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Abstract
Spin-lattice relaxation time ($T_1$) is an important biomarker in cardiac parametric mapping for characterizing myocardial tissue and diagnosing cardiomyopathies. Conventional Modified Look-Locker Inversion Recovery (MOLLI) acquires 11 breath-hold baseline images with interleaved rest periods to ensure mapping accuracy. However, prolonged scanning can be challenging for patients with poor breathholds, often leading to motion artifacts that degrade image quality. In addition, $T_1$ mapping requires a voxel-wise nonlinear fitting to a signal recovery model involving an iterative estimation process. Recent studies have proposed deep-learning approaches for rapid $T_1$ mapping using shortened sequences to reduce acquisition time for patient comfort. Nevertheless, existing methods overlook important physics constraints, limiting interpretability and generalization. In this work, we present an accelerated, end-to-end $T_1$ mapping framework leveraging Physics-Informed Neural Ordinary Differential Equations (ODEs) to model temporal dynamics and address these challenges. Our method achieves high-accuracy $T_1$ estimation from a sparse subset of baseline images and ensures efficient null index estimation at the test time. Specifically, we develop a continuous-time LSTM-ODE model to enable selective Look-Locker (LL) data acquisition with arbitrary time lags. Experimental results show superior performance in $T_1$ estimation for both native and post-contrast sequences and demonstrate the strong benefit of our physics-based formulation over direct data-driven $T_1$ priors.
Links to Paper and Supplementary Materials
Main Paper (Open Access Version): https://papers.miccai.org/miccai-2025/paper/4475_paper.pdf
SharedIt Link: Not yet available
SpringerLink (DOI): Not yet available
Supplementary Material: Not Submitted
Link to the Code Repository
https://github.com/nunomiguelc18/pinn-node-cardiac-t1mapping
Link to the Dataset(s)
N/A
BibTex
@InProceedings{CapNun_PhysicsInformed_MICCAI2025,
author = { Capitão, Nuno and Zhang, Yi and Zhao, Yidong and Tao, Qian},
title = { { Physics-Informed Neural ODEs for Temporal Dynamics Modeling in Cardiac T1 Mapping } },
booktitle = {proceedings of Medical Image Computing and Computer Assisted Intervention -- MICCAI 2025},
year = {2025},
publisher = {Springer Nature Switzerland},
volume = {LNCS 15960},
month = {September},
}
Reviews
Review #1
- Please describe the contribution of the paper
The main contirbution is inclduing temporal dynamics modeling within deep-neural-networks for cardiac T1 mapping from small number of T1 weighted images. Conventional imaging use 11 or 8 T1W images. Recent deep learning methods aimed to reduce the number of required T1W images. Yet, these methods are mostly data driven and do not consider the temporal dynamics. The proposed approach introduce temporal dynamics by means of including the gradient between the subsequent T1W images as part of the loss function.
- Please list the major strengths of the paper: you should highlight a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
- Innovative integration of the temproal dynamics in deep-neural-networks for T1 mapping.
- Rigorous evaluation for both in distribution and out of distribution data.
- Clear presentation.
- Please list the major weaknesses of the paper. Please provide details: for instance, if you state that a formulation, way of using data, demonstration of clinical feasibility, or application is not novel, then you must provide specific references to prior work.
- There are no details for groud-truth generation beyond using the LM algorithm. How it was initialized? That may introduce bias toward the neural networks trained with this data compared to the TRF.
- Authors did not use community recommended colormaps (https://pubmed.ncbi.nlm.nih.gov/39415361/).
- A 17 AHA segment analysis of the estimation error for each algorithm would be more informative.
- All methods here are essentially pixel-wise, CNN that leverage spatial corrleations should be also compared as these probably will provide more accurate results.
Minor: Eq. 1: I would recommend the use of A dn B instead of k as these are more common in the literature.
- Please rate the clarity and organization of this paper
Satisfactory
- Please comment on the reproducibility of the paper. Please be aware that providing code and data is a plus, but not a requirement for acceptance.
The submission does not mention open access to source code or data but provides a clear and detailed description of the algorithm to ensure reproducibility.
- Optional: If you have any additional comments to share with the authors, please provide them here. Please also refer to our Reviewer’s guide on what makes a good review and pay specific attention to the different assessment criteria for the different paper categories: https://conferences.miccai.org/2025/en/REVIEWER-GUIDELINES.html
N/A
- Rate the paper on a scale of 1-6, 6 being the strongest (6-4: accept; 3-1: reject). Please use the entire range of the distribution. Spreading the score helps create a distribution for decision-making.
(4) Weak Accept — could be accepted, dependent on rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
An ok paper, not very innovative. Clear impact is not demonstrated as authors did not compare to CNN and reference is not of high quality.
- Reviewer confidence
Very confident (4)
- [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.
N/A
- [Post rebuttal] Please justify your final decision from above.
N/A
Review #2
- Please describe the contribution of the paper
Adapting neural ODE-based physics informed neural network for cardiac T1 mapping.
- Please list the major strengths of the paper: you should highlight a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
Both mathematical modeling and the PINN framework are novel. The advantage of the proposed method is demonstrated with experimental results.
- Please list the major weaknesses of the paper. Please provide details: for instance, if you state that a formulation, way of using data, demonstration of clinical feasibility, or application is not novel, then you must provide specific references to prior work.
The proposed networks demonstrated higher fitting standard deviations, potential compromise of precision, which may require further optimizations.
- Please rate the clarity and organization of this paper
Satisfactory
- Please comment on the reproducibility of the paper. Please be aware that providing code and data is a plus, but not a requirement for acceptance.
The authors claimed to release the source code and/or dataset upon acceptance of the submission.
- Optional: If you have any additional comments to share with the authors, please provide them here. Please also refer to our Reviewer’s guide on what makes a good review and pay specific attention to the different assessment criteria for the different paper categories: https://conferences.miccai.org/2025/en/REVIEWER-GUIDELINES.html
Table 1 includes some errors in terms of bold (best) and underlined (second best) methods. The authors need to check the accuracy. Quality of Figure 3 needs to be improved.
- Rate the paper on a scale of 1-6, 6 being the strongest (6-4: accept; 3-1: reject). Please use the entire range of the distribution. Spreading the score helps create a distribution for decision-making.
(5) Accept — should be accepted, independent of rebuttal
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
Despite the sub-optimal performance in terms of precision, the proposed concept is novel.
- Reviewer confidence
Confident but not absolutely certain (3)
- [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.
N/A
- [Post rebuttal] Please justify your final decision from above.
N/A
Review #3
- Please describe the contribution of the paper
The work proposes and evaluates a novel deep learning-based method for T1 mapping using the popular MOLLI sequence. The method has two key novelties: (i) a physics-informed loss based on the physical signal recovery mode for the sequence; (ii) a continuous-time LSTM-ODE (CT-LSTM-ODE) model is employed so that the model is inherently suitable for training and evaluation on data with non-uniform temporal sampling.
- Please list the major strengths of the paper: you should highlight a novel formulation, an original way to use data, demonstration of clinical feasibility, a novel application, a particularly strong evaluation, or anything else that is a strong aspect of this work. Please provide details, for instance, if a method is novel, explain what aspect is novel and why this is interesting.
The motivation and formulation of both the physics-based loss terms and the use of the CT-ODE-based model is clear, concise, and novel. This is an excellent application of CT-ODE models. The description of the datasets and implementation are sufficiently thorough, and the suggestion of public release of the code is greatly appreciated.
The evaluation of the model is also nicely explained and well-thought out. The inclusion of pre- and post-contrast sequences into the validation and testing sets is a nice test for generalization capability. The integration of MyoMapNet’s architecture into the physics-based training framework is a very interesting ablation study.
- Please list the major weaknesses of the paper. Please provide details: for instance, if you state that a formulation, way of using data, demonstration of clinical feasibility, or application is not novel, then you must provide specific references to prior work.
The benefits of the proposed approach over existing DL methods, e.g., MMNet or T1N, and the non-learning TRF fitting algorithm are not particularly clear. Table 1 shows that for LL4-5, the proposed method has the smallest mean bias (in both the native and post-gd datasets), however the same trend is not true for LL3. Additionally, the TRF method consistently achieves the lowest (or at least second-lowest) SD. Fig. 3 seems to echo the results of Table 1. Furthermore, it isn’t clear how “best” is defined for the Mean Bias columns in Table 1. This should be added to the table caption.
As their presented, the results of the TRF method seem to bring to question the usefulness of DL models for T1 quantification. For instance, are DL-based methods faster than methods such as TRF? How do the inference times of the three methods compare? It is clear that the physics-based loss if very helpful for polarity correction, but this and other benefits should be emphasized.
While the choice of the ODE-based model seems natural given the differential loss term, the work doesn’t compare other CT model architectures, e.g., rotory embedding, or close-form continuous-time neural networks (https://www.nature.com/articles/s42256-022-00556-7). parameters,
- Please rate the clarity and organization of this paper
Good
- Please comment on the reproducibility of the paper. Please be aware that providing code and data is a plus, but not a requirement for acceptance.
The authors claimed to release the source code and/or dataset upon acceptance of the submission.
- Optional: If you have any additional comments to share with the authors, please provide them here. Please also refer to our Reviewer’s guide on what makes a good review and pay specific attention to the different assessment criteria for the different paper categories: https://conferences.miccai.org/2025/en/REVIEWER-GUIDELINES.html
N/A
- Rate the paper on a scale of 1-6, 6 being the strongest (6-4: accept; 3-1: reject). Please use the entire range of the distribution. Spreading the score helps create a distribution for decision-making.
(6) Strong Accept — must be accepted due to excellence
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
The quality and presentation of the work is excellent, and the proposed methods are novel and interesting.
- Reviewer confidence
Confident but not absolutely certain (3)
- [Post rebuttal] After reading the authors’ rebuttal, please state your final opinion of the paper.
N/A
- [Post rebuttal] Please justify your final decision from above.
N/A
Author Feedback
We sincerely thank the reviewers for their constructive feedback. It is encouraging to see that our integration of physics-based principles into deep learning methods for qMRI was positively received. Below, we address the questions and concerns raised by each reviewer:
Reviewer 1:
- We thank the reviewer for pointing out the lack of clarity regarding how the LM algorithm is initialized. We will include the relevant details in the camera-ready version and in our open-source GitHub repository.
- We also appreciate the reviewer’s comments regarding CNN-based architectures. However, we would like to clarify that, in our view, CNNs may lead to suboptimal accuracy due to the smoothing effect of convolution operations. This is supported by the benchmark paper we reference (T1Net — Le JV et al., Med Phys, 2022), where a CNN-based U-Net was compared to pixel-wise estimation and consistently showed inferior results.
Reviewer 2:
- We thank the reviewer for highlighting the lack of clarity regarding the criteria used to select the best models. We will include this clarification in the caption of the camera-ready version.
- We agree with the comment on the absence of comparisons with other CT model architectures. The primary goal of this work was to validate our physics-based methodology against existing approaches in the literature. We acknowledge that the current model may be suboptimal, and exploring alternative architectures will be part of future work.
- Regarding the wall-clock time comparison between methods, our model is indeed slower than a standard RNN (e.g., T1Net), as it includes a NODE hidden state integration at each sequential step. MLP-based models (e.g., MyoMapNet) are expected to be the fastest due to their simplicity. TRF can be significantly slower but is a more complex case, as its O(n^4) complexity is highly hardware-dependent. It can be accelerated through parallelization, but it is also constrained by the number of available CPU cores.
Reviewer 3:
- Similar to Reviewer 2, we thank the reviewer for pointing out that the criteria for selecting the best models were unclear. This will be reviewed and clarified in the camera-ready version.
We hope our responses have addressed your concerns and clarified the keys aspects of our work.
Meta-Review
Meta-review #1
- Your recommendation
Provisional Accept
- If your recommendation is “Provisional Reject”, then summarize the factors that went into this decision. In case you deviate from the reviewers’ recommendations, explain in detail the reasons why. You do not need to provide a justification for a recommendation of “Provisional Accept” or “Invite for Rebuttal”.
All reviewers have provided positive reviews highlighting the novelty and contributions of this submission.