Three-dimensional synthetic blood vessel generation using stochastic L-systems

Transcription

1 Three-dimensional synthetic blood vessel generation using stochastic L-systems Miguel A. Galarreta-Valverde a, Maysa M. G. Macedo a, Choukri Mekkaoui b and Marcel P. Jackowski* a a Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil; b Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Boston, MA, USA. ABSTRACT Segmentation of blood vessels from magnetic resonance angiography (MRA) or computed tomography angiography (CTA) images is a complex process that usually takes a lot of computational resources. Also, most vascular segmentation and detection algorithms do not work properly due to the wide architectural variability of the blood vessels. Thus, the construction of convincing synthetic vascular trees makes it possible to validate new segmentation methodologies. In this work, an extension to the traditional Lindenmayer system (L-system) that generates synthetic 3D blood vessels by adding stochastic rules and parameters to the grammar is proposed. Towards this aim, we implement a parser and a generator of L-systems whose grammars simulate natural features of real vessels such as the bifurcation angle, average length and diameter, as well as vascular anomalies, such as aneurysms and stenoses. The resulting expressions are then used to create synthetic vessel images that mimic MRA and CTA images. In addition, this methodology allows for vessel growth to be limited by arbitrary 3D surfaces, and the vessel intensity profile can be tailored to match real angiographic intensities. Keywords: L -system, stochastic L-system, synthetic vessel, 3D blood vessel. 1. DESCRIPTION OF PURPOSE This paper aims to improve previous work 1, 2 concerning the generation of synthetic blood vessels by L-systems to produce more realistic synthetic blood vessels in 3D by adding stochastic and parametric rules to the L-system grammar. We show that convincing angiographic volumetric images can be obtained using our methodology. 2. INTRODUCTION Magnetic resonance angiography (MRA) or computed tomography angiography (CTA) images allows for a detailed analysis of blood vessels. However, due to its three-dimensional nature, powerful tools are needed to support radiologists in making the right diagnoses. For this purpose, in some cases, it is necessary to segment vessels from angiography images. Although much research in recent years has focused on solving this task, it still remains an open subject. In order to validate these techniques, a comparison between the segmentation results and ground-truth data must be performed. This information is easier to obtain from a synthetic vessel image than from a physical phantom, which may not be straightforward to build, possibly requiring specific acquisition parameters. The synthesis of artificial vessels using a mathematical model offers advantages such as acquisition time, cost and the flexibility to create vessels with different growth configurations and the possibility to incorporate anomalies. Synthetic blood vessels could also help in the simulation of surgeries, decreasing the risks involved with invasive real procedures. 1 While Lindenmayer systems 3 (L-systems) were originally developed to model cellular interactions, they have been used to generate synthetic blood vessels. 1, 2 However, these synthetic vessels were confined to 2D and their properties did not resemble those measured from real MR or CT angiographic images. Extensions to 3D have been proposed, but have been specific to modeling plant growth. 4 In this work, we propose a methodology that extends the traditional L-system grammar with stochastic and parametric rules which can be used to synthesize angiographic images that simulate real CT or MR angiography datasets. * mjack@ime.usp.br, Telephone: +55 (11) Medical Imaging 2013: Image Processing, edited by Sebastien Ourselin, David R. Haynor, Proc. of SPIE Vol. 8669, 86691I 2013 SPIE CCC code: /13/$18 doi: / Proc. of SPIE Vol I-1

2 3. METHODS Our methodology is divided into three steps: (i) string generation, (ii) synthetic vessel generation and (iii) discretization. The first step requires the grammar as input, and generates a string that represents the execution of the grammar in a given iteration. This string represents a sequence of instructions, where each character has an associated action. These actions are executed by the second step in order to generate a sequence of points with their respective properties, such as vessel diameter. This set of points and their properties compose our synthetic blood vessel meta-data. The discretization step finally creates the synthetic image by connecting the previously generated points while adding volumetric information to mimic a real angiographic image. 3.1 Grammar Definition Following formal L-system definitions, 4, 5 we define a rule as Name successor. An L-system will replace symbols according to its rules. For example, the rule F af b, will generate afb in the first iteration and aafbb in the second iteration. A stochastic L-system is one that contains at least one rule with an associated probability. For example, a grammar with the next two rules, F : 0.5 af b and F : 0.5 a, will generate afb or a in the first iteration, each with a probability of 0.5. Subsequently, one could get aab, aafbb or a in the second iteration. Our rules are composed by symbols whose respective actions are: f(length, diam): goes forward by length units in the direction vector and record vessel diameter diam at that point; +(angle) and -(angle): rotate the direction vector clockwise or counterclockwise by angle degrees on the plane defined by the current direction vector and a perpendicular vector to it, /(angle) and *(angle): rotate the current plane about the direction vector by angle degrees, [ : stores the current state formed by the diameter and position on the stack, ] : restores the last saved state from the stack and allows the generation of a new branch. Finally, { and } define the limits of a vessel segment, which is used in the discretization step. 3.2 String Generation In order to produce the final string whose actions will generate a synthetic vessel tree, intermediate functions are generated for each grammatical rule using a lexical and syntactic analyzer, this sequence is shown in the Figure 1. (String generation Grammar syntactic analyzer String of instructions Python functions intermediate functions generator Figure 1: Vessel string generation sequence Lexical and syntactic analyzer The task of a lexical analyzer is to produce tokens from a string given as an input. 6 These tokens will be used by the syntactic analyzer to generate an object code for each rule. An example for a very simple rule using the grammar is F(d0) f[+(th1)f(d1)]-(th2)f(d2). The corresponding object code will be generated as a function in the Python language, as follows: 1 def F(n, d0 ) : 2 i f n>0: 3 params=c a l c u l a t e B i f u r c a t i o n ( d0 ) 4 return f + [ + +( +s t r ( params [ th1 ] )+ ) +F(n 1,params [ d1 ] )+ ] + ( + 5 s t r ( params [ th2 ] )+ ) +F(n 1,params [ d2 ] ) 6 else : return F Proc. of SPIE Vol I-2

3 In this function, the parameter n represents the number of prescribed iterations and d0 is the diameter in the main vessel segment. The function calculatebifurcation uses d0 and vessel diameter definitions from previous studies 2 to define values for parameters d1 and d2, which are the diameters of the segments after a bifurcation, and th1 and th2, which represent the corresponding rotation angles, respectively. In the case of stochastic grammars, rules contain probabilities associated with their occurrence. Hence, an additional function is created for each group of rules. As an example, for a grammar formed by the two rules F(d0):0.3 f and F(d0):0.7 ff, the additional function created is: 1 def F(n, d0 ) : 2 r=random. random ( ) 3 i f r >=0.0 and r <0.3: return F1(n, d0 ) 4 i f r >=0.3 and r <1.0: return F2(n, d0 ) Once these functions are generated, the first rule known as the axiom, is executed. It expands each of the non-terminal character by the rule that defines it. The resulting string obtained after execution of a sample rule given a fixed number of iterations is shown the table below. Iteration Resulting string for the rule F(d0) f(d0)[+(th1)f(d1)]-(th2)f(d2) 1 f[+(43.47)f][-(31.58)f] 2 f[+(53.57)f[+(39.98)f][-(34.98)f]][-(22.24)f[+(65.38)f][-(12.39)f]] 3 f[+(37.46)f[+(38.94)f[+(18.5)f][-(57.87)f]][-(36.0)f[+(37.47)f][-(37.47)f]]][- (37.47)f[+(37.47)f[+(40.32)F][-(34.64)F]][-(37.47)f[+(55.6)F][-(20.46)F]]] 3.3 Synthetic vessel generation To obtain the synthetic vessels, we use the string generated from the previous step by splitting it into tokens and applying the respective actions. The instructions and their respective parameters are: -(angle), +(angle), *(angle), /(angle) and f(length, diameter). If one of the parameters is omitted, a default value is obtained by using definitions from previous studies. 2 For example, the string, +(25) is a token that represents a clockwise rotation by 25 degrees. After that, each token is executed iteratively according to the function that defines its behavior. However, the execution of this string may exceed the physical space where the vessel is allowed to grow. Therefore, before creating each point, a verification is performed to confirm whether or not the point is contained within the allotted space. If a point happens to fall outside this space, which is represented by a bounding surface, a heuristic approach is used to determine the closest point that falls inside the surface. 3.4 Discretization To generate a synthetic angiographic image, the final image size and spacing must be known. In addition, synthetic vessel points created by the previous step need to be scaled according to the desired image size. If the generated vessel trajectories are enclosed by a box defined by points A 0 and A 1, the size of the final image is ( X, Y, Z), and the desired spacing is (E x, E y, E z ) for each point P 1 (x, y, z) the correspoding scaled point P 2 is given by: ( P1x A 0x P 2 = X P1y A 0y, Y P1z A 0z, Z ). A 1x A 0x E x A 1y A 0y E y A 1z A 0z E z Once that all the points are normalized, a vessel skeleton is initially created by using the points combined with a modification of Bresenham s algorithm for 3D. When a vessel segment is formed by sub-segments, a B-Spline interpolation function is used to obtain intermediate points, producing smoother curves while avoiding abrupt angles. Proc. of SPIE Vol I-3

4 3.4.1 Intensity distributions A spherical kernel with a sigmoid or Gaussian intensity distribution is used to convolve each one of the vessel skeletons. Note that this intensity distribution is user selectable and mimic the vessel intensity profiles of real CT and MR angiographies. The kernel size is proportional to the vessel diamater and are parametrized as follows: I p = e d2 2σ 2, with σ = radius/2, radius is the value of the radius in the vessel segment. distribution is 1 P (d) = 1 + e, γ(d β) the values for γ and β with better experimental results are γ = 12/radius and β = 0.7 radius. The equation for the sigmoid (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 2: (a)-(l) MIP projections of generated synthetic images. (g) simulation of the basic structure of kidneys, (j) simulation of aneurysms and (k) and (l) simulation of the estenosis. 4. RESULTS We have generated a synthetic angiographic image database of more than 40 images along with their respective grammars. A few of the images from this database are depicted by the maximum intensity projection (MIP) and are shown in the Figure 2. Figure 3 shows two perspectives of two datasets generated with spatial constraints, a liver, and a thigh segment, respectively. To create the constraint surfaces, a liver, 7, 8 a thigh and a femoral bone were segmented from real CT images. The time required to generate the images showed in Figure 2 using 10 iterations and without constraints was less than two minutes for each image, on a machine with a dual-core Proc. of SPIE Vol I-4

5 processor of 1596 Mhz and 8 GB of RAM. When constraints are on, the computational cost depends on the number of points of the bounding the surfaces. Thus, the vessel tree created within the liver surface containing a total of took 55 minutes for 11 iterations; and the vascular tree within the thigh surface took 10 minutes for 9 iterations. Resulting image sizes were 512x512x212, and 270x230x230 pixels, respectively. (a) (b) (c) (d) (e) (f) (g) (h) Figure 3: Different perspectives of synthetic blood vessels generated using the liver and thigh as physical constraints. (a), (b), (c) and (d) show the trajectories of the vessel segments (red) and the limiting surfaces (orange). (d), (e), (f) and (g) show two MIP projections of the resulting 3D angiographic images. 5. NEW OR BREAKTHROUGH WORK TO BE PRESENTED To the best of our knowledge, this is the first methodology that provides three-dimensional image results that simulate intensity profiles of real angiography images determined from a stochastic L-system grammar. Previous work on synthetic blood vessel generation 1 using L-systems were limited to 2D and did not produce volumetric angiographic images as a result. In addition, the grammar can be parameterized to generate organ-specific blood vessels and also allows for physical constraints to limit their growth. 6. DISCUSSION The main motivation for this work was to generate volumetric images of vessels with known properties (i.e. diameters) to help validate blood vessel segmentation algorithms. The synthetic images presented here show the flexibility of this methodology, and the diversity of the artificial vessel trees that can be generated by simply altering grammar rules and associated parameters. It is important to point out that, in order to achieve the realistic-looking vessels in this work, distinct grammars and parameters had to be used. This often required tedious and intricate handwork and parameter tuning to match the properties of real vascular trees. Hence, future work will examine the application of machine learning techniques to infer rules and parameters automatically from real angiographic images. Proc. of SPIE Vol I-5

6 7. CONCLUSIONS We have developed a methodology that generates realistic-looking synthetic blood vessels in three dimensions, taking into account arbitrary surfaces as physical constraints. The resulting volumetric images are visually convincing and the methodology is flexible enough to synthesize a wide diversity of vascular trees, and associated anomalies. We believe that this work may be of help in validating vascular segmentation algorithms using CT and MR images and may be useful as a tool for virtual surgery. REFERENCES [1] Liu, X., Liu, H., Hao, A., and Zhao, Q., Simulation of blood vessels for surgery simulators, in [2010 International Conference on Machine Vision and Human-machine Interface], , IEEE (2010). [2] Zamir, M., Arterial branching within the confines of fractal L-system formalism, The Journal of general physiology 118(3), 267 (2001). [3] Lindenmayer, A., Mathematical models for cellular interactions in development I. filaments with one-sided inputs, Journal of Theoretical Biology 18(3), (1968). [4] Qi, H., Qiu, R., and Jia, J., L-system based interactive and lightweight web3d tree modeling, in [Proceedings of the 10th International Conference on Virtual Reality Continuum and Its Applications in Industry], , ACM (2011). [5] Samal, A., Peterson, B., and Holliday, D., Recognition of plants using a stochastic L-system model, Journal of Electronic Imaging 11, 50 (2002). [6] Xiao, X. and Xu, Y., The design and implementation of c-like language interpreter, in [Intelligence Information Processing and Trusted Computing (IPTC), nd International Symposium on], , IEEE (2011). [7] Heimann, T. and et al., I. W., Comparison and Evaluation of Methods for Liver Segmentation from CT datasets, IEEE Transactions on Medical Imaging 28(8), (2009). [8] Dário, O., Raul, F., and Mauro, C., Segmentation of liver, its vessels and lesions from CT images for surgical planning, BioMedical Engineering OnLine 10(1), 1 23 (2011). Proc. of SPIE Vol I-6