Generating Cerebral Vessel Trees of Acute Ischemic Stroke Patients using Conditional Set-Diffusion

Authors: thijs p kuipers
praneeta r konduri
charles b l majoie
henk a marquering
erik j bekkers
p  kuipers
Venue: MIDL_io_2024_Conference
Published: 2024-01-30 11:36:57

Abstract: The advancements in computational modeling and simulations have facilitated the emergence of in-silico clinical trials (ISCTs). ISCTs are valuable in developing and evaluating novel treatments targeting acute ischemic stroke (AIS), a prominent contributor to both mortality and disability rates. However, obtaining large populations of accurate anatomical structures that are required as input to ISCTs is labor-intensive and time-consuming. In this work, we propose and evaluate diffusion-based generative modeling and set transformers to generate a population of synthetic intracranial vessel tree centerlines with associated radii and vessel types. We condition our model on the presence of an occlusion in the middle cerebral artery, a frequently occurring occlusion location in AIS patients. Our analysis of generated synthetic populations shows that our model accurately produces diverse and realistic cerebral vessel trees that represent the geometric characteristics of the real population.

Main paper: 
 introduction: 
 Advancements in computational modeling and simulations have enabled the development of in-silico clinical trials (ISCTs). Combined with machine learning methods, ISCTs can provide valuable insights into new medical treatments and devices. ISCTs allow for a reduction of resources spent on developing new devices and treatments by simulating their performance on virtual patient populations, reserving actual clinical trials only for the most encouraging candidates (Konduri et al., 2020; Miller et al., 2023; Viceconti et al., 2016) .ISCTs can aid in the development of novel treatments for acute ischemic stroke (AIS), one of the leading causes of mortality and disability (Phipps and Cronin, 2020) . AIS occurs when a thrombotic occlusion reduces the blood flow in a cerebral vessel. Treatment of large vessel occlusions, for example, in the interior cerebral artery (ICA), the middle cerebral artery (MCA) M1 and M2 segments, or the anterior cerebral artery (ACA), involves mechanical thrombectomy, for which in-silico treatment models have been developed and validated (Luraghi et al., 2021) . The training of in-silico treatment models benefits from large populations of patient-specific information derived from radiological images, including segmentations of cerebral vessels and characteristics of the occluding thrombus. Such data enhances the generalizability and robustness of the trials. However, obtaining this data is often expensive and time-consuming. With the advances in generative modeling, synthetic patient populations that can be conditioned on certain auxiliary variables of interest are becoming increasingly popular. Conditioning provides several advantages, such as aligning synthetic data to real-world patient data or customization for specific patient groups.Various methods have been introduced for generating vessel geometry. Bridio et al. (2023) uses statistical shape modeling (SSM) to generate cerebral vessel anatomies described by their centerlines and diameters. However, SSM approaches are linear models, limiting their ability to model complex and large variations in vessel geometry and topology (Kalaie et al., 2023) . Furthermore, conditioning SSM approaches on auxiliary variables is not straightforward (Bannister et al., 2022) . Danu et al. (2019) use deep generative voxelbased models for generating vessels. However, voxel volumes have limited resolution due to their high computational cost, making it difficult to convert them to a mesh that is suitable for ISCTs. Wolterink et al. (2018) , instead, uses a conditional generative adversarial network (GAN) (Goodfellow et al., 2020) vessel centerlines, which is not limited by resolution. Nevertheless, it assumes a fixed topology, i.e., centerline points have a fixed ordering, and therefore cannot generate structures with bifurcations.In this work, we present a deep generative diffusion-based method for generating vessel geometry. Diffusion (Ho et al., 2020) has emerged as one of the most powerful generative methods. We address the limitations of voxel-based methods and fixed topology by generating vessel tree centerlines with arbitrary topologies using a conditional set transformer-based architecture (Lee et al., 2019) . By representing vessel trees as point clouds, or equivalently, fully connected graphs, our model is not constrained by topology. Since we are interested in computational simulation of stroke treatment for large vessel occlusions, our approach generates vessel tree centerlines with corresponding radii and associated ICA, ACA, M1, and M2 vessel types. However, since our method supports arbitrary toplogy, it is not limited to these specific vessels. The model is conditioned on the presence of an M1 occlusion. We evaluate the synthetic vessels by comparing their geometric characteristics with those observed in clinical data. However, since set-diffusion output is unordered, we present a post-processing centerline-sequencing algorithm for ordering the generated centerlines.We summarize our contributions as follows. We use diffusion with a set-transformer architecture to generate topology free conditional vessel tree centerlines. A simple postprocessing step is used to sequence the unordered centerlines for use in further downstream tasks. Our evaluation shows that our method can generate realistic and unique vessel trees. 
method: 
 Section 2.1 describes the used dataset and how we represent the centerlines for training the diffusion model. The diffusion process and its set-transformer backbone is described in 2.2. We present the sequencing algorithm that orders the synthetic centerlines sampled from the diffusion model in Section 2.3. An overview of the entire generative process is presented in Figure 1 . We make our code publicly available here. In this work, we used data from the MR CLEAN Registry, an ongoing, prospective, observational, multicenter study from 16 EVT-capable hospitals in the Netherlands. The dataset consists of 110 patients with an occlusion in the M1 artery, in either the left or right side of the brain, and sufficient image and segmentation quality. We segmented the intracranial arteries using a vessel segmentation algorithm developed within StrokeViewer (NICO.LAB, Amsterdam). The centerline and geometry characteristics (radius, tortuosity, and bifurcation angle) extraction and artery segment labeling were performed with a semi-automated software: iCAFE(© 2016-2018 University of Washington. Used with permission) (Chen et al., 2018) . The vessel trees include ICA, ACA, and the MCA (M1 and M1 segments) on the contralateral hemisphere (without occlusion). To increase the number of data samples, we mirrored vessel trees with occlusion on the left side to the right side. Hence, our dataset consists of 181 training samples (95 with and 86 without an M1 occlusion) and 39 testing samples (20 with and 19 without occlusion).The vessel trees are parameterized by their centerline, i.e., a set of points x i = (c i , h i ) where c i contain the point coordinates and h i the point features, such as the radius and vessel type. The vessel type is a categorical feature represented as a one-hot encoding. Similar to Hoogeboom et al. (2022) , we multiply the categorical features by 0.1 to stimulate the denoising process to first emphasize the shape of the centerline before segmenting it. Note that in practice, x i is the concatenation of c i , h i . We sample 256 equidistantly spaced centerline points using linear interpolation that are scaled down by a factor of 24 to be approximately within the range of [-1, 1] and have a standard deviation of 0.5, as this is expected by the EDM (Karras et al., 2022) diffusion formulation.The diffusion model consists of three parts. The forward diffusion process adds noise to the input. A denoising function aims to remove the added noise to reconstruct the original input. The reverse diffusion process synthesizes a vessel tree by iteratively denoising noise from the unit Gaussian distribution. The reverse diffusion process requires 18 steps to synthesize a centerline vessel tree.Diffusion Process We use the EDM formulations introduced by Karras et al. (2022) . EDM drastically simplifies the forward diffusion process and the training of the denoising function. Sampling from the diffusion model is also significantly faster, requiring only 18 steps compared to the hundreds of steps such as in (Hoogeboom et al., 2022) .Denoising Objective Given a centerline point x i = (c i , h i ) and noise n i , the objective of the denoising function is to map the diffused input back to the original input. The amount of noise n i added to the input is determined by the noise level σ. As σ increases, the noisy input increasingly resembles unit Gaussian noise. Formally, the denoising objective isE n i ∼N (0,σ 2 I) 1 N N i (F θ (x i + n i , σ, C) -x i ) 2 .(1)Here, F θ is the denoising function with learnable parameters θ. During training, F θ is conditioned on the noise level σ and any optional auxiliary conditional information C. We use learnable embedding vectors for conditioning on the presence of an M1-occlusion and parameterize. The denoising function F θ is modeled by a set-transformer.Cross Attention Our denoising network consists of a series of cross-attention blocks.The attention mechanism allows elements in the input to pass information to each other while being permutation equivariant (Vaswani et al., 2017) . Given matrices X ∈ R N ×L and Y ∈ R M ×P , with rows denoting individual set elements, we formulate cross-attention asEQUATIONwhere W Q ∈ R L×d and W K , W V ∈ R P ×d are learnable parameters mapping X and Y to sets of queries, keys, and values respectively. In the case where X = Y, Equation (2) becomes self-attention.The set transformer consists of a series of cross-attention blocks. Each cross-attention block consists of three components. First, self-attention is applied where centerline elements exchange information with each other. Next, conditional information is incorporated via cross-attention, serving as an effective conditioning mechanism (Rombach et al., 2022) . In the case of unconditional generation, the cross-attention layer becomes a self-attention layer. Finally, an inverse-bottleneck feed-forward network performs channel mixing. Adaptive layer normalization is applied before each component to inject the noise levels. Pre-normalization in the transformer architecture improves gradient stability, reducing training time and the need for hyperparameter tuning (Xiong et al., 2020) .Our generative model generates an unordered set of centerline points. We turn the unordered sets into ordered and cleaned-up connected centerline segments with the following three-stage post-processing algorithm.(1) Noise Reduction Noise mainly occurs if points are far away from the centerline or if points form clusters. Points that have a nearest-neighbor distance (nn-distance) larger than four times the average nn-distance are removed. Clusters, generally occurring at bifurcations, are reduced by applying furthest-point sampling.(2) Sequencing Sequencing starts with an empty sequence s to which points are added to the end. We define the last point added to the sequence as the endpoint and the remaining points as candidate points. The candidate point with the minimum distance to the endpoint, weighted by the current direction of the sequence is chosen as the next endpoint. Given the current endpoint and direction x cur and d cur , and a candidate point x i with direction d i , the direction-weighted distance d is calculated asEQUATION)where α determines the importance of the current direction. The directions are calculated as d cur = (x prior -x cur )/||x prior -x cur || and d i = (x i -x cur )/||x i -x cur ||, where x prior is the point prior to x cur in the sequence. The point chosen as the initial endpoint is given as the point x that maximizes the average pairwise inner product between the directions from x to its k nearest neighbors. Intuitively, the point where its nearest neighbors have similar directions is likely one of the end points of the sequence.(3) Segment Merging Individual vessel segments are merged by calculating a common bifurcation point. Given an endpoint x belonging to segment s, its nearest neighbors from the remaining segments are candidate bifurcation points. Candidate points with distances greater than 4 times the average nn-distance of s are discarded. The common bifurcation point is the average of the remaining candidate points.  
results and evaluation: 
 We train the diffusion model to generate cerebral vessel tree centerlines of 256 points, conditioned on an M1 occlusion. The training and sampling details are given in Section 3.1. In Section 3.2, we assess the geometric characteristics of the synthesized vessels, as well as model diversity. We quantitatively analyze the distributions of the geometric characteristics in Section 3.3.We use the default hyperparameters from EDM (Karras et al., 2022) for the diffusion process, as tuning these did yield better results. The transformer architecture consists of 6 cross-attention blocks with 8 attention heads and a hidden dimension of 64 trained on centerlines. The model is trained for 2500 epochs with a batch size of 16 using the AdamW (Loshchilov and Hutter, 2017) optimizer. We use a cosine annealing learning rate schedule, with an initial learning rate of 1e -7 that increases to 1e -4 in the first 100 epochs, after which it decreases back to 1e -7 . We found these hyperparameters to result in accurate generative results while preventing the model from overfitting, which we validated on the held-out test set.For evaluating the generative model, we sample a synthetic population of 200 samples, 100 of which contain an M1 occlusion. All synthetic samples are processed by the sequencing algorithm. Of the 200 samples, the sequencing algorithm failed on 7 M2 segments due to too much noise. We attribute this due to the large variance in M2 geometry combined with the small training set. For the sequencing algorithm, we set α = 0.25 and k = 5. With these parameters, the algorithm successfully processes the train and test sets.The quality of the synthetic vessels is assessed by comparing the distributions of tortuosity, average radius of the ICA, ACA, M1, and M2 and ICA-M1 bifurcation angle between the synthetic data and the real training population in Figure 3 . We observe that the distributions are specific to each vessel type and the presence of an M1-occlusion. Furthermore, the synthesized vessels represent the distributions of the geometric characteristics of the training set. This suggests that the model successfully captures the geometry of the vessel trees. Noteworthy, the model does not assign the wrong vessel type to the centerline points, e.g., assigning type ICA to a point belonging to the M1. We also assess the diversity of the synthesized population compared to the training set. We analyze the diversity by finding the closest sample from the training set for each sample in the synthetic population using the Chamfer distance. We observe that the model generates both diverse samples and samples that more closely resemble the training set, see Appendix A. Examples of generated vessels are shown in Figure 2 .We further analyze the distributions from Figure 3 by measuring their difference using the Kolmogorov-Smirnov (KS) test. When we compare the top and bottom rows of Figure 4 , we observe that the differences between distributions between the synthetic vessels and the training set are similar to the differences within the training set itself. We also observe that the distributions of the vessels in the training set are specific to each vessel type and the presence of an M1-occlusion. The same specific distributions are seen in the synthetic population, which aligns with our observations in Section 3.2. This indicates that the distributions generated by the model are not arbitrary, but rather that the model successfully captures the specific geometries and conditioning from the training set. 
discussion and conclusion: 
 In this work, we presented a method for conditionally generating cerebral vessel trees with diffusion and a set-transformer backbone architecture. Our model generates vessel tree geometry parameterized by centerline points with associated radius and vessel type. Our model can generate complex and diverse cerebral vessel tree centerlines. In addition, the Our experiments showed that our model can generate complex vessel trees that share the geometric characteristics of the training set. We observed that the model accurately captures the geometric differences between the individual vessel segments. The model can be effectively conditioned on auxiliary variables that affect vessel geometry, which we tested with the presence of an occlusion in the M1 vessel. Moreover, the model's generations are diverse and do not simply mimic the geometry from the training set. This suggests our model has learned an accurate distribution of cerebral vessel geometry.It is important to consider that our evaluation does not directly validate the usefulness of our model on downstream tasks. However, centerline-based in-silico stroke treatment models have been developed and validated using the same dataset as our study (Luraghi et al., 2021; Miller et al., 2023) . Since our model generates samples that are comparable to the real patient data, we are confident of the usefulness of our model for such computational models. In future studies, we will experiment on the required resolution and variation of the generated vessel trees to validate their usefulness for downstream tasks.In conclusion, we showed that diffusion with a set-transformer architecture is a capable solution for conditionally generating realistic and diverse cerebral vessel tree geometry. 
 
 Appendix:  


Decision: Reviewers appreciated both the methodological originality of the approach as well as the quality of the generation of vasculature itself which seems realistic. The reviewers also appreciated the clarity of the paper.

Reviews: review: paper_summary: paper_summary: This article addresses the problem of generating realistic synthetic intracranial vascular centerlines, with or without occlusion in the M1 artery. The authors propose addressing this problem using a diffusion model with a set transformer, which takes as input the set of points of the intracranial centerlines (including radius and vessel branch information) while enforcing permutation invariance. The diffusion model can be conditioned by the presence or absence of an occlusion in the M1 artery.
Sampling in the diffusion model yields a set of centerline points without any relation between each other. The authors propose a post-processing algorithm to recover topology information by linking the points together and finding the corresponding bifurcation points.

They trained and evaluated their model on a private dataset of 110 patients. They validated their approach by comparing the distribution of geometric properties of their simulated vascular trees (radius and tortuosity of specific vessels) with the same distributions of the training set, with and without occlusion. They claim that their generated vascular trees share the same properties per vessel branch as the training set and that the conditioning affects the vascular properties as expected.
main_review: main_review: I thank the authors for having carefully reply to my questions and comments. 
The description of the approach seems clearer now. 
I still have doubts regarding the choice of the point cloud instead of a graph, particularly concerning the scalability and complexity of generating larger vascular trees. However, the authors have provided adequate justification for their choice, which satisfies my initial question. Consequently, I maintain my original rating and believe that the article is suitable for publication in MIDL.
strength_weakness: strength_weakness: - The description of the proposed approach may lack clarity and accessibility for individuals familiar with transformers and diffusion models but not experts in the most recent developments.
- The authors generate sets of points instead of centerlines, necessitating post-processing to recover the graph-like structure. The positioning of this work in relation to the graph diffusion model literature, which would seem more natural, is absent.
- The article lacks a summary figure describing the complete proposed approach.
- The generated vascular tree only contains a few vessels, and it is not clear whether this is a limitation of the proposed approach or done intentionally.
questions: questions: - The input of the approach consists of centerlines, which could be more appropriately represented by a graph rather than a set of points. The authors should provide commentary on their decision to use the set of point representation instead of employing a graph diffusion model.
- It remains unclear how the set transformer and the diffusion process interact. Is the set transformer the backbone architecture of the diffusion model ?  It seems that it is not the case, as Figure 1 outputs the predicted centerlines instead of the predicted noise. Alternatively, does the diffusion model have a classic UNet backbone, and the set transformer is trained separately by sampling in the diffusion model to perform denoising ? The authors should clarify this aspect and add an overview figure of their approach.
- The authors should explicitly state their contributions in the introduction section.
- Quantitative results on the distribution of Chamfer distances should be added to the article to support the observations made in section 3.2. These results could be included in the appendix.
paper_summary: paper_summary: This paper proposes to use a diffusion-based generative model and set transformers to generate a population of synthetic intracranial vessel tree centerlines with associated radii and vessel types. The model is conditioned on the presence of an occlusion in the middle cerebral artery, a frequently occurring location of occlusion in patients with Acute Ischemic Stroke (AIS). In addition to the set diffusion based generative model, the paper also uses a centerline sequencing technique to re-order the produced segments.

They evaluate the synthetically generated data against real data from the MRClean Registry (used during training) by comparing derivates of centerline co-ordinates and features. By examining the geometric properties of the generate vessels, they show that their model can accurately generate distributions of realistic looking cerebral vessel trees and reflect the geometric characteristics within real populations.
main_review: main_review: I am satisfied with the author's rebuttal and it addresses the points I raised in the review adequately. This is an interesting proof of concept application and I would support the acceptance of the paper into the MIDL program.
strength_weakness: strength_weakness: 1. It is not clear how the hyperparameters for the model are set in Section 3.1 since no validation data is mentioned. Similarly, how are the parameters for the diffusion model/set transformer determined?

2. While the analysis presented supports the claim that the model generates realistic looking samples (that match distributional characteristics for the training set), it is not clear whether the model generalizes in the sense of use for downstream applications. Please elaborate.

3. There are no comparisons performed against methods that currently exist in literature for this problem, more discussion here would be welcome.
questions: questions: Please refer to weaknesses, particularly the points on the potential use-cases where the model may find appropriate use. Please also provide additional discussion on how the model compares to existing techniques used to address this problem.
paper_summary: paper_summary: The authors propose a DDPM-based approach for generating cerebral vessel trees. Vessel centerlines are generated in a denoising process, using a transformer as the backbone network, with some post-processing steps to produce 'joined-up' trees. The model is conditioned on the presence of M1 occlusion. The authors assess generated samples by comparing their geometric characteristics, such as bifurcation angles, to those found in real data.
main_review: main_review: I'm happy in general with the authors' answers and will keep my rating and weak accept. The authors were not able to address my questions about downstream validation of the usefulness of the generated trees, but I accept that is probably out of the scope of a conference paper with limited space, and think this paper is good overall.
strength_weakness: strength_weakness: My primary concern is the question of how good these generated trees would need to be to be useful in the in-silico clinical trials that motivate the work. While the authors show the generated trees follow similar distributions to those of real trees, we don't know how similar they need to be to be useful. It would have been nice to have some downstream validation of the generated trees on some sort of task to show these generated trees are 'good enough'.
questions: questions: Is there any way the authors can validate the downstream usefulness of these generated trees?

I would like some responses to the questions about setup and architectures raised above, too.

Comments: title: Response to Rebuttal
Rebuttal - addressing individual reviewer's feedback
Rebuttal - addressing individual reviewer's feedback
Rebuttal - addressing individual reviewer's feedback.
Rebuttal
comment: I would like to thank the authors for taking the time to read through my comments and address them. I appreciate their effort in incorporating additional explanation to support the contribution and have no further concerns. Since I was leaning towards accepting the paper before the rebuttal, I stand by this decision.
**Questions To Address In The Rebuttal:**

* *“Is there any way the authors can validate the downstream usefulness of these generated trees?”*

  **ANSWER:** We thank the reviewer for their feedback. Our proof-concept study aimed to evaluate the use of conditional set-diffusion for conditionally generating realistic vessel centerline data. The centerline data used in this study has been used to develop and validate in-silico stroke treatment models [Luraghi et al, 2021; Miller et al, 2023]. We are confident that our generated vessel centerlines can be utilized for such downstream tasks, since our model has proven to generate realistic synthetic data that is comparable to real patient data. Nevertheless, further experiments are required to learn about the required resolution and variation of our model  to generalize for downstream tasks. However, given the proof-of-concept nature of this study, performing such experiments remains outside of the scope of this study. We have now added these limitations and our intended future plans to perform the mentioned experiments  to the discussion section of our manuscript.

**Detailed Comments:**

* *“As far as I am aware, it is typical to standardize the input to DDPMs so they have a variance of 1, not 0.5, as claimed in the paper. (How important the difference is in practice is another question)”.*

  **ANSWER:** We thank the reviewer for their comment. In our study, we use the default settings as formulated in the EDM [Karras, et al.] paper for our diffusion model. Hence, we set the std to 0.5. This std refers to the std of the target dataset, which does not need to be 1. The initial distribution from which noise is sampled does have an std of 1 (unit Gaussian). We now mention the 0.5 default from EDM in the method section 2.1.

* *“Equation 1 suggests that noise is entirely additive. Generally in DDPMs the input is multiplied by a factor that is dependent on the timestep t, and also has a random noise component added. Can the authors clarify if their setup differs from the standard setup?”*

  **ANSWER:** In the EDM [Karras, et al.] formulations, noise is indeed additive to the inputs. the noise is sampled from the Gaussian distribution N(0, sigma^2 * I), where sigma is the current noise level. Here, noise level can be interpreted as the time step from the general DDPM formulation. Note that some additional operations are performed after the noise has been added to the input, for example to take into account the standard deviation of the target (data) distribution (0.5 in our case). However, to keep the text accessible, we opted to not include these formulations in the current manuscript and rather reference the EDM paper that presents the full diffusion formulation. We added a more explicit reference to EDM for the diffusion hyperparameter setup in Results section 3.1.

* *“It is unclear to me if the set transformer is a new contribution by the authors or an existing architecture. As far as I can tell it is a standard transformer with no positional embedding.”*

  **ANSWER:** The set transformer is indeed a transformer without positional embeddings and it is not a new contribution. Instead, the novelty of our method stems from utilizing the permutation equivariance and unordered point sets for topology-free vessel tree generation. We highlighted this in the introduction section of our manuscript. 

* *“It might actually be useful to include positional embedding in the transformer on the sequence of coordinates. This might allow the model to generate ordered sequences' centreline points and reduce the need for the sequencing post-processing step. Have the authors tried this approach?”*

  **ANSWER:** Ideally, we would like to have the model output an ordered set of points, so no post-processing would be required. One of the main design goals of our model was to be general, i.e., the model should be able to handle centerline trees with arbitrary topology, e.g., arbitrary number of branches, number of points per vessel segment, etc. However, such topology-free trees have no defined ordering of points, and treating them as a point cloud, i.e., a permutation equivariant fully connected graph, seemed more natural. Besides, under the assumption that the model generates well-defined lines with equidistantly spaced points, sequencing these lines is quite a straightforward task. We also hypothesize that making the model free of topological restrictions would allow it to better generalize to the geometry of the vessel tree. Nonetheless, in future studies, it is definitely interesting to experiment with having the model generate an ordered output, perhaps using a learned permutation matrix. We added our motivations for using unordered point sets in the introduction section.
**Questions To Address In The Rebuttal:**

* *“It is not clear how the hyperparameters for the model are set in Section 3.1, ... . Similarly, how are the parameters for the diffusion model/set transformer determined?”*

  **ANSWER**: We thank the reviewer for their feedback and we agree that a description of hyperparameter choices was lacking. Since we found that tuning the hyperparameters of the diffusion formulation did not improve our results, we set all diffusion hyperparameters to the default from the EDM [Karras et al, 2023] paper. We found that lowering the number of attention heads from 8 and the feature dimension from 64 drastically impacted the performance. Increasing these did not increase performance, but did result in overfitting. We added the motivations for these choices in the Results section 3.1.

* *“... it is not clear whether the model generalizes in the sense of use for downstream applications. Please elaborate.”*

  **ANSWER**: We thank the reviewer for their comment. In this study, we aimed to evaluate the use of conditional diffusion set models to generate realistic and conditioned vessel centerline data. Notably, the centerline data used in this study has been used to develop and validate computational stroke treatment simulation models [Luraghi et al, 2021, Miller et al, 2023]. Since our generated data has proven realistic and comparable with patient data, we are confident that vessel centerlines generated by our model can be utilized in the downstream tasks. Nevertheless, further experiments are necessary to learn about the required resolution and variation to generalize our model for such downstream tasks. However, given the proof-of-concept nature of this study to generate conditional vessel centerline data, performing such experiments remains outside the scope of this study. We have now added these limitations and future plans to the discussion section of our manuscript.

* *“There are no comparisons performed against methods that currently exist in literature for this problem, more discussion here would be welcome.”*

  **ANSWER**: To the best of our knowledge, we are the first to generate vessel centerline trees with arbitrary topology. Prior work either generates vessels without bifurcation (fixed topology), or cannot be conditioned on auxiliary variables. On the other hand, our qualitative and quantitative evaluation showed that the vessel centerlines generated from our model match those from the training set, while being unique implying that our model does not simply copy from the training set. 

**Detailed Comments:**

* *“How is the directional flow estimate obtained in Eq. 4?”*

  **ANSWER**: d_cur refers to the direction the centerline is currently moving in, and is calculated as the normalized difference between the point prior to the current last point, and the last point itself in the sequence.  The directions d_i from the last point in the sequence to the candidate points, are calculated as the normalized difference between candidate points and the last point added to the sequence. During sequence initialization, the sequence consists of only 1 point, so no directional information is used to determine the next point. We added the formulas for the directions to the revised manuscript.


* *“Have the authors performed any analysis on the failure modes of the model leading to non-convergence in the M1 occlusion cases ...”*

  **ANSWER**: We have indeed analyzed the failure cases of our model. All failure cases occurred  in the case where no M1 occlusion was present, specifically, in the synthesis of the M2 vessel.  A failure occurs when the post processing algorithm either cannot connect the individual points or when the vessel segment cannot be connected to the M1 segment. This could be explained by the inconsistencies in the M2 ground truth, especially in defining the end of the vessel segment, due to the inherent variation in the anatomy of M2, and the small training set. This could cause the M2 vessel to sometimes not converge, resulting in a noisy centerline which cannot be post-processed. We added our analysis to the Results 3.1.

* *“As mentioned in the discussion, how do you qualify whether the generated centerlines are free of artifacts?”*

  **ANSWER**: We qualify the synthetic sample as valid, or artifact-free, if the sequencing algorithm is able to reconstruct the centerline ordering, and the geometric characteristics of this ordering fall within the distribution of the training set. Artifacts generally present themselves in two ways: non-convergence of the generated centerline (noisy centerlines) or gaps in the generated centerlines. Since the post-processing step assumes points to be distributed equidistantly along the centerline, noisy distributed points and gaps in the centerline result in the failure of the post-processing step to form a connected vessel centerline tree. We revise this statement to highlight these details in the revised manuscript.
**Questions To Address Rebuttal:**

* *“... The authors should provide commentary on their decision to use the set of point representation instead of employing a graph diffusion model?”*. 

    **ANSWER:** We thank the reviewer for their question. By representing the vessel trees as point clouds, we essentially treat them as a fully connected graph through the message passing framework in the self-attention mechanism. Unlike graph representations that limit the model to only the neighboring points, the point cloud representation enables the model to learn about the geometry on a global scale and makes our model applicable to trees with arbitrary topology. We highlighted our motivations for using point sets in the introduction.

* *“It remains unclear how the set transformer and the diffusion process interact. … The authors should clarify this aspect and add an overview figure of their approach.”*

    **ANSWER:** Thank you for addressing these issues. The transformer model is indeed the backbone of the diffusion process, modeling the denoising function.. The forward diffusion process adds noise to the point cloud, and the transformer is tasked to denoise the point cloud. During sampling, the initial noise is sampled from the unit Gaussian and iteratively denoised to form the centerline point cloud that includes. We apply the centerline sequencing post-processing step to obtain the final ordered vessel tree. This has been clarified in the method section, as well as in the new overview Figure 1.

* *“Missing contribution section in the introduction and the distributions of chamfer distances.”*

    **ANSWER:** We added the contribution section to the introduction and the distributions of the chamfer distances to the appendix section A.

**Weaknesses**:

For the first two points, please refer to the answers for the rebuttal questions above.

* *“The article lacks a summary figure describing the complete proposed approach.”*

    **ANSWER:** In line with revising the methodology to make the description of the current method more clear, we have revised Figure 1 to have it represent an accurate representation of the proposed method.

* *“The generated vascular tree only contains a few vessels, and it is not clear whether this is a limitation of the proposed approach or done intentionally.”*

    **ANSWER:** We thank the reviewer for their remark. Large vessel occlusions most frequently occur in the arteries of the anterior circulation such as ICA, ACA, M1, and M2. To increase the usefulness of the centerline data generated by our model to be relevant for downstream tasks such as in computational models of stroke treatment simulation, we chose to focus on the few larger arteries. Nonetheless, since we consider vessel trees as point clouds, our model is not constrained by specific topology. We highlighted our choices in the introduction section.

**Detailed Comments:**

* *“How are d_cur and d_i calculated?”*

    **ANSWER:** d_cur refers to the direction the centerline is currently moving in, and is calculated as the normalized difference between the point prior to the current last point, and the last point itself in the sequence.  The directions d_i from the last point in the sequence to the candidate points, are calculated as the normalized difference between candidate points and the last point added to the sequence. During sequence initialization, the sequence consists of only 1 point, so no directional information is used to determine the next point. We added the formulas for calculating the directions in the revised manuscript.
 
* *“With the proposed segment merging approach, it seems that more than two segments can be merged. Is this the case? If so, is this behavior desired, and does it happen in practice?”*

    **ANSWER:** It is indeed the case that more than two segments can be merged. This is intentional, since we wanted the sequencing algorithm to be general, meaning it should work with any tree topology – similarly to the diffusion model itself. It is, for example, possible for trees to have tri-furcations, or even ‘higher-furcations’. However, our data only contains bifurcations, and we did not see the merging algorithm merge more than two segments.

* *“The authors have stated, “Noteworthy, the model never incorrectly classifies a vessel segment.” … The authors should provide more details on this sentence.”*

    **ANSWER:** We thank the reviewer for pointing this out. Indeed, ‘classifying’ is an incorrect terminology. At the start of the reverse diffusion process, the xyz location, radius, and vessel type are unknown. Henc, during the reverse diffusion process, the model has to find the right xyz locations and radius for this point, as well as assign the right type to it. We observed that our model does not incorrectly assign a vessel type to a point, for example, assign type M1 to a point that has been placed in the ICA segment. We have clarified our statement in the results Section 3.2.
We would like to thank the reviewers for their time and valuable feedback which has allowed us to improve our work. We are also happy to hear that our work was found to be of interest and  well-written. In the following paragraphs, we summarize the feedback of the reviewers along with our answers and changes to the manuscript. We then address the questions raised by each reviewer in detail. For all specific changes made to the manuscript to address the feedback of the reviewers, we refer to the revised manuscript, where all changes are highlighted in red.

All reviewers raised minor questions regarding the model architecture and study design choices, specifically regarding the interaction of the set-transformer and the diffusion process, the choice of points without positional embeddings instead of graphs, and the selection of hyperparameters. In essence, the diffusion process is tasked with denoising a set of randomly sampled points to form a centerline tree. The set-transformer performs this denoising. We made these details more explicit in the manuscript, and added these details to an overview figure (Figure 1). The centerline trees are represented by an unordered set of points, as this makes our approach topology-free, and learn about the global geometry. We added these motivations in our manuscript. We used the default diffusion parameters from EDM [Karras et al, 2022], as we found that tuning these parameters did not improve our results. We made our hyperparameter choices more explicit in the manuscript. The hyperparameters of the transformer were tuned to decrease model run-time, and prevent overfitting without impacting our results. This was tested on the held-out test set. We added our motivation for the chosen hyperparameters to the manuscript.

Reviewers uaH9 and bt7k asked about validating and the usefulness of the generated data for downstream tasks. Our model generates vessel centerline data that can serve as input to computational treatment simulations, such as the virtual treatment procedure proposed by [Luraghi et al, 2021; Miller et al, 2023], which was developed using the same dataset as our study. Since our synthetic data generated by our model is proven to be comparable to real-world patients, we are confident that our model is useful for such computational treatment models. Nonetheless, we agree that additional  experiments on the resolution and variation of the synthetic data are necessary to validate the usefulness for downstream tasks. However, given the proof-of-concept nature of this study, we are unfortunately unable to perform such experiments. We have therefore added these limitations and future plans to the discussion of our manuscript.

Lastly, reviewers uaH9 and iwJM highlighted missing details from our manuscript on comparisons with literature, a contribution section in our Introduction, and missing Chamfer distance distributions to determine uniqueness quality of our model. To the best of our knowledge, we are the first to introduce a conditional topology free centerline vessel tree generative method. We do note that our quantitative and qualitative analysis are in line with existing work [Bridio et al, 2023; Wolterink et al, 2018] on generating vessel centerlines. If the reviewer has any particular existing work in mind, we would gladly incorporate it into the manuscript. We have now incorporated the missing contribution section in the introduction and added an Appendix section where we present the Chamfer distance distributions.

**References:**

* Giulia Luraghi, Sara Bridio, Jose Felix Rodriguez Matas, Gabriele Dubini, Nikki Boodt, Frank JH Gijsen, Aad van der Lugt, Behrooz Fereidoonnezhad, Kevin M Moerman, Patrick McGarry, et al. The first virtual patient-specific thrombectomy procedure. Journal of Biomechanics, 126:110622, 2021.


* Miller, C., Konduri, P., Bridio, S., Luraghi, G., Terreros, N. A., Boodt, N., ... & Hoekstra, A. (2023). In silico thrombectomy trials for acute ischemic stroke. Computer Methods and Programs in Biomedicine, 228, 107244.

* Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-based generative models. Advances in Neural Information Processing Systems, 35:26565–26577, 2022.

* Jelmer M Wolterink, Tim Leiner, and Ivana Isgum. Blood vessel geometry synthesis using generative adversarial networks. arXiv preprint arXiv:1804.04381, 2018.

* Sara Bridio, Giulia Luraghi, Anna Ramella, Jose Felix Rodriguez Matas, Gabriele Dubini, Claudio A Luisi, Michael Neidlin, Praneeta Konduri, Nerea Arrarte Terreros, Henk A Marquering, et al. Generation of a virtual cohort of patients for in silico trials of acute ischemic stroke treatments. Applied Sciences, 13(18):10074, 2023.