Past exam of the medical physics course of the University of Cambridge

Historical Part III paper. Exact student-answer availability for 2012 remains unverified; papers with advertised student answers are skipped. The public original is linked from each paper.

Source: Cambridge examination PDF, physical pages 477–478; printed paper 230.

Biology and clinical applications are deferred in this draft. The generic mathematical, detector and wave-physics parts below are included.

TODO: Biology deferred at the user's request: the cancer-treatment account, clinical dose choices, treatment planning and clinical interpretation of the numerical model.

For the specified energy-loss model, multiply $dE/dx=-k/E$ by $2E$ and integrate from the entrance:The positive stopping power is $S=-dE/dx=k/E$. The model reaches zero kinetic energy atAt the entrance,Half a centimeter before the formal endpoint, $E^2=2k(0.5\mathrm{cm})=770\mathrm{Me}{\mathrm{V}}^2$, soThe signed derivatives are the negatives of the quoted stopping powers. This inverse-energy stopping-power model deposits energy more strongly toward the endpoint. Its divergent endpoint derivative is integrable, giving finite total deposited energy $E_0$; a real stopping curve includes finite energy spread, range straggling, scattering and a more accurate low-energy stopping law. These numbers are consequences of the supplied ideal equation.

TODO: Biology deferred at the user's request: clinical contrast interpretation, applications, and the head-injury MRI-versus-CT comparison.

The generic physics of magnetic resonance imaging combines a main static magnet, radiofrequency transmit and receive coils, controlled magnetic field gradients, signal electronics and reconstruction. A transverse radiofrequency pulse excites nuclear spins near their Larmor frequency, ${\omega }_0=\gamma B_0$. The evolving transverse magnetization induces a received signal; phase coherence and relaxation determine its time dependence.

Gradients encode position into phase. In a simplified homogeneous receive-sensitivity model,Here $\mathbf{k}$ is in cycles per unit length and $\gamma$ is in radians per second per tesla. Echo sign changes, pulse sequence timing and receive sensitivity belong in a more complete signal model. Sampling this MRI k-space and reconstructing its data produces an image; the measured samples are not already real-space pixel values.

The physical signal depends on spin density, longitudinal recovery and transverse dephasing. Timing changes their weighting. Static field inhomogeneity adds reversible dephasing beyond intrinsic transverse relaxation, motivating echo techniques and distinguishing $T_2^{\ast }$ from $T_2$. Those are physical signal mechanisms; their biological interpretation remains deferred.

TODO: Biology deferred at the user's request: clinical registration applications and their validation requirements.

Generic image registration estimates a coordinate transformation aligning corresponding features in two images. A rigid transform permits translation and rotation, an affine transform additionally permits linear deformation, and a nonrigid model permits spatially varying displacement. The transformation class expresses assumptions about what changes between the acquisitions.

One can minimize a discrepancy between $I_1(T(\mathbf{x}))$ and $I_2(\mathbf{x})$, using interpolated image values and an explicit optimization procedure. Squared intensity differences suit some comparable acquisitions; mutual information can compare statistical dependence when image contrasts differ. Landmarks, feature descriptors or additional geometric constraints provide other information. Smoothness or other regularization controls overly flexible transformations. A favorable objective value does not by itself prove correct feature correspondence.

TODO: Biology deferred at the user's request: biological radiation effects and clinical risk assessment.

In pulse-echo ultrasound, a transducer emits a pulse and records returning echoes. For a straight round-trip path in a medium of assumed sound speed $c$, an echo delay $t$ gives depth $z=ct/2$. Reflections depend on acoustic-impedance changes, while attenuation, scattering and propagation geometry affect the received amplitude.

An A-mode ultrasound scan displays echo amplitude against inferred depth along one beam. A B-mode ultrasound scan maps received amplitude to brightness and assembles many beam directions or translated beam lines into a spatial image. An M-mode ultrasound scan repeats one beam line and displays its depth profile versus time, making motion along that line visible. Thus the display axes and acquisition pattern differ even when the pulse-echo principle is shared.

For a pulse lasting ${\tau }_p$, a simple axial-separation estimate is $c{\tau }_p/2$. Lateral resolution depends on aperture, wavelength, focusing and depth. Shorter pulses and broader bandwidth can improve axial resolution; an assumed sound speed or a multiple-reflection path can instead misplace an echo. Brightness is not a direct universal map of composition.

Doppler ultrasound uses frequency or pulse-to-pulse phase changes from moving scatterers. In a simple backscatter geometry, $\Delta f\simeq 2f_{0}v\cos \theta /c$. Inferring speed requires the beam-to-motion angle; pulsed sampling can alias frequencies, and an unknown angle leaves a velocity-projection ambiguity. This describes generic acoustic acquisition rather than a clinical diagnosis.

TODO: Biology deferred at the user's request: biological tracer behavior and clinical PET applications.

The generic physics of positron emission tomography uses electron-positron annihilation. A slowly moving electron-positron pair predominantly produces two nearly opposite photons with energy close to $m_e{c}^2=511\mathrm{keV}$ each. A ring of detectors records approximately simultaneous interactions; a coincidence identifies a coincidence line of response along which an annihilation could have occurred. Finite Positron travel before annihilation and nonzero pair momentum limit how directly this line identifies the original emission location.

Many such measurements constrain an emission distribution through a tomography model. Attenuation, scattered coincidences, unrelated random coincidences, finite detector resolution and dead time alter the observed counts and require calibration or modeling. With useful time-of-flight information, the arrival-time difference further localizes position along the line: a timing uncertainty $\delta t$ corresponds to a position scale $c\delta t/2$. The coincidence data therefore provide spatial constraints, rather than each event directly supplying a complete three-dimensional image point.