This document discusses coronary blood flow and functional assessment of stenosis. It begins by introducing concepts of fluid mechanics applied to blood flow. It then summarizes different methods to assess stenosis functionally, including quantitative coronary angiography (QCA), fractional flow reserve (FFR), and instantaneous wave-free ratio (iFR). The key assumptions and limitations of FFR are discussed, particularly regarding the measurement of distal pressure and simplification of fluid dynamics. Virtual FFR is also introduced as a computational fluid dynamics (CFD) approach to model blood flow and calculate pressure without some of the assumptions of FFR.

1. Coronary blood flow primer for the
interventional cardiologist
Rosaire Mongrain
Professor
Department of Mechanical Engineering

2. Disclosure
I have no conflict of interest to declare

3. Claude Navier (1785-1836) & Georges Stokes (1819-1903)
derived the equation of motion for a viscous fluid:
fpV
Dt
DV
maF   2
Jean Louis Poiseuille (1799-1869) obtained his analytical
solution:
)1( 2
2
max
R
r
VV 
Viscous effects
1. Biofluids Background

4. 2. Functional assessment of stenosis: QCA
Ds/Dn or As/An
Diameter or Area
Longitudinal axis distance [cm]
0.0 0.5 1.0 1.5 2.0
IntraluminalPressure[mmHg] 50
60
70
80
90
100
110
36%
51%
75%
91%
Recovery
Drop
(disturbances)
∆𝑃 = 𝐶𝑣 𝑉 + 𝐶𝑒 𝑉2
V=Q/A
Functional (Gould)
Anatomical
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Losses become dominant as the
degree of stenosis increases
CFD calculations

5. The conceptual jump is an analogy between hydraulic and
electrical circuit (underlying assumptions ?)
FLOW PRESSURE RESISTANCE
3. Functional assessment of Stenosis: FFR
- Pao aortic pressure [mmHg], - Pd distal stenosis pressure [mmHg], - Pv venous pressure [mmHg],
- Rmyo,n healthy myocardial resistance and Rmyo,d diseased myocardial resistance [mmHg∙s∙ml-1]
Healthy
Diseased
Conceptual Jump

6. 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 ∝
𝑆𝑢𝑝𝑝𝑙𝑦
𝐷𝑒𝑚𝑎𝑛𝑑 𝑂𝑥𝑦𝑔𝑒𝑛
Pappano, A.J., and Wier, W.G., (2012) Cardiovascular physiology, Elsevier/Mosby, Philadelphia, PA.
4. FFR Simplification (Rmyo,d / Rmyo,n)
ao
d
P
P
FFR 

7. • In going from fluid mechanics to electrical analogs, complex
phenomena are reduced to a simple global linear resistance R
• The non-linear effects of phenomena like flow separation,
vortices, recirculation, turbulence are all lumped into a unique
resistance value R
4. Implicit assumptions and limitations
Recirculation
Flow asymmetryFlow asymmetry
Secondary flow

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

9. 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 
5. FFR: The measurement of Pd ?

10. As a result, the measurement of Pd is affected by a certain error due to
local dynamic pressure gradients in the post-stenotic region
5. FFR: The measurement of Pd ?
ao
d
P
P
FFR 
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

11. • The transition from Flow to Pressure requires a hydraulic model
• The electrical analog is conceptual jump (simplification) of fluid
mechanics
• Neglecting the venous pressure Pv
• 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)
11
5. The measurement of FFR
(Summary):

12. 6. iFR assumption
• In the case of iFR the simplifications are implicit (the needed
linear relationship is justified with the free wave diastolic
zone)
Linear hypothesis
(and linear ratio)
When stenosis becomes
significant, the non linear
phenomena become more
important
The corresponding myocardial
resistance is always higher than
the hyperemic value

13. 7. Virtual FFR
1. generation of the geometry
2. meshing of the computational domain
3. application of fluid properties
4. application of boundary conditions
5. solution of the flow field in time and space
6. processing of the simulations results.
http://www.hippokratia.gr/index.php/cur
rent/1129-patient-specific-arterial-
system-flow-oscillation
Frattolin et al. (2015),
Comput. Biol. Med.

14. • Efficiently model global dynamics of large and complex cardiovascular domains
Fluid Domain Electrical Domain
Calculated Parameters
Pressure Voltage
Flow Current
Domain Properties
Frictional/viscous resistance Resistance
Arterial compliance Capacitance
Fluid inertia Inductance
Governing Equations:
with:
7. Virtual FFR
(Lumped Model 3D + 0D)

15. 15
7. Virtual FFR
(Results)