses revealed that under a very small blade tip clearance gap about 20 μm such as in cases 2 and 3, the shear stress becomes very high. James et al.26 reported that for a clearance gap in the range of 75 μm to 215 μm, there was minimal range of hemolysis. This may be attributed to the reduced dwell time of the cells through the gap and the exclusion of the red blood cells in such tiny gap regions as described by Fåhraeus.27 Because there are no blood-damage models directly applicable to the cases investigated in the present study, the estimations for blood damage are not reported here.
A novel CFD-based design optimization system, which integrates the three-dimensional inverse design approaches, parameterized geometry models, automatic mesh generator, and mathematical blood damage models, is presented in this study. The system enables a very efficient improvement and optimization for the impeller geometry and other components in arbitrary types of pumps. It has been applied to our miniature pediatric centrifugal blood pump, PediaFlow, to eliminate the undesirable adverse leakage flow in the back clearance gap. CFD analyses revealed that a very complex three-dimensional flow structure with significant retrograde flow occurs in the back clearance gap for the original design. Through design optimization including extensive CFD analyses, we found that secondary blades located along the back or extended to the side surfaces of the rotor have the capacity to reduce and eliminate the retrograde flow in the back clearance gap. With 18 geometrically optimized secondary blades with 90-degree blade angles at the radial part and a 135-degree blade trailing angle, a net antegrade flow with minimal zones of retrograde flow was achieved. The CFD analyses further revealed that the secondary blades can more effectively balance axial thrusts compared to the original design without secondary blades.
Because there are no blood-damage models applicable to the present problem, the estimation of shear-induced blood damage is not presented in this report. However, blood-shearing experiments for flow through these very narrow gaps are being conducted by our group, and these results will be reported in a future publication.
This study was supported in part by NIH Contract HHSN268200448192C (NO1-HV-48192), “Pediatric Circulatory Support,” to the University of Pittsburgh. The authors wish to thank Dr. Marina Kameneva for very helpful discussions regarding hemodynamics and hemorheology in narrow gaps.
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