1pt

11institutetext:  Consejo Nacional de Investigaciones Científicas y Técnicas,
Argentina 22institutetext: Universidad Nacional del Sur, Bahía Blanca,
Argentina 33institutetext: Universidad Torcuato Di Tella, Buenos Aires,
Argentina
33email<EMAIL_ADDRESS>
# VesselVAE: Recursive Variational Autoencoders for 3D Blood Vessel Synthesis

Paula Feldman 1 Consejo Nacional de Investigaciones Científicas y Técnicas,
Argentina 13Universidad Nacional del Sur, Bahía Blanca, Argentina
<EMAIL_ADDRESS>Miguel Fainstein 3Universidad Nacional del Sur,
Bahía Blanca, Argentina
<EMAIL_ADDRESS>Viviana Siless 3Universidad Nacional del Sur,
Bahía Blanca, Argentina
<EMAIL_ADDRESS>Claudio Delrieux 1 Consejo Nacional de
Investigaciones Científicas y Técnicas, Argentina 12Universidad Torcuato Di
Tella, Buenos Aires, Argentina 2 Emmanuel Iarussi 1 Consejo Nacional de
Investigaciones Científicas y Técnicas, Argentina 13Universidad Nacional del
Sur, Bahía Blanca, Argentina
<EMAIL_ADDRESS>1 Consejo Nacional de Investigaciones Científicas
y Técnicas, Argentina 12Universidad Torcuato Di Tella, Buenos Aires, Argentina
22Universidad Torcuato Di Tella, Buenos Aires, Argentina 22Universidad
Torcuato Di Tella, Buenos Aires, Argentina 21 Consejo Nacional de
Investigaciones Científicas y Técnicas, Argentina 13Universidad Nacional del
Sur, Bahía Blanca, Argentina
<EMAIL_ADDRESS>1 Consejo Nacional de Investigaciones Científicas
y Técnicas, Argentina 12Universidad Torcuato Di Tella, Buenos Aires, Argentina
2

###### Abstract

We present a data-driven generative framework for synthesizing blood vessel 3D
geometry. This is a challenging task due to the complexity of vascular
systems, which are highly variating in shape, size, and structure. Existing
model-based methods provide some degree of control and variation in the
structures produced, but fail to capture the diversity of actual anatomical
data. We developed VesselVAE, a recursive variational Neural Network that
fully exploits the hierarchical organization of the vessel and learns a low-
dimensional manifold encoding branch connectivity along with geometry features
describing the target surface. After training, the VesselVAE latent space can
be sampled to generate new vessel geometries. To the best of our knowledge,
this work is the first to utilize this technique for synthesizing blood
vessels. We achieve similarities of synthetic and real data for radius (.97),
length (.95), and tortuosity (.96). By leveraging the power of deep neural
networks, we generate 3D models of blood vessels that are both accurate and
diverse, which is crucial for medical and surgical training, hemodynamic
simulations, and many other purposes.

###### Keywords:

Vascular 3D model Generative modeling Neural Networks.

##  1 Introduction

Accurate 3D models of blood vessels are increasingly required for several
purposes in Medicine and Science [25]. These meshes are typically generated
using either image segmentation or synthetic methods. Despite significant
advances in vessel segmentation [26], reconstructing thin features accurately
from medical images remains challenging [2]. Manual editing of vessel geometry
is a tedious and error prone task that requires expert medical knowledge,
which explains the scarcity of curated datasets. As a result, several methods
have been developed to adequately synthesize blood vessel geometry [29].

Within the existing literature on generating vascular 3D models, we identified
two primary types of algorithms: fractal-based, and space-filling algorithms.
Fractal-based algorithms use a set of fixed rules that include different
branching parameters, such as the ratio of asymmetry in arterial bifurcations
and the relationship between the diameter of the vessel and the flow [7, 33].
On the other hand, space-filling algorithms allow the blood vessels to grow
into a specific perfusion volume while aligning with hemodynamic laws and
constraints on the formation of blood vessels [9, 25, 22, 17, 21]. Although
these _model-based_ methods provide some degree of control and variation in
the structures produced, they often fail to capture the diversity of real
anatomical data.

In recent years, deep neural networks led to the development of powerful
generative models [30], such as Generative Adversarial Networks [8, 12] and
Diffusion Models [11], which produced groundbreaking performance in many
applications, ranging from image and video synthesis to molecular design.
These advances have inspired the creation of novel network architectures to
model 3D shapes using voxel representations [28], point clouds [31], signed
distance functions [19], and polygonal meshes [18]. In particular, and close
to our aim, Wolterink et al. [27] propose a GAN model capable of generating
coronary artery anatomies. However, this model is limited to generating
single-channel blood vessels and thus does not support the generation of more
complex, tree-like vessel topologies.

In this work we propose a novel _data-driven_ framework named VesselVAE for
synthesizing blood vessel geometry. Our generative framework is based on a
Recursive variational Neural Network (RvNN), that has been applied in various
contexts, including natural language [24, 23], shape semantics modeling [14,
15], and document layout generation [20]. In contrast to previous data-driven
methods, our recursive network fully exploits the hierarchical organization of
the vessel and learns a low-dimensional manifold encoding branch connectivity
along with geometry features describing the target surface. Once trained, the
VesselVAE latent space is sampled to generate new vessel geometries. To the
best of our knowledge, this work is the first to synthesize multi-branch blood
vessel trees by learning from real data. Experiments show that synth and real
blood vessel geometries are highly similar: radius (.97), length (.95), and
tortuosity (.96).

##  2 Methods

Figure 1: Top: Overview of the Recursive variational Neural Network for
synthesizing blood vessel structures. The architecture follows an Encoder-
Decoder framework which can handle the hierarchical tree representation of the
vessels. VesselVAE learns to generate the topology and attributes for each
node in the tree, which is then used to synthesize 3D meshes. Bottom: Layers
of the Encoder and Decoder networks comprising branches of fully-connected
layers followed by leaky ReLU activations. Notice that right/left Enc-MLPs of
the Encoder only execute when the incoming tree requires it. Similarly, the
Decoder only uses right/left Dec-MLPs when the Node Classifier predicts
bifurcations.

Input. The network input is a binary tree representation of the blood vessel
3D geometry. Formally, each tree is defined as a tuple $(T,\mathcal{E})$,
where $T$ is the set of nodes, and $\mathcal{E}$ is the set of directed edges
connecting a pair of nodes ${(n,m)}$, with $n,m\in T$. In order to encode a 3D
model into this representation, vessel segments $V$ are parameterized by a
central axis consisting of ordered points in Euclidean space:
$V={v_{1},v_{2},\ldots,v_{N}}$ and a radius $r$, assuming a piece-wise tubular
vessel for simplicity. We then construct the binary tree as a set of nodes
$T={n_{1},n_{2},\ldots,n_{N}}$, where each node $n_{i}$ represents a vessel
segment $v$ and contains an attribute vector
$\mathbf{x}_{i}=[x_{i},y_{i},z_{i},r_{i}]\in\mathbb{R}^{4}$ with the
coordinates of the corresponding point and its radius $r_{i}$. See Section 3
for details.

Network architecture. The proposed generative model is a Recursive variational
Neural Network (RvNN) consisting of two main components: the Encoder ($Enc$)
and the Decoder ($Dec$) networks. The role of the Encoder is to transform a
tree structure into a hierarchical encoding situated on the learned manifold.
On the other hand, the Decoder network is capable of sampling from this
encoded space utilizing these samples to decode tree structures, as depicted
in Fig. 1. The encoding and decoding processes are achieved through a depth-
first traversal of the tree, where each node is combined with its parent node
in a recursive manner. The model outputs a hierarchy of vessel branches, where
each internal node in the hierarchy is represented by a vector encoding its
own attributes and the information of all subsequent nodes in the tree.

Within the RvNN Decoder network there are two essential components: the Node
Classifier ($Cls$) and the Features Decoder Multi-Layer Perceptron (Features
Dec-MLP). The Node Classifier discerns the type an encoded node should be
decoded into, whether a leaf node or an internal node with one or two
bifurcations. This is implemented as a multi-layer perceptron trained to
predict a three-category bifurcation probability with an encoded vector as
input. Complementing the Node Classifier, the Features Dec-MLP is responsible
for reconstructing the attributes of each node, specifically its position and
radius. Furthermore, two additional components, the Right and Left Dec-MLP,
are in charge of recursively decoding the next encoded vector in the tree
hierarchy. These decoder branches do not always execute, and depend on the
decision made by the classifier. If the Node Classifier predicts one
bifurcation for a node, a right child is assumed by default.

In addition to the core architecture, our model is further augmented with
three auxiliary, shallow, fully-connected neural networks: $f_{\mu}$,
$f_{\sigma}$, and $g_{z}$. Positioned before the RvNN bottleneck, the
$f_{\mu}$ and $f_{\sigma}$ networks shape the distribution of the latent space
where encoded tree structures lie. Conversely, the $g_{z}$ network, situated
after the bottleneck, facilitates the decoding of latent variables, aiding the
Decoder network in the reconstruction of tree structures. Collectively, these
supplementary networks streamline the data transformation process through the
model. All activation functions used in the network are leaky ReLUs. See the
Appendix for other implementation details.

Objective. Our generative model is trained to learn a probability distribution
over the latent space that can be used to generate new blood vessel segments.
After encoding, the decoder takes samples from a multivariate Gaussian
distribution: $z_{s}(x)\sim N(\mu,\sigma)$ with $\mu=f_{\mu}(Enc(x))$ and
$\sigma=f_{\sigma}(Enc(x))$, where $Enc$ is the recursive encoder and
$f_{\mu},f_{\sigma}$ are two fully-connected neural networks. In order to
recover the feature vectors $\mathbf{x}$ for each node along with the tree
topology, we simultaneously train the regression network (Features Dec-MLP in
Fig. 1) on a reconstruction objective $L_{recon}$, and the Node Classifier
using $L_{topo}$. Additionally, in line with the general framework proposed by
$\beta$-VAE [10], we incorporated a Kullback-Leibler (KL) divergence term
encouraging the distribution $p(z_{s}(x))$ over all training samples $x$ to
move closer to the prior of the standard normal distribution $p(z)$. We
therefore minimize the following equation:

$L=L_{recon}+\alpha L_{topo}+\gamma L_{KL},$ (1)

where the reconstruction loss is defined as
$L_{recon}=\left\|Dec\left(z_{s}(x)\right)-x\right\|_{2}$, the Kullback-
Leibler divergence loss is
$L_{\mathrm{KL}}=D_{\mathrm{KL}}\left(p\left(z_{s}(x)\right)\|p(z)\right)$,
and the topology objective is a three-class cross entropy loss
$L_{topo}=\Sigma_{c=1}^{3}x_{c}\log(Cls(Dec(x))_{c})$. Notice that $x_{c}$ is
a binary indicator (0 or 1) for the true class of the sample $x$.
Specifically, $x_{c}=1$ if the sample belongs to class $c$ and 0 otherwise.
$Cls(Dec(x))_{c}$ is the predicted probability of the sample $x$ belonging to
class $c$ (zero, one, or two bifurcations), as output by the classifier. Here,
$Dec(x)$ denotes the encoded-decoded node representation of the input sample
$x$.

3D mesh synthesis. Several algorithms have been proposed in the literature to
generate a surface 3D mesh from a tree-structured centerline [29]. For
simplicity and efficiency, we chose the approach described in [6], which
produces good quality meshes from centerlines with a low sample rate. The
implemented method iterates through the points in the curve generating a
coarse quadrilateral mesh along the segments and joints. The centerline
sampling step is crucial for a successful reconstruction outcome. Thus, our
re-sampling is not equispaced but rather changes with curvature and radius
along the centerline, increasing the frequency of sampling near high-curvature
regions. This results in a better quality and more accurate mesh. Finally,
Catmull-Clark subdivision algorithm [5] is used to increase mesh resolution
and smooth out the surface.

##  3 Experimental Setup

Figure 2: Dataset and pre-processing overview: The raw meshes from the IntraA
3D collection undergo pre-processing using the VMTK toolkit. This step is
crucial for extracting centerlines and cross-sections from the meshes, which
are then used to construct their binary tree representations.

Materials. We trained our networks using a subset of the open-access IntrA
dataset 111https://github.com/intra3d2019/IntrA published by Yang et al. in
2020 [32]. This subset consisted of 1694 healthy vessel segments reconstructed
from 2D MRA images of patients. We converted 3D meshes into a binary tree
representation and used the _network extraction_ script from the VMTK toolkit
222http://www.vmtk.org/vmtkscripts/vmtknetworkextraction to extract the
centerline coordinates of each vessel model. The centerline points were
determined based on the ratio between the sphere step and the local maximum
radius, which was computed using the advancement ratio specified by the user.
The radius of the blood vessel conduit at each centerline sample was
determined using the computed cross-sections assuming a maximal circular shape
(See Figure 2). To improve computational efficiency during recursive tree
traversal, we implemented an algorithm that balances each tree by identifying
a new root. We additionally trimmed trees to a depth of ten in our
experiments. This decision reflects a balance between the computational
demands of depth-first tree traversal in each training step and the complexity
of the training meshes. Trees with higher depth and non-binary bifurcations or
loops were excluded from our study. However, non-binary trees can be converted
into binary trees and it is possible to train with deeper trees at the expense
of higher computational costs. Ultimately, we were able to obtain 700 binary
trees from the original meshes using this approach.

Implementation details. For the centerline extraction, we set the advancement
ratio in the VMTK script to $1.05$. The script can sometimes produce multiple
cross-sections at centerline bifurcations. In those cases, we selected the
sample with the lowest radius, which ensures proper alignment with the
centerline principal direction. All attributes were normalized to a range of
$[0,1]$. For the mesh reconstruction we used 4 iterations of Catmull-Clark
subdivision algorithm. The data pre-processing pipeline and network code were
implemented in Python and PyTorch Framework.

Training. In all stages, we set the batch size to 10 and used the ADAM
optimizer with $\beta_{1}$ = 0.9, $\beta_{2}$ = 0.999, and a learning rate of
$1\times 10^{-4}$. We set $\alpha=.3$ and $\gamma=.001$ for Equation 1 in our
experiments. To enhance computation speed, we implemented dynamic batching
[16], which groups together operations involving input trees of dissimilar
shapes and different nodes within a single input graph. It takes approximately
12 hours to train our models on a workstation equipped with an NVIDIA A100
GPU, 80GB VRAM, and 256GB RAM. However, the memory footprint during training
is very small ($\leq$1GB) due to the use of a lightweight tree representation.
This means that the amount of memory required to store and manipulate our
training data structures is minimal. During training, we ensure that the
reconstructed tree structure aligns with the original tree’s structure, rather
than relying solely on the classifier’s predictions. We train the classifier
using a cross-entropy loss that compares its predictions to the actual values
from the original tree. Since the number of nodes in each class is unbalanced,
we scale the weight given to each class in the cross-entropy loss using the
inverse of each class count. During preliminary experiments, we observed that
accurately classifying nodes closer to the tree root is critical. This is
because a miss-classification of top nodes has a cascading effect on all
subsequent nodes in the tree (i.e. skip reconstructing a branch). To account
for this, we introduce a weighting scheme that assigns a weight to each node’s
cross-entropy loss based on the number of total child nodes. This weight is
then normalized by the total number of nodes in the tree.

Metrics. We defined a set of metrics to evaluate our trained network’s
performance. By using these metrics, we can determine how well the generated
3D models of blood vessels match the original dataset distribution, as well as
the diversity of the generated output. The chosen metrics have been widely
used in the field of blood vessel 3D modeling, and have shown to provide
reliable and accurate quantification of blood vessels main characteristics
[13, 3]. We analyzed the tortuosity per branch, the vessel centerline total
length, and the average radius of the tree. Tortuosity distance metric [4] is
a widely used metric in the field of blood vessel analysis (mainly because of
its clinical importance). It measures the amount of twistiness in each branch
of the vessel. The vessel total length and average radius were used in
previous work to distinguish healthy vasculature from cancerous malformations.
Finally, in order to measure the distance across distributions for each
metric, we compute the cosine similarity.

##  4 Results

Figure 3: (a) shows the histograms of total length, average radius and
tortuosity per branch for both, real and synthetic samples. (b) shows a visual
comparison among our method and two baselines [27, 9].

We conducted both quantitative and qualitative analyses to evaluate the
model’s performance. For the quantitative analyses, we implemented a set of
metrics commonly used for characterizing blood vessels. We computed histograms
of the radius, total length, and tortuosity for the real blood vessel set and
the generated set (700 samples) in Figure 3 (a). The distributions are aligned
and consistent. We measured the closeness of histograms with the cosine
similarity by projecting the distribution into a vector of $n$-dimensional
space ($n$ is the number of bins in the histogram). Since our points are
positive, the results range from 0 to 1. We obtain a radius cosine similarity
of .97, a total length cosine similarity of .95, and a tortuosity cosine
similarity of .96. Results show high similarities between histograms
demonstrating that generated blood vessels are realistic. Given the
differences with the baselines generated topologies, for a fair comparison, we
limited our evaluation to a visual inspection of the meshes.

The qualitative analyses consisted of a visual evaluation of the reconstructed
outputs provided by the decoder network. We visually compared them to state-
of-the-art methods in Figure 3 (b). The method described in Wolterink et al.
[2018] [27] is able to generate realistic blood vessels but without branches
and the method described in Hamarneh et al. [2010] [9] is capable of
generating branches with straight shapes, missing on realistic modeling. In
contrast, our method is capable of generating realistic blood vessels
containing branches, with smooth varying radius, lengths, and tortuosity.

##  5 Conclusions

We have presented a novel approach for synthesizing blood vessel models using
a variational recursive autoencoder. Our method enables efficient encoding and
decoding of binary tree structures and produces high-quality synthesized
models. In the future, we aim to explore combinations of our approach with
representing surfaces by the zero level set in a differentiable implicit
neural representation (INR) [1]. This could lead to more accurate and
efficient modeling of blood vessels and potentially other non-tree-like
structures such as capillary networks. Since the presented framework would
require significant adaptations to accommodate such complex topologies,
exploring this problem would certainly be an interesting direction for future
research. Additionally, the generated geometries might show self-
intersections. In the future, we would like to incorporate restrictions into
the generative model to avoid such artifacts. Overall, we believe that our
proposed approach holds great promise for advancing 3D blood vessel geometry
synthesis and contributing to the development of new clinical tools for
healthcare professionals.

###  5.0.1 Acknowledgements

This project was supported by grants from Salesforce, USA (Einstein AI 2020),
National Scientific and Technical Research Council (CONICET), Argentina (PIP
2021-2023 GI - 11220200102981CO), and Universidad Torcuato Di Tella,
Argentina.

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Supplementary Materials: VesselVAE: Recursive Variational Autoencoders for 3D
Blood Vessel Synthesis

Paula Feldman Miguel Fainstein Viviana Siless Claudio Delrieux Emmanuel
Iarussi

Table 1: Our proposed method consists of two primary components: a recursive Encoder and a recursive Decoder. All layers in each component are fully connected with leaky rectified linear unit (Leaky ReLU) activations. The Classifier is a sub-network inside the Decoder that predicts a label for each node in the input graph. Layer name | Input shape | Output shape | Layer name | Input shape | Output shape
---|---|---|---|---|---
Encoder | Decoder
fc1 | 4 | 512 | fc1 | 128 | 256
fc2 | 512 | 128 | fc_left1 | 256 | 256
right_fc1 | 128 | 512 | fc_left2 | 256 | 128
right_fc2 | 512 | 128 | fc_right1 | 256 | 256
left_fc1 | 128 | 512 | fc_right2 | 256 | 128
left_fc2 | 512 | 128 | fc2 | 256 | 128
fc3 | 256 | 128 | fc3 | 128 | 4
Sample Encoder ($f$) | Sample Decoder ($g$)
fc1 | 128 | 512 | fc1 | 128 | 256
fc2mu | 512 | 128 | fc2 | 256 | 128
fc2var | 512 | 128 | - | - | -
| Classifier
- | - | - | fc1 | 128 | 256
- | - | - | fc2 | 256 | 256
- | - | - | fc3 | 256 | 3
Figure 4: (a) Additional renders of blood vessels generated using VesselVAE.
Our approach is capable of generating diverse and intricate vessel structures,
including variations in thickness, branching patterns, and curvatures. These
outputs closely resemble real anatomical structures and demonstrate the
effectiveness of our neural network architecture and training procedures. (b)
Limitations of our method: While VesselVAE is able to generate a wide variety
of complex structures, it may occasionally struggle to reproduce realistic
data. For example, the sample at the top of the figure features an extremely
thin segment that may not occur in real blood vessels. Additionally, the mesh
reconstruction algorithm employed by our method can sometimes produce vessels
with self-intersections, which are not physically plausible in biological
systems.