Physics-Informed Implicit Neural Representations for Improved Myocardial Perfusion MRI Quantification

This paper presents a physics-informed neural network (PINN) framework with spatiotemporal implicit neural representations (INRs) for quantitative myocardial perfusion MRI. The method extends prior voxelwise PINN approaches by representing the dynamic MR signal and kinetic parameter maps as continuous functions of space and time using sinusoidal representation networks (SIRENs). The framework consists of three coupled INRs: one for the arterial input function (AIF), one for the myocardial tissue concentration, and one for the kinetic parameters (Fp, vp, ve, PS). These are coupled through the two-compartment exchange model (2CXM), enforced as a physics loss in the analytical convolution form. The kinetic parameter INR is the central novelty, as it constrains parameters to a continuous spatial manifold, providing implicit spatial regularization through weight sharing across voxels. The method was evaluated on a digital reference object (DRO) and realistic Bloch-equation simulated CMR data with cardiac motion and undersampling. On the DRO, the proposed method achieved the lowest mean NMSE (0.09 vs 0.13 for voxelwise PINN and 0.17 for NLLS) and highest mean SSIM (0.63 vs 0.53 and 0.47), with particularly strong improvements for Fp and PS. On simulated CMR data, it outperformed NLLS across all four kinetic parameters, with an overall median NMSE of 0.212 vs 0.884 for NLLS, and produced spatially smooth parameter maps that correctly captured perfusion deficits. The authors conclude that spatiotemporal PINNs offer a more accurate and robust alternative to existing approaches, with the physics constraints acting as regularization against noise, motion, and undersampling artefacts. Limitations include evaluation only in simulation and the need to specify SIREN frequencies, though these can be fixed per acquisition protocol.

Improvements to AI systems:

1. Add a spatiotemporal implicit neural representation (INR) module that encodes both spatial coordinates (x, y, z) and time (t) into a single continuous function, using sinusoidal activation functions (SIREN) to capture high-frequency variations in dynamic signals.

2. Implement a multi-head coupled INR architecture where three separate neural networks share a common latent space: one head outputs the arterial input function (AIF), one outputs tissue concentration, and one outputs kinetic parameter maps (Fp, vp, ve, PS). This forces the network to learn consistent representations across all outputs.

3. Replace voxelwise parameter fitting with a global spatial manifold constraint by having the kinetic parameter INR take spatial coordinates as input and output all parameters simultaneously. This implicitly regularizes neighboring voxels to have similar parameter values, reducing noise sensitivity.

4. Integrate a physics-based loss term that enforces the two-compartment exchange model (2CXM) in its analytical convolution form, rather than as a differential equation. This makes the physics constraint differentiable and stable during training, acting as a hard regularizer against noise and undersampling artifacts.

5. Add a frequency-adaptive initialization scheme for SIREN layers, where the frequency of the sinusoidal activations is automatically tuned per acquisition protocol (e.g., based on temporal resolution and spatial resolution) to avoid spectral bias.

6. Implement a joint training strategy that alternates between fitting the signal INR (to match measured MR data) and fitting the parameter INR (to satisfy the physics model), with a weighting schedule that gradually increases the physics loss weight to stabilize early training.

What the improved AI system can do:

• Reconstruct quantitative myocardial perfusion maps (blood flow Fp, plasma volume vp, extracellular volume ve, permeability-surface area product PS) directly from undersampled, motion-corrupted dynamic MRI data without per-voxel curve fitting.

• Produce spatially smooth, physiologically plausible parameter maps that preserve perfusion deficits (e.g., ischemic regions) even when the raw signal-to-noise ratio is low, outperforming traditional nonlinear least squares (NLLS) by 4x in median normalized mean squared error (0.212 vs 0.884) on simulated cardiac MRI.

• Generalize across different acquisition protocols by automatically adjusting the INR’s frequency parameters, removing the need for manual tuning per dataset.

• Provide uncertainty estimates implicitly via the spatial manifold (voxels with inconsistent neighboring parameters are flagged), enabling automated quality control of perfusion maps.

• Operate in a fully self-supervised manner—no labeled training data required—by leveraging the physics model as the only supervision signal, making it applicable to new patients or scanners without retraining.

Quantifying myocardial perfusion from cardiac magnetic resonance (CMR) can be achieved by fitting tracer-kinetic models to the dynamic contrast-enhanced MR data. However, fitting the observed data with multi-compartment exchange models, which describe the evolution of the contrast agent in the tissue, to estimate perfusion parameters is a challenging inverse problem that is sensitive to noise and acquisition variability. Previously, physics-informed neural networks (PINNs) have been proposed as an alternative to conventional non-linear least squares fitting methods with promising results for quantitative perfusion CMR. In this work, we extend the previously proposed PINN framework with spatiotemporal implicit neural representations (INRs) to represent the MR signal as a continuous spatiotemporal function and to improve the accuracy, smoothness, and physical consistency of the PINN model. In realistic simulated CMR datasets, our proposed PINN with INRs demonstrates improved robustness and parameter estimation accuracy over the previously established methods. The code is available at https://github.com/q-cardIA/pinn-inr.