On the representation of turbulent stresses for computing blood damage

Abstract

Computational prediction of blood damage has become a crucial tool for evaluating blood-wetted medical devices and pathological hemodynamics. A difficulty arises in predicting blood damage under turbulent flow conditions because the total stress is indeterminate. Common practice uses the Reynolds stress as an estimation of the total stress causing damage to the blood cells. This study investigates the error introduced by making this substitution, and further shows that energy dissipation is a more appropriate metric of blood trauma.

Introduction

The Food and Drug Administration evaluates medical devices based on the criteria of safety and efficacy. The anchor of safety is biocompatibility, which in the case of blood-wetted medical devices is dictated by the absence of shear-induced blood trauma, thrombosis, and infection [2], [5], [41]. One of the greatest prevailing challenges in the development of blood-wetted cardiovascular devices is the elimination of unintended flow-induced trauma to the blood, namely hemolysis (the destruction of blood cells) and shear-activation of platelets [7], [9], [56].

Because of the cellular composition of blood and the complexity of the flow fields in these devices, it has been impractical to develop a microstructural model of cellular damage. Consequently, investigators over the past several decades have relied upon homogenized models [4], [7], [8], [14], [17], [28], [58]; mostly empirical relationships between the shear history and an experimentally measured indicator of cell trauma (such as hemoglobin liberated from the red blood cell into the plasma). These empirical models are generally of the form:

$$D=C{\tau }^{\alpha }{t}_{\exp }^{\beta }\text{,}$$

where D is an index of damage, τ is the scalar shear stress, t_exp is the exposure time, and C, α, and β are coefficients [6], [9], [10], [14], [23], [43], [56] (see Table 1). In addition, there are two theoretical models of significance, the first by Richardson [37]:

$$D=C\eta {\dot{\gamma }}^2{t}_{\exp }=\frac{C}{\eta }{\tau }^2{t}_{\exp }\text{,}$$

and a second by Yeleswarapu et al. [58], who in turn were motivated by Kachanov [25]:

$$\dot{D}={\tau }^2\frac{C}{(1-D{)}^{\delta }}\text{,}$$

where η is the viscosity, $\dot{\gamma }$ is the scalar shear rate, and δ is a model constant.

The unknown constants have been determined through various controlled experiments, intentionally designed to cause cellular trauma. The preponderance of these experiments has been laminar, viscometric flows. However, blood flow in cardiovascular devices occurs in both the laminar and turbulent regimes.

Prediction of hemolysis in turbulent flow, therefore, requires an appropriate constitutive model to determine turbulent stresses. The most accurate approaches used in engineering applications include Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES), which for geometries of practical importance are prohibitive. It is therefore common to employ time-averaged equations that describe the mean motion. The most frequently used approach to model turbulent flow in blood-wetted devices has been the Reynolds Averaged Navier–Stokes (RANS) equations.¹ However, the formulation of the RANS results in the loss of the small-scale turbulent features of velocity and stress. To account for these lost features, it is common for bioengineers to incorporate the so-called the Reynolds Stress as a surrogate for the turbulent stresses² when computing blood trauma [9], [12], [29], [32], [44], [50], [53], [54], [55]. With this approach the k-ε model [1] and the algebraic stress model (ASM) have been introduced [40]. However, it has been suggested that the assumptions leading to the k-ε model are inadequate for the modeling of turbulence in highly swirling flows ([21], [22], [42], [47]). Even with the various modifications made for the k-ε model, it is still not capable of describing secondary flows in noncircular ducts and it cannot predict non-zero normal-Reynolds-stress differences ([45], [46], [49].

This study seeks to investigate the errors introduced by using the Reynolds stress as a surrogate for turbulent stress through arithmetic manipulation of the Navier–Stokes and RANS equations.