Basic Principles and Applications of Fractal Geometry

Analytical and Quantitative Cytology and Histology ®

f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D Basic Principles and Applications of Fractal Geometry in Pathology A Review

Pranab Dey, MBBS, M.D., M.I.A.C., MRCPath

The basic principles and prospects of fractal geometry in pathology are promising. All articles found with a PubMed search with the keywords fractal geometry and FD and related to pathology were reviewed. All fractal objects have FDs, commonly calculated with box counting. Fractal geometry has been applied to measure the irregularities of nuclear and glandular margins to distinguish malignant lesions from benign ones, to measure the infiltrative margin of a malignant tumor, to assess tumor angiogenesis and to measure the distribution of collagen in tissue. Fractal geometry has also been applied to assess the irregular distribution of chromatin in malignant cells. Biologic model formation is possible with fractal geometry. In the future, fractal geometry may help with the diagnosis, understanding of pathogenesis and management of lesions. It may also provide new insights into disease processes. (Anal Quant Cytol Histol 2005;27:000–000) Keywords: fractals, image cytometry, morphometry.

Irregularity and complexity are the main features of every biologic system, including human tissues, cells and subcellular components. These 2 properties of organized biologic matter cannot be quantified by means of classical Euclidean geometry. For the first time, Mandelbrot formalized the concept of fractal dimension (FD) when he tried to identify the irregularities of the coastline of Britain.1 He introduced the term fractal geometry and with it attempted to explain the behavior of chaos in nature. Unlike Euclidean geometry, fractal geometry deals with objects in noninteger dimensions. Fractal geometry is a powerful tool for describing the irregular but ordered shape of many natural objects. The word fractal is derived from the Latin word fractus, meaning an irregular surface, such as that of a broken stone. Basically, a fractal is a rough geometric figure that has 2 properties: First, most magnified images of fractals are essentially indistinguishable from the unmagnified images. This property of invariance under a change of scale is called self-similarity. Second, fractals have FDs (FDs). The concept of fractals has a great influence on pure and applied mathematics and on other aspects of life. However, there has been relatively slow progress in applying fractal geometry in image cytometry and pathology. This review briefly discusses the basic principles, methods of calculation and

From the Department of Cytology, Post Graduate Institute of Medical Education and Research, Chandigarh, India. Dr. Dey is Associate Professor.

Address correspondence to: Pranab Dey, MBBS, M.D., M.I.A.C., MRCPath, Department of Cytology, Post Graduate Institute of Medical Education and Research, Chandigarh 160012, India ([email protected]). Financial Disclosure: The author has no connection to any companies or products mentioned in this article.

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f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D future applications of fractal geometry in the field of pathology. What Is Fractal Geometry?

The geometry taught in school is Euclidean. The world of Euclidean geometry is made up of objects that exist in integer dimensions, single dimensional points, 1-dimensional lines and curves, 2-dimensional plane figures such as circles and squares, and 3dimensional, solid objects such as spheres and cubes. While a straight line has a dimension of exactly 1, a fractal curve will have a dimension between 1 and 2, depending on how much space it takes up as it curves and twists. The more a fractal fills up a plane, the closer it approaches 2 dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between 2 and 3. Hence, a fractal landscape composed of a rough surface area with many average-sized hills would be much closer to the third dimension. What Is the Need for Fractal Geometry?

In pathology we see complex, irregular and ramified lesions, such as tumors, vascular ramifications and irregular shapes of malignant cells. A tumor has an irregular and complex shape, and its infiltrative margin is difficult to describe objectively with Euclidean geometry. Similarly, it is easy to make subjective comments on irregular, complex glandular shapes in atypical complex hyperplasia of the endometrium. However, can we objectively assess the complexity of the gland margin with Euclidean geometry? We can easily measure the area and volume of a cube, but how do we measure the dimensions inside a lung, kidney or brain? Angiogenesis in a tumor is a critical event in tumor progression. Can we measure the complexity of angiogenesis with Euclidean geometry? Nuclear margin irregularity and chromatin texture are important findings in diagnosing malignant cells on light microscopy. Can traditional image morphometry based on Euclidean geometry accurately measure these findings? Recently there has been rapidly growing evidence that fractal geometry may be promising in solving many of these issues.

cates the dimension. Now consider a 2-dimensional square. With a magnification of 2 we obtain 4 identical shapes in both of them, with 22 = 4; hence, 2 indicates FD. Considering a 3-dimensional cube and doubling its length, breadth and height, we shall obtain 8 identical cubes, or 23 = 8; 3 indicates FD. A clear pattern is thus evident. If one takes the magnification and raises it to the power of dimension, the number of shapes is obtained. Therefore, ED = N, where E stands for magnification, D for dimension and N for the number of identical shapes. Applying logarithms, D = logN/logE.

FD can also be calculated with the geometric method. A true fractal has an infinite amount of detail. This means that magnifying it adds additional detail, which increases the overall size. In nonfractals, however, the size always stays the same (Figure 1). We can easily calculate FD using the slope of this line; it can be done by using the simple formula: FD = slope+1.

This geometric method is very useful for calculating the dimensions of natural irregular shapes, such as those of coasts, borderlines and clouds. Techniques of Measuring FD

There are various ways of measuring FD: 1. Modified pixel dilatation 2. Perimeter–area method 3. Ruler counting method 4. Box counting method.

Calculation of FD

One of the simplest ways to calculate FD is by taking advantage of self-similarity. For example, suppose we have a 1-dimensional line segment. If we look at it with a magnification of 2, we will see 2 identical line segments, with 21 = 2, where 1 indi-

Figure 1 The length of the line increases with the increment of magnification in a fractal line.

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f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D a negative slope that equals –D (FD). Box Counting FD

Figure 2 Boxes of various pixel lengths are superimposed on the digitized image of an object/cell, and the number of boxes touching the object/cell is counted each time to measure the box counting FD.

This is the most commonly used method for detection of FD. Box counting FD is a simple and reproducible way of measuring FD. In this method, boxes of different pixel length are generated, and this grid is superimposed on the digitized image of a cell or object to be measured (Figure 2). The number of boxes touching the peripheral margin of the cell/object is counted, and finally a graph is made with log (box number)/1/log (box size). Linear regression analysis is done, and the slope of the line is calculated for the measurement of FD (Figure 3). Certain precautions should be taken during box counting FD measurement: 1. The grid interval should not exceed the size of the object to be measured, the nuclear diameter. 2. The grid interval should not be less than the pixel width. 3. Two grid intervals determine a slope. The intervals must be far enough apart to avoid measurement errors. 4. More than 2 grid intervals determine a slope by regression analysis. More grid intervals are better but may encroach on the limits of measurement.

Modified Pixel Dilatation Method

Application of Fractal Geometry in Pathology

In this method the digital image of the object is processed by sequential pixel dilatation by a pixel array so that the analyzed picture is enlarged. During the dilatation, the areas of the element (En) and of the picture transformed by dilatation (Sn) are calculated. By plotting En, Sn/En on a log/log scale, a slope is obtained. FD is subsequently calculated from the slope.

Fractal geometry has recently been applied in pathology. A search of all articles in PubMed using the keywords fractal geometry and fractal dimension uncovered the articles related to pathology.

Perimeter–Area Method

To calculate perimeter–area dimension, it is necessary to measure the perimeter and the area with the boxes of different side length, d. Logarithm of area number in the vertical axis versus logarithm of perimeter on the horizontal axis should be drawn. Slope of the line is measured to calculate FD. Ruler Counting Method

This method is convenient for estimating FD of a jagged, self-similar line. The number of steps Nd taken by a divider of length d is calculated, and the log of Nd versus the log of d is plotted. If the line is fractal, then this plot will follow a straight line with

Figure 3 Log-log graph of 1/box size and number of boxes touching the outline of the object/cell is plotted. The slope of the line is calculated (y = ax +b, a is slope of line) to measure FD.

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f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D Nuclear Margin Irregularity of Cells

Nuclear morphometric data, such as nuclear area, diameter and perimeter, have been applied by various workers to distinguish between benign and malignant cells in cytologic smears.2,3 These attempts have been partly successful. Nuclear margin irregularity is an important feature of malignant cells. There has been little attempt to objectively assess the irregularity of nuclear margins by traditional morphometry. Various workers measured FD of the cell nuclei of cervical intraepithelial neoplasia (CIN), lymphoid cells of lymphoblastic leukemia, hairy cell leukemia and breast carcinomas and the effect of steroid hormones on breast carcinoma cell lines.4-10 FDs of the nuclei of CIN 1, 2 and 3 are significantly different.4 There also is a significant difference in FD of nuclei of normal and atypical lymphocytes.5 FD of the cell contour of human T lymphocytes from normal donors and the hairy leukemia cells were measured, and it was shown that FD correlates with the structural complexity of the individual cell contour.6 We measured FD of malignant and benign breast cells on cytologic smears.9 We also measured FD of benign and malignant cells on cervical smears.10 We noted that FD was significantly different in malignant and benign cells. We suggested further application of fractal geometry to discriminate benign and malignant cells in other lesions. In all the above studies, box counting FD was applied to measure FD of cell nuclei. There was no attempt in any of these studies to compare traditional image morphometry and fractal geometry to distinguish cells of different groups. It would be more informative if the findings of Euclidean geometry and fractal geometry were compared in the same lesions in the same cell. Glandular Abnormality

Gland margin irregularity can be a useful discriminating factor in different groups of lesions. FD has been measured in colorectal polyps, serrated adenomas, gallbladder adenocarcinomas and endometrial hyperplasia.11-14 Cross et al11 measured FD of different groups of colorectal polyps and showed that FD is a better way of quantitating the polyp’s shape and is a useful morphometric discriminant between different groups of polyps. In the case of gallbladder carcinomas, there was fractal regularity in the distribution of glandlike structures.14 We recently attempted to calculate FD of the

glands of simple hyperplasia, complex atypical hyperplasia and carcinoma of the endometrial glands with box counting FD.12 We noted significant differences between FD of simple hyperplasia versus complex atypical hyperplasia and carcinoma. The numbers of cases and fields examined in this study were low. In the future, a large number of cases should be studied to reach in a more valid conclusion. Tumor Stromal Interphase

Infiltrative margins of neoplastic and dysplastic epithelial cells in the stroma may provide useful information related to prognosis and diagnosis. It is difficult to assess objectively the complex nature of the epithelial and stromal interphase by light microscopy. Various workers15-18 tried to measure FD of the epithelial and stromal junctional margins and concluded that it may successfully reflect the complex nature of the epithelial connective tissue interphase and that the measurement of FD may help to differentiate premalignant and malignant oral growths.18

Quantification of Collagen

Various workers have attempted to quantify the irregular pattern of fibrosis with fractal geometry.19-22 FD provides a reproducible, rapid and inexpensive method of measuring the irregular distribution of collagen in the liver.22 FD is independent of the amount of material on the slide to be measured. Tumor Metastasis

It has been shown that the distribution of the normal hemopoietic cells in the bone marrow is fractal.23 There is loss of FD when bone marrow spaces are infiltrated and replaced by metastatic cells.23 Moatamed et al23 measured the cellular areas of the bone marrow in a successively doubled area of concentric circles to measure FD. Malignant cells and normal hemopoietic cells were considered the cellular area and were not distinguished in this study. This study of FD of bone marrow was unique. In the future, individual hemopoietic cells can be stained by immunocytochemistry, and FD of the distribution of individual components of hemopoietic cells may provide useful information for understanding pathologic conditions in bone marrow. Angiogenesis

Angiogenesis is a complex pattern, and many

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f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D workers attempted to assess objectively this complex architectural pattern in various lesions with FD.24-29 Tumor vasculature is more chaotic than normal vasculature. Gazit et al28 showed that tumor vessels have higher FD than do normal arteries and veins. Heymans et al29 measured FD of microvasculature of cutaneous melanoma. They quantified the degree of randomness with respect to vascular distribution with FD. Sabo et al27 evaluated the utility of measuring microvessel FD (MFD) as a parameter of architectural microvascular complexity in renal cell carcinoma (RCC). Fractal analysis of the microvascular network of RCC was performed, and the MFD was computed in each case. The results showed that measurement of MFD was a useful marker of tumor microvascular complexity and provided important prognostic information. Fractal and Chromatin Pattern

Chromatin texture is one of the important features for detection of malignant cells. Fractal geometry may help to assess the distribution of chromatin of benign and malignant cells. Einstein et al30 found a statistically significant difference between benign and malignant cells in lacunarity, a fractal property characterizing the size of the holes or gaps in chromatin. It was demonstrated that chromatin appearance in the breast epithelial cell’s nuclear image is fractal and that probably the 3-dimensional chromatin structure in these cells also has fractal properties. Neuropathology

The branching processes of neurons and astroglial cells are fractal.31,32 The whole brain is probably a fractal object. This is related to the folding and compartmentalization of the brain, which developed during the process of evolution.33 The complex branching pattern of peripheral nerves can also be quantified with fractal geometry.34 Bone Pathology

Histomorphometry of bone biopsy specimens is often performed in cases of metabolic bone disease with Euclidean parameters. The box counting FD of trabecular bone is significantly greater than the topologic dimension,35,36 and there is a fractal element in the trabecular structure of bone. It is also claimed that the methods of measurement of FD are accurate and reproducible. In fact, FD highly correlates with conventional Euclidean measurements, such as area and volume.35,36

Biologic Modeling

Fractal images can be generated by multiple iterations of relatively simple equations with graphing of the results. Various workers attempted to model biologic and pathologic processes.37-39 Landini and Rippin37 generated a cell growth model in which the cells were capable of displacing the adjacent populations. Aon et al38 developed a model of cellular cytoplasm based on percolation clusters, a type of random fractal structure. Fractal geometry was also helpful in generating a vascular model of the organ system.39 In the future, this model probably will be a helpful tool for oncologists in introducing antiangiogenic therapy and also understanding vascular pathologies.40 Discussion

There are several advantages of fractal geometry over Euclidean geometry. Shrinkage and expansion of a specimen will not affect the measurement of FD if the artifacts act equally in all directions. Thus, FD of specimens can be compared in different laboratories even if the specimens are processed in different batches in different laboratories. In comparison, integral dimensions, such as length and area, are very sensitive to processing artifact. Another major advantage of fractal geometric analysis is that the fractal component of dimension is retained when a fractal object is projected to a lower-order dimension41; for example, a 3dimensional renal arterial tree projected as only a 2-dimensional radiographic film will retain its fractal character. Traditional image morphometry uses both interactive and automated morphometry to measure various morphometric features, such as diameter and perimeter. Fractal geometry measures only FD, which is an integer or noninteger number. Interactive morphometry may be needed initially to draw the object on a computer screen; that is a timeconsuming process and may not be reproducible totally. Automated image morphometry is necessary to obtain the outline of the object. Special stains may be helpful to in this respect. After identification of the object of interest, one can measure FD. Again, an automated technique is needed for rapid and reproducible measurement of FD. There are various problems related to fractal geometry. In the case of measurement of FD from an image, one should be careful about the edgeprocessing functions, such as binary noise reduc-

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f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D tion and nonlinear sharpening filter. These may alter the peripheral complex boundaries of the digitized image and the FD of the object. A fractal object will have FD, and there are various ways to measure it. FD may be different in the same object measured by different techniques. A single FD of an object may not be an adequate descriptor, and multiple FDs measured at different magnifications may be a more comprehensive methodology. High-resolution images are necessary to measure FD of an object. Another important point is that there are both upper and lower limits to the size range over which the fractals in nature are indeed fractal. Measurement of FD is time consuming and tedious. One has to be computer literate and well acquainted with the field of mathematics for calculation and interpretation of fractal geometry. Prospects and Conclusion

Fractal geometry may be used to understand molecular events in the cell and cellular differentiation. A gene can contribute to the emergence of more than just 1 phenotypic trait, and a phenotypic trait can be expressed by > 1 gene. This implies the lack of a proportional relationship between input and outcome and forming complexity. Nonlinearity within the space of cellular molecular events underlies the existence of a fractal structure within a member of metabolic processes and pattern of tissue growth, which is measured experimentally as FD. In the future, fractal geometry may be applied to better understanding molecular events and cellular differentiation.42 The concept of fractal geometry is simple, and with a simple digital image analyzer, one can measure FD of the object of interest. However, FD is not a complete descriptor of the shape of an object. Therefore, other morphometric data should also be included for more accurate interpretation in category assignment. In the future, fractal geometry will probably help in classifying different diseases, differentiating benign and malignant cells, quantifying fibrosis and angiogenesis, and understanding tumorigenesis. Biologic modeling may help us to understand various physiologic processes and to plan treatment.

and cytologic grade of breast carcinomas. Anal Quant Cytol Histol 2000:22;483–485

3. Rajesh L, Dey P, Joshi K: Automated image morphometry of lobular carcinoma. Anal Quant Cytol Histol 2002;24:81–84

4. Sedivy R, Windischberger C, Svozil K, Moser E, Breitenecker G: Fractal analysis: An objective method for identifying atypical nuclei in dysplastic lesions of the cervix uteri. Gynaecol Oncol 1999;75:78–83 5. Niwa Y, Yoshida T, Moriwake K, Okamoto T, Kuramoto C, Yamazaki E, Matsuda T, Masutani T, Okamoto Y: Determination of FD using digital nuclear image of lymphocyte. Rinsho Byori 2002;50:702–705

6. Nonnenmacher TF, Bauman G, Barth A, Lossa GA: Digital image analysis of self-similar cell profiles. Int J Biomed Comput 1994;37:131–138 7. Lossa GA, Bauman G, Nonnenmacher TF: FD of pericellular membranes in human lymphocytes and lymphoblastic leukemia cells. Pathol Res Pract 1992;188:680–686 8. Lossa GA, Graber R, Bauman G, Nonnenmacher TF: Effects of steroid hormones on nuclear membrane and membranebound heterochromatin from breast cancer cells evaluated by fractal morphometry. Anal Quant Cytol Histol 1999;21: 430–436 9. Dey P, Mohanty SK: FD of breast lesions on cytology smear. Diagn Cytopathol 2003;29:85–86

10. Ohri S, Dey P, Nijhawan R: FD on aspiration cytology smears of breast and cervical lesions. Anal Quant Cytol Histol 2004;26:109–112 11. Cross S, Bury J, Silcocks P, Stephenson T, Cotton D: Fractal geometric analysis of colorectal polyps. J Pathol 1994;172: 317–323 12. Dey P, Rajesh L: FD of endometrial carcinoma. Anal Quant Cytol Histol 2004;26:113–116

13. Iwabuchia M, Endoh M, Hiwatashi N, Kimouchi Y, Shimosegawa T, Masuda T, Moriya T, Sasano H: Three-dimensional reconstruction and fractal geometric analysis of serrated adenoma. Jpn J Cancer Res 2002;93:259–266

14. Waliszewski P: Distribution of gland-like structures in human gallbladder adenocarcinomas possesses FD. J Surg Oncol 1999;71:189–195 15. Abu Fid R, Landini G: Quantification of the global and local complexity of the epithelial-connective tissue interface of normal, dysplastic, and neoplastic oral mucosae using digital imaging. Pathol Res Pract 2003;199:475–482 16. Landini G, Hirayama Y, Li TJ, Kitano M: Increased fractal complexity of the epithelial-connective tissue interface in the tongue of 4NQO-treated rats. Pathol Res Pract 2000;196: 251–258

17. Landini G, Rippin JW: How important is tumour shape? Quantification of the epithelial-connective tissue interface in oral lesions using local connected FD analysis. J Pathol 1996;179:210–217

1. Mandelbrot BB: How long is the coastline of Britain? Statistical self similarity and FD. Science 1967;156:636–638

18. Landini G, Rippin JW: FDs of the epithelial-connective tissue interfaces in premalignant and malignant epithelial lesions of the floor of the mouth. Anal Quant Cytol Histol 1993;15: 144–149

2. Dey P, Ghoshal S, Pattari S: Nuclear image morphometry

19. Dioguardi N, Franceschini B, Aletti G, Grizzi F: FD rectified

References

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f e o t o a r c P i l e p g u a D P T f O e o t N o a r o c P i l D e p g u a P TD f O o N o r o c P i l D e p g u a P TD O N o D meter for quantification of liver fibrosis and other irregular microscopic objects. Anal Quant Cytol Histol 2003;25:312– 320

30. Einstein AJ, Wu HS, Sanchez M, Gill J: Fractal characterization of chromatin appearance for diagnosis in breast cytology. J Pathol 1998;185: 366–381

20. Soda G, Nardoni S, Bosco D, Grizzi F, Dioguardi N, Melis M: Fractal analysis of liver fibrosis. Pathologica 2003;95:98–102

31. Mckinnon RD, Smith C, Behar T: Distinct effects of bFGF and PDGF on oligodendrocytes progenitor cells. Glia 1993;7:245– 254

21. Moal F, Chappard D, Wang J, Vuillemin E, Michalak-Provost S, Rousselet MC, Oberti F, Cales P: FD can distinguish models and pharmacologic changes in liver fibrosis in rats. Hepatology 2002;36:840–849 22. Dioguardi N, Grizzi F, Bossi P, Roncalli M: Fractal and spectral dimension analysis of liver fibrosis in needle biopsy specimens. Anal Quant Cytol Histol 1999;21:262–266 23. Moatamed F, Sahimi M, Naeim F: FD of the bone marrow in metastatic lesions. Hum Pathol 1998;29:299–303

24. De Felice C, Bianciardi G, Dielo L, Latini G, Parrini S: Abnormal oral vascular network geometric complexity in EhlersDanlos syndrome.Oral Surg Oral Med Oral Pathol Oral Radiol Endod 2004;98:429–434 25. De Felice C, Latini G, Bianciardi G, Parrini S, Fadda GM, Marini M, Laurini RN, Kopotic RJ: Abnormal vascular network complexity: A new phenotypic marker in hereditary non-polyposis colorectal cancer syndrome. Gut 2003;52: 1764–1767

26. Oczeretko E, Juczewska M, Kasacka I: Fractal geometric analysis of lung cancer angiogenic patterns. Folia Histochem Cytobiol (suppl 2) 2001;39:75–76

27. Sabo E, Boltenko A, Sova Y, Stein A, Kleinhaus S, Resnick MB. Microscopic analysis and significance of vascular architectural complexity in renal cell carcinoma. Clin Cancer Res 2001;7:533–537 28. Gazit Y, Berk DA, Leunig M, Baxter LT, Jain RK: Scale-invariant behavior and vascular network formation in normal and tumor tissue. Phys Rev Lett 1995;75:2428–2431 29. Heymans O, Blacher S, Brouers F, Pierard GE: Fractal quantification of microvasculature heterogeneity in cutaneous melanoma. Dermatology (Basel) 1999;198:212–217

32. Reichenbach A, Siegel A, Senitz D: A comparative fractal analysis of various mammalian astroglial cell types. Neuroimage 1992;1:69–77

33. Hofman MA: The fractal geometry of convoluted brain. J Hirnforsch 1991;32:103–111 34. Aross GA, Dimitru PP: A fractal characterization of the type II fiber distribution in the extensor digitorium longus and soleus muscle of the adult rat. Muscle Nerve 1995;18;961–968 35. Fazzalari NL, Parkinson IH: FD and architecture of trabecular bone. J Pathol 1996;178:100–105

36. Parkinson IH, Fazzalari NL: Cancellous bone structure analysis using image analysis. Aust Phys Eng Sci Med 1994; 17:64–70

37. Landini G, Rippin JW: Fractal fragmentation in replicative systems. Fractals 1993;1:239–246 38. Aon MA, Cortassa S: On the fractal nature of cytoplasm. FEBS Lett 1994;344:1–4

39. Landini G, Misson G: Simulation of corneal neovascularization by inverted diffusion limited aggregation. Invest Ophthalmol Vis Sci 1993;34:1872–1875 40. Baish JW, Jain RK: Fractals and cancer. Cancer Res 2000; 60, 3683–3688

41. Mayhew TH: The new stereological method for interpreting functional morphology from slices of cells and organs. Expl Physiol 1991;76:639–665 42. Waliszewski P, Molski M, Konarski J: On the relationship between fractal geometry of space and time in which a system of interacting cells and dynamics of gene expression exist. Acta Biochim Pol 2001;48:209–220