Practical implications of the erroneous treatment of exposure time in the Eulerian hemolysis power law model.

BACKGROUND: Eulerian and Lagrangian power-law formulations are both widely used for computational fluid dynamics (CFD) to predict flow-induced hemolysis in blood-contacting medical devices. Both are based on the same empirical power-law correlation between hemolysis and the shear stress and exposure time. In the Lagrangian approach, blood damage is predicted by tracking both the stress and exposure time along a finite number of pathlines in the domain. In the Eulerian approach, a scalar transport equation is solved for a time-linearized damage index within the entire domain. Previous analytical work has demonstrated that there is a fundamental problem with the treatment of exposure time in the Eulerian model formulation such that the only condition under which the model correctly represents the true exposure time is in a flow field with no streamwise velocity variation. However, the practical implications of this limitation have yet to be thoroughly investigated. METHODS: In this study, we demonstrate the inaccuracy of Eulerian hemolysis power-law model predictions due to the erroneous treatment of exposure time by systematically considering four benchmark test cases with increasing degrees of flow acceleration: Poiseuille flow through a straight tube, inclined Couette flow, and flow through a converging tube with two different convergence ratios. RESULTS: Compared with Lagrangian predictions, we show that, as flow acceleration becomes more pronounced, the resultant inaccuracy in the Eulerian predictions increases. For the inclined Couette flow case, there is a small degree of flow acceleration that yields a discrepancy in the range of 10% between Lagrangian and Eulerian predictions. For flows with a larger degree of acceleration, as occurs in the converging tube flow cases, the discrepancy is considerably larger (up to 257%). CONCLUSION: The inaccuracy of hemolysis predictions due to the erroneous treatment of exposure time in the Eulerian power-law model can be significant when there is streamwise velocity variation in the flow field. These results may partially explain the extremely large variability in CFD hemolysis predictions reported in the literature between Lagrangian and Eulerian models.

On the Discretization of the Power-Law Hemolysis Model.

Flow-induced hemolysis remains a concern for blood-contacting devices, and computer-based prediction of hemolysis could facilitate faster and more economical refinement of such devices. While evaluation of convergence of velocity fields obtained by computational fluid dynamics (CFD) simulations has become conventional, convergence of hemolysis calculations is also essential. In this paper, convergence of the power-law hemolysis model is compared for simple flows, including pathlines with exponentially increasing and decreasing stress, in gradually expanding and contracting Couette flows, in a sudden radial expansion and in the Food and Drug Administration (FDA) channel. In the exponential cases, convergence along a pathline required from one to tens of thousands of timesteps, depending on the exponent. Greater timesteps were required for rapidly increasing (large exponent) stress and for rapidly decreasing (small exponent) stress. Example pathlines in the Couette flows could be fit with exponential curves, and convergence behavior followed the trends identified from the exponential cases. More complex flows, such as in the radial expansion and the FDA channel, increase the likelihood of encountering problematic pathlines. For the exponential cases, comparison of converged hemolysis values with analytical solutions demonstrated that the error of the converged solution may exceed 10% for both rapidly decreasing and rapidly increasing stress.

Evaluation of energy dissipation rate as a predictor of mechanical blood damage.

A long-standing goal in the field of biofluid mechanics has been to reliably predict hemolysis across the wide range of flows that can occur in prosthetic cardiovascular devices. A scalar representation of the complex three-dimensional fluid stresses that are exerted on cells is an attractive alternative for the simplicity that it lends to the computations. The appropriateness of the commonly used von-Mises-like scalar stress as a universal hemolysis scaling parameter was previously evaluated, finding that erythrocyte membrane tensions calculated for laminar shear and extensional flows and for three cases of turbulent flow were widely divergent for the same value of scalar stress. The same techniques are applied in this study to laminar and turbulent flows that each have the same energy dissipation rate. Results showed that agreement of membrane tension between laminar shear and turbulent shear inside an eddy was improved relative to the common scalar stress cases, but disagreement between laminar shear and laminar extension remained the same and disagreement between laminar shear and other turbulent flows increased. It is therefore concluded that energy dissipation rate alone is also likely not sufficient to universally scale blood damage across the range of different flows that can be encountered clinically.

Extending the Power-Law Hemolysis Model to Complex Flows.

Hemolysis (damage to red blood cells) is a long-standing problem in blood contacting devices, and its prediction has been the goal of considerable research. The most popular model relating hemolysis to fluid stresses is the power-law model, which was developed from experiments in pure shear only. In the absence of better data, this model has been extended to more complex flows by replacing the shear stress in the power-law equation with a von Mises-like scalar stress. While the validity of the scalar stress also remains to be confirmed, inconsistencies exist in its application, in particular, two forms that vary by a factor of 2 have been used. This article will clarify the proper extension of the power law to complex flows in a way that maintains correct results in the limit of pure shear.

Mechanisms of tracer transport in cerebral perivascular spaces.

Tracers infused into the brain appear to be transported along channels surrounding cerebral blood vessels. Bulk fluid flow has been hypothesized in paravascular "glymphatic" channels (outer space between the pial membrane and astrocyte endfeet), as well as in the periarterial space (inner space between smooth muscle cells). The plausibility of net flow in these channels due to steady and oscillatory pressures is reviewed, as is that of transport by oscillatory shear-enhanced dispersion in the absence of net flow. Models including 1D branching networks of annular channels and an expanded compartmental model for humans both predict that flow driven by physiologic steady pressure differences is unlikely in both periarterial and paraarterial spaces, whether the spaces are open or filled with porous media. One exception is that a small additional steady pressure difference could drive paraarterial flow if the space is open. The potential that the tracer injection itself could present such a pressure difference is outlined. Oscillatory (peristaltic) wall motion alone has been found to be insufficient to drive significant forward flow. However, a number of hypothesized mechanisms that have yet to be experimentally verified in the brain may create directional flow in combination with wall motion. Shear-augmented dispersion due to oscillatory pressure in channels with a range of porosity has been modeled analytically. Enhancement of axial dispersion is small for periarterial channels. In open paraarterial channels, dispersion enhancement with optimal lateral mixing is large enough that it may explain observed tracer transport without net forward fluid flow.

Deformation of human red blood cells in extensional flow through a hyperbolic contraction.

Flow-induced damage to red blood cells has been an issue of considerable importance since the introduction of the first cardiovascular devices. Early blood damage prediction models were based on measurements of damage by shear stress only. Subsequently, these models were extrapolated to include other components of the fluid stress tensor. However, the expanded models were not validated by measurements of damage in response to the added types of stress. Recent investigations have proposed that extensional stress might be more damaging to red cells than shear stress. In this study, experiments were conducted to compare human red cell deformation under laminar extensional stress versus laminar shear stress. It was found that the deformation caused by shear stress is matched by that produced by an extensional stress that is approximately 34 times smaller. Assuming that blood damage scales directly with cell deformation, this result indicates that mechanistic blood damage prediction models should weigh extensional stress more than shear stress.

Modeling and prediction of flow-induced hemolysis: a review.

Despite decades of research related to hemolysis, the accuracy of prediction algorithms for complex flows leaves much to be desired. Fundamental questions remain about how different types of fluid stresses translate to red cell membrane failure. While cellular- and molecular-level simulations hold promise, spatial resolution to such small scales is computationally intensive. This review summarizes approaches to continuum-level modeling of hemolysis, a method that is likely to be useful well into the future for design of typical cardiovascular devices. Weaknesses are revealed for the Eulerian method of hemolysis prediction and for the linearized damage function. Wide variations in scaling of red cell membrane tension are demonstrated with different types of fluid stresses when the scalar fluid stress is the same, as well as when the energy dissipation rate is the same. New experimental data are needed for red cell damage in simple flows with controlled levels of different types of stresses, including laminar shear, laminar extension (normal), turbulent shear, and turbulent extension. Such data can be curve-fit to create more universal continuum-level models and can serve to validate numerical simulations.

Is bulk flow plausible in perivascular, paravascular and paravenous channels?

BACKGROUND: Transport of solutes has been observed in the spaces surrounding cerebral arteries and veins. Indeed, transport has been found in opposite directions in two different spaces around arteries. These findings have motivated hypotheses of bulk flow within these spaces. The glymphatic circulation hypothesis involves flow of cerebrospinal fluid from the cortical subarachnoid space to the parenchyma along the paraarterial (extramural, Virchow-Robin) space around arteries, and return flow to the cerebrospinal fluid (CSF) space via paravenous channels. The second hypothesis involves flow of interstitial fluid from the parenchyma to lymphatic vessels along basement membranes between arterial smooth muscle cells. METHODS: This article evaluates the plausibility of steady, pressure-driven flow in these channels with one-dimensional branching models. RESULTS: According to the models, the hydraulic resistance of arterial basement membranes is too large to accommodate estimated interstitial perfusion of the brain, unless the flow empties to lymphatic ducts after only several generations (still within the parenchyma). The estimated pressure drops required to drive paraarterial and paravenous flows of the same magnitude are not large, but paravenous flow back to the CSF space means that the total pressure difference driving both flows is limited to local pressure differences among the different CSF compartments, which are estimated to be small. CONCLUSIONS: Periarterial flow and glymphatic circulation driven by steady pressure are both found to be implausible, given current estimates of anatomical and fluid dynamic parameters.