imaging of polymer cast of normal microvasculature, showing simple, organized arrangement of arterioles, capillaries, and venules. Vasa vasorum of rat carotid.
CHARACTERIZATION OF TUMOR ANGIOGENESIS USING FRACTAL MEASURES Loretta Ichim1,2, Radu Dobrescu1 1
Politehnica University of Bucharest, Faculty of Automatic Control and Computers, 313 Spl. IndependenŃei, Bucharest, Romania 2 “Ştefan S. Nicolau” Institute of Virology, Romanian Academy 285 Şos. Mihai Bravu, Bucharest, Romania
Abstract: Tumors vascular networks are different from normal vascular networks, but the mechanisms underlying these differences are not known. Underlying these mechanisms may be key to improving the efficacy of the treatment of tumor. We studied possibility to apply two types of fractal measure: fractal dimension and succolarity for characterizing medical images. We find that fractal dimension value to the normal vasculature is smaller than the results for the tumor vasculature. Moreover, for applying succolarity the results do not vary considerable with the direction in normal vasculature, while for tumor vasculature the curves differs significantly. In conclusion, the results obtained show that the fractal measure is an important tool for analyzing medical images. Keywords: angiogenesis, fractal dimension, succolarity, tumor vasculature,
1. INTRODUCTION Tumor evolution is a most complex process involving many different phenomena. Understanding the dynamics of cancer growth is one of the great challenges of modern science. Angiogenesis is the process, by which tumors induce blood vessels from the host tissue to sprout capillary tips, which migrate towards and ultimately penetrate the tumor, providing it with a circulating blood supply and, therefore, an almost limitless source of nutrients. The vascular growth phase, which follows angiogenesis, is marked by a rapid increase in cell proliferation and is usually accompanied by an increase in the pressure at the centre of the tumor. This may be sufficient to occlude blood vessels and, thereby, to restrict drug delivery to the tumor. Tumor vasculature has long been known to be more chaotic in appearance than normal vasculature. Now that angiogenesis has been identified as a critical event in tumor progression and as a potential target for treatment (Folkman, 1995) , there is an increasing need to understand the origins and consequences of
the abnormal vascular architectures found in tumors (Baish, et al., 2000). The formation of new vessels, or angiogenesis, is a key process in the growth and metastasis of tumors. If the tumor is unable to generate its own blood supply it will not increase in size beyond a diameter of approximately a millimeter. While our understanding of the process of angiogenesis has been greatly enhanced on the molecular level over the last two decades (Jain, et al., 1997), there has been little progress in understanding the angiogenic process on the organ and tissue levels. In particular, it is unclear why tumor vascular networks look so different from normal vascular networks although presumably the same growth factors and inhibitors are involved in their formation. Little is known about the determinants of vascular network formation and architecture – the way in which vessels are arranged and interconnected. This knowledge is important in understanding the differences between physiologic and pathophysiologic angiogenesis, and in designing interventions that modify angiogenesis. The main goal of this research is to characterize medical images of growing tumor vasculature using
two types of fractal measures: fractal dimension and succolarity, for better understanding of vascular morphology.
2. RELATED WORK ON TUMOR ANGIOGENESIS MODELING Despite substantial progress in characterizing the mechanisms that control tumor angiogenesis, the dynamical correlation between tumor growth and neovascularization is still not fully understood. In tumors, vessels may form via different processes: sprouting, intussusception, co-option, vascular looping and cancer stem-like cell differentiation (Jin, 2012). It is difficult to discriminate between these various processes by analysis of fixed tumor material. In order to contribute at a better description of the biological process, several continuum or discrete models for tumor angiogenesis were used. In continuum models, only the distribution of endothelial cells is considered while vascular networks are not included. Chaplain et al. (1996, 1997) then presented two- and three-dimensional models of tumor angiogenesis using a ‘hybrid’ by combining both discrete and continuum methods, using for models partial differential equations for the concentration of the tumor angiogenesis factor (TAF). Alarcon et al. (2003) then developed a mathematical model which showed the influence of blood flow and red blood cell heterogeneity on tumor growth and angiogenesis. Furthermore, Zheng et al. (2005) have computed tumor angiogenesis using a formulation similar to Chaplain, but they modeled the nutrient distribution assuming a spatial variation of TAF within viable and necrotic tumor regions. In this context, we note also Gazit et al. (1995) as one of the first who employed fractal theory to compute the vessel networks that surround a tumor and computed the hemodynamics within these vessel. The fractal approach was then sustained by the works of Grizzi et al. (2005), Dobrescu (2009), Shim et al. (2010), Jurczyszyn et al. (2012)
decreasing orders of magnitude, and so the form of their component parts is similar to that of the whole: this property is called self-similarity. Unlike geometrical self-similarity, which only concerns mathematical fractal objects in which every smaller piece is an exact duplicate of the whole (e.g. Koch's snowflake curve, Sierpinski's triangle and Menger's sponge), statistical self-similarity concerns all complex anatomical systems, including tumor vasculature.
3. MATERIALS AND METHODS The real images used here were: fig. 1 a) representing microscopic imaging of normal and angiogenic blood vessels. Scanning electron microscopic (SEM) imaging of polymer cast of normal microvasculature, showing simple, organized arrangement of arterioles, capillaries, and venules. Vasa vasorum of rat carotid sinus; and (b) SEM image of cast of tumor microvasculature, showing disorganization and lack of conventional hierarchy of blood vessels. Arterioles, capillaries, and venules are not identifiable as such. The images characterize a xenograft of human tumor in nude mouse (McDonald, et al., 2003).
a)
2. GEOMETRICAL PROPERTIES OF A VASCULAR NETWORK The human vascular system can be geometrically depicted as a complex fractal network of vessels that irregularly branch with a systematic reduction in their length and diameter (Grizzi, et al., 2005). The vascularization of tumors is a dynamic process. Vascular networks are fractal. Fractal objects are mainly characterized by four properties: a) the irregularity of their shape; b) the self-similarity of their structure; c) their non-integer or fractal dimension; and d) scaling, which means that the measured properties depends on the scale at which they are measured (Grizzi, et al., 2005). One particular feature of fractal objects is that the schemas defining them are continuously repeated at
b) Fig.1. Microscopic imaging of: a) normal and b) angiogenic blood vessels The input images used for this analysis are grayscales images and have 380 x 340 pixels. 2.1. Fractal dimension approach Several comprehensive reviews of the use of fractal dimensions in pathology have recently appeared in
the literature. There is a growing literature that shows fractals to be useful measure of the pathologies of the tumor border, cellular/nuclear morphology, and vascular architecture (Baish, et al., 2000). Fractal dimension has been well studied; a great number of approaches have been presented to extract it from images. The method used to calculate the fractal dimension for the vascular images was the box-counting algorithm (Block, et al., 1990) 2.2. Succolarity approach Succolarity analysis is a complimentary approach to fractal and lacunarity analysis. It measures the percolation or draining capacity of the image, i.e., how much a given fluid can flow though this image (Mandelbrot, 1982). Melo and Conci (Melo, et al., 20011) suggested that this type of analysis is very useful when the input texture has direction and/or flow information associated with it e.g. drainage patterns. Succolar fractals include the filaments that would have allowed percolation, i.e., the amount of interconnected pixels in drainage textures. These textures consist of two types of pixels, i.e. empty spaces and impenetrable mass i.e. drainage. The first approach to calculate this fractal measure was provided by Melo and Conci using approach similar to the box counting adapted to the notions of succolarity The degree of percolation of an image (how much a given fluid can flow through this image) can be measured through Succolarity, a fractal parameter. The succolarity of a binary image is defined as: ,
∑ , ∑ ,
where ‘dir’ denotes direction; BS(n) where n is the number of possible divisions of a binary image in boxes. The occupation percentage (OP) is defined as, for each box size, k, the sum of the multiplications of the (OP(BS(k)), where k is a number from 1 to n, by the pressure PR(BS(k), pc), where pc is the position on x or y of the centroid of the box on the scale of pressure) applied to the box are calculated. Therefore for any binary decomposed images of f(x, y), the succolarity can be obtained (Melo, et al., 2008) .
4. IMAGE PROCESSING The pre-processing is useful to increase the quality of the images emphasizing some characteristics that are interesting to a particular method. The both methods implemented here require binary images as input. The medical images of microvasculature used here employ a supervised approach to generate the binary version of the images. This was done to guarantee that the binary images resemble the particular features on the original image.
For the fractal dimension, after threshold, the image edges are also detected before being used in the computing method. Mathematical morphology was used to find the contour of these images.
5. RESULTS AND DISCUSSION Figure 2 (a) represent de binary version of image in figure 1 (a), after the threshold step by 105 and the boundaries of the binary image were obtained. Figure 2 (b) represent de binary version of image in figure 2 (b), after the threshold step by 151 and the boundaries of the binary image were obtained. The method to obtain the contour used is the internal contour of mathematical morphology, named morphological gradient.
a)
b)
Fig. 2. Edges of images in fig. 1 obtained through mathematical morphology: a) normal and b) angiogenic blood vessels Using a software package we computed fractal dimension values (Df) based on box-counting algorithm.
Fig. 3. Results of the Fractal dimension of the image of fig. 2(a) For the image in figure 2 (a) the observed result in fig 3 was Df = 1.6527 corresponds to the analysis of normal image.
obtained when the original images was analyzed horizontally: left to right (l2r) and right to left (r2l) and vertically: top to bottom (t2b) and bottom to top (b2t). The next images in fig. 6 show the intermediate aspects generated during the running of the method proposed to calculate the succolarity of the input images (tumor vasculature). t2b
b2t
l2r
r2l
Fig. 4. Results of the Fractal dimension of the image of fig. 2(b) The results observed for the image of the figure 2 (b) presents Df = 1.7429, as can be seen in figure 4 and corresponds to microvasculature tumor. The value of fractal dimension to the normal microvasculature is smaller than the results for the tumor microvasculature. This is because figure 2 (b) presents more complex edges distributed through the image, causing a bigger occurrence of occupied boxes for this image. To demonstrate that succolarity, like the other fractal measure, is a powerful method to characterize real images, an application of the succolarity to vascular diagnosis. Fig. 6. Intermediate image for the direction t2b, b2t, l2r, r2l (tumor vasculature). t2b
b2t
The numerical values of succolarity in the plots are represented in tables 1 and 2 to better illustrate the difference on the results. Table 1 Numerical results of the succolarity of image with normal vasculature
l2r
r2l
Dividing factor d 8 4 2
Succolarity (σ) b2t 0.282 0.2807 0.2753
t2b 0.2844 0.2822 0.2792
l2r 0.2932 0.2929 0.2946
r2l 0.2925 0.2914 0.2915
Table 2 Numerical results of the succolarity of image with tumor vasculature
Fig. 5. Intermediate image for the direction t2b, b2t, l2r, r2l (normal vasculature).
Fig. 5, show the intermediate images generated during the executing of the method proposed to calculate the succolarity of the input images (normal vasculature). More exactly, four images were
Dividing factor d 124 62 31 4 2
Succolarity (σ) b2t 0.4101 0.4101 0.4101 0.4078 0.3969
t2b 0.4065 0.4065 0.4065 0.4066 0.3993
l2r 0.3699 0.3699 0.3699 0.3733 0.381
r2l 0.3748 0.3748 0.3747 0.3763 0.3814
These results are shown in the log × log plots of the succolarity in figs. 7 and 8. 3,8
Succolarity
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range of the morphological variability of vasculature that can be produced in nature, thus increasing its diagnostic importance in cancer research. As future work, we intend to collect more real medical images for testing the potential of these proposed techniques.
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AKNOWLEDGEMENTS 3,2
This work was supported by FP7 project REGPOT 2010 - 1, ERRIC – Empowering Romanian Research on Intelligent Information Technologies.
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REFERENCES Fig. 7. Results of succolarity of the image with normal vasculature 3,78 3,76 3,74
Succolarity
3,72 3,70 3,68 3,66 3,64 3,62 3,60 3,58 0
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Fig. 8. Results of succolarity of the image with tumor vasculature Figure 7 demonstrate that, the results do not vary considerable with the direction, while in figure 8 the direction used on the evaluation presents great impact on the results. 6. CONCLUSIONS The fractal dimension is already largely used to characterize patterns on a great number and kinds of medical applications. The use on this work of the box-counting method is only to confirm the idea that fractal dimension is a good approach for texture analysis. The fractal dimension values of tumor vasculature obtained are consistent with those presented by other researchers in their studies. Succolarity can be used as a new feature, not only for the type of images applied here, but to generic images, that present some information associated with direction or flow. The results of the experiments on medical images show that the method combining fractal dimension and succolarity measures is very useful to integrate other characteristics on pattern recognition processes. The other advantage of the method is to be simple, easy and fast to be calculated. The potential broad applicability of the proposed quantitative index makes it possible to explore the
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