Blood Vessel Geometry Synthesis using Generative Adversarial Networks
BBlood Vessel Geometry Synthesis using GenerativeAdversarial Networks
Jelmer M. Wolterink ∗ Image Sciences InstituteUMC UtrechtUtrecht, The Netherlands
Tim Leiner
Department of RadiologyUMC UtrechtUtrecht, The Netherlands
Ivana Išgum
Image Sciences InstituteUMC UtrechtUtrecht, The Netherlands
Abstract
Computationally synthesized blood vessels can be used for training and evaluationof medical image analysis applications. We propose a deep generative model tosynthesize blood vessel geometries, with an application to coronary arteries incardiac CT angiography (CCTA).In the proposed method, a Wasserstein generative adversarial network (GAN)consisting of a generator and a discriminator network is trained. While the generatortries to synthesize realistic blood vessel geometries, the discriminator tries todistinguish synthesized geometries from those of real blood vessels. Both real andsynthesized blood vessel geometries are parametrized as 1D signals based on thecentral vessel axis. The generator can optionally be provided with an attributevector to synthesize vessels with particular characteristics.The GAN was optimized using a reference database with parametrizations of4,412 real coronary artery geometries extracted from CCTA scans. After training,plausible coronary artery geometries could be synthesized based on random vectorssampled from a latent space. A qualitative analysis showed strong similaritiesbetween real and synthesized coronary arteries. A detailed analysis of the latentspace showed that the diversity present in coronary artery anatomy was accuratelycaptured by the generator.Results show that Wasserstein generative adversarial networks can be used tosynthesize blood vessel geometries.
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 atheroscleroticplaque or stenosis is a clinically valuable tool for diagnosis and prognosis of coronary artery disease[1]. Consequently, quantitative analysis methods for CCTA have long been a topic of interest inmedical image analysis. Recently, powerful but data-hungry machine learning methods for CCTAanalysis have been proposed [2, 3]. Training of such algorithms requires large and diverse trainingdata 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 trainingdata. An alternative to data augmentation is the synthesis of completely new data. For coronaryartery stenosis detection, such data would consist of geometric models of the coronary artery lumen,along with corresponding CCTA images. To synthesize vessel geometries, model-based methods ∗ [email protected] a r X i v : . [ c s . C V ] A p r igure 1: Overview of the proposed method for coronary artery synthesis. The generative adversarialnetwork consists of a generator network G that transforms noise vectors z from a latent probabilitydistribution p z into a 4-channel 1D parametrization of a coronary artery. These representations arecompared with those of real arteries in a discriminator network D , which tries to predict a high scorefor real arteries and a low score for synthesized arteries. In addition, the generator an discriminatorcan be conditioned on an attribute vector y containing information about the real or synthetic samples.have previously been proposed, in which tube-like vessel structures are synthesized based on a set ofheuristics [4]. These vessel structures could then be transformed into corresponding CT images usingstandard assumptions about CT image formation. Although model-based geometry synthesis allowsfor a certain amount of control over the synthesized anatomies, the underlying heuristics of suchmethods may fail to capture the anatomical diversity of real coronary arteries. Hence, in this work wepropose data-driven blood vessel geometry synthesis in contrast to previously proposed model-based approaches. We train a model that aims to accurately capture the data distribution of real coronaryarteries. For this we use a generative adversarial network (GAN) ([5], Fig. 1). GANs have previouslybeen used in medical image analysis for tasks such as noise reduction in CT [6], segmentation [7],visual feature attribution [8], cross-modality image synthesis [9]. In terms of vessel structures, GANshave mostly been used to synthesize 2D retinal vessel maps [10].The goal of this work is to generate plausible 3D blood vessel shapes. In computer vision, volumetricGANs have been used to synthesize meshes or voxelizations of 3D objects such as chairs and cars [11].A drawback of this approach is that there is no guarantee that the generated object is contiguous; theremay be holes or fragments. This is undesirable for vessel shapes. In addition, the size of the generatedvoxelization may be limited to e.g. × × voxels as in [11]. In this work, we overcome thisproblem by casting the 3D vessel shape into a 1D parametrization using a set of primitives [12].Instead of synthesizing a 3D volume, we synthesize a 4-channel 1D sequence representing the vesselcentral axis. Previously proposed GANs for sequence synthesis have used recurrent neural networksfor both the generator and discriminator [13]. In contrast, in our GAN we use convolutional neuralnetworks (CNNs) with large receptive fields, motivated by recent applications of CNNs to longsequences [14, 15].This work has the following contributions. First, we propose to use an efficient parametrizationof blood vessels for generative models. Second, we show how a GAN can be trained to obtaina transformation between a latent space p z and the space of plausible coronary artery anatomies.Third, we provide a detailed analysis of synthesized vessels and the latent space from which they aresampled. We use a data set consisting of 4,412 real coronary artery centerlines with radius measurements.These centerlines were semi-automatically extracted from a data set of 50 clinically acquired cardiacCCTA images using a deep-learning based tracking method [16]. Each centerline was extractedbased on a manually annotated seed. Seeds were placed approximately equidistantly (10 mm) inthe coronary arteries. Because multiple seeds could be placed in the same coronary artery and onecenterline was extracted from each seed, the training set can potentially contain overlapping arteries.2igure 2: Blood vessels are parametrized according to their central axis, which is represented asa sequence of points v , v , . . . , v N , where each point v i is a 4-channel vector of x , y and z coordinates and a corresponding radius size r , assuming a piece-wise tubular vessel. While blood vessels are intrinsically three-dimensional, we here simplify the representation ofboth real and synthesized vessels by using a standard 1D parametrization of the central vessel axis[17, 18]. Each vessel V is parametrized by a central axis or centerline consisting of ordered points: V = { v , v , . . . , v N } (Fig. 2). Each point v i ∈ V is characterized by an x , y and z coordinate inEuclidean space as well as a vessel radius r (in mm), assuming a piece-wise tubular vessel. Thissubstantially reduces the complexity of the synthesis task. We use the convention that the first point v of a vessel is always located at the coronary ostium. We propose to synthesize blood vessel models using a generative adversarial network (Fig. 1). Thegenerative adversarial network consists of a generator network G that can transform a noise vector z sampled from a distribution p z into a 4-channel 1D parametrization G ( z ) of a coronary artery. Thediscriminator network D compares synthesized coronary artery geometries to real arteries sampledfrom p data 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 y as additional input, whichcontains characteristics of the coronary artery that is synthesized. The GAN consists of a generator network G which is trained to synthesize blood vessels, and adiscriminator network D which is trained to distinguish real from synthesized samples. In the originalGAN formulation [5], the generator and discriminator jointly optimize an objective function min G max D V ( D ) ( D, G ) = E x ∼ p data [log D ( x )] + E z ∼ p z [log (1 − D ( G ( z )))] , (1)where x is a sample drawn from the real data distribution p data and z is a sample drawn from thenoise distribution p z . The discriminator tries to maximize this objective function, while the generatortries to minimize it.We here use a GAN-variant in which the generator tries to minimize the Wasserstein distance betweenthe real data distribution p data and the distribution of synthesized samples, i.e. the amount of workrequired to transform the synthesized sample distribution into the distribution of real samples [19].The advantage over the loss function in Eq. 1 is that stronger gradients are provided to the generatorby the discriminator at each time step, even when the discriminator can easily distinguish syntheticfrom real samples. To meet the requirement of a 1-Lipschitz discriminator function as posed in[19], we here enforce a penalty on the gradients between the two distributions [20]. Hence, the fullobjective becomes min G max D ∈D V ( D ) ( D, G ) = E x ∼ p data [ D ( x )] − E z ∼ p z [ D ( G ( z ))] − λ E ˆ x ∼ p ˆ x [( (cid:107)∇ ˆ x D (ˆ x ) (cid:107) − ] , (2)3igure 3: CNN architectures of the generator G and discriminator D . The generator uses transposedconvolutions in each layer to increase the length of the sequence. The input layer consists of aunit-length sequence with m channels, intermediate layers have 64 channels, and the final layer hasfour channels which represent x , y , z and r for each vessel point v i . The discriminator’s architecturemirrors that of the generator, with strided convolutions to compress the sequence into a single scalarprediction. The sequence length N and the number of layers l relate to each other as N = 2 l − .The generator and discriminator optionally take a c -channel attribute vector y as input.where p ˆ x is the distribution of points along straight lines between pairs of samples in p data and p z ,and D is the set of 1-Lipschitz functions. The gradient penalty is weighted by a factor λ , which weset to 10.0 as in [20]. 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. Forexample, given some labels in the training set, we may want to synthesize only left or right coronaryarteries, or only vessels with a particular length. We here capture these features in an attribute vectorthat is provided to both the generator and discriminator network, in the form of an additional inputchannel [21]. The objective then becomes min G max D ∈D V ( D ) ( D, G ) = E x ∼ p data [ D ( x | y )] − E z ∼ p z [ D ( G ( z | y ) | y )] − λ E ˆ x ∼ p ˆ x [( (cid:107)∇ ˆ x D (ˆ x ) (cid:107) − ] , (3)where y is the attribute vector on which G and D are conditioned. The GAN contains two neural networks: the discriminator network G and the discriminator or criticnetwork D . Both G and D are convolutional neural networks operating on sequences of points, i.e.1D signals. Fig. 3 shows the CNN architectures used for G and D .Instead of directly synthesizing the x , y , z -coordinates in Euclidean space, we let the generatorpredict for each point the displacement with respect to the previous point. This simplifies the signalthat the generator has to synthesize, as displacements for x , y and z are arranged around 0.0. If thelength of a training sample is < N , we use zero-filling up to N for all four output channels. Thisfacilitates synthesis of vessels with varying lengths. The location of a vessel point v i in Euclideanspace can be easily retrieved by accumulating all displacements up to point v i .The generator G uses transposed, or fractionally-strided, convolutions to rapidly increase the lengthof a sequence from 1 to N . Each convolutional layer consists of a kernel with width 3 and stride2. Hence, the sequence length N after layer l equals l − . The input to the generator networkis an m -channel sequence with length 1, with m being the dimensionality of the latent probabilitydistribution p z from which noise is sampled. In this work p z is a spherical Gaussian distributionin m dimensions. In the case of conditioning on an attribute vector (Eq. 3), the generator G takes c additional channels, one per attribute. Intermediate layers contain 64 channels. The output layercontains 4 channels for the x , y and z -displacements, as well as the radius of the vessel along thecentral axis.The discriminator mirrors the generator CNN architecture and rapidly reduces the length of thesequence representation from N to 1. The number of convolutional layers l is equal to that in G .4 a) G ( z ) , l = 7 , N = 255 (b) G ( z ) , l = 9 , N = 1023 (c) G ( z ) , l = 11 , N = 4095 (d) Closest to G ( z ) (e) Closest to G ( z ) (f) Closest to G ( z ) Figure 4: The top row shows volume renderings of coronary artery geometries (in red) synthesizedat random points z , z and z sampled from the distribution p z . The bottom row shows, for eachsynthesized artery, the closest real coronary artery geometry in the training data set in terms ofHausdorff distance. Ascending aorta (in white) shown for visualization purposes.Instead of using transposed convolutions, the discriminator CNN uses standard convolutions withwidth 3 and stride 2. The number of input channels for the discriminator is 4: x , y , z and r . Incase 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 thegenerator.Both the discriminator and the generator use leaky rectified linear units in all layers except for thefinal layer to stimulate easier gradient flow [22]. In both networks, the final layer uses a linearactivation function. The GAN was trained by alternating updates for D and G . Parameters of D were updated accordingto Eq. 2 using one mini-batch of 64 real samples and one mini-batch of 64 synthetic samples.Parameters of G were updated based on the response provided by D to a mini-batch of 64 syntheticsamples. For each update of G , we ran five updates of D to make sure that the discriminator wasstrong enough. All parameters were optimized using two separate Adam optimizers, both with a withlearning rate of α = 0 . [23]. The method was implemented in PyTorch and all experiments wereperformed 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 generatorcould 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 theability of the GAN to generate long sequences, we resampled the vessels in the data set to 0.1 mm,5 a) (b)(c) Figure 5: Histograms showing the distribution of vessel length (in mm), volume (in mm ) andtortuosity (using the distance metric [24]) among real and synthesized vessels. (a) α = 0 (b) α = (c) α = (d) α = (e) α = (f) α = (g) α = (h) α = (i) α = 1 Figure 6: Vessels obtained by linearly interpolating between two points z and z in the latent spacedetermined by p z . While traversing from z ( α = 0 ) to z ( α = 1 ), the vessel takes on differentorientations, shapes and lengths.0.25 mm and 1.0 mm. For the lowest resolution, i.e. 1.0 mm, we trained a model with l = 7 layersand a maximum sequence length N = 255 . For resolution 0.25 mm we trained a deeper networkat l = 9 and N = 1023 . Finally, for the highest resolution, i.e. 0.1 mm, we trained a model with l = 11 layers and a maximum sequence length N = 4095 . In all cases, the dimensionality of thelatent space p z was m = 3 . Fig. 4 shows a randomly sampled coronary artery geometry for eachof these three models. For each of these generated vessels, we identified the closest real coronarysample based on Hausdorff distance. We find that while there are real samples that are close to thesynthesized samples, there are still differences between the real and synthetic samples. This suggeststhat the synthesized samples do not have an exact matches in the real data distribution, and that thegenerator learns to synthesize new and unseen samples. More samples are provided online .To assess whether synthesized vessels have similar properties as real vessels, we synthesized 4,412random vessels and computed their length (in mm), volume (in mm ) and tortuosity (using thedistance metric [24]). Fig. 5 shows how these statistics compare to those of real samples in thetraining distribution, showing strong overlap between real and synthesized samples. The generator G samples coronary artery geometries from the m -dimensional probability distribution p z . Different locations in the latent space determined by p z correspond to different synthesized vessel More images and movies are available online at: https://tinyurl.com/y99uqk8t a) Lengths of unseen vessels mapped to p z (b) Labels of unseen vessels mapped to p z Figure 7: Unseen vessel samples mapped to the latent space obtained by GAN training. (a) Markercolor indicates length of vessels. (b) Points have been manually labeled as belonging to the left orright coronary artery tree. Marker color indicates whether the vessel belongs to the left coronaryartery tree (LCA) or the right coronary artery tree (RCA). Without supervision, the GAN has assignedparts of the latent space to represent either short or long, or left or right coronary arteries.geometries. To investigate the contents of this latent space, we perform an interpolation in whichwe linearly interpolate between two input points z , z ∈ p z . Hence, we obtain several samples G (ˆ z ) , where ˆ z = (1 − α ) z + α z , ≤ α ≤ . Fig. 6 shows the generated vessels when using theseinterpolated points as input for G . This example shows that while traversing the latent space, thegenerator can consecutively synthesize right coronary arteries (Figs. 6a and 6b), left coronary arteries(Figs. 6c, 6d, 6e, 6f, 6g), and again right coronary arteries (Fig. 6h and 6i). Moreover, different pointsin the latent space correspond to different vessel length and tortuosity. More samples are providedonline .To investigate whether the learned latent space contains structure that generalizes to new data, wetrained a GAN using vessels from 45 out of 50 patients. We then mapped the vessels belonging to theremaining 5 patients to the latent space by finding the location z for which G ( z ) had the smallestHausdorff distance to the vessel. Fig. 7a shows how vessels with different lengths are mapped todifferent locations in the latent space p z . Moreover, we manually labeled vessels as belonging tothe left or right coronary artery tree and identified their location in p z . Fig. 7b shows how differentareas of the latent space correspond to left and right coronary arteries. However, during training thegenerator has never been provided with artery labels, and the separation has thus been obtained in apurely unsupervised manner. Such a mapping could potentially be used for automatic labeling ofextracted vessels. While results in the previous section showed that the latent space p z contains some structure, withoutknowing exactly what this structure is we can not query the trained GAN for vessels with particularcharacteristics. To overcome this, we trained a conditional GAN in which the attribute vector y contained the desired vessel length. The GAN was optimized to generate diverse and plausiblesamples, while meeting the requirement that the output of the generator matches the desired length.Fig. 8 shows the result of querying the trained generator G 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 rowwe have fixed the latent input vector z and only varied the conditional input vector y to reflect thedesired vessel length. More samples are provided online .The results highlight some of the characteristics of the trained GAN. First of all, samples are quitedifferent for different points in latent space. Shorter vessels (50, 100, 150 mm) are in this case alwayssampled from the left coronary artery tree. Vessels with length 200 mm are sampled from either theleft coronary tree (rows 1 and 2) or the right coronary tree (row 3). The model has learned that verylong vessels (250 mm) are more likely to originate at the right coronary ostium. Hence, while the7 a) G ( z ) , 50 mm (b) G ( z ) , 100 mm (c) G ( z ) , 150 mm (d) G ( z ) , 200 mm (e) G ( z ) , 250 mm(f) G ( z ) , 50 mm (g) G ( z ) , 100 mm (h) G ( z ) , 150 mm (i) G ( z ) , 200 mm (j) G ( z ) , 250 mm(k) G ( z ) , 50 mm (l) G ( z ) , 100 mm (m) G ( z ) , 150 mm (n) G ( z ) , 200 mm (o) G ( z ) , 250 mm Figure 8: Vessels synthesized with different lengths using a conditional GAN. The rows correspondto different fixed points z , z , z in the latent space determined by p z . The columns correspondto different conditional inputs y to the generator representing the desired length. Depending on thelength and the latent vector, the generator synthesizes left or right coronary arteries.location in latent space is fixed, the generator prefers to sample a right coronary artery in all threecases. We have presented a generative method for the synthesis of blood vessel geometries. The generativemodel learns a low-dimensional latent space that represents the geometry of full coronary arteries.From this low-dimensional 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 particularcharacteristics.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 fromthe latent space p z . Hence, the model efficiently captures the data present in the training set, yet isable to generate samples that are different from those that is has seen during training. Synthesizedgeometries could potentially be used to augment training data for discriminative machine learningmethods, e.g. those studying flow in blood vessels [25].The results have shown that the generative method is able to synthesize realistic vessel models usinga data-driven approach. In previously published work, model-based methods have been proposed forvessel synthesis. The proposed method could complement such model-based methods, by introducingrealistic variations to the model-based output. In the work by Hamarneh et al., synthesized vesselgeometries were transformed into corresponding CT images using assumptions about tissue densities[4]. In future work, we will investigate if the current model can be extended to include such synthesis.8ne of the advantages of model-based over data-driven methods is increased control over thesynthesized data. However, we found that the GAN organized the latent space into different areasdepending on vessel characteristics. Furthermore, we showed that we could control characteristics ofthe synthesized data using a conditional GAN. We were able to synthesize vessels having differentlengths, and to condition the generator network on vessel lengths. In future work, this generativemodel can be extended to include more vessel characteristics, such as presence and severity ofstenosis, vessel location and tortuosity. This requires labeled training samples. In addition, thegenerator could be encouraged to pass through (user-)indicated key points, thereby allowing morecontrol over the location of the synthesized vessel.In this work, 3D volumes were synthesized using a set of primitives, namely assuming a locallytubular model for vessels. This substantially simplifies the synthesis task, while yielding contiguous3D volumes. A similar approach could be used for synthesis of other tubular structures, such asother vessels or airways. In future work, we will extend the set of primitives to include trees andbifurcations, e.g. using graph representations.In conclusion, we have found that a Wasserstein generative adversarial network can be used tosynthesize diverse and realistic blood vessel geometries. Acknowledgments
This work is part of the research programme Deep Learning for Medical Image Analysis with projectnumber P15-26, 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 XpGPU used for this research.
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