## Stenosis Flow-Dependence

#### Straight to the Heart

• The Gorlin formula doesn't give us the physical stenosis area because there is a missing piece of information, the value of the discharge coefficient ($$\large C_D$$).
• "Flow dependence" seems to be a loosely applied term referring to the fact that the value of a stenosis index changes depending on the flow rate.  This has significant clinical impact since we can't prescribe the flow rate in a clinical setting so that apparent stenosis "severity" may be difficult to determine or define.  However "flow dependence" means different things for different indices
• The "degree" of flow dependence of various indices can be interpreted from well-founded mathematical principles that often are not appreciated in the clinical-medical literature. The Bernoulli equation demonstrates that the pressure gradient, $$\large \Delta p$$, is roughly proportional to squared flow rate, $$\large q^2$$.  We say mathematically that it is Order $$\large q^2$$; it increases quadratically and boundlessly with flow rate and so is highly (utterly) flow dependent.
• As indices of stenosis severity, the peak flow velocity ($$\large V_p$$ as determined from a Doppler measurement) and the stenosis "resistance" ($$\large R = \Delta p/q$$) are both Order $$\large q$$.  These too are highly flow dependent but of a different Order than the gradient.  ( $$\large V_p$$ is ostensibly independent of body size whereas $$\large R$$ and flow area are not; this is a separate issue.)
• The Gorlin area is Order 1 with respect to flow rate, but is NOT independent of flow rate.  Order 1 does NOT mean that the value is 1; obviously ALL the stenosis indices depend on the orifice size -- the stenosis severity -- or they wouldn't be indices at all.  The stenosis area is "indexed" for flow rate but is not independent of flow rate; it is NOT indexed for body size.  Regardless of flow rate, the Gorlin area has an upper bound; it can't be larger than the physical stenosis area.  This is not to say that the value obtained in the cath lab couldn't be greater than the physical area due to measurement glitches, location of pressure determinations,  use of the incorrect Gorlin constant (pervasive), and a host of other technical considerations.
• ALL hemodynamic indices of stenosis severity (determined from hemodynamic measurements) MUST be flow dependent because they are Reynolds number dependent; the Reynolds number changes with flow rate.  The Reynolds number is the ratio of inertial to viscous forces inherent to the flow and governs the "shape" of the flow, e.g. boundary layer thicknesses and velocity contraction as the flow traverses the stenosis, even for a fixed geometry.   The flow dependent aspects of the Gorlin area are embodied by the discharge coefficient, $$\large C_D$$ which depends on the Reynolds number (specifically) and also the SPECIFIC geometry of the stenosis.   $$\large C_D$$ embodies everything about the shape of the flow,  separate from the simplistic geometry (physical areas).
• Recognition of flow dependence of the Gorlin area led some investigators to propose the valve "resistance" as an index which seemed to be less flow dependent than Gorlin for the restricted range of flow rate of these studies.  However the resistance formula does NOT embody the physics of stenosis flow. The resistance formula applied to a stenosis entails an empirical regression for $$\large C_D$$ ($$\large C_D \sim k \sqrt{\Delta p}$$); this regression is particular to the individual study or studies and implies directly that the Gorlin area can be greater than the physical area (NO).  The resistance regression simply fails when applied to other data available in the literature.  Investigators proposing the resistance did not appear to recognize that $$\large C_D$$ embodies the flow-dependent aspects of a stenosis and implied that the "Gorlin constant" depends on flow rate.
• The interesting aspect of the resistance proposal is that it relates changes in $$\large C_D$$ to the pressure gradient, the latter obviously depending on both flow rate and stenosis severity.  This is in keeping with ALL the available stenosis literature which suggests that $$\large C_D$$ is most variable at low flow rate and for mild stenosis severity (i.e. low $$\large \Delta p$$).  Basically  the value of $$\large C_D$$ is low at a low flow rate (Gorlin area much lower than physical area) and gradually increases with flow rate (Reynolds number) to a plateau value ($$\sim 0.7$$ - $$0.9$$) that's specific to the geometry.  The range over which this occurs is quite dependent on the severity of the stenosis; the more mild the stenosis, the more gradual the change (larger range of flow or Reynolds number to reach the plateau).  One troublesome clinical issue is that it's difficult to recognize the flow dependence of any hemodynamic stenosis index on an individual basis since it requires determinations for a range of flow values.
• Adhering loosely to established engineering principles, I attempted to reconcile these studies by developing an empirical mathematical expression that describes the transition of $$\large C_D$$ across the flow range and encompassing the stenosis severity.  I failed.
• Engineers employ the physical principles of restricted orifice flow to measure flow rate, i.e. as a flow meter.  These are accepted, well-verified practices in hydraulics engineering, but I don't know if they can be extended to our clinical problems because of the range of applicable Reynolds numbers are different.

In this section we'll go over some results of the numerical "experiments" presented thus far, review some bench top results from the literature, and also discuss the meaning and origins of "flow dependence".   I started working on this problem back in the 90s when I felt that the medical literature had taken a "bad hop" with regard to this topic.  I think it's reasonably well straightened out in present day writings, but again the issue of flow dependence was addressed long ago by Colin Clark and I don't have anything to say that hasn't been known for a long time.  I will present data from Clark, Voelker et al, and Cannon et al that I've lifted from their publications to bolster my position.  Before proceeding, however, I want to clarify that this article deals with results (numerical and bench top) where the physical stenosis area DOES NOT CHANGE.  Sure, the physical stenosis area might/does actually change in a clinical setting and that's one of many reasons why the cath lab is no place to learn physics; there are too many uncontrolled variables.  If you're inclined to Jump to Conclusions on this topic, they're listed at the bottom of the page.

### What is "Flow Dependence"?

Let's start by getting on the same page here.   The steady flow Bernoulli equation (sans gravitational terms) appears as:

$$\Large \Delta p = \frac{\rho}{2} \left[ |\textbf{u}_2|^2 - |\textbf{u}_1|^2 \right] + L_{\mu}$$

For background on the setup and what the symbols stand for, please see the PREVIOUS article.  With appropriate definition of the (Gorlin) effective stenosis area, $$A_e$$, the Bernoulli equation is rewritten as:

$$\Large \Delta p = \frac{\rho}{2} \frac{q^2}{A_e^2} = \frac{\rho}{2} \frac{q^2}{C_D^2 A_p^2}$$

$$\Delta p$$ is the pressure "gradient", $$q$$ is the flow rate, and $$\rho$$ is the density of the blood.  The underpinnings of this equation were shown in exasperating detail in the PREVIOUS article.  The Gorlin formula is derived by rearranging the above to solve for $$A_e$$.  The point to recognize is that this is still the Bernoulli equation, just expressed in a different form.  It's not a regression, there's no guesswork.  It describes the physics of stenosis flow.

### Now

So the gradient, $$\Delta p$$ is one of the quintessential descriptors of stenosis severity.  Is $$\Delta p$$ flow dependent?  ...

$$\Large \Delta p = \frac{\rho}{2} \frac{q^2}{A_e^2} = \frac{\rho}{2} \frac{q^2}{C_D^2 A_p^2}$$

Well OF COURSE it is! (read the equation)  This physical equation (not a regression) for $$\Delta p$$ shows that it is determined by something that multiplies $$q^2$$.  In mathematical terms we say that $$\Delta p$$ is ORDER $$q^2$$, ($$O(q^2)$$).  This has a very specific meaning explored elsewhere (follow the link), but the equation indicates that $$\Delta p$$ increases (quadraticallywithout bound with increasing $$q$$; in fact the rate of increase in $$\Delta p$$ also increases with increasing $$q$$!  Forever! (until the model is no longer applicable).

OK. Somebody says that the stenosis "resistance" is not flow dependent, or is less flow dependent than Gorlin area.  Here's the "resistance":

$$\Large \frac{\Delta p}{q} = R= \frac{\rho}{2} \frac{q}{A_e^2} = \frac{\rho}{2} \frac{q}{C_D^2 A_{p}^2}$$

Here's the area:

$$\Large A_e = C_D A_p = q \sqrt{\frac{\rho}{2\Delta p}}$$

So it looks like the valve "resistance" is ORDER $$q$$ ($$O(q)$$), again increasing (linearly) without bound with increasing flow rate.  It doesn't go to infinity nearly as fast as the pressure gradient ($$O(q^2)$$), but it's NOT flow independent.  NOTE: You'll never get me to use the term "resistance" with a straight face when talking about a stenosis.  It's a misapplication of the term and causes no end of confusion.  More on this throughout...

What about the Gorlin area?  That's got a $$q$$ in the equation; must be ORDER $$q$$ also??  NOPE.  The $$q$$ in the Gorlin formula is divided by $$\sqrt{\Delta p/\rho}$$.  From the Bernoulli equation we can see that $$q^2$$ and $$\Delta p$$ are the same ORDER (at least they are for Reynolds numbers significantly greater than 1).  The result of the Gorlin formula is ..... well it's ORDER $$A$$ (area) or squared length ($$O(L^2)$$).   That's obvious enough from looking at the fact that it's used to compute an area, $$A_e = C_D A_p$$; the Gorlin area is the discharge coefficient multiplying the physical area.  For the situation where area is not an independent variable, the discharge coefficient is ORDER 1 ($$O(1)$$)!!  That DOES NOT mean it is equal to 1 or even that it's a constant.  It just means that it's going nowhere in a hurry.

Now the literature includes discussions of "flow dependence" of the "Gorlin constant" and this is erroneous as we've seen PREVIOUSLY.  The constant IS a constant for this particular planet unless you want to get into variability of the gravitational constant depending on your location on earth or relativistic frames of reference.  What we saw in the previous article is that the fluid element trajectories ( thereby affecting viscous losses, velocity at the vena contracta, recirculation region, etc.) are all dependent on the Reynolds number SPECIFICALLY.  Yes, flow rate has a bearing on the Reynolds number.  But the next group of figures are recapped to show how the flow changes with Reynolds number at the same flow rate for a particular geometry.  Here are some velocity contours (see above or the PREVIOUS article for orientation, text with each figure describes the conditions including Reynolds number).

And here are some velocity vectors from the same flow solution.

So we see that the "shape" of the flow changes significantly with Reynolds number.

In terms of the basic variables of flow rate and pressure gradient, all of these flow nuances are embodied by the discharge coefficient (also contraction coefficient and loss coefficient).  These coefficients are all ORDER 1.  They can change a bit over the range of Reynolds numbers but they are well constrained and DO NOT head off for infinity with increasing flow rate.  As a matter of fact, all the coefficients must lie between $$0$$ and $$1$$ (well, almost).

With regard to stenosis flow, "flow dependence" of the pressure "gradient" and the Gorlin effective area are 2 entirely different animals.  So we must always specify for discussion - flow dependence of WHAT.   The pressure gradient is of ORDER $$q^2$$ and grows quadratically in an unbounded fashion with increasing flow rate.  The Gorlin area is ORDER 1 (or ORDER $$A$$ for area).  While it exhibits flow rate dependence specifically through the Reynolds number and discharge coefficient, it's NOT going anywhere fast.  In fact the Gorlin area is bounded by (does not exceed) the physical area.  For clinical purposes we will see that "flow dependence" of the Gorlin area occurs primarily at low Reynolds numbers and is more prominent for mild stenoses.  There are reports suggesting that the Gorlin area may exceed the physical area, but this is not physically plausible and can only be due to measurement error, misuse of conversion constants, misapplication of the simplified formula to a very mild stenosis, et al.

### Discharge, Contraction, and Loss Coefficients

In a PREVIOUS article the mathematical/physical origins of the discharge ($$C_D$$), loss ($$C_L$$), and contraction ($$C_C$$)  coefficients were given in nauseating detail.  I'm now going to try to recap the gist of that whole article in < 20 lines with minimal equations (you have not idea how difficult this is for me).

1. The Gorlin formula computes an effective valve area ($$A_e$$) from a measured pressure gradient ($$\Delta p$$) and flow rate ($$q$$); I will use the terms "Gorlin area" and "effective area" interchangeably.  Think of accelerating the blood from $$0$$ to the velocity $$|\textbf{u}|$$ that you would get from the Simplified Bernoulli equation, $$\Delta p = \rho |\textbf{u}|^2/2$$, but with $$\Delta p$$ specified instead of velocity.  Rearranging the SBE gives $$|\textbf{u}| = \sqrt{2 \Delta p/\rho}$$.  For the measured flow rate ($$q$$), $$A_e$$ is the area that would result in this average velocity, $$A_e = q/|\textbf{u}|$$.
2. When we use the Gorlin formula clinically, we employ time-averaged values for $$\Delta p$$ and $$q$$; this complicates the interpretations given here somewhat but the essential concepts hold (see the PREVIOUS article).
3. The measured $$\Delta p$$ is always greater than what would be required to accelerate the blood to the average velocity at the physical stenosis, $$q/A_{p2}$$; I'm going to call $$q/A_{p2}$$ the "average stenosis velocity".  $$\Delta p$$ is greater because some of the pressure is "lost" to friction/heat, but particularly because of velocity "contraction".  Velocity at the vena contracta is greater than the average stenosis velocity due to specifics of the way the flow occurs (see figures above).  The blood gets accelerated more than would be expected from a simplistic representation of the geometry.
4. The additional $$\Delta p$$, i.e. in excess of that required to accelerate the flow to the average stenosis velocity, is embodied by the discharge coefficient.  The average stenosis velocity is $$q/A_{p2}$$, but $$\Delta p$$ is the result of accelerating the flow to velocity $$q/A_e$$ where $$A_e$$ is less than $$A_{p2}$$.  The discharge coefficient can be thought of as a fraction of the physical area, $$C_D = A_e/A_{p2}$$ (if the stenosis is severe enough).
5. The additional acceleration, i.e. to get to $$|\textbf{u}_2|$$ instead of just $$q/A_{p2}$$, is encapsulated in a contraction coefficient, $$C_C$$.  The average stenosis velocity is $$q/A_{p2}$$, but there's additional acceleration (and pressure gradient) to accelerate the flow to the actual velocity at the vena contracta, $$|\textbf{u}_2|=q/A_u$$; I'm calling $$A_u$$ a velocity area..  The contraction coefficient can be thought of as a fraction of the physical area, $$A_u/A_{p2}$$ (if the stenosis is severe enough).
6. Part of $$\Delta p$$ may be due to pressure loss resulting from friction.  This would be the actual (measured) $$\Delta p$$ minus the part of $$\Delta p$$ due to the acceleration to the vena contracta velocity. In applying the Gorlin formula we're actually equating the pressure loss to an acceleration, i.e. an additional area reduction beyond the velocity area.  The loss coefficient, $$C_L$$, can be thought of as a fraction of the velocity area, $$A_e = C_L A_u$$. (God that's awful.)

We saw in the PREVIOUS article that $$C_D$$ embodies all aspects of the flow, apart from simplistic geometric specifications, that result in the actual pressure gradient and flow.  Here is what I previously called the Engineering Gorlin formula:

$$\Large A_e = C_D A_{p2}= q \sqrt{\frac{\rho}{2 \Delta p}} \sqrt{1-\beta^4}$$

$$\beta$$ is the ratio of the physical stenosis diameter to the parent conduit diameter and can usually be ignored for sufficiently severe stenoses (small $$\beta$$) leading to the following simplification:

$$\Large A_e = C_D A_{p2} = q \sqrt{\frac{\rho}{2\Delta p}}$$

You should recognize this last as the Gorlin formula although I haven't included the averaging process for pusatility so it's for a steady flow situation. I've also left out the stuff about gravity, height of a water or mercury column, 44.3, etc.; all that stuff is just a way of expressing $$\Delta p$$ that's confused the H out of people over the decades.  In what follows, I'm plotting discharge, loss, and contraction coefficients resulting from computational studies (CFD).  I've intentionally included some pretty mild stenoses, so I'm going to HAVE to use the Engineering formula for the numerical solutions.  Here is a plot of $$C_D$$ for a wide range of orifice plate problems - a wide range of stenosis severity; the legend indicates the value of $$\beta$$ (low value of $$\beta$$ is a severe stenosis, approaching 1.0 is no stenosis at all).

Here are the data plotted out to Reynolds number 5000 (gets pretty boring at higher $$Re$$, not much change in coefficient values out there).

I've intentionally focused on the relatively flow dependent segment at the lower Reynolds numbers.  As you see:

1. $$C_D$$ lies between $$0$$ and $$1$$.  Remember: $$A_e$$ is a fraction of the physical stenosis area, $$A_{p2}$$.  $$C_D$$ is the value of the fraction.  It's what you multiply the physical stenosis area by to get the effective area.
2. $$C_D$$ is Reynolds number dependent (i.e. flow dependent).  It is SPECIFICALLY the thing about the Gorlin formula that is flow dependent because it embodies all of the nuances of the flow that change with Reynolds number.
3. The curves exhibit some basic comparable features: there is a transition zone at sufficiently low Reynolds number (or flow rate) where $$C_D$$ is increasing to a plateau value that remains relatively constant for higher Reynolds numbers.  The plateau value might increase or decrease a little, but it can't increase above $$1.0$$. (Obviously I don't mean plateau in a literal sense; it's apparent in the above figure that the value of $$C_D$$ can decrease somewhat with increasing Reynolds number.)
4. This is a plot of flow dependence of the Gorlin formula for a wide range of flow rates and stenosis severity for the simple orifice plate geometry.  It shows the basics of HOW the Gorlin area depends on flow rate (Reynolds number).

So WHY is the Gorlin area flow dependent?  In terms of what we have to work with there are 3 potential reasons:

1. The physical area of the stenosis actually changes.  Sure! Maybe! But not in the above where area was fixed.  I'm here to point out the other reasons so we can all understand them.
2. The nature of the flow causes flow dependence of the contraction; i.e. the flow physics leads to variation in how much the vena contracta velocity exceeds the average stenosis velocity.
3. The nature of the flow causes flow dependence of the pressure "loss" (due to viscosity); i.e. the flow physics leads to variation in how much of the pressure change is due to friction versus acceleration.  This one is pretty much a foregone conclusion and we'll explore the effect shortly.  A decreasing Reynolds number means that the viscous forces in the flow are increasing relative to the inertial (acceleration) forces.   This is why we need the Reynolds number to study fluid mechanics.

In the following plots of the contraction and loss coefficients, I've use the center streamline of the flow to make the calculations.  While $$C_D$$ is NOT dependent on the choice of streamline (it's just about the same $$\Delta p$$ for all of them), the streamlines from inlet to vena contracta have differing amounts of energy loss; so the division of $$\Delta p$$ between acceleration and loss varies with the choice of the streamline.  (I don't know if fluids engineers get balled up with this; Ans: NO.)  In a later article, I'll illustrate how we can define a velocity that encompasses ALL the streamlines to express the conservation of energy principle for the whole tube (warning: it's rather mathy).

In conjunction with CFD solution plots above, this plot of the contraction coefficient illustrates a big part of WHY the Gorlin area is flow dependent.  In general we see that the contraction increases from a relatively low flow value (low Reynold number) to a plateau value at higher flow rate.  The actual values vary with stenosis severity also ($$\beta$$).  I wouldn't say that this is intuitive, but I'll give some rules of thumb for thinking about it shortly.

Next is a plot of the loss coefficient for all the solutions out to a Reynolds number of 500.

This shows a value that's pretty close to $$1.0$$ as soon as we get out of the low Reynolds number doldrums.  Again this is the center streamline we're talking about, the one you're trying to find with your Doppler interrogation.  A value of $$1.0$$ for $$C_L$$, implying no significant energy loss on the centerline, is just what the doctor ordered.  That's telling us that the Simplified Bernoulli equation is a good approximation as long as we're not working on a case with an "excessively" low flow rate (Reynolds number).  If the Reynolds number is too low, the pressure gradient has a significant contribution from energy loss and the SBE assumptions are violated.  We'll still get an effective orifice area from the Gorlin formula, but the area has less and less to do with acceleration through an orifice as the Reynolds number gets very low.  I'm not sure how much of a practical matter this is; I would think this is VERY low flow but we'll see that the literature is conflicting.

Also shown on this plot are some values for $$C_L$$ at higher Reynolds numbers (in the plateau section) slightly above $$1.0$$.  Can that happen?! Well I didn't think so for a long time but apparently that was a misconception on my part! For flow in the cardiovascular system I (almost) always think of friction and viscosity as causing a loss of energy.  While we CAN'T have a NET GAIN of energy.  But we CAN have energy transferred via friction from one streamline to another so that total energy increases on one of the streamlines at the expense of another.  I don't think there's a special name for this; it's just that shear work is done on the "lucky" streamline (receiving the energy) by the adjacent one.   Please refer to a previous digression for details.

Next I'll show some data from the literature to illustrate the consistencies, inconsistencies, and perhaps why there has been a dichotomy in the interpretations.  The following plot represents discharge coefficients from 3 steady flow bench top studies by  Clark.  Data were digitized from the publication and may not be exact.

Each of these models is for a stenosis with circular cross-section and abrupt geometry. The legend indicates the stenosis severity, $$\beta$$ which is the diameter ratio of the stenosis to parent conduit; a smaller number is a more severe stenosis.  The plots exhibit general characteristics that were evident from the computational studies; $$C_D$$ increases from a low value at low Reynolds number to a relatively constant upper value at high Reynolds number.  This plot differs from the numerical studies above in that the transition of $$C_D$$ towards a maximal plateau value is much more gradual.  Let's say they are more flow dependent; i.e. the value of $$C_D$$  transitions over a wider range of Reynolds numbers.  It won't do us much good to talk about how much $$C_D$$ changes over the full range of flow; it's always going to be a number in the $$0.6$$ to $$0.9$$ range.  So flow dependence really comes down to wider range of flow rates (Reynolds numbers) to transition to the "plateau" value.

It's also quite apparent here that the transition is much more gradual for the milder stenosis; a mild stenosis exhibits greater flow dependence, other things being equal.  We can't determine what the upper value of $$C_D$$ might be for the mild stenosis; it's still climbing at the highest Reynolds numbers tested.  However we do know that it's not going above $$1.0$$.

Next are discharge coefficients from a bench top model study by Voelker et al.  Four models are illustrated having an abrupt, orifice plate geometry with circular cross section; values of $$\beta$$, the diameter ratio, are shown in the legend where smaller $$\beta$$ implies a worse stenosis.  This study incorporated pulsatile flow with oscillatory flow parameters adjusted in an attempt to mimic circumstances of medical interest.

Once again we see a transition through the low flow (low Reynolds number) section of the plot to a plateau value at high flow.  For these flow models, it looks like they all have about the same plateau value, $$0.7-0.8$$.  The milder stenoses have a longer transition (more flow dependent).

To further understand flow dependence, it is informative to analyze the functional form of the loss coefficient; you'll need to review the idea of function ORDER for this to make good sense.  As shown above, the loss coefficient has the following form:

$$\Large C_L = \sqrt{ \frac{\rho q^2}{\rho q^2+2 A_{uE}^2 L_{\mu}}} = \sqrt{\frac{ \rho \left[|\textbf{u}_2|^2-|\textbf{u}_1|^2\right] } {2L_{\mu} + \rho \left[|\textbf{u}_2|^2-|\textbf{u}_1|^2\right]} }$$

If the loss term, $$L_{\mu}$$, is ORDER $$q$$ as theory predicts (think of Poiseuille flow where $$\Delta p = q/R$$), then the loss coefficient can be estimated as:

$$\Large C_L \approx \sqrt{ \frac{ q^2}{ q^2+ q \; q_t} }$$

Here $$q_t$$ is a "transition" flow rate.  At high flow rates ($$q >> q_t$$), the $$q^2$$ term dominates the function and it approaches $$1.0$$ (it is ORDER 1).  At low flow rates ($$q << q_t$$), the function is approximated as $$\sqrt{q/q_t}$$; we can see that the function is ORDER $$\sqrt{q}$$. When $$q$$ equals the transition rate, we have $$C_L = 1/\sqrt{2}$$.  While we can't really guess at the form of the contraction coefficient, which changes with the Reynolds number and aspects of the flow geometry, we know it is ORDER 1 and for the moment we'll just call it a constant.  We could just as easily express the relationship in terms of the Reynolds number, with $$Re_t$$ as the transition value; this is likely more appropriate:

$$\Large C_L \approx \sqrt{\frac{Re^2}{Re^2+Re \;Re_t}}$$

Then the following plot depicts a series of contrived curves for $$C_D$$ with the value of $$C_C$$ constant ($$0.8$$) and a range of values for $$q_t$$ shown in the legend.

Now it's necessary to understand that all these curves are the same; the only difference between them is the scale of the ordinate ($$x-$$ axis).  This is readily apparent by making a variable substitution that renders the equation into a non-dimensional form.  Here's the variable substitution:

$$\Large q = \hat{q} q_t$$, i.e. $$\Large \hat{q} = \frac{q}{q_t}$$

where $$\hat{q}$$ is a rescaled, nondimensional version of the flow variable.  The functional form of the discharge coefficient is then:

$$\Large C_D \approx C \sqrt{\frac{\hat{q}^2}{\hat{q}+\hat{q}^2}}$$

$$C$$ is the constant, high flow rate value of the discharge coefficient, set to $$0.8$$ for the illustrative figure. (We're allowed to add $$\hat{q}$$ and $$\hat{q}^2$$ in this situation because they have the same physical units, i.e. nondimensional.) Have no doubt that all of these curves are headed for $$0.8$$ if we look at a large enough value of $$\hat{q}$$ (all curves approach $$0.8$$ asymptotically).   This would work the same way if we had the Reynolds number for the independent variable (as we should).

Note: While the above explanation for choosing $$C_D \sim \sqrt{X^2/(X^2 + X \; X_t)}$$might look like there's some science to it, I might as well admit here that this is empirical; it's really just a function that exhibits some basic features of the observed data.  It's clear from plots of my own CFD studies that much of the change in $$C_D$$ is due to the changing contraction coefficient for which I have no quantitative guidelines to propose. I want it to be clear that I'm not proposing that this is the formula for the discharge coefficient but we'll see that this is still a useful exercise that may shed some light on the flow dependence issue.  Engineers have done a lot of work trying to come up with empirical formulas that will predict $$C_D$$ for a wide range of circumstances.  Please refer to my DISCLAIMER on this topic and the Reader-Harris/Gallagher equation for the discharge coefficient of an orifice plate.  The problem with the R-H/G equation is that it doesn't seem to apply for the relatively low Reynolds number situation we have for the cardiovascular system.  I did come across a statement that there is currently no general formula to predict $$C_D$$ at low Reynolds; I don't know if that's the current state of the art (anyone with knowledge about this please send help).

### Stenosis Resistance

Here I'd like to explain how I think we ended up with resistance, $$R=\Delta p/q$$ applied to stenosis flow in the clinical literature.  But let me give you the bottom line first.  I've spoken to cardiologists about this who stated that the Gorlin formula and stenosis resistance are the "same thing".  That may actually be true from a clinical standpoint, i.e. the 2 might be the same in terms of their ability to distinguish patient groups.  However this article is about physical principles and these are NOT equivalent representations.  You can't have it both ways at the same time in the world of physics except in quantum mechanics (that was wry humor).   Bernoulli-Gorlin is always correct (with stipulations); resistance (alone) does NOT apply to flows where fluid accelerations are significant. Here I'll try reconcile the 2 realms based on the physics.

Cannon et al did some bench top flow experiments for a range of stenosis severity and proposed that the "Gorlin constant" wasn't a constant but varied with pressure gradient in such a way that the pressure - flow relationship was best described by the resistance formula.  We'll get back to this latter statement, but let's start analyzing the situation by looking at an obvious plot from that report, pressure gradient against flow rate (reconstructed from available plots and data).  If the premise is true, then plotting $$\Delta p$$ against $$q$$ should yield straight lines through the origin.  Here are the data as best I could lift them from the publication:

Linear regressions through the origin are included for each stenosis in black.  What do ya think?

I have to admit, these look close enough to straight lines that I can't fault the conclusion.  I'll comment however that the range of flow values appears to be somewhat limited; watch for the range of the flow variable from other reports below.  The Cannon investigators used a flow source to set the flow rate through the apparatus; you can see that flows occur at the same fixed, obvious values for every stenosis.  (That will lead to a limited range of pressure gradient for the mild stenoses.) Now let's take a look at someone else's data.  Here's Voelker et al:

The legend shows the value of $$\beta$$ which is the physical diameter ratio, stenosis diameter to parent conduit diameter (smaller value of $$\beta$$ implies a more severe stenosis).  With a more extensive data range, I hope you can appreciate the fact that these data do NOT represent lines through the origin (they MUST have a 0.0 intercept for the resistance formula to be applicable).  Fitting the data to a parabola (Bernoulli-Gorlin) cures a lot of ills:

Here are Clark's data (as close as I could lift them from the report):

Again Clark's data do NOT support the stenosis flow resistance formula. The mild stenosis (diameter ratio $$\beta = 0.5$$, red data) looks like it might be rather linear, and that's a clue as to what is happening here.  So we've got studies with conflicting results. At the least, however, the approximation of a linear relationship between pressure gradient and stenosis flow requires further scrutiny.   I won't bother you with data of this format from my numerical studies (very parabolic).

Here's my explanation for this.  We've seen that $$C_D$$ depends on the Reynolds number and therefore flow rate. If data are from the "flow-dependent" segment of the relationship (low flow), $$C_D$$ is (could be) ORDER $$\sqrt{q}$$ (or $$\sqrt{Re}$$).  At low flow, this offsets the natural dependency of $$\Delta p$$ on $$q^2$$ and the relationship between $$\Delta p$$ and $$q$$ appears roughly linear.  An empirical relationship between $$C_D$$ and flow rate (or Reynolds number) was shown above where $$q_t$$ is a "transition" flow rate:

$$\Large C_D \sim \sqrt{\frac{q^2}{q^2 + q\; q_t}}$$

Using this formula to approximate the value of $$C_D$$, the relationship between $$\Delta p$$ and $$q$$ appears as follows with $$q_t = 250$$.

For this plot pair, we're looking at the "low flow", highly flow-dependent region of the relationship ($$C_D$$ changing markedly with flow rate).  The maximal value of $$C_D$$ for the above plots was set to $$0.8$$, i.e. the lower plot approaches that value asymptotically at sufficiently high flow rates (much higher values for $$q$$ than shown).  It's apparent that the relationship between $$\Delta p$$ and $$q$$ is linear to good approximation.

However if  $$C_D$$ is principally "flow independent", i.e. at "high" flow, then the relationship between $$\Delta p$$ and $$q$$ is essentially parabolic:

Here the value of $$q_t$$ was $$2.5$$ and the discharge coefficient is near it's "final" upper value (set to $$0.8$$ for the illustration) for much of the flow range.  The pressure-flow relationship is primarily parabolic due to the squared velocity term of the Bernoulli equation. REMEMBER that the curve (empirical function) shown for $$C_D$$ below is THE SAME as the one above (and below); the only difference is the degree of "compression" of the ordinate. There is actually a least squares fit parabola (black line) superimposed on the pressure-flow plot above.

For completeness, an intermediate value of $$q_t = 25$$ is shown below.

The relationship between $$\Delta p$$ and $$q$$ is clearly curved here, and a parabola fits it perfectly, but it might be difficult to tell that this relationship is not a straight line without the pristine "data" of this contrived illustration.

Flow-dependence of the discharge coefficient at a sufficiently low flow rate (Reynold number) can result in a relatively linear pressure-flow relationship for a stenosis. However for a sufficient range of the flow variable, including sufficiently high Reynolds numbers, we will always find a roughly parabolic relationship between pressure gradient and flow rate.

### Making Sense of it All

As previously noted Cannon et al  used a bench top apparatus with a range of model stenosis severity to compute variations in the "Gorlin constant".  Here are their data as near as I could lift from the publication:

In the above the legend represents the cross-sectional area of the physical stenosis.  Taken in whole, the authors felt that these data could be represented by a sideways parabola, i.e. the "Gorlin constant" is approximated by $$k \sqrt{\Delta p}$$.  Please note:

1. My plot results from digitizing the publication and performing some calculations to try to make sense of it.  Please consult the original publication for the original plot.
2. The authors  did not realize (apparently) that they were actually trying to evaluated changes in the discharge coefficient  (multiplied by a constant).  The Gorlin formula includes a constant units conversion factor (sort of) that is misunderstood throughout the clinical literature as $$44.3$$ (for use with pressure units of cm H20; $$51.6$$ is the correct value for pressure in mmHg.)  It's the discharge coefficient that varies, not the physical constant.
3. The data are from a limited range of flow rate. Consequently they did not recognize that the data MUST exhibit an upper bound and that $$k \sqrt{\Delta p}$$ is unsuitable as a functional form (except perhaps for a restricted range of data).
4. The publication did not include sufficient information for determination of the Reynolds numbers involved (I did try).

If $$C_D$$ actually were proportional to $$\Delta p$$,  we'd have the following happy occurrence:

$$\Large A_p = \frac{q}{C_D} \sqrt{\frac{\rho}{2 \Delta p}} = \frac{q}{k \sqrt{\Delta p}} \sqrt{\frac{\rho}{2 \Delta p}} \sim \frac{q}{\Delta p}$$

That would mean that the physical area, $$A_p$$ is inversely proportional to the stenosis "resistance", $$\Delta p/q$$ and we'd have a good way to estimate physical stenosis area.  But here's the problem; the discharge coefficient might be roughly proportional to $$\sqrt{\Delta p}$$ for some restricted range of flow (at low flow rate), but it cannot behave that way in general.  Below are plots suggesting how $$C_D$$ depends on your choice of flow variable ($$X$$).  The blue is $$C_D \sim \sqrt{X}$$, red is $$C_D \sim \sqrt{X^2/(X^2+X \; X_t)}$$.  Being proportional to $$\sqrt{X}$$ the blue curve increases without bound, regardless of your choice of $$X$$. It implies that $$C_D$$ can be greater than 1.0 and that the effective area can be greater than the physical one (no can do).  The red curve, like $$C_D$$, has a strict upper bound; although also empirical, it doesn't exhibit the impossible behavior.

The notion that the resistance formula is "less flow dependent" than the Gorlin area is not valid except for a restricted range of data.

Now what's interesting about the Cannon report is that their approach seems to rationalize the discharge coefficient data from stenoses of differing severity.  We have no right AT ALL to expect or hope that the variation of $$C_D$$ between different stenoses would be a function of $$\Delta p$$ !  In fact we know a priori that it cannot be.   There is a fundamental axiom of fluid dynamics (physics in general) called the Buckingham $$\Pi$$ Theorem which predicts the number of dimensionless parameters that govern the physics ( pressure gradient is not dimensionless ).  That is why you can't have a conversation with a fluids engineer without having the Reynolds number come up.  To my way of thinking, it's a wonder that these investigators ever chose to plot the "Gorlin Constant" against $$\Delta p$$ in the first place.

Given this background, I wanted to see what would happen if $$C_D$$ from other publications/experiments were plotted against $$\Delta p$$.  Here's Voelker et al:

Here's Clark:

The legends show the value of $$\beta$$ in these studies (small $$\beta$$, worse stenosis).  The first point you can see is that these data cannot be described as by $$k \sqrt{\Delta p}$$.  The data reach a plateau and cannot increase above the limiting value of $$1.0$$ as we've already determined.  Secondly, it's no trick to describe the data once it has reached the plateau.   Our problem is to somehow make sense of what is happening to $$C_D$$ in the flow dependent section of the data where it transitions from a low value to the plateau.  While the flow dependent data appear to fall on a single curve in the Cannon report, this approximation doesn't work for data from other reports.

Still there is something compelling about this apparent relationship and I sought to try to make sense of it with some analysis.  (I hope they don't revoke my engineering degree for this, but I'll probably retire soon anyway and won't need it.) While $$\Delta p$$ is due to both the inertial and viscous forces  I reasoned that $$\Delta p$$ stems primarily from acceleration (inertial forces) for a sufficiently severe stenosis.  By plotting against $$\Delta p$$, the Cannon report in essence is a plot $$C_D$$ against a measure of the inertial force.  Maybe we can fix this and come up with a dimensionless parameter that would do a better job than $$\Delta p$$?  What we need is a ratio of inertial to viscous forces, not for fixed geometry (like the Reynolds number), but for the range of geometries inherent to the range of stenosis severity.  For starters, here is an estimate of the pressure change due to the acceleration through the constriction:

$$\Large \Delta p \approx \frac{\rho}{2} \frac{q^2}{A_2^2} \sqrt{1-\beta^4} = \frac{\rho}{2} |\textbf{u}_1|^2 \frac{\sqrt{1-\beta^4}}{\beta^4}$$

The above is simply the pressure change required to accelerate the fluid due simplistically to the physical area change alone, i.e. neglecting the specifics embodied by the discharge coefficient. The pressure change due to the viscous flow is more difficult; it depends on how the fluid layers shear against each other and consequently on the specifics of the flow.  If we think of the Poiseuille resistance we have $$\Delta p = q \; 8\mu l/\pi r^4$$.  We know that this formula will NOT characterize the pressure (energy) losses for stenosis flow.  Poiseuille flow does not occur at a stenosis and we also have the problem of determining what to use for the length ($$l$$) and the radius ($$r$$) in the formula.   However this exercise is not about calculating the pressure loss.  We're only trying to scale  the loss relative to the acceleration in a mathematical sense.  To this end I decided to incorporate the Poiseuille relationship and try the geometric average of the 2 diameters for $$l$$ and also for the $$r^4$$ in the denominator, i.e. $$l = \sqrt{d_1 d_2}$$ and $$r^4 = \sqrt{d_1^2 d_2^2}/16$$ where $$d_1$$ and $$d_2$$ are the diameters of the parent and physical stenosis respectively.

$$\Large \Delta p \approx \frac{8 \mu \sqrt{d_1 d_2}}{\pi \frac{1}{16} d_1^2 d_2^2} q = \frac{32 \mu |\textbf{u}_1| \sqrt{\beta}}{d_1 \beta^2}$$

Then the ratio of pressure change, inertial to viscous:

$$\Large \frac{\rho |\textbf{u}_1| d_1}{64 \mu} \frac{\sqrt{1-\beta^4}}{\beta ^{\frac{5}{2}}} \sim Re \frac{\sqrt{1-\beta^4}}{\beta^{\frac{5}{2}}}$$

In the final analysis, the constants ($$64$$, $$\pi$$) are omitted since they don't contribute anything of import.  So this parameter has a ring truth to it; it's the Reynolds number multiplied by a dimensionless quantity that embodies the severity of the stenosis through  $$\beta$$.  Note that the numerator $$\sqrt{1-\beta^4}$$ doesn't do much; we saw previously that this number is pretty close to $$1$$ for sufficiently severe stenoses.  I'm assuming this is witchcraft to the person reading this, so let me spell this out in practical terms.  Cannon et al  plotted $$C_D$$ (essentially) against $$\Delta p$$ (without having much reason for doing so).  $$\Delta p$$ depends on the flow rate and the stenosis severity.  These investigators felt that this rationalized the data so that $$C_D$$ could be expressed as a single-valued function of $$\Delta p$$ ($$C_D \sim k \sqrt{\Delta p}$$.  However this doesn't work for data from other reports; we need to find a new ordinate that does a better job, $$Re \sqrt{1-\beta^4}/\beta^{5/2}$$  So think of this principally as the Reynolds number multiplied by a number, $$1/\beta^{5/2}$$, that rescales the ordinate depending on the stenosis severity.  Here's what that looks like using data from Voelker et al:

If we're on the right track, we might hope that this conglomerate data from different stenosis severity cases might describe a single curve.  I think this looks better that the plot above with $$C_D$$ plotted against $$\Delta p$$.  To go one final step, we can admit that the value of the exponent for $$\beta$$ ( in the denominator ) is really pure guesswork.  We can leave it as an open parameter and try to perform a statistical procedure to estimate it's value.

$$\Large X \equiv Re \frac{ \sqrt{1-\beta^4}}{\beta^n}$$

$$\Large C_D = C_m \sqrt{\frac{X^2}{X^2 + X_t X}} = C_m \sqrt{\frac{\frac{Re^2 (1-\beta^4)}{\beta^{2n}}}{\frac{Re^2 (1-\beta^4)}{\beta^{2n}}+X_t \frac{Re \sqrt{1-\beta^4}}{\beta^{n}}}}$$

This looks like H, but most of the information ($$Re$$, $$\beta$$) is available from the studies.  Our problem is to estimate $$C_m$$ (the plateau value of the discharge coefficient), $$X_t$$ (transition value for $$X$$), and $$n$$, the exponent of $$\beta$$ (Levenber-Marquardt / damped least squares. Thank-you Dr Wolfram / Mathematica!).  And the result for the Voelker et al data:

Here the ordinate is the previously noted $$X = Re \sqrt{1-\beta^2}/\beta^n$$ where the regression value of $$n = 2.42$$.  Parameter estimates for the curve fit, $$C_D = C_m \sqrt{X^2/(X^2 + X_t X)}$$ were $$C_m = 0.8825$$ and $$X_t = 8943$$.  And here is data from the study by Clark:

The "regression" here is not that but the result of  "hand-fitted" regression parameters.  Actual least-squares regression parameters exhibit unsupportable values (e.g. $$C_m > 1.0$$) which is not really surprising given the relatively short range available for the independent variable and lack of overlap for the 3 stenoses.

### Are We Done Yet?

I've mucked with this problem off and on for years and the bottom line is that I haven't  found a suitable way of unifying the data from these various studies of stenosis flow. None of the bench studies discussed here was designed to allow determination of parameters for prediction of $$C_D$$ for which we would need a lot more data and a systematic approach.  My own numerical studies don't adequately predict the outcome of the bench experiments and I haven't been able to determine the source of the discrepancies.  In particular, the bench experiments in the literature are much more flow dependent than the CFD results; differences between studies of Clark, Voelker et al, and my own CFD results are dramatic.  Computational studies should be able to predict the outcome of the "wind tunnel" and this article depicts transition Reynolds numbers ("flow dependence") that differ wildly between the bench top experiments (Clark, Voelker et al) and the numerical ones despite comparable stenosis severity. Sure, the specific stenosis geometry varied between the studies.  However varying the stenosis geometry considerably for the CFD studies didn't reproduce the degree of flow dependence seen at the bench (See NEXT article). The Cannon et al study is excluded from the comparisons since the Reynolds numbers can't be estimated from the report.  Suffice it to say that their discharge coefficient data appear to be limited to the highly flow dependent region and the conclusion that $$C_D \approx k \sqrt{\Delta p}$$ (implying a roughly linear relationship between $$\Delta p$$ and $$q$$) was premature.  Stenosis flow "resistance" is untenable as a functional form for the pressure-flow relationship through a stenosis, but this doesn't necessarily disqualify it as a clinical index.

There is a great deal written on this topic in the engineering literature; engineers commonly use a precisely defined flow constriction geometry as a means of determining flow rate by measuring pressure gradient. The  Reader-Harris/Gallagher equation is an example where empirical methods have been used to estimate $$C_D$$ for a range of "stenosis" severity (simple orifice plate geometry) but this doesn't appear to be applicable to the low Reynolds flows suitable for clinical applications.

1. For clinical purposes, flow dependence of the Gorlin area can be characterized as the range of flows (Reynolds numbers) over which the discharge coefficient undergoes significant variation.   In general terms, the Gorlin area increases with increasing flow rate regardless of whether the physical stenosis area does.
2. Flow dependence of the Gorlin area is most prominent at lower Reynolds numbers (flow rate).
3. Flow dependence occurs over a wider range of flows for a less severe stenosis. (Milder stenoses are more flow dependent).
4. The discharge coefficient is the specific flow dependent aspect of the Gorlin formula and the available literature does not suggest a coherent method for predicting it as a function of stenosis severity (or geometry) short of well-designed bench experiments or computational studies. Stenosis "resistance" is not the answer.

Cannon SR, Richards KL, Crawford M. Hydraulic estimation of stenotic orifice area: A correction of the Gorlin formula. Circulation 1985;71:1170–1178

Cannon SR, Richards KL, Crawford MH, et al. Inadequacy of the Gorlin formula for predicting prosthetic valve area. Am J Cardiol 1988;62:113–116.

Clark C. The fluid mechanics of aortic stenosis—I. Theory and steady flow experiments. J Biomech 1976;9:521–528.

Voelker W, Reul H, Nienhaus G, et al. Comparison of valvular resistance, stroke work loss, and Gorlin valve area for quantification of aortic stenosis. An in vitro study in a pulsatile aortic flow model. Circulation 1995;91:1196–2204.