Question

You are examining bloodflow in a blood vessel using Doppler sonography. Original frequency of ultrasound emitted by the device is 2.50 MHz. After mixing the incoming frequency reflected from a bloodstream in the blood vessel with the original frequency of the device, the device produces a beat frequency of 974 Hz. You are asked to calculate the diameter of the blood vessel if the volumetric flow rate of the blood stream in the blood vessel is measured to be 5×10-6m3/s. Use the speed of sound in human tissue as 1540 m/s.

Answer 1

freqeuncy sent, f = 2.50 M Hz
beat frequency = 974 Hz
frequency received, f' = 2.50 M Hz + 974 Hz

= 2.50*10^6 + 974 Hz
speed of sound in blood, v_sound = 1540 m/s

let v is the speed of blood

we know,

f' = f*(v_sound + v)/(v_sound - v)

f'*(v_sound - v) = f*(v_sound + v)

f'*v_sound - f'*v = f*v_sound + f*v

f'v_sound - f*v_sound = f*v + f'*v

v_sound*(f' - f) = v*(f' + f)

v = v_sound*(f' - f)/(f' + f)

= 1540*974/(2.50*10^6 + 974 + 2.50*10^6)

= 0.30 m/s

let d is the diameter of the blood vessel

volume flow rate, Q = A*v

Q = (pi*d^2/4)*v

==> d^2 = 4*Q/(pi*v)

d = sqrt(4*Q/(pi*v))

= sqrt(4*5*10^-6/(pi*0.3))

= 4.61*10^-3 m (or) 4.61 mm <<<<<<<---------------Answer