The fractal dimension of chromatin - a potential molecular marker for carcinogenesis, tumor progression and prognosis

ABSTRACT

Introduction: Fractality is omnipresent in medicine and life sciences. In particular, the fractal principle is found simultaneously at different organization levels of the cell nucleus. The aim of this review is to show whether fractal characteristics of chromatin could be related to tumor pathology and pathophysiology.

Areas covered: This review provides an overview of the application of fractal measurements of chromatin or DNA for the characterization of physiological or pathological processes, in particular for the detection of preneoplastic changes, the characterization of tumor progression, the differential diagnosis between neoplasms and for prognosis. We used a network-based literature research strategy, i.e. after a systematic investigation by key-words, we looked for all citations (and the citations to these citations) of the selected papers in Scopus and Webofscience.

Expert opinion: The fractal dimension (FD) increases during carcinogenesis, thus permitting the diagnosis of malignancy. In various malignant tumors, a higher FD or diminished goodness-of-fit of its regression line indicates a more aggressive behavior and worse prognosis. Applying new spectral techniques, the chromatin FD can be estimated at scales below the light microscopic resolution. The latter also permits the examination of live cells and studies on field carcinogenesis and chemoprophylaxis.

KEYWORDS:

• Fractal dimension
• goodness-of-fit
• lacunarity
• succolarity
• prognostic factor

1. Introduction – fractals in biology and life sciences

Fractality is a mathematical construct related to self-similarity of an object or process over a range of scales [1–6]. The Euclidean geometry is an idealistic abstraction of nature. Whereas many natural objects are highly irregular, fractals are self-similar, i.e. they reveal a pattern which is repeatedly observed at different magnification, or, in other words, are built up of nested copies of the whole object.

The estimation for the fractal dimension (FD) is made comparing the topological dimension with the object´s space-filling properties [6]. Its value is between 1 and 2 for an irregular bi-dimensional object and between 2 and 3 for a 3D structure. In the last case, usually a ‘rough’ landscape-like surface fills more or less a 3D cube. A higher FD is equivalent to a more intensively folded and wrinkled object with larger surface. For FD estimation, we measure a feature of an object at a certain scale, for instance the number of intercepts with boxes. Then, this measurement is repeated at different scales and a so-called log–log diagram is built up with the x-axis representing the logarithmic values of the scaling unit and the y-axis the logarithmic number of intercepts with the structuring element. The data points can be well approximated by a linear regression line and its slope defines the FD. In theory, there is no specially defined scale for the measurement of a fractal, i.e. fractals are scale-independent and an ‘Ideal,’ mathematical fractal is self-similar over an unlimited range. Biological objects or processes in real life, however, are self-similar only within a limited range, also called the ‘scaling window,’ with an upper and a lower limit, defining the range wherein the fractal characteristic can be observed [3,4,6].

An object or process can be described by many measurement variables characterizing different features, which all may reveal fractal characteristics. Thus, different FDs may be present in the same object at the same time at different levels of analysis.

Ideal mathematical fractals can be simulated by iteration of a motif. Biological objects are often constructed in a similar way, but usually there is an additional influence of random effects, i.e. the structure of smaller parts is similar, but not identical to the whole structure. A broccoli head illustrates natural fractal growth with an iterative branching pattern at different scales. At a more detailed examination, however, we can see that in different parts and scales, the pattern is very similar, but not exactly the same (Figure 1).

Figure 1. A broccoli head as an example of a natural object with fractal characteristics regarding the branching pattern and the surface relief, which both repeat their pattern at different scales, but with some imperfections which are due to additional influences (‘noise’) modifying the ideal image during growth. Both lower pictures at higher magnification illustrate the limit of the scaling window, i.e. the end of the typical branching pattern.

Fractal analyses can also be done with time series and therefore characterize physiological processes [5]. Fractals can be found in the structure and function of brain [7–12], heart [13–17], lung [3,18], skeletal muscle [19], bone [20–22], blood components [23], eye [24–27], and placenta [28].

The fractal concept can also be applied to every kind of science, for instance biochemistry [29,30], pharmaceutical sciences [31], botany and ecology [32,33], sociology [34], economy [35], and music [36]. Even a fractal theory of the whole universe is under discussion [37].

The fractal concept has improved our understanding of many physiological phenomena [3–6,13,38–46], for instance, metabolic rate, intracellular bioenergetic dynamics, drug clearance, population genetics, tissue organization, and tumor growth. Fractal structures and processes are highly preserved during evolution due to several advantages:

1. Complex structures can be built in a parsimonious way, for instance by repetition of a simple process (program) many times, as found in branching.

2. Fractal structures are economic, e.g. fractal branching creates short distances for transport in a complex structure.

3. Fractal folding of surfaces permits to increase the surface area inside a 3D structure keeping its volume constant.

4. Physiological systems with power-law organization (revealing linearity in a log–log diagram) can adapt more rapidly to challenges from the environment, i.e. gain more stability for the organism.

5. Allometric scaling relations can be derived from a general model based on the assumption that nutrients are transported through a space-filling fractal. An example is the three-quarter power relationship between the metabolic rate and the mass of an organism. This model is able to predict structural and functional characteristics in vertebrates, invertebrates, and plants [47].

6. Fractal structures are basic conditions for self-organization. Life has been interpreted as an organizational system or phase of nonliving matter originated by evolution and self-organization [6,41,46].

2. Fractality – a construction principle of the cell nucleus

Regarding the omnipresence of fractals, it would not be a surprise to find fractality in the cell nucleus at different levels of organization. In 1989, Takahashi [48] suggested to use the FD as an estimate for chromatin condensation, prophesizing that a fractal theory of chromatin would find an application in the study of cancer cells. Later on, many different investigations followed the concept of a fractal organization of the cell nucleus [6,49–126].

Small angle neutron scattering demonstrated fractal organization in chicken erythrocyte nuclei, but with different dimensions according to the scaling window: FD = 2.4 for the range between 15 and 400 nm and FD = 2.9 above that range, and also a fractal nuclear protein organization (FD approximately 2.3) [61,62]. For the scaling window of 2–100 nm, a fractal model of chromatin was suggested with an FD = 2.6 for euchromatin and FD = 2.2 for heterochromatin [85,87]. The so-called Hi-C technique revealed a power-law scaling for intrachromosomal contact probabilities and gave origin to the concept of the ‘fractal globule’ [58]. In this model, a knot-free, very densely packed polymer conformation with a diameter of about 1 μm and an FD close to 3 is formed by nucleotide crumpling with the advantage of an easy and rapid unfolding without self-crossings or entanglements (Figure 2). Results of investigations with super-resolution microscopy are considered to be consistent with the expected properties of the fractal globule model [127].

Figure 2. Simplified scheme of the fractal globule formation. In the upper part the unfolded chromatide fibre with different gene regions. In the lower part the beginning of the fractal globule formation by crumpling provoking approximation of distant genes without entanglements or knots, thus permitting rapid unfolding.

Chromatin fibers can interact within a fractal globule [86] (Figure 2). It is believed that these fractal globules are packed together in chromosome territories with globules from adjacent chromosomes forming interdigitating surfaces. Bancaud et al. summarized different models of chromatin configuration, which support the concept of fractal architecture of the nuclear DNA mass distribution at length scales larger than 300 nm [87]. Other authors, however, claimed to have found inconsistencies in the fractal globule model [55,89,90,91]. Another theory, the so-called strings and binders switch model, provides a more general concept, which joins together the scaling properties of the chromatin folding process and fractal features of chromatin with domain formation and looping out, considering the ‘fractal-globule’ model as one of many possible transitory conformations [88].

There is clear evidence that the primary DNA sequence has a fractal structure with species-specific and evolutionary features both at the molecular organization level and as a whole [93].

Computing the distance distributions between loci of the histone proteins, a power law was found, which permitted to measure the correlation FD of chromatin with FD = 2.7. The latter was interpreted as an expression of a dynamically maintained nonequilibrium state. The authors postulated a high local scale mobility of the chromatin structure, but a stable organization at larger scales, which is sufficiently strong to prevent a shift from the fractal organization (in an out-of-equilibrium state) to the equilibrium state during the cell cycle [94].

A study on more than 3000 cancer specimens showed the presence of a fractal chromatin organization also in neoplasias and the relevance of this configuration for the formation of chromosomal alterations [74].

In summary, fractality is simultaneously present at different levels of nuclear organization. Now, we have to ask, whether all these levels are interrelated, for instance, whether a fractal arrangement of the DNA base sequence could lead to the emergence of a 3D fractal structure of chromatin. One hypothesis suggests that this might be due to the fractal features of fluctuating proportions of nucleotide bases [76]. Other authors discuss the possibility that ‘repeat pair interactions’ might provoke clusters in the 3D chromatin space, thus modulating higher order structures. Since repetitive DNA elements are considered to coordinate chromatin folding [95] and since repetitive DNA sequences from transposable elements are distributed according to power laws for inter repeat distances [96], we may deduce that the fractal organization of the DNA sequence probably contributes to the emergence of the 3D fractal structure [79].

The fractal organization in different levels of the nuclear architecture influences the gene regulatory network functions. The dynamics between genes and their transcription factors, as well as the gene expression time series, have fractal characteristics [128].

3. Techniques to measure FD of chromatin in conventional light microscopy

In this article, we will discuss the measurement of fractal characteristics of stained histologic sections or cytologic preparations. The resulting images are representations of the 3D dye-binding features to chromatin of fixed nuclei [97–149].

About two decades before the formulation of the fractal globule hypothesis, classical morphologists had already described a fractal organization of nuclear chromatin. In the majority of these investigations, microscopic photographs were binarized at a certain gray value threshold, resulting in FDs between 1 and 2. The main disadvantage for this procedure is that the cutoff point for thresholding may influence FD values, provoking loss of information [6]. Therefore, some authors prefer to estimate FDs in pseudo-3D images (also called 2.5D images), where the z coordinate represents the pixel´s gray level (Figure 3). The FD can now be estimated with the help of three-dimensional structuring elements of varying size. In this case, FD estimates ‘self-affinity’ rather than ‘self-similarity,’ because the z axis represents another variable (gray level) than the x and y axes (topography) (Figure 3).

Figure 3. Three virtual dye-stained nuclei created ‘in silico’ with different chromatin texture subjected to transformation into pseudo-tree-dimensional (2.5D) landscape- like images but using the gray-value for the z-axis.Please note the increasing space-filling character when moving from image a to images b and c, accompanied by an increase in the FDs. Images had been subjected to three different fractal estimation techniques: box-counting [153], or according to Minkowski-Bouligand [115] or Einstein [98]. Please note that besides the absolute values being different, the trend to higher FDs for more space-filling images is independent of the technique.

In the lower third we see the three nuclei after binarizing according to the Otsu-algorithm with the respective box-counting fractal dimension and lacunarity values. Please note, that for the calculi the inverse images (with black background) were used.

Besides the box-counting method and its variations, there are other techniques to estimate the FD [150, 151].

In the Richardson method, the perimeter of the object of interest is measured using rulers with different lengths. FD is estimated by the slope of the regression line in the bi-logarithmic plot built up by perimeter and ruler length. In the dilation technique, the object is dilated by a disk with varying radius r and the logarithm of the total area of the dilation A(r) is plotted against the logarithm of the disk radius r. In practice, this area corresponds to the number of pixels at a distance at most r from the object of interest. The dimension is given by N − α, where N is the Euclidean dimension of the space surrounding the object and α is the slope of the straight line fit to the bi-logarithmic curve. Another geometrical approach is the Einstein method [99] applying the difference between the extreme values assumed by the image function within a circle with radius r. This circle is centered at each point of the surface and the variation in each point is summed up, resulting in the total variation V(r). Again, the dimension is provided by the slope of a straight line fit to the curve log(r) × log(V(r)).

Another kind of approach is based on the fractional Brownian motion, usually employing variograms or Fourier transform. The variograms are based on the difference between values at a specific distance when such distance is varied. The Fourier method computes the Fourier transform of the image and obtains the dimension from the slope of a straight line fit to the curve of the logarithm of the Fourier magnitude against the logarithm of the frequency.

Considering that in real life perfectly fractal-shaped structures do not exist, an ideal scaling in the log–log plot may not be present. In other words, the points may not lie perfectly on the calculated, ideal straight regression line, but somehow close to it (Figure 4). This property can be quantified by measuring the ‘goodness-of-fit’ represented by the R² value of the Pearson correlation coefficient R between the real and the estimated y values in the diagram. For an ideal and ‘perfect’ fractal, R² is 1.0, but real-world fractals have values below that.

Figure 4. Blast of a patient with acute myeloid leukemia.

On the left, cytologic preparation after May-Grünwald-Giemsa staining. Interactive segmentation of the nucleus and transformation into gray values. On the right, Pseudo-3D representation. FD is estimated in the log-log-plot from the slope of the ideal linear regression line obtained by curve fitting. X-axis represents the logarithmic values of the inverse values of the size of the structuring element. The y axis shows the logarithmic values of the intercepts. Goodness-of-fit describes how close the real data points are to the ideal mathematical regression line.

4. Fractal changes of chromatin in different physiologic and pathologic conditions

4.1. Fractal changes of chromatin during normal development at the light microscopic scale

Nuclear FDs increased during mesenchymal stem-cell differentiation [152] but decreased during transition from normal myeloblasts to promyelocytes [130].

In cardiomyocytes, macula densa cells, and splenic follicular cells, chromatin FD values were reported to decrease during postnatal development [6,117,132] and the same was shown for hepatocytes during lifetime aging [108]. Alterations of the nuclear architecture during physiologic development, growth, and aging may reflect epigenomic changes and a topological redistribution, accompanied by the FD of the global nuclear architecture. After UV irradiation of cell cultures, nuclear FD significantly decreases in early apoptosis. In this experiment, fractality changes occurred earlier than the detection of DNA fragmentation and cell membrane permeabilization, which are commonly used for the diagnosis of apoptosis [153]. In a study on carcinoma tissue, the FD of mitotic nuclei was significantly lower than that of their non-mitotic counterparts, due to the chromatin condensation during metaphase [154].

4.2. Fractal changes of chromatin for the characterization of neoplastic tissue on the light microscopic level

Several investigations using routine histological or cytological preparations have shown higher chromatin FD in neoplastic tissue than in corresponding normal tissue. This has been demonstrated for dog lymphoma nuclei compared with normal B-cell progenitors [126], cells of chronic lymphocytic leukemia, follicular lymphoma and diffuse large B-cell lymphoma versus cells of normal lymph nodes [111], or chronic lymphocytic leukemia (B-CLL) cells compared with normal peripheral blood lymphocytes [155].

The nuclear FD of oropharyngeal carcinoma [145, 156] and hepatocarcinoma [157] is significantly higher than that of corresponding normal tissues.

The same was true for malignant epithelial cells in urinary cytology [158], in human and canine breast tissue and human cervical lesions [159,160,161]. In another study on breast cytology, only the ‘worst’ FDs but not the mean FD values were significantly higher in breast cancer cases [162].

Nuclear FD permits to distinguish different tumor subtypes, e.g. in basal cell carcinomas [113] or leukemias [163].

5. Fractal changes during carcinogenesis and tumor progression

Exposure of human blood cells in culture to ionizing irradiation increased the FD of nuclear chromatin [67]. During carcinogenesis and tumor progression, FD of nuclear chromatin increased in biopsies [135] and cytologic smears from the uterine cervix [142,143], as well as in material from precancerous oral lesions [109] (Table 1). In one study on oral cytology, the authors claimed to have found a significant difference between the FDs of low- and high-grade dysplasias, but the reported difference is extremely small [164].

Table 1. Fractal dimension as a marker of pre-neoplasia and tumor progression using light microscopy.

Tissue	Image	Results	Ref.
In cytologic preparations
Cervix	Binarized	FD increasing from normal to HSIL	[143]
Cervix	Pseudo-3D	FD increasing from normal to HSIL	[142]
In histological preparations
Cervix	Binarized	FD increasing from normal to CIN3	[135]
Oral mucosa	Binarized	FD increasing from normal to dysplasia and to carcinoma	[109]
Pancreas	Binarized	FD higher in non-resectable than resectable adenocarcinomas	[146]
Thyroid	Pseudo-3D	FD increases with the size of follicular adenomas	[112]

FD: Fractal dimension; HSIL: high-grade lesions.

In histological routine preparations, the FD of nuclei was higher in unresectable than in resectable adenocarcinomas of the pancreas [146]. In summary, we can state that usually chromatin FD increases with tumor progression. For this reason, the positive correlation between the size of follicular adenomas and nuclear FD was thought to be a potential indicator of an adenoma–carcinoma sequence in this gland [112].

Since the segmentation of individual nuclei in tissue sections is considered to be cumbersome, several research groups developed different ways for fractal analysis by using whole images where nuclei had been identified but not isolated individually for separate FD measurements. The ‘global’ FD values comprise FDs of several nuclei, the spatial relation, and eventually cytoplasmic features. Interestingly, their results are comparable with the aforementioned investigations on chromatin of individualized nuclei, i.e. the FDs increase with tumor progression [101,107,109,138,146,165].

6. Fractality of chromatin in as a prognostic factor at light microscopy

Since FD increases during the progression of intraepithelial malignant clones to invasive carcinoma and further tumor evolution, it is obvious to postulate that in a tumor biopsy higher chromatin, FDs should indicate more aggressive behavior with shorter survival.

Several studies have corroborated this theory. The FD of nuclear chromatin is a negative prognostic factor for survival measured in histological sections of malignant melanomas of the skin [166] and in cytologic preparations of small cell neuroendocrine carcinoma of the lung [104].

Moreover, the fractality of the nuclear chromatin texture is also important in hematologic neoplasias (Table 2). In a preliminary investigation, based on only very few patients, some relation between the FD of routinely stained leukemic blasts and the clinical outcome has been suggested [111]. In subsequent studies on multiple myelomas [115], acute B-precursor leukemia [116], and acute myeloid leukemia [167], the variable ‘goodness-of-fit’ of the regression line was a more important prognostic factor than FD (slope of the regression).

Table 2. Fractality as a prognostic marker using light microscopy.

Neoplasia	Image	Results	Ref.
In cytologic preparations
Myeloma	Pseudo-3D	FD only in univariate analysis, goodness-of-fit in multivariate, favorable survival	[115]
AML	Pseudo-3D	Goodness-of-fit in multivariate, favorable survival	[167]
B-ALL	Pseudo-3D	Goodness-of-fit in multivariate, favorable survival	[116]
Small-cell neuroendocrine carcinoma of lung	Pseudo-3D	FD, multivariate, adverse survival	[104]
In histological preparations
Cutaneous melanoma	Pseudo-3D	FD, multivariate, adverse survival	[166]
Meningioma	Binarized	FD (univariate, adverse, recurrence)	[119]

AML: Acute myeloid leukemia; B-ALL: acute precursor B lymphoblastic leukemia.

Higher nuclear FDs in histological slides of meningiomas indicated a tendency to faster recurrence in the univariate Cox model, but no longer in the multivariate analysis [119].

In some prognostic factor investigations [98,109], whole images harboring several nuclei were submitted to an FD analysis, as described in the article earlier. Again, with increasing ‘global’ FDs, patients had worse prognosis. Although these results are similar to those based on single cell analyses, they are more difficult to interpret, since the ‘global’ FD represents chromatin and tissue texture characteristics simultaneously.

6.1. Light microscopic fractal changes of chromatin in field carcinogenesis

It had been shown by light microscopy that in normal appearing diploid tissue adjacent to prostate carcinomas, the nuclear texture is significantly different to that of tissue around hyperplastic nodules. The discrimination algorithm was based on three components, two of them being fractal-derived variables: more precisely the mean FD and the standard deviation of the fractal area [168]. In a similar way, nuclear FD was significantly higher in histologically normal epithelium adjacent to oropharyngeal cancer than in the normal contralateral mucosa, but lower than that of carcinoma cells [156].

6.2. Fractal measurements of chromatin at the nanoscale level in different pathologic conditions

Transmission electron microscopic and modern spectroscopic techniques, such as partial wave spectroscopy and inverse spectroscopic optical coherence tomography, are methods to estimate fractal features in scales beyond the limits of light microscopy [68,69,70, 169–172].

Several fractal features estimated from the peripheral chromatin in transmission electron microscopy images of mouse liver cell nuclei were able to distinguish between normal cells, hyperplastic nodules, and carcinoma [121]. Interestingly, the outer 25–30% of the cell nuclei contained relevant information about the differences between the entities. In diabetes type 2 patients, fasting blood glucose levels were negatively correlated with FDs of lymphocyte chromatin measured in electron microscopic images. Moreover, after treatment with metformin, FD of chromatin dropped significantly [131].

Studies on electron microscopic chromatin images showed a decrease of FD during apoptosis of human breast cancer cells [173]. In an investigation on nuclear contours in electron microscopy images, the higher FD of mycosis fungoides cells permitted differentiation from benign lymphoid cells [174].

In summary, the fractal studies based on electron microscopic images corroborate the findings made by light microscopy.

Partial wave spectroscopy at nanoscale dimension enables to observe chromatin changes represented by the mass FD during early stages of carcinogenesis, when these alterations are still not detectable by the human observer, with light microscopy. Again, with progression of carcinogenesis, the mass FD increased in the same way as reported in previous articles on studies using light microscopy. The important difference is that the scaling window of spectral techniques is approximately between 20 and 200 nm, whereas that of light microscopy usually has a range between 400 and 20,000 nm.

7. Fractality estimated in virtual DNA walk plots

An elegant way to determine fractality of DNA is to transform the sequence of the bases in a ‘random walk’ of a point in a 1D, 2D, or 3D space. For example, in the 2D-plot method, the sequence of the four bases deflects a point up, down, right, or left according to a previously established rule, thus creating a planar Brownian-motion-like trajectory. In a second step, this virtual image is subjected to fractal analyses. Adaptions for 2D and 3D of this model have been described [3,52,175,176]. Since damaged DNA sequences showed higher FDs than normal DNA, this method is considered to be potentially useful for early detection of lung cancer or other types of neoplasia [3,52,175,176].

This method can not only reveal mutations on a global level but also changes of sequences between wild-type and cancer cells on a micro level and is considered to give important information about potential functional consequences of these mutations [175].

8. Complementary techniques for texture analysis of chromatin

8.1. Fractality versus lacunarity and succolarity

Benoit Mandelbrot [177] created not only the concept of fractals and FD but also two further complementary texture descriptors: lacunarity [49,117,132,133,153,177–185] and succolarity [177,186]. In mathematical terms, lacunarity is defined as the exponent of a lacunarity measure at different scales, being the quotient between the second moment of the point distribution and the squared first moment of the same distribution. Thus, lacunarity describes in a global manner the size and distribution of gaps, their spatial heterogeneity, and the degree of deviation of the image from translational invariance. Higher lacunarity is equivalent to marked heterogeneity of gap distribution with a more coarse and clumped appearance and a higher sensitivity to rotation, i.e. the image changes more after rotation than an image with low lacunarity.

Several methods can be employed to calculate this feature [178]. The most popular one is ‘gliding box’ in the binarized [187] image, where a box sweeps the image domain counting the number of points falling within the box at each position. By changing the box size, one can again construct a curve of the lacunarity measure in log–log scale and compute it from the slope of the fitted straight line. The differential box-counting approach replaces the number of points in the lacunarity measure by the variation (difference between the minimum and maximum value within a box). The three-term local quadrant variance method divides the gliding window into three blocks and uses the variance between the number of points in each block to compose the lacunarity measure.

There are only few studies where lacunarity was investigated for the evaluation of chromatin (Table 3). It is interesting to note that in these investigations, lacunarity changes (increasing) were always inverse to alterations of FD (decreasing), such as apoptosis [153], postnatal development, or aging [108,117,132] besides nuclear incorporation of nanoparticles [179,182]. This is, of course, not a general rule, as we can see in Figure 3, where FD and lacunarity are not related.

Table 3. Lacunarity and fractal changes of chromatin in different disease models.

Disease model	Chromatin fractal dimension	Chromatin lacunarity	Ref.
Incorporation of silver nanoparticles in buccal epithelial nuclei	Decreased	Increased	[182]
Incorporation of iron oxide nanoparticles in buccal epithelial nuclei	Decreased	Increased	[179]
Aging of hepatocytes	Decreased	Increased	[108]
Postnatal changes in nuclei of spleen follicular cells	Decreased	Increased	[132]
Postnatal changes of macula densa cell nuclei	Decreased	Increased	[117]
UV-induced glioma cell apoptosis	Decreased	Increased	[153]
UV-irradiated neutrophil granulocytes	Decreased	Increased	[181]
Hyper-osmotic stress in chondrocytes	Not reported	Increased	[184]
Adrenal fasciculate cell nuclei	Direct correlation with circularity	No correlation with circularity	[133]

Succolarity is defined as a measure of the capacity of percolation when the binary image is interpreted as a hypothetical physical barrier through which a fluid intends to cross. Its value is obtained by the normalized product between the area of the flooded space and the fluid pressure. Succolarity indicates the capacity of a fluid to cross the set after binarizing the image and defining, e.g. the white pixels as obstacles, whereas the liquid flows in the black ones. Succolarity evaluates the degree of the percolation (penetration) capacity of a hypothetic fluid in a defined direction and thus estimates connectivity and intercommunication [177].

The most popular approach to compute succolarity uses similar ideas to those employed for box counting. The image is partitioned into a grid of boxes with varying size r. The succolarity is given by the summation of O(r) × P(r) for all possible values of r divided by the summation of P(r), where O is the spatial occupation and P is the fluid pressure. Succolarity was only rarely applied as a metric for DNA [49].

In summary, speaking in an intuitive manner, FD estimates how much objects occupy the underlying metric space, lacunarity the size and arrangements of gaps, and succolarity the capacity of a hypothetic fluid to cross the set.

8.2. Gray-level co-occurrence matrix

Fractal descriptors can be combined with other methods of image analysis [106,114,131,132,162,167,168]. A widely used statistical method of texture analysis, which is not related to the fractal concept, is the gray-level co-occurrence matrix, also known as gray-level spatial dependence matrix. It quantifies the frequency of combinations of neighboring pixels with certain gray values (e.g. the image contains 40 times the pixel pair with the gray values 87–126). In a second step, various statistical variables are extracted describing different aspects of the relationships between pixels, such as local variations, joint probability, sum of squared elements, or entropy. In comparison with fractal features, gray-level co-occurrence matrix variables may have inferior [132] or superior capacity for discrimination [162]. This depends always on the specific set of images under investigation. We are not aware of general rules which could predict which kind of texture analysis would be the best way for an efficient and parcimonious classification or prognostication in a specific experimental situation. Whereas the fractal measurements of chromatin correspond to projections of real existing fractal structures or processes in the cell nucleus, descriptors of the co-occurrence matrix are mostly abstract mathematical concepts (for instance ‘second angular moment’) without clearly defined biological equivalent.

9. Expert opinion

Changes of the chromatin FD indicate widespread modifications of the nuclear architecture. Isolated alterations, such as the  FLT3 mutation with internal tandem duplication (FLT3-ITD) in acute promyelocytic leukemia, may not be  clearly detectable by fractal measures  [163].

We would expect that all pathological states with widespread changes of the chromatin architecture should also modify the FD. For example, in various carcinomas, nuclei with overexpression of satellite transcripts may provoke global alterations in heterochromatin silencing [188], or, as in small-cell cancer cells, Nfib amplification provokes increased chromatin accessibility in many intergenic regions [189]. If our hypotheses were true, changes of the chromatin FD should be detectable at the light microscopic level in both situations.

Our literature review permits the conclusion that the FD of nuclear chromatin increases during carcinogenesis. In fully established malignant neoplasias, FD is positively correlated with aggressiveness and negatively with survival. When this is not the case, the goodness-of-fit of the regression curve may be of prognostic relevance, lower values indicating bad prognosis. It is astonishing that these conclusions are based on studies of different laboratories with varying staining and fractal estimation techniques, using histologic or cytologic material. Regarding the variation of techniques, several theoretical and experimental studies demonstrated that 2D projection of a nucleus still represents well the fractal properties of the 3D organization [6]. Absolute FD values cannot, of course, be compared between studies from different laboratories, but inside a single investigation. For routine implementation, however, rigorous standardization and application in different centers would be necessary.

Carcinogenesis and tumor progression are due to both genomic and epigenomic alterations provoking changes of the 3D organization of the nucleus. They comprise single-nucleotide substitutions, gene fusions, deletions, insertions, rearrangements, copy-number alterations, chromosome deletions, or duplications, with tens of thousands of additional mutations termed passengers. Common alterations are the increased activity of oncogenic pro-growth pathways with simultaneously inactivated tumor suppressors [6,69].

In malignant neoplasias, epigenetic modifications are found, such as global hypomethylation and foci of hypermethylation [6]. During carcinogenesis and tumor progression, these changes increase in number. The nucleus acquires dark chromatin spots with ‘clear’ hypomethylated surroundings, resulting in an increasing number of chromatin areas of varying staining intensity. Additional gene amplifications and alterations of the chromosomal positions could form more irregularly folded borders between the darker (inactive) and lighter (active) chromatin, thus creating a ‘rougher’ and more space-filling surface, i.e. a higher FD [6].

An alternative hypothesis has been proposed, based on changes in the hetero-euchromatin balance. Every pathologist knows the diagnostic value of heterochromatin changes in neoplastic nuclei. The increase in chromatin FD is attributed to alterations of the proportions between eu- and heterochromatin, which have different FDs [6] (Figure 5).

Figure 5. Design of a routinely dye-stained nucleus with darker areas of more compact heterochromatin and the clearer areas of euchromatin.

Simplified representation of a zig-zag arranged chromatin fibres composed of many nucleosomes with DNA wrapped around histone protein cores. Many dyes stain histone proteins. Therefore, heterochromatin is intensively stained and ‘open’ euchromatin areas with less histone proteins, which allows more transcription activity, bind fewer dye molecules and appear clearer in the light microscopic image.

A more refined theory is based on findings with transmission electron microscopy, partial wave spectroscopy, and inverse spectroscopic optical coherence tomography in fixed material and live cells [69,70]. According to this theory, the nuclear nanostructure becomes more heterogeneous in the early stages of carcinogenesis. The increased heterogeneity of the physical chromatin structure is paralleled by a higher FD, which correlates with an increased heterogeneity of gene networks with both higher chromatin accessibility and compaction heterogeneity. Higher FDs are associated with increased tumor aggressiveness and worse prognosis. With rising FDs, there is an increase in the accessible surface area and the variations of local chromatin compaction, both with competing effects on the global gene expression, i.e. simultaneous gene activation and repression. Expression of some genes is related to the FD. These genes regulate cellular pathways, in particular glucose metabolism and suppress mitochondrial activity, thus provoking a shift toward glycolysis as FD increases.

The variable ‘goodness-of-fit’ is a very robust texture analysis feature, since it is much more independent of staining variations than the FD [114]. This may in part explain why in some studies only this variable and not the FD was associated with survival. The observation that a chromatin configuration approximated by an ideal fractal (high value of R²) has a better prognosis than a structure with a lower goodness-of-fit may have the following explanation: during initial carcinogenesis, there is an increase in heterochromatin content and clump size together with a transition of the spatial distribution from a fractal to a stretched exponential function in the 20–200-nm scaling window. We may hypothesize that these nanoscale changes might have some influence on the higher 3D configuration with changes also in the >400 nm scale window and a shift to a ‘lesser perfect fractal configuration,’ with lower goodness-of-fit. Therefore, we postulate that differences in the R² values are equivalent to some shift away from the ideal fractal configuration and modify the transcription activity of the nucleus [69].

This concept of the heterogeneity of chromatin packing gives the possibility to measure therapeutic effects by describing the mass FD. Chemotherapy can regulate fluctuations of chromatin density in vitro, lower the global transcriptional activity, and thus kill tumor cells [63].

In that way, the fractal concept is a fundamental part of the theoretical framework that may be used in this new kind of therapy approach and mass fractal measurements in vivo may accompany chemotherapy or even predict therapy success.

In summary, the fractal concept is useful for the evaluation of carcinogenesis, tumor progression, chemoprophylaxis, and therapy, applying traditional light microscopy or recent spectroscopic techniques with nanoscale resolution.

9.1. Five-year view

The use of the FD for chromatin evaluation is interesting, since it is a way to describe genetic and epigenetic changes in a global way without focusing on specific genes or chromosomes. For the measurement of nuclear fractal features in routine slides, we need a light microscope with 100× oil immersion objective, a commercial high-resolution photographic equipment, and a software for segmentation and extraction of fractal features. The whole process from taking pictures until getting the computational data takes about 60–90 min for 100 cells for a trained operator. The use of slide scanners has some drawbacks, since we need 100× oil immersion objectives and uncompressed images. Fractal features are sensitive to every kind of non-lossless image compression [190] and noise [191]. An important advantage of fractal features is their robustness against even larger variations of the manually segmented nuclear area [192]. For the routine use in pathology or cytology laboratories, larger studies on the influence of the equipment and inter- and intralaboratory variability would be necessary in order to elaborate standardization rules.

Several researchers choose to use the whole image for FD estimation of histological slides with or without some segmentation, which is an easier method. The problem is that these approaches evaluate nuclear chromatin, eventually cytoplasmatic structure and, mostly, the intercell relations. From a pragmatic point of view, these approaches seem to work, but a theoretical interpretation of the chromatin fractal features is more difficult or impossible and thus does not contribute to the understanding of the nuclear pathophysiology. Unless all these drawbacks are resolved, the ‘classical’ pathologists and cytologists may hesitate to use fractal measurement of individualized nuclei in routine work.

On the other hand, recent research in biophysics and biomedical engineering presented new quantitative imaging methods in the sub-light-microscopic, nanoscale dimension. One central variable in all these studies is the estimation of the mass FD as a marker of global 3D changes of the nucleus. These new technologies have corroborated former theories based on classical light microscopy studies. Moreover, the mass FD can be measured in live cells, and so, applications are possible in endoscopic supravital biopsy material. Moreover, with an estimation of the FD in the range of 20–200 nm, very subtle chromatin alterations can be seen, when light microscopic changes are still not visible.

Therefore, these new methods are able to study carcinogenesis and, more importantly, the application and efficiency of chemoprophylaxis. The latter is a new area that will get important pulses from these technical innovations still at the beginning [172].

The mass fractal measurement of the nanoscale structure of chromatin is rather fast and takes only a few seconds for up to 100 cells within a single field of view (personal communication: e-mail from Vadim Backman 2–11–2019).

It would also be interesting to examine macroscopically manifest tumors and see whether the mass FD could predict in small biopsies (or even ‘in situ’ during endoscopy) the presence of lymph node metastases, disease-free or overall survival, or therapy response. Furthermore, an application in cytologic smears should be investigated. Probably user-friendly devices will be commercially available in future [172].

We are convinced that in all these areas, the fractal concept will be important for diagnosis, prognosis, and prediction of therapy success in the next years.

Article Highlights

• The fractal concept has improved our understanding of many biological structures and physiological phenomena, in particular of the cell nucleus, where fractality is simultaneously present at different organization levels.

• Stained histologic sections or cytologic preparations reveal fractal characteristics of chromatin regardless of the staining or analysis techniques used at a scale between 400 and 20 000 nm on light microscopic examination.

• Fractal characteristics of chromatin can also be shown in a range between 20 and 200 nm, by more sophisticated methods, such as electron microscopy or spectral analysis techniques, the latter also permitting the examination of live cells.

• The fractal dimension of chromatin rises during carcinogenesis and tumor progression, and these changes are detectable even at the beginning of carcinogenesis.

• Neoplasms with higher fractal dimension of chromatin or a lower goodness-of fit of its regression curve are more aggressive and show worse clinical outcome.

• The measurement of the mass fractal dimension is a promising marker for the early detection of carcinogenesis and can be applied in chemoprophylaxis studies.

• An increased mass fractal dimension is considered equivalent to a larger area of accessible chromatin and increased variation of chromatin compactation in the nucleus, with simultaneous gene activation or repression, causing carcinogenesis and tumor progression.

Declaration of interest

The authors have no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.

Reviewer Disclosures

Peer reviewers on this manuscript have no relevant financial relationships or otherwise to disclose.

Acknowledgments

The author would like to thank Adilson A Piaza, Mário M. da Silva e Mercedes F. Santos for assistance with the creation of the illustrations, Fernanda A Borges da Silva for secretarial service and Prof.Dr.Lorand-Metze for critical comments.

Thanks to Dr.V. Backman, Dr. J. Jabalee and Dr M.Guillaud for providing additional technical details of their studies.

Additional information

Funding

The authors received grants from the National Council of Technological and Scientific Development CNPq (project 308975/2014-6, project 301480/2016-8) and 309910/2018-8), and the State of São Paulo Research Foundation (FAPESP) (Proc. 2016/16060-0).