Online Appendix for the following 52/17 October 21 JACC article
TITLE: Myocardial Capillaries and Coronary Flow Reserve
AUTHORS: Sanjiv Kaul, MD, FACC, Ananda R. Jayaweera, PhD
APPENDIX
The effect of a decrease in mean capillary density (MCD) on CFR can be
calculated using the equation:
MCD′/MCD = n′/n = Rc/R′c
(1)
where n and Rc denotes capillary number and resistance, respectively, and primes denote
idiopathic dilated cardiomyopathy (IDC) values1.
The effect of decrease in capillary diameter (D) on CFR can be calculated by the
equation:
Rc/R′c = (D′/D)4
(2)
where again primes denote IDC values1.
Resting coronary flow Qr=ΔP/(Rav + Rc) where ΔP is the coronary driving
pressure and Rav is the combined arteriolar and venous resistance. When n′ decreases
with IDC, R′c increases but is compensated by an equal decrease in Rav keeping Qr
unchanged (Q′r=Qr). Hence, R′av + R′c=Rav + Rc. This compensation occurs until 7080% of the capillaries are destroyed (i.e. until R′c=5·Rc) at which point Rav assumes its
minimum value Ravmin. Hence Ra + Rc=Ravmin + 5·Rc from which we get:
Ravmin=Ra – 4·Rc.
(3)
Hyperemic flow Qh=ΔP/(Ravmin + Rc). Substituting for Ravmin from above we get
Qh=ΔP/(Ra – 3·Rc).
Hence, CFR=Qh/Qr= (Ra + Rc)/(Ra – 3·Rc)
= (Ra/Rc + 1)/(Ra/Rc – 3).
(4)
If we assume that normal CFR ≈ 5 then from the above equation we get Ra/Rc ≈ 4.
Hyperemic flow in the presence of IDC is
Q′h=ΔP/(Ravmin + R′c).
(5)
Substituting for R′c and Ravmin from equations 1 and 3 respectively we get
Q′h=ΔP/{Rav + (MCD/MCD′ - 4)·Rc}.
(6)
Hence CFR in the presence of IDC is:
CFR′=(Rav + Rc)/{Rav + (MCD/MCD′ - 4)·Rc}
= (Rav/Rc + 1)/{Rav/Rc + MCD/MCD′ - 4}.
(7)
Dividing equation 7 by equation 4 we get:
CFR′/CFR=(Rav/Rc – 3)/ {Rav/Rc + MCD/MCD′ - 4}.
(8)
If we substitute Rav/Rc ≈ 4 from above in equation 8 and simplify we get:
CFR′/CFR ≈ MCD′/MCD.
(9)
Thus, the % decrease in MCD equals the % decrease in CFR (blue line in Figure 1).
Even if CFR <5, CFR′/CFR will decrease with reduction in MCD′/MCD.
When capillary D decreases, the same derivations apply with (D′/D)4 replacing
MCD′/MCD.
Hence, CFR′/CFR ≈ (D′/D)4 ,
(10)
which implies that CFR decreases more rapidly with decrease in capillary D than with
decrease in MCD (Figure 2).
The effect of a combined decrease in both MCD and capillary D on CFR (Figure
2) can be derived by altering both variables simultaneously. From equation 1, the
decrease of MCD to MCD′ will result in Rc changing to (MCD/MCD′)Rc, while D
decreasing to D′ will further change it to (MCD/MCD′)(D/D′)4·Rc according to equation
2. Substituting this expression for R′c in equation 5 and proceeding in a similar fashion
we get:
CFR′/CFR ≈ (MCD′/MCD )(D′/D)4 ,
(11)