It's Not Just About Stenosis Area



Straight to the Heart

  • The Gorlin effective orifice area depends on both the physical orifice area and the Reynolds number, but that's not all.  The hemodynamic area is affected also by any and all geometric factors that influence the "shape" of the flow through the orifice.  This includes the specifics of both axial and cross-sectional shape.  NOTE: If you don't think the shape would have much affect on how the flow will occur, DON'T get into an airplane!
  • An abrupt change in physical area results in an abrupt change in flow geometry, tending to cause a larger component of flow velocity perpendicular to the axial direction.  Abrupt geometry leads to greater velocity contraction  (smaller contraction coefficient, \(\large C_C\), which means a smaller effective orifice).
  • A circular orifice has the smallest perimeter for a given cross-sectional area.  This also translates to a less abrupt change in flow geometry and (potentially) a reduction in viscous energy loss upstream of the vena contracta.  An oval or slit orifice geometry, or a multi-lobulated slit (e.g. reminiscent of fuse aortic cusps) are examples of increasing orifice perimeter length relative to area and tend to result in a smaller hemodynamic area for a given physical area.  The hydraulic diameter ( 4 times the area divided by the perimeter) is sometimes utilized in engineering to define the orifice size for non-circular cross-sectional geometry.  
  • The discharge coefficient, \(\large C_D\), embodies ALL of these geometric factors and is a means by which they can be distilled and quantified.
  • Just as the Gorlin formula doesn't compute the physical stenosis area, knowledge of the latter is not sufficient to fully imply the hemodynamic effect of the stenosis.  My view is that the heart experiences the hemodynamics, not the physical orifice.  However it may be quite important to know as much as possible about the physical construction of the stenosis in a clinical setting, e.g. to fix the darn thing.
  • A stenosis results in a loss of hydraulic energy and requires the heart to generate greater (transmural) pressure to offset this loss.  An under-appreciated aspect of stenosis flow is that the diseased valve or outflow tract isn't the principle location of the loss -- it's downstream of the vena contracta.  ("Falling out of a skyscraper doesn't hurt; it's the sudden stop at the bottom.")  It's the abrupt emergence of the flow jet into the aorta that sets the stage for the shear layer, turbulence, and energy loss -- the cost to the heart.  While this doesn't help us much in a clinical setting, it may augment your understanding of stenosis physics to recognize that we could go a long way towards "fixing" a diseased valve by altering the downstream geometry to a (much) more gradual return to normal cross-sectional area. (Good luck with that.) 



 For quite some time the "Holy Grail" for determination of stenosis severity seemed to be the valve (stenosis) cross-sectional area -- the physical area.  My interpretation of the past literature is that there was an assumption that the physical area is what was provided by Gorlin formula; the previous articles here have reviewed that fact that this latter is not the case.  While this article may provide some insights as to why this may be a vain quest, the discussion also is a microcosm for a general clinical question that I feel doesn't receive enough attention.  

Why do you want to know that!?

The physical stenosis area sounds like a good thing to know; it seems like it ought to be independent of hemodynamic circumstances and therefore separate from the myriad of factors which might affect the outcome of a disease and the need for a treatment.  But does the heart "know" what the physical area is?  How would it know? Is an area of 8 cm2 good enough ....... for a whale?  Does "critical stenosis area" depend on the heart that has to push the blood through it?  I was holding forth on this topic to a surgeon one time who impressed me with my own ignorance.  When asked why it was necessary to know the physical stenosis area he replied (to the effect) that you need to know that the thing you're going to stick in there (for valve replacement) is bigger than what's already there.  (Yeah, I guess that would be pretty important.) But even as he imparted this revelation to me, I was thinking that what we'd really like to know is that the effective area of the replacement is greater than the valve in place.

To help think rationally about the above question, I often play a game with myself - I imagine that I already have the gadget that would give the exact value of the physical stenosis area (or ventricular volume, or ejection fraction, or \(\tau\), 3D stress-strain, etc.)  Would I be happy then?  Would it accomplish the goal I wanted it for in the first place?  (Could I retire from proceeds from the patent?!)  I want a Maserati also, but the answers to such questions for me are usually "Naw".  I think most of us in medical fields are interested not only in caring for our patients, but also understanding as deeply as possible how and why the thing works as it does -- to be able to explain what has happened and even predict outcomes with sufficient prior knowledge.  This is very much the purview of engineers also who need to be able to predict that the airplane will fly (for example) before it's built and actually how fast it will go, it's fuel consumption, etc. etc.  It's my fond wish that some of these pages might inspire more of the medical folks, engineers, physicists, and other scientists to work together on some of the problems and questions in medicine.  The world has certainly changed in that direction since I started out.  (That sounds like the end of my book, but no.)

Effect of Longitudinal Shape






 Effect of Orifice Cross-Sectional Shape


Effect of Entrance Velocity Profile






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