Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S10 / 2

For a megavoltage X-ray beam, the surface absorbed dose is below the subsurface maximum. Photons generate energetic secondary electrons, mainly through Compton scattering; the downstream electron fluence initially builds up until approximate charged-particle equilibrium is established. Beyond this buildup region, decreasing primary photon fluence, scattering and geometric spreading make the depth-dose curve fall gradually. A penetrating photon beam has a substantial exit dose.

A nearly monoenergetic proton beam deposits a moderate entrance dose, then a sharp Bragg peak near the end of its range. As nonrelativistic protons slow, their electronic stopping power rises approximately as $1/v^2$ over the relevant part of the track. Once the protons stop, the dose falls steeply. Range straggling and energy spread broaden the peak; nuclear interactions add a smaller nonideal tail, so an exact hard cutoff is an idealization. Superposing appropriate incident energies produces a spread-out Bragg peak covering an extended target.^[ref]

The label MV for an X-ray treatment beam specifies the nominal accelerating potential of the electron accelerator. Its bremsstrahlung beam is a broad photon energy spectrum with an endpoint of order $eV$, not a monoenergetic beam whose individual photons all carry the quoted numerical energy. Proton therapy beam energies can instead be specified directly in MeV per incident proton.

A passively scattered proton therapy beam uses distinct devices to control lateral spread, range and distal shape. The lower panel of the figure gives a functional schematic; the exact ordering varies among beamlines. An energy selector or range shifter sets the maximum range. A stepped range-modulation wheel or ridge filter gives a weighted collection of proton energies, forming the spread-out Bragg peak. One or more scattering foils broaden the narrow beam; a double-scattering system can improve lateral uniformity. Ionization chambers monitor delivered fluence and position. A patient-specific aperture or collimator blocks unwanted lateral regions. A shaped range compensator makes the remaining range vary across the field, matching the distal target boundary. Their effects must be combined: lateral collimation alone cannot flatten the depth distribution, and an energy modulator alone cannot set the lateral contour.^[ref]

Use the linear-quadratic radiobiology model, independent fractions with complete inter-fraction repair, and no tumour repopulation. For $N$ fractions of dose $d$, the logarithmic cell kill is $N(\alpha d+\beta d^2)$. Dividing by $\alpha$ defines the biologically effective dose,For the tumour, the original schedule gives $26(2)(1+2/10)=\boxed{62.4{\mathrm{Gy}}_{10}}$. A $3\mathrm{Gy}$ fraction contributes $3(1+3/10)=3.9{\mathrm{Gy}}_{10}$, so precisely $62.4/3.9=\boxed{16}$ such fractions give the same tumour biologically effective dose. The physical total dose changes from $52$ to $48\mathrm{Gy}$.

For the specified normal-tissue $\alpha /\beta =3\mathrm{Gy}$, the original and altered schedules giveThus the normal-tissue model predicts an increase of $9.33{\mathrm{Gy}}_3$, or about $10.8\%$, despite the smaller total physical dose. The smaller normal-tissue $\alpha /\beta$ makes increasing fraction size relatively more costly. These are comparisons within the source's hypothetical radiobiological model and equal-dose geometry, not individualized treatment prescriptions.^[ref]