Phase 3 · Signal Encoding & Pulse Sequence Engineering
Signal Acquisition, K-Space & Pulse Sequence Dynamics
The mathematics of spatial frequency encoding: Magnetic resonance k-space trajectories, ultrasound plane-wave beamforming, electrophysiological spatial sampling, and ultrafast acquisition for motion-vulnerable pediatric subjects.
Course: IDNE 701 (Fall 2026)
Date: Oct 16, 2026 (Week 8)
Duration: 80 Minutes
Instructor: Mark Bolding, PhD
01. The Generalized Spatial Encoding Equation
In tomography and non-invasive neural sensing, our goal is to reconstruct an underlying spatial object property $\rho(\vec{r})$ (e.g., spin density, acoustic reflectivity, or bioelectric source distribution) from a time series of physical sensor measurements $S(t)$.
By applying spatial gradient fields $\vec{G}(t) = \nabla B_z$, the phase of transverse magnetization accumulates spatially according to:
$$\vec{k}(t) = \frac{\gamma}{2\pi} \int_0^t \vec{G}(\tau) \, d\tau \quad (\text{cm}^{-1} \text{ or } \text{m}^{-1})$$
Equation 1: K-space spatial frequency coordinate trajectory
The received RF baseband voltage across the entire imaging volume $V$ is governed by the Fourier transform of the effective object magnetization:
$$S(t) = \int_V \rho(\vec{r}) e^{-i 2\pi \vec{k}(t) \cdot \vec{r}} \, d\vec{r}$$
Equation 2: The MR Signal Equation (K-space formulation)
02. Trajectory Engineering: Cartesian, EPI & Spiral
How we navigate k-space dictates scan time, acoustic noise, gradient slew demands, and vulnerability to subject movement.
📐 Cartesian Spin-Warp
Acquires one line of k-space per RF excitation ($TR$). Extremely robust to off-resonance and field inhomogeneities, but prohibitively slow ($T_{\text{scan}} = TR \times N_y$). Incompatible with uncooperative or un-sedated pediatric patients.
⚡ Echo Planar Imaging (EPI)
Traverses the entirety of 2D k-space following a single RF excitation using a high-speed blipped gradient raster ($T_{\text{acq}} \sim 30 - 60\,\text{ms}$). Enables single-shot functional BOLD and diffusion neuroimaging, but suffers from severe geometric distortion near air-tissue interfaces ($\Delta B_0$).
🌀 Spiral & Radial Trajectories
Continuous oversampling of the k-space center ($k = 0$) on every readout trajectory. Provides inherent self-navigation and motion robustness. Off-resonance causes blurring rather than spatial shearing.
03. Plane-Wave Coherent Compounding in Ultrasound (fUS)
Traditional focused ultrasound scans line-by-line, achieving frame rates of only $30 - 50\,\text{Hz}$—far too slow to capture microvascular transient hemodynamics without aliasing pulsatile wall motion.
In ultrafast plane-wave imaging, the entire aperture emits unfocused plane waves tilted at angles $\alpha_m$:
$$\tau_i(\alpha_m) = \frac{x_i \sin\alpha_m}{c}$$
Equation 3: Element firing delay for steered plane wave emission
By transmitting at pulse repetition frequencies up to $10 - 20\,\text{kHz}$ and coherently summing backscattered synthetic echoes across tilted angles, ultrafast ultrasound boosts contrast and SNR while delivering power Doppler frames at $500 - 1000\,\text{Hz}$.
04. Grand Challenge 06: Pulse Sequence Adaptation for Neonates
👶 Acoustic Noise, Slew Limits & Fontanelle Sampling
Designing pulse sequences and sampling schemes for neonates requires specialized acoustic, thermal, and mechanical adaptations:
🔇 Acoustic Noise Limits (Quiet Sequences)
High-slew gradient switching in standard adult EPI produces acoustic noise up to $115 - 120\,\text{dBA}$. In premature neonates, this risks cochlear damage and autonomic decompensation. Quiet Sequences utilize sinusoidal or continuous radial readouts to restrict gradient $dG/dt$, keeping acoustic levels below $80\,\text{dBA}$.
⚡ High Heart Rate & Short $T_1$
Neonatal resting heart rate is $120 - 160\,\text{bpm}$ ($RR \approx 400\,\text{ms}$). Cardiac gating must accommodate extremely narrow diastolic resting phases. Furthermore, unmyelinated white matter exhibits prolonged $T_1$ and $T_2$ relaxation times ($T_1 \sim 1800 - 2200\,\text{ms}$ at 3T), demanding customized repetition times ($TR$).
🪟 Transfontanellar Acoustic Sectoring
In transfontanellar fUS, the physical footprint of the anterior fontanelle is approximately $2 \times 2\,\text{cm}$. Ultrasound phased arrays must steer wide sector angles ($\pm 45^\circ$) through this narrow acoustic aperture to reconstruct coronal and sagittal volumetric slabs spanning the deep germinal matrix and basal ganglia.