Scientists and sailors: may your compass be true

In my opinion, progress in science is usually made by dropping assumptions.

David Bohm, FRS, as quoted from first-hand conversation by Paul Davies, in About Time: Einstein’s Unfinished Revolution

Fata Morgana is is a complex form of superior mirage that is seen in a narrow band right above the horizon. It is an Italian term named after the Arthurian sorceress Morgan le Fay, from a belief that these mirages, often seen in the Strait of Messina, were fairy castles in the air or false land created by her witchcraft to lure sailors to their deaths.

Scientists, like sailors, are sometimes lured by the mirage of assumptions during the process of deduction. To answer causal questions in science, scientists use deduction to build a conjecture, hypothesis or theory and test it. Assumptions are logical simplifications of facts; they intend to to make the relationship between observations intuitive and physically plausible to facilitate further investigation. The history of science is a history of the rise and fall of assumptions, from the Galilean planetary motion theory to the standard atomic model.

It was not until 1950s that scientists had a mathematical description of blood flow in arteries. In 1955, J.R. Womersley published his seminal work on the dynamics of arterial blood flow and described the relationship between velocity and pressure in cardiac waveforms. He used a brilliant combination of experiments and mathematical derivation to devise what was later known as the Womersley solution of the Navier-Stokes equation. His work became the founding principle for vascular hemodynamics in the following decades.

During the same era, research on turbulence has been making long strides towards a unified theory. The works of Kolmogorov and Obukhov were translated to English around 1954. Kraichnan developed his theory by the early 1960s. By the early 1970s, the world had a unified theory for isotropic homogenous turbulence; entirely established on the assumptions made by Osborne Reynolds in the late 19th century. Reynolds described transition from laminar to fully developed turbulence in Newtonian steady flows subjected to finite perturbations. Along a century, the fluid mechanics community developed a universal description of turbulence, however, for flows that are far from being similar to that of blood in arteries!

Reynolds demonstrated that a steady laminar pipe flow would lose stability and transits into turbulent state when the ratio between inertia and viscosity (i.e. Reynolds number) exceeds 2300. Transition happens when a flow accumulates kinetic energy budget to a certain limit where it becomes unstable to finite disturbances. The flow starts to lose its kinetic energy in what is known as the cascade process, develop complex nonlinear formations known as vortices and eventually enter a chaotic state marked by sensitive dependence to initial conditions. The entire theory that we have about turbulence is based on numerous assumptions, accumulated through the scientific journey from 19th to 20th century.

The assumptions made in modern hemodynamics theory can be summed up under one statement: researchers believe that multiharmonic pulsatile shear-dependent flow in flexible tubes should be treated similar to steady shear-independent flow in rigid tubes. Therefore, it is believed that blood flow regime in arteries should be characterized based on the Reynolds criteria, and turbulence should be described using the Kolmogorov-Obukhov statistical theory of turbulence. Literature records list thousands of research articles, addressing all and every vascular disease, established on such assumptions. It has become a grounded belief that laminar flow promotes vascular health while turbulent flow compromises it. The assumptions have completely ignored the multiharmonic pulsatile nature of blood flow and reduced it to a binary problem and answer.

Do harmonics matter?

In 2013, a group from the University of Virginia led by Professor Brett Blackman published a paradigm-shifting study in Nature Communications. The group used simplified experimental setup to raise the question about the role of blood flow harmonics in endothelial cells biology. Effective blood flow harmonics range from 0.1 to 12 Hz with metabolic bandwidth that can reach up to 50 Hz. They evidently showed that harmonics regulate endothelial cells inflammatory phenotype. Since these harmonics exist independent from Reynolds number, they are regime-independent. The study confirmed and extended earlier works of Dai et al (2004) and Himburg et al (2007). But how to characterize harmonics in laminar flow differently from such in turbulent flow?

In 2020, Saqr et al published the first complete evidence that physiologic blood flow is turbulent. They used in vivo measurements, analytical solutions of the Womersley flow model, chaos theory and nonlinear stability theory to explain why multiharmonic pulsatile flow cannot be treated as laminar flow – in the sense of Reynolds criteria and Kolmogorov-Obukhov theory of turbulence. They evidently showed that blood flow in arteries possesses three characteristics of turbulence namely kinetic energy cascade, global instability and sensitive dependence on initial conditions.

There is sufficient evidence, now, to revisit the principles of the vascular hemodynamics theory. Not only because of its illusive assumptions, but also because of its poor achievements. It is unable to help us understand endothelial cells mechanobiology and is not useful in characterizing disease nor in clinical perspectives. The regime-based classification of hemodynamic environments, that is based on Reynolds criteria, should be now abolished. The properties and features of turbulence can provide better theoretical frame and more inclusive markers. A new future of hemodynamics research starts now!