In our recent ultrasound refresher course, I tried to give a talk on the vagaries of stenosis graduation (mainly) for extracranial stenoses. The gist of the talk is outlined in the following notes.

## The Bernoulli principle

While Bernoulli’s equation is rather intricate, the underlying principle of conservation of flow along a stenosed tube is simple. Consider a tube with a short stenosis with laminar flow of a Newton fluid. Then the bigger area A1 multiplied with the flow velocity v1 (take psv for simplicity, although this is not really correct) should be the same as the smaller area A2 multiplied with v2. Thus the increase of flow is proportional of A1 / A2, so that the reduction to a third of the area leads to an increase by the factor 3 of the flow velocity, a reduction to a tenth leads to ten times the flow velocity in the stenosis.

## Adding friction

Since we usually don’t observe velocities higher than 5 m/sec, there must be a limiting principle and this is the resistance offered by the stenosis, which reduces flow in the whole vessel. This resistance can be approximated by the law of Hagen-Poiseuille and is proportional to the length of the stenosis, the inverse of r^4 and the inverse of viscosity. Again, this only holds for laminar flow and the case of blood offers some more complications, but the core message is: the longer the stenosis the higher the flow reduction. Also the flow reduction grows much more with decreasing vessel diameter than the flow velocity increase by Bernoulli’s principle can compensate. Very tight and long stenoses show a flow velocity reduction despite there high grade. If you find a psv of only 2,4m/sec this might therefore mean either a mere medium grade stenosis or a very tight stenosis (near occlusion).

## Spencer’s curve

Taking these two principles into account, Spencer and Reid (in their brilliant 1979 stroke article) deduced the famous curve now known as Spencer’s curve (see Alexandrov’s papers for a more detailled exposition).

Since at the time duplex sonography was technically not feasible, the Spencer curve is based on the theoretical assumption of a 2 mm stenosis and thus does not correct for the length of the stenosis (as well as the other factors mentioned below). This explains why the cw-doppler-data in their paper does not really fit the theoretical model. Still it is the best approximation we have to a theoretical foundation of stenosis quantification.

## Measuring diameters

### Diameter instead of area

Most of the studies have been done with angiographic imaging of stenoses or ultrasound measurements of the stenosis diameter rather than the area. Of course, the area could easily be calculated from the diameter (r^2 pi) if it were a circle, but then it isn’t. In ultrasound we could measure the area itself (if no shadowing artifacts are present), yet no one does really. Therefore you should remember that most of calculations of the stenosis area from the diameter are systematically invalid.

### Where, when and how to measure diameters

For some absurd reason, Europeans kept to the local stenosis degree (i.e., diameter of perfused lumen divided by the original vessel diameter), at least in their ECST trial, while the NASCET trial used the more reasonable distal stenosis degree (i.e., minimal lumen diameter divided by non-stenosed distal ICA diameter). Since the ICA bulb varies in its bulbiness and distends with age and blood pressure (and with stenosis degree), the local stenosis degree is usually a shot in the dark. The distal (NASCET) degree suffers from pseudoocclusion, i.e., collapse of the distal vessel in very high grade stenosis. All attempts to calculate the NASCET stenosis degree from ECST and vice versa are irrational.

Being faithful to both traditions, I measure all the lumina (minimal lumen, original vessel lumen, distal lumen). The only really interesting number is the residual diameter (combined with the length of the minimal lumen), because this determines the hemodynamic compromise.

## Other important factors

### Viscosity

Neither is blood a Newton fluid, nor is all the viscosity (rarely measured today) explained by the hematocrit alone. Yet the hematocrit is an important number to factor into your interpretation of ultrasound data. You should note it.

### Turbulence

Every vessel wall abnormality leads to small perturbations of flow and thus turbulence. Turbulence reduces anterograde flow and thus reduces the distal pressure after a stenosis. While very hard to quantify it is essential to mention turbulent flow when you see it. Remember that to distinguish retrograde (i.e. turbulent) flow from flow increase with aliasing you need to look at the color bar on the side of your duplex image, noting that flow increase jumps over the upper limit of the color spectrum while retrograd flow (usually) passes the black zero flow region.

### Elasticity

decreases over the lifetime, even more if severe hypertension or calcification of vessel walls is present. This, again, is very hard to quantify, but often easy to recognize qualitatively in your duplex image, when you recognize the pulsation of the vessel wall. Reduced elasticity has to lead to increased flow velocities.

### Pulsation

Since blood flow is not continuous but pulsatile and this in varying shapes, we should really be using mean flows in our stenosis calculations. This has historically not been done. As a consequence, valve abnormalities (aortic stenosis, insufficiency) have to be factored in, when we try to calculate the stenosis degree from peak systolic velocities.

### Blood pressure and atrial fibrillation

The (pulsating) blood pressure is the driving force of cerebral blood flow, trying to overcome venous and intracerebral pressure as well as the distal blood pressure offered by collaterals (see below). At least, you should note the blood pressure and relativize your stenosis graduation in cases of extreme values. When the patient has atrial fibrillation, you probably should use an “average” heartbeat rather than the extreme values. But bear in mind that absolute arrhythmia is a risk factor for arterioarterial embolization in itself.

### The geometry of the stenosis

usually is far from being that of a tube. Rather, the blood flow curves around plaques, rotating and hitting small plaques on the distal wall. Again, the effects are impossible to quantify, but at least the geometry should be noted. The shape of the stenosis area should be remarked upon, if it isn’t circular.

### The role of collaterals

The pressure difference along a stenosis is not only determined by the resistance of the stenosis itself, but also by the collateral blood flow which leads to an increase of distal pressure (mostly but not only in diastole). This leads to a reduction of the blood flow velocity in the case of good collateralization, thus also reduced flow velocities.

## The danger of a stenosis

In the neurovascular clinic, I try to estimate four risks of a stenosis: the hemodynamic risk (what happens if the stenosis were to increase?), the embolic risk (how high is the risk of an embolic event from the stenosis?), risk factors and other risks.

### Hemodynamic risk

The hemodynamic risk is determined by

- the current hemodynamic compromise (jet flow velocity, CCA flow vs. ICA flow, pulsatilities, MCA flow, CO2 reserve)
- the dynamics of the stenosis (how long has this been going on? was there time to develop collaterals?)
- the completeness of the circle of Willis (variations such as A1 hypoplasia, Pcomm hypoplasia)
- secondary stenoses in the collateral circulation.

The problem is that we cannot foresee whether the stenosis will be slowly progressive or suddenly close up (as in plaque rupture). At least in asymptomatic stenoses I require CT- or MR-angiography to determine the completeness of the circle of Willis.

### Embolic risk

The embolic risk is determined by

- Plaque morphology and
- Plaque type
- Whether the atherosclerotic process is active or burnt out.

As in coronaries, it is not reasonable to revascularise every severe asymptomatic stenosis. But in a patient where the overall atherosclerotic process is currently active (after an NSTEMI, say), we can expect the plaques to rupture. - Previous embolic events can be noted on MRI.
- Emboli detection
- Is the anti-platelet medication working? Multiplate or similar tests.

### Risk factors

- Did the patient stop smoking? How long ago?
- Can we use high dose statins in this patient? Statins are highly effective against plaque deterioration, but also have serious side effects (less exercise tolerance, diabetes, muscle problems), especially in the high doses we like to use for severe stenosis.
- Is the patient’s blood pressure controllable? Note that severe stenosis lead to very labile blood pressures as one of the most important sensors of the system is damped.
- Exercise? Although this has not been studied properly in carotid artery stenosis, I surmise that health by fitness should improve the prognosis of carotid artery stenosis.

### Other risks

- Central or mixed sleep apnea syndrome – very prevalent among ICA stenosis patients, leading to a bag of systemic problems, not the least being poor blood pressure control.
- Bad blood pressure control (see above)
- Development of secondary stenoses in the collateral vessels (contralateral, ECA, …)
- Development of a collateral rete with its danger of bleeding

## Summary

I don’t see any better physical theories coming. Also, we can never expect better data than NASCET and this is a bad foundation. Therefore you have to tackle all the complexities outlined above and refrain from simplifying an ICA stenosis to a mere number (always the worst approach).

## References

- Spencer and Reid’s famous and fabulous 1979 stroke paper
- Alexandrov’s discussion of the paper and it’s legacy
- The german ICA ultrasound guidelines (which do make the mistake to try to find a number for every stenosis, but otherwise are quite good). International guidelines more or less follow the same ideology.