Parallel Coil Networks (PCNs) Overview

• Parallel Coil Networks (PCNs) are systems of multiple coils arranged strategically to capture or generate electromagnetic fields, serving critical roles in MRI reconstruction and field uniformity optimization.
• PCNs leverage implicit coil-weighting and hybrid classical-deep learning architectures to combine multi-coil data efficiently while enforcing data consistency during image reconstruction.
• Experimental benchmarks reveal that PCNs achieve high PSNR and SSIM in accelerated MRI, and physical PCN designs markedly enhance field uniformity in shielded environments.

A Parallel Coil Network (PCN) is a computational or physical system composed of multiple, spatially arranged coils operating in a coordinated manner—either for electromagnetic field generation or in multi-channel MRI data acquisition and image reconstruction. In computational imaging, PCNs specifically refer to neural network architectures that process measurements from parallel MRI receiver coils, typically eschewing explicit sensitivity map estimation by learning coil-weighted image representations directly. In physical electromagnetic design, PCNs encompass arrays of coils within shielded environments, optimized for spatial uniformity or specialized field distributions. The following sections present a comprehensive overview of theoretical, algorithmic, experimental, and application aspects of Parallel Coil Networks as documented in the scientific literature.

1. Mathematical Modeling of Parallel Coil Networks

In MRI applications, the canonical parallel coil measurement model considers $N_c$ receiver coils, each characterized by a spatially varying complex sensitivity map $S_c(x)$, where $c=1,\ldots,N_c$. The measurement process for an image $x \in \mathbb{C}^{N^2}$ (discrete 2D) is:

$y_c = M F (S_c x) + n_c$

where $F$ is the 2D discrete Fourier transform (DFT), $M$ is a binary undersampling mask applied in k-space (frequency domain), and $n_c$ is measurement noise. The aggregate system can be summarized as:

$y = M F S x + n$

with $y$ stacking all coil measurements and $S$ the block-diagonal coil sensitivity operator (Sriram et al., 2019).

In field generation (e.g., inside magnetic shields), the spatial superposition principle governs composite fields:

$$B^{\mathrm{PCN}}(r) = \sum_{j=1}^{N_c} B^{\mathrm{coil},j}(r)$$

where $B^{\mathrm{coil},j}$ is the field from coil $j$ at position $r$. For shielded environments, correction factors for finite conductivity, geometry, and mutual coupling (mirror-image coefficients, reaction factors) are required (Liu et al., 2020).

2. Algorithmic Frameworks and Architectures

Two principal PCN algorithmic paradigms in MRI emerge:

• Implicit Coil-Weighting Networks: The PCN learns to combine complex-valued coil images via convolutional neural networks with multiple channels corresponding to the coils. No explicit estimation or usage of sensitivity maps occurs; instead, coil combinations are internalized via learned convolutional filters. Architectures are typically U-Nets or Down-Up Networks (DUNs) with $Q$ complex channels ($Q=N_c$), incorporating multi-scale skip connections and deep residual units (Schlemper et al., 2019, Hammernik et al., 2019).
• Hybrid Classical-Deep Composites: Networks such as GrappaNet embed classical k-space interpolation layers (e.g., GRAPPA convolution) as differentiable operators within deep network pipelines. The system alternates between learned units (e.g., U-Nets) and scan-specific or physics-based operations, leveraging both data-driven and model-based elements, and keeping all layers differentiable for end-to-end training (Sriram et al., 2019).

The standard PCN reconstruction pipeline involves:

1. Input: Under-sampled multi-coil k-space data.
2. Channel-wise processing: Each coil is treated as a separate channel in convolutional networks.
3. Learned coil fusion: The network learns implicit weight maps for coil image combination.
4. Data consistency: Incorporation of analytically tractable data consistency (DC) layers, such as gradient-descent, proximal mapping, or variable splitting, ensures conformance with acquired data.
5. Output: The final image is typically derived by root-sum-of-squares (RSS) or via a learned combiner network.

3. Training Schemes and Loss Functions

PCNs are trained with combinations of image fidelity, structural similarity, and, occasionally, adversarial objectives. For pure supervised approaches:

• Base loss: $$\ell_\mathrm{base}(x_\mathrm{rec}, x_\mathrm{ref}) = \mathrm{SSIM}(m \odot |x_\mathrm{rec}|, m \odot |x_\mathrm{ref}|) + \lambda \|m \odot |x_\mathrm{rec}| - m \odot |x_\mathrm{ref}|\|_1$$, where $m$ is a foreground mask (Hammernik et al., 2019, Schlemper et al., 2019).
• GAN fine-tuning: A least-squares GAN component to encourage sharper texture—$\ell_\mathrm{LSGAN}$ is used alongside $\ell_\mathrm{base}$.
• Semi-supervised adaptation: At inference, network parameters θ can be further refined on new, label-free data by solving

$$\min_{\theta} \frac{1}{2} \|A x_{\theta} - y\|_2^2 + \alpha \max(\mathrm{SSIM}(|x_\theta|, |x_\mathrm{rec}|) - \beta, 0)^2$$

where $x_\mathrm{rec}$ is the network’s initial output and the SSIM term anchors the solution to previously learned topology (Schlemper et al., 2019).

The entire pipeline is differentiable, with analytic gradients for both learned and physics-based layers, enabling backpropagation through the entire system, including classical GRAPPA or RAKI interpolators (Sriram et al., 2019).

4. Experimental Performance and Benchmarking

Comprehensive benchmarking demonstrates that PCNs can achieve competitive or superior performance versus explicit sensitivity-map-based approaches and classical compressed sensing. Key quantitative metrics include normalized mean squared error (NMSE), peak signal-to-noise ratio (PSNR), and structural similarity index (SSIM).

For the fastMRI knee dataset (4× acceleration, multi-coil):

• GrappaNet (PCN with explicit GRAPPA layer): PSNR $\approx$ 40.7 dB, SSIM $\approx$ 0.957.
• Pure U-Net (single-coil): PSNR $\approx$ 39.6 dB, SSIM $\approx$ 0.949.
• Σ-net (ensemble including PCN and SN): PSNR $\approx$ 39.57 dB, SSIM $\approx$ 0.9205 (R=4) (Schlemper et al., 2019, Hammernik et al., 2019, Sriram et al., 2019).

Ablation studies reveal that implicit coil weighting in PCNs performs comparably to explicit sensitivity-based fusion; ensembling (e.g., in Σ-net) further increases robustness and can preserve high quantitative metrics while enhancing perceptual texture via adversarial fine-tuning.

For dynamic MRI (cine), unsupervised per-coil networks with a learned CNN combiner demonstrate:

• 4× acceleration: Proposed PCN, PSNR $\approx$ 36.1 dB, SSIM $\approx$ 0.945, runtime $\approx$ 0.5 s/volume, outperforming k-t FOCUSS, k-t SLR, and low-rank models by several dB and substantial SSIM margin (Ke et al., 2019).

5. Physical PCNs: Field Uniformity and Optimization

In electromagnetic field engineering, physical PCNs (e.g., for magnetically shielded rooms) comprise parallel arrays of coils designed for optimized spatial uniformity, often inside high-permeability enclosures. The three-step modeling approach consists of:

1. Superposition of coil fields in free space.
2. Application of improved mirror-image coefficients per coil per image, accounting for finite shield plate thickness and permeability.
3. Multiplication by spatially-varying reaction factors $R(x,y,z)$, computed once via finite element methods (FEM) for a given shield geometry (Liu et al., 2020).

This yields highly accurate predictions of field uniformity without resorting to full 3D FEM for each configuration.

Quantitative results: For two 2.75 m square coils in the BMSR-2 shield, relative variation $\Delta B/B_0 = 6$ ppm (max-min over 5 cm), compared to 36 ppm for a Helmholtz pair in the same environment, demonstrating a 5.4-fold improvement through optimized PCN spacing.

The methodology generalizes to arbitrary $N$-coil networks, with field evaluation and optimization tractable via precomputed coefficient tables and analytic superposition.

6. Implicit Versus Explicit Coil Sensitivity Estimation

The defining distinction between PCNs and Sensitivity Networks (SNs) is the approach to coil combination:

• PCN (implicit): The network directly learns the coil combination as an internal function of its convolutional filters without access to or explicit estimation of $S_c(x)$. Weight-maps per coil, $w_c(x)$, emerge implicitly for the final output $m(x) = \sum_{c=1}^Q w_c(x)m_c(x)$, but are neither hand-crafted nor normalized (Schlemper et al., 2019, Hammernik et al., 2019).
• SN (explicit): The network relies on precomputed sensitivity maps (e.g., from ESPIRiT or JSENSE), using them for explicit pixelwise re-weighting. Standalone PCNs are simpler to deploy and less vulnerable to failures of map estimation, although they may be less robust to changes in coil configuration or SNR without retraining.

Empirically, both paradigms achieve similar quantitative metrics, and ensembling across both types further improves robustness (Schlemper et al., 2019, Hammernik et al., 2019).

7. Extensions and Future Directions

Possible generalizations of the PCN paradigm include:

• Replacement of fixed classical coil fusion (e.g., GRAPPA) with learned, scan-specific k-space CNNs (as in RAKI) (Sriram et al., 2019).
• Incorporation of deeper and multi-scale combiners, such as multi-coil attention modules, to enhance cross-coil synergy (Ke et al., 2019).
• Self-supervised training regimes, leveraging k-space consistency constraints for improved generalization and scan specificity without ground-truth labels (Schlemper et al., 2019).
• Hybrid networks combining explicit coil sensitivity estimation with PCN modules, or incorporating explicit uncertainty modeling of learned coil weights.

In field-engineering, acceleration of field summation through fast multipole or FFT-based algorithms provides scalability to large networks. The reaction-factor formalism enables rapid prototyping of PCN layouts for instrument design (Liu et al., 2020).

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References:

• (Sriram et al., 2019) "GrappaNet: Combining Parallel Imaging with Deep Learning for Multi-Coil MRI Reconstruction"
• (Schlemper et al., 2019) "$\Sigma$-net: Ensembled Iterative Deep Neural Networks for Accelerated Parallel MR Image Reconstruction"
• (Hammernik et al., 2019) "$\Sigma$-net: Systematic Evaluation of Iterative Deep Neural Networks for Fast Parallel MR Image Reconstruction"
• (Ke et al., 2019) "An Unsupervised Deep Learning Method for Multi-coil Cine MRI"
• (Liu et al., 2020) "A Three-step Model for Optimizing Coil Spacings Inside Cuboid-shaped Magnetic Shields"
• (Plonka et al., 19 Mar 2024) "MOCCA: A Fast Algorithm for Parallel MRI Reconstruction Using Model Based Coil Calibration"