It is observed that a substantial reduction in the lumen of a major coronary art
ID: 3162725 • Letter: I
Question
It is observed that a substantial reduction in the lumen of a major coronary artery is necessary' before there is a substantial reduction in flow through that artery. The following problem provides an explanation for this observation based on a Pikeville flow model and electrical analog. The resistance of the major branch is observed to be about 10% of the total resistance to flow; 90% of the resistance is provided by the smaller arteries which branch off from the main artery. Let R be the resistance of the main artery. Then the resistance terminating the main artery is about 9R. What must be the increase in the resistance, R, of the main branch in order for flow through it to be cut by 55%? Supposed the increase in R is due to a stenosis of the lumen over 10% of the length of the main artery. Find the decrease in radius at the stenosis region to account for the increased resistance. In other words, find r_1/r_0 where L is the length of the main branch R_o is the radius of the main branch lumen 0.1 L is the length of the stenotic region r_1 is the radius of the lumen in the stenotic region.Explanation / Answer
GIven Resistance of main artery = R
Resistance of the rest of the blood vesels = 9R
Rest of the blood vessels are to be considered one resistor, in series with the main artery
A. initial current ( flow) = I
initial resistance = R + 9R = 10 R
initial voltage ( pressure head) = V
V= I(9R)
final current ( flow) = I - 0.55I = 0.45I
let new resistance be R'
R'*0.44I = V = 9IR
R' = 20.4545R
now, R' = 9R + (R + dR) [ where dR is increase in resistance of main artery]
20.4545R = 10R + dR
dR = 10.4545R
B. From poiseuiili's equation we know
Resistance of a tube ( in electrical analogy) R = 8*mu*L/n^2 *pi*r^4(q*)2
so its proportional to Length, and inversly proportional to the 4th power of the rafdius
so, initial Resistance, R = k*L/Ro^4
final resistance, R + dR = 11.4545R = k*0.9L/Ro^4 + k*0.1L/r^4 [ where r is the new radius of the stenotic region]
11.4545k*L/Ro^4 = k*0.9L/Ro^4 + k*0.1L/r^4
11.4545/Ro^4 = 1/r^4
r/Ro = 1/11.4545 = 0.0873
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