Solution

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,

ω_{0} = γ 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,

s (t) = \int ρ (r) e^{- 2 πi k (t) \cdot r} d^{3} r, k (t) = \frac{γ}{2 π} \int^{t} G (t^{'}) d t^{'} .

Here

k

is in cycles per unit length and

γ

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}^{*}

from

T_{2}

. Those are physical signal mechanisms; their biological interpretation remains deferred.

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