Question: from intermediate physics for medicine and biology section 187 problem...
From intermediate physics for medicine and biology
In solving this problem, you will develop a simple model for estimating the radio-frequency energy absorption in a patient undergoing an MRI procedure.
(a) Consider a uniform conductor with electrical con- ductivity σ. If it is subject to a changing magnetic field B1(t) = B1 cos(ω0t), apply Eq. 8.19 to a circular path of radius R at right angles to the field to show that the electric field at radius R has amplitude E0 = Rω0B1/2. (Because this is proportional to R, the model gives the skin dose, along the path for which R is largest.)
(b) Use Ohm’s law in the form j = σE to show that the time average power dissipated per unit volume of material is p = σE02/2 = σR2ω02B12/8 and that if the mass density of the material is ρ, the specific absorption rate (SAR) or dose rate is SAR = σR2ω02B12/8ρ.
(c) If the radio-frequency signal is not continuous but is pulsed, show that this must be modified by the “duty cycle” factor ∆t/TR, where ∆t is the pulse duration and TR is the repetition period.
(d) Combine these results with the fact that rotation through an angle θ (usually π or π/2) in time ∆t requires B1 = 2θ/γ∆t and that ω0 = γB0, to obtain SAR = ( 1 / T R ∆ t ) ( σ / 2 ρ ) ( R 2 / 4 ) B 02 θ 2 .
(e) Use typical values for the human body—R = 0.17 m, σ = 0.3 S m−1—to evaluate this expression for a π/2 pulse.
(f) For B0 = 0.5 T and SAR< 0.4 W kg−1 determine the minimum value of ∆t for TR = 1 s. Also find B1.
(g) For 180◦ pulses, what is the dose in Gy? (This should not be compared to an x-ray dose because this is nonionizing radiation.)