Interactive MRI physics01 — 09

You don’t photograph a body.
You encode it.

Start with RF excitation and relaxation, then follow gradients as they turn position into frequency and phase—from body and local coils to an oblique trajectory through k-space and an image reconstructed line by line.

Start the live scan ↘

B₀: 3.0 T
γ̄ / ¹H: 42.58 MHz/T
model: Cartesian GRE

ƒEquation ?meaning · units · live what-if · why care ◎Tap object / explain viewidentify one 3D object in place or open the complete visual key ↔Change a controloutputs and cause/effect notes update together

Interactive 3D scanner

Scanner coordinate frame

↔ Drag to rotate

THE CENTRAL RELATION

s(k) = ∫ ρ(r) e^{−i2π k·r} dr

The received signal at one k-space coordinate is a weighted sum from the entire excited object.

01 / FROM FIELD TO POSITION

A controlled tilt in the magnetic field.

A gradient coil adds a small, nearly linear variation to the much larger main field. Proton frequency now carries an address: where a spin sits determines how fast its phase turns.

B₀ − ΔB field strength → B₀ + ΔB

x = +8.0 cm

Live field model

¹H at 3 T

LOCAL FIELD

B_z(r,t) = B₀ + G(t) · r

Gradient strength25.0 mT/m −400+40 mT/m

Spin position+8.0 cm −12 cmisocenter+12 cm

ΔB+2.00 mT

Δf = γ̄ΔB+85.15 kHz

local f127.825 MHz

i The gradients are tiny beside B₀, but they are switched quickly and precisely. The audible knocking in MRI is gradient hardware moving under Lorentz force.

02 / RF TRANSMIT, RELAXATION & RECEIVE

Excite broadly.
Listen locally.

The transmit coil creates the rotating B₁⁺ field that tips magnetization. After excitation, the scanner stops transmitting and receive coils detect the tiny voltage induced by precessing transverse magnetization. A common 1.5 T / 3 T workflow is body-coil transmit with a close local array receiving.

Three-dimensional RF transmit and receive model

TRANSMIT WINDOW · B₁⁺127.73 MHz

BODY COIL → B₁⁺ → SPINS α = 92.0° Local receive elements are detuned while the body coil transmits.

Drag to orbit · mint cage is body RF · coral loops are local elements

TRANSMIT FLIP ANGLE

α = γ ∫ B₁⁺(t) dt

RECEIVED VOLTAGE

v_Rx(t) ∝ −dΦ_M/dt

Transmit B₁⁺ amplitude6.0 µT 2 µTRF field strength18 µT

RF pulse duration1.00 ms 0.20 mspulse area3.00 ms

Local coil distance3.0 cm close · 1 cmreceive geometryfar · 12 cm

Visible local array elements

RF waveformpower amplifier

→

body coilB₁⁺ transmit

→

magnetizationprecessing spins

→

local arrayB₁⁻ receive

→

preampsADC / k-space

BODY TRANSMIT · LOCAL RECEIVE

The large built-in body coil drives a broad, comparatively uniform B₁⁺ field. During transmit the nearby receive array is actively detuned; during reception the transmitter is isolated and the local elements feed low-noise preamplifiers.

predicted rectangular-pulse flip92.0°

receive sensitivity proxy1.00× local

excitation coveragebroad / uniform

receive coveragelocal · 8 channels

transmit safety focuswhole-body SAR

switch stateRx array detuned

The sensitivity number is a normalized geometry teaching proxy, not a scanner specification. Actual B₁ fields, SNR, coupling, noise covariance, and SAR require coil- and patient-specific electromagnetic measurements.

B₁⁺

Transmit field

RF power at the Larmor frequency rotates magnetization. Amplitude and pulse duration set flip angle; spatial B₁⁺ variation makes flip angle nonuniform.

B₁⁻

Receive sensitivity

Precessing transverse magnetization induces a tiny voltage. Close local elements couple strongly to nearby anatomy and admit less distant noise.

T/R

Isolation matters

Receive-only elements are detuned during the high-power transmit pulse. The transmit path is then isolated while low-noise preamplifiers listen for the echo.

Predicted relative signalSPIN ECHO · TR 500 / TE 15 ms

fatgraywhiteCSF

Recovery before RF / decay before echoT₁ RECOVERY + T₂ DECAY

IDEAL SPIN-ECHO SIGNAL

S = ρ(1 − e^−TR/T₁)e^−TE/T₂

The 180° pulse refocuses static dephasing, so ideal echo amplitude follows T₂ rather than T₂*.

TR · repetition time500 ms 200 msrecovery + scan time6000 ms

TE · echo time15 ms 5 mstransverse decay200 ms

Flip angle α90° fixed 1°spin echo fixes 90°60°

TISSUE A · MINT TISSUE B · CORAL

WHITE MATTERρ 0.70 · T₁ 850 · T₂ 80 · T₂* 55 ms

CSF / FLUIDρ 1.00 · T₁ 4000 · T₂ 2000 · T₂* 400 ms

One repetition90° → 180° → ECHO → NEXT 90°

SHORT TR · SHORT TE

Short TR samples tissues before full longitudinal recovery, strengthening T₁ differences. Short TE limits T₂ decay, so the current spin echo is predominantly T₁ weighted.

white matter signal0.280

CSF / fluid signal0.114

pairwise contrast42.2%

dominant weightingT₁ weighted

2D scan-time proxy1:04 · 128 lines

echo fraction TE / TR3.0%

Signals use simplified steady-state equations and illustrative relaxation values near 3 T. Real contrast also depends on sequence details, RF profiles, echo trains, magnetization transfer, flow, diffusion, coils, reconstruction, pathology, and field strength.

03 / THREE ORTHOGONAL CONTROLS

Same physics.
Three jobs.

“X, Y, and Z” are logical axes chosen by the pulse sequence. The scanner combines all three physical coil sets to rotate an oblique slice, so readout does not have to mean anatomical left–right.

GX · ON DURING ADC

Position becomes frequency.

During signal readout, Gx makes spins at different x positions precess at different frequencies. Sampling through time walks continuously across one row of k-space.

FREQUENCY OFFSETΔf(x) = γ̄ G_x x

gradient actionContinuous plateau

moves throughk_x within a line

resolution set by±k_x,max

NAMING DECODER

“Logical” names the job. “Physical” names the coil.

The prescribed image can tilt anywhere inside the scanner, so read, phase, and slice cannot always be permanently assigned to physical X, Y, and Z.

READduring ADC · traverses kx PHASEbefore ADC · selects ky SLICEwith RF · selects z / kz

Rplane rotation→

physical Gxphysical Gyphysical Gz

WHY “LOGICAL”?

A role in the image coordinate system

Logical axes rotate with the prescribed image plane. “Read” could use only Gx in an axial scan, but a tilted scan may synthesize it from Gx + Gy + Gz.

WHY CARE · It prevents anatomical directions from being confused with fixed coil hardware.

WHY “READOUT”?

The receiver reads while this gradient is on

During ADC, the read gradient makes position along one image axis appear as different receive frequencies and moves continuously across one k-space row.

WHY CARE · Its FOV, sample rate, and kmax control pixel size, bandwidth, chemical-shift displacement, and read-direction distortion.

WHY “PHASE”?

A brief gradient leaves phase behind

The phase gradient switches off before ADC, but spins retain a position-dependent phase. Repeating different signed areas selects different ky rows.

WHY CARE · The number of phase steps drives scan time; phase FOV controls wrap, and this direction is often most motion-sensitive.

WHY “SLICE”?

RF frequency plus a gradient chooses a slab

With Gslice on, resonance frequency maps to distance normal to the plane. The RF passband excites only a thickness; in 3D, the same logical direction can instead be phase encoded as kz.

WHY CARE · It determines coverage, partial volume, slice gap or crosstalk, SNR, and 3D partition resolution.

Three-dimensional oblique coordinate model

logical read / phase / slice frame fixed physical Gx / Gy / Gz frame

LOGICAL READOUT GRADIENT +30.0 mT/m Gx +30.0 · Gy 0.0 · Gz 0.0 mT/m

Drag to orbit · plane normal is logical slice Z

COORDINATE TRANSFORM

G_physical = R · G_logical

Columns of R are the physical directions of logical readout, phase, and slice. Because R is orthonormal, the three logical axes remain mutually perpendicular.

X tilt0° −90°physical X rotation+90°

Y tilt0° −90°physical Y rotation+90°

In-plane rotation0° −90°about slice normal+90°

Logical amplitude+30.0 mT/m −70reverse at 0+70 mT/m

Rotation matrix R · physical rows by logical columns
physical	Read	Phase	Slice
Gx	+1.000	0.000	0.000
Gy	0.000	+1.000	0.000
Gz	0.000	0.000	+1.000

PHYSICAL AMPLIFIER COMMANDS±40 mT/m per-axis model

GX+30.0 mT/m

GY0.0 mT/m

GZ0.0 mT/m

AXES ALIGNED

In an axial prescription, logical readout aligns with the physical X gradient. Only the Gx amplifier is needed for this +30.0 mT/m readout plateau.

logical vector magnitude30.0 mT/m

safe logical maximum40.0 mT/m

limiting physical channelGX · 75%

Ideal per-axis amplitude example. Real systems also constrain slew rate, duty cycle, peripheral nerve stimulation, and vector-dependent safety limits.

04 / GRADIENT MOMENT & SPIN PHASE

The area under G
becomes phase.

Gradient amplitude alone does not set a k-space coordinate. Its signed time integral does. Build one lobe, add an opposite rewinder, and watch a three-dimensional spin ensemble wind into a phase pattern—or return coherently to k = 0.

Three-dimensional spin phase ensemble

transverse magnetization vectors equal-phase planes

GX MOMENT k_x = +170.3 m⁻¹ Phase is drawn in the transverse x–y plane; B₀ points along physical z.

Drag to orbit · scroll to zoom

ZEROTH GRADIENT MOMENT

M₀(t) = ∫₀^t G(t′) dt′

POSITION IN K-SPACE

k(t) = γ̄ M₀(t)

SPIN PHASE

φ(r,t) = 2π k(t) · r

Signed gradient area and accumulated kUNBALANCED

Gradient amplitude+10.0 mT/m −300+30 mT/m

Encoding duration0.40 ms 00.601.20 ms

Opposite rewinder area0% none100% · k = 0120%

COHERENT SIGNAL FROM THE COLUMN8.9%

The positive Gx area moves the sample to +kx. Spins separated along x retain different phases after the lobe turns off, so their vector sum is small.

net M₀+4.00 mT·ms/m

k coordinate+170.3 m⁻¹

phase across 20 mm+3.41 turns

Area, not height

A weak gradient held longer can create the same phase slope and k-space displacement as a short, strong gradient.

Polarity sets direction

Changing the sign of G reverses the phase ramp and moves to the opposite side of k-space along the selected logical axis.

Balanced area refocuses

For stationary spins in this ideal model, an equal opposite lobe cancels M₀. The phase ramp unwinds and the coherent signal returns at k = 0.

05 / LIVE CARTESIAN ACQUISITION

Watch k-space
become an image.

Each TR selects a slice, applies one phase-encoding moment, then samples an echo under the readout gradient. Repeat with a new Gy area and the raw data matrix fills one row at a time.

Every spin’s complex contributionUNIFORM PHASE AT k = 0

Color is the phase of ρ(x,y)e^−i2πk·r. The receiver adds every colored contribution into one complex number.

The sample’s k-space addresskx 0 · ky 0

k_x

k_y

LOG |S(k)|LOWHIGH

The background is the phantom’s exact Fourier spectrum. The coral ring marks the single receiver sample inspected at left.

ONE RECEIVER SAMPLE

S(k_x,k_y) = ∫∫ ρ(x,y)e^{−i2π(k_xx+k_yy)} dxdy

one k-space coordinate is a weighted sum from the entire excited slice

Readout coordinate kx0 Δk · 0.0 m⁻¹ −32 Δkcenter+31 Δk

Phase coordinate ky0 Δk · 0.0 m⁻¹ −32 Δkcenter+31 Δk

COHERENT DC SAMPLE

At k = 0 the encoding phase is identical everywhere. All positive spin density adds coherently, producing the large center coefficient that represents the object’s average signal—not a center pixel.

|S(k)| / |S(0)|100.00%

sample phase0.0°

normalized complex sample+1.000 + i0.000

required gradient momentMx 0.000 · My 0.000

phase cycles / FOVX 0 · Y 0

encoded wavelengthuniform phase

GY PREPHASEky = 0 set · Gx / ADC idle

0% READY · SELECT k OR SWEEP

RF, gradients, receiver, and signalTE 6 ms · TR 30 ms

The event block is expanded so short RF and gradient operations remain visible; the broken segment compresses idle recovery before the next RF pulse.

Integrated gradient pathkx 0.0 · ky 0.0 m⁻¹

Violet is unsampled prephasing. Mint is the portion stored by the ADC. The yellow ring is kx = 0—the echo center on this ky row.

RF + GZ · SLICE EXCITATION

BODY TX ON · RX ARRAY DETUNED

+GZ · RF BAND SELECTS Z

The transmit coil tips spins in the RF-selected slice. No in-plane k-space sample is recorded yet.

TIME IN TR0.00 / 30 ms

K-SPACE COORDINATEkx 0.0 · ky 0.0 m⁻¹

ADC STATEOFF · 0 / 64 SAMPLES

RECEIVE MAGNITUDE0.0% · BEFORE ECHO

THE ACQUISITION CHAIN

k(t) = γ̄∫G(τ)dτ → S(k) → ℱ⁻¹

gradients choose the Fourier address; the ADC stores the complex receiver voltage only while its gate is open

Readout FOV220 mm 160sets Δk + pixel320 mm

Total sample rate128 kHz 64receiver bandwidth256 kHz

Phase-encode row−16 Δk −32Gy moment+31

Echo time TE6 ms 5RF center → echo25 ms

Repetition time TR30 ms 15next RF + scan time100 ms

T₂* envelope55 ms 20illustrative tissue160 ms

REFERENCE PRESCRIPTION

A 220 mm FOV sampled at 64 readout points produces 3.44 mm pixels and Δk = 4.55 m⁻¹. Scrub the timeline or change a parameter to see the dependent quantities move together.

Δk = 1 / FOV4.55 m⁻¹

nominal kx,max145.5 m⁻¹

ideal Δx = FOV / 643.44 mm

ADC dwell / window7.81 µs / 0.50 ms

readout Gx13.7 mT/m

Gy moment−1.708 mT·ms/m

echo T₂* envelope89.7%

64-line scan proxy1.92 s

Ideal one-line-per-TR Cartesian GRE model with 64 complex samples and no ramp sampling. “Total sample rate” is used here because vendor bandwidth displays may instead report Hz/pixel. The T₂* value controls only the transverse envelope; use the TE/TR lab above for the full spoiled-GRE steady state.

Scrub one repetition0%

0 / 64 0 / 64 ADC SAMPLES

BEFORE THE FIRST TR

Frequency is not amplitude.

Open the definitions below, then press Run. This explanation will follow RF transmit, gradients, receive sampling, and reconstruction through each repetition.

RF carrier frequency≈ 127.73 MHz at 3 T+

How fast the transmit field oscillates and the received voltage alternates. It is tuned near the ¹H Larmor resonance. It may be offset or shaped to select a slice, but it is not the waveform height.

RF amplitude · B₁⁺µT · pulse envelope+

How strong the transmit field is. Together with pulse duration it sets flip angle. The pulse diagram draws this slow envelope; it cannot display 127 million carrier cycles per second at this scale.

Gradient · Gx, Gy, GzmT/m · field slope+

Not a radio wave. A gradient slightly changes the local Larmor frequency with position. Its signed area sets phase and k-space position; during readout its amplitude sets travel speed through k-space.

Received signalcomplex voltage · I + iQ+

The coil detects a tiny RF voltage near the carrier. The receiver removes that fast carrier and stores a complex sample: magnitude says how much coherent signal arrived and phase preserves spatial encoding.

Image brightness|inverse Fourier transform|+

Not raw RF amplitude and not a k-space location. Reconstruction combines every acquired complex sample, and magnitude display maps the resulting voxel signal to brightness.

Pulse sequence / one TRTR 01 / 64

ExciteSelect zSet kyTraverse kx

Raw signal / k-spacekx −32 · ky +31

k_x

k_y

Brightness shows log signal magnitude. Position in k-space is spatial frequency—not a location in the head.

Fourier reconstruction0% DATA

Δx = 3.4 mm

Press run to acquire the first phase-encoded line.

Near k = 0

Contrast & broad shape

Slow spatial variation. High signal energy. Acquired at the echo center.

Large |k|

Edges & fine detail

Rapid spatial variation. Extending farther raises the ideal resolution limit.

Sample spacing Δk

Field of view

Closer k-space samples encode a wider unaliased FOV: FOV = 1 / Δk.

06 / TRAJECTORY STUDIO

Gradients draw
the path.

A gradient is velocity in k-space. Hold it constant and k moves in a straight line; reverse it and the path turns back; rotate two gradients together and the path spirals. Compare five encoding strategies generated from that one rule.

Interactive k-space trajectory

commanded k(t) delayed response

CURRENT SAMPLE k = (−1.00, −1.00, 0.00)

Drag to orbit · scroll to zoom

MULTI-SHOT · RECTILINEAR

One echo, one row.

A prephaser moves to −kx. The readout gradient crosses one horizontal line while ADC samples. A new Gy area selects the next ky on the following TR.

THE TRAJECTORY LAWDirection of G sets direction of travel. Its magnitude sets speed in m⁻¹/s.

dkdt= γ̄ G(t)

Representative gradient commandSHOT 01 / 15

acquisition0%

Simulated Gx hardware delay0 µs ideal10 µs20 µs

Show ADC samples

i Delayed or filtered gradient response means the scanner may sample somewhere other than the commanded coordinate. Non-Cartesian reconstruction therefore often uses a measured trajectory.

coverage / excitation1 k-space row

center crossingsonce per echo

reconstructiondirect FFT

G(t)

Gradient amplitude

Controls instantaneous speed through k-space. Stronger readout gradient covers more spatial frequency per unit time.

dG/dt

Slew rate

Limits how sharply a path can turn. Fast switching also drives acoustic noise and peripheral nerve-stimulation constraints.

ADC(t)

Sampling window

The path may move while the receiver is off. Only coordinates visited during ADC become acquired data samples.

07 / RESOLUTION LAB

Shape the voxel
in all three axes.

Resolution is not a “megapixel” setting. It follows from field of view, sample count, and how far the acquisition reaches in k-space. Smaller voxels usually cost signal-to-noise, scan time, or both.

3D ENCODED VOLUME1.72 × 1.72 × 5.00 mm

Drag to orbit · highlighted cell is one voxel

Protocol builder

XReadoutfrequency

FOV220 mm

matrix128

1.72 mm

YPhaseencoded

FOV220 mm

matrix128

1.72 mm

ZSliceRF selected

RF BW2.0 kHz

Gz9.4 mT/m

5.00 mm

voxel volume14.77 mm³relative SNR proxy 1.00×

k-space extent±291 / ±291 m⁻¹Z uses slice profile

minimum phase encodes128≈ 25.6 s at TR 200 ms

REFERENCE PRESCRIPTIONWhat changed—and what it costs

This 220 mm, 128 × 128 reference produces 1.72 mm in-plane sampling. Move any control to see its direct consequence for detail, signal, and encoding time.

in-plane pixel area 1.00× signal proxy 1.00× phase-time proxy 1.00×

PIXEL / VOXEL SIZE

Δx = FOV_x / N_x ≈ 1 / (2k_x,max)

FIELD OF VIEW

FOV_x = 1 / Δk_x

2D SLICE THICKNESS

Δz = BW_RF / (γ̄ |G_z|)

Three-dimensional RF-selected slice

LOWER FREQUENCYHIGHER FREQUENCY

RF-SELECTED SLAB 4.70 mm center z = 0.0 mm · axial plane

Drag to orbit · highlighted rings fall inside the transmit passband

FREQUENCY ADDRESS

Δf(z) = γ̄ G_z z

SELECTED THICKNESS

Δz = BW_RF / (γ̄ |G_z|)

Frequency-to-position map+GZ · CENTERED RF

Slice gradient |Gz|10.0 mT/m 41730 mT/m

RF transmit bandwidth2.0 kHz 0.53.256.0 kHz

RF center-frequency offset0.0 kHz −20on-resonance+20 kHz

CENTERED AXIAL SLICE

Gz maps position to frequency. The 2.0 kHz RF passband intersects that line around z = 0, exciting an ideal 4.70 mm slab.

frequency slope0.426 kHz/mm

RF passband−1.0 … +1.0 kHz

slice center0.0 mm

ideal thickness4.70 mm

BW_RF ↑

Thicker slab

A wider transmit band matches a larger interval of positions on the same frequency slope.

|G_z| ↑

Thinner slab

A steeper frequency slope maps the same RF bandwidth onto a narrower spatial interval.

f_RF ↕

Move the slice

Changing the RF center frequency moves the intersection without changing the ideal thickness.

08 / SAMPLING & POINT RESPONSE

Change k-space.
The image answers.

Spatial resolution, field of view, ringing, and wraparound are different consequences of how k-space is sampled. Apply the operations directly to the complex data and inspect both the reconstructed image and the point-spread function that produced it.

Sampling prescription

CUTOFF / RESOLUTION

Δx ≈ 1 / (2k_x,max)

SPACING / FIELD OF VIEW

FOV_x = 1 / Δk_x

Retained k-space extent100% kmax 25% · blurred60%100% · full

Uniform sampling interval

k-space weighting

FULL REFERENCE

All encoded frequencies are retained at the native Δk. The PSF approaches one image pixel and no coherent wrap replicas are introduced.

nominal Δx × Δy3.44 × 3.44 mm

effective FOV220 × 220 mm

PSF FWHM1.0 × 1.0 px

retained samples4,096 / 4,096

Applied sampling mask100% RETAINED

k_yk_x

Mint points contribute to reconstruction. Coral lines are inside the selected extent but skipped by uniform undersampling.

Image-space PSFℱ⁻¹{MASK}

positivenegative

The central lobe sets effective resolution; sidelobes produce ringing or displaced wrap replicas.

Resulting reconstructionREFERENCE

Full k-space extent and native spacing preserve the simulated image’s available detail and field of view.

Truncate extent

Removing outer k-space lowers the spatial-frequency cutoff. Image space is convolved with a broader sinc-like response: detail softens and sharp boundaries ring.

Increase spacing

Keeping every Rth point multiplies the effective Δk by R and reduces the encoded FOV by R. Repeated PSF peaks fold distant anatomy into the displayed FOV.

Apodize

A Hann or Hamming taper suppresses PSF sidelobes and Gibbs ringing, but broadens the central lobe. A calmer boundary is purchased with effective resolution.

09 / THE COMPLETE CHAIN

From current waveform to one image voxel.

01
Gradient current
Amplifiers drive X, Y, Z coil windings. Current and geometry create a controlled field slope in T/m.
I(t) → G(t)
02
Spin phase
The local field changes Larmor frequency. Accumulated phase records the gradient’s area through time.
φ(r,t) = γ r·∫G dt
03
k-space sample
The receiver sums every transverse spin at the current spatial-frequency coordinate.
k(t) = γ̄∫G dt
04
Image estimate
An inverse Fourier transform separates the superposed spatial frequencies into locations.
ρ̂(r) = ℱ⁻¹{s(k)}

REFERENCE DESK

Continue into the physics.

Clinical overviewK-space in the clinic↗ Open reviewCMR physics for clinicians↗ ISMRM courseSpatial encoding in MRI↗ Trajectory reviewEPI and spiral imaging↗ Signal derivationGradient phase and k-space↗ Cartesian acquisitionPrephasing, readout, and ADC trajectories↗ Oblique coordinatesLogical-to-physical gradient rotation↗ RF coil reviewBody transmit and local receive↗ Relaxation modelSpoiled GRE steady-state signal↗ Slice-selection modelRF bandwidth, Gz, and slice position↗ Artifact reviewTruncation, aliasing, and filtering↗