1 Introduction
The quantitative analysis of medical images showing blood vessels has many important applications. For example, the analysis of coronary CT angiography (CCTA) for the detection of atherosclerotic plaque or stenosis is a clinically valuable tool for diagnosis and prognosis of coronary artery disease Leip14
. Consequently, quantitative analysis methods for CCTA have long been a topic of interest in medical image analysis. Recently, powerful but datahungry machine learning methods for CCTA analysis have been proposed
Wolt16 ; Guls16 . Training of such algorithms requires large and diverse training data sets with accurate ground truths.One commonly used technique to enlarge the amount of available training data is data augmentation, in which predetermined transformations are applied to the already available training data. However, this is likely to only provide machine learning algorithms with transformations of the available training data. An alternative to data augmentation is the synthesis of completely new data. For coronary artery stenosis detection, such data would consist of geometric models of the coronary artery lumen, along with corresponding CCTA images. To synthesize vessel geometries, modelbased methods have previously been proposed, in which tubelike vessel structures are synthesized based on a set of heuristics
Hama10 . These vessel structures could then be transformed into corresponding CT images using standard assumptions about CT image formation. Although modelbased geometry synthesis allows for a certain amount of control over the synthesized anatomies, the underlying heuristics of such methods may fail to capture the anatomical diversity of real coronary arteries. Hence, in this work we propose datadriven blood vessel geometry synthesis in contrast to previously proposed modelbased approaches. We train a model that aims to accurately capture the data distribution of real coronary arteries. For this we use a generative adversarial network (GAN) (Good14 , Fig. 1). GANs have previously been used in medical image analysis for tasks such as noise reduction in CT Wolt17 , segmentation Moes17 , visual feature attribution Baum17 , crossmodality image synthesis Wolt17b . In terms of vessel structures, GANs have mostly been used to synthesize 2D retinal vessel maps Cost17 .The goal of this work is to generate plausible 3D blood vessel shapes. In computer vision, volumetric GANs have been used to synthesize meshes or voxelizations of 3D objects such as chairs and cars
Wu16 . A drawback of this approach is that there is no guarantee that the generated object is contiguous; there may be holes or fragments. This is undesirable for vessel shapes. In addition, the size of the generated voxelization may be limited to e.g. voxels as in Wu16 . In this work, we overcome this problem by casting the 3D vessel shape into a 1D parametrization using a set of primitives Zou17. Instead of synthesizing a 3D volume, we synthesize a 4channel 1D sequence representing the vessel central axis. Previously proposed GANs for sequence synthesis have used recurrent neural networks for both the generator and discriminator
Este17. In contrast, in our GAN we use convolutional neural networks (CNNs) with large receptive fields, motivated by recent applications of CNNs to long sequences
Oord16 ; Bai18 .This work has the following contributions. First, we propose to use an efficient parametrization of blood vessels for generative models. Second, we show how a GAN can be trained to obtain a transformation between a latent space and the space of plausible coronary artery anatomies. Third, we provide a detailed analysis of synthesized vessels and the latent space from which they are sampled.
2 Data
2.1 Training Data
We use a data set consisting of 4,412 real coronary artery centerlines with radius measurements. These centerlines were semiautomatically extracted from a data set of 50 clinically acquired cardiac CCTA images using a deeplearning based tracking method
WoltThesis . Each centerline was extracted based on a manually annotated seed. Seeds were placed approximately equidistantly (10 mm) in the coronary arteries. Because multiple seeds could be placed in the same coronary artery and one centerline was extracted from each seed, the training set can potentially contain overlapping arteries.2.2 Blood Vessel Parametrization
While blood vessels are intrinsically threedimensional, we here simplify the representation of both real and synthesized vessels by using a standard 1D parametrization of the central vessel axis Scha09 ; Lesa09 . Each vessel is parametrized by a central axis or centerline consisting of ordered points: (Fig. 2). Each point is characterized by an , and coordinate in Euclidean space as well as a vessel radius (in mm), assuming a piecewise tubular vessel. This substantially reduces the complexity of the synthesis task. We use the convention that the first point of a vessel is always located at the coronary ostium.
3 Method
We propose to synthesize blood vessel models using a generative adversarial network (Fig. 1). The generative adversarial network consists of a generator network that can transform a noise vector sampled from a distribution into a 4channel 1D parametrization of a coronary artery. The discriminator network compares synthesized coronary artery geometries to real arteries sampled from and tries to predict a high score for true arteries and a low score for synthesized arteries. The discriminator and generator can optionally take an attribute vector as additional input, which contains characteristics of the coronary artery that is synthesized.
3.1 Generative Adversarial Network
The GAN consists of a generator network which is trained to synthesize blood vessels, and a discriminator network which is trained to distinguish real from synthesized samples. In the original GAN formulation Good14 , the generator and discriminator jointly optimize an objective function
(1) 
where is a sample drawn from the real data distribution and is a sample drawn from the noise distribution . The discriminator tries to maximize this objective function, while the generator tries to minimize it.
We here use a GANvariant in which the generator tries to minimize the Wasserstein distance between the real data distribution and the distribution of synthesized samples, i.e. the amount of work required to transform the synthesized sample distribution into the distribution of real samples Arjo17
. The advantage over the loss function in Eq.
1 is that stronger gradients are provided to the generator by the discriminator at each time step, even when the discriminator can easily distinguish synthetic from real samples. To meet the requirement of a 1Lipschitz discriminator function as posed in Arjo17 , we here enforce a penalty on the gradients between the two distributions Gulj17 . Hence, the full objective becomes(2) 
where is the distribution of points along straight lines between pairs of samples in and , and is the set of 1Lipschitz functions. The gradient penalty is weighted by a factor , which we set to 10.0 as in Gulj17 .
3.2 Conditional Training
Training the GAN using the objective function in Eq. 2 allows us to sample a wide variety of vessels. However, this provides very little control over the actual appearance and characteristics of the vessels. In some cases, we may be interested in synthesis of only vessels with particular characteristics. For example, given some labels in the training set, we may want to synthesize only left or right coronary arteries, or only vessels with a particular length. We here capture these features in an attribute vector that is provided to both the generator and discriminator network, in the form of an additional input channel Mirz14 . The objective then becomes
(3) 
where is the attribute vector on which and are conditioned.
3.3 Network architectures
The GAN contains two neural networks: the discriminator network and the discriminator or critic network . Both and are convolutional neural networks operating on sequences of points, i.e. 1D signals. Fig. 3 shows the CNN architectures used for and .
Instead of directly synthesizing the , , coordinates in Euclidean space, we let the generator predict for each point the displacement with respect to the previous point. This simplifies the signal that the generator has to synthesize, as displacements for , and are arranged around 0.0. If the length of a training sample is , we use zerofilling up to for all four output channels. This facilitates synthesis of vessels with varying lengths. The location of a vessel point in Euclidean space can be easily retrieved by accumulating all displacements up to point .
The generator
uses transposed, or fractionallystrided, convolutions to rapidly increase the length of a sequence from 1 to
. Each convolutional layer consists of a kernel with width 3 and stride 2. Hence, the sequence length after layer equals . The input to the generator network is an channel sequence with length 1, with being the dimensionality of the latent probability distribution from which noise is sampled. In this workis a spherical Gaussian distribution in
dimensions. In the case of conditioning on an attribute vector (Eq. 3), the generator takes additional channels, one per attribute. Intermediate layers contain 64 channels. The output layer contains 4 channels for the , and displacements, as well as the radius of the vessel along the central axis.The discriminator mirrors the generator CNN architecture and rapidly reduces the length of the sequence representation from to 1. The number of convolutional layers is equal to that in . Instead of using transposed convolutions, the discriminator CNN uses standard convolutions with width 3 and stride 2. The number of input channels for the discriminator is 4: , , and . In case of conditioning, the number of input channels is supplemented with one channel per condition. The output consists of a single scalar value, while intermediate layers have 64 channels, as in the generator.
Both the discriminator and the generator use leaky rectified linear units in all layers except for the final layer to stimulate easier gradient flow
Radf15. In both networks, the final layer uses a linear activation function.
4 Experiments and Results
The GAN was trained by alternating updates for and . Parameters of were updated according to Eq. 2 using one minibatch of 64 real samples and one minibatch of 64 synthetic samples. Parameters of were updated based on the response provided by to a minibatch of 64 synthetic samples. For each update of , we ran five updates of to make sure that the discriminator was strong enough. All parameters were optimized using two separate Adam optimizers, both with a with learning rate of King15
. The method was implemented in PyTorch and all experiments were performed on a single NVIDIA Titan Xp GPU. GANs were trained for a total of 200,000 iterations. Training took around 5 hours, while synthesis of 5,000 vessel geometries using a trained generator could be performed in less than a second.
The GAN was trained using the training set of 4,412 real samples described in Sec. 2. To test the ability of the GAN to generate long sequences, we resampled the vessels in the data set to 0.1 mm, 0.25 mm and 1.0 mm. For the lowest resolution, i.e. 1.0 mm, we trained a model with layers and a maximum sequence length . For resolution 0.25 mm we trained a deeper network at and . Finally, for the highest resolution, i.e. 0.1 mm, we trained a model with layers and a maximum sequence length . In all cases, the dimensionality of the latent space was . Fig. 10 shows a randomly sampled coronary artery geometry for each of these three models. For each of these generated vessels, we identified the closest real coronary sample based on Hausdorff distance. We find that while there are real samples that are close to the synthesized samples, there are still differences between the real and synthetic samples. This suggests that the synthesized samples do not have an exact matches in the real data distribution, and that the generator learns to synthesize new and unseen samples. More samples are provided online^{1}^{1}1More images and movies are available online at: https://tinyurl.com/y99uqk8t.
To assess whether synthesized vessels have similar properties as real vessels, we synthesized 4,412 random vessels and computed their length (in mm), volume (in mm) and tortuosity (using the distance metric Bull03 ). Fig. 14 shows how these statistics compare to those of real samples in the training distribution, showing strong overlap between real and synthesized samples.
Vessels obtained by linearly interpolating between two points
and in the latent space determined by . While traversing from () to (), the vessel takes on different orientations, shapes and lengths.4.1 Exploring the Latent Space
The generator samples coronary artery geometries from the dimensional probability distribution . Different locations in the latent space determined by correspond to different synthesized vessel geometries. To investigate the contents of this latent space, we perform an interpolation in which we linearly interpolate between two input points . Hence, we obtain several samples , where . Fig. 24 shows the generated vessels when using these interpolated points as input for . This example shows that while traversing the latent space, the generator can consecutively synthesize right coronary arteries (Figs. (a)a and (b)b), left coronary arteries (Figs. (c)c, (d)d, (e)e, (f)f, (g)g), and again right coronary arteries (Fig. (h)h and (i)i). Moreover, different points in the latent space correspond to different vessel length and tortuosity. More samples are provided online^{1}^{1}footnotemark: 1.
To investigate whether the learned latent space contains structure that generalizes to new data, we trained a GAN using vessels from 45 out of 50 patients. We then mapped the vessels belonging to the remaining 5 patients to the latent space by finding the location for which had the smallest Hausdorff distance to the vessel. Fig. (a)a shows how vessels with different lengths are mapped to different locations in the latent space . Moreover, we manually labeled vessels as belonging to the left or right coronary artery tree and identified their location in . Fig. (b)b shows how different areas of the latent space correspond to left and right coronary arteries. However, during training the generator has never been provided with artery labels, and the separation has thus been obtained in a purely unsupervised manner. Such a mapping could potentially be used for automatic labeling of extracted vessels.
4.2 Conditioning
While results in the previous section showed that the latent space contains some structure, without knowing exactly what this structure is we can not query the trained GAN for vessels with particular characteristics. To overcome this, we trained a conditional GAN in which the attribute vector contained the desired vessel length. The GAN was optimized to generate diverse and plausible samples, while meeting the requirement that the output of the generator matches the desired length. Fig. 43 shows the result of querying the trained generator with attribute vectors for 50, 100, 150, 200 or 250 mm. The different rows correspond to different locations in latent space, i.e. for each row we have fixed the latent input vector and only varied the conditional input vector to reflect the desired vessel length. More samples are provided online^{1}^{1}footnotemark: 1.
The results highlight some of the characteristics of the trained GAN. First of all, samples are quite different for different points in latent space. Shorter vessels (50, 100, 150 mm) are in this case always sampled from the left coronary artery tree. Vessels with length 200 mm are sampled from either the left coronary tree (rows 1 and 2) or the right coronary tree (row 3). The model has learned that very long vessels (250 mm) are more likely to originate at the right coronary ostium. Hence, while the location in latent space is fixed, the generator prefers to sample a right coronary artery in all three cases.
5 Discussion and Conclusion
We have presented a generative method for the synthesis of blood vessel geometries. The generative model learns a lowdimensional latent space that represents the geometry of full coronary arteries. From this lowdimensional latent space a wide range of coronary arteries can be sampled. In addition, our experiments showed how the model can be constrained to only synthesize vessels with particular characteristics.
We found that the synthesized coronary arteries shared statistical properties with the training data set, but that the model also allowed us to synthesize new coronary artery geometries by sampling from the latent space . Hence, the model efficiently captures the data present in the training set, yet is able to generate samples that are different from those that is has seen during training. Synthesized geometries could potentially be used to augment training data for discriminative machine learning methods, e.g. those studying flow in blood vessels Itu16 .
The results have shown that the generative method is able to synthesize realistic vessel models using a datadriven approach. In previously published work, modelbased methods have been proposed for vessel synthesis. The proposed method could complement such modelbased methods, by introducing realistic variations to the modelbased output. In the work by Hamarneh et al., synthesized vessel geometries were transformed into corresponding CT images using assumptions about tissue densities Hama10 . In future work, we will investigate if the current model can be extended to include such synthesis.
One of the advantages of modelbased over datadriven methods is increased control over the synthesized data. However, we found that the GAN organized the latent space into different areas depending on vessel characteristics. Furthermore, we showed that we could control characteristics of the synthesized data using a conditional GAN. We were able to synthesize vessels having different lengths, and to condition the generator network on vessel lengths. In future work, this generative model can be extended to include more vessel characteristics, such as presence and severity of stenosis, vessel location and tortuosity. This requires labeled training samples. In addition, the generator could be encouraged to pass through (user)indicated key points, thereby allowing more control over the location of the synthesized vessel.
In this work, 3D volumes were synthesized using a set of primitives, namely assuming a locally tubular model for vessels. This substantially simplifies the synthesis task, while yielding contiguous 3D volumes. A similar approach could be used for synthesis of other tubular structures, such as other vessels or airways. In future work, we will extend the set of primitives to include trees and bifurcations, e.g. using graph representations.
In conclusion, we have found that a Wasserstein generative adversarial network can be used to synthesize diverse and realistic blood vessel geometries.
Acknowledgments
This work is part of the research programme Deep Learning for Medical Image Analysis with project number P1526, which is partly financed by the Netherlands Organisation for Scientific Research (NWO) and Philips Healthcare.
We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.
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