Heuristic optimization of impeller sidewall gaps-based on the bees algorithm for a centrifugal blood pump by CFD

Abstract

Optimization studies on blood pumps that require complex designs are gradually increasing in number. The essential design criteria of centrifugal blood pump are minimum shear stress with maximal efficiency. The geometry design of impeller sidewall gaps (blade tip clearance, axial gap, radial gap) is highly effective with regard to these two criteria. Therefore, unlike methods such as trial and error, the optimal dimensions of these gaps should be adjusted via a heuristic method, giving more effective results. In this study, the optimal gaps that can ensure these two design criteria with The Bees Algorithm (BA), which is a population-based heuristic method, are investigated. Firstly, a Computational Fluid Dynamics (CFD) analysis of sample pump models, which are selected according to the orthogonal array and pre-designed with different gaps, are performed. The dimensions of the gaps are optimized through this mathematical model. The simulation results for the improved pump model are nearly identical to those predicted by the BA. The improved pump model, as designed with the optimal gap dimensions so obtained, is able to meet the design criteria better than all existing sample pumps. Thanks to the optimal gap dimensions, it has been observed that compared to average values, it has provided a 42% reduction in aWSS and a 20% increase in efficiency. Moreover, original an approach to the design of impeller sidewall gaps was developed. The results show that computational costs have been significantly reduced by using the BA in blood pump geometry design.

Keywords

Centrifugal blood pump CFD optimization the bees algorithm impeller sidewall gaps

Introduction

Heart failure (HF) affects nearly 26 million people worldwide.¹ HF is a significant and growing medical and economic problem with high prevalence rates worldwide.^2,3 It is not only a disease of the elderly, but also a public health problem that occurs across all ages and genders, and concerns all segments of society⁴ HF does have an increasing prevalence with age,⁵ so the number of patients suffering from HF is predicted to increase dramatically over the next decade due to the aging population.⁶ People with HF, which is increasing every year around the world, impose a growing burden on health systems.⁷ The worldwide economic burden of HF is estimated at $108 billion annually.⁸ HF is a disease condition in which the heart cannot adequately pump the blood needed by the tissues.⁹ Heart support devices including blood pumps help patients with HF by providing temporary hemodynamic support until recovery or heart transplantation or by otherwise providing destination treatment.^10,11 However, the use of heart support devices is limited because of the high development costs.¹² For this reason, shape optimization of blood pumps is important because it allows for a reduction in high simulation and experimental costs in a short time before clinical application.

These devices operate at a high and wide rotation speed.¹³ This high rotational speed and the narrow channels between impeller and pump housing cause shear stresses.¹⁴ There is a relationship between shear stress and blood damage. In rotational blood pumps, shear stress is the main cause of blood damage.¹⁵ To predicting such damage, the ability to evaluate the shear stress is paramount.¹⁶ Faghih and Sharp¹⁷ described how red blood cells are deformed under different shear stresses. In many studies in the literature the effects of the clearance gap size is considered to be the most important parameter affecting blood damage.^18,19 Moreover, Wiegmann et al.²⁰ found that the size of the clearance gap has significant effects not only on blood damage, but also on the hydraulic efficiency of the pump. They examined the clearance gap size at four different points in the range 50–500 µm finding that as the clearance increases, hydraulic efficiency output decreases. Although studies have been conducted on these effects of impeller sidewall gaps, there is little work in the literature on the optimization of blood pumps.^21–24 In those few published studies, an improved version of the investigated pump is presented instead of the optimized version with heuristic or metaheuristic algorithms.^25,26 Zhu et al.²¹ presented a study using 14 variable points to optimize the camber line of the diffuser blade. Ghadimi et al.²³ studied the use of a genetic algorithm model, introducing an innovate simultaneous optimization of the impeller and the volute. Tesch and Kaczorowska-Ditrich²⁴ investigated wall shear stresses related to angular velocities in axial flow blood pumps via the differential evolution method.

In this study, we first find the optimum sizes of the impeller sidewall gaps of a centrifugal blood pump via the heuristic BA. The CFD-based optimization of the centrifugal blood pump impeller and volute geometries still includes major challenges when comparing an automatic search with a heuristic algorithm. The main novelty of this work, as different from previous studies, is that the optimal impeller sidewall gaps search is performed using a popular heuristic algorithm via a mathematical model instead of the manual and limited variation methods.

Pump modelling

Preliminary design

In this study, a centrifugal-type pump was specified to work on, because this form of pump has flatter H-Q characteristics, and wider range of peak efficiency than axial pumps for medical applications.^25,27 The pump is a rotary pump with continuous flow since continuous flow pumps have certain advantages such as; lower blood damage, smaller size, lower filling volume, better transportability and absence of spallation.¹⁵ The base pump is a modified version of a LVAD designed by Incebay and Yapici²⁸ which is designed to reach 100 mm-Hg pressure rise at a 5 L/min flow rate.²⁹ The base pump was formed by cutting the shroud of the impeller of the previous pump to examine the effect of the blade tip clearance on hydraulic efficiency and aWSS.

In order to simplify the procedures, the effect of exposure time is neglected and only wall shear stress is considered. Since the application to be made in this optimization study is comparison of the pump samples, WSS was considered to be sufficient to this end. In this way, what would otherwise be an intensive workload is reduced dramatically and optimal solutions are obtained more efficiently.

Fixed design parameters for the impeller and volute are calculated according to industrial design method.³⁰ Fixed design parameters are given in Table 1. These parameters are used to generate the base pump.

Table 1.

Fixed design parameters of the base pump.

Fixed design parameters	Symbol	Value
Number of vanes	Z	6
Inlet diameter	D ₁	15 mm
Outlet diameter	D ₂	36 mm
Vane inlet height	b ₁	6.5 mm
Vane outlet height	b ₂	1.6 mm
Vane inlet angle	β₁	13°
Vane outlet angle	β₂	26°
Vane thickness	e	0.5 mm

Geometry parametrization

The impeller sidewall gaps shown in Figure 1(b) are determined as optimization parameters of the study. Four different possibilities were used for each gap, namely 50, 100, 150, and 300 µm. Four levels for the optimization parameters are given in Table 2. Three optimization parameters with four levels give total pump population of 4³ = 64. To reduce the simulation time for all 64 pumps, 16 sample pumps are selected by using L₁₆ orthogonal array. Solid models of each of these sample pumps are generated and simulated.

Figure 1.

(a) All fluid domains and (b) geometry representation of impeller sidewall gaps.

Table 2.

Levels of optimization parameters.

Optimization parameters	Symbol	Dimension	Levels
Optimization parameters	Symbol	Dimension	1	2	3	4
Blade tip clearance	e _t	µm	50	100	150	300
Axial gap	e _a	µm	50	100	150	300
Radial gap	e _r	µm	50	100	150	300

CFD simulations

Fluent 19.0 (Ansys Inc.) is used for simulations. The fluid domain for computational simulations is shown Figure 1(a) Fluid regions inside the impeller and volute are meshed, whilst solid zones are suppressed. Mesh elements are tetrahedral and the element order is linear. Face meshing is used at the channel walls of the impeller sidewall gaps and the minimum mesh size is set to be 50 µm. A mesh sensitivity test was performed from which it could be seen that increasing the cell count to more than 1.7 million did not affect the solution in any meaningful manner. Even though blood is a non-Newtonian fluid, many researchers model blood in a Newtonian fluid, and which has a 0.0035 Pa.s viscosity and 1050 kg/m³ density.^31,32 The energy equation is not activated since the effects of heat are neglected. For the inlet, the mass flow rate is set to be 0.0875 kg/s. For the turbulence model, k-ω SST is used in a similar manner to many other researchers.³³ The convergence criteria are set as 10⁻⁵. For discretization, the Coupled Algorithm was selected and all equations set for Second Order Upwind.³⁴ CFD analyses for all sample pump models were carried out at a 3300 rpm rotation speed since the base pump meets the design criteria (5 L/min flow rate 100 mm-Hg pressure rise) at this speed. The pump speed and flow rate could also be design parameters in the optimization procedure, but have been fixed to reduce the computation cost.

The parameters used to determine the pump performance are calculated as follows:

Mechanical power;

P_{mech} = T * ω

(1)

Fluid power;

P_{h y d r} = Δ P_{t} * Q

(2)

Pump hydraulic efficiency;

η_{h} = \frac{P_{h y d r}}{P_{mech}}

(3)

Hydraulic efficiency is a function of mechanical power, P_mech, which is multiplication of torque, T (Nm), and angular velocity ω (rad/s) and fluid power P_hydr, which is multiplication of pressure head, ∆Pt (mmHg) and flow rate, Q (m³/s).

The results of aWSS (Pa) and hydraulic efficiency for 16 sample pump models are shown in Table 3.

Table 3.

aWSS (Pa) and hydraulic efficiency (%) results for the 16 sample pump models.

No	Optimization parameters			aWSS (Pa)	Hydraulic efficiency (%)
No	e_t (µm)	e_a (µm)	e_r (µm)	aWSS (Pa)	Hydraulic efficiency (%)
1	50	50	50	193.82	36.74
2	50	100	100	129.68	42.39
3	50	150	150	113.95	44.07
4	50	300	300	82.43	47.91
5	100	50	100	205.45	35.71
6	100	100	150	141.09	41.36
7	100	150	300	125.75	42.92
8	100	300	50	93.26	46.85
9	150	150	150	216.51	34.41
10	150	100	300	153.40	39.59
11	150	150	50	136.60	41.45
12	150	300	100	105.03	45.11
13	300	50	300	216.01	30.03
14	300	100	50	152.65	35.10
15	300	150	100	136.45	36.48
16	300	300	150	104.63	39.83
Average				144.17	39.98

Mathematical model

The Minitab program was used for the derivation of mathematical equation. The CFD simulation results for the 16 sample pumps (Table 3) are defined in Minitab. While nonlinear regression analysis was performed for aWSS, linear regression analysis was performed for efficiency. For aWSS, the regression analysis is performed with data obtained from simulations, and a mathematical equation (as shown in equation (4)) is obtained. It is observed that there is good agreement between the simulation results and the results obtained from the mathematical equation (R² = 0.93) Similarly a regression analysis is conducted for efficiency and a second mathematical equation (as shown in equation (5)) is obtained. There is again good agreement between the simulation results and the those obtained from the mathematical equation for efficiency. (R² = 0.91)

τ_{a w s s} = e_{t} * e_{a} (K_{1} + K_{2} * e_{t} + K_{3} * e_{t} * e_{a} + K_{4} * e_{t} * e_{r})

(4)

η_{h} = C_{1} + C_{2} * e_{t} + C_{3} * e_{a} + C_{4} * e_{r}

(5)

In these equations, the coefficients can be evaluated as K₁ = 3.53392, K₂ = −0.30597, K₃ = 0.00991, K₄ = 0.0000983, C₁ = 38.05, C₂ = −0.02978, C₃ = 0.04125, and C₄ = 0.00074, as determined from regression analysis.

Optimization of the BA

The BA, developed by Pham et al.³⁵ is a population-based, heuristic search algorithm that mimics the resource (water, nectar etc.) search behavior of honey bees and the sharing of information (direction, distance, and amount of the resource etc.) with each other. The BA is both a local and global optimization method combined with a random neighborhood search. In the algorithm, the population consists of scout and recruit bees that perform different tasks. Scouts bees search for new resources independently without using any information, while recruit bees transfer information about the resource to other bees in the hive with a movement that called “waggle dancing.” For further details, the reader is referred to Pham et al.’s.^36,37 studies. Optimization approaches with the BA are still applied in the solution of actual engineering problems.³⁸

In addition, the authors have conducted many other studies on controller design with the BA. To select appropriate BA parameters, the basic approaches and the previous studies of the authors are used.^39–43 The determination of the values of the parameters in different configurations does not have a significant effect on obtaining optimal results; rather, the main purpose is to design the function in accordance with the objective function of the optimization. These parameters are generally determined by trial-and-error methods based on experience, taking into account the computing load and time.⁴⁴ The BA requires a number of parameters to be set as given Table 4. The general flow-chart of this study and the pseudo code for the BA are showed in Figure 2.

Table 4.

The BA Parameters.

The BA parameters	Value	Description
n	200	Number of scout bees
m	50	Number of sites selected out of n visited sites
e	20	Number of best sites out of m selected sites
nep	5	Number of bees recruited for best e sites
nsp	10	Number of bees recruited for the other (m-e) selected sites
ngh	1	Initial size of patches ngh which includes site and its neighborhood
itr	300	Iteration number (stopping criteria)

Figure 2.

The BA optimization flow chart.

The objective function convergence performances of the BA algorithm are graphically illustrated in Figure 3. As seen obviously, the BA algorithm has well-converged solutions at the end of a global and local search. The bees algorithm is programmed in MATLAB and run on an Intel(R) Core (TM) i7–10875H CPU 2.30 GHz PC with 16.0 GB memory.

Figure 3.

The performances on reducing the cost function of the BA.

The main motivation of the optimization study is to be able to determine the optimal gaps that can guarantee the highest hydraulic efficiency despite the least aWSS. The objective function, which has critical importance in achieving this aim, has been designed in such a way that its reduction will reduce the aWSS and increase hydraulic efficiency. The performance criterion for BA optimization is to minimize the proposed objective function as quickly as possible. Moreover, in order for a unit change in aWSS and hydraulic efficiency to affect the objective function at the same rate (or desired rate), these two variables must be weighted relative to each other with constant coefficients, taking into account the possible ranges values. Thus, even small changes in aWSS or hydraulic efficiency values act on scale to the objective function over iterations. The cost function (J) as determined based on the authors’ knowledge and achieved and described in previous papers^39–43 is:

J = \frac{w_{1} \times a W S S}{w_{2} \times η_{h}}

(6)

The weighted gain constants (w₁ = 45, w₂ = 145) are determined by taking into account the average aWSS and hydraulic efficiency (Table 3) of the 16 sample pump models, and it is desired that the effect of aWSS on the objective function is slightly greater. The proposed objective function is more successful in terms of optimal results than traditional gap design methods.

Results

Simulations on the sample pump models showed that increasing axial gap (e_a), decreases aWSS and increases hydraulic efficiency as seen in Table 3. Both of these changes are attempts at meeting both design criteria. This situation can be explained in the sense that as the distance between stationary wall of the volute and moving wall of the impeller increases, friction effects caused by shear stress decrease dramatically. Hence, the axial gap can be as wide as possible within the limits of the pump’s dimensions. According to the regression analysis, the radial gap (e_r) has very little effect on aWSS and efficiency relative to the other gaps. Most effective gap on aWSS is blade tip clearance (e_t). As the clearance increases, so does aWSS and as the clearance value decreases, the hydraulic efficiency increases. The pressure head decreases significantly as the blade tip clearance increases. This significantly affects the hydraulic efficiency of the pump model. Similarly, Kim et al.⁴⁵ found that as the clearance gap increases from 50 to 200 µm, the shear rate increases. Also Wiegmann et al.²⁰ found that as clearance gap increases, low wall shear stress area decreases. Furthermore, as blade tip clearance increases, hydraulic efficiency decreases, because the increase in the tip clearance decreases the pump head.

The optimization study gives an improved pump with optimal sidewall gaps as seen in Table 5: e_t = 55.6 µm, e_a = 298.5 µm, and e_r = 182.4 µm. The estimated aWSS of the improved pump model is 83.4 Pa and hydraulic efficiency is 48.8% with a cost function that converged to 0.62 from 2.28 at the end. This improved pump’s CAD model was generated and simulated using Fluent 19.0. In the CFD simulations, the hydraulic efficiency of the improved pump was found to be 48.02% and aWSS was found to be 82.29 Pa which are nearly the same as predicted by the BA.

Table 5.

The performances on reducing the cost function of the BA.

Iterations number	Blade tip clearance (μm)	Axial gap (μm)	Radial gap (μm)	aWSS (Pa)	Hydraulic efficiency (%)	Cost function (J)
1	265.9	53.9	92.2	268.8	32.4	2.28
20	134.1	76.3	293.6	167.4	37.4	1.23
40	123.6	106.4	154.4	143.3	38.8	1.01
60	216.1	170.2	219.3	123.3	38.7	0.88
80	58.2	150.3	61.0	114.2	42.5	0.81
100	178.6	237.7	89.8	102.1	42.6	0.75
120	106.6	225.4	259.0	91.3	44.3	0.72
140	120.9	274.8	201.8	87.7	45.9	0.68
160	56.7	295.8	120.0	85.1	48.6	0.63
180	57.8	296.9	200.5	84.5	48.7	0.63
200	55.6	298.5	182.4	83.4	48.8	0.62
225	55.6	298.5	182.4	83.4	48.8	0.62
250	55.6	298.5	182.4	83.4	48.8	0.62
300	55.6	298.5	182.4	83.4	48.8	0.62

In Figure 4, the highest and lowest values of the aWSS and efficiency performances of the sample pumps models (Table 3) are compared with performance of the proposed improved pump according to classified blade tip clearance.

Figure 4.

The distribution of wall shear stress at different blade tip clearance (left column), hydraulic efficiencies and aWSS in the highest and lowest wall volute fluid domain according to blade tip clearance. (right column).

In terms of aWSS and hydraulic efficiency, according to the results shown in Table 3, the fourth pump gave the best performance. As can be clearly understood from Figure 4, although the performance of the improved pump model with optimal impeller sidewall gaps is superior to the highest performance that can be offered by the existing sample models, the performance of the fourth pump and the improved pump are very close.

The shape optimization of the blood pump is a challenging problem due to the high simulation cost.²³ While it could take days to gain simulation results for the entire pump population, optimal solutions found using a metaheuristic search algorithm can quickly and effectively give results in just minutes. Thus, compared to CFD analysis, which can take weeks, with less computational cost, faster foresight about the pump can be provided and the cost and time of experimental studies can be reduced. For this study, the convergence performance time of the objective function of the BA algorithm is less than 5 min with 11% CPU utilization capacity (approximately 85% in CFD analysis). Therefore, the improved pump using the optimization algorithm offers an incomparably fast and effective solution in terms of temporal costs.

Thus, the improved pump model ensures a reduction in aWSS from approximately 144 to 82 Pa and an increase in hydraulic efficiency from approximately 40% to 48%, as related to the average result for the sample pump models.

Conclusion

This study investigates the effects of optimal three impeller sidewall gaps on aWSS and efficiency which are the main performance indicator for the blood pump. An improved pump model with optimal sidewall gaps is proposed by carrying out the optimization of a mathematical model which is obtained from the CFD simulation data of 16 different sample pump models.

The results obtained from the optimization procedure are extremely similar to those of the CFD analyses. The CFD simulation results performed on the new improved pump model prove that adjusting the dimension of the gap using a population-based heuristic the BA assures improvement.

The optimal pump design ensures a 42% reduction in aWSS and 20% increase in efficiency in comparison with averages values of existing sample pumps which are designed with more limited conventional techniques. The results clearly indicate that the proposed method of using the BA in the design of the blood pump provides a marked reduction in computational cost. The BA can be used as a useful approach in the shape optimization of the blood pump.

Moreover, the optimization procedure proposed in this study can be implemented for various pump and fluid types (e.g. axial pump, non-Newtonian blood modelling) to improve performance through more realistic studies. In future studies, the results obtained here could be compared through the use of different algorithms.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ahmet Onder

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