\section{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 \cite{konduri2020silico,miller2023silico, viceconti2016silico}.
% ISCTs benefit from access to large virtual populations that include a wide variety of 

ISCTs can aid in the development of novel treatments for acute ischemic stroke (AIS), one of the leading causes of mortality and disability \cite{phipps2020management}. 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 \cite{luraghi2021first}. 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.
% We propose to use deep generative modeling to generate 
% We hypothesize that generative modeling can be used to develop synthetic and clinically accurate cerebral vessel trees containing ICA, MCA, and ACA vessel segments that can be conditioned on auxiliary variables of interest such as the presence of occluding thrombi.

Various methods have been introduced for generating vessel geometry. \citet{bridio2023generation} 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 \cite{kalaie2023geometric}. 
Furthermore, conditioning SSM approaches on auxiliary variables is not straightforward \cite{bannister2022deep}. \citet{danu2019deep} use deep generative voxel-based 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.
\citet{wolterink2018blood}, instead, uses a conditional generative adversarial network (GAN) \cite{goodfellow2020generative} 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.

% Statistical Shape Modelling (SSM) has been widely used for generating anatomical structures \cite{gooya2017mixture, cosentino2020statistical, lamecker2002statistical}. For example, \citet{bridio2023generation} proposes an SSM approach using principal component analysis (PCA) to synthesize cerebral vessel anatomies described by their centerlines and diameters. However, SSM approaches generally linearly project anatomical structures to lower dimensional subspaces, which can limit their ability to capture the variation in the data (expressivity) \cite{kalaie2023geometric}. Furthermore, conditioning on auxiliary variables, such as the presence of an occlusion, is also not straightforward \cite{bannister2022deep}. 

% On the other hand, deep generative models have better expressivity due to their non-linearity and can be conditioned with auxiliary variables in a straightforward way. \cite{bouritsas2019neural}. For example, variational autoencoders \cite{kingma2013auto} and generative adversarial networks (GANs) \cite{goodfellow2020generative} have been used to model 3D vessel geometry through voxelization \cite{danu2019deep}. However, in the 3D setting, voxelization methods are limited to lower resolutions due to high computational requirements. \citet{wolterink2018blood} generate coronary vessel centerlines using GANs. While effective, neither \citet{wolterink2018blood} nor \citet{danu2019deep} can generate vessel anatomy that includes bifurcations. More recently, stable diffusion \cite{ho2020denoising} has emerged as one of the most powerful generative methods, even outperforming GANs in certain cases \cite{dhariwal2021diffusion}. Recent work pairs diffusion with set-transformers \cite{lee2019set} to leverage their powerful conditioning abilities for conditionally generating high-quality geometry \cite{zhang20233dshape2vecset}.

In this work, we present a deep generative diffusion-based method for generating vessel geometry. Diffusion \cite{ho2020denoising} 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 \cite{lee2019set}. 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 post-processing 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. 

% Our code is available \href{here}{upon publication}.
% Our approach, based on a set-transformer architecture \cite{lee2019set}, overcomes the limitations of fixed topology and directly generates vessel geometry with arbitrary topologies. Furthermore, it can be effectively conditioned on variables of interest.
 
% In this work, we present and evaluate diffusion with a set-transformer to generate synthetic vessel trees centerlines with corresponding radii, associated vessel type (ICA, M1, M2, or ACA), and presence of an M1 occlusion. We also evaluate the accuracy of the synthetic vessels by comparing their geometric characteristics with those observed in clinical data. However, since set-diffusion models draw samples in an unordered pattern, we propose a straightforward centerline sequencing algorithm that orders the separate vessel segments. Our code is available \href{here}{upon publication}.





% DL methods

% Out approach and contributions



% CEnterliens are an efficient parameterization of vessel geometry.