Basics of Image acquisition =========================== Topics overview --------------- - **k‑space trajectories**: Cartesian, radial, spiral, PROPELLER, 3‑D shells - **Encoding**: frequency‑encode, phase‑encode, slice‑select; partial Fourier - **Sampling parameters**: FOV, matrix size, receiver bandwidth, dwell time, echo train length - **Readout schemes**: single‑shot, segmented, turbo factor / ETL - **Averaging & noise**: NEX/NSA, parallel‑receive SNR behavior - **Reconstruction**: Fourier transform, gridding, NUFFT, density compensation - **Coil combination**: sum‑of‑squares, adaptive combine, ESPIRiT maps - **Parallel imaging**: SENSE, GRAPPA, CAIPIRINHA - **Compressed sensing / low‑rank**: k‑t, XD‑GRASP - **Corrections**: motion/phase, eddy‑currents, :math:`B_0` drift - **Artifact mitigation & QC**: ghosts, Gibbs ringing, geometric distortion; basic checks Historical notes (FID and early NMR) ------------------------------------ - **1946 (CW NMR)** — Bloch, Purcell et al. used fixed‑frequency :math:`B_1` and **swept** :math:`B_0` through resonance, observing “resonance absorption” / “nuclear induction signal.” - **1948** — Bloembergen, Purcell, Pound reported small oscillations (“**wiggles**”) flanking the main absorption after passing resonance. - **Hahn (pulsed NMR)** — Held :math:`B_0` constant, applied pulsed :math:`B_1` at the Larmor frequency; observed the **free induction decay (FID)**. MRI differs from other modalities because the **signal is generated by the patient**; localization identifies **where** signals originate. Spatial localization -------------------- Three components: - **Slice selection (SS)** - **Frequency encoding (FE)** - **Phase encoding (PE)** .. important:: Having two gradients **on at the same time** means the **vector sum** of gradients; it is not equivalent to turning them on independently. Slice selection ^^^^^^^^^^^^^^^ - Apply :math:`G_{\text{slice}}` during the RF pulse. RF **bandwidth** and gradient **strength** set the **slice thickness**: - Wider RF bandwidth or weaker gradient → **thicker** slice. - Narrower RF bandwidth or stronger gradient → **thinner** slice. - After the RF, apply an **opposite‑polarity rephasing lobe** on :math:`G_{\text{slice}}` to undo slice‑selection dephasing within the slice. Frequency encoding (readout) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - During signal acquisition at :math:`\mathrm{TE}`, apply :math:`G_{\text{read}}` so position maps to frequency. - The ADC samples with **dwell time** :math:`\Delta t`; receiver bandwidth :math:`\mathrm{RBW} \approx 1/\Delta t` (per complex sample). - **1‑D Fourier transform** along readout reconstructs the spatial profile in the read direction. Phase encoding ^^^^^^^^^^^^^^ - A brief gradient :math:`G_{\text{phase}}` before readout imparts a **position‑dependent phase** that persists (spins see the same :math:`B_0` before/after the lobe). - Repeating the sequence with **different PE amplitudes** fills different **rows of k‑space**. k‑space formalism ----------------- Define k‑space coordinates from gradients: .. math:: \mathbf{k}(t) = \frac{\gamma}{2\pi} \int_0^{t} \mathbf{G}(\tau)\, d\tau \quad\Rightarrow\quad s(t) = \int \rho(\mathbf{r})\, e^{-i 2\pi \mathbf{k}(t)\cdot \mathbf{r}}\, d\mathbf{r} Key properties: - **Central k‑space** (low spatial frequencies) encodes **contrast**; acquired with smaller PE amplitudes, higher signal coherence. - **Outer k‑space** (high spatial frequencies) encodes **edges / resolution**; larger PE amplitudes. - **Symmetries** permit **partial Fourier** (half‑scan) if conjugate symmetry holds after phase correction (e.g., homodyne, POCS). Sampling & imaging relations ---------------------------- Let readout **matrix** be :math:`N_x`, **FOV** be :math:`\mathrm{FOV}_x`, and read gradient be constant during acquisition. - k‑space sample spacing: :math:`\Delta k_x = 1/\mathrm{FOV}_x` - Resolution: :math:`\Delta x \approx \mathrm{FOV}_x / N_x` - Maximum spatial frequency: :math:`k_{x,\max} \approx N_x/(2\,\mathrm{FOV}_x)` - Dwell time vs bandwidth: :math:`\Delta t \approx 1/\mathrm{RBW}` (per complex sample) Readout schemes --------------- - **Single‑shot**: entire k‑space (or a plane/segment) in one shot (e.g., EPI); fast, sensitive to distortions and :math:`T_2^*`. - **Segmented**: k‑space split across shots; reduced distortion, longer scan. - **Turbo/fast spin echo (FSE/TSE)**: multiple echoes per TR; **echo train length (ETL)** / turbo factor is number of k‑space lines per TR. Averaging & noise ----------------- - **NEX/NSA** (number of excitations / signal averages): SNR scales as :math:`\sqrt{\text{NEX}}`. - **Parallel receive arrays**: SNR depends on coil geometry and **g‑factor**; combination method matters. Coil combination ---------------- - **Sum‑of‑squares (SoS)**: simple, robust magnitude combination. - **Adaptive combine**: SNR‑optimal with estimated noise covariance / sensitivity. - **ESPIRiT / sensitivity maps**: explicit coil sensitivity modeling for parallel imaging. Parallel imaging ---------------- - **SENSE**: image‑domain unfolding using sensitivity maps. - **GRAPPA**: k‑space interpolation using autocalibration lines (ACS). - **CAIPIRINHA**: controlled aliasing pattern to improve conditioning / g‑factor in 2‑D/3‑D. Compressed sensing & low‑rank ----------------------------- - **CS (k‑t)**: incoherent undersampling + sparsity (e.g., temporal TV, wavelets). - **Low‑rank / subspace**: e.g., **XD‑GRASP** for motion‑resolved reconstruction. Non‑Cartesian recon ------------------- - **Radial / spiral / PROPELLER / 3‑D shells** require **gridding** or **NUFFT** with **density compensation** before FFT. - Trajectory calibration and system delays affect image quality; correct with field probes, trajectory measurement, or self‑navigation. Corrections & QC ---------------- - **Motion/phase correction**: navigator echoes, PROPELLER, retrospective registration, phase stabilization. - **Eddy‑current correction**: pre‑emphasis, post‑hoc modeling, bipolar gradients. - **:math:`B_0` drift correction**: reference navigators, field monitoring. - **Artifacts**: - **Ghosting** (EPI Nyquist): odd/even echo phase mismatch; calibrate/phase‑correct. - **Gibbs ringing**: insufficient high‑frequency sampling; mitigate with apodization or higher matrix. - **Geometric distortion**: :math:`B_0` inhomogeneity in EPI; reduce bandwidth per pixel in phase‑encode, use parallel imaging, field maps. - **Basic QC**: check noise level, ghost‑to‑signal ratio, NPS/PSF behavior, gradient timing, and k‑space center stability. Workflow summary ---------------- 1. **Design** trajectory and encoding (SS, PE table, readout). 2. **Acquire** data with appropriate sampling (dwell time, RBW, FOV, matrix). 3. **Preprocess** (DC/phase corrections, eddy/:math:`B_0` drift fixes). 4. **Reconstruct**: - Cartesian: FFT (with coil combine / parallel imaging). - Non‑Cartesian: gridding/NUFFT + FFT (then coil combine, PI, CS/low‑rank as needed). 5. **Validate/QC** and mitigate artifacts where necessary.