On the representation of turbulent stresses for computing blood damage
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:where D is an index of damage, τ is the scalar shear stress, texp 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]:and a second by Yeleswarapu et al. [58], who in turn were motivated by Kachanov [25]:where η is the viscosity, 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.
Section snippets
Mathematical preliminaries
Assuming blood to behave as a single-phase and incompressible fluid, the conservative form of the linear momentum equation can be represented as:and conservation of mass as:where u(x) is the velocity field, p(x) is the pressure, ρ is the density, and f represents the body forces. The Navier–Stokes equation, which is the foundation of the Reynolds Averaged Equations, assumes a constitutive relation of the form:where η is a constant related to
Analysis of error in blood trauma estimation due to the Reynolds stress assumption
The error introduced by the use of the Reynolds stress as a surrogate for turbulent stress, i.e.may be investigated by comparison to the exact expression in Eq. (10). It is observed that the mean of τ′, in Eq. (10), is the zero tensor. Therefore, Eq. (16) can hold only if κ is identically zero for all time or all components of the Reynolds stress are zero. The former case implies that there cannot be a direct relationship between the Reynolds and total stress. The latter case is only
Energy dissipation for modeling blood trauma
Using independent methods Jones [24] and later Quinlan and Dooley [33] proposed that turbulent stresses are better approximated through the use of turbulent energy dissipation. This approach entails the use of a scalar norm for the total stress tensor (see [15], [16]) commonly practiced in failure mechanics [52].
Beginning once again with the assumption of a Newtonian fluid, and expressing the constitutive relationship Eq. (6) as:wherewhich can be decomposed as:
Analysis of error of damage predictions
Without loss of generality, for the purpose of error analysis, the collection of prevailing blood damage models may be represented as a power law equation of the following form:where D is a (damage) function with dependence on exposure time, texp, and viscosity, η. The exponent of shear stress is chosen here to be identically equal to 2, which agrees with the statistical accuracy of several empirical models [6], [10], [43], [56] and two theoretical blood damage models [37], [58].
Discussion
Standard closure models for turbulent flow of blood are insufficient to accurately calculate the total stress experienced by the cells suspended therein. Direct numerical simulation is so computationally expensive that only the simplest of geometries can be evaluated. This difficulty arises from the fact that the RANS, with its associated closure models, were developed to study large-scale flow features [31], whereas hemolysis is induced by stresses acting on the micro-scale [33]. The most
Acknowledgements
The authors express their gratitude to Dr. K.R. Rajagopal for his tutelage, inspiration, dedication, and pursuit of perfection.
This work was funded in part by the National Institutes of Health (R01 HL089456-01A2).
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