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Abstract
In diffusion MRI (dMRI), a uniform single or multiple shell sampling scheme is typically required for data acquisition in q-space, because uniform spherical sampling offers the advantage of capturing more information using fewer samples, leading to superior reconstruction results. Uniform sampling problems can be categorized into continuous and discrete types. While most existing sampling methods focus on the continuous problem that is to design spherical samples continuously from single or multiple shells, this paper primarily investigates two discrete optimization problems, i.e., 1) optimizing the polarity of an existing scheme (P-P), and 2) optimizing the ordering of an existing scheme (P-O). Existing approaches for these two problems mainly rely on greedy algorithms, simulated annealing, and exhaustive search, which fail to obtain global optima within a reasonable timeframe. We propose several Mixed Integer Linear Programming (MILP) based methods to address these problems. To the best of our knowledge, this is the first work that solves these two discrete problems using MILP to obtain global optimal or sufficiently good solutions in 10 minutes. Experiments performed on single and multiple shells demonstrate that our MILP methods can achieve larger separation angles and lower electrostatic energy, resulting better reconstruction results, compared with existing approaches in commonly used software (i.e., CAMINO and MRtrix).
Links to Paper and Supplementary Materials
Main Paper (Open Access Version): https://papers.miccai.org/miccai-2024/paper/2540_paper.pdf
SharedIt Link: https://rdcu.be/dV1On
SpringerLink (DOI): https://doi.org/10.1007/978-3-031-72069-7_30
Supplementary Material: N/A
Link to the Code Repository
N/A
Link to the Dataset(s)
N/A
BibTex
@InProceedings{Zha_Mixed_MICCAI2024,
author = { Zhang, Si-Miao and Wang, Jing and Wang, Yi-Xuan and Liu, Tao and Zhu, Haogang and Zhang, Han and Cheng, Jian},
title = { { Mixed Integer Linear Programming for Discrete Sampling Scheme Design in Diffusion MRI } },
booktitle = {proceedings of Medical Image Computing and Computer Assisted Intervention -- MICCAI 2024},
year = {2024},
publisher = {Springer Nature Switzerland},
volume = {LNCS 15002},
month = {October},
page = {313 -- 322}
}
Reviews
Review #1
- Please describe the contribution of the paper
This paper introduced a mixed integrer linear programming method to optimize the sampling scheme in diffusion MRI. The proposed method optimizes the polarity of an existing scheme and the order of the existing scheme.
- Please list the main strengths of the paper; you should write about 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 proposed optimization objective functions and optimization methods are novel.
- Please list the main weaknesses of the paper. Please provide details, for instance, if you think a method is not novel, explain why and provide a reference to prior work.
While this paper is interesting in the technical aspect, the problem considered in this paper do not seem to be very important from a practical aspect. First, the method still relies on an initial set of gradient directions. Second, the objective functions and evaluation metrics are mainly motivated from the mathematical or physical aspects. The dependence of diffusion MRI measures on the sampling scheme was not examined.
- Please rate the clarity and organization of this paper
Very 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.
- Do you have any additional comments regarding the paper’s reproducibility?
N/A
- Please provide detailed and constructive comments for the authors. Please also refer to our Reviewer’s guide on what makes a good review. Pay specific attention to the different assessment criteria for the different paper categories (MIC, CAI, Clinical Translation of Methodology, Health Equity): https://conferences.miccai.org/2024/en/REVIEWER-GUIDELINES.html
1) It would be help to show the distribution of the gradient directions on a single shell or multiple shells for visualization. 2) Some notations are not defined such as covering radius. 3) The difference in diffusion MRI metrics related to different sampling scheme, such as DTI, DKI and NODDI, should be compared. 4) There seems to be a typo in (3a), x_s,i,s,j-> s_s,i,t,j.
- 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
Weak Reject — could be rejected, dependent on rebuttal (3)
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
Although the mathmatical formulations are interesting, the practical relance of the proposed method for diffusion MRI estimation is not clear.
- Reviewer confidence
Very confident (4)
- [Post rebuttal] After reading the author’s rebuttal, state your overall opinion of the paper if it has been changed
N/A
- [Post rebuttal] Please justify your decision
N/A
Review #2
- Please describe the contribution of the paper
Summary: In this paper, the authors proposed a learning programming-based method for designing the sampling scheme in diffusion MRI, which includes the design of the polarity and the ordering of a scheme. The method is tested using the multi-shell Human Connectome Project data.
- Please list the main strengths of the paper; you should write about 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.
1.The authors provided detailed experiments, which are convincing to show that the proposed method is better than other methods. 2.The mathematical derivation in this work is thorough and interesting.
- Please list the main weaknesses of the paper. Please provide details, for instance, if you think a method is not novel, explain why and provide a reference to prior work.
1.For the method, the authors need to clarify the following things: 1)In the paragraph above Equation 4(a), the authors uses an approximation that ((cos(\theta_k) + 1) / 2) ^ (1/2) = (cos(\theta_k) + 1) / 2. However, from Figure 1, we know that \theta_k can be larger than 15 degrees. Is this approximation still close to the actual value? 2)The author designed a method to optimize the ordering of the scheme. However, will the magnitude field change too much so that there will be larger artifacts like the eddy current?
2.For the experiments and results, I have the following concerns: 1)The authors need to provide a quantitative analysis of the accuracy of the reconstructed Fiber Orientation Distributions (FODs). Now, they only show one example. 2)Do the authors only use one subject in their experiments? 3)For the presented example of FODs, although the proposed method is better than other methods at regions in the red boxes, the FODs outside the red boxes seem worse than the other methods. For example, the FODs at the bottom part of the images contain more peaks than the ground truth. The authors need to explain that. 4)Figure 3 shows that the proposed method is worse than other methods at the combined shell condition; why are the FODs better than other methods? 5)Figure 2 shows that when the number of samples is about 100, the distance between the proposed method and other methods is the largest. Why did the authors use 60 in the experiments? 6)For the CAMINO method in Figure 2 (c) and (d), why is there a sudden increase at around 100 samples? 7)The experiments on the ordering of a scheme are only done on existing HCP data; authors should use the scheme in actual data acquisition. 8)For the optimization of the polarity of a scheme, the authors should show a figure on the change of polarity before and after the optimization.
- Please rate the clarity and organization of this paper
Very 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 mention open access to source code or data but provides a clear and detailed description of the algorithm to ensure reproducibility.
- Do you have any additional comments regarding the paper’s reproducibility?
N/A
- Please provide detailed and constructive comments for the authors. Please also refer to our Reviewer’s guide on what makes a good review. Pay specific attention to the different assessment criteria for the different paper categories (MIC, CAI, Clinical Translation of Methodology, Health Equity): https://conferences.miccai.org/2024/en/REVIEWER-GUIDELINES.html
The main weaknesses are the experiment results and the explanation of methods. I will reconsider my rating in the rebuttal.
- 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
Weak Accept — could be accepted, dependent on rebuttal (4)
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
Authors need to clarify my concerns.
- Reviewer confidence
Somewhat confident (2)
- [Post rebuttal] After reading the author’s rebuttal, state your overall opinion of the paper if it has been changed
Weak Accept — could be accepted, dependent on rebuttal (4)
- [Post rebuttal] Please justify your decision
The authors have answered all my questions.
Review #3
- Please describe the contribution of the paper
The paper presents a novel application of Mixed Integer Linear Programming (MILP) to optimize discrete sampling schemes in diffusion MRI (dMRI), focusing on optimizing polarity and ordering of samples. This methodology aims to achieve global optima in solution quality within reasonable computational times, specifically targeting improvements in electrostatic energy and angular separation in dMRI data reconstruction.
- Please list the main strengths of the paper; you should write about 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.
- MILP allows for a systematic and rigorous approach to finding the global optimum or sufficiently good solutions within a bounded time (10 minutes, as mentioned). This application shows advantages over greedy algorithms, simulated annealing, or exhaustive searches.
- While MILP has been used in various domains, its application to optimize both the polarity and the ordering of sampling schemes in the specific field of dMRI is original.
- Please list the main weaknesses of the paper. Please provide details, for instance, if you think a method is not novel, explain why and provide a reference to prior work.
- As you mentioned, continuous and discrete types of problems, what is the main difference in terms of algorithm design between these two types? As MILP has already been applied to continuous types for dMRI sampling, what adaptation have you made towards discrete types?
- The paper primarily compares the proposed MILP methods with a couple of existing methods, namely those implemented in CAMINO and MRtrix software. While these are standard tools, the comparative analysis could be broader, including various optimization algorithms.
- There is limited direct evaluation of reconstructed results, e.g., qualitative and quantitative comparison of MRI images of certain diffusion gradients.
- 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.
- Do you have any additional comments regarding the paper’s reproducibility?
none
- Please provide detailed and constructive comments for the authors. Please also refer to our Reviewer’s guide on what makes a good review. Pay specific attention to the different assessment criteria for the different paper categories (MIC, CAI, Clinical Translation of Methodology, Health Equity): https://conferences.miccai.org/2024/en/REVIEWER-GUIDELINES.html
- It might need a clearer comparison between continuous and discrete types of uniform sampling problems and explain why you chose to deal with discrete types.
- Simplify the equations and improve the clarity of the algorithm. May consider to put the inference part to supplementary materials.
- Including additional comparisons with a wider range of existing methods could further validate the superiority of the MILP approach.
- Demonstrating more examples of qualitative (visual) comparisons of reconstruction results.
- 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
Weak Accept — could be accepted, dependent on rebuttal (4)
- Please justify your recommendation. What were the major factors that led you to your overall score for this paper?
MILP’s systematic approach and novelty in dMRI sampling optimization are notable strengths, surpassing traditional heuristic methods. However, the paper lacks clarity on the algorithmic differences between continuous and discrete problems in dMRI sampling optimization, requiring further elaboration. Additionally, broader comparative analyses beyond CAMINO and MRtrix methods would strengthen the validation of MILP’s superiority. Although the paper provides clear algorithmic descriptions, simplification, and clarity enhancements are suggested, possibly relocating detailed inference procedures to supplementary materials. Including more qualitative comparisons of reconstruction, results would enhance the paper’s comprehensiveness.
- Reviewer confidence
Very confident (4)
- [Post rebuttal] After reading the author’s rebuttal, state your overall opinion of the paper if it has been changed
Accept — should be accepted, independent of rebuttal (5)
- [Post rebuttal] Please justify your decision
Thank you for the rebuttal. I believe the paper has merits in developing a new sampling scheme for diffusion MRI. I believe most concerns are addressed. I would be grateful if the authors could expand the introduction to further clarify the motivation and contribution.
Author Feedback
We thank all reviewers (R1, R4, R6) for the constructive comments. All reviewers acknowledged that our method is mathematically sound and novel, as “objective functions and optimization methods are novel” (R1), “the mathematical derivation in this work is thorough and interesting” (R4), “MILP’s systematic approach and novelty are notable strengths” (R6).
The main concerns:
- Continuous v.s. discrete sampling scheme problems.
- Take single shell case as an example. Continuous problem is to optimize a direction scheme {u_i} in the continuous sphere S^2 (Jones MRM 1999, Caruyer CDMRI 2011, Cheng TMI 2018). Discrete problem is to optimize a subsampling set (Cheng TMI 2018), optimize ordering (Cook JMRI 2007, Deriche MIA 2009) and polarity (Tournier NI 2019), of an existing scheme {u_i}. This paper is to propose MILP to optimize ordering and polarity of an existing scheme.
- As shown in gen_scheme in MRTrix, a practical scheme design is to first apply continuous optimization to obtain a scheme, then apply discrete optimization to optimize ordering and polarity of the scheme.
- Therefore, discrete optimization of ordering and polarity aims to work with continuous optimization, and it is based on an existing scheme. Thus, using the initialization of ordering and polarity from the existing scheme is necessary and reasonable.
- Practical aspects and quantitative experimental results.
- Objective functions based on electrostatic energy or covering radius of existing methods (Jones 1999, Caruyer 2011, Cheng 2018, etc) are all motivated from the mathematical or physical aspects. These objective functions are based on the assumption that a good sampling scheme should have large angular separation such that the reconstruction has large angular resolution and good rotational invariance. These methods have been widely used in dMRI domain (in HCP, in MRtrix and CAMINO). Our method follows this research line. One advantage of math derived objective functions is that they work with arbitrary diffusion models and reconstruction methods.
- Some other existing methods consider objective functions based on reconstruction results. However, these methods then reply on specific diffusion model (e.g., DTI, NODDI) and specific reconstruction methods (e.g., least squares). It limits their applications.
- Following existing papers, we evaluate our methods based on math derived metrics (electrostatic energy and covering radius), and reconstruction results. Fig. 4 shows reconstruction results (fODF, peak, GFA) of partial scanned data from various ordering methods, where the background shows the GFA map of reconstructed fODFs of the spherical deconvolution method. Compared with the results of the full HCP scheme (270 samples), our MILP with 60 samples obtains more closer results than the methods in CAMINO and MRTrix, although there are still some reconstruction errors in some voxels as noticed by R4.
- More visualization and reconstruction experiments on more diffusion metrics are not included in the paper due to limited space.
- Reproducibility. Codes will be released after acceptance.
Additional responses.
To R1: Q1: Covering radius is not defined. A1: We give a simple explanation (i.e. minimal angular separation) in introduction. We will add a mathematical formula in final paper. Q2: A typo in equation (3a). A2: Thank you very much. We will correct it in the final paper.
To R4: Q1: Approximation in Equation 4(a) may not be accurate. A1: We agree that the approximation of packing density may be inaccurate, but it is necessary to convert the optimization into MILP. Q2: Will there be larger artifacts like the eddy current. A2: Optimizing the ordering mainly focus on obtaining uniform distributed subsets upon interruption. We could better reduce eddy current after optimizing the polarity. Q3: Worse covering radius in the combined shell. Why better fODF? A3: MILP obtains better angular separation in each shell, and multi-shell SD is used.
Meta-Review
Meta-review #1
- After you have reviewed the rebuttal and updated reviews, please provide your recommendation based on all reviews and the authors’ rebuttal.
Accept
- Please justify your recommendation. You may optionally write justifications for ‘accepts’, but are expected to write a justification for ‘rejects’
The authors’ rebuttal has comprehensively addressed reviewers’ concerns within the limited space. Notable, the practical relance of the proposed method for diffusion MRI estimation is still confused.
- What is the rank of this paper among all your rebuttal papers? Use a number between 1/n (best paper in your stack) and n/n (worst paper in your stack of n papers). If this paper is among the bottom 30% of your stack, feel free to use NR (not ranked).
The authors’ rebuttal has comprehensively addressed reviewers’ concerns within the limited space. Notable, the practical relance of the proposed method for diffusion MRI estimation is still confused.
Meta-review #2
- After you have reviewed the rebuttal and updated reviews, please provide your recommendation based on all reviews and the authors’ rebuttal.
Accept
- Please justify your recommendation. You may optionally write justifications for ‘accepts’, but are expected to write a justification for ‘rejects’
This is certainly an interesting paper in terms of technical contribution. Although the experiment design may be improved, I believe a more comprehensive version of the work can be completed in a journal paper.
- What is the rank of this paper among all your rebuttal papers? Use a number between 1/n (best paper in your stack) and n/n (worst paper in your stack of n papers). If this paper is among the bottom 30% of your stack, feel free to use NR (not ranked).
This is certainly an interesting paper in terms of technical contribution. Although the experiment design may be improved, I believe a more comprehensive version of the work can be completed in a journal paper.