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1.Introduction and MotivationThe automatic classification and recognition of images obtained from various microscopic applications is a task of paramount importance to disciplines ranging from materials science to cell biology. This paper focuses on the morphology of the cytoskeleton, made visible by epifluorescence microscopy, processed by some methods that extract features (morpho-logical descriptors) and classified by means of multivariate statistics. 1.1.CytoskeletonThe cytoskeleton is a network of proteins that structurally and dynamically organize the cytoplasm of living cells.1 It is composed of three major structural elements: microtubules, intermediate filaments, and microfilaments, each consisting of polymers of protein subunits. The cytoskeleton is responsible for the maintenance of cell architecture, shape, and internal organization. For example, it enables the transport of organelles and vescicles. Microtubules and actin filaments play a role in mitosis, cell signaling, and motility.2 A well-organized cytoskeletal network exhibits, as in the top tile of Fig. 1 and in Fig. 2 , bundles of microtubules radiating from the microtubule organizing center (MTOC), located near the cell nucleus, toward the cell periphery. Fig. 1Outline of the experiment design. All images show microtubules of cultured rate hepatocytes obtained by immunolocalization and epifluorescence microscopy. , tile from the untreated (control) set; , tile from treatment at the highest concentration for the longest time ; ; tile from the recovery experiment; and ; tile from an intermediate treatment. In tiles and the dark round area corresponds to the nucleus, in proximity of which the MTOC is located. Square side . ![]() In general, alterations of cytoskeletal functions are related to (may be the cause or the effect of) a broad spectrum of biochemical reactions such as adenosine triphosphate (ATP) depletion, disruption of intracellular ion homeostasis, thiol oxidation, and phosphorylation,3, 4, 5, 6, 7 all of which have major repercussions on cell functionality. Many substances are known to directly or indirectly interact with cytoskeletal constituents and cause damage: metals,8, 9 herbicides and fungicides,3, 10 the neurotoxin4 MPTP, as well as natural toxins.5 In morphological terms, damage to the microtubule network consists of structural disorganization, inhibition of assembly, disassembly, or depolymerization. All of these patterns can be visualized by means of suitable techniques11 (e.g., Sec. 3). In turn, morphological alterations of the cytoskeleton reflect functional disturbances of the whole cell, e.g., altered transport of very low density lipoproteins in hepatocytes,12 neurodegeneration,13, 14, 15 and cell transformation.16 For this last reason the cytoskeleton has become one of the preferred targets of anticancer drugs.17, 18, 19, 20, 21, 22 Therefore cytoskeletal morphology is a valuable indicator of cell functionality. In this context, morphological analysis and classification of cytoskeletal images can assist in estimating the degree of cell injury. 1.2.Image ClassificationImage classification can be either dichotomic or of “continuous” type. The former is meant to provide yes/no answers, the latter is expected to rank images according to some predefined properties. Moreover, image classification paradigms differ by the type of data processed. Raw images can be supplied as such to a black box algorithm, which is requested to sort out similarities and rank images accordingly.23, 24 Instead, the paradigm implemented in this paper starts with the extraction of a few quantitative morphological descriptors, also called indicators or features, believed to effectively translate structural, hence functional information. Features were extracted by the following methods: contour (Secs. 5.2, 6.1) and mass fractal analysis (Secs. 5.3, 6.2); spatial differentiation (Secs. 5.4, 6.3); and “spectrum enhancement,” a nonlinear filter in the Fourier domain (Secs. 5.5, 6.4). Descriptor selection is based on known function-morphology relations and driven by the properties of the available images (Sec. 4.1). Classification and quantitative assessment then relies on the four standard steps:
These stages are all reflected in the experiment design (Sec. 4) and the developments described in the following, where the eventual application is the quantitative description of cytoskeletal morphology affected by damage (Sec. 9.1) or undergoing recovery (Sec. 9.2). 2.Related WorkVery roughly speaking, mathematical models of the morphology and organization of the cytoskeleton have served two different purposes: either to simulate cytoskeletal dynamics or to analyze a specific process of interest. Although a comprehensive overview of prior work is beyond the scope of this paper, some results concerning modeling and simulation deserve to be mentioned, namely, those by Dufort and Lumsden,25 who developed a cellular automaton model; by Edelstein-Keshet and Ermentrout,26 who interpreted actin-filament length distribution in a lamellipodium; and by Aon and Cortassa,27 who investigated the influence of cytoskeletal organization and dynamics on cell biochemistry. Moreover, fractal models have been proposed in a variety of situations involving the cytoskeleton.28, 29 On the process analysis side, as early as 1992 and within a program aimed at extracting features from moving cells, Lifshitz30 developed a microtubule tracking algorithm that relied on model matching, and Thomason 31 related the mass fractal dimension to cytoskeleton rearrangement dynamics in cardiac muscle. Spatial differentiation and edge detection have been applied, e.g., by Karlon, 32 who measured cell alignment and cytoskeletal organization by means of a gradient method; Knight 33 applied a Sobel filter to obtain comprehensive results about the quantitative morphology of cytoskeletal organization. Fourier analysis was used, e.g., by McGough and Josephs34 to quantify the structure of erythrocyte spectrin and by Petroll 35 to describe the orientation of stress fibers. To the best of our knowledge Kohler 36 were the first to compare arc-averaged power spectral profiles of cytoskeletal images in the reciprocal domain. In fact, image comparison based on arc-averaged power spectra (improperly called “radially averaged”) is a well-established procedure37 derived from robust (rotation-invariant) target detection. Other methods of pattern analysis such as hierarchical feature vector matching have been applied to confocal microscope images of living cells to quantify cytoskeletal deformation.38 More generally, the supervised or unsupervised classification and recognition of subcellular structures are strategic tasks in cell biology and proteomics,39, 40 which have far reaching consequences in diagnostics and health care. This paper is an account of the results obtained so far by the authors. The earliest results appeared in some conference proceedings.41, 42, 43, 44, 45 3.Materials and Experimental MethodsExperiments that yielded the images of interest involved primary cultures of rat hepatocytes. The latter cells were obtained by the modified method of collagenase perfusion.3 Male Sprague-Dawley rats (weighing ) were anesthetized by intraperitoneal (ip) injection of sodium penthobarbital/kg body weight. The abdomen, previously washed with alcohol, was opened and the portal vein cannulated, the liver was perfused with Hank’s balanced salt solution (HBSS)— buffer ( NaCl, KCl, , , glucose, phenol red, 7.5% w/v , pH 7.4) for and then for with HBSS buffer containing collagenase ( medium). The liver was placed in a culture dish containing William’s E medium (Sigma Chemical Company, St. Louis, Missouri, USA) supplemented with 0.2% bovine serum albumine (BSA), dexamethasone, bovine pancreas insulin, and gentamycin sulphate and decapsulated. The suspension was filtered through cotton gauze and washed twice with culture medium by sedimentation. Cell viability was determined by trypan blue exclusion and was . Hepatocytes were plated on collagen-coated glass coverslips at density with William’s E medium supplemented with 5% fetal calf serum (FCS). Collagen isolated from rat tail was supplied by Sigma Chemical Co. (St. Louis, Missouri). Twenty-four hours after plating, some cells were processed right away for tubulin visualization and formed the “negative control” set. Some more were exposed to the fungicide Benomyl™ (ICI Soplant S.p.A., 98% purity) dissolved in dimethylsulphoxide (DMSO: 1% v/v final concentration) at a given concentration and for a given time and formed the various “treated” sets. The “recovery” experiment consisted of replacing the Benomyl™-contaminated medium by the standard one and letting the culture incubate for more. Tubulin was made visible through the following steps. Cells were washed in phosphate-buffered saline (PBS: NaCl, KCl, , , pH 7.4) and fixed for at room temperature with 3% formaldehyde in PBS. After rinsing with PBS, cells were permeabilized first with high performance liquid chromatography (HPLC)-grade methanol then with HPLC-grade acetone both at for a few seconds, then incubated with a blocking solution made of BSA for . Standard indirect immunofluorescence staining was performed: the primary anti- -tubulin antibody (Amersham, Amersham Buks., United Kingdom) was diluted 1:100 in and the cells incubated in a humid atmosphere for at . After rinsing with , fluorescent staining was performed for at with the secondary antibody (Amersham, Amersham Buks., United Kingdom), namely, Texas Red-conjugated goat anti-mouse immunoglobulin (Ig) (1:50 in ). After the final rinsing with PBS and distilled water, the coverslips were mounted and viewed in a Zeiss Axioplan™ microscope equipped with epifluorescence optics at magnification and a numerical aperture of 1.40 in oil. The filter set consisted of the bandpass BP 546 tuned for excitation wavelengths ranging from , a bandstop FT 580, a dichroic beamsplitter and a lowpass LP 590, which transmits fluorescence radiation at and higher. Photographs were taken with Kodak T-max 400™ film. All epifluorescence images were digitized at a resolution of and saved in BMP, gray scale format such that 0 corresponded to black and 255 to white, as a consequence they had the same magnification . Since spectral analysis relied on the discrete Fourier transform (Sec. 5.5), square tiles had to be cut out of the original images by supplying the upper left corner coordinates to the image readin function. The size of a tile was . Original images usually contained more than one cell. Therefore, tiles were cut in such a way as to include the border areas of the cell aggregate, where microtubules were best visible in each set. This rule applied to all tiles except the one labeled (Fig. 6 in Sec. 6.4), taken from the perinuclear region of an untreated cell. In that area, the organization of microtubule bundles could not be evaluated; context-independent visual observation was insufficient to tell whether the image came from a control cell or from a treated one, regardless of dose. Fig. 6Perinuclear tile cut from a control image. Context-free classification of this tile is impossible, even to a human expert. It is classified as an outlier (Sec. 7). ![]() 4.Classifier Specifications and Design4.1.Required Morphological DescriptorsSince experiments were carried out in different situations, the average fluorescence yield affecting a given image varied. Moreover, the photograph sets of a given experiment were developed and printed at different times. As a consequence, images had to be analyzed by methods exhibiting the least possible sensitivity to the average fluorescence intensity and to the absolute relation between exposure and film density. The features to be extracted from the images for classification had to translate into quantitative terms those properties that a human expert believes to characterize cytoskeletal organization or loss thereof. In morphological terms, said features depend on image structure (leading features) and, to some extent, on texture (fine details). The fully developed microtubules of a normal cytoskeleton are known to have indented contours, whereas an altered cytoskeleton has a smoother contour. One morphological descriptor known to quantify indentation is the contour fractal dimension, denoted by (Secs. 5.2, 6.1). The number of microtubule bundles per unit area, i.e., the “mass density,” of a normal cytoskeleton is lower than that of an altered one. A suitable descriptor, suggested, e.g., by the morphological analysis of dendritic materials,46 is the mass fractal dimension (Secs. 5.3, 6.2). Microtubule bundles give rise to more or less visible edges in a gray-scale image. Global, direction-independent edge indicators are therefore necessary, such as the total variation (TV) and a suitable norm of the Laplace operator . Both descriptors are computed in the direct domain, i.e., by (discrete) spatial differentiation (Secs. 5.4, 6.3). Finally, a set of descriptors was required that
Separation of image structure from texture is one of the fundamental tasks of image understanding, which was formalized by the Osher-Rudin paradigm (Chap. 1 of Ref. 47). In this application, structure observed in a cytoskeleton is due to organized microtubule bundles, whereas (straight or curly) isolated tubules and image noise contribute to texture. The diameter of a microtubule is . If straight bundles are formed by, say, 5 to 20 microtubules, the image will contain structures of spatial period ranging from . The corresponding fundamental spatial frequency will range from . Hence the analysis of the power spectrum shall be tuned to that frequency domain. The implementation of these requirements by means of “spectrum enhancement” is described in Secs. 5.5, 6.4. Note also that Table 1 summarizes the symbols used in this paper. Table 1List of symbols.
4.2.Classification Experiment DesignThe classification experiment was designed to implement all three stages, i.e., training, validation and recognition. Figure 1 provides examples of the image types and an outline and Table 2 the complete list of processed tile sets. Untreated cells, which underwent all sample preparation stages described in the previous section except exposure to Benomyl™, served as negative control: they yielded comparable images divided into the and tile sets for experimental purpose. The and sets were derived from treatment with Benomyl™ at experimental conditions( for ) classified as “extreme” according to the morphological and biochemical evidence given by Ref. 3. Cells from intermediate treatments gave rise to other tile sets, e.g., ( for ). Cells from recovery after the treatment yielded the tile set. The and sets were used for classifier training (Sec. 7) as well as the tile (Sec. 3 and Fig. 6 in Sec. 6.4). The and tiles were used for classifier validation (Sec. 8). Sets , , , and were used in the recognition stage to rank the effects of intermediate concentrations and exposure times on morphology (Sec. 9.1). The analysis of set is presented in Sec. 9.2. Table 2Image sets forming the experiment design.
NA stands for not applicable. 5.Methods of Morphological AnalysisAn image is conveniently modeled by the discrete counterpart of a nonnegative function of position with support in the open set , the tile. 5.1.Preprocessing and Optimal ThresholdingGiven the heterogeneity of experimental conditions and of photographic processing (Sec. 4.1), independence of the absolute fluorescence intensity could be achieved by normalizing the gray-scale histogram of each tile separately. Histogram normalization is an affine transformation of into according to where and depend on the specific tile.The extraction of a few morphological descriptors required image thresholding, e.g., for binarization or background subtraction. Several methods exist that determine a binary threshold according to some optimality criterion. In this paper, the optimum threshold for each tile was determined to maximize cross-correlation between the original gray scale and the black-and-white image binarized according to the threshold . Formally, The algorithm, described by Kindratenko,48 is reported in Appendix A. Adaptive methods that determine position-dependent thresholds were not implemented to avoid introducing statistical heterogeneity between images.5.2.Contour Fractal AnalysisThe purpose of contour fractal analysis is to estimate the corresponding fractal dimension . A typical algorithm consists of the following eight stages: (1) thresholding (Sec. 5.1); (2) determination of the Férét diameter ; (3) contour tracking; (4) estimation of the perimeter by different values of the yardstick ; ranging from to ; (5) contour correction wherever applicable; (6) construction of the versus Richardson plot; (7) determination of the morphological threshold ; and (8) linear interpolation of the plot in . Some of the computer algorithms can be found in Ref. 48. As shown in Appendix B, Proposition 1, the value of estimated from an image binarized by the of Eq. 2 is invariant with respect to affine transformations of such as histogram normalization [Eq. 1]. As a consequence is an intrinsic property of the analyzed tile, not affected by absolute gray levels. 5.3.Mass Fractal AnalysisAmong the methods that estimate the mass fractal dimension , box counting46 is the most straightforward. It also operates on a binarized image and relies on regression of a Richardson plot. Details are left to the implementation (Sec. 6.2). 5.4.Direct Methods (Spatial Differentiation)Although all methods are applied to a discrete context, a continuum setting now simplifies the description. By assuming the Sobolev space of functions absolutely integrable in together with their derivatives up to the second, the following subdomain was definedwhere is a suitable threshold. Heuristically, is an estimate of peak background noise, hence in is above such noise level. The averaged total variationand 1 norm of the Laplacian,where is the area of , can serve as direction-independent edge indicators, as specified in Sec. 4.1. Obviously, they are affected by affine gray-scale transformations. By letting [Eq. 1], where is directly proportional to of Eq. 2, and defining , then and one can immediately verify that the norms scale according to5.5.“Spectrum Enhancement”The spectrum enhancement algorithm, suggested by the experimental results from a coherent optical image processor many years ago49 and specifically developed41, 50, 51, 52, 53 for image classification in the past , meets all the specifications listed in Sec. 4.1. Let denote a square of side-length and consider an image, i.e., a function , , which is continuous on the surface of the torus obtained by glueing the opposite sides of together. One way of obtaining such a from an image defined in a square tile of side-length is the application of the twofold reflection. Next let be discretized by a square grid of step-length . Let be the spatial frequency vector. Then the discrete Fourier transform of is defined in the square In the implementation below , corresponding to , hence,As a consequence of the continuity of on , exhibits no “cross-artifact” (e.g., Chap. 4 of Ref. 46).Represent in polar coordinates , where is the wave number, and is the polar angle such that . Denote by the power spectral density. Due to the symmetry of , the main axes of inertia of (those that diagonalize the inertia tensor) coincide with and . Let denote an arc symmetric with respect to either axis. Then the normalized, arc-averaged spectral density profile is the function of alone defined in (cycles/image) according to where is the length of , and obviously for any nondegenerate image. The value of is provided in Sec. 7.Let be a model spectral density. For example, choose where is the model exponent. Then the enhanced spectrum is defined byIntuitively, the function represents deviations of from a given model.Among the properties of the enhanced spectrum the following two are worth mentioning. Proposition 2. In the interval , is invariant with respect to scaling transformations such that in Eq. 1, applied to the gray levels of the input image. This is a straightforward consequence of normalization [Eq. 10]. As a result, the enhanced spectrum reflects the intrinsic properties of the analyzed tile. The second property is a relation between and spatial differentiation of . Proposition 3 (in words). When is an integer, the enhanced spectrum corresponds to evaluating all spatial derivatives of order of the image, taking their Fourier transforms, forming a linear combination of the squared moduli, adding , taking the logarithm and averaging over . The formal statement and other remarks are provided in Appendix C. From a practical point of view, conversion of to the log scale [Eq. 10] amplifies the contribution of structured but weakly emitting microtubules. Averaging over makes independent of bundle direction. Comparison between the asymptotics of the given power spectrum and a model of the class has been implied by previous studies involving visual processing.54 The emphasis herewith is on the low-frequency behavior of for example in , aimed at enhancing image structure and assessing the relative weight of texture. Given the scale factor of Eq. 9, the approximate spatial frequency range to which microtubule bundles shall contribute most in terms of power spectral density is The spectrum enhancement code was based on the fast Fourier transform and singular value decomposition taken from Ref. 55 and on additional authors’ own feature extraction functions. It was written in C and run in conjunction with other pieces of software.6.Extraction of Morphological Descriptors6.1.Contour Fractal DimensionContour correction, step 5 of Sec. 5.2, consisted of subtracting from the independent length of the straight segments along the boundary of the square before building the Richardson plot. Otherwise would have been underestimated. The threshold (step 7) was determined once for all from the Richardson plot of a contour of known dimension. This is a yardstick value that separates the structural part (large , highly oscillatory graph) of the plot from the textural one (small , smoother graph). The Richardson plots yielded by the tiles of Figs. 2 and 3 are shown by Fig. 4 . Contour correction was applied to the former plot. The latter tile, instead, which showed a whole cytoskeleton, did not need any correction. The estimated values of for tiles, including that of Fig. 2, were affected by the deep contour indentations due to filaments. The smoother contours of tiles (Fig. 3) yielded lower values, as expected. 6.2.Mass Fractal DimensionThe was estimated by box counting46 on each previously normalized and binarized tile. The software used was Benoit™ 1.3 of TruSoft Inc.56 Typically, the box scaling ratio was 1.3 and the grid orientation angle was stepped by . The values of tile set-averaged and and their standard deviations are shown in Table 3 . Table 3Set averages of descriptors used by the implemented classifier.
Units of measurement, wherever defined, are omitted, Standard deviations are shown between parentheses.
DFP
,
DFM
,
TVη
, and
Lη
are strictly positive, whereas
A
,
q′(0)
, and
c
can be negative, zero or positive. 6.3.Total Variation and Laplacian NormThe relation between and mentioned at the end of Sec. 5.4 was where the coefficient 0.8 was found during classifier training (Sec. 7). The set-averaged values of and and their standard deviations are listed by Table 3. The signed, domain-averaged Laplacian was used in early training attempts (Sec. 8.1) and later dismissed.6.4.Descriptors of the Enhanced SpectrumFrom here on the wave number is assumed to be discrete and span the interval . The enhanced spectrum can be regarded as the signature of a given tile, and as such, the graph is an array of first-level morphological descriptors. The decision was made to extract fewer second-level descriptors according to some criterion. For any exponent exhibits a trend over which jumps and oscillations are superimposed. In fact, the latter are amplified with respect to those of . For this reason the value was reassigned according to and was interpolated in by a polynomial of suitable degree . Four parameters eventually affected : the axis of inertia ( : major axis, : minor axis) about which integration was carried out [Eq. 10], the arc , the model exponent , and the degree . Plots obtained from some tiles used in classifier training with , , , and are shown in Fig. 5 . In agreement with the estimate of Eq. 13, enhanced spectra from the set generally exhibited a local maximum in /image as shown, e.g., by conṯ3b (Fig. 5). One exception was tile conṯ1a. None of the tiles (plots be502̱3a and be502̱1a of Fig. 5) had such maximum; in fact, in the same frequency range. The plot conṯx of the perinuclear tile seemed to deviate from all other plots of either set. Tile itself is shown in Fig. 6 . Fig. 5Interpolated enhanced spectra of representative tiles used in classifier training. Integration is about the major axis of inertia over an arc of ; model exponent ; degree of polynomial . Plots conṯ1a and conṯ3b represent the set, plots be502̱1a and be502̱3a represent the set; and plot conṯx is the perinuclear tile (see Fig. 6). Local maxima in the spatial frequency range /image correspond to structured microtubule bundles of a normal cytoskeleton. ![]() The selection of second-level descriptors was a part of classifier training (Sec. 7) and initially involved the following five quantities:
The values of and were also determined in the training stage. As one can infer from the last three columns of Table 3, the first three descriptors, , , and were eventually selected for image classification. 7.Classifier TrainingThe data used at this stage came from 8 tiles, 8 tiles, and , hence a total of tiles were analyzed. By examining morphological descriptors one by one (univariate analysis) it was found that none of them was capable of discriminating the three classes. For example, according to two tiles were clustered with the set. The same misclassification error was caused by choosing alone. As a consequence, descriptor fusion had to be attempted: the chosen method was principal components analysis (PCA), which is one of the most relevant procedures in unsupervised factor extraction (e.g., Chap. 5 of Ref. 57). The goal of training was to maximize discrimination between classes in the sense to be specified in the following. This was achieved by finding optimal (or suboptimal) values of the following: The raw feature matrix consisted of rows and columns. Each column was formed by the descriptors of a tile. Before applying PCA the entries of were normalized rowwise: the sample mean , , of each descriptor over the training set was subtracted and the difference divided by the sample standard deviation . The normalized feature (column) vectors denoted by , , formed the normalized feature matrix . As is well known, PCA constructs the linear space spanned by principal components, i.e., the eigenvectors of the covariance matrix generated from . Eigenvectors are labeled after the corresponding eigenvalues, arranged in a nonincreasing sequence. Then PCA projects the s onto a subspace, the dimension of which is no greater than min . The quotient is the cumulated percent variance or information content carried, or explained by the first eigenvalues. Each of the tiles is represented by a -dimensional vector , . For ease of interpretation the result is displayed in the plane of the first two principal components. Eventually, to each tile in the training set there corresponds a point in .Discrimination between classes , , and was rated by drawing the minimum spanning tree (MST) (Ref. 58 and algorithm AS13 in Ref. 59) in . The two longest branches, and , of the MST by definition divide the whole set of points into three subsets, say , , and . If each of these subsets contains points belonging to one class only, i.e., there is a one-to-one correspondence between the a posteriori assignment and the a priori classes of belonging , then classification is MST-correct and as such accepted. Optimization of the preceding items 1 to 3 was performed by running cluster analysis (e.g., Chap. 8 of Ref. 57), varying and the parameter values within reasonable bounds, running PCA, and looking at the ordered branch list of the MST. After several attempts the following values were determined: In the first place, an MST-correct classification was attained: joined to a tile, thus classifying as an “outlier.” Moreover, the distance between the set centroids and was found to be and the fractions of explained variance were , , and . Since it was decided to let and to form the matrixto be used later in validation (Sec. 8.2) and recognition (Sec. 9). The classification result is shown by Fig. 7a , where the rectangle circumscribed to the training tiles,is drawn.Fig. 7(a) Training set represented in the principal plane, where set, filled boxes; set, filled triangles; (diamond), the perinuclear tile; and and , two longest branches of the MST. The spread of both the and the sets along is slightly larger than that along . The first principal component explains 58% of the sample variance, the first two together explain 75% and the first three 89%; , rectangle circumscribed to the tiles. (b) Factor loadings of the seven morphological descriptors extracted by the methods of Sec. 6. Four descriptors correlate negatively with and are less correlated to . The total variation is more negatively correlated with than with . Mass fractal dimension is positively correlated with . The Laplacian is highly anticorrelated with and uncorrelated to . The two fractal dimensions carry independent pieces of information, whereas a high mutual correlation characterizes the three spectral descriptors , , and . ![]() The factor loadings of the morphological descriptors are shown by Fig. 7b. The caption carries some comments. Factor loadings provide, in part, the morphological interpretation of the principal components themselves and justify the position of some tiles in . For example, stands out for its high , therefore it will lie in the lower half-plane far away from the origin. All tiles have a larger , therefore they will end up in the left half-plane. All tiles have a larger , and are therefore located in the right half-plane. Those tiles that have larger , , and lie in the second quadrant. All of these deductions are easily verified by inspecting Fig. 7a. PCA was carried out by SPSS™ (Statistical Package for the Social Sciences), version 11. 8.Classifier Sensitivity and Validation8.1.Sensitivity to DescriptorsIn an early training attempt,44 the set such that , was formed by using the same parameters as in the first line of Eq. 16, with one exception: was computed at the fixed binary threshold, 127. The resulting representation in is shown by Fig. 8 . Classification was MST-correct and yielded . More comments are provided in the figure caption. The factor loadings also exhibited a pattern that, modulo reflections of both axes, resembled that of Fig. 7b.Fig. 8Classification of the set (filled boxes), the set (filled triangles), and by descriptor set . Modulo reflections of both axes, the pattern is similar to that of Fig. 7a, in spite of the different number ( instead of 7) and list of descriptors [Eq. 19 instead of Eq. 16]. Note the different scales of and . The first principal component explains 41% of the sample variance, the first two together explain 75%, and the first three 87%. ![]() The sensitivity of the classification to changes in the descriptor sets (from to ) can be now quantified as follows. The set differs from by three descriptors out of eight. The situation in can be summarized by the following quotients: , , and , where dist(∙,∙) is the distance between two clusters (minimum distance between two points belonging to either cluster), diam(∙) is the diameter of a cluster, and serves as a yardstick in the plane. The values are given by Table 4 . Ideally would be close to 1, and the other two will be much smaller than 1. Neither nor provide the “ideal” classification. However, the focus is on classifier sensitivity: therefore, a comparison between and has to be made in relative terms. The normalized distance between clusters is almost insensitive to the descriptor set: this means sets and are as “separable” by , as they were by . Instead, the normalized diameters of both clusters and are more sensitive to the replacement of by , because they increase by 26 and 13%, respectively; makes both training sets look “more heterogeneous” than they were under . From this numerical experiment, one can conclude that the classifier is relatively “stable.” Further evidence of classifier “stability” is provided by the values of , , given in the caption of Fig. 8; the first three principal components derived from either or carry the same amount of information. Table 4Indicators of classifier sensitivity with respect to descriptor sets.
Note
dCT
serves as a yardstick in the
{z1;z2}
plane. Classifier sensitivity to descriptors has to be assessed in relative, not absolute, terms. In going from
D8
to
D7
the normalized distance between clusters.
[dist(C,T502h)∕dCT]
(column 2), in almost unchanged, whereas the normalized diameters
[diam(.)∕dCT]
of both clusters
C
(column 3) and
T502h
(column 4) increase by 26 and 13% respectively. One can conclude that the classifier is relatively “stable,” Details are provided in the text. 8.2.Internal ValidationInternal validation consisted of submitting to the trained classifier some new tiles belonging to either class, then projecting the corresponding feature vectors onto by means of of Eq. 17 and finally computing sensitivities and specificities. In detail, the 8 and 8 tiles defined in Sec. 4 were selected from boundary areas of the available images. Unlike those used in training, some and tiles could not be easily classified visually. For example Figs. 9a and 9b exhibit “borderline” morphologies; at the top right of Fig. 9a, near the cell nucleus, is a dense mass of tubulin, which looks disorganized, whereas the boundary tubules of Fig. 9b are sufficiently structured as in a control cytoskeleton. Fig. 9(a) Tile of the validation set, where the dense, poorly structured microtubules on the upper right, close to the perinuclear region, look treated. Misclassification is expected, as shown in Fig. 10b. Square side . (b) Tile of the validation set, where microtubules on the boundary, although randomly oriented, exhibit sufficient structure as in a control cytoskeleton. Misclassification is expected, as shown in Fig. 10b. Square side . ![]() Fig. 10(a) Centroids of the training sets (gray) and of the validation sets (black). The displacement of the centroid with respect to the centroid is relatively small. (b) Validation result. Tiles of the set are represented by filled boxes, those of the set by filled triangles. Training tiles (Sec. 7) are shown by gray boxes and triangles respectively. Not all tiles are on the left half-plane as well as not all tiles are on the right one. ![]() Descriptors were extracted by the procedure outlined in Sec. 7. By assumption, the training tiles represented their respective sets, therefore normalization consisted of subtracting the and dividing by the computed in the preceding. The normalized feature vectors , , were determined for each tile of the validation sets. The were projected onto according to The validation result in terms of centroids is displayed by Fig. 10a . The relevant distances between centroids are and , which can provide the confidence bounds of subsequent recognition results. In relative terms, using as a yardstick again, the quotients and quantify the “stability” of the classifier.In terms of individual tiles the result is shown by Fig. 10b. References to a few images are also displayed. The diameters of both and clusters are larger than those of and . Two tiles, one of which is Fig. 9a, are positioned closer to the set and one tile [Fig. 9b] is closer to the set. The coordinates of these three tiles [Fig. 10b], which deviate from those of the remaining tiles, can be explained by means of the loadings of the morphological indicators [Fig. 7b]. For example, the tile of Fig. 9b has , the highest value in the set, comparable to the and set averages (Table 3). Since is negatively correlated to , this property suffices to position the tile in the left half-plane. About the tile of Fig. 9a, its enhanced spectrum, like that of a tile, has no local maximum. The presence or absence of the latter affects all three indicators , , and , which negatively correlate to . Therefore, this tile shall lie away from the centroid of the -(training) set and possibly in the right half-plane. If is regarded as a feature plane, then inspection of Fig. 10b suggests the training sample is separable; there is a straight line, e.g., , that divides the and sets. Each tile of the validation set could be classified accordingly. The confusion matrix was formed and the sensitivity and specificity values of Table 5 were obtained. Table 5Sensitivity and specificity of the classifier in validation mode.
8.3.Discriminant Function AnalysisClassification is said to be supervised whenever the class to which a feature vector belongs is known beforehand. Given classes, discriminant analysis (DA for short; e.g., Chap. 3 of Ref. 57) implements supervised classification by looking for affine combinations, called discriminant functions, of the features (the morphological descriptors), which maximize separation between classes and minimize within-class scatter. Three classes of belonging were defined: (1) the singleton , (2) made of 8 and 8 tiles, and (3) made of 8 and 8 tiles, hence . The corresponding raw feature matrix was normalized rowwise as in Sec. 7 and DA carried out both by SPSS™ and by the LDA module of Q-Parvus.60 The classification (or “confusion”) matrix was constructed. Since 14 tiles were assigned to the class and 2 to the class, whereas and all tiles were correctly classified, the relevant sensitivity (normalized row-wise sums of entries of ) and specificity (normalized column-wise sums of entries of ) of Table 6 resulted. A cutout of the discriminant functions plane is shown by Fig. 11 . For the purpose of classifier training and validation, DA answers the question of how suitable are the given morphological descriptors to discriminate between the given classes. Fig. 11Discriminant analysis where the and sets are made of training and validation tiles together; (boxes) and (triangles) tiles are represented in a plane where the discriminant functions form the reference frame. The tile is located further away from the origin. ![]() Table 6Sensitivity and specificity obtained from discriminant analysis.
Sensitivities and specificities shall all be greater than 50%, regardless of . In the ideal case, they are all equal to 100%, which corresponds to a diagonal . From the entries of Table 6 one deduced that information stored in the chosen descriptor set, , could adequately and reliably tell from tiles, a necessary condition for going over to the next, more challenging tasks: recognizing and ranking tiles from other experiments. 9.Quantitative Estimates of Structural Damage and RecoveryThe , , , , and sets were submitted to the trained classifier with the aim of representing their elements in and, in case of sensible results, quantifying cytoskeletal organization. Feature vectors were normalized and projected onto , as in Sec. 8.2. 9.1.Effects of Different TreatmentsProjection onto scattered the tiles almost evenly in the rectangle , as shown by Fig. 12 . Their centroid lay at . Tiles of the set (Fig. 9 of Ref. 45) were mostly aggregated in the area. Their centroid lay at . Fig. 12tiles (filled triangles) in the plane. Tiles derived from exposure to for . Training (gray boxes) and tiles (gray triangles) are shown for reference. ![]() Classification of tiles derived from exposure at is represented by Fig. 13 , which pertains to the set. Comments are included in the caption. Results for the set were reported in Fig. 10 of Ref. 45. The centroid coordinates were and , respectively. Fig. 13Tiles (filled disks) in the plane. Tiles derived from exposure to for . Training (gray boxes) and tiles (gray triangles) are shown for reference. As compared to tiles, tiles are more scattered. Some are closer to the half-plane. The tile placed at the extreme left exhibited the typical microtubule structure of a control cytoskeleton. ![]() 9.2.Quantitative Estimate of Structural Recovery after TreatmentUndisputable experimental evidence3 showed that cytoskeletal structures are capable of recovering after exposure to toxic substances, provided that dose does not exceed some threshold. Visual inspection of images (Sec. 3) suggested that cytoskeletal damage had to be reversible, at least in part. The first quantitative result about structural recovery was obtained by the classifier (Ref. 44 and Sec. 8.2). Herewith, the classifier (Sec. 7) was applied to the tiles and yielded the result of Fig. 14 . Only 3 out of 12 tiles are located in the half-plane and the remainder in the half-plane. Fig. 14Tiles (filled diamonds) in the plane. This result provides evidence of 0.73 structural recovery after extreme treatment ( for ). Tiles and are shown for reference. ![]() From the coordinates of centroids, , , and one can form the quotient and estimate tile-set-averaged cytoskeletal organization therefrom. Namely, corresponds to normal structure and to a fully functional cytoskeleton , whereas corresponds to the most severe damage of the whole experimental design . From Eq. 21,where the worst case confidence interval has been determined from the spread of the centroid coordinates (Sec. 8.2). The estimate compares to 0.77, determined by the classifier.44 This may be regarded as further evidence of classifier stability in the recognition mode.9.3.Ranking by CentroidsTo summarize the results, the centroids of all sets, except those used in validation, are displayed by Fig. 15 . According to the coordinate alone, the sets form the sequence where the order of the last three terms shall not be taken for final, because the confidence bounds of Sec. 8.2 apply.Fig. 15Set centroids, where containing the centroids and circumscribed to all classified tiles. Comments are provided in Sec. 10.3. The tile is not shown. ![]() 10.DiscussionThe relevance of microtubule dynamics in living cells and the sensitivity of cytoskeletal organization to anticancer drugs, xenobiotics, and pathological conditions have provided the basic motivation for this paper. Further motivation at the practical level has come from the abundance of images of cyto-skeletal microtubules obtained during previous experiments.3 In general, the heterogeneity of biological response and the variability of experimental results even under controlled conditions pose a challenge to quantitative classification by simplistic methods intended to support and/or replace visual assessment. The latter has been the only way to judge the degree of cytoskeletal organization for a long time. However, conclusions have always been influenced by the operator. 10.1.Image Formation, Capture, and AnalysisAll images were obtained from an ordinary epifluorescence microscope, acquired from photographic film developed and printed after each experiment. Therefore,
On one hand, the wide dynamic range of film obviously was an advantage because it made weakly emitting microtubules visible. On the other hand, parameter variability would have prevented classification based on absolute fluorescence values. The threshold of Eq. 2 (which implies Proposition 1, Appendix B) and the spectrum enhancement algorithm of Secs. 5.5, 6.4 (to which Proposition 2 applies) were selected as countermeasures for 1 and 2. The opportunity of replacing ordinary by confocal microscopy in the morphological analysis of the cytoskeleton is still a matter of discussion; in Ref. 38, confocal images were successfully processed; however, there are situations61 where ordinary microscopy has been more informative. In principle the whole procedure, from analysis to classification, can be applied to images acquired by different microscopes and sensors. 10.2.Meaning of Some Morphological DescriptorsThe estimated values of the fractal dimensions and agree with expectations (Sec. 4.1) and are easily interpreted accordingly. Fully developed microtubules of the control set do have indented contours (higher ) and lower mass density . The loss of organization caused by treatment (sets , , and ) yields a more compact cytoskeleton (higher ) with a smoother contour (lower ). The two descriptors from direct methods (Secs. 5.4, 6.3) are linked to visual properties of images. For example, the well organized microtubules of cells (Fig. 2) suggest a higher value of total variation than that of cells (Fig. 3), which is in agreement with Table 3. The norm of the Laplacian, , is uncorrelated to [Fig. 7b] and uniquely characterizes : microtubules around the MTOC, hence in tile , appear entangled under an ordinary, nonconfocal microscope, and as such do not exhibit any symmetry. This justifies the very large of and the anticorrelation of to . If classes are ranked by increasing one obtains the following sequence (Table 3) Similarly, ranking by yieldsBoth and have thus determined the positioning of the centroid in the lower half-plane and of the and centroids in the upper one. However, a detailed explanation of how microtubule arrangement affects and remains beyond reach.In spectrum enhancement (Secs. 5.5, 6.4) the subtraction of [Eq. 11] was the key feature of the algorithm. Namely, the subtraction of from [Eq. 12] represents the deviation of from the proposed asymptotics. Moreover, polynomial interpolation reduced the sensitivity of morphological indicators to the oscillations of . Separation of structure from texture is context dependent. In this application the values of either or in the interval /image (approximately in the case of Figs. 2 and 3) represented image structure, whereas those in corresponded to those textural details, which contributed to classification. In particular, local maxima of in /image were originated by high contrast bundles of microtubules, typical of control cells, not observed in damaged cytoskeletal proteins (Fig. 3). Since the selected degree of is relatively low , it is obvious that a local maximum to the right of the origin correlates with both and . If said maximum is peaked enough, then the of Sec. 6.4 is positive (lines and of Table 3). In the sets and all three descriptors have opposite signs. The simultaneous occurrence of these properties explains why , and have very close factor loadings [Fig. 7b]. The angle-averaged spectra shown in Refs. 36 (logarithmic intensity scale) and 37 (linear scale) can be compared to of Eq. 10. They do exhibit local maxima even without enhancement. The remarks in this section should help in justifying the classification result of Fig. 7a. 10.3.Overall Classifier PerformanceThe statistical significance of the described classification results is supported by the following arguments and figures. Section 8 provided the necessary evidence of classifier “stability” and validity for continuing on to applications. Recognition of a relatively large set of new tiles (67) in Sec. 9 were essentially an extended validation test of a classifier trained by a total of tiles: namely, the , , , , and centroids (Fig. 15) all belong to the rectangle , i.e., they fall within the strip defined by the and centroids after training. Moreover, none of the recognized tiles falls outside the rectangle and 46 out of 67, i.e., 68.6% fall within of Eq. 18. These results should not be taken for granted and prove that training based on a set of 17 tiles has captured enough of cytoskeletal morphology. The correspondence between the gross a priori threefold distinction and the a posteriori ranking of centroids (Fig. 15) has been established in terms of , which explains 58% of the sample variance.10.4.Innovative FeaturesAll image analysis methods quoted in Sec. 2 represented significant advances toward automated classification. However, each of them relied on only one type of morphological descriptor. Instead, this paper implemented the fusion of heterogeneous morphological descriptors: fractal analysis, direct methods, and spectrum enhancement together have led to the fingerprint of an image. Moreover, spectrum enhancement is a relatively new algorithm. It was introduced by the first author and so far it has been successfully applied to classify scattering patterns50 and electron microscope images of other materials such as ceramic nanoaggregates51, 52 and tire tread particles.53 Its first application to images of cytoskeleta was presented in Ref. 41. The third innovative feature of this paper is the application of the trained classifier to rank structural damage and quantify recovery. 10.5.Possible DevelopmentsAlthough multivariate statistics is a standard tool in image classification and pattern recognition, feature extraction remains task specific and deserves additional effort. Therefore, some improvements could be brought in to increase classifier throughput and to speed up the learning phase:
On the applications side, the analysis of image sets coming from different experiments on different cell lines would outline the “operating region” of the whole approach. 11.ConclusionImages of cytoskeletal microtubules obtained from the immunofluorescence microscopy of primary culture rat hepatocytes were processed. Contour and mass fractal analysis, direct methods, and spectrum enhancement were designed and tuned to make the extracted morphological descriptors insensitive to absolute fluorescence intensities. Data fusion for classification was achieved by means of multivariate analysis. Validation was supplemented by discriminant functions analysis. Finally, the classifier was applied to ranking structural damage and quantifying recovery. The innovative features of the classifier described here, other than the application of spectrum enhancement, are the concurrent implementation of different analysis methods, the fusion of morphological descriptors by multivariate statistics, and the application of the classifier to images from various experiments. Differences in cytoskeletal morphology were translated into the selected descriptors. The procedure developed and implemented here may lead to automatic image recognition. Namely, the method can become a tool for testing cytotoxicity and for extracting quantitative information about intracellular alterations of various origins. AppendicesAppendix AThe optimum threshold is defined as follows. Let the image be the discrete counterpart of a nonnegative function of position with support in the open set , the tile, which takes values in the interval . Denote by a threshold, , and by the binarized image. Let denote the relative frequency by which the gray level of is and define the below and above threshold means , , of according to Next define the first- and second-order moments of and :and letThe cross-correlation function of is defined byAs a consequence the optimum threshold is the value that maximizes [Eq. 2].Appendix BProposition 1. is invariant with respect to affine transformations of defined by Eq. 1. Proof. Let be the transformed threshold, denote the binarized counterpart of , and let , , , , and denote the corresponding moments. By applying the affine transformation of Eq. 1 it is immediate to show that the moments relate by and so forth. As a consequence the standard deviations of Eq. 28, scale according toand therefore the correlations can be shown to comply withFinally, the subdomains and satisfySince , i.e., the contours are the same, one concludes that the estimated contour fractal dimensions coincide.Appendix CLet be the Dirac measure supported at integer coordinates in the plane. As a consequence of periodicity, the Fourier transform of is the tempered distributionwith the ranges of and specified by Eq. 34. The integral over of Eq. 10 therefore is understood as a sum over the grid nodes where the distribution is supported. The normalized power spectral density derived from Eq. 35 isProposition 3 (formally). Assume the image is not degenerate, the derivatives of up to a suitable order exist as tempered distributions and the model exponent satisfies , integer.
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