\section{Discussion and Conclusion}

% \todo{What have we presented in this work.}\\

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 model labels each point with its corresponding vessel type. Furthermore, our simple sequencing algorithm is effective at connecting the individual points and vessel segments. Since the sequencing algorithm assumes equidistantly spaced points, this suggests the generated centerlines are relatively artifact-free.

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 \cite{luraghi2021first, miller2023silico}. 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.

% As such, our approach allows for the generation of diverse and customizable synthetic populations of segmented cerebral vessel geometry that can be used for in-silico clinical trials (ISCTs).
% further experiments on the required resolution and variation of the generated centerline data are necessary to validate the usefulness of the generated data for downstream tasks. 


% \revision{In future studies, we will experiment on the required resolution and variation of the generated vessel trees to validate their usefulness of for downstream tasks. We also want to experiment with generating geometry through implicit neural representations, such as in \cite{zhang20233dshape2vecset}, and expand the conditioning aspect to include patient demographic parameters for generating highly customizable synthetic populations.}

% It is important to consider that the centerline parameterization is a simplification of the vessel geometry. For instance, the cross-sectional area of a vessel is (usually) not perfectly circular. In future work, we want to address this limitation and directly generate the vessel surface geometry. Recent work shows promising results in conditionally generating complex and high-fidelity geometry using implicit neural representations (INRs). \citet{zhang20233dshape2vecset} generate INRs through latent diffusion that can be effectively conditioned on a wide range of modalities. We also want to expand upon the conditioning aspect of our model and include patient demographic parameters. Since cross-attention allows for conditioning on combinations of variables, this will make it possible to generate highly customizable synthetic populations.

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.



% \todo{What do the results mean}\\



% \todo{Address shortcoming, e.g., centerlines as a simplification of true vessel geometry.}\\

% \todo{Future work - condition on patient demographic variables and synthesize vessel geometry more directly through INRs.}\\

% \todo{Final conclusion.}\\