A patient with a mean aortic flow speed of (Chapter 5) is scanned

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Radiation physics and diagnostic imaging · Nuclear medicine, dosimetry and radiotherapy

Life at Thermal Energy, Ch Echoes, precession and the therapeutic beam

Problem

A patient with a mean aortic flow speed of

0.17\,\mathrm{m\,s^{-1}}

(Chapter 5) is scanned, treated and modelled in three ways. Take

c_s = 1540\,\mathrm{m\,s^{-1}}

k_BT/e = 26.7\,\mathrm{mV}

at body temperature (the voltage scale of a Nernst potential, Chapter 4), and the nuclear spin level gap at

1.5\,\mathrm{T}

\Delta E/k_BT = 9.9\times10^{-6}

. ((a)) An ultrasound probe at

f = 4\,\mathrm{MHz}

is held at

\theta = 45^\circ

to the aorta. Calculate the Doppler shift

\Delta f

. ((b)) The potassium Nernst potential of this patient's cells has magnitude

|E_{\rm K}| = 90\,\mathrm{mV}

. At that equilibrium, the Boltzmann factor

e^{-ze|E_{\rm K}|/k_BT}

for an ion of valence

z=1

(Chapter 4) is the ratio of the smaller potassium concentration to the larger. Compute this factor, and the Boltzmann factor

e^{-\Delta E/k_BT}

for a proton spin at

1.5\,\mathrm{T}

. By what factor do the two exponents

ze|E_{\rm K}|/k_BT

and

\Delta E/k_BT

differ? ((c)) The patient's tumour (

\alpha_{\rm LQ} = 0.3\,\mathrm{Gy^{-1}}

\beta_{\rm LQ} = 0.03\,\mathrm{Gy^{-2}}

) and late-responding tissue (

\alpha_{\rm LQ} = 0.15\,\mathrm{Gy^{-1}}

\beta_{\rm LQ} = 0.05\,\mathrm{Gy^{-2}}

) are treated with

20

fractions of

2.5\,\mathrm{Gy}

. Using

\ln\mathcal{S} = -(\alpha_{\rm LQ}\mathcal{D} + \beta_{\rm LQ}\mathcal{D}^2)

per fraction, calculate the surviving fraction

\mathcal{S}

of each after the full course.

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