This example shows how we can use calculus to find rates of change in biology.
Poisuille’s law can be applied to blood flow in capillaries, arteries, and veins.
The law of laminar flow states that: v = (P/4nL)(R^2-r^2) In this equation n is the viscosity of the blood and P is the pressure difference between the ends of the tube. If P and L are constant, than v is a function of r with a domain of [0,R]
(In the following equations r2 stands for r with a subscript of 2 and r1 stands for r with a subscript of 1). The average rate of change of velocity as blood moves from r = r1 outward to r =r2 equals the change in v divided by the change in r = [v(r2) -v(r1)]/[r2-r1]
If we let the change in r as it approaches zero, we get a velocity gradient , that is the instantaneous rate of change of velocity with respect to r: Velocity gradient = limit as the change in r approaches 0 of the change in v divided by the change in r, which equals dv/dt
If we use the equation for the law of laminar flow, and then find out the derivative of it, which is: dv/dr = (P / 4nL)(0 - 2r) = - (Pr) / (2nL)
For one of the smaller human arteries we can take n = 0.027, R= 0.008 cm, L= 2 cm, and P = 4000 dynes/cm^2, which gives:
V = [4000 / (4(0.027)2)](0.000064 - r^2)
Which equals approximately 1.85 x 10^4 (6.4 x 10^-5 - r^2)
At r = 0.002 cm the blood is flowing at a speed of:
v(0.002) = approximately 1.85 x 10^4 (64 x 10^-6 - 4 x 10^-6)
=1.11 cm/s
The velocity gradient at that point is:
dv/dr when r = 0.002 = -[4000(0.002)]/[2(0.027)2 = approximatly -74 (cm/s)/cm
When we change the units from centimeters to micrometers we can see what the formula really does better.
1cm = 10,000 µm
So now the radius of the artery is 80 µm. The velocity at the central axis is 11,850 µm/s, and it decreases to 11,110 µm/s at a distance of r=20 µm. dv/dr = -74 (µm/s)/µm This tells us that when r = 20 µm, the velocity is decreasing at a rate of about 74 µm/s for each micrometer that we move away from the center.
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