Description AC7-03

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Turbulent Blood Flow in a Ventricular Assist Device

Application Challenge AC7-03   © copyright ERCOFTAC 2021

Description

Introduction

Ventricular Assist Devices (VADs) are implanted in patients with severe heart failure. Today, nearly all VADs are designed as turbomachinery, since they have a higher power density as pulsatile pumps, and therefore can be implanted within the human body.

By using Computational Fluid Dynamics (CFD), VADs must be designed and optimised in such a way that they reproduce a physiological pressure increase in order to sufficiently supply the body with enough blood flow. Furthermore, they must be designed in order to guarantee that the blood, which passes the VAD, is not damaged due to non-physiological flow conditions (high shear stresses, stagnation areas, high turbulent kinetic energy (TKE) regions, ...) in the device.

The CFD simulation in a VAD can be challenging, since the inflow is laminar and all turbulence is produced within the pump and decays shortly after the pump outlet. Furthermore, the pump Reynolds number $Re_{p}={u_{2}D_{2}}/{\nu }$ is small with $Re_{p}=10^{3}-10^{4}$ compared to industrial pumps ($Re_{p}\approx 10^{6}$), and transition might occur.

In this respect, the aim of this study is to investigate the suitability of different URANS methods (with different turbulence models and solver settings) for the flow computation in an axial VAD. Here, both fluid mechanical parameters, such as the pump characteristics and velocity fields, as well as haemodynamic parameters, such as the haemolysis index MIH or stagnation zones, are investigated. The flow fields of the URANS simulations will be compared with a highly turbulence-resolving large-eddy simulation, which represents the reference case for comparison. Furthermore, the influence of the grid resolution in the URANS computations will also be investigated based on a extended grid study.

Relevance to Industrial Sector

The flow computation in a Ventricular Assist Device (VAD) is an important procedure for the VAD design and optimization in the pre-clinical evaluation. The aim of these CFD studies is, on the one side, to guarantee that the VAD offers an physiological relevant pressure increase at the chosen desing point to sufficiently support the VAD patient. On the other side, haemodynamical parameter must be evaluated in these studies. Here, it is important that the CFD reflects relevant regions for potential blood damage or thrombi formation, so that these regions can be minimised in the optimization procedure. Additionally, CFD is important for VAD studies in order to compare different designs to find the pump with the highest haemocompatibility (lowest blood damage).

When the VAD designer is able to find a good VAD design by CFD, some amount of in-vitro (experimental test of pump performance or hemolysis [red blood cell damage]) and in-vivo testing (animal trials) might be reduced.

Design or Assessment Parameters

The main assessment parameters for this AC are:

• Pressure increase via the VAD (pressure head) $H$

• Hydraulic efficiency of the pump $\eta _{i}$

• equivalent (scalar) shear stress $\tau _{s}$

• Modified index of hemolysis $MIH$

• Volumetric analysis of stress thresholds $I_{\tau _{s}\geq threshold}$

• Evaluation of stagnation areas based on the wall shear stress $\tau _{w}$

The rationale behind these parameters and details of how to calculate each of these are describe below.

• Pressure head

The pressure increase via a VAD is typically defined in millimeters of mercury $[mmHg]$:

$H={\frac {\sum _{outlet}\langle p_{tot}\rangle \cdot {\dot {m}}_{cell}}{\sum _{outlet}{\dot {m}}_{cell}}}-{\frac {\sum _{inlet}\langle p_{tot}\rangle \cdot {\dot {m}}_{cell}}{\sum _{inlet}{\dot {m}}_{cell}}}\qquad \qquad \qquad \qquad (1)$

with $p_{tot}$ as the time-averaged total pressure, which is massflow-averaged at the outlet and inlet.

• Hydraulic efficiency

$\eta _{i}={\frac {H\cdot Q}{M\cdot 2\pi n}}\qquad \qquad \qquad \qquad \qquad \qquad \qquad (2)$

$Q$ denotes the flow rate and $n$ the rotational speed. The blade torque $M$ is determined at the rotating surfaces of the impeller by accounting the surface pressures $p_{sf}$ and surface shear stresses $\tau _{sf}$ (e.g. the impeller rotates around the z-axis, S - denotes the impeller surface):

$M_{z}={\big [}\int _{S}{r_{s}\times (n_{s}\langle p_{sf}\rangle dS)}+\int _{S}{r_{s}\times (t_{s}\langle \tau _{sf}\rangle dS)}{\big ]}\cdot e_{z}\qquad \qquad \qquad (3)$

with:

• $t_{s}$ - unit parallel vector
• $n_{s}$ - unit vector parallel to $dS$
• $e_{z}$ - unit vector parallel to the z-axis

• Equivalent shear stress

The equivalent shear stress $\tau _{s}$ is the parameter, by which numerical blood damage is assessed in a VADs flow simulation. The basis for this parameter derives from the shear stress tensor $\tau _{ij}$, which is built with the velocity gradients in a flow field.

$\tau _{ij}=\mu ({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}})\qquad \qquad \qquad \qquad \qquad \qquad \qquad (4)$

For the blood damage prediction in complex, 3D flows this viscous stress tensor is reduced to a scalar representation, the equivalent shear stress $\tau _{s}$. Most numerical VAD simulation use a formulation based on the second invariant of the rate-of-strain tensor $S_{ij}$:

$\tau _{s}=\mu {\sqrt {2\cdot S_{ij}S_{ij}}}$ with $S_{ij}=0.5({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}})\qquad \qquad \qquad \qquad \qquad \qquad \qquad (5)$

The expression within the square-root in Eq.5 can be further connected to a flow variable, the energy dissipation in the flow $\epsilon =2\nu S_{ij}S_{ij}$. From this, an effective shear stress can be derived from Eq.5, which connects the effectiv stress $\tau _{eff}$ with the computed dissipation rates in the time-averaged flow, namely the direct dissipation $\epsilon _{dir}=2\nu \langle S_{ij}\rangle \langle S_{ij}\rangle$, the resolved turbulent dissipation $\epsilon _{turb}=2\nu \langle S_{ij}^{'}S_{ij}^{'}\rangle$ and the modelled turbulent dissipation from the turbulence model $\epsilon _{mod}=2\langle \nu _{t}S_{ij}S_{ij}\rangle$:

$\tau _{eff}={\sqrt {\rho \mu (\epsilon _{dir}+\epsilon _{turb}+\epsilon _{mod})}}\qquad \qquad \qquad \qquad \qquad \qquad \qquad (6)$

• Modified hemolysis index

The modified index of hemoylsis $MIH$ is a widely used parameter to numerically assess hemolysis in VADs. A large number of numerical haemolysis prediction models are based on power laws. These models establish a direct relationship between hemolysis $MIH$, an equivalent stress $\tau _{eff}$ and the exposure time $t$. Three experimentally determined constants $(\alpha ;\beta ;C)$ are used to link the parameters:

$MIH=H\cdot 10^{6}=C\tau _{eff}^{\alpha }t^{\beta }\cdot 10^{6}\qquad \qquad \qquad \qquad \qquad \qquad \qquad (7)$

Because of the non-linear relationship between the hemolysis value $H(\tau _{eff};t)$ and suspension time $t$, Eq. 7 is transformed for a numerical implementation and written as a transport equation.

${\frac {\partial H^{1/\beta }}{\partial t}}+u_{i}{\frac {\partial H^{1/\beta }}{\partial x_{j}}}=C^{1/\beta }\tau _{eff}^{\alpha /\beta }\qquad \qquad \qquad \qquad \qquad \qquad \qquad (8)$

If it is now assumed that the hemolysis is time-independent and homogeneous in space, Eq.8 can be further simplified to a (temporally and spatially) averaged hemolysis value ${\underline {H^{1/\beta }}}$

$\int _{V}{(u_{i}{\frac {\partial H^{1/\beta }}{\partial x_{j}}}})dV=\int _{V}{(u_{i}\cdot n_{i}\cdot \partial H^{1/\beta }})dS\qquad \qquad \qquad \qquad \qquad \qquad \qquad (8)$

Flow Physics and Fluid Dynamics Data

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Contributed by: B. Torner — University of Rostock, Germany

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© copyright ERCOFTAC 2021