Fractals and their contribution to biology and medicine

Fractals and their contribution to biology and medicine

by G. A. Losa, Switzerland

Gabriele A. LOSA, PhD
Fellow Member of the European
Academy of Sciences, Institute of Scientific Interdisciplinary
Studies (ISIS), Locarno

The term Fractal coined by Mandelbrot from the Latin adjective fractus (fragmented, irregular) derives from the Latin verb frangere, meaning to break, to create irregular fragments. To be called fractals, biological and/ or natural objects must fulfill a certain number of theoretic and methodological criteria including a high level of organization, shape irregularity, functional morphological and temporal auto-similarity, scale invariance, iterative pathways, and a non-integer peculiar fractal dimension [FD]. Whereas mathematical objects are deterministic invariant and self-similar over an unlimited range of scales, biological components and morphological structures are iterated entities statistically self-similar only within a fractal domain called “scaling window,” ie, only within this scaling window can the scale-invariant (fractal) properties of an irregular object of finite size be observed. The latter needs to be experimentally established for each element, while the scaling range has to account for at least two orders of magnitude. The application of the fractal principle is very valuable for measuring dimensional properties and spatial parameters of irregular biological structures, for understanding the architectural/ morphological organization of living tissues and organs, and for achieving an objective comparison among complex morphogenetic changes occurring through the development of physiological, pathologic, and neoplastic processes. Emphasis will be laid on the fractal contribution to the knowledge of cell membranes, hematological tumors, cell tissue cancers, and brain tissues in healthy and diseased states.

Medicographia. 2012;34:365-374 (see French abstract on page 374)

The Fractal Geometry of Nature, Benoît Mandelbrot’s masterpiece, has provided a novel epistemological framework for interpreting real life and the natural world in a way that avoids any subjective view.1 Founded upon a body of well-defined laws and coherent principles, including those derived from chaos theory,2 fractal geometry allows the recognition and quantitative description of complex shapes, living forms, biologic tissues, and organized patterns of morphologic features correlated through a broad network of functional interactions and metabolic processes that shape adaptive responses and make the process of life possible. Obviously, this is in opposition to the ancient, conventional vision based on Euclidean geometry and widely adopted concepts, such as homeostasis, linearity, smoothness, and thermodynamic reversibility, which stems from a more intuitive—but artificially ideal—view of reality. In the chapter of his work entitled Epilog: The Path to Fractals, Benoit Mandelbrot wrote “The reader knows well that the probability distribution of fractals is hyperbolic, and that the study of fractals is rife with other power law relationships.” Although Mandelbrot’s famous seminal paper on statistical self-similarity and fractal dimension dates back to 1967,3 and the first coherent essay on fractal geometry was published 35 years ago,4 it is worth here recalling exactly how and when the ‘‘heuristic introduction’’ of this innovative discipline occurred or, more vividly expressed, when ‘‘the irruption of fractal geometry’’ into the life sciences such as biology and medicine actually took place.5

Although there no precise date can be given, it is generally agreed that fractal geometry was introduced during the ‘‘golden age’’ of cell biology—that is, between the 1960s and 1990s, under the impulse of Swiss and French groups.6,7 It was discovered that biologic elements, unlike deterministic mathematical structures, express statistical self-similar patterns and fractal properties within a defined interval of scales, termed “scaling window,” in which the relationship between the observation scale and the measured size or length of the object can be established and defined as the fractal dimension (FD).8 The fractal dimension of a biological component remains constant within the scaling window and serves to quantify variations in length, area, or volume with changes in the size of the measuring scale. However, concrete “fractality” exists only when the experimental scaling range encompasses at least two orders of magnitude, namely, spans two decades on the logarithmic scale axis. Data spanning several decades of scale have been previously reported in many other fields: thus, defining a “scaling range” appears an inescapable requisite for assessing the fractality of every biological element. This emphasizes Mandelbrot’s statement “fractals are not a panacea; they are not everywhere.”9

To conclude, the fractal dimension is a statistical measure that correlates the morphological structural complexity of cellular components and biological tissues.10 Fractal dimension is also a numerical descriptor that measures qualitative morphological traits and self-similar properties of biological elements. Recourse to the principles of fractal geometry has revealed that most biological elements, whether at cellular, tissue, or organ level, have self-similar structures within a defined scaling domain that can be characterized by means of the fractal dimension.

Figure 1
Figure 1. Changes of surface density estimates for outer and inner mitochondrial membranes, with increased magnification.

After reference 6: Paumgartner D et al. J Microsc. 1981;121:51-63.
© 1981, The Royal Microscopical Society.

Cell membranes and organelles

Application of fractal geometry to cell biology stemmed from the discovery that cellular membrane systems had fractal properties. What started it all was the uncertainty of observations regarding the extent of cell membranes in the liver, as findings from morphometry studies of liver cell membranes by various laboratories failed to match. This triggered much debate as to which of these estimates was correct, and whether liver cells contained 6 or 11 m2 of membranes per cm3, quite a significant difference. This cast doubt on the reliability of stereological methods, since they yielded conflicting results when measurements were made under different magnifications of the electron microscope. Ultimately, it was found that the estimates of surface density of liver cell membranes increased with increased resolution.6 Mandelbrot suggested that these results were attributable to a scaling effect, analogous to the ‘‘Coast of Britain effect.’’3 This explained why measurements of liver cell membranes at higher magnification yielded higher values than at lower magnification.6,11 This scaling effect applies mainly to cellular membranes with a folded surface or an indented profile, such as the inner mitochondrial membrane or the rough endoplasmic reticulum (ER). In fact, the surface density estimate of rough membranes was found to be increased with increasing magnification, while the surface density measure of the smooth outer mitochondrial membrane and of the smooth ER counterpart was only slightly affected by the resolution effect (Figure 1).6

Figure 2. Electron microscopy view of human lymphocytes.

A. Healthy human suppressor T lymphocyte [CD8] haracterized by a wrinkled cell surface. Magnification: 18 400×. B. Human lymphoblasts
of acute leukemia (T-ALL), characterized by a smooth cell surface and a low fractal dimension. Magnification 18 400×. After reference 23: Losa GA et al. Pathol Res Pract. 1992;188:680-686. © 2011, Elsevier GmbH.

Fractal analysis proved particularly useful with regard to electron microscopy for the objective investigation of fine cytoplasmic structures and the organization of various types of chromatin, nuclear components, and other subcellular organelles, both in normal and pathological tissues and cell cultures. Thus, external nuclear membranes (ENM) and nuclear membrane-bound heterochromatin (NMBHC) domains of human breast cancer MCF-7 cells briefly triggered by steroid hormones, such as 17β-estradiol or dexamethasone, were shown to undergo ultra-structural changes at the beginning of growth, which were quantified by their fractal dimensions. Indeed, after a very short treatment (5 min) with 17β-estradiol (1 nM), the ultrastructural irregularity or the DNA unfolding of the NMBHC domain was significantly enhanced as documented by an increase its fractal dimension, whereas with dexamethasone (1 nM) it was reduced. Neither steroid significantly modified the ENM ultrastructure.12

This fractal tool has also been employed to document the feasibility of using ultra-structural changes in cell surface and nuclear inter(eu)chromatin to assess the early phases of apoptosis (programmed cell death) induced in human breast cancer SKBR-3 cells by the ionophore calcimycin. The ultra-structural changes that involved a loss in heterochromatin irregularity due to its increased condensation quantified by a lower fractal dimension were evident well before the detection of conventional cell markers, which were only measurable during the active phases of apoptosis.13 Similarly, it was shown that the nuclear complexity of human healthy lymphocytes14 underwent reduction during the apoptotic process.15 Measuring the fractal dimension of euchromatin and heterochromatin nuclear domains helped discriminate lymphoid cells found in mycosis fungoides from those in chronic dermatitis.16

In histology and cytology, fractal morphometry applied to microscopic examination of cell nuclei and nuclear components has greatly improved the understanding of cell behavior and the diagnosis and prognosis of various disease states.17,18 Quantification of nuclear chromatin organization by fractal morphometry is used to evaluate the degree of malignancy in human breast cytology19 and in aspiration cytology smears of cervical lesions.20 Recent studies targeting the periphery of cell nuclei have shown fractal properties, making it possible to classify early ovarian cancers and even to distinguish normal from malignant liver cells.21,22

Leukemia and hematological malignancies

Application of fractal morphometry to nonsolid cancers came later, when human leukemia cells of lymphoid and/or myeloid origin were characterized on electron microscopic images through quantitative measurement of membrane surface properties that could be correlated with specific phenotype markers. Cells isolated ex vivo from the blood of humans with acute T-lymphoid leukemia revealed pericellular membranes with a nearly smooth outline as documented by fractal dimension values significantly lower than those found for pericellular membranes of healthy blood cells. Healthy lymphocytes of B-cell lineage had a fractal dimension FD (1.20) significantly different from that of lymphocytes of T-cell lineage, ie, CD4-T helper (1.17) and CD8-T suppressor (1.23) cells (Figure 2).23 Unexpectedly, strongly proliferating T-lymphoid leukemic cells were found to possess a plasma membrane characterized by a low FD value (1.10), close to the FD value measured on the plasma membrane of in vitro growing lymphoblasts derived from mature T-lymphocytes triggered by phytohemagglutinin (PHA), a mitogenic lectin. About 80% of acute leukemia subtypes of the B-cell lineage (c-ALL and pre-B undifferentiated phenotype) showed plasma membranes with FDs ranging from 1.12 to 1.17, below the FD of the plasma membrane of differentiated B-lymphocytes. The remaining cases (20%) of acute lymphoblastic B-leukemia showed a more convoluted cell surface with FD values of up to 1.24. Cells from hairy-cell leukemia, a chronic type of human leukemia, with a highly convoluted plasma membrane morphology and a completely dif- ferent surface phenotype displayed the highest FD, between 1.32-1.36.24 The fractal dimension of scale-invariant self-similar chromatin was measured in nuclei of blasts isolated from patients suffering from acute leukemia of the precursor B lymphoblastic type (B-ALL). The increase in FD, together with the accentuated coarseness of the nuclear surface, reflects significant changes in the DNA methylation pattern usually localized in heterochromatin nuclear regions and therefore was regarded as a bad prognostic factor for these patients.25

The usefulness of fractal analysis to assess the hematological cell phenotype and to define a clinical group was confirmed some 20 years later by Mashiah et al.26 These authors used conventional slide preparations to analyze “nuclei contours” of cells belonging to the B lineage, ie, normal and reactive lymphocytes and lymphoid cells isolated from patients with chronic lymphocytic leukemia (CLL), follicular lymphoma (FL), and diffuse large B-cell lymphoma (DLBCL). They found that the fractal dimensions of perinuclear membranes were significantly different between the groups and all correlated with their biological properties, ie, that reactive lymphocytes (FD= 1.20) were situated between CLL (FD=1.25) and normal cells (FD=1.13), while aggressive lymphoma cells had a significantly higher fractal dimension ranging from 1.23 (FL) to 1.31 (DLBCL). By comparing data from the latter papers dealing with hematological malignancies, it turned out that cells isolated from patients with different types of leukemia and/or lymphoma have nuclear chromatin with roughness or complexity (high FD value) increasing with increasing degrees of aggressiveness and malignancy, whereas pericellular membranes acted inversely and looked smoother (low FD value) in cells having a high degree of malignancy. One could infer that hematological tumors did not undergo uniform neoplastic transformations, but rather manifest a great number of metabolic and phenotype changes that imply either an increasing or a decreasing complexity of the morphological surface and an altered organization of cell components mainly dependent upon the cytotype under investigation. This contrasts with the behavior of several cell colonies of breast cancer origin and experimental tumors, which were observed to obey the same dynamics of proliferation and growth and display contours with self-similar fractal features when submitted to scaling analysis.27

Cancer tissues

For an objective description of neoplastic and pathologic traits of cell tissues by the fractal approach, a main condition is the experimental definition of a scaling interval rather than a unique dimensional scale selected a priori. A critical reading of the literature shows that such a distinctive characteristic is insufficiently taken into account and inadequately applied in many investigations, as exemplified by Baish and Jain:

These views are typically interpreted in a qualitative manner by clinicians trained to classify abnormal features such as structural irregularities or high indices of mitosis. A more quanti tative and hopefully more reproducible approach, which may serve as a useful adjunct to trained observers, is to analyze images with computational tools. Herein lies the potential of fractal analysis as a morphometric measure of the irregular structures typical of tumor growth.28

Among the most promising fields of investigation, for which fractal geometry provides an original approach and fractal dimension represents more than an additional geometrical parameter or just “a useful adjunct,”28 are cell heterogeneity; architectural organization of tissues tumor; parenchymal border; cellular/nuclearmorphology; and developmental and morphogenetic processes in tissues and organs in healthy, pathologic, or tumor conditions, and the pathologies of the vascular architecture. Tumor grading on histology specimens (a measure of the degree of cellular differentiation) is difficult to assess because tumors often consist of a heterogeneous mixture of cells with varying degrees of irregularity as well as local variations in cellular differentiation.

Measuring the fractal dimension could aid pathologists in grading heterogeneity and in determining the spatial extent of poorly differentiated regions of breast and prostate tumors.29 Cell heterogeneity, known to contribute in a determinant way to the histological grading of human breast cancer, has been examined by means of geostatistics and the Hurst fractal parameter.30 Several examples seem to indicate that the occurrence of morphogenetic dynamics, the emergence of complex patterns, and the architectural organization of active tissues and tumor masses may be driven by constructive mechanisms related to fractal principles, including deterministic and/or random iteration of constituent units with varying degrees of self-similarity, scaling properties, and form conservation.31 Preservation of tissue architecture and cell polarity of organs and the eventual restoration of organized traits in tumor tissues, deconstructed and deregulated at various levels, is an emerging field of interest since it has been observed that biological entities organize with their own degrees of structural and behavioral complexity and develop on different spatial and time scales.32-35 Stromal tissue has a major role in the control and regulation of physiological processes, in modulating tumorigenesis36-38 and eventually in inducing cancers to revert to normal tissues.39

Fractal dimension has been used as a characterization parameter of premalignant and malignant epithelial lesionsof the floor of the mouth in humans40 and of architectural changes of the epithelial connective tissue interface (ECTI) of the buccal mucosa during aging (Figure 3).41

The outline roughness and the internal irregularity of collagen extracellular matrix examined on biopsy specimens of chronic liver diseases were evaluated by the fractal approach, which yielded a reliable measure much useful in describing these two qualitative properties of the liver matrix.42

Figure 3
Figure 3. Histological sections representing normal oral mucosa
from different age-groups.

Aging of the oral mucosa does not seem to affect significantly the irregularity of
the epithelial connective tissue interface [ECTI] as documented by close fractal
dimension (FD) values in the table below. A. Image of the epithelium; B. Hematoxylin component of A; C. Segmented epithelial compartment; D. Segmented epithelial compartment after logical operation between A and C. Magnification 20×. After reference 41: Eid RA et al. Riv Biol–Biol Forum. 2008;101:129-158. © 2008, Tilgher-Genova.

Figure 4
Figure 4. Microscopic view of canine ribbon type trichoblastoma of medusoid pattern with an irregular contour. Magnification 40×.

After reference 43: Losa GA et al. Connect Tissue Res. 2009;50:28-29. © 2009, Informa Healthcare, London.

Fractal morphometry has provided quantitative information concerning the link between molecular, cellular, and tissue changes during the development of canine tumors (Figure 4).43 The onset of fundamental phenomena such as development, growth, and cell death during different stages of carcinogenesis and cell differentiation, ie, from mesenchymal to smooth muscle cells, has been adequately investigated by fractal geometry as recently reported.44,45 One highly promising approach appears to be a combination of fractal analysis, to provide a quantitative description of shapes, with radiographic imaging, which has the ability to discriminate malignant from benign tumor masses, as well as from normal tissue structures.46 The computed FD of the contour of amassmay be useful for characterizing shape and gray-scale complexity, which may vary between benign masses and malignant tumors in mammograms (Figure 5).47

Figure 5
Figure 5. Fractal analysis of contours of breast masses in mammograms. Thirty-seven benign breast masses and 20 malignant tumors were ranked by their fractal dimension FD estimated by the 1-D ruler method. B: benign, M: malignant.

After reference 47: Rangayyan RM et al. J Digit Imaging. 2007;20(3)223-237. © 2006 by SCAR (Society for ComputerApplications in Radiology).

Fractal theory has provided the basis for a unique software platform program, which has been developed for use in conjunction with magnetic resonance imaging (MRI), and has shown great promise in the early diagnosis and treatment of breast cancer. In a recent study where this advanced method was applied, more than 30% of the patients were shown to have additional tumors in the same breast, and in almost 10% of cases tumors were shown also to be present in the other breast.48 Image analysis combined with fractal analysis has been applied to describe changes in the actin cytoskeleton of neonatal cardiac fibroblasts responding to mechanical stress.49

Brain, brain diseases, and neural tissue

The evolutionary concourse of two major events, “the tremendous expansion and the differentiation of the neocortex,” as reported by De Felipe, has contributed to the development of the human brain (Figure 6).50

Today, modern neurosciences recognize the presence of fractal properties in brain at various levels, ie, anatomical, functional, pathological, molecular, and epigenetic, but not so long ago there was no analytical method able to objectively describe the complexity of biological systems such as the brain. The intricacy of mammalian brain folds led Mandelbrot to argue that “A quantitative study of such folding is beyond standard geometry, but fits beautifully in fractal geometry.”1 At that time however, there was no certainty about the brain’s geometry or about neuron branching. Anatomical-histological evidence that the complexity of the plane-filling maze formed from dendrites of neural Purkinje cells of cerebellum was more reduced in nonammalian species than in mammals led Mandelbrot to comment: “It would be very nice if this corresponded to a decrease in D (fractal dimension), but the notion that neurons are fractals remains conjectural.”1 Since then, a wealth of investigations have documented the fractal organization of the brain and nervous tissue system, and the implication of fractals for neurosciences has been unambiguously affirmed.51,52 The brain consists of distinct anatomical areas formed by nervous tissue mainly composed of neurons and glial cells of distinct types. Neurons contain the axon (a long cytoplasmic process associated with the cell body that communicates with target organs), and the dendrites (shorter cytoplasmic processes off the cell body that allow communication between neurons),while glial cells of various types have a structural role as a net via their branched and unbranched protoplasmic processes (Figure 7).53

Figure 6
Figure 6. Increase in brain size and the maturation of cortical circuits.

The maturation of mental processes and motor skills is associated with an approximately 4-fold enlargement in brain size. A,B. Photographs of the brains of a 1-month and 6-year-old-child. Increase in the complexity is clearly evident in the drawings of Golgi stained cortical neurons from the cerebral cortex of a 1-month-old child (C.“Pars triangularis of gyrus frontalis inferior”; D.“Orbital gyrus”) and 6-year-old child (E.
“Pars triangularis of gyrus frontalis inferior”; F. “Orbital gyrus”). Scale bar for (A,B): 2 cm.
After reference 50: De Felipe J. Front Neuroanat. 2011;5:1-16. © 2011, Frontiers Media, SA.

These anatomical, morphological, and physiological properties combine to create the brain’s complexity, which can only be modeled by a supercomputer.54 The growth and morphological differentiation of spinal cord neurons in culture and the degree of dendritic branching of thalamic and retinal neurons were among the first applications of fractal analysis.55 Further studies have confirmed that the fractal dimension correlates with the increase in morphological complexity and neuronal maturity (Figure 8).56-59 Fractal analysis was applied to anatomical/histological images and high-resolution magnetic resonance images in order to quantify the developmental complexity of the human cerebral cortex, the alterations in diseased brain with epilepsy, schizophrenia, stroke, multiple sclerosis, cerebellar degeneration, and the morphological differentiation of the peripheral nervous system.60 In the normal human retina, blood vessels or vascular trees exhibited an FD of 1.7, the same fractal dimension found for a diffusion- limited growth process, a finding that may have implications for the understanding of the embryological development of the retinal vascular system.61 Lastly, it has been shown that the quantitative evaluation of the surface fractal dimension may allow not only to measure the complex geometrical architecture, but also to model the development and growth of tumor neovascular systems and explore the morphological variability of vasculatures in nature, in particular the microvasculature of normal and adenomatous pituitary tissue.62

Figure 7
Figure 7. Fractal analysis of the three main types of human astrocytes.

Grouped on columns are the original highresolution images, the binary silhouettes and the outline masks together with the corresponding fractal dimension (FD) values. After reference 53: Pirici D et al. Rom J Morphol Embryol. 2009;50(3):381-390. © 2009, Romanian Academy Publishing House.

Figure 8
Figure 8. Application of the box-counting method to a dendritic branching pattern.

A. The whole image is covered with a set of squares and squares that cover dendrites are counted. B. Log-log plot between number of squares (N) and square size (r) is fitted by a straight line. Fractal dimension FD = 1.415. R is the correlation coefficient.
After reference 57: Milosevic NT and others. J Theor Biol. 2009;259:142-150. © 2009, Elsevier.


Irregularity and self-similarity under scale changes are the main attributes of the morphologic complexity of cells and tissues, both normal and pathologic. In other words, the shape of a self-similar object does not change when scales of measure change because any part of it might be similar to the original object. Size and geometric parameters of an irregular object, however, differ when inspected at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how to analyze irregular shapes and structures in the physical and biological sciences thanks to the discovery by Mandelbrot of a practical geometry of nature called fractal geometry, and the continuous improvements in computational capabilities. The application of the principles of fractal geometry, unlike the conventional Euclidean geometry developed for describing regular and ideal geometric shapes practically unknown in nature, enables one to measure the fractal dimension, contour length, surface area, and other dimensional parameters of almost all irregular and complex biological tissues. During the past decade, a large amount of experimental evidence has accumulated showing that even in biomedical sciences fractal patterns could be observed within a “scaling window,” a condition to be experimentally established for each tissue element. Fractal dimension is a quantitative descriptor that can be used alone to identify the ultra-structural features of distinct cell components and other tissues sharing different morphological traits and functional peculiarities. Through several examples borrowed from the recent literature, we have highlighted the application of the fractal approach to measuring irregular self-similar features in normal and pathologic cells and tissues with a high degree of organized complexity, and also its potential role in the reassessing the morphological information for a deeper insight into, and understanding of, the biology of tumor and normal tissues. In the end, the fractal approach will enable not only to avoid any approximation or simplification in analyzing real shapes and hence to describe irregular morphologic components and ultra-structural features as they are, but also to evidence every modification in time through a quantitative comparison. _

1. Mandelbrot BB. The Fractal Geometry of Nature. San Francisco, CA:WH Freeman & Co; 1982.
2. Prigogine I. Les Lois du Chaos. Paris, France: Flammarion; 1997.
3. Mandelbrot BB. How long is the coast of Britain ? Statistical self-similarity and fractional dimension. Science. 1967;155:636-640.
4. Mandelbrot BB. Fractals: Form, Chance and Dimension. San Francisco, CA: WH Freeman & Co; 1977.
5. Belaubre G. L’Irruption des Géométries Fractales Dans les Sciences. Paris, France: Éditions de l’Académie Européenne Interdisciplinaire des Sciences (AEIS); 2006.
6. Paumgartner D, Losa GA, Weibel ER. Resolution effect on the stereological estimation of surface and volume and its dimensions and its interpretation in terms of fractal dimensions. J Microscopy. 1981;121:51-63.
7. Rigaut JP. An empirical formulation relating boundary length to resolution in specimens showing “non-ideally fractal’’ dimensions. J. Microsc. 1984;13:41- 54.
8. Losa GA, Nonnenmacher TF. Self-similarity and fractal irregularity in pathologic tissues. Mod Pathol. 1996;9:174-182.
9. Mandelbrot BB. Is Nature fractal ? Science. 1998;279:783-784.
10. Nonnenmacher TF, Baumann G, Barth A, Losa GA. Digital image analysis of self-similar cell profiles. Int J Biomed Comput. 1994;37:131-138.
11. Weibel ER. Design of Biological Organisms and Fractal Geometry. In: Nonnenmacher TF, Losa GA, Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 1994;1:68-85.
12. Losa GA, Graber R, Baumann G, Nonnenmacher TF. Steroid hormones modify nuclear heterochromatin structure and plasma membrane enzyme of MCF-7 cells. A combined fractal, electron microscopic and enzymatic analysis. Eur J Histochem. 1998;42:1-9.
13. Losa GA, Castelli C. Nuclear patterns of human breast cancer cells during apoptosis: characterization by fractal dimension and (GLCM) co-occurrence matrix statistics. Cell Tissue Res. 2005;322:257-267.
14. Santoro R,Marinelli F, Turchetti G, et al. Fractal analysis of chromatinduring apoptosis. In: Losa GA, Merlini D, Nonnenmacher TF,Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 2002;3:220-225.
15. Marinelli F, Santoro R, Maraldi NM. Fractal analysis of heterochromatin nuclear domains in lymphocytes. In: Losa GA, Merlini D, Nonnenmacher TF,Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 1998;2:77-84.
16. Bianciardi G, Miracco C, Santi MD, et al. Fractal dimension of lymphocytic nuclear membrane in Mycosis fungoides and chronic dermatitis. In: Losa GA, Merlini D, Nonnenmacher TF,Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 2002;3:150-155.
17. When B, Jacob W, Van de Wouwer, et al. Fractal dimension, form and shape factors for the quantification of nuclear signature profiles. In: Losa GA, Merlini D, Nonnenmacher TF, Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 2002;3:47-54.
18. Muniandy SV, Stanlas J. Modelling of chromatin morphologies in breast cancer cells undergoing apoptosis using generalized Cauchy Field. Comput Med Imaging Graph. 2008;32:631-637.
19. Einstein AJ,Wu HS, Sanchez M, Gil J. Fractal characterization of chromatin appearance for diagnosis in breast cytology. J Pathol. 1998;185:366-381.
20. Ohri S, Dey P, Nijhawan R. Fractal dimension in aspiration cytology smears of breast and cervical lesions. Anal Quant Cytol Histol. 2004;26:109-112.
21. Nielsen B, Albregtsen F, Danielsen HE. Fractal signature vectors and lacunarity class distance matrices to extract new adaptive texture features from cell nuclei. In: Losa GA, Merlini D, Nonnenmacher TF, Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 2002;3:55-65.
22. Nielsen B, Albregtsen F, Danielsen HE. Fractal analysis of monolayer cell nuclei from two different prognostic classes of early ovarian cancer. In: Losa GA, Merlini D, Nonnenmacher TF, Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 2005;4.
23. Losa GA, Baumann G, Nonnenmacher TF. Fractal dimension of pericellularmembranes in human lymphocytes and lymphoblastic leukemia cells. Pathol Res Pract. 1992;188:680-686.
24. Losa GA. Fractal morphometry of cell complexity. Biol Forum. 2002;95:239-258.
25. Adam RL, Silva RC, Pereira FG, et al. The fractal dimension of nuclear chromatin as a prognostic factor in acute precursor B lymphoblastic leukemia. Cell Oncol. 2006;28:55-59.
26. Mashiah A, Wolach O, Sandbank J, et al. Lymphoma and leukemia cells possess fractal dimensions that correlate with their interpretation in terms of fractal biological features. Acta Haematol. 2008;119:142-150.
27. Bru´ A, Albertos S, Subiza JL, et al; The universal dynamics of tumor growth. Biophys J. 2003;85:2948-2961.
28. Baish J, Jain RK. Fractals and cancer. Cancer Res. 2000;60:3683-3688.
29. Tambasco M, Magliocco AM. Relationship between tumor grade and computed architectural complexity in breast cancer specimens. Hum Pathol. 2008; 39(12):1859-1860.
30. Sharifi-Salamatian V, Pesquet-Popescu B, Simony-Lafontaine J, Rigaut JP. Index for spatial heterogeneity in breast cancer. J Microscopy. 2004;216(2):110- 122.
31. Landini G. Pattern complexity in organogenesis and carcinogenesis. In: Losa GA, Merlini D, Nonnenmacher TF,Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 2002;3:3-13.
32. Russo J, Linch H, Russo JH. Mammary gland architecture as a determining factor in the susceptibility of the human breast to cancer. Breast J. 2001;7: 278-291.
33. Bissell MJ, Rizki A, Mian IS. Tissue architecture: the ultimate regulator of breast epithelial function. Curr Opin Cell Biol. 2003;15:753-762.
34. Grizzi F, Chiriva-Internati M. The complexity of anatomical systems. Theor Biol Med Model. 2005;2:26-35.
35. Nelson CS, Jean RP, Tan JL, et al. Emergent patterns of growth controlled by multicellular form and mechanics. Proc Natl Acad Sci U S A. 2005;102(33): 11594-11599.
36. Kim JB, Stein R, O’Hare MJ. Tumour-stromal interactions in breast cancer: the role of stroma in tumourigenesis. Tumor Biol. 2005;26:173-185.
37. Provenzano P, Kevin P, Eliceiri W, et al. Collagen reorganization at the tumorstromal interface facilitates local invasion. BMC Med. 2006;4:38-53.
38. Beck AH, Espinosa I, Gilks CB, et al. The fibromatosis signature defines a robust stromal response in breast carcinoma. Lab Invest. 2008;88:591-601.
39. Ingber DE. Can cancer be reversed by engineering the tumor microenvironment. Semin Cancer Biol. 2008;18(5):356-364.
40. Landini G, Rippin JW. Fractal dimension of epithelial connective tissue interfaces in premalignant and malignant epithelial lesions of the floor of the mouth. Anal Quant Cytol Histol. 1993;3(9):1159-1165.
41. Eid RA, Sawair F, Saku T, Landini G. Architectural changes associated with ageing of the normal oral buccal mucosa. Proceedings Fifth International Symposium: Fractals in Biology and Medicine. Riv Biol–Biol Forum. 2008;101:131- 158.
42. Grizzi F, Ceva-Grimaldi G, Dioguardi N. Fractal geometry as useful tool for quantifying irregular lesions in human liver biopsy specimen. Ital J Anat Embryol. 2001;106:337-346.
43. Losa GA, De Vico G, Cataldi M, et al. Contribution of connective and epithelial tissue components to the morphologic organization of canine trichoblastoma. Connect Tissue Res. 2009;50:28-29.
44. Roy HK, Iversen P, Hart J, Liu Y, et al. Down-regulation of SNAIL suppresses MIN mouse tumourigenesis: modulation of apoptosis, proliferation, and fractal dimension. Mol Cancer Ther. 2004;15:144-151.
45. Ghazanfari S, Tafazzoli-Shadpour M, Shokrgozar MA, et al. Analysis of alterations in morphologic characteristics of mesenchymal stem cells by mechanical stimulation during differentiation into smooth muscle cells. Yakhteh Med J. 2010;12:73-80.
46. Li H, Giger ML, Olopade OI, Li L. Fractal analysis of mammographic parenchymal patterns in breast cancer risk assessment. Acad Radiol. 2007;14:513-521.
47. Rangayyan RM, Nguyen TM. Fractal analysis of contours of breast masses in mammograms. J Digit Imaging. 2007;20(3):223-237.
48. Wiener JI, Schilling KJ, Adami C, Obuchowski NA. Assessment of suspected breast cancer by MRI: a prospective clinical trial using a combined kinetic and morphologic analysis. Am J Radiol. 2005;184:878-886.
49. Fuseler JW, Millette CF, Davis JM, CarverW. Fractal and image analysis of morphological changes in the actin cytoskeleton of neonatal cardiac fibroblasts in response to mechanical stretch. Microsc Microanal. 2007;13:133-143.
50. De Felipe J. The evolution of the brain, the human nature of cortical circuits, and the intellectual creativity. Front Neuroanat. 2011;5:1-16.
51. Werner G. Fractals in the nervous system: conceptual implications for theoretical neuroscience. Front Physiol. 2010;1:1-28.
52. Losa GA, Di Ieva A, Grizzi F, De Vico G. On the fractal nature of nervous cell system. FrontNeuroanat. 2011;5:1-2.
53. Pirici D,Mogoanta L,Margaritescu O, et al. Fractal analysis of astrocytes in stroke and dementia. Rom J Morphol Embryol. 2009;50(3):381-390.
54. Markram, H. The blue brain project. Neuroscience. 2006;7:153-160.
55. Smith TG Jr. A fractal analysis of morphological differentiation of spinal cord neurons in cell culture. In: Nonnenmacher TF, Losa GA, Weibel ER, eds. Fractals in Biology and Medicine. Basel, Switzerland: Birkhäuser Press; 1994;1: 211-220.
56. Bernard F, Bossu JL, Gaillard S. Identification of living oligodendrocyte developmental stages by fractal analysis of cell morphology. J Neurosci Res. 2001; 65:439-445.
57. Milosevic NT, Ristanovic D, Jelinek HF, Rajkovic K. Quantitative analysis of dendritic morphology of the alpha and delta retinal ganglions cells in the rat: a cell classification study. J Theor Biol. 2009;259:142-150.
58. Milosevic NT, Ristanovich D. Fractality of dendritic arborization of spinal cord neurons. Neurosci Lett. 2006;396:172-176.
59. Jelinek HF, Milosevic NT, Ristanovich D. Fractal dimension as a tool for classification of rat retinal ganglion cells. Riv Biol—Biol Forum. 2008;101(1):146-150.
60. King RD, Brown B, Hwang M, et al. Fractal dimension analysis of the cortical ribbon in mild Alzheimer’s disease. Neuroimage. 2010;53:471-479.
61. Masters BR. Fractal analysis of the vascular tree in the human retina. Annu Rev Biomed Eng. 2004;6:427-452.
62. Di Ieva A. Angioarchitectural morphometrics of brain tumors: are there any potential histopathological biomarkers? Microvasc Res. 2010;80(3):522-533.

Keywords: brain; brain disease; form invariance; fractal dimension [FD]; fractal geometry; irregular morphology; leukemia; nervous tissue; non-Euclidean dimension; scaling window; self-similarity; solid tumor