Influence of turbulent shear stresses on the numerical blood damage prediction in a ventricular assist device

Abstract

The blood damage prediction in rotary blood pumps is an important procedure to evaluate the hemocompatibility of such systems. Blood damage is caused by shear stresses to the blood cells and their exposure times. The total impact of an equivalent shear stress can only be taken into account when turbulent stresses are included in the blood damage prediction. The aim of this article was to analyze the influence of the turbulent stresses on the damage prediction in a rotary blood pump’s flow. Therefore, the flow in a research blood pump was computed using large eddy simulations. A highly turbulence-resolving setup was used in order to directly resolve most of the computed stresses. The simulations were performed at the design point and an operation point with lower flow rate. Blood damage was predicted using three damage models (volumetric analysis of exceeded stress thresholds, hemolysis transport equation, and hemolysis approximation via volume integral) and two shear stress definitions (with and without turbulent stresses). For both simulations, turbulent stresses are the dominant stresses away from the walls. Here, they act in a range between 9 and 50 Pa. Nonetheless, the mean stresses in the proximity of the walls reach levels, which are one order of magnitude higher. Due to this, the turbulent stresses have a small impact on the results of the hemolysis prediction. Yet, turbulent stresses should be included in the damage prediction, since they belong to the total equivalent stress definition and could impact the damage on proteins or platelets.

Keywords

Computational fluid dynamics large eddy simulation rotary blood pump equivalent shear stress turbulent stress turbulence dissipation

Introduction

Ventricular assist devices (VADs) are technical solutions to improve the life situation of patients with severe heart failure. Nowadays, most VADs are designed as rotary flow pumps. These pumps consist of a rotary and a stationary part, where non-physiologically high shear stresses occur. These shear stresses $τ$ and their exposure times $t_{e x p}$ cause damage to the blood cells, which lead to serious adverse events in a VAD patient.¹ Therefore, investigations of VADs regarding their blood damage potential is of utmost importance, since the VAD therapy has become the most promising treatment for severe heart failure.²

Regarding this, flow simulations (computational fluid dynamics (CFD)) combined with blood damage prediction models are commonly used to estimate the cell damage in VADs, for example, to minimize the blood damage within the design process. This is mainly done by evaluating the computed shear stresses and their exposure times to the blood cells.

If those shear stresses are treated by Reynolds’ decomposition, the total, instantaneous, viscous shear stress tensor (bold symbols represent tensors) can be expressed by

τ_{t o t} = 〈 τ 〉 + τ^{'}

(1)

where $τ_{t o t}$ is the spatial- and temporal-dependent total shear stress, $< τ >$ denotes the mean shear stress, derived from the mean (time-averaged) flow field

〈 τ 〉 = τ_{m e a n} = μ (\frac{\partial 〈 u_{i} 〉}{\partial x_{j}} + \frac{\partial 〈 u_{j} 〉}{\partial x_{i}}) = 2 μ 〈 S_{i j} 〉

(2)

and $τ^{'}$ is termed as the fluctuating, turbulent shear stress³

τ^{'} = μ (\frac{\partial {u^{'}}_{i}}{\partial x_{j}} + \frac{\partial {u^{'}}_{j}}{\partial x_{i}}) = 2 μ {S^{'}}_{i j}

(3)

In the aforementioned equations, $u_{i}$ and $x_{i}$ are the ith components of the velocity and the position vector, $S_{i j}$ is the strain-rate tensor, and µ denotes the dynamic viscosity.

In this context, the question how turbulence damages blood cells is still not answered.⁴ Kameneva et al.⁵ showed in their pipe flow experiments that a turbulent flow could have a more harmful influence on hemolysis (damage to red blood cells (RBCs)) than a comparable laminar flow with similar wall shear stresses. Quinlan⁶ analyzed the experiment conducted by Kameneva et al. and revealed that the mean shear stresses $τ_{m e a n}$ in the experiment, taken alone, are not suitable to predict hemolysis, since these stresses were higher in the laminar pipe flow than in the comparable, turbulent case.

Morshed et al.⁷ used an energy balance in the pipe flow to show that there is a strong correlation between the flow variable total dissipation rate $ε_{t o t}$ and blood damage. Furthermore, Hund et al.³ showed that the energy dissipation can be connected to the (scalar) norm of the shear stress tensor. The total dissipation rate $ε_{t o t}$ characterizes the transfer of kinetic energy into internal energy in an instantaneous or time-averaged flow field. This transfer is caused by high-velocity gradients in the flow.⁸

In fact, many researchers have suggested to take the total dissipation rate $ε_{t o t}$ or a correlated flow quantity to study flow-induced blood damage. Some workers proposed to use the Kolmogorov length scale as a predictor for blood damage.^9–11 Jones, Li et al., and Yen et al. considered the scalar, turbulent viscous shear stress (TVSS), which can be built with the time-averaged turbulent dissipation rate $< ε_{t u r b} >$ , the viscosity µ, and the density $ρ$

〈 τ^{'} 〉 = T V S S = \sqrt{\frac{1}{2} ρ μ 〈 ε_{t u r b} 〉}

(4)

to evaluate blood cell damage.⁹,¹²,¹³ Although it was shown by these authors that this quantity seems to be more appropriate to assess blood damage than the Reynolds stresses, it is still questionable, why the theoretically or experimentally determined hemolysis thresholds of the TVSS seem to be lower than the laminar stress thresholds. Jones suggested that this is due to the treatment of TVSS in a time-averaged framework. He argued that the instantaneous (time-dependent) shear stresses which correlate with the instantaneous dissipation rates, were expected to be much higher and could cause damage to blood cells. The treatment of the instantaneous shear stress $τ_{t o t}$ as an indicator for mechanical stress on blood cells was also advised by Ge et al.¹⁴

Despite the many unknown issues, it is certain that turbulence has an impact on the blood cells. Hence, the stresses from the turbulent field should generally not be disregarded in the numerical damage prediction in a VAD’s flow.¹⁵,¹⁶

Nowadays, the flow simulations in VADs are commonly conducted using unsteady Reynolds-averaged Navier–Stokes (URANS) methods, in which a great part of the turbulent field is modeled.¹⁷ Furthermore, the viscous shear stresses in the flow field are commonly analyzed using equivalent, scalar representations,¹⁶,^18–25 which are based on the second invariant $I I_{S} = \sqrt{2 S_{i j} S_{i j}}$ of the strain-rate tensor $S_{i j}$ .³ These are, for example, the stress definition of Bludszuweit,²⁶ the van-Mises stresses²⁷ as well as the direct use of the second invariant²⁰ in the equivalent stress definition. It is shown in section “Methods” that these equivalent stress representations have three components in an unsteady flow simulation as follows: a mean, a resolved turbulent, and a modeled turbulent part. These parts exist in the instantaneous as well as the time-averaged flow field. All three components have to be considered in order to account for the total equivalent shear stress.

But, many CFD studies do not include all components in their investigation. Often, the contribution from the modeled turbulent stress is neglected, while the resolved ones are included.^18–22 By doing so, the impact of the turbulent stresses on the total equivalent stress is just partly considered in the blood damage estimation.

On this account, the first aim of this article is to state the three components of an equivalent shear stress definition. It is shown that the resulting total equivalent stress can be linked to a physical flow variable, the total dissipation rate $ε_{t o t}$ .

The second aim of this article is to quantitatively analyze the influence of the turbulent stresses on the blood damage prediction. With the derived total equivalent shear stress, three damage models were applied to the flow to numerically predict the blood trauma in the VAD. Afterward, the damage prediction was performed a second time with an equivalent stress definition, where just mean stresses were considered and the resolved and modeled turbulent stresses were omitted. Then, the percentage differences between both results were used to investigate the influence of the turbulent stresses on the damage prediction.

The simulations were performed using the large eddy simulation (LES) method. Idea of this method is to resolve the greatest part of turbulence (at least 90% of the turbulent kinetic energy)²⁸ directly in the flow field, while a small share of turbulence is modeled by a turbulence model. It is shown in this article that the performed simulations, in fact, directly resolve for the greatest share of the computed turbulent stresses. To realize this, a numerical setup with a spatially and temporally high discretization was used, which is explained subsequently.

Methods

Computational setup

Pump geometry

The flow field of an axial, rotary flow pump was investigated. This VAD has been designed in our institute. The design was inspired by axial flow VADs, which are currently in clinical use. The design principles are briefly explained as follows.

After choosing the inner (11 mm) and outer (16 mm) diameter, meridian lines were placed between hub and casing to set the blade angles for a chosen nominal operation point $(Q = 4.5 L / \min, n = 7900 r / \min)$ . Afterward, the wrap angle of the blades was adjusted to obtain an even blade progression. After that, the blade thickness and gap height (120 µm) were included. The outlet guide vanes were set that the swirl is reduced as much as possible. Finally, the coupled rotor and stator were adjusted using CFD, until the pump achieved a pressure head of $H = (70_{\dots} 80) mmHg$ and a hydraulic efficiency of $η_{h} \approx 33 %$ at the nominal operation point.

The VAD within the computational domain is illustrated in Figure 1(a). It consists of a two-bladed rotor, an inlet guide vane with five, slightly bended blades, to apply a counter-swirl, which theoretically leads to an increased pressure head for a specific rotational speed, and a three-bladed outlet guide vane.

Figure 1.

Numerical setup: (a) computational domain and (b) sections of the LES mesh. Left: surface mesh on the hub and the blades. Right: cut-plane through the volume mesh near the rotor’s leading edge.

This VAD was designed for research solely. Hence, a bearing concept for a clinical use was not considered in the design concept. Furthermore, no axial gaps between the rotor and the stators were included in the original design.

The aim of the design was not to build a “perfect” implantable VAD with an optimal flow behavior, but rather a pump, in which the same flow pattern can occur as in real implanted VADs. In these pumps, the inflow angle cannot be optimal during the entire operation time due to the varying input from the remaining heart. Thus, non-optimal inflow angles and flow paths were deliberately accepted at the nominal operation point.

Meshing

The mesh was created in ICEM CFD 18 (Canonsburg, PA, USA) using a block structure with hexahedral elements. Parts of the mesh are displayed in Figure 1(b). The mesh has a size of approximately 100M elements. For mesh construction, literature recommendations for wall-resolving LES²⁹,³⁰ were applied. In doing so, a near-wall mesh was built, which fits the upper limits of $Δ_{x}^{+} = (Δ_{x} \cdot {(τ_{w} / ρ)}^{0.5}) / ν = 50$ and $Δ_{z}^{+} = 15$ in the flow- and span-wise direction, respectively. The dimensionless mesh distance $Δ^{+}$ is calculated by the viscosity $ν$ , density $ρ$ , and the wall shear stress $τ_{w}$ . The first wall node had a dimensionless wall distance of $y_{1}^{+} \leq 1$ and the mesh growth factor was $r_{g} = 1.05$ . Higher $y_{1}^{+}$ values of ≈ $1.5$ were tolerated at small locations of the leading edges to prevent disproportionally larger mesh sizes. Grid angles were larger than 23° and maximum aspect ratios were about 40 in the rotor. The gap between the rotor and the casing was discretized with 28 and 74 elements for the height and width.

CFD model and simulation setup

ANSYS CFX 17 (Canonsburg, PA, USA) was used to solve the Navier–Stokes equations by a finite volume method.³¹ The differential equations were discretized by a bounded central difference scheme in space and a second-order scheme in time. The rotary and stationary parts of the VAD were connected by transient sliding interfaces. Blood was modeled as a Newtonian fluid with a viscosity of $μ = 3.5 mPa s$ and a density of $ρ = 1050 {kg / m}^{3}$ . This simplification is adequate in blood pumps, since the non-Newtonian character of blood is negligible at high shear rates.³²

The simulations were performed at two operation points with flow rates of $Q = 4.5 and 2.5 L / \min$ . A VAD operates in various flow conditions, under residual cardiac pulse. Hence, it seems reasonable to consider more than one operation point for this study.

The aforementioned flow rates were defined at the outlet of the computational domain and a zero total pressure was set at the inlet. The dynamic Smagorinsky model was applied for turbulence modeling. This model is advantageous compared to standard Smagorinsky models for VAD simulations, since its application is automatically valid for laminar flow regimes and no near-wall corrections are needed.³³

No turbulent inflow conditions were defined at the inlet of the computational domain. Theoretically, the inflow field at $Q = 4.5 L / \min$ can be transitional, since the inflow Reynolds number, based on the area-averaged velocity and inlet diameter, is $R e = 2537$ . This value is comparable to the critical Reynolds number of ≈ $2300$ for pipe flows. But, the development of turbulent flow structures is strongly dependent on flow disturbances in this Reynolds number region.³⁴ This means that no turbulent effects are present in a pipe flow without disturbances, even beyond the critical Reynolds number. A straight, hydraulically smooth pipe was assumed for the inlet of the simulation. Hence, no turbulent phenomena are expected in this region.

The rotor speed was 7900 r/min, and the time step equals a rotational increment of 1°. Time-averaging was done for 10 revolutions after Root Mean Square (RMS) residuals were smaller than $5 \times 10^{- 5}$ and monitored variables, like velocities at certain points, the pressure head, or the hydraulic efficiency, showed a statistically steady behavior.

Experimental examination of pressure heads

The pressure heads via the VAD were measured for comparison with the numerically determined pressure heads. The VAD casing and cannulas were built by milling of two acrylic blocks (Figure 2(a)). The hydraulic components in both blocks, namely, the guide vanes and the rotor, were also constructed by milling of acrylic material. All wetted parts of this measurement cell were hand polished and have tolerances of ≈10 µm. Furthermore, axial gaps of 100 µm had to unavoidably be included between the rotary and stationary parts in the measurement cell, which were not considered in the numerical flow simulations. A motor shaft was integrated into the experimental setup through the outlet guide vane and the outflow cannula. Preliminarily performed URANS simulations indicated that the rotating shaft has a neglectable influence on the pressure measurements. The shaft is fixed through three ball bearings in the acryl block as well as in the outlet guide vane. An electric motor (3564K024B; Faulhaber, Schönaich, Germany) was used as the drive. The measurement cell was included in a test bench (Figure 2(b)). The measurement cell and test bench were constructed according to the guidelines of the DIN EN ISO 9906:2012 for hydraulic measurements in turbomachines.³⁵ The pressures were measured by two relative pressure sensors (sensor 1: DMU 0,25 ES; Landefeld, Kassel, Germany; sensor 2: P-31-R; WIKA, Klingenberg, Germany) at four positions along the perimeter in the outflow cannula behind and in the inflow cannula before the VAD. A water–glycerol mixture was used as the working fluid. The mixture was adjusted in the way that a dynamic viscosity of $μ = 3.5 mPa s$ was achieved.

Figure 2.

Experimental setup: (a) measurement cell and (b) test bench.

Methods for the analysis

Viscous shear stress formulation and consideration of turbulent stresses

The viscous shear stress is a tensorial quantity and is derived from the velocity gradients of the flow. In complex medical devices, it is common to replace the tensor by an equivalent shear stress formulation.²⁴

For analyzing the blood damage in VADs, a stress definition based on the second invariant of the strain-rate tensor $S_{i j}$ is often used

τ_{s s r} = μ \sqrt{2 S_{i j} S_{i j}} w i t h S_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})

(5)

The term within the square root can be directly related to the total dissipation rate $ε_{t o t}$ of an instantaneous flow field²⁹

ε_{t o t} = 2 ν S_{i j} S_{i j}

(6)

By implementing this variable in the stress formulation, assuming a statistically steady flow field and applying the Reynolds decomposition, $τ_{s s r}$ can be written as

\begin{array}{l} τ_{s s r} = \sqrt{ρ μ ε_{t o t}} \\ = \sqrt{ρ μ (2 ν 〈 S_{i j} 〉 〈 S_{i j} 〉 + 4 ν 〈 S_{i j} 〉 {S^{'}}_{i j} + 2 ν {S^{'}}_{i j} {S^{'}}_{i j})} \end{array}

(7)

In the design process of a VAD using numerical simulations, the flow field is often not only analyzed in an instantaneous but also in a time-averaged manner. This can be done by averaging equation (7) within the square root (as proposed by Hund et al.).³

\begin{array}{l} 〈 τ_{s s r} 〉 = τ_{t o t} \\ = \sqrt{ρ μ 〈 ε_{t o t} 〉} \\ = \sqrt{ρ μ (2 ν 〈 S_{i j} 〉 〈 S_{i j} 〉 + 2 ν 〈 {S^{'}}_{i j} {S^{'}}_{i j} 〉)} \\ = \sqrt{ρ μ (ϵ_{d i r} + 〈 ε_{t u r b} 〉)} \end{array}

(8)

The crossterm $4 ν < S_{i j} > {S^{'}}_{i j}$ from equation (7) is removed in equation (8) due to the averaging process.³ The resulting time-averaged, total equivalent stress from equation (8) can be correlated with two flow variables. The first one is the direct dissipation $ϵ_{d i r}$ , which describes the energy loss within the flow due to time-averaged velocity gradients.⁸ The second term is the time-averaged, turbulent dissipation $< ε_{t u r b} >$ and represents the energy loss due to fluctuating velocity gradients in the time-averaged flow.

In numerical simulations, the equivalent stress from equations (7) and (8) could only be obtained properly in direct numerical simulations (DNS). But, in most flow simulations within VADs, the fluctuating turbulent flow is entirely or partially modeled by turbulence models. Hence, the turbulent stress component from a turbulence model must be taken into account when analyzing the total equivalent stress field. Following the connection between shear stresses and total dissipation rate from equation (7), the modeled turbulent dissipation rate $ε_{m o d}$ from the respective turbulence model must be inserted into the stress formulation. For the instantaneous, total equivalent stress $τ_{s s r},$ which will be used for the hemolysis estimation via transport equations, this reads as follows

\begin{array}{l} τ_{s s r} = \sqrt{ρ μ (ε_{t o t} + ε_{m o d})} \\ = \sqrt{ρ μ (ϵ_{d i r} + 4 ν 〈 S_{i j} 〉 {S^{'}}_{i j} + 2 ν {S^{'}}_{i j} {S^{'}}_{i j} + ε_{m o d})} \end{array}

(9a)

and for the time-averaged, total equivalent stress $< τ_{s s r} >$ , the formula is

\begin{array}{l} 〈 τ_{s s r} 〉 = \sqrt{ρ μ (〈 ε_{t o t} 〉 + 〈 ε_{m o d} 〉)}) \\ = \sqrt{ρ μ (ϵ_{d i r} + 〈 ε_{t u r b} 〉 + 〈 ε_{m o d} 〉)} \end{array}

(9b)

As shown in the aforementioned equation, the total equivalent stress definition contains three components. The contribution from the mean flow (marked in red), from the resolved turbulent flow (marked in blue), and from the modeled turbulent flow field (marked in green).

The direct dissipation rate $ϵ_{d i r}$ is defined as follows

ϵ_{d i r} = 2 ν 〈 S_{i j} 〉 〈 S_{i j} 〉

(10)

This variable is connected with the mean shear stress $τ_{m e a n}$ (equation (11)), which is also present in a steady laminar flow and origins from the velocity gradients of the mean (time-averaged) flow. The blue- and green-colored terms in equations (9a) and (b) arise from the resolved and modeled velocity fluctuations and can be assigned as the contribution of the turbulence to the total equivalent stress formulation. Both these terms are caused by turbulent fluctuating gradients and can consequently be designated as turbulent stresses.

To investigate the influence of these turbulent stresses on the total equivalent stress field and subsequent on the damage prediction, the mean shear stress formulation $τ_{m e a n}$ was built, which excludes turbulent, fluctuating components

τ_{m e a n} = \sqrt{ρ μ ϵ_{d i r}} = μ \sqrt{2 〈 S_{i j} 〉 〈 S_{i j} 〉}

(11)

With the equivalent stress formulations from equations (9a) and (b) and (11), the blood damage was predicted within the VAD.

Numerical models for the blood damage prediction

Three blood damage prediction models in an Eulerian manner were applied with the aforementioned equivalent stress definitions to estimate the influence of turbulent stresses on the damage prediction.

For the first damage model, the equivalent stresses were taken from the instantaneous field (equation (9a)) and were compared against the mean stresses from equation (11). The first model predicts a hemolysis value $H$ by a scalar transport equation, where the equation computes the convective transport and the production of plasma-free hemoglobin within the flow²⁰

\frac{\partial H_{L}}{\partial t} + \frac{\partial u_{i} H_{L}}{\partial x_{i}} = S (1 - H_{L}), with

(12a)

S = C^{\frac{1}{β}} τ^{\frac{α}{β}}

(12b)

H = H_{L}^{β}

(12c)

In equation (12a), $H_{L}$ denotes a linearized, scalar variable, $H_{L} = H^{(1 / β)}$ , where $H$ represents the instantaneous hemolysis value, which is the ratio of released hemoglobin to the total hemoglobin in the blood volume at a certain time and space. A source term $S$ for the production of plasma-free hemoglobin is present on the right-hand side (RHS) of the transport equation. The empirical constants $C,$ $α$ , and $β$ were taken from the experiments of Zhang et al.³⁶ for this study. The constants C, α, and $β = 1.228 \times 10^{- 7}, 1.9918 and 0.6606$ , respectively. The representative hemolysis value of the whole VAD was determined by calculating the instantaneous, mass flow-averaged hemolysis value $H$ at the VAD’s outlet. In order to grant a better overview, the hemolysis value was modified to an MIH-value (modified index of hemolysis), $M I H = 10^{6} \cdot H,$ ³⁷ in section “Results and discussion.”

The idea of solving a transport equation for the numerical hemolysis estimation was first introduced by Garon and Farinas.³⁷ They also proposed a further, numerical damage model, where hemolysis can be quickly approximated by predicting a simplified, averaged hemolysis value $\bar{H}$ . This variable can be obtained, by assuming a time-independent flow field and integrating equation (12a) over the domain volume $V$

\int_{V} \frac{\partial u_{i} H_{L}}{\partial x_{i}} d V = \int_{V} S d V

(13)

Since the original formulation of Garon and Farinas was used in this article, the term $(1 - H_{L})$ from equation (12a) was not considered in equation (13). Turning the volume integral into a surface integral, equation (13) can be transferred to

\int_{A} u_{i} \cdot n_{i} H_{L} d A = \int_{V} S d V

(14)

with the surface normal vector $n_{i}$ and the outlet cross-sectional area $A$ of the computational domain. Using the assumption that $H_{L}$ is zero at the domain’s inlet, we obtain the (spatial and temporal) average hemolysis value $\bar{H}$ with equations (15a) and (b)

\bar{H_{L}} = \frac{1}{Q} \int_{V} C^{\frac{1}{β}} τ^{\frac{α}{β}} d V

(15a)

\bar{H} = {\bar{H}}_{L}^{β}

(15b)

where $Q$ denotes the VAD’s flow rate. For blood damage prediction using this model, the equivalent stresses were calculated from the time-averaged flow field only. Hence, equations (9b) and (11) were used. As with the transport equation, the average hemolysis value was transformed into an average MIH value, $\bar{M I H} = 10^{6} \cdot \bar{H}$ .³⁷

Due to its simplicity and many assumptions, this second hemolysis model lacks in accuracy and is just a rough estimator of the real hemolysis process. Nevertheless, it has its justification in the design process of VADs, where lots of simulations are analyzed and compared against each other and the computation of additional transport equations could be too time-consuming.

With the third applied damage model, volumes within the VAD $I_{τ > τ_{t h r e s}} = V_{τ > τ_{t h r e s}} / V_{t o t a l}$ were computed, where certain shear stress thresholds ( $τ > τ_{t h r e s}$ ) are exceeded. For this model, the time-averaged flow field has to be examined using the equivalent stresses of equations (9b) and (11). The thresholds can be related to three different types of blood damage. According to Fraser et al.,³⁸ a threshold of 9 Pa indicates a potential cleavage of the von-Willebrand factor (vWF). Shear stresses above 50 Pa are assumed to correlate with the activation of platelets, which are associated with pump thrombosis.³⁹ Furthermore, the threshold for hemolysis is set to 150 Pa in the volumetric analysis. With this procedure, a comparison of different VAD designs regarding different types of blood cell damage is feasible.¹⁶

Power loss analysis for the verification

It was shown that the considered equivalent stresses can be directly correlated with the total energy dissipation $(ε_{t o t} + ε_{m o d})$ . To evaluate, if the dissipation rates are properly computed, the authors developed a verification method, called the power loss analysis (PLA), which was presented at first in the studies by Torner et al.⁴⁰ and Konnigk et al.⁴¹ The PLA gives information whether the simulation with its mesh resolution, numerical schemes, and turbulence modeling is capable of resolving the computed dissipative losses.

A detailed description of this method is given in the study by Torner et al.⁴⁰ To explain it briefly, the total power loss $P_{l o s s}$ via the VAD is calculated using two equations. The first one is a result of the difference between the blade’s drive power $P_{b l a d e s}$ and the increase in hydraulic power ∆

P_{L o s s, 1} = P_{b l a d e s} - Δ P_{F l u i d} = M_{B l a d e s} \cdot ω - Δ p_{t o t} \cdot Q

(16)

Equation (16) is built with the blades moment $M_{b l a d e s}$ and angular velocity $ω$ at the rotating surfaces, the flow rate $Q$ as well as the total pressure increase ∆ $p_{t o t}$ from the inlet to the outlet boundary of the computational domain. The total power loss $P_{l o s s}$ can also be calculated by summing up all dissipative losses within the flow domain, see equation (17)

\begin{array}{l} P_{L o s s, 2} = ρ (\int_{V} ϵ_{d i r} d V + \int_{V} 〈 ε_{t u r b} 〉 d V + \int_{V} 〈 ε_{m o d} 〉 d V) \\ = ρ (\int_{V} ν 〈 \frac{\partial u_{i}}{\partial x_{j}} 〉 (〈 \frac{\partial u_{i}}{\partial x_{j}} 〉 + 〈 \frac{\partial u_{j}}{\partial x_{i}} 〉) d V) \\ + ρ (\int_{V} ν 〈 \frac{\partial u_{'}^{i}}{\partial x_{j}} (\frac{\partial u_{'}^{i}}{\partial x_{j}} + \frac{\partial u_{'}^{j}}{\partial x_{i}}) 〉 d V) \\ + ρ (\int_{V} 〈 ν_{t} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) \frac{\partial u_{i}}{\partial x_{j}} 〉 d V) \end{array}

(17)

From a physical point of view, equations (16) and (17) should lead to the same power loss $P_{L o s s}$ . If this is fulfilled, it can be verified that the equivalent stresses are properly computed.

Results and discussion

Experimental examination of the pressure heads

The measured characteristic curve for $n = 7900 r / \min$ and in the range of $0 - 4.5 L / \min$ is shown in Figure 3. In addition, the numerically assessed pressure heads are shown in Figure 3. A good agreement for both operation points can be seen. The relative differences between the simulated and measured pressure heads are 5.3% (2.5 L/min) and 0.2% (4.5 L/min), respectively.

Figure 3.

Experimentally assessed characteristic curve and comparison with the computed pressure heads from the large eddy simulation (LES).

Verification of the computed equivalent shear stresses

The results of the PLA are shown in Table 1. Similar LES results were also presented and comprehensively discussed in the study by Torner et al.⁴⁰ Therefore, only a short summary of the verification is given.

Table 1.

Power loss analysis (PLA) results for both operation points.

	$Q = 4.5 L / \min$	$Q = 2.5 L / \min$
$P_{b l a d e s}$	$2.244 W$	$2.038 W$
∆ $P_{F l u i d}$	$0.744 W$	$0.604 W$
$P_{L o s s, 1}$	$1.50 W$	$1.434 W$
$ρ \int_{V} ϵ_{d i r} d V$	$1.173 W (83 %)$	$0.986 W (75 %)$
$ρ \int_{V}^{} 〈 ε_{t u r b} 〉 d V$	$0.211 W (15 %)$	$0.288 W (22 %)$
$ρ \int_{V} 〈 ε_{m o d} 〉 d V$	$0.022 W (2 %)$	$0.034 W (3 %)$
$P_{L o s s, 2}$	$1.406 W (100 %)$	$1.308 W (100 %)$
$\frac{P_{L o s s, 1} - P_{L o s s, 2}}{P_{L o s s, 1}}$	$6.26 %$	$8.78 %$

$P_{l o s s}$ is the total power loss. $P_{b l a d e s}$ and ∆ $P_{F l u i d}$ are the blade’s drive power and the increase in hydraulic power. Also, the values and the percentage shares to the total power loss of the direct dissipation $ϵ_{d i r}$ , the resolved turbulent $ε_{t u r b}$ , and modeled turbulent dissipation $ε_{m o d}$ are included in the table.

As it can be seen from the table, the direct dissipation $ϵ_{d i r}$ contributes to the greatest dissipative loss share with 75% ( $Q = 2.5 L / \min$ ) and 83% ( $Q = 4.5 L / \min$ ), respectively. The losses from turbulent dissipation $< ε_{t u r b} > + < ε_{m o d} >$ are less, with 15%–22% directly resolved dissipative losses and 2%–3% modeled losses.

The relatively small deviations of around 6% and 9% let us conclude that the LES computations are generally capable of reflecting the majority of the flow losses and significant portions of the considered equivalent shear stresses are adequately resolved from a global perspective.

It can be seen from the PLA results that the “shear-related” losses $(P_{l o s s, 2})$ are underpredicted compared to the “hydraulic” losses $(P_{l o s s, 1})$ . This deviation was found to be even greater for URANS simulation in the study by Torner et al.⁴⁰ One reason is the spatially and temporally limited mesh resolution, which dampens the mean gradients near the wall as well as the resolution of turbulent structures up to the Kolmogorov scale. These turbulent structures possess turbulent shear stresses, which are not being computed. Other aspects are improper turbulence models (in a sense of adding modeled dissipation to the flow) and too dissipative numerical schemes. In this context, blood damage will be greatly underpredicted in a simulation with too small meshes,⁴² too large time sttif, or improper turbulence models,⁴⁰ even when an appropriate blood damage model is applied.

Influence of turbulent stresses on blood damage prediction results

The results for the two operation points are shown in Tables 2 and 3. The values on the right hand side (RHS) of the tables present the percentage difference between the respective total equivalent shear stress from equations (9a) and (b) and the mean shear stress $τ_{m e a n}$ (equation (11)).

Table 2.

Blood damage prediction results for the operation point with $Q = 2.5 L / \min$ .

$Q = 2.5 L / \min$	$τ_{s s r}$ (equation (9a))	$〈 τ_{s s r} 〉$ (equation (9b))	$τ_{m e a n}$ (equation (11))	$\frac{(τ_{s s r} - τ_{m e a n})}{τ_{m e a n}} \times 100 %$	$\frac{(〈 τ_{s s r} 〉 - τ_{m e a n})}{τ_{m e a n}} \times 100 %$
$I_{τ > 9 Pa}$ (%)	–	8.41	3.58	–	134.8
$I_{τ > 50 Pa}$ (%)	–	0.87	0.69	–	25.8
$I_{τ > 150 Pa}$ (%)	–	0.125	0.120	–	4.2
$\bar{M I H}$ (equation (15))	–	83.4	74.1	–	12.6
$M I H$ (equation (12))	76.1	–	66.6	14.3	–

MIH: modified index of hemolysis.

$I_{τ}$ is the volume with equivalent stresses above the prescribed threshold in relation to the total domain volume. $M I H$ and $\bar{M I H}$ are the hemolysis results from the respective hemolysis estimation model. $τ_{s s r}$ and $〈 τ_{s s r} 〉$ denote the total equivalent shear stress (with turbulent stresses) in the instantaneous and time-averaged flow field. $τ_{m e a n}$ denotes the mean shear stress (without turbulent stresses).

Table 3.

Blood damage prediction results for the operation point with $Q = 4.5 L / \min$ .

$Q = 4.5 L / \min$	$τ_{s s r}$ (equation (9a))	$〈 τ_{s s r} 〉$ (equation (9b))	$τ_{m e a n}$ (equation (11))	$\frac{(τ_{s s r} - τ_{m e a n})}{τ_{m e a n}} \times 100 %$	$\frac{(〈 τ_{s s r} 〉 - τ_{m e a n})}{τ_{m e a n}} \times 100 %$
$I_{τ > 9 Pa}$ (%)	–	11.34	5.49	–	106.4
$I_{τ > 50 Pa}$ (%)	–	0.97	0.85	–	13.8
$I_{τ > 150 Pa}$ (%)	–	0.134	0.130	–	2.7
$\bar{M I H}$ (equation (15))	–	59.1	55.7	–	6.3
$M I H$ (equation (12))	56.3	–	52.4	7.4	–

MIH: modified index of hemolysis.

The time-averaged flow was examined in the volumetric analysis $(I_{τ > τ_{t h r e s}})$ . It can be seen from the tables that the turbulent stresses have an influence on the calculated volumes to varying degrees. A high contribution of 106.4%–134.8% is noticeable for the threshold of 9 Pa. Here, disregarding turbulent stresses leads to volumes, which are only half the size as those calculated with the total equivalent stresses $< τ_{s s r} >$ . For moderate shear stresses (>50 Pa), the influence of turbulent stresses is significant with relative differences between 13.8%–25.8%. For high stresses (>150 Pa), the contribution is small with differences between 2.7% and 4.2%.

These results suggest that the turbulent stresses are frequently influential between the stress levels of 9 and 150 Pa in the VAD. For a more detailed analysis of the flow field, the equivalent stress fields are shown in Figures 4 and 5. All considered stress definitions (equations (9a) and (b) and (11)) were examined in three different, cylindrical blade-to-blade views through the rotor and outlet guide vane. The blade-to-blade views were located at radius ratios $r / R_{c}$ of 0.2, 0.5, and 0.8, where $R_{c}$ denotes the radius of the pump’s casing. The fields were plotted using a contour plot in a logarithmic range between 9 and 150 Pa.

Figure 4.

Equivalent shear stress distributions at the walls and in the flow field in a cylindrical cut-plane through the rotor and a part of the outlet guide vane at different radius ratios. Simulation with a flow rate of $Q = 2.5 L / \min$ . The relation $r / R_{c}$ is the radius ratio, where $R_{c}$ is the casing radius. $τ_{s s r}$ and $〈 τ_{s s r} 〉$ denote the total equivalent shear stress (with turbulent stresses) in the instantaneous and time-averaged flow field. $τ_{m e a n}$ denotes the mean shear stress (without turbulent stresses). Please note that the scale is logarithmic.

Figure 5.

Equivalent shear stress distributions at the walls and in the flow field in a cylindrical cut-plane through the rotor and a part of the outlet guide vane at different radius ratios. Simulation with a flow rate of $Q = 4.5 L / \min .$ The relation $r / R_{c}$ is the radius ratio, where $R_{c}$ is the casing radius. $τ_{s s r}$ and $〈 τ_{s s r} 〉$ denote the total equivalent shear stress (with turbulent stresses) in the instantaneous and time-averaged flow field. $τ_{m e a n}$ denotes the mean shear stress (without turbulent stresses). Please note that the scale is logarithmic.

In all figures, it can be seen that the highest stresses occur in the proximity to the walls within the boundary layer, where $ϵ_{d i r}$ and $τ_{m e a n}$ , respectively, dominate. Here, stresses higher than 150 Pa occur at the whole rotor blade and at large parts of the guide vanes. Contrarily, away from the walls, $ϵ_{d i r}$ adds a significant share to the total equivalent stresses $< τ_{s s r} >$ only at few positions. In the wall-distant region of the flow, the turbulent part $(< ε_{t u r b} > + < ε_{m o d} >)$ is the main contributor to the total equivalent stress $< τ_{s s r} >$ . These turbulent stresses origin from turbulent flow structures, which arise in the VAD’s flow due to turbulence at the walls, flow separations, mixing layers behind the rotating impeller as well as secondary flow structures like the gap vortex or horseshoe vortices near the rotor’s hub. The described turbulent flow phenomena are universal and therefore occur in every operation point to varying degree in every rotary blood pump, for example, due to the rotor–stator interaction and secondary flow structures.⁴³

The global contribution of turbulent stresses can be evaluated by analyzing the average hemolysis index $\bar{M I H}$ from equation (15) in Tables 2 and 3. The deviation between the average hemolysis indices from the mean stresses $τ_{m e a n}$ (equation (11)) and total stresses $< τ_{s s r} >$ (equation (9b)) is 6.3% and 12.6%.

It is noticeable that the $\bar{M I H}$ results are not strongly dependent on the turbulent stresses for the two simulations. This can be explained by the different contribution from mean and turbulent stresses to this value. Mathematically, equation (15) is like an exponentiated integration over the whole stress field, similar to the PLA. In the PLA, it was detected that the direct dissipation $ϵ_{d i r}$ accounts for the greatest loss share for both operation points with over 70%–80%. Since $ϵ_{d i r}$ is also a part of the total equivalent stress definition, it dominates the global stress field and therefore the average $\bar{M I H}$ value too.

Finally, the influence of turbulent stresses on the blood damage prediction is analyzed by calculating the relative deviation between the instantaneous hemolysis values $M I H$ from the transport equation (12a) in Tables 2 and 3. These results were determined with the instantaneous total stress $τ_{s s r}$ (equation (9a)) and the mean stress $τ_{m e a n}$ (equation (11)). Relative deviations between 7.4% and 14.3% are observable in the tables. As in the case of the averaged hemolysis value from equation (15), the contributions from the turbulent stresses $(4 ν < S_{i j} > {S^{'}}_{i j} + 2 ν {S^{'}}_{i j} {S^{'}}_{i j} + ε_{m o d})$ to the predicted hemolysis value are again small compared to the influence of the dominant stress component $ϵ_{d i r}$ near the walls.

This instantaneous hemolysis prediction via transport equation was performed using a “stress-based” damage model, where the instantaneous hemolysis is a direct function of the equivalent stresses,²⁰ whereas the biophysical aspects of RBCs (shape deformation, relaxation, viscoelastic effects) are not considered. According to Pauli et al.,²⁴ this simplification can lead to a potential risk for overpredicting hemolysis, since the shape of an RBC does not deform immediately and rather takes time to adapt to the turbulent flow field. The instantaneous turbulent stresses in the wall-distant field act only in a relatively small range and only for a short time in our analysis. In this context, the influence of the turbulent stresses on the hemolysis estimation in our VAD could be possibly even lower, when the morphological properties of RBCs are taken into account. But this is just a presumption and has to be proven in further studies.

Limitations

A comparison between experiments and CFD was performed only by analyzing the pressure heads. Although the validation of pressure heads is not as valuable as a full validation of the velocity or turbulent kinetic energy field, it is an appropriate and commonly used validation method¹⁶,¹⁸,²³,⁴⁴ to analyze the suitability of a numerical model. The validation of the H–Q curve holds as a first step of flow validation, since an incorrectly computed flow field will also lead to incorrectly computed pressure heads.

In this context, the used acrylic model can theoretically be used for optical flow measurements. Unfortunately, these measurements have not been performed up to now, but are planned for future work.

Just “stress-based” damage prediction models were used to estimate and evaluate the blood trauma. It would be interesting to apply a “strain-based” model and analyze the effect of the turbulent flow also in these models. Unfortunately, it was not possible for the authors to implement a complex, “strain-based” model into the used commercial solver.

Summary and conclusion

This study dealt with the influence of turbulent stresses on the numerical blood damage prediction in two operation points of an axial rotary blood pump.

In a first step, a total equivalent stress definition $τ_{s s r}$ for an unsteady flow simulation was derived. This total equivalent stress is based on the invariant of the strain-rate tensor $S_{i j}$ and consists of three components: a mean, a resolved turbulent, and a modeled turbulent part. All parts must be considered in an unsteady flow simulation with turbulence modeling (URANS, LES, and hybrid approaches) in order to account for the total equivalent stress, regardless of the type of mechanical circulatory support system.

In addition, another shear stress definition, $τ_{m e a n}$ , was considered, where the turbulent stress parts were excluded. Afterward, the relative differences in the blood damage prediction between the total stresses and mean stresses were determined, to indicate the influence of turbulent stresses.

This influence was investigated using a volumetric analysis of different stress thresholds and a simplified, average hemolysis value, wherefore the time-averaged flow fields were examined. It could be shown that large fields of turbulent stresses are present distant from the walls and act in a range between 9 and 50 Pa. According to the study by Fraser et al.,³⁸ this is the potential range, where vWF proteins or platelets could be damaged. Here, a disregard of turbulent stresses would lead to an underprediction in the volumetric analysis and potential sources for blood damage may not be identified in the flow field. Hence, it is recommended to include turbulent stresses in all numerical analyses which use evaluation methods like the volumetric analysis.

Nonetheless, the dominant component in the total equivalent stress formulation is not the turbulent stress, but the mean stress $τ_{m e a n}$ . This term origins from the mean velocity gradients near the wall and correlates with the direct dissipation rate $ϵ_{d i r}$ . The amplitudes of the mean stresses at the wall exceed the hemolysis threshold (>150 Pa) in all places of the rotor surface and lead to a high impact on the computed hemolysis values.

A transport equation for the hemolysis prediction was also analyzed in the instantaneous flow field. Here, the damage prediction results from the instantaneous total stresses $τ_{s s r}$ were compared to the results from the mean stresses $τ_{m e a n}$ . Again, the mean stresses dominate the instantaneous hemolysis value and relative deviations between 7.4% and 14.3% are observable, when the turbulent stress part $(4 ν < S_{i j} > {S^{'}}_{i j} + 2 ν {S^{'}}_{i j} {S^{'}}_{i j} + ε_{m o d})$ is omitted in the equivalent stress definition. The conclusion of this finding is that the used equivalent turbulent stresses have a small impact on the state-of-the-art hemolysis prediction via transport equations in the analyzed VAD. The turbulent stresses are too small and act too shortly in the flow to potentially damage RBCs to a large extent.

Nevertheless, turbulent stresses should always be included into the damage model when an equivalent stress definition based on the strain-rate tensor invariant is used, since these stresses belong to the total equivalent stress formulation and could be more significant in other flow cases.

Nonetheless, this research cannot answer the question how and to which extent the turbulent stresses impact true blood trauma. It rather indicates how the contribution from turbulent stresses can be numerically included in an equivalent stress definition (which is based on the invariant of the strain-rate tensor) to account for the total flow field. Due to the correlation between the total equivalent stress and the total dissipation rate, there exists a physical basis to balance the considered turbulent stresses against the mean stresses. Due to this, the turbulent stresses are not overestimated as with the use of Reynolds stresses and seem more appropriate for flow simulations in the author’s opinion.

Turbulence has an effect on blood damage, and therefore, the turbulent stresses must be and are already partly considered in the damage prediction in blood pumps. Turbulent stresses will always be present in axial or radial rotary blood pumps, since secondary flow structures and free shear layers trigger the production of turbulent kinetic energy. Therefore, the authors are of the opinion that turbulent stresses have to be considered, for example, in the pump’s design process, to identify all potential sources of damage, in spite of the fact that the true mechanism of blood trauma in turbulent flow conditions is still unknown.

Footnotes

Acknowledgements

The authors acknowledge the North-German Supercomputing Alliance (HLRN) for providing HPC resources that have contributed to the research results reported in this article. We want to thank Dr Guenther Steffen and Sebastian Hallier, MSc, who have made an immense contribution to the creation of this article.

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.

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