## Stenosis Flow

#### Straight to the Heart

• Stenosis flow is a relevant intersection of fluid dynamics and the clinical evaluation of disease.
• Misconceptions abound as to how the flow occurs at a stenosis, why various indices are "flow-dependent", and what we actually determine by employing equations like Gorlin, Bernoulli, or continuity.  A stenosis is NOT a "resistance".
• Understanding the physics of stenosis flow will enhance your understanding of the clinical problems.

Flow through a restrictive orifice, whether a stenosis or regurgitant valve, is one of the most relevant intersections of fluid dynamic theory and clinical cardiology.  Direct application of physical principles has greatly aided clinical interpretations, beginning with the Gorlin studies and including the application of the simplified Bernoulli equation to Doppler studies.  Yet incorporation of these principles has been slow and imperfect.   Pertinent clinical studies largely recapitulate concepts and findings that have long been available in the  fluid dynamics literature; in some cases they have served to muddy the waters where clarity was available.  My perspective is that little has been added to this topic since the seminal studies of Colin Clark (apologies to others who have contributed significantly).

The fluid mechanics of aortic stenosis—I. Theory and steady flow experiments.  Clark C. Journal of Biomechanics. Volume 9, Issue 8, 1976, Pages 521-528
The fluid mechanics of aortic stenosis—II. Unsteady flow experiments. Clark C. Journal of Biomechanics. Volume 9, Issue 9, 1976, Pages 567-568,

Clark contributed several other papers on the subject also. The problem is that these papers do not reside in the clinical literature and are written in a language that is foreign to many clinicians – that of mathematical equations that express physical principles.  It is my hope that the following might supply sufficient background and provoke interest that access to those papers will improve.

I will also attempt to provide insights on the topic with the aid of a technique not available to Clark – computational fluid dynamics (CFD).   Subsequent webpages will display a wide range of figures generated from approximate solutions of the Navier-Stokes equations that describe flow behavior for Newtonian fluids.  The solutions shown are for simplified geometries, not the complex, dynamical anatomy that typifies the mammalian heart.  The solutions shown here were computed using homegrown software and it remains a possibility that the software contains bugs that I will never have the time to find and fix.  Rather than pack up and go home, I will point out some of the inaccuracies in the solutions as a learning opportunity, to show you why they are implausible.  Hence the CFD solutions may be considered as illustrations of concepts, not to be construed as accurate renditions of specific circumstances in the circulation.  Since it's assumed that you may not be familiar with CFD, I'll point out that accurate CFD solutions are employed ubiquitously today in a host of engineering endeavors such as designing aircraft, watercraft, landcraft, turbomachinery, etc. etc.

For those in the know, my software employs a so-called high-resolution finite volume method for incompressible flow with a small stencil pressure projection method for enforcement of mass conservation.  It utilizes the venerable $$k-\epsilon$$ turbulence model which I concede is likely inadequate for the situation at hand.  I owe my understanding of these techniques to Dr. Kenneth Powell (University Of Michigan Aerospace Engineering) and hope I will not embarrass him by dropping his name and putting this stuff out there for public scrutiny.  If you know anything about CFD or fluids engineering, please go elsewhere. I'm just a citizen scientist trying to bridge the gap between physical and medical science.

The intention here is to understand the essence of stenosis flow physics, not the nuances resulting from highly individualized geometry.  To that end, many of the solutions will be shown in a 2 dimensional format that occurs naturally from geometries that have an axis of symmetry.  For example, this figure:

Corresponds to the 3 dimensional geometry suggested by this figure:

In both these cases, fluid flow enters from the left-hand side of the figure and must pass through an orifice plate, essentially a partition with a circular orifice.  These particular figures happen to show isovelocity contour lines with velocity also depicted in color. Hence the location of the vena contracta is clearly indicated just downstream of the stenosis orifice plate.

Note that CFD stenosis flow data are available for download, manipulation, and visualization at this website.  The software is explained and demonstrated on THIS page and sample problems including stenosis flow solutions can be downloaded HERE.  Others (engineers) have employed CFD to study flow through constrictions and you wouldn't need to look far to find better than what's here.

### History and Disclaimer

I don't think I have to worry about any fluids engineers reading this, but I'd like to make it clear that the real source for this topic is the engineering literature.  Fluids engineers intentionally place constrictions in piping to control or measure flow rate; cardiologists are stuck with the constriction but employ the same physical principles as the engineers to try to evaluate the "severity" of the constriction.  The landmark paper by Gorlin & Gorlin was principally an application of Torricelli's law to evaluate stenosis severity (among other contributions, Torricelli invented the barometer).  I encourage all cardiologists to check out the original Gorlin paper where you will note that one of the authors has an MSE (master of science in engineering).  Whether they said it that way or not, a component of the paper consists of attempting to determine an appropriate empirical value for a thing called the discharge coefficient, $$C_D$$, that I'll discuss in nauseating detail in subsequent articles.  I think the notion was that we might be able to use a fixed value for $$C_D$$, or maybe one for aortic stenosis, one for mitral, etc.  The notion that $$C_D$$ is a fixed value seems to have gotten stuck in the collective cardiology mind and there seemed to be a bit of a kerfuffle back in the 90s upon finding that the area computed by the Gorlin formula changes with flow rate - that the formula is "flow-dependent".

But this has been known for a long time and investigated fully.  While stenotic cardiac valves are not necessarily of fixed cross-sectional area, the discharge coefficient is well-known to change with flow rate or, more specifically, Reynolds number; $$C_D$$ is specifically the flow-dependent aspect of the Gorlin formula. If you want to start looking at this, check out the Wiki entry on flow through an Orifice Plate. Where the Gorlins wanted to find an acceptable fixed number for $$C_D$$, below is a formula used by engineers to estimate it under specified circumstances (Reader-Harris/Gallagher equation for an orifice plate):

$$\Large C_D = 0.5961+0.0261\beta^2-0.216\beta^8+0.000521 (\frac{10^6\beta}{Re_D})^{0.7}$$ $$\Large +(0.0188+0.0063 A) \beta^{3.5} (\frac{10^6}{Re_D})^{0.3}$$ $$\Large + (0.043+0.080e^{-10{L_1}}-0.123e^{-7{L_1}})(1-0.11A)\frac{\beta^4}{1-\beta^4}-0.031(M'_2-0.8{M'_2}^{1.1})\beta^{1.3}$$

$$\Large \beta$$ is a ratio of diameters (constriction orifice to parent pipe)

$$\Large Re_D$$ is the Reynolds number for flow in the parent pipe

$$\Large A = \left(\frac{19000 \beta}{Re_D}\right)^{0.8}$$

$$\Large D$$ is the diameter of the parent pipe

$$\Large M_2' = \frac{2 L_2'}{Re_D}$$

This is a formula devised from exhaustive experimentation to estimate the value of $$C_D$$ for a simplistic geometry!  $$C_D$$ is not a constant! The Gorlin formula is flow-dependent!  In the above, $$L_1$$ and $$L_2'$$ have to do with the exact placement of the "tappings", locations for pressure measurement (follow the link for more information).   The configuration and shape of the orifice plate as well as all aspects of the piping and measurements are standardized by the American Society of Mechanical Engineers.  The point is that flow through a stenosis (constriction) is a much studied topic and there are people all over the world who know a lot about it.  I'm not one of them; I'm just trying to make the topic more available.