Physiologic Blood Flow is Turbulent

Full text (Open-Access): https://www.nature.com/articles/s41598-020-72309-8

The pioneering work of J.R. Womersley1 and his coworkers2,3 essentially laid the foundation of modern hemodynamics research. By developing a time-dependent one-dimensional exact solution of the Navier-Stokes equation, Womersley showed that blood flow in main arteries can be described by a Fourier decomposition of the cardiac harmonics3,4. This work has been later extended to account for wall elasticity5,6 and non-Newtonian blood viscosity7. The Womersley flow model (WFM) became the founding principle upon which modern blood hemodynamics studies are based. Researchers have assumed, based on WFM, that blood flow is essentially laminar, and transition to turbulence (or presence of disturbed flow, which is a poorly defined hemodynamic term often used in medical research) shifts the blood hemodynamics leading to the initiation of vascular diseases such as brain aneurysms or atherosclerosis8,9. Such disease association is attributed to the mechano-sensing properties of endothelial cells that make them responsive to various flow properties10. Therefore, the accurate identification of blood flow regimes is an essential step in characterizing the hemodynamic patterns that govern endothelial cells mechanobiology9. We have previously shown how the inaccurate assumptions of blood viscosity and the misinterpretation of wall shear stress (WSS) 9,11 have impacted intracranial aneurysm research, leading to inconsistently varying and contradicting results9. Moreover, we have recently shown that turbulence exists in pulsatile multiharmonic flow of mean Reynolds number in idealized model of intracranial aneurysm flow using particle imaging velocimetry (PIV)12. Similar transitional and turbulent regimes were detected in intracranial aneurysm by other research groups in vitro13 and in silico14-18. In a seminal article, Jain et al18 showed that at peak systolic conditions, intracranial aneurysm exhibit random velocity fluctuations and kinetic energy cascade at in the parent artery. Subsequently, Jain et al19 argued that such complex flow patterns may alter our understanding of aneurysm growth and rupture. Turbulence casts complexity not only on the hemodynamics of intracranial aneurysm, but also on such of carotid occlusive disease20,21. It is established that disturbed flow is associated with carotid stenosis22 and atherosclerosis23. While the exact nature, characteristics and regime of such disturbed flow is ambiguous in literature8,24, numerous studies investigated its parametric relationship with the diseases progression. Kefayati et al25 showed that turbulence intensity varies considerably with the variation of carotid stenosis severity in vitro using particle image velocimetry. Grinberg et al26 found mixed states of laminar and turbulent flows downstream the stenosis using high-resolution CFD model. To that end, it can be argued that the contemporary theory of non-laminar flow regimes (i.e. transitional/turbulent/disturbed) are considered to be associated with pathologic conditions while physiologic condition is believed to coincide with laminar and stable flow regime. The purpose of this article is to introduce an alternate paradigm where blood flow is seen as turbulent in both physiologic and pathologic states. However, the notion and interpretation of turbulent flow in this alternate paradigm is different from such of fully developed turbulent flow that are commonly perceived in literature through Reynolds criteria and Kolmogorov-Obukhov statistical theory of isotropic homogenous turbulence. This is thoroughly discussed later in this article.

The methodology and approach used in our article are schematically illustrated in the following figure. Blood flow is represented using two sets of data: the first set is obtained from the exact solution of Navier-Stokes equation as represented by the WFM. The second set is obtained from DU measurements of the carotid artery of healthy volunteers (two of the authors).  The exact solution of WFM was established using boundary conditions from the Hae-Mod open access database (http://haemod.uk/) to offer reproducible evidence of our work. The difference between the two datasets is obvious; WFM describes ideal flow while DUS measurements describe real physiologic flow. By ideal we mean the flow described by WFM is one-dimensional, Newtonian, single phase and is bounded by rigid circular wall. Real physiologic flow is three-dimensional, non-Newtonian, multi-phase, and is bounded by elastic semi-circular wall. The purpose of the present work is to demonstrate that the existence of turbulence in physiologic blood flow is inherited from its simplest mathematical formulation. To that end, the purpose of using DUS measurements is to show that the existence of turbulence, as shown by the exact solution, can also be demonstrated in vivo. First, Lyapunov exponents were calculated for both datasets using the open-source code provided by Wolf et al27. Then, a criteria for hydrodynamic stability of multiharmonic pulsatile flow was derived from the Reynolds-Orr equation and used to examine the global instability of blood flow, as it is represented by both datasets. Finally, the kinetic energy cascade from both datasets was analyzed in space and frequency domains.

Schematic illustration of the methodology and approach used in the present work. Space–time velocity fields of physiologic blood flow were obtained from (a) exact solution of the Navier–Stokes equation as described by Womersley flow model with physiologically realistic boundary conditions from HaeMod database and (b) in vivo Doppler ultrasound measurements of healthy volunteers. Physiologic blood flow was tested for three characteristics of turbulence that are (c) sensitive dependence on initial conditions, (d) global instability and (e) turbulent kinetic energy cascade. Three methods were used to test such characteristics namely Lyapunov exponents test, hydrodynamic stability analysis and calculation of kinetic energy cascade in space and frequency domains.

REFERENCES

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