                    IDNE 701 · Week 8 Lecture
                

Signal Acquisition & K-Space Dynamics

Pulse Sequence Design, Spatial Frequency Trajectories & Plane-Wave Ultrasound

📄 Lecture Notes 📅 Course Syllabus

The Unified Spatial Encoding Model

Linking spatial gradient magnetic fields to the spatial frequency domain.

$$S(t) = \int_V \rho(\vec{r}) e^{-i 2\pi \vec{k}(t) \cdot \vec{r}} \, d\vec{r} \quad \text{where} \quad \vec{k}(t) = \frac{\gamma}{2\pi} \int_0^t \vec{G}(\tau) \, d\tau$$

The MR Signal Equation: Continuous Fourier integral of effective object spin density

🎯 Spatial Resolution (Field of View)

Step Size: $\Delta k = \frac{1}{\text{FOV}}$ (avoids spatial aliasing).

Maximum Extent: $k_{\text{max}} = \frac{1}{2\Delta x} \implies \Delta x = \frac{1}{2 k_{\text{max}}}$.

⚡ Sampling Speed & Gradient Slew

Traversing k-space requires dynamic magnetic gradient ramps limited by maximum amplitude ($G_{\text{max}}$) and slew rate ($SR = dG/dt$).

K-Space Trajectory Architectures

Cartesian Spin-Warp

• One $k_y$ line per TR.

• Rectilinear FFT reconstruction.

• Highly robust to $\Delta B_0$.

• Slow: Minutes per volume.

Echo Planar (EPI)

• Full 2D raster in single excitation.

• Acquisition: $30 - 60\,\text{ms}$.

• High acoustic gradient chatter.

• Geometric distortion near sinuses.

Spiral & Radial

• Self-navigating center oversampling.

• Inherent motion robustness.

• Non-Cartesian gridding needed.

• Ideal for un-sedated pediatric scans.

Ultrafast Plane-Wave Ultrasound (fUS)

Replacing focused line scans with coherently compounded plane-wave transmissions.

✈️ Steered Plane Wave Emission

All array elements fire simultaneously with a linear phase tilt:

$$\tau_i(\alpha_m) = \frac{x_i \sin\alpha_m}{c}$$

Emits unfocused planar waves across angles $\alpha_m \in [-15^\circ, +15^\circ]$ at PRF up to $15\,\text{kHz}$.

⚡ Frame Rates: 500–1000 Hz

Coherent compounding across 11–21 angles yields high-contrast, synthetic-focus images with $100\times$ the frame rate of clinical ultrasound.

Crucial for microvascular power Doppler imaging without aliasing cardiac pulse waves.

Grand Challenge Focus: Neonatal Sequence Design

🔇 Acoustic Safety (Quiet MR)

Standard EPI gradients create loud acoustic banging ($> 110\,\text{dBA}$).

Neonatal protocols use Quiet Sequences with sinusoidal gradient derating, holding noise below $80\,\text{dBA}$ to protect fragile infant hearing.

❤️ Rapid Neonatal Heart Rates

Neonatal resting heart rate is $120 - 160\,\text{bpm}$ ($RR \sim 400\,\text{ms}$).

Cardiac gating must operate within extremely narrow diastolic rest periods to avoid pulsatile blurring in germinal matrix vessels.

🪟 Fontanelle Aperture Limits

The anterior fontanelle is approximately $2 \times 2\,\text{cm}$.

Transfontanellar arrays require wide sector beam steering ($\pm 45^\circ$) to cover the entire coronal slab of the neonatal brain from a compact footprint.

Looking Ahead to Week 9

Next: Reconstruction Pipelines & Machine Learning Biomarker Extraction

📄 Read Lecture Notes 📅 Week 9 Syllabus Outline

                    IDNE 701: Introduction to Neuroengineering · Department of Biomedical Engineering
                