This document discusses fractional flow reserve (FFR) and instantaneous wave-free ratio (iFR), which are used to assess the severity of a stenosis. It notes that calculating FFR requires making assumptions and simplifications from a fluid mechanics perspective. Specifically, it assumes hydraulic models, neglects venous pressure, assumes equal myocardial resistances, and faces challenges in measuring distal pressure due to dynamic pressure effects, which are more problematic at higher flow rates. The document suggests iFR measurement of distal pressure could be even more inaccurate due to typically higher velocities during the wave-free period.

1. Bio-­Fluid Mechanics and
Fractional Flow Reserve
Rosaire Mongrain
Olivier F. Bertrand

2. Disclosure
I have no conflict of interest to declare

3. 1. Assessment of a Stenosis
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Geometrical/anatomical Functional (Gould)
V=Q/A
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Taylor Series Expansion of the Pressure Gradient Function:

4. 1. Assessment of Stenosis Severity: FFR
HealthyDiseased

5. Conceptual Jump
The conceptual jump is an analogy between hydraulic and
electrical circuit
FLOW PRESSURE RESISTANCE
1. Assessment of Stenosis Severity: FFR

6. Thus the pressure based calculation of FFR requires:
-­‐ Pao aortic pressure [mmHg],
-­‐ Pd distal stenosis pressure [mmHg],
-­‐ Pv venous pressure [mmHg],
-­‐ Rmyo,n healthy myocardial resistance [mmHg·∙s·∙ml-­‐1],
-­‐ Rmyo,d diseased myocardial resistance [mmHg·∙s·∙ml-­‐1]
1. Assessment of Stenosis Severity: FFR

7. The concept is appealing as the healthy myocardial blood flow is
unknown
Since the average venous pressure (right atrial pressure) is
approximately 5mmHg (De Bruyne 1995), it can be assumed to
be negligible (Pijls 1994) and we get:
However, given that the FFR is a ratio, neglecting the venous
pressure may, in certain situations, induce non-­negligible errors
Neglecting Pv could be significant in high Pv patients
(hemodialysis up 10-­12 mmHg)
nmyoao
dmyod
RP
RP
FFR
,
,
/
/

2. FFR Simplification (Pv)

8. The FFR calculation is further simplified by assuming that the diseased
and healthy myocardial resistances become equal under induced maximal
hyperaemic conditions (in the absence of microvascular disease)
Autoregulation of coronary blood
flow based on perfusion pressure
During maximum physiological dilation,
there is a linear relationship between
the perfusion pressure and the
coronary blood flow
coronary blood flow
Pappano, A.J., and Wier, W.G., (2012) Cardiovascular physiology, Elsevier/Mosby, Philadelphia, PA.
As such, in a clinical setting, under
induced hyperaemic conditions (through
the injection of a vasodilatory agent),
the FFR can be approximated by:
2. FFR Simplification (Rmyo,d – Rmyo,n)

9. 2. FFR Simplification (Rmyo,d – Rmyo,n)
Autoregulation of coronary
blood flow based on perfusion
pressure
ao
d
P
P
FFR 
It should be noted that the hydraulic
resistance corresponds to the
inverse of the slope of auto-­
regulation curve.
In other words, the steeper the slope
the less resistance.
In principle, under hyperemic
conditions, the slope would be
steeper (even lower resistance but
for a relatively narrow coronary
perfusion pressure).

10. • In fact, there is another implicit conceptual jump in using
hydraulics
• Hydraulics is a model lumping of fluid mechanics
• In going from fluid mechanics to hydraulics, complex local
phenomena are reduced to a simple global resistance R
• The effects of phenomena like flow separation, vortices,
recirculation, turbulence are all lumped into a unique
resistance value R
3. FFR: The measurement of Pd ?
ao
d
P
P
FFR 

11. Recirculation
3. FFR: The measurement of Pd ?

12. Obviously, the post-­‐stenotic has to be avoided for the measurement of Pd.
Where is it fine from a bio-­‐fluids point of view to do the measurement ?
?
3. FFR: The measurement of Pd ?
Referring to a cartoon representation of a stenosis (Wong 1986):
ao
d
P
P
FFR 

13. From a bio-­‐fluids perspective, we refer to the notion of velocity field recovery
In other words, where does the flow become undisturbed past the stenosis ?
This is answered with the equation: L/D ~ 0.06 Re (empirical for laminar flow)
with L the length before the flow is redeveloped, D the vessel diameter and
Re the Reynolds number:
(Re ~ 250 for a normal vessel, Re ~ 500-­‐600 for mid range
stenosis and Re > 1000 for a severe stenosis)
For a mid range stenosis normal vessel, L ~ 0.06 X 500 Re X 3.5 mm ~ 10 cm !
This would be impractical for side branches

DU0
Re 
3. FFR: The measurement of Pd ?

14. As a result, the measurement of Pd is affected by a certain error due to
local dynamic pressure gradients in the post-­stenotic region
It should be noted that these dynamic pressure gradients are very
sensitive to geometry changes (this includes the presence of
wires, catheters, wall deformations, side branches)
Also the higher the velocity, the higher the problematic of dynamic
pressure gradients
In fact, for a fluid, the notion of resistance is linked not only to its
viscosity but also to the flow dynamics
3. FFR: The measurement of Pd ?
ao
d
P
P
FFR 

15. Viscous kinetic
The flow also influences the Resistance. In other words, the Resistance
is flow dependent.
This can be viewed with the Gould equation:
Flow dependent Resistance
3. FFR: The measurement of Pd ?

16. 4. What about iFR ?
However, as discussed above, from a bio-­fluid perspective, higher
velocities are associated with higher flow disturbances and higher flow
dependent stenosis resistance depence
This, in principle, would make the measurement of Pd more
problematic to dynamic pressure gradient effects
In general, the flow velocity is
higher and pressure is lower
over the wave free period
In principle, it results in lower
microvascular resistance during
the free period in comparison to
the complete cardiac cycle
https://en.wikipedia.org/wiki/Instantaneous_wave-­‐free_ratio#/media/File:Coronary_flow_velocity_and_microvascular_resistance_over_the_wave-­‐free_period.jpg

17. • The transition from Flow to Pressure requires a hydraulic model
• The hydraulic model is a simplification of fluid mechanics
• Neglecting the venous pressure Pv may not always be precise
• The equality of the myocardial bed resistances is an assumption
• The measure of the distal pressure Pd can be problematic for
several reasons (dynamic pressure effects)
• From a bio-­‐fluids point of view, the measurement of Pd could be
more problematic with IFR (dynamic pressure effects)
16
5. The measurement of FFR
(Summary):