Magnetic Resonance Imaging (MRI)

MRI is arguably the most physically elegant imaging modality in routine clinical use. It produces high-resolution, soft-tissue-contrast images of the body's interior using no ionising radiation, exploiting instead a quantum-mechanical property of the hydrogen nucleus: nuclear-spin angular momentum and its precession in an applied magnetic field. The physics begins with nuclear magnetic resonance (NMR) — a phenomenon first observed in 1946 by Bloch and Purcell, who shared the 1952 Nobel Prize — and ends with computer-reconstructed tomographic images. This lesson rebuilds MRI from the proton-precession picture up through the role of gradient coils, T1 and T2 relaxation, and image reconstruction, and contrasts MRI's strengths and weaknesses with the X-ray and ultrasound modalities covered elsewhere in the option.

Spec mapping: This lesson sits under AQA 7408 section 3.10.4 (Non-ionising imaging — MRI). It covers the nuclear-magnetic-resonance phenomenon for protons placed in a strong static magnetic field B_0; the Larmor precession frequency f_L = γB_0 / (2π) and its numerical value for protons in fields of clinical strength (typically 0.5-3 T); the role of a resonant radiofrequency pulse in tipping spin populations; the relaxation processes T1 (longitudinal / spin-lattice) and T2 (transverse / spin-spin) and their use as image-contrast mechanisms; spatial encoding using linear field gradients superimposed on B_0; Fourier-transform reconstruction of the image; and a comparison of MRI's advantages (non-ionising, exquisite soft-tissue contrast, multiplanar) with its disadvantages (cost, scan time, contraindications). (Refer to the official AQA specification document for exact wording.)

Synoptic links:

• Section 3.2 (quantum phenomena): the two-level spin-up / spin-down structure of a proton in a magnetic field is one of the few cleanly two-level quantum systems treated at A-Level, and the resonance condition hf = γħB_0 is a direct application of E = hf. MRI is one of the very few medical modalities for which a quantum-mechanical interpretation is genuinely necessary.
• Section 3.7.5 (magnetic fields): the force on a moving charge and the precession of a magnetic moment in a field both follow from the underlying μ × B torque. Larmor precession in MRI is the analogue of the cyclotron motion in the magnetic-fields course.
• Section 3.10.5 (X-ray imaging) and 3.10.4 (ultrasound): MRI is the third member of the major imaging trio. Synoptic exam questions routinely ask candidates to compare modalities for a specific clinical question — bone (X-ray), soft tissue (MRI), motion (ultrasound, fluoroscopy).

---

The Physics of Nuclear-Spin Resonance

The hydrogen nucleus — a single proton — possesses an intrinsic angular momentum (spin) of magnitude ℏ/2 and an associated magnetic moment μ. When placed in an external magnetic field B_0, the proton's magnetic moment cannot be parallel to the field at all times: the field exerts a torque on μ that causes it to precess about B_0 at a characteristic frequency, the Larmor frequency:

f_L = (γ B_0) / (2π)

where γ is the gyromagnetic ratio of the proton. For the proton, γ/(2π) ≈ 42.6 MHz T⁻¹. So:

B_0	f_L (proton)
0.5 T	~21.3 MHz
1.0 T	~42.6 MHz
1.5 T	~63.9 MHz
3.0 T	~127.7 MHz

Clinical scanners operate at 1.5 T or 3 T almost exclusively; research scanners push to 7 T and beyond. The numerical Larmor frequency lies in the radiofrequency (RF) range — convenient because RF electronics is mature, and because RF at these frequencies penetrates the body cleanly with little tissue absorption at MRI power levels.

Spin Populations and Net Magnetisation

In the quantum-mechanical description, the proton in field B_0 has two allowed spin states: spin-up (lower energy, μ aligned with B_0) and spin-down (higher energy, μ anti-aligned). The energy gap between them is

ΔE = γ ℏ B_0 = h f_L

At thermal equilibrium the relative populations follow the Boltzmann distribution. At 1.5 T and body temperature (310 K), the excess of spin-up over spin-down is tiny — about 5 in a million. But because a 1-cm³ tissue voxel contains around 10²³ protons, this tiny excess sums to a measurable bulk net magnetisation M_0 aligned with B_0. M_0 is the macroscopic quantity that MRI manipulates and detects.

The RF Pulse: Tipping the Magnetisation

A short, oscillating magnetic field B_1 applied at the Larmor frequency, perpendicular to B_0, exchanges energy resonantly with the spin system. In the rotating frame of reference (rotating at f_L about B_0), B_1 appears static; the magnetisation M precesses about B_1 in this frame. The duration and amplitude of the RF pulse can be chosen to tip M by 90° (so that it ends up in the transverse plane, perpendicular to B_0), 180° (so that it is inverted), or any chosen angle.

After a 90° pulse, M lies entirely in the transverse (x-y) plane and precesses about B_0 at the Larmor frequency. This rotating transverse magnetisation induces a voltage in a receiver coil — the free-induction-decay (FID) signal that is the raw output of an MRI experiment.

---

T1 and T2 Relaxation

Once tipped, the magnetisation does not stay in the transverse plane indefinitely. Two distinct relaxation processes return it to the equilibrium configuration.

T1 — longitudinal relaxation (spin-lattice). The component of M along B_0 (M_z) regrows from 0 back to M_0. The protons release energy to the surrounding lattice (the molecular environment) through dipolar coupling. The recovery is exponential with time constant T1.

T2 — transverse relaxation (spin-spin). The component of M perpendicular to B_0 (M_xy) decays from its initial post-pulse value back to 0. Individual proton spins lose phase coherence with each other because each experiences a slightly different local magnetic field. The decay is exponential with time constant T2.

T1 and T2 are independent in principle. In practice T1 ≥ T2 always (for biological tissues T1 is typically several times T2). Both depend strongly on the tissue environment — the molecular tumbling rate, the proximity of paramagnetic species, the water-binding state.

Tissue (at 1.5 T)	T1 (ms)	T2 (ms)
Cerebrospinal fluid	~2,500	~1,500
Fat	~250	~80
White matter (brain)	~780	~90
Grey matter (brain)	~920	~100
Muscle	~870	~50
Liver	~490	~40

These values are illustrative textbook ranges; precise values depend on field strength, temperature and measurement technique.

Image Contrast: T1-Weighted and T2-Weighted Images

By choosing the timing of repeated RF pulses — specifically the repetition time (TR) between successive 90° pulses and the echo time (TE) between a 90° pulse and the moment of signal acquisition — the radiographer controls which relaxation process dominates the image contrast.

T1-weighted image: short TR (~500 ms) and short TE (~10 ms). Tissues with short T1 (e.g. fat) appear bright; tissues with long T1 (e.g. CSF) appear dark. Good for anatomical clarity.

T2-weighted image: long TR (~2000 ms) and long TE (~80-100 ms). Tissues with long T2 (e.g. CSF, oedema) appear bright; tissues with short T2 appear dark. Excellent for pathology — most pathological tissue has increased water content and bright on T2.

The same patient can therefore be imaged twice in succession with quite different pulse sequences, producing complementary T1 and T2 images that together drive the diagnostic conclusion.

Worked Example 1 — Larmor Frequency

Calculate the Larmor frequency for protons in a clinical 3.0 T scanner. Take γ/(2π) = 42.58 MHz T⁻¹.

f_L = γ B_0 / (2π) = 42.58 × 3.0 = 127.7 MHz.

This sits in the FM-radio range — well above audio, well below microwave. Standard 3 T scanner electronics are tuned to this band.

---

Spatial Encoding: Gradient Coils

Pure NMR (Larmor precession of all protons at the same frequency in a uniform field) gives you a single integrated signal from the whole sample — useful for chemistry but not for imaging. Magnetic Resonance Imaging adds three pairs of gradient coils that superimpose linear magnetic-field gradients on B_0 in each of the x, y and z directions:

B(r) = B_0 + G · r

where G is the (small) gradient vector. With a gradient applied, protons at different positions experience different total fields and therefore precess at different Larmor frequencies. The spatial information is frequency-encoded into the signal.

The standard MRI pulse sequence applies gradients in three coordinated steps:

1. Slice-select gradient in the z-direction during the RF pulse. Only protons in a thin slice whose Larmor frequency matches the RF pulse frequency are tipped; the rest are unaffected. This isolates a 2-D plane through the patient.
2. Phase-encoding gradient in the y-direction for a brief period between the RF pulse and acquisition. Protons in different rows of the selected slice acquire different phase shifts.
3. Frequency-encoding gradient in the x-direction during signal acquisition. Protons in different columns precess at different frequencies as the signal is recorded.

The acquired raw data is the k-space representation: a 2-D Fourier transform of the image. A standard image is reconstructed by 2-D inverse Fourier transform of k-space. The sequence is repeated with different phase-encoding strengths to fill k-space line by line — typically 256 lines for a 256 × 256 image, with each line taking one TR. Total acquisition time = TR × number of phase encodings ≈ 1-5 minutes for a single anatomical sequence.

---

MRI Block Diagram

graph LR
    A["Superconducting<br/>magnet B_0 ~1.5/3 T"] --> B["Gradient coils<br/>x, y, z"]
    B --> C["RF transmit coil<br/>tipping pulse"]
    C --> D["Patient<br/>(protons precess)"]
    D --> E["RF receive coil<br/>FID signal"]
    E --> F["ADC + k-space store"]
    F --> G["2-D inverse<br/>Fourier transform"]
    G --> H["Image"]

    style A fill:#2980b9,color:#fff
    style G fill:#27ae60,color:#fff

---

Advantages, Disadvantages and Contraindications

Advantage	Why
Non-ionising	No cumulative radiation dose; safe for repeated imaging and pregnancy (with caveats)
Soft-tissue contrast	T1/T2 differences exceed X-ray attenuation differences for soft tissues
Multiplanar	Slice can be in any plane (axial, coronal, sagittal, oblique)
Functional imaging	fMRI exploits BOLD contrast for brain activity mapping
No bone artefact	Bone gives little MR signal but does not block imaging of nearby soft tissue

Disadvantage	Why
Cost	Capital cost £1-3M; running cost dominated by liquid-He cryogen
Scan time	Minutes per sequence; multiple sequences per study; total often 20-60 min
Patient experience	Loud (gradient coils Lorentz-force the coil structure into vibration); claustrophobic for some
Contraindications	Pacemakers (older), ferromagnetic implants and aneurysm clips, some shrapnel, cochlear implants (mostly)
Limited bone detail	Cortical bone has very low water signal — CT is the modality of choice for fractures
Susceptibility artefacts	Iron, dental amalgam, surgical clips distort the local field

The contraindication list is the single most important practical concern: a ferromagnetic implant in a strong field can be torqued and translated with potentially catastrophic consequences. MRI safety screening before every scan is non-negotiable.

Worked Example 2 — A T1 vs T2 Decision

A patient is suspected of a stroke (cerebral infarct) within the past 24 hours. Should T1- or T2-weighted imaging be the first sequence chosen? Justify briefly.

Acute infarction produces cytotoxic oedema — intracellular water accumulation in the affected territory. The affected tissue has prolonged T2. On a T2-weighted image, prolonged T2 appears bright, so the lesion stands out against the darker surrounding healthy tissue. In practice, modern stroke protocols start with diffusion-weighted imaging (DWI), which is even more sensitive to cytotoxic oedema, but on the question as posed the answer is T2-weighted (or FLAIR — a T2-weighted variant that suppresses CSF signal for cleaner detection of subtle lesions).

Worked Example: Larmor Frequency at 1.5 T vs 3.0 T and the Field-Strength Trade-Off

A clinical MRI scanner operates at B₀ = 1.5 T. The hospital is considering an upgrade to a 3.0 T scanner. Use γ/(2π) = 42.58 MHz T⁻¹ for the proton.

(a) Larmor frequency at 1.5 T.

f₀ = (γ / 2π) × B₀ = 42.58 × 1.5 = 63.87 MHz ≈ 63.9 MHz.

(b) Larmor frequency at 3.0 T.

f₀ = 42.58 × 3.0 = 127.74 MHz ≈ 127.7 MHz.

The frequency scales linearly with B₀, as expected — doubling the field doubles the precession frequency. This is in the VHF / lower UHF radio band; the RF transmit and receive coils, and all associated electronics, must be re-engineered for the new operating frequency on any field-strength upgrade.

(c) Signal-to-noise ratio and clinical preference for higher fields.