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Abstract
Magnetic Resonance Elastography (MRE) is a non-invasive imaging technique that estimates tissue elasticity using Magnetic Resonance Imaging. The conventional approach for elasticity reconstruction in MRE involves solving an inverse problem through numerical methods such as Helmholtz inversion and the finite element method. However, these techniques suffer from noise sensitivity and high computational costs due to iterative optimization. Recently, Physics-Informed Neural Networks (PINNs) have been studied for tissue elasticity reconstruction, integrating physical constraints into deep learning models. While PINNs improve noise resistance, they require a separate network to be trained for each instance, resulting in a computationally inefficient training. In this study, we introduce an operator learning-based approach to tissue elasticity reconstruction, which learns a generalized mapping from input measurements to tissue elasticity. This method enables simultaneous learning across multiple instances, significantly improving computational efficiency. Experimental results using box and abdomen simulation data show that our approach achieves superior reconstruction performance and robustness to noise.
Links to Paper and Supplementary Materials
Main Paper (Open Access Version): https://papers.miccai.org/miccai-2025/paper/2595_paper.pdf
SharedIt Link: Not yet available
SpringerLink (DOI): Not yet available
Supplementary Material: Not Submitted
Link to the Code Repository
https://github.com/youjin-DDAI/MRE-Hyper
Link to the Dataset(s)
https://bioqic-apps.charite.de/downloads
BibTex
@InProceedings{KimYou_PhysicsInformed_MICCAI2025,
author = { Kim, Youjin and Lee, Jae Yong and Kwon, Junseok},
title = { { Physics-Informed Neural Operators for Tissue Elasticity Reconstruction } },
booktitle = {proceedings of Medical Image Computing and Computer Assisted Intervention -- MICCAI 2025},
year = {2025},
publisher = {Springer Nature Switzerland},
volume = {LNCS 15969},
month = {September},
page = {380 -- 389}
}
Reviews
Review #1
- Please describe the contribution of the paper
The paper presets an approach to solving the inverse problem of computing tissue elasticity from magnetic resonance elastography imaging. A novel operator learning approach is proposed based on a combination of two HyperDeepONet [11] networks to infer tissue elasticity. The method is compared against traditional PDE and FEM solvers of the inverse problem and with a PINN that has very similar structure with the proposed method. Results show that the proposed method is comparable with the PINN method and better than the traditional methods.
- 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.
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the paper presented an elegant solution to the MRE reconstruction problem. It builds and extends a previous approach (ref [20]) bye replacing the PINN with operator learning networks. The new model has a clear advantage by not having to be retrained for new instances.
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solid experiments are presented based on two datasets showing good performance of the method and excellent robustness to noise
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- 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.
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the work is incremental, building on ref [20] and [11]. Similar to [20] the method first predicts displacements and next elasticity. The PINN from [20] are replaces with the HyperDeepONet networks.
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the method is tested only on simulated data; to the merit of the paper, several levels of noise is added and tested in a comparative manner, showing good robustness to noise. Is there any way to validate the method in real clinical data ?
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- 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 submission has provided an anonymized link to the source code, dataset, or any other dependencies.
- 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
- for the abdomen data - how is the GT tissue elasticity assigned ?
- noise - Gaussian noise at five different levels to the wave images - is this realistic for the wave images ? a discussion on the type of possible noise present in those images could be provided
- why is CTE error used ? is this a standard in estimating elasticity ? MSE could have been used instead.
- there is no discussion on how the boundary conditions are incorporated in the approach; is it though the domain y ?
- 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?
The paper presents an interesting solution to a difficult problem. It is well validated on synthetic simulations. Nevertheless is mainly incremental.
- Reviewer confidence
Somewhat confident (2)
- [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
The manuscript presents a deep physics-informed deep operator learning for tissue elasticity reconstruction.
- 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 results are promissing
- The novelty has value and suitable for MICCAI
- The method is compared with recent state of art methods
- The method addresses less explored area of physics-informed learning
- 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.
-dataset was limited -some details are missing (please see the suggestion comment) -They could have provided the ablation experiment to see the effect of physics-informed part -The networks were traine on the same dataset but different frequency. It would be interesting to see the performance when they are trained and evaluated on different datasets.
- 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 submission has provided an anonymized link to the source code, dataset, or any other dependencies.
- 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
1- In table 1, what is the frequency the networks were trained on? were they trained on single frequency? 2- In robustness to noise section, did the networks trained on noisy samples?
- 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?
The idea and application would be suitable for MICCAI community but there are a few weaknesses. It seems to me the strength slightly outweighs the weaknesses.
- 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 authors present a novel method for addressing the inverse problem in MRE reconstruction. By applying neural operators, they achieve superior performance and enhanced robustness compared to PINNs, Helmholtz inversion, and FEM.
- 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.
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The authors introduce the first operator learning approach for tissue elasticity reconstruction.
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The authors conduct a comprehensive evaluation of noise effects.
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- 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.
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The related work section is somewhat repetitive with the introduction and could be consolidated.
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The equations are missing numbering.
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In the initial equations, theta is not defined and is used for both \N_Trunk and N_Branch. Does this imply that they share the same parameter?
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The authors employ two HyperDeepONets, arguing that the standard DeepONet is limited to a linear combination of the trunk and branch networks. However, the results demonstrating that two DeepONets are indeed insufficient are missing.
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What is the reason for presenting mu and CTE instead of simple metrics like MSE?
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All results are lacking basic statistical tests or, at a minimum, standard error values.
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The solwest method takes 20 seconds; why is this speed considered insufficient for clinical practice?
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Why is the FEM method not shown in Figure 6?
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- 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 submission does not provide sufficient information for 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?
The paper presents valuable findings, yet it lacks a fundamental statistical evaluation despite the method’s strong and robust performance.
Publishing the code would significantly enhance reproducibility.
- 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 time and constructive suggestions. Their comments will be considered to improve the clarity and completeness for the final version. We provide responses to the reviewers’ comments below.
Experimental Setup and Dataset Limitations # R1, R2 (1) Dataset Limitation Due to the difficulty of obtaining paired MRE wavefields and ground-truth(GT) elasticity maps from clinical cases—mainly due to privacy, cost, and limited public datasets—we conducted all experiments on two simulation datasets: FEM box and FEM abdomen. Despite this limitation, the model’s consistent performance across different geometries and noise levels suggests strong potential for real-world applicability. (2) Realism of Noise For noise robustness, we followed [20]’s protocol by adding Gaussian noise at five levels to the input wavefields. While this may not fully reflect clinical noise, Gaussian noise is a standard and interpretable proxy in inverse problem studies. Exploring more realistic noise models remains an important direction for future work.
Training detail #R1, R2 (1) GT elasticity in the FEM-Abdomen dataset The FEM-Abdomen dataset assigns GT elasticity based on predefined mechanical properties of each segmented tissue as detailed in [1]. (2) Treatment of boundary conditions Our model learns the physical behavior of tissue elasticity by minimizing the PDE residual loss across the entire spatial domain. While we do not explicitly impose boundary conditions, training with the governing equations provides sufficient regularization. (3) Frequency handling during training (Table 1) In our proposed MRE-Hyper, a single model was trained on wavefields from all available frequencies, allowing it to perform evaluation across frequencies. In contrast, MRE-PINN trains and evaluates separate models for each frequency. (4) Noise robustness training setting As the R2 correctly noted, we trained our models exclusively on noisy wavefields, and performed evaluation using similarly noisy inputs. The models were first optimized to reconstruct these noisy wavefields and then used them as input for elasticity reconstruction. This setup reflects realistic clinical conditions where measurement noise is unavoidable.
Justification of Methodological Choices #R1, R2, R3 (1) Lack of MSE metric We followed the evaluation protocol of the prior work MRE-PINN [20], which uses CTE and μ-correlation to assess the quality of elasticity reconstructions. These metrics are preferred over MSE because they better reflect the contrast and structural consistency in elastograms. (2) Lack of Ablation Study for the Physics-Informed part In our setting, we assume that GT elasticity (μ) is unavailable during training, reflecting realistic clinical scenarios where such information is typically inaccessible. Therefore, our model incorporates physics-informed loss terms to guide learning without direct supervision. Without the physics-informed component, training would not be possible. (3) Why use HyperDeepONet instead of DeepONet? In our internal experiments, models using HyperDeepONets consistently outperformed standard DeepONets in elasticity reconstruction tasks. As HyperDeepONet showed better performance in our evaluations, we chose to proceed with the more effective architecture for the main study and did not include the alternative results due to space constraints.
Additional Clarification #R3 (1) Use of θ We used θ to denote learnable parameters in a general sense when introducing DeepONet, as stated in the paper. The trunk and branch networks do not share parameters; each has its own independently learned θ. (2) Noise robustness of FEM The reason FEM results are omitted from Figure 6 is the same as stated for Figure 5 in the paper: even with minimal noise, FEM-based methods exhibited severe performance degradation, making them unsuitable for meaningful visualization.
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”.
This paper proposes to use operator learning approach for magnetic resonance elastography (MRE) reconstruction using dual HyperDeepONet architectures. The method addresses a critical inverse problem in medical imaging by offering a more efficient alternative to traditional PDE solvers, FEM, and even physics-informed neural networks (PINNs). The reviewers generally agree that this is a timely and valuable contribution to the MICCAI community.
All three reviewers rate the paper as a weak accept, citing complementary strengths. Reviewer #1 highlights the method’s robustness to noise, noting it builds incrementally on previous work but with important practical advantages—such as eliminating the need to retrain for new cases. Reviewer #2 mentioned the novelty and value of the application within the MICCAI scope, and Reviewer #3 emphasises the robustness and methodological advancement in using operator learning for elasticity estimation.
The main limitations identified are: 1) the work is tested only on simulated data; 2) a lack of more extensive statistical evaluation (e.g., confidence intervals or error bars); and 3) missing details about certain design choices such as loss functions and boundary condition handling. These are valid concerns, but do not overshadow the promising nature of the approach or the quality of the experimental results. Moreover, it would be great to enhance the discussion on the difference wrt similar works where use PINNs for Elasticity imaging, and what would be the advatange over other existing Nueral Operator techniques when addressing this application.
The authors are encouraged to incorporate reviewer feedback during the camera-ready revision to address remaining concerns.