Interactive MRI physics01 — 09
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.
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S(k) = ∫ ρeff(r) e−i2π k·r dr
In plain words: treat the excited body as many tiny, equal-size regions. Each region contributes some receive-coil voltage. Gradients give those contributions a calculable phase—where they sit around one repeating cycle. At address k, the receiver adds every region and stores one pair of signed numbers, I and Q, called S(k). That pair describes the whole excited region; it is not one image pixel. Tap the equation for every symbol, unit, zero case, and a live numerical example.SEARCHABLE MRI DICTIONARY · 88 / 88 TERMS HAVE THE COMPLETE BEGINNER EXPLANATION
Every result starts without assumed MRI knowledge, then lets you go deeper into a labeled visual, scanner behavior, clinical consequence, numerical example, and connected terms.
BEGINNER-FIRST LEARNING · CHOOSE HOW FAR TO ZOOM
The same three-level control appears as a short entry card at every major topic. It changes the explanation layer—not the MRI physics or any simulated parameter.
START SIMPLEEach chapter now begins with one plain-language mental model. Use “Zoom in” on any card when you are ready.
THE WHOLE MRI EXAM · NOT ONLY RF, GRADIENTS, OR K-SPACE
An MRI scanner is a coordinated system: safety screening, a continuously energized main magnet, a moving patient table, shim and gradient coils, RF transmit/receive, monitoring, sequence control, digitization, corrections, reconstruction, storage, and display.
Technologists screen the patient and every entering object, position anatomy and coils, choose a landmark, provide hearing protection and an alarm device, and use MR-conditional monitoring or gating when required.
The main magnet supplies B₀ continuously. The cryostat thermally supports the superconducting system; site infrastructure, shielding, and emergency procedures manage fields and rare abnormal events.
Fast survey images establish geometry. Field mapping and shim adjustments improve uniformity; reference scans can estimate coil sensitivity, center frequency, transmit behavior, and reconstruction corrections.
The controller schedules RF synthesizers/amplifiers, gradient amplifiers, transmit/receive protection, receiver bandwidth, and ADC sampling. Timing—not a single component—defines the acquisition.
Raw multi-channel I/Q data may be corrected for sampling and hardware behavior, calibrated, reconstructed, combined across coils, filtered, scaled, and packaged with geometry and protocol metadata.
The console and downstream systems show reconstructed images with orientation, scaling, annotations, and metadata. A displayed gray level is the end of a long weighted chain—not a direct photograph or universal tissue unit.
ρ EXPLAINED WITHOUT HIDDEN FACTORS · SOURCE → ECHO → COIL → IMAGE
Textbooks often reuse ρ for both proton density and the already-weighted signal distribution. This lab keeps those quantities separate, names every factor in the teaching product, and states the reference behind every “relative” number.
Most clinical proton MRI signal comes from hydrogen nuclei in mobile water and fat. Hydrogen locked in very rigid material can lose transverse coherence before the receiver can sample it; air has very few hydrogen nuclei. “Visible” never means visible light.
Here, an equal-size reference voxel with ρH = 1.00, complete recovery, no T₂* loss, a 90° excitation, and receive sensitivity 1.00 has ρeff = 1.00. Scanner gain and display windowing can rescale all image numbers, so there is no universal brightness or volt value.
One k-space measurement is not a pixel. After all complex samples are reconstructed, the ideal local complex value is proportional to ρeff. A magnitude image displays its size, usually after coil combination, scaling, filtering, and window/level.
Local mobile-¹H source, one T₁ recovery term, one T₂* survival term, ideal sin α excitation, and one relative receive-sensitivity number.
Flow/inflow, diffusion gradients, magnetization transfer, chemical exchange, contrast agents, fat/water phase, B₀ and B₁ nonuniformity, motion, multi-echo history, receive-channel combination, filters, noise, gain, and display window/level. These are named here rather than hidden inside “other factors,” but they are deliberately held out of this five-factor lab.
Brighter does not automatically mean “more protons.” First ask whether sequence timing, excitation, coil position, pathology, reconstruction, or display scaling also changed.
COORDINATE PRIMER · FIX THE NAMES BEFORE ENCODING
An axial example often pairs read with Gx, phase with Gy, and slice with Gz—but that pairing is not a law. Rotate the prescribed image plane and the scanner synthesizes each logical job by firing two or three physical gradient coils together.
WHY CARE · Confusing these naming systems can make an oblique image plane, artifact direction, or amplifier limit look wrong.[Gx Gy Gz]ᵀ = [+1.000 0.000 0.000]ᵀ × +40.0 mT/m
Each coefficient is a unitless direction cosine. Superscript ᵀ means “write this row as a column vector” (transpose)—it does not mean tesla here. Multiplying by +40.0 mT/m gives a real physical-coil command.01 / FROM FIELD TO POSITION
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.
Bz(r,t) = B0 + G(t) · r
f0 = γ̄B0
42.58 MHz/T × 3.0 T = 127.73 MHzAt fixed G, position, B₁⁺, and pulse duration, changing B₀ retunes the MHz carrier. The gradient offset, ideal flip angle, and dk/dt stay unchanged.
Scope: this control retunes the field/RF frequency model. The TE/TR tissue constants remain the explicitly stated illustrative 3 T model; real coil fields, SAR, and relaxation are field-dependent.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
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.
RF WAVEFORM DECODER · THREE DIFFERENT SHAPES, THREE DIFFERENT JOBS
The plotted field goes from a positive peak, through zero and a negative peak, back to the next positive peak: one complete 360° oscillation. It is a cycle of the RF magnetic field—not a proton traveling in a circle.
The slow outline connecting the peaks of thousands of fast carrier cycles. It is not a second radio wave and not a container; it summarizes how strongly the transmitter is driven over time.
The size of the complex voltage after the receiver mathematically removes the MHz carrier. Phase is stored too, even though this row shows only non-negative magnitude.
Before demodulation there is one rapidly alternating voltage. The receiver compares it with two synchronized reference waves; low-pass filtering leaves two slower signed voltages that preserve the signal’s magnitude and phase.
Drag to orbit · mint cage is body RF · coral loops are local elements
α = γ ∫ B₁⁺(t) dt
vRx(t) ∝ −dΦM/dt
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.
The sensitivity number is a normalized geometry teaching proxy, not a scanner specification. The shared B₀ control retunes the displayed ¹H carrier; it does not rescale this idealized coil geometry, B₁ field, SNR, relaxation, or SAR model. Actual values require coil- and patient-specific electromagnetic measurements.
RF power at the Larmor frequency rotates magnetization. Amplitude and pulse duration set flip angle; spatial B₁⁺ variation makes flip angle nonuniform.
Precessing transverse magnetization induces a tiny voltage. Close local elements couple strongly to nearby anatomy and admit less distant noise.
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.
S = ρH(1 − e−TR/T₁)e−TE/T₂
The 180° pulse refocuses static dephasing, so ideal echo amplitude follows T₂ rather than T₂*.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.
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
“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
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.
They are not two different kinds of magnetism. Both logical jobs use a gradient, and either gradient creates position-dependent frequency offsets while it is on. Their timing relative to the receiver is what makes their stored information different.
Called “readout” because the receiver’s ADC reads many samples while the positive read gradient is on.
Called “phase” because the lobe ends before ADC, yet its position-dependent phase pattern remains during readout.
The squares are k-space addresses, not anatomy pixels. One TR normally acquires one horizontal row in this simplified 2D Cartesian example.
Logical axes belong to the prescribed image plane. Physical Gx, Gy, and Gz are fixed hardware channels. The scanner rotates the two logical commands into coil currents; the anatomy and image jobs do not have to line up with the bore.
The RF pulse tips magnetization so a receive signal can exist. RF transmit is not readout: the receiver is protected and ADC is closed during excitation. Neither in-plane logical gradient job has encoded a k-space row yet.
A stronger read gradient crosses k-space faster. With dwell time fixed, that changes sample spacing and therefore FOV; with the acquisition prescription adjusted, it also affects bandwidth and distortion.
WHY CARE · This job sets read-direction sampling, bandwidth, chemical-shift displacement, and distortion behavior.The signed area under the phase gradient sets ky. Changing that area on successive TRs chooses different rows; it does not move to a literal y location in the patient.
WHY CARE · Phase steps strongly affect scan time, phase FOV, wrap, motion ghosts, and the direction of many artifacts.A magnetic-field gradient changes local precession frequency. “Frequency encoding” and “phase encoding” describe how the sequence uses the accumulated effect, not two different gradient mechanisms.
Readout samples continuously while Gread is on. Phase encoding applies a lobe before sampling, closes it, and carries the retained phase ramp into the readout window.
Readout collects many kx points in one ADC window. Conventional 2D Cartesian imaging repeats the TR with a new phase area to cover many ky rows, so the phase loop often dominates scan time.
MODEL BOUNDARY · Events are separated here so each cause is visible. Real pulse sequences often overlap the read prephaser and phase-encode lobe, use tens to thousands of samples, include finite ramps and spoilers, and may collect multiple ky lines per TR with echo trains or segmented readouts.
Drag to orbit · plane normal is logical slice Z
Gphysical = R · Glogical
| 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 |
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.
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
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.
Drag to orbit · scroll to zoom
M0(t) = ∫0t G(t′) dt′
k(t) = γ̄ M0(t)
φ(r,t) = 2π k(t) · r
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.
A weak gradient held longer can create the same phase slope and k-space displacement as a short, strong gradient.
Changing the sign of G reverses the phase ramp and moves to the opposite side of k-space along the selected logical axis.
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
This is an interactive acquisition—not a prerecorded video and not an image being uncovered. Start with an intentionally empty reconstruction; each TR adds complex Fourier data from the whole slice, and the image is recalculated.
A k-space “address” is not a place in the patient and has no width. It is a number attached to one measured complex sample. That number states how many gradient-created phase cycles occur per metre across the object for that sample.
It labels the phase pattern used when one whole-object I/Q sample was measured. It does not select a body point.
For example, 9.10 and 13.65 m⁻¹ are neighbors when no planned address lies between them.
13.65 − 9.10 = 4.55 m⁻¹. An address itself has no size; Δk is only the numerical separation.
One cycle is a full 360° change in the gradient-created phase pattern, not one RF carrier oscillation and not a proton orbit.
Color is the phase of ρeff(x,y)e−i2πk·r. Here ρeff means the local echo signal after the sequence and receive-coil factors explained in the rho lab. The receiver adds every colored contribution into one complex number.
The background is the phantom’s exact Fourier spectrum. The coral ring marks the single receiver sample inspected at left.
S(kx,ky) = ∫∫ ρeff(x,y)e−i2π(kxx+kyy) dxdy
one k-space coordinate is a weighted sum from the entire excited sliceAt 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.
The event block is expanded so short RF and gradient operations remain visible; the broken segment compresses idle recovery before the next RF pulse.
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.
The transmit coil tips spins in the RF-selected slice. No in-plane k-space sample is recorded yet.
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 openA 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.
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. Here T₂* controls only the slow decay of transverse signal magnitude with time; use the TE/TR lab above for the full spoiled-GRE steady state.
Open the definitions below, then press Run. This explanation will follow RF transmit, gradients, receive sampling, and reconstruction through each repetition.
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.
How strong the transmit field is. Together with pulse duration it sets flip angle. The drawn outline is simply B₁⁺ strength versus time; it cannot display the tens to hundreds of millions of carrier cycles per second at this scale.
Not a radio wave. A gradient slightly changes local Larmor frequency with position. Its signed area sets retained phase and the k address; while it remains on, its amplitude sets how many inverse metres that numerical address changes per second.
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.
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.
The reconstruction canvas is intentionally empty. Press Run or drag the bottom progress slider; it controls the simulated acquisition and recalculates the image from only the samples acquired so far.
THE RULEA k-space sample does not paint one image pixel. Every acquired complex coefficient changes the calculation of every reconstructed pixel.
Each acquired cell stores S = I + iQ. Its displayed brightness is = log(1 + √(I² + Q²)); this compresses the preview only. Position in k-space is spatial frequency—not a location in the head.
0 samples: the canvas is deliberately empty because no image can be reconstructed yet.
Near k = 0
Large |k|
Sample spacing Δk
06 / TRAJECTORY STUDIO
Here is a spatial-phase address, not a physical position or a moving particle. A gradient changes that address over time: hold one component constant and the address follows a straight line; reverse it and the address turns back; vary two components together and the address can spiral. Compare five encoding strategies built from that rule.
k = 100 m⁻¹ means the gradient-created phase pattern completes 100 turns per metre. Therefore two fixed positions 10 mm apart differ by one full 360° turn at that instant.
The marker is the current data address where ADC stores a whole-object complex sample. It is not a proton, voxel, anatomical location, RF carrier cycle, or signal amplitude.
dk/dt = γ̄G(t)
dk/dt is the rate at which the spatial-phase address changes, in m⁻¹/s. G(t) is the field slope in T/m. Positive G moves toward +k, negative G toward −k, and G = 0 holds the address still.
WHY CARE · The visited extent sets potential detail, spacing sets FOV, and a wrong address from delay or miscalibration creates blur, ghosts, or geometric distortion.
Drag to orbit · scroll to zoom
MULTI-SHOT · RECTILINEAR
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.
dkdt= γ̄ G(t)
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.
Controls how rapidly the numerical k address changes. A stronger readout gradient covers more spatial-frequency address per unit time; nothing physically flies through the patient.
Limits how sharply a path can turn. Fast switching also drives acoustic noise and peripheral nerve-stimulation constraints.
The path may move while the receiver is off. Only coordinates visited during ADC become acquired data samples.
07 / RESOLUTION LAB
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.
Drag to orbit · highlighted cell is one voxel
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.
Δx = FOVx / Nx ≈ 1 / (2kx,max)
FOVx = 1 / Δkx
Δz = BWRF / (γ̄ |Gz|)
Drag to orbit · highlighted rings fall inside the transmit passband
Δf(z) = γ̄ Gz z
Δz = BWRF / (γ̄ |Gz|)
Gz maps position to frequency. The 2.0 kHz RF passband intersects that line around z = 0, exciting an ideal 4.70 mm slab.
A wider transmit band matches a larger interval of positions on the same frequency slope.
A steeper frequency slope maps the same RF bandwidth onto a narrower spatial interval.
Changing the RF center frequency moves the intersection without changing the ideal thickness.
08 / SAMPLING & POINT RESPONSE
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.
Δx ≈ 1 / (2kx,max)
FOVx = 1 / Δkx
All encoded frequencies are retained at the native Δk. The PSF approaches one image pixel and no coherent wrap replicas are introduced.
Mint points contribute to reconstruction. Coral lines are inside the selected extent but skipped by uniform undersampling.
The central lobe sets effective resolution; sidelobes produce ringing or displaced wrap replicas.
Full k-space extent and native spacing preserve the simulated image’s available detail and field of view.
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.
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.
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
Amplifiers drive X, Y, Z coil windings. Current and geometry create a controlled field slope in T/m.
I(t) → G(t)The local field changes Larmor frequency. Accumulated phase records the gradient’s area through time.
φ(r,t) = γ r·∫G dtThe receiver sums every transverse spin at the current spatial-frequency coordinate.
k(t) = γ̄∫G dtAn inverse Fourier transform separates the superposed spatial frequencies into locations.
ρ̂(r) = ℱ⁻¹{S(k)}REFERENCE DESK
STEP 1 OF 16
The scanner’s main field aligns proton magnetization. Gradient coils add controlled spatial slopes.