AUTOMATIC QUANTIFICATION OF IN VITRO ENDOTHELIAL CELL NETWORKS USING MATHEMATICAL MORPHOLOGY Jes´us Angulo Centre de Morphologie Math´ematique, Ecole des Mines de Paris, France
Sabine Matou Department of Biological Sciences, The Manchester Metropolitan University, UK
[email protected] ;http://cmm.ensmp.fr/∼angulo
[email protected]
ABSTRACT Experiments of in vitro angiogenesis are important tools for studying both the mechanisms of formation of new blood vessels and the potential development of therapeutic strategies to modulate neovascularisation (screening of new pharmacological molecules). One of the most frequently used angiogenesis assays is the culture of endothelial cells on a reconstituted basement membrane named Matrigel, since the cells constitute a capillary-like network which can be quantified by image analysis. In this paper, a global, robust and fully automated methodology is proposed to segment and quantify in vitro endothelial cell networks from greyscale images using operators from mathematical morphology. Experimental results allow us to draw a discussion of the pertinence of the alternative morphological parameters to evaluate the characteristics of the cell culture. KEY WORDS biomedical image processing, endothelial cell angiogenesis, mathematical morphology, granulometry, top-hat transformation, watershed, morphological wavefront
1 Introduction Angiogenesis, formation of new blood vessels from preexisting ones, is a complex process involved in embryonic implantation. This process also contributes to the dissemination of solid tumor growth via metastasis formation or to the development of angioproliferative diseases (psoriasis, diabetic retinopathy, rheumatoid arthritis) [7]. Endothelial cells play a central role in angiogenesis process. Under an angiogenic factor effect, activated endothelial cells degrade the underlying basement membrane, migrate into surrounding vascular tissue toward the angiogenic factor and proliferate to increase the length of neo-vessel formed after cell differentiation. Many in vitro and in vivo assays have been set up for the study of angiogenesis. In vitro angiogenesis assays make it possible to interpret results in a short time at lower cost and system complexity than in vivo assays [2]. Quantification of angiogenesis assays has been proposed in the literature according to different approaches to compare the morphological and geometric features of several types of cells and types of supports [4] [8]. Among in vitro angiogenesis assays the most frequently used, the re-
constituted basement membrane (Matrigel), is the fast angiogenesis assay that could be accompanied by an accurate and fast quantification analysis. In the contact with Matrigel, endothelial cells undergo morphological differentiation: they assume bipolar cytoplasmic elongations and spindle shapes, align and associate with one another to form tubular-like structures; they can also clump together into cell aggregates and branch points characterized by interconnecting segments. At the end of this cell differentiation step (about 18-24 hours) and according to experimental conditions, these tubular-like structures may organize into a capillary-like network [9]. Mathematical morphology is a non-linear image processing approach which is based on the application of lattice theory to spatial structures [12]. This technique is proven to be a very powerful tool in microscopic image analysis [1]. We present in this paper the methodology developed for quantitative analysing the tubular-like networks of in vitro endothelial cells cultured on a Matrigel support. The method deals with a completely automatic approach which provides a powerful descriptor, by taking into account all the morphological features which describe the structures. In particular, we have established an automatic quantitative analysis to evaluate a modulator effect of a sulphated exopolysaccharide on FGF-2-induced in vitro angiogenesis [10], according to different parameters. Experimental results allow us to draw a discussion of the pertinence of the alternative morphological parameters to evaluate the characteristics of the cell culture. More than 50 images have been satisfactory processed using the present approach, showing the robustness of the algorithms.
2 Overview to the approach Figure 1 shows the example of a culture of endothelial cells in a Matrigel support. The image corresponds to the preparation after fixing and staining with Giemsa and illustrates the different morphological phenomena observed: cell aggregates, interconnecting segments, branch points and capillary-like network structure. By these regularities, it is possible to develop a quantitative analysis based on image processing techniques. To quantify automatically the morphology of an endothelial cell network we have developed an approach into two main steps. The automated image analysis is based on the pro-
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Figure 1. Example of image to be processed (culture of endothelial cells in a Matrigel support) illustrating the different morphological phenomena of the tubular-like cell network.
cessing of the original image to obtain intermediate images which undergo the second phase, the measurement of the parameters and the morphological quantification. The image analysis starts with the extraction of the cellular structures by means of a top-hat transformation, follows by the separation of the tubular structures and the cell aggregates. An interpolation algorithm yields a reconstituted closed network and the skeleton of the tubes is also calculated by a thinning transformation. Then from these transformed images, different kinds of quantitative parameters are calculated. On the one hand, the isolated cells and the cell aggregates are described both by a series of scalar size/shape parameters and by two curves of multi-scale aggregation. On the other hand, the quantification of the tubular-like structures implies the determination of the following geometric and morphologic parameters: (i) series of scalar size/shape parameters, (ii) morphological distribution of length, (iii) geodesic parameters such as branch points and extremities and (iv) spatial organisation of the reconstituted closed network.
3 Automated image analysis 3.1 Endothelial network extraction Let Iculture be the original gray level image of a culture of endothelial cells. In this image the cytological structures appear as dark upon a bright background relatively variable (shading effect). In order to filter out the noise and the small mistakes of digitalisation the first step is a Gaussian filter of size n × n, Gn×n (typically n = 3), followed by an opening of size s1 which removes the small bright particles in the tubes (they could “disconnect” the 0 structures which belongs to the same tube), i.e., Iculture = γs1 (G3×3 (Iculture )), such that the value of s1 must be smaller than the typical diameter for a tube; experimentally we have fix s1 = 4. The top-hat transformation is a powerful operator which permits the detection of contrasted objects on nonuniform backgrounds [11]. The black (or dual) tophat is the residue of a closing ϕ(I) and the initial image I for extracting dark structures. Therefore, from 0 the image Iculture , a black top-hat, ρ− s2 enhances the en00 0 dothelial network structures, Iculture = ρ− s2 (Iculture ) =
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Figure 2. Extraction of the image structures corresponding to the endothelial network: (a) original image Iculture , (b) 0 opening of size 4, Iculture = γ4 (G3×3 (Iculture )), (c) dual 00 0 top-hat of size 20, Iculture = ρ− 20 (Iculture ), (d) binary im00 age after thresholding, Inet = T[40,255] (Iculture ) (here is shown the negative).
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ϕs2 (Iculture ) − Iculture , where s2 corresponds to the size of the biggest cellular structure which can be found. After different tests, the value has been experimentally fixed to s2 = 20. The structuring element B for the above pre00 sented transformations is a circle. On the image Iculture a thresholding operation T[u,tmax ] is performed to provide the binary mask of the endothelial network (the interval [u, tmax = 255] determines the set of gray levels associ00 ated with the object of interest): Inet = T[u,tmax ] (Iculture ); the choice of the threshold value is not so critical (the tophat facilitates just the thresholding), e.g. u = 40. In the image 2 is depicted an example for extracting the endothelial network by means of the present algorithm. Systematic tests on a selection of images have shown that the approach is quite robust.
3.2 Separation of tubular structures / cell aggregates In this second stage of the segmentation, the aim is the separation of the tubular-like structures (which constitute properly the endothelial network) on the one hand, and the isolated cells and the cellular clusters or aggregates (which have no elongated shape) on the other hand. More precisely, the goal is to classify the structures of Inet according to two criteria: (1) size (the size of the connected component X is calculated by its surface area or number of pixels, A(X)); (2) shape (based on the form factor or circularity index of a set X, i.e. F F (X) = P (X)2 /4πA(X), equals to 1 for a circle-shaped object and increasing when the shape becomes elongated or irregular). The isolated cells and the cellular clusters are structures with a surface area smaller than a threshold given, usize ; or structures with a larger surface area but with an index of circularity larger than a threshold fixed, ucircular . The endothelial tubes (and the lengthened clusters or quasi-tubes) are the structures with a surface larger than usize and with an index of circularity smaller than ucircular . See in figure 3(a) an example of classification of the connected components in a binary image with the tubular structures, Itubes , and
another image with the isolated cells and the cell aggregates, Icells . After an exhaustive analysis on a representative selection of images, we found that for our images the application of the values usize = 200 and ucircular = 1.5 provides good results for classifying the structures.
3.3 Skeleton of tubes by thinning The thinning θB (X) is a morphological operator which is the subtraction between the input image and the hit-or-miss transformation ηB=(B 0 ,B 00 ) with the structuring elements B 0 and B 00 (B 0 ∩ B 00 = ∅), i.e., θB (X) = X \ ηB (X) = X \ (εB 0 (X) ∩ εB 00 (X c )). The result is an image preserving the pixels which center contains the pattern specified by B 0 and B 00 marked as zero and removing the pixels which satisfies the pattern given by the structuring elements B 0 and B 00 . The skeleton by thinning, T hinnθ , is the application until stability (or nθ times) of the thinning operator using a sequence of structuring elements B 0 and B 00 , generated by successive rotations of a pattern. See in [12] details on the suitable series of patterns to be used to obtain homotopic skeletons.
(partition the space) is very interesting to know the biological/physical phenomena implied in the process of cellular differentiation. For instance, one could be interested by the state of formation of the network or by the number and the average surface of the polygons, etc. Consequently, it seems important to be able to evaluate which would have been the natural evolution of the cellular structures if the network would have continued to be formed and also which is the current state (after fixing the cellular culture) compared to the supposed final one. From the mathematical morphology viewpoint, we can regard that as the interpolation of the network structures.
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Figure 4. Interpolation of the tubular-like image structures: (a) initial image Itubes (negative), (b) filtered dis0 tance function, Idist−tubes , (c) watershed lines superposed on the original structures, Iinterpol−tubes (d) result of the interpolation of the tubular network on the culture image.
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Figure 3. Binary processing of the endothelial network (here are shown the negative images). Separation of the tubular structures and the cell aggregates using the criteria of size and shape (uarea = 200 and uF F = 1.5): (a1) Inet , (a2) Icells , (a3) Itubes . Morphological skeleton by homotopic thinning of the tubes: (b) Isk−tubes = T hinnθ (Itubes ).
Note that the skeleton is calculated only for the endothelial tubes: Isk−tubes = T hinnθ (Itubes ). In practice, the size nθ = 20 is sufficient for the structures which one can find on these networks. In the figure 3(b) is given the example of application of the algorithm after separation of the tubular structures. The skeleton Isk−tubes allows us the calculation of several parameters for the tubes.
3.4 Interpolation of partial tubular network On the images of endothelial cell networks (see the original ones in figure 5), we can observe that the tubular-like structures of cells have the tendency to be organised into polygonal structures which lead on a network. This geometrical organisation in a network which divides the space
We propose the application of the watershed transformation [3] on the distance function [6] for the interpolation of the tubular structures (this is the dual algorithm of the classic one for the separation of overlapping particles). This approach was satisfactory used for extracting the 3D structures in polyurethane foam [5]. The common methodology is quite simple and is achieved in three steps (see in figure 4 an example of interpolation): 1. Computation of the distance function of the negative c of the binary image, i.e., Idist−tubes = distItubes . The maxima (positive peaks) of the function distance are the centers of the quasi-polygons (the zones between the tubes) and the minima (valleys) are associated to the tubular structures. 2. Simplification of the distance function by means of an opening by reconstruction using as marker the function obtained by subtracting a constant 0 value h from the distance function: Idist−tubes = γ rec (Idist−tubes , Idist−tubes − h). The maxima of the distance function after filtering will depend on the value of h and these maxima are important as markers for the watershed line which interpolates the structures (ideally, we need a maximum for each center of polygon). We found that using h = 5 the results are satisfactory. 3. Determination of the watershed for the negative of the simplified distance function: Iinterpol−tubes = 0 0 W shed((Idist−tubes )c , M in((Idist−tubes )c )). This
transformation provides a means of deciding if the beginning of a structure must be prolonged to be linked with another neighbouring structure (that is, if this edge creates a minimum of the distance).
P µk = n (n − µ)k P S(I, n). The influence zone of a connected component Xi of X is the set of points of the plane that are closer to Xi than to any other component of X. The SKIZ(X) is then defined as the boundary of all zones of influence [12]. Anti−Granulometry Curve of Cells 35
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4 Measured parameters and morphological quantification
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4.1 Quantification of cell aggregates Using the binary connected components of image Icells and in order to quantify the cell size/shape, classical global parameters can be calculated. It is more interesting to calculate what is the distribution and spatial organisation (dispersion) of the isolated cells and the clusters compared to the tubular-like structures. To do that, we propose a study of cell aggregation based on computing a granulometry combined with the skeleton of influence zones, SKIZ. A granulometry is the study of the size distributions of the objects of an image, formally defined as a family of openings of increasing size. Moreover, granulometries by closings (or antigranulometry) can also be defined as families of increasing closings. Performing the granulometric analysis of an image I is equivalent to mapping each opening of size n with a measure M(γn (I)) of the opened image (M(I) is the area in the binary case and the volume in the gray scale case). The size distribution or pattern spectrum of I is then defined as the following (normalised) mapping n+1 (I)) P S(I, n) = M(γn (I))−M(γ , n ≥ 0. It maps each M(I) size n to some measure of the bright image structures with this size (loss of bright image structures between two successive openings); i.e., . P S(I, n) is a probability density function (a histogram): a large impulse in the pattern spectrum at a given scale indicates the presence of many image structures at that scale. It is also possible to use standard probabilistic definitions to compute the moments of P S. P The first moment µ is given by µ = µ1 = n nP S(I, n), the k-th pattern spectrum moment, k ≥ 2, is computed as
In figure 6(a) is given an example of aggregation analysis. Starting from this morphological muti-scale decomposition it is possible to define two curves. The Anti-Granulometry Curve of Cells, P S(Icells , −n), which corresponds directly to the pattern spectrum by closings: n → P S(Icells , −n) = (M (ϕn (Icells ))−M (ϕn−nstep (Icells )))/(M (ϕ0 (Icells ))), associating to each size n the increasing surface taken up by the cells aggregates (40 ≤ n ≤ 160). A second curve is derived from the n images SKIZ(ϕn (Icells )). The Aggregation Curve of Cell Regions, AGCR(Icells , n) is obtained by computing for each n the standard deviation of area of influence regions, i.e., AGCR(Icells , n) = σASKIZ (n). This curve yields a compact description of dispersion in the aggregation (peaks correspond to maximal disorder). The aggregation curves are very useful to compare different cell populations obtained according to the concentration of the molecule to be tested, see figure 7.
4.2 Quantification of tubular structures In order to quantify the tubes, a granulometry analysis can be applied again. In fact, the most indicated transformation to be used is a linear opening of size n which is obtained by supremum of openings according to several dio o o o rections, i.e., γnL = γn0 ∨ γn45 ∨ γn90 ∨ γn315 . Then, the associated granulometry or Linear Pattern Spectrum of Tubes, LP S(Itubes , n), allows describing length distribution of tubular structures: n → LP S(Itubes , n) =
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L (M (γnL (Itubes )) − M (γn+1 (Itubes )))/(M (γ0L (Itubes ))), with 10 ≤ n ≤ 150. In figure 8 is shown a comparative for different cell cultures of LP S(Itubes , n) as well as the derived moments (mean and variance) very useful to estimate compactly the length distribution of tubes.
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Figure 9. Geodesic description of tubular structure: (a) end-points and triple-points from skeleton by thinning, (b) two examples of extremities and branch points from geodesic distance function.
Besides the previous length parameters, for this type of tubular network is interesting to quantify other intrinsic features to evaluate the complexity of their ramified structure. These characteristics are mainly the extremity points and the branch points of the particles. A classical way to obtain these two features consists in, starting from the skeleton of the tubes, obtaining the end-points and the triple-points by means of the hit-or-miss transformation,
see example in figure 9(a). However, we prefer to use the technique of morphological wavefronts [13], which consists in considering the geodesic distance as an isotropic wave, with a origin at the most distant extremity, and to study the evolution of the connected components (branching points) as well as the maxima of the distance function (extremity points). The most important advantage of this method with respect to the skeleton is the passage that we can make from a binary object to a weighted graph which describes its characteristics; with this second more compact representation, the analysis of the object can be done effectively. All details concerning the implementation can be found in [1]. In figure 9(b) are shown two examples of extremities/branches obtained by this technique. Then, from a quantitative viewpoint, we propose three parameters for the endothelial cell networks: (1) Normalised Numbends (for all tubes of the network with reber of Ends, N spect to the surface), (2) Normalised Number of Branch bbranches , (3) Tubule Mean Geodesic Length FacPoints, N PNtubes −1
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tor, ρtubes = i=0 , where ρ = πL Ntubes 4S such that L is the geodesic length of the tube (obtained from the morphological wavefront) and S its surface. Finally, starting from the result of the morphological interpolation Iinterpol−tubes , we propose a last set of parameters to characterise the progression of differentiation cellular process consider as associated to the spatial network organisation. The result of this polygonal partition of space Iinterpol−tubes is very similar to the result of a SKIZ; and mathematically, it is rather close to a very known probabilistic model of space partition (the partition of Vorono¨ı). However, to quantify the spatial partitions obtained by interpolation we propose to use simple scalar pabpolygons , rameters: (1) Normalised Number of Polygons, N b (2) Normalised Length of Polygons, Lpolygons , (3) NetA(Inet ∧Iinterpol−tubes ) work Progression Factor, N P F = , A(Iinterpol−tubes ) (4) Polygon Mean Area, Apolygons , (5) Polygon Standard Deviation Area, σApolygons . Figure 10 summarises the values of geodesic parameters and parameters to evaluate spatial progression for the interpolated networks for different examples of cell cultures. These two sets of parameters are especially useful to quantify the effect of a molecule on the cells.
5 Conclusions and perspectives In this paper we presented a set of algorithms developed for quantitative analysing of the differentiation in tubularlike networks of in vitro endothelial cells cultured on a Matrigel support. We have shown that the most interesting parameters to quantify this kind of structures are the morphological distributions of lengths, the geodesic parameters (branch points and extremity points), the spatial distribution of the isolated cells and the spatial organisation of the tubes into polygonal networks. Obviously, the present approach can be used to study the kinetics of the dynamic process of migration of the endothelial cells. Concerning
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[2] Auerbach R, Lewis R, Shinners B, Kubai L and Akhtar N. Angiogenesis assays: a critical overview (Review). Clin Chem. 2003 ; 49(1):32-40.
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Figure 10. (a) Summary of the values of geodesic parameters for the tubular networks: (1) normalised number of ends, (2) normalised number of branch points, (3) tubule mean geodesic length factor. (b) Summary of the values of spatial progression for the interpolated networks: (1) normalised number of polygons, (2) normalised length of network, (3) network progression factor, (4) polygon mean area, (5) polygon standard deviation area. The first bar is the negative control, the second one FGF-2 alone and the other bars corresponds to the EPS concentration.
the perspectives of this study, we believe that the next stage to be considered is the development a morphological-based mathematical model for the simulation of the cellular organisation into tubular-like networks. We have proved the interest of the approach for an improved quantifying (compared with a manual quantification) to evaluate the angiogenic modulator effect of a molecule and in order to determine the optimal concentration. In addition, the usefulness of this descriptor or a part of the present algorithms is not limited to study the assays of endothelial cells on Matrigel. Other angiogenesis assays as the rat aortic ring or in general, other experimental models of proliferation of biological structures in tubular-like networks, branching phenomena, etc. could be objectively quantified by means of the suggested techniques.
Acknowledgements Exopolysaccharide was provided by IFREMER. We would like to thank Prof. Georges Flandrin who favors this scientific collaboration.
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