Localize and prescribe first; then use the same quantitative sequence controls to acquire T₁, proton-density, T₂, FLAIR, DWI, or GRE contrast.
Theme 03 · Signal, contrast and motion
Run a visual exam from coil setup and localizers through slice prescription, sequence contrast, modeled k-space, and reconstruction—then inspect the physics underneath.
Choose receive hardware, acquire three-plane localizers, prescribe and reconstruct synthetic T1/PD/T2/FLAIR/DWI/GRE anatomy, then explore echo formation, steady states, perfusion, flow, angiography, and diffusion.
03 / SIGNAL, CONTRAST & MOTION
Wait for recovery.
Encode displacement.
Start with the integrated virtual exam: choose receive hardware, acquire localizers, prescribe a slice, tune T₁/PD/T₂/FLAIR/DWI/GRE controls, and reconstruct modeled k-space. TE, TR, flip angle, and RF history shape the available signal. BOLD fMRI turns a delayed vascular oxygenation consequence into T₂*-weighted time-series change; ASL magnetically labels arterial blood to estimate perfusion; TOF turns replacement of saturated blood into bright-vessel magnitude; phase contrast turns coherent motion into signed velocity; diffusion sensitization probes microscopic displacement. These are distinct mechanisms—not interchangeable maps.
BOLD fMRI is delayed and indirect: a vascular state changes susceptibility-related T₂* weighting, then EPI samples volumes at TR.
ASL couples label duration, ATT, PLD, T₁ survival, control−label difference, and model-based CBF.
TOF couples slab-normal transport to spoiled-GRE history, source magnitude, directional preparation, and MIP.
Phase contrast subtracts two complex measurements, decodes one component, and exposes VENC aliasing and area integration.
Microscopic displacement between lobes leaves a phase distribution; a wider distribution reduces the coherent echo.
Repeat directions to estimate anisotropy; one direction and one b-value are not a tensor fit or tract map.
3D ECHO FORMATION · ONE VOXEL IN THE ROTATING FRAME
A 180° pulse can reverse static phase spread.
It cannot rewind true T₂ loss.
Follow the same transverse ensemble through excitation, dephasing, a refocusing action, rephasing, the echo, and renewed dephasing. Switch mechanisms without changing the tissue or field spread.
01 / 06 · EXCITATION COMPLETESTART WITH COHERENT TRANSVERSE PHASE
A 90° excitation is represented as complete. Every visible spin packet begins along +X′ in a frame rotating near the carrier; the fast MHz carrier itself is removed.
3D
The rotating-frame echo model needs WebGL. The synchronized timeline, equations, metrics, and controls remain available.
Measured time stays 2DSPIN ECHO · 180° AT 40.0 ms · ECHO AT 80 ms
The pulse or gradient command is a cause. The phase fan converges later. Vertical screen position is a teaching state—not another scanner coordinate.
exciterefocus at 50% · echo at 100%after
This teaching scrubber changes the instant being inspected. It does not change the prescribed TE, tissue, or field spread.
20 ms180° / reversal at TE ÷ 2160 ms
Longer TE gives both irreversible T₂ relaxation and unrefocused static offsets more time to reduce the gradient echo. The ideal spin echo removes the static-offset term only at its echo center.
0 HzLorentzian reversible offset distribution20 Hz
A wider static frequency distribution fans phase faster. The 180° pulse reverses its ordering; a gradient-polarity reversal does not.
IDEAL SPIN ECHO AT TE
SSE(TE) / S₀ = e−TE/T₂
The 180° pulse removes the static-offset phase factor at TE; it does not restore the T₂ envelope.
DECLARED LORENTZIAN FIELD-SPREAD MODEL
1/T₂* = 1/T₂ + 1/T₂′ · T₂′ = 1/(πΔfFWHM)
1/T₂* = 1/80 + 1/79.6 ms → T₂* = 39.9 msCURRENT TIME / ACTION0.0 ms · excitation completeall visible packet phases begin aligned
INTRINSIC COHERENCE ENVELOPE1.000T₂ survival at the inspected time
REVERSIBLE STATIC COHERENCE1.000static Δf phase factor in spin-echo mode
MODELED CURRENT |S| / S₀1.000analytic envelope; finite visible arrows are a sampling cue
IDEAL SPIN ECHO AT TE0.368T₂ loss remains after static refocusing
GRADIENT ECHO AT SAME TE0.135T₂* includes unrefocused static spread
DERIVED T₂′ / T₂*79.6 / 39.9 msreversible-only / combined apparent constant
SIGNAL RECOVERED BY IDEAL 180°+0.233SE − GRE at the same TE; not recovered T₂ loss
WHY THE TWO ECHOES DIFFER ·At TE 80 ms, intrinsic T₂ leaves 36.8%. A 4.0 Hz static spread reduces the gradient echo to 13.5%, while an ideal 180° pulse removes that static factor at the spin-echo center.
Static frequency offsets keep their sign.
Fast packets move ahead and slow packets fall behind. An ideal 180° pulse swaps that phase ordering, so continued evolution brings them together at TE.
True T₂ coherence keeps decaying.
The visible packet arrows shorten as the modeled coherent contribution survives. The refocusing pulse changes future phase evolution; it does not add lost coherence or reset time.
Opposite gradient area is not a 180° pulse.
A gradient echo cancels the deliberate gradient moment at TE but leaves static B₀ offsets accumulating, so its ideal envelope follows T₂*.
VIRTUAL MRI EXAM · SYNTHETIC TEACHING WORKSTATION
Choose receive hardware, localize, prescribe a slice, then acquire it.
Review the 3D RF + Gz slice-selection microscope →LIVE SEQUENCE · T₁ SPIN ECHOTR 500 · TE 15 ms
Start with a study · then change one thingILLUSTRATIVE TEACHING STATES · NOT PROTOCOLS
HEAD TEACHING STARTBrain, ventricles, and posterior fossa. Acquire the synthetic localizers, prescribe, then run the virtual scan.
Interactive 3D coil and slice setup
COIL SETUPHEAD / NECK
3D LAYERS
Receive hardware · representative named examplesSELECT ONE SETUP
HEAD / NECK 2020 receive channelsopen-face head helmet with neck extension
- TRANSMIT
- built-in body transmit
- RECEIVE
- circumferential head and upper-neck receive array
- CENTER SENSITIVITY
- 1.00× relative
- UNIFORMITY PROXY
- 0%
- MEAN RECEIVE PROXY
- 0.00× effective
- PARALLEL IMAGING
- not simulated
Receive proxy = mean analytic sensitivity over one central axial elliptical mask. Nominal channel count is a hardware label, not a multiplier. A posterior partner changes the modeled sensitivity term. This does not predict clinical SNR or usable acceleration.
THREE-PLANE LOCALIZER / SCOUTNOT YET ACQUIRED
Acquire the survey first. Localizers reveal internal synthetic anatomy relative to scanner coordinates; the external landmark alone does not prescribe a slice.
ONE 3D MODEL, THREE REPARAMETERISATIONSAll three surveys and every prescribed slice sample the same code-native 3D tissue field in patient millimetres: R/L positive toward L, A/P positive toward P, H/F positive toward F, measured from each region's modelled isocentre. Axial reads (R/L, A/P) at a fixed H/F, sagittal reads (A/P, H/F) at a fixed R/L, coronal reads (R/L, H/F) at a fixed A/P. R→L runs left to right; axial A→P and sagittal/coronal H→F run top to bottom. Positive slice position points toward L, P, or F for sagittal, coronal, or axial prescriptions, and its limit is that region's own half-extent along the selected normal. Each survey is a soft-maximum (power-mean, p = 3) projection of tissue proton density ρ through a 15 mm slab. Every displayed scout voxel uses 2 × 2 in-plane midpoint samples and seven through-slab samples—28 samples total. It is a declared projection, not a simulated pulse sequence, so it carries no TR, TE, TI, flip angle, relaxation, diffusion, coil weighting, or noise. Survey brightness uses a declared display window (ρ 0.56–1.02, γ 0.90); the prescription overlay is drawn in true millimetres at each survey's own scale.
Slice prescriptionWAITING FOR LOCALIZER
−110H/F axis+110 mm
−45°relative teaching rotation+45°
1 mmthrough-plane averaging12 mm
1stack coverage32
0not tissue removed8 mm
100read × phase coverage520 mm
PIXEL1.72 × 1.72 mm
VOXEL14.8 mm³
RELATIVE SNR1.00× preset
THROUGH-PLANE DETAIL4.0 mm samples
STACK COVERAGE71 mm
SCAN-TIME PROXY1:04
ONE-CHANGE EXPERIMENTScompare with this study's starting state
AT THE TEACHING STARTChange slice thickness, FOV, matrix, or NEX. This readout separates resolution, SNR, coverage, and time so “brighter” is not mistaken for “better.”
CONTRAST FOR THE NEXT ACQUISITIONThese buttons drive the same anatomy and quantitative controls below.Fine-tune TR / TE / TI / b / ADC / flip ↓
Physiologic motion + triggeringLINE-WISE FOURIER PHASE MODEL
35cycles per minute180
0phase-encode direction30 mm
5%of each physiology cycle80%
0%timing jitter + rejected windows40%
- CYCLE
- —
- LINES / WINDOW
- all lines
- RESIDUAL MOTION
- 0.0 mm
- WALL-CLOCK PROXY
- 1:04
NO PERIODIC MOTIONThe k-space rows remain mutually consistent. Choose Cardiac or Respiratory, then compare gating on and off in the reconstructed image.
VIRTUAL ACQUISITION · FOURIER RECONSTRUCTION FROM ACQUIRED LINESREADY AFTER LOCALIZER + PRESCRIPTIONEach acquired phase-encode line adds modeled complex k-space values for the selected center slice; the evolving image is recalculated from only those lines, with unacquired lines set to zero.
CONTROLS CHANGEDThis result preserves the acquired state. Run the scan again to apply the new sequence or prescription.
WHAT IS QUANTITATIVETR, TE, TI, flip angle, b/ADC signal equations; FOV/matrix pixel size; voxel volume; slice-stack coverage; one-line-per-TR base time; SNR scaling with voxel volume and √NEX while other factors are held fixed; the exact Fourier translation phase applied to each modeled kᵧ line; the Fourier transform of the displayed center-slice synthetic source; and the plane geometry itself — every survey, overlay, embedded 3D slice, and acquired image is the same 3D field sampled through one millimetre coordinate map.
WHAT IS ILLUSTRATIVEThe ten study buttons are starting states, not protocols. Analytic coil sensitivity, deterministic complex k-space noise, the cardiac/respiratory waveform, trigger acceptance and wall-clock proxy, whole-slice rigid displacement, procedural anatomy, derived display shells, localizer projection/window, partial-volume quadrature, preview resolution, per-region scene scale, nominal channel labels, and compressed playback are teaching models. Slice thickness can improve this model's SNR and increase through-plane averaging; fixed display windowing means it does not guarantee a brighter image.
WHAT IS OMITTEDThe other prescribed slices and repeated NEX lines are not separately reconstructed. The anatomy is hand-declared—not segmented, measured, atlas-derived, registered, diagnostic, or population-normal. Motion is one rigid phase-axis translation, not organ deformation, flow, temporal cine reconstruction, navigator design, arrhythmia classification, prospective controller behavior, or a patient-specific quiet window. Also omitted: measured B₀/B₁/coil maps, channel noise covariance, bandwidth changes, parallel reconstruction, safety screening/limits, calibration, and vendor protocol logic. Nothing here recommends a clinical protocol.
Shared synthetic axial anatomy · live signalT₁ SPIN ECHO · TR 500 / TE 15 ms
ANATOMY PROBE · SAME LOCATION THROUGH EVERY SEQUENCE
RESTRICTED-DIFFUSION CORE
ρ 0.92 · T₁ 1500 · T₂ 120 ms · ADC 0.45 × 10⁻³ mm²/s
current relative signal 0.000
fatgraywhiteCSFT₂ lesionrestricted core
Recovery before RF / decay before echoT₁ RECOVERY + T₂ DECAY
IDEAL SPIN-ECHO SIGNAL
S = ρH(1 − e−TR/T₁)e−TE/T₂
The 180° pulse refocuses static dephasing, so ideal echo amplitude follows T₂ rather than T₂*.
200 msrecovery + scan time6000 ms
5 mstransverse decay200 ms
1°spin echo fixes 90°60°
0choose signed Mz before readout5000 ms
b 0S/S₀ = e−b·ADCb 2000
0.20synthetic target only2.00 × 10⁻³ mm²/s
WHITE MATTERρ 0.70 · T₁ 850 · T₂ 80 · T₂* 55 ms
CSF / FLUIDρ 1.00 · T₁ 4000 · T₂ 2000 · T₂* 400 ms
One repetition · event spacing compressed90° → 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%
MODEL BOUNDARY · SYNTHETIC ANATOMY, NOT A PATIENT OR PROTOCOL RECOMMENDATION. Every pixel belongs to one ideal nonexchanging tissue compartment with illustrative values near 3 T. Spin echo uses ρ(1−e−TR/T₁)e−TE/T₂; FLAIR adds one ideal inversion-prepared repeated cycle; DWI multiplies the same spin-echo baseline by e−b·ADC; GRE uses the spoiled steady state and T₂*. The grayscale uses one fixed relative-signal display mapping, so numeric comparisons remain available without per-preset auto-windowing. Real anatomy and contrast also depend on field strength, sequence family, echo trains and k-space ordering, inversion and fat-suppression profiles, diffusion direction, T₂ shine-through, perfusion, flow, exchange, coils, noise, artifacts, reconstruction, pathology, safety limits, and vendor timing constraints.
ADVANCED 3D MODEL · T₁ SATURATION ACROSS REPETITIONS
One RF pulse is simple.
The history creates the steady state.
Follow one ideally spoiled gradient-echo experiment pulse by pulse. The vector shortens inside a reference sphere, the history plot records the exact recurrence, and the Ernst protractor shows which flip angle maximizes ideal steady-state signal for the selected tissue and TR.
ROTATING FRAME · Z ∥ B₀ · NORMALIZED M / M₀
Before RF · repetition 1
The experiment begins at equilibrium: Mz⁻ = 1 before the first RF pulse. The translucent sphere is only the M₀ reference boundary.
The quantitative history plot remains available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
3D LADDER · ONE SHARED SCALE
Mz⁻ pre-RF
Mz⁺ post-RF
Mxy⁺ immediate
S(TE) readout
Each column is one repetition. Ideal spoiling sets coherent Mxy to zero before the next RF pulse.
Quantitative repetition history
2D PLOT IS THE NUMERIC SOURCE OF TRUTH
tissue A · Mz⁻ before RF
tissue A · Mxy just after RF
tissue B · Mz⁻ before RF
tissue B · Mxy just after RF
Important: these curves show magnetization state, not TE-weighted signal. TE scales the echo readout after each pulse; it does not change convergence toward the spoiled-GRE steady state.
1°near WM Ernst angle60°
5 msT₁ recovery between tips100 ms
1 mssignal only · not convergence19 ms
IDEALLY SPOILED GRE · DISCRETE LONGITUDINAL RECURRENCE
Mn+1− = 1 − (1 − Mn− cosα)E₁
E₁ = e−TR/T₁ · fixed point Mss− = (1 − E₁)/(1 − E₁ cosα) · αE = cos−1(E₁)WHITE MATTER · CURRENT Mz⁻1.000pulse 1 · 100% of initial error remains
steady Mz⁻0.521within 5% by pulse 67 · beyond rail
Ernst angle · tissue A12.4°TR 20 ms · T₁ 850 ms
selected α − Ernst−0.4°near the ideal signal maximum
Mxy immediately after RF0.208state before T₂* weighting
echo signal at TE0.133ρ × Mxy × e⁻ᵀᴱ/ᵀ²*
The first repetition starts at equilibrium. Later pulses begin from the recovery left by every earlier pulse.
Instantaneous ideal RF creates Mxy = Mz⁻ sinα and leaves Mz⁺ = Mz⁻ cosα.
T₂* reduces the measured echo. The recurrence then applies coherent Mxy → 0 before the next pulse without claiming a gradient- or RF-spoiling mechanism.
During TR, Mz returns toward M₀. Repeating the same map approaches one fixed point at rate E₁ cosα.
OPEN A TERM
MODEL BOUNDARY · IDEALLY RF-SPOILED GRE, NOT A FULL BLOCH OR bSSFP SIMULATOR
One normalized isochromat per tissue is shown in the rotating frame, with instantaneous nonselective RF, perfect loss of coherent transverse magnetization between repetitions, monoexponential T₁ recovery, and magnitude-only T₂* echo weighting. T₁ recovery during the short TE is folded into the repetition map. The model excludes coherent/balanced SSFP, RF phase cycling details, B₁ error, slice profile, inflow and flow, magnetization transfer, diffusion, exchange, multiple compartments, off-resonance banding, imaging gradients, noise, and reconstruction. Z/B₀ is the spin-frame reference here, not a patient-orientation claim.
ADVANCED 3D MODEL · INVERSION RECOVERY FAMILY
Invert first.
Wait for one tissue to disappear.
An inversion preparation sends longitudinal magnetization below zero. Tissues recover at different T₁ rates, so a readout placed at the right inversion time can null fat, null fluid, or preserve the sign that ordinary magnitude display folds away.
Efficiency controls the effective longitudinal inversion; it is not drawn as a fictitious hard-pulse angle.
The finite-TR fixed cycle shifts that crossing from the familiar long-TR T₁ ln 2 result.
A phase-sensitive display preserves negative-versus-positive recovery; magnitude folds both sides above zero.
LONGITUDINAL FRAME · NORMALIZED M / M₀ · MATERIALS, NOT ANATOMY
Compare tissue-specific zero crossings
The five declared materials rise through the shared zero plane at different times; fat is at its finite-TR crossing near TI 243 ms.
05 / 05
Mz +0.001
fat · TI 243 ms
The signed TI-response plot remains available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Signed longitudinal state + displayed echo versus TI
EXACT NUMERIC RESPONSE · CURRENT TR AND READOUT
white matter
gray matter
CSF / fluid
fat
custom lesion
target / comparator echo
Upper plot: signed Mz immediately before the readout. Lower plot: the chosen signed or magnitude echo after flip-angle, proton-density, and TE weighting.
inversionnear fat nullreadout
800sets next preparation history12000 ms
0T₂ spin-echo survival180 ms
0.50 · saturationcos θeff = 1 − 2η1.00 · ideal
5°tip Mz into measurable Mxy90°
300same declared PD / T₂ / T₂*4500 ms
EXACT SINGLE-COMPARTMENT PRE-READOUT STATE · REPEATED REGIME
Mz(TI) = 1 − cinvETR − (1 − cinv)ETI1 − cinvcos α ETR
cinv = 1 − 2η · ETI = e−TI/T₁ · S = ρ Mz(TI) sin α e−TE/T₂(or T₂*)FAT · MAGNITUDE ECHO0.000spin echo / T₂ · TE 60 ms
signed Mz at TI+0.001pre-readout · above zero
exact target null243 msfinite-TR fixed cycle · long-TR 243 ms
distance from null0 mstarget is nulled
target − comparator−0.163WM +0.163
cycle closure< 10⁻¹²exact repeated fixed point
η = 1 maps +Mz to −Mz. Lower η reduces that effective longitudinal displacement and changes the null.
Fat crosses zero before white matter, gray matter, and CSF in this declared 3 T teaching table.
The readout flip converts Mz into Mxy. Spin echo uses T₂; gradient echo uses T₂* in this model.
Signed mode retains polarity; magnitude mode applies an absolute value after the same physical signal calculation.
OPEN A TERM
MODEL SOURCES
Boudreau et al. 2017 · general Bloch expression
Redpath & Smith 2014 · finite-TR null timing
Bydder & Young 1985 · STIR
Kellman et al. 2002 · PSIR
MODEL BOUNDARY · IDEAL SINGLE-COMPARTMENT INVERSION RECOVERY, NOT A PROTOCOL DESIGNER
Every material is one normalized, nonexchanging compartment with monoexponential T₁, T₂, and T₂* constants. Inversion and readout are instantaneous and spatially uniform. η is an effective longitudinal action, not a literal pulse angle or a model of adiabatic-pulse dynamics. The repeated mode contains one readout per TR with exact fixed-cycle history; fully recovered mode forces Mpre = M₀. The colored lanes and tiles are declared material samples, not anatomy or a diagnostic image. “STIR” and “FLAIR” presets isolate their nulling mechanism; they are not scanner protocols. “Signed / PSIR-like” preserves ideal signal polarity but does not model a reference acquisition, phase errors, or PSIR reconstruction. Excluded effects include slice profile, B₁ variation, finite pulse duration, magnetization transfer, exchange, multi-component fat, CSF flow, FSE echo trains, stimulated echoes, k-space ordering, coils, noise, motion, SAR, parallel imaging, and vendor timing constraints.
ADVANCED 3D MODEL · FSE / TSE / RARE · EXTENDED PHASE GRAPH
One excitation.
Many echoes, many pathways.
A fast spin-echo train is not one spin echo copied many times. Every refocusing pulse mixes transverse and longitudinal configuration states. Those coherent and stimulated-echo pathways determine each echo; assigning different echoes to phase-encode lines then changes contrast, point spread, and scan time.
EPG order k records accumulated gradient phase area. It is dimensionless pathway bookkeeping—not an imaging ky coordinate.
Sub-180° pulses store part of the history longitudinally and recall it later as stimulated echoes.
The resulting complex echo weighting is an MTF; its inverse transform is the phase-direction PSF.
TWO DIFFERENT SPACES ARE SHOWN—NEVER IDENTIFY THEM
EPG CONFIGURATION ORDER k tracks dephasing pathways inside one modeled compartment. IMAGING ky is a commanded spatial-frequency address acquired on one echo. The dashed bridge is a data-flow link, not an equality.
HARD-PULSE EPG · SIGNED COMPLEX STATES · SEPARATE GROUPED 1D ky ORDER
Echo 8 · coherent pathways meet at F₀
The receiver samples only F₀ at 80 ms. Other transverse and longitudinal orders preserve hidden history for later echoes.
The exact echo-train, MTF, and PSF plots remain available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Echo train + refocusing schedule
NORMALIZED |F₀| BEFORE ρ · DIRECT T₂ PATH IS A REFERENCE, NOT A CEILING
normalized pathway |F₀| · before proton density ρ
normalized direct T₂ path · before ρ
commanded refocusing angle
Declared echo-to-ky mapping → complex MTF → PSF
SEPARATE IMAGING SPACE · 128 PHASE LINES
echo-index bands
normalized |W(ky)|
|inverse DFT W|
source line
weighted magnitude
48 shots for 128 ky lines32
5 mstrain duration 160 ms20 ms
50°constant hard pulses180°
first echoTE 80 mslast echo
early effective TEeffective TE 80 mslate effective TE
1000does not alter this single-train signal6000 ms
REFOCUSING PHASE
FLIP SCHEDULE
GROUPED 1D PHASE-ENCODE VIEW ORDER
HARD-PULSE EPG STATE TRANSITION + DECLARED IMAGING WEIGHT
Ωkafter RF = T(α,φ)Ωkbefore RF · Fk+ → Fk+1+
Sn = ρF0(n·ESP) · W(ky) = Se(ky)/maxe|Se| · PSF = DFT−1{W}SELECTED ρ-SCALED COMPLEX ECHO |S|0.368phase −90.0° · echo 8
ρ-scaled direct T₂-path reference0.368EPG − direct +0.000
effective echo time80 msky zero |S| 0.368
longitudinal pathway energy0.0001 displayed configuration
PSF RMS width3.55 px90% energy width 3 px
largest PSF sidelobe30.3%|W| 0.153–1.000
scan time · 128 ky lines24.0 s8 shots · 6.25% of single-echo
numerical closure< 10⁻¹²Z symmetry + zero discarded state
Relaxation scales F states by E₂ and Z states by E₁; recovery adds only to equilibrium Z₀.
A 180° CPMG pulse swaps transverse pathways. Lower angles also create Z storage and later stimulated echoes.
The receiver sees the signed complex sum at order zero, not the hidden higher-order states individually.
Center ky largely sets effective contrast; variation over ky determines the PSF, blur, ringing, and possible phase modulation.
OPEN A TERM
MODEL SOURCES
Hennig 1988 · low-angle multiecho pathways
Weigel 2015 · EPG formalism
Hennig et al. 2004 · variable refocusing angles
Busse et al. 2006 · prescribed signal evolution
Constable & Gore 1992 · echo train and image blur
MODEL BOUNDARY · IDEAL SINGLE-COMPARTMENT HARD-PULSE EPG, NOT A VENDOR SEQUENCE OR SAFETY CALCULATOR
The EPG engine is on-resonance, nonselective, single-compartment, and monoexponential. RF pulses are instantaneous and spatially uniform; every half-echo interval receives one identical positive unit gradient-order shift. CPMG and CP specify ideal RF-axis relationships. The “declared ramp” is a transparent exponential 180°-to-floor teaching schedule—not a vendor optimizer, prescribed-signal algorithm, or claim of an executable protocol. The imaging panel separately maps the solved echo train into declared grouped one-dimensional phase-encode bands across 128 lines. It is not the EPG configuration axis, a universal view order, or a two-dimensional anatomy reconstruction. TR is used only for scan-time accounting because every displayed train begins at equilibrium; inter-train saturation is excluded. Also excluded are slice profile and selective-pulse crushers, B₁/B₀ distributions, finite RF duration, chemical shift, multiple compartments, exchange, magnetization transfer, diffusion, flow, motion, noise, coils, parallel imaging, partial Fourier, echo sharing, ramp sampling, gradient limits, SAR calculation, and vendor-specific constraints. Signal, MTF, PSF, and timing values are educational simulations, not clinical recommendations.
ADVANCED 3D MODEL · COHERENT BALANCED STEADY STATE
Balance every gradient.
Keep every transverse history.
The spoiled-GRE lesson deliberately destroys transverse coherence. Here the net gradient area returns to zero every TR, so coherence survives, off-resonance phase accumulates, and the repeating vector cycle creates bright passbands separated by dark signal nulls.
The next pulse remembers mainly longitudinal recovery.
The next RF pulse receives a coherent 3D vector with longitudinal and transverse history.
Signal nulls repeat every 1/TR and move when RF phase cycling changes.
RF-PHASE-ALIGNED FRAME · TE = TR / 2 · NORMALIZED M / M₀
Closed steady-state cycle · central passband
The analytic fixed point repeats after free precession, T₁/T₂ relaxation, RF phase advance, and the next ideal RF rotation.
04 / 05
|S| 0.103
analytic repeating state
The quantitative frequency profile remains available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Frequency response + receiver phase
EXACT NUMERIC COMPANION · SAME SOLVES AS 3D
tissue A · single phase cycle
tissue B · dashed comparison
four-cycle RMS when enabled
tissue A receiver phase
Read magnitude and phase together: a stopband is a signal minimum with an abrupt phase transition, not a dark stripe painted at a fixed location.
pulse 1analytic fixed point is selectedpulse 80
−250 Hzcentral passband+250 Hz
5°signal, contrast, RF demand90°
2 msband spacing 200 Hz12 ms
better shimlinear teaching field mapwider offset
RF PHASE INCREMENT Δφ PER TR
DISPLAYED MAGNITUDE
EXACT SINGLE-COMPARTMENT FIXED POINT · RF-ALIGNED FRAME
M+ss = Rx(α)[D(β)M+ss + c]
β = 2πΔf·TR − Δφ · E₁ = e−TR/T₁ · E₂ = e−TR/T₂ · solve (I − RxD)M = RxcWHITE MATTER · ECHO MAGNITUDE0.103single 180° cycle · TE 2.50 ms
receiver phase−180.0°RF-aligned convention
band spacing 1/TR200.0 Hznearest null ±100.0 Hz
distance to nearest null100.0 Hzinside a passband
post-RF vector[0.000, −0.151, 0.352]cycle closure error < 10⁻¹²
transient errorfixed pointnot assumed after one pulse
Position-dependent gradient phase is rewound within each TR. Residual off-resonance phase is not.
Both transverse coherence and longitudinal recovery enter the following RF rotation, so the state is three-dimensional.
The steady state is solved exactly; the optional pulse history shows why it is not established after the first excitation.
RF phase cycling translates the periodic response. Separate phase-cycled acquisitions can reduce, but do not prevent, the underlying off-resonance sensitivity.
OPEN A TERM
MODEL SOURCES
Scheffler 2003 · transient bSSFP
Nayak et al. 2007 · band spacing
Plähn et al. 2024 · phase convention
MODEL BOUNDARY · IDEAL SINGLE-COMPARTMENT bSSFP, NOT A CLINICAL SEQUENCE DESIGNER
Each displayed frequency uses one normalized isochromat with monoexponential T₁/T₂ relaxation, instantaneous nonselective RF, a constant phase increment, and perfectly zero net gradient area each TR. TE is fixed at TR/2. Vectors use an RF-phase-aligned receiver frame that distributes the programmed phase increment uniformly through each TR; the physical laboratory-frame azimuth is not plotted. The 3D frequency–I/Q manifold maps a declared linear off-resonance ramp to independent steady states: frequency is the long axis, I/Q occupy each perpendicular plane, and radius is relative echo magnitude. These are data coordinates, not anatomy, a measured B₀ map, or a reconstructed patient image. Four-cycle RMS combines four separate magnitude acquisitions and has no single coherent magnetization vector. The model excludes intravoxel frequency distributions, slice profile, B₁ error, finite RF duration, eddy-current imbalance, gradient first-moment flow effects, motion, exchange, magnetization transfer, diffusion, chemical multi-peaks, noise, coil sensitivity, k-space ordering, transient catalyzation, SAR calculation, and reconstruction. Real bSSFP names and implementations vary by vendor.
ADVANCED 3D BOLD fMRI · NEURAL INPUT → HEMODYNAMICS → T₂* SIGNAL
The scanner does not measure neurons firing.
It measures a delayed vascular consequence.
Drive one declared cortical region, follow the extended Balloon-model states, and watch changing deoxyhemoglobin alter local susceptibility, spin coherence, gradient-echo magnitude, and the sampled EPI time series. Neural input, physiology, MR contrast, acquisition, and statistical interpretation remain separate objects.
Three-dimensional BOLD neurovascular and T₂-star signal model
01 · DECLARED NEURAL INPUT
A task box drives one modeled cortical region.
The yellow activity cue is an input to the model. It is not a measured neuron, a BOLD value, or an activation-statistic result.
CONTINUOUS FORWARD MODEL + DISCRETE VOLUME SAMPLES
t = 0.0 s · baseline · volume 0
LATEST ACQUIRED SYNTHETIC AXIAL EPI VOLUME
No sampled signal change at baseline
The grayscale background is fixed synthetic anatomy. The colored patch displays the latest TR-sampled volume, while the chart cursor retains the continuous state; neither is a statistical activation map or patient image.
READ THE 3D CHAIN · The translucent brain and voxel are spatial context. Yellow pulses declare modeled neural input. Mint vessel flow and coral deoxyhemoglobin beads are hidden physiological states. Distorted field rings and spin arrows expose an MR consequence that is normally invisible. The final volume cards are discrete samples taken at TR; they are not neurons, blood cells, or proof of activation. Drag to orbit, use the visible camera controls, or tap an object for its exact meaning.
ADVANCED 3D ARTERIAL SPIN LABELING · RF LABEL → TRANSIT → DELIVERY → CBF
Turn arterial blood into a tracer.
Then subtract away almost everything else.
Run one quantitative pseudo-continuous ASL experiment from the neck to a cortical voxel. A control/label RF preparation creates the difference, arterial transit delays delivery, blood T₁ erases part of the tag, and a paired subtraction exposes a signal only a few tenths of one percent of M₀ before the consensus single-PLD equation estimates cerebral blood flow.
Three-dimensional ASL labeling, transit, subtraction, and perfusion model
01 · CONTROL / LABEL PREPARATION
RF and gradient pulses mark arterial blood below the brain.
The label condition inverts the flowing arterial ensemble relative to its control partner. This is magnetic preparation, not an injected substance.
ONE LABEL TRAIN · ARRIVAL WINDOW · ACQUISITION
t = 0.00 s · label train begins · no delivery yet
PAIRED SYNTHETIC VOLUMES · IDENTICAL DISPLAY SCALE
Control and label are almost identical before subtraction.
The control and label anatomy share one magnitude scale. Their difference uses a separately declared amplified display so the tiny modeled perfusion signal is visible; the CBF panel applies the displayed equation to that synthetic difference.
READ THE 3D CHAIN · The lower plane represents a pCASL RF-plus-gradient preparation. Moving beads are synchronized spin-packet cues: coral means label difference, fading toward gray as T₁ erases it. The vessel tree is schematic physical transport; the highlighted cortical voxel is the modeled delivery region. Control and label cards are separate acquisitions, while the difference card is a calculation. Drag to orbit, use the camera controls, or tap any object for its exact role and boundary.
ADVANCED 3D TIME-OF-FLIGHT MRA · TRANSPORT × REPEATED SPOILED GRE
Fresh blood enters bright.
RF history spends that advantage.
Follow one constant-velocity blood cohort across a hard 3D excitation slab. Physical transport decides where each RF event meets the cohort; the exact spoiled-GRE recurrence decides its remaining Mz; and a calculated maximum-intensity projection shows how that depth-dependent signal becomes a bright-blood angiographic display.
01 · TRANSPORT THROUGH THE SLAB
Normal velocity sets RF encounters per transit.
The rings mark where one synchronized cohort meets successive whole-slab RF pulses. They are time samples, not separate RF sheets.
Three-dimensional TOF inflow and saturation model
EXACT STROBOSCOPIC DEPTH PROFILE · DISTANCE FROM ENTRY
5.00 mm per TR · 12 RF encounters · distal blood/background 2.5×
flowing blood signal
stationary background signal
cohort position at an RF event
Why the steps are real in this model: the plot freezes a continuous plug-flow column at one RF instant. Spins that entered during the same TR share an RF count. A real voxel, slice profile, pulsatile velocity distribution, and asynchronous entry can smooth this ideal stroboscopic pattern.
READ THE 3D CAUSAL CHAIN · Each stage button replaces the central geometry so unrelated spaces cannot compete for room. The transparent box is a selected slab, not scanner hardware. A physical vessel crosses it at the chosen angle; vector length and color encode pre-RF Mz; rings locate one cohort at successive RF instants; the coral plane is an ideal 90° preparation outside the positive slab face; and Volume → MIP keeps the calculated source stack visibly separate from its max-over-Y projection. Drag to orbit, zoom, or tap any object for its exact meaning.
ADVANCED 3D PHASE-CONTRAST FLOW · BALANCED M₀, OPPOSITE M₁
Stationary phase cancels.
Velocity survives the subtraction.
Follow one steady through-plane velocity component from a transparent vessel, through two opposite bipolar encodings, into two measured I + iQ vectors and a wrapped velocity map. The physical flow, gradient moments, phase subtraction, VENC, aliasing, and integrated volume flow all share one exact state.
01 · PHYSICAL FLOW PROFILE
The center of the ideal laminar profile moves fastest.
Velocity is physical displacement per second along the vessel. The arrows are not gradient force, electron current, or k-space motion.
Three-dimensional phase-contrast velocity-encoding model
TWO RECTANGULAR BIPOLAR ENCODINGS · TIME IS HORIZONTALM₀,A = M₀,B = 0 · ΔM₁ +28.8 mT·ms²/m
READ THE 3D CAUSAL CHAIN · Each stage button replaces the central geometry. The tube contains the physical velocity profile. The two depth-separated moment volumes are acquisition-time diagrams—not gradients located in the vessel. The volumetric I/Q clocks show SA, SB, conjugation, and their wrapped product. The final relief surfaces use height and hue from the same signed 61 × 61 velocity arrays integrated for mL/s; violet marks wrapped samples. Drag to orbit, use visible zoom controls, or tap an object for its exact meaning.
ADVANCED 3D DIFFUSION EXPERIMENT · PULSED-GRADIENT SPIN ECHO
Stationary spins refocus. Moving water keeps a phase memory.
Orbit a magnified displacement-space model. The first gradient labels position with phase; the 180° pulse reverses that phase ordering; the matched second lobe cancels it only if a spin has not changed position along the selected direction.
06 / 06 · ECHO
MEASURE THE PHASE-DISPERSED ECHO
The stationary-position phase has cancelled. Displacement along the gradient leaves residual phase, so the vector sum is smaller than the b = 0 reference.
Three-dimensional diffusion encoding model
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object to identify it · molecular displacement is magnified, not voxel scale
0both matched lobes80 mT/m
4 msarea per lobe30 ms
20 ms minleading-edge interval80 ms
GRADIENT WEIGHTING32.6 mT/m, δ 20 ms, and Δ 40 ms produce b = 1014 s/mm² in this ideal rectangular-lobe model.
−180°rotate in XY+180°
−90°tilt toward Z+90°
DIRECTIONAL QUESTIONg = [+1.00, +0.00, +0.00], parallel to the tensor’s fixed X principal axis.
0.10tensor eigenvalue3.00
0.10two equal transverse values3.00
AXIALLY SYMMETRIC TEACHING TENSORD∥ is 3.75× D⊥. The ellipsoid is longer along physical X; it is a voxel-scale statistical model, not a drawn axon.
RECTANGULAR-LOBE MODEL
b = (γGδ)²(Δ − δ/3)
S(b,g) / S₀ = e−bDapp(g)
γ uses radians per second per tesla inside b. Dapp = gᵀDg probes one direction through the tensor. The effective finite-lobe diffusion-time term is Δ − δ/3 = 33.3 ms.b-value1014 s/mm²gradient timing and amplitude
apparent diffusion Dapp1.500 × 10⁻³ mm²/stensor projected onto g
Gaussian phase spread σφ1.74 rad√(2bDapp)
ideal diffusion signal S/S₀0.21878.2% attenuated before other contrast and noise
G along g creates a position-dependent frequency offset. Integrating for δ writes phase proportional to g·r.
Between the lobe centers, molecules change position statistically. The 3D paths magnify micrometre-scale displacement so direction can be seen.
After the 180° reversal, the matched lobe cancels stationary phase. Motion along g leaves phase proportional to the projected displacement.
The coil receives one vector sum. Broad residual phases cancel more, so larger bDapp lowers the ideal coherent echo.
QUANTITATIVE 3D DTI · MULTI-DIRECTION ACQUISITION → SIX-PARAMETER FIT
One direction measures one projection. A direction set estimates a tensor.
Continue directly from the pulsed-gradient experiment. Every diffusion-weighted direction supplies one value of gᵀDg. Combine non-collinear measurements, fit all six independent tensor elements, diagonalize D, and inspect where a single tensor fails.
Interactive 3D diffusion-tensor acquisition and fit
01 · DECLARED DIFFUSION SOURCE
A rotated Gaussian tensor defines directional diffusion.
Compare displacement spread √λ, directional ADC gᵀDg, and signal exp(−b gᵀDg). Each panel uses its own linear display scale, so compare shape and direction—not absolute size.
DIRECTION-BY-DIRECTION LEDGER
Measurement 01 / 12 · g = [+0.707, +0.707, 0.000]
The selected axis contributes one row to the log-linear design matrix. The opposite direction is equivalent in this ideal tensor model.
first axisone acquired DWIlast axis
READ THE 3D SPACES · The source and fitted ellipsoids live in one voxel’s physical coordinate frame. Gradient arrows are encoding directions, not water motion. The direction shell is a measurement design, not k-space. Drag to orbit, zoom, or tap an object for its exact role.