Fractals in Biology and Medicine

Angela Downs-Rigaut,  Triel-sur-Seine, May 2005

            Jean Paul Rigaut                                    
                               1944 - 2005

 Jean Paul was a born researcher: when he was 3 years old, his mother, worried by the hours he spent examining leaves and other natural objects, took him to a psychiatrist, who reassured her that he was completely normal - simply highly intelligent!  He was however unhappy with the commonly-used French term chercheur  and - on official forms - frequently (and somewhat provocatively) wrote savant, a now little-used word which he felt more accurately described his activity as a research scientist. When he was seven, his father's job took the family to London where they lived for the next five years.  There, Jean Paul attended the French Lycée and became bilingual - a chance which served him well in later years. Both on arriving in England and on returning to France, he somehow managed to skip a whole year, and thus obtained son baccalauréat at the tender age of 15, having missed out on all formal teaching of grammar, despite which he wrote in exemplary French - and English!  Despite his undeniable research leanings, he began by studying medicine, judged by his family to be more respectable and lucrative than science (he was apparently recognised as the youngest-ever student to enrol at the Faculty of Medicine in Paris).  He never regretted this choice, but once his medical studies were completed - and in parallel with his work as a hospital doctor - he began his formal scientific studies (degree in Biochemistry, University Paris 7, 1972; diploma in Embryology, University Paris 6, 1974; Ph D in Biology, University Paris North, 1978).


During these latter years, he was working as a doctor in the public sector in the northern suburbs of Paris (département Seine-Saint-Denis).   His chosen specialities were Maternity and Neonatology.  He delivered thousands of babies and performed routine examinations of new-borns in two large suburban hospitals (at Saint-Denis and Aulnay-sous-Bois).  His medical thesis (1970) concerned the screening of new-borns for potential hip dislocations through systematic clinical examination, and the subsequent treatment of affected infants who could almost invariably be  cured by wearing specially designed abduction knickers for the first few months of life.  He concluded that "only neonatal screening in maternity units can provide the possibility of eradicating the consequences of disclocable hip disease".  He continued his work as a neonatologist in Aulnay long after accepting a permanent scientific position, first at University Paris Nord (in 1980) and subsequently with the national health and medical research institute INSERM.  It was however one of his great regrets that his recommendations concerning early neonatal screening for hip dislocations were not implemented by the state, and it is perhaps partly for this reason that he finally ceased all clinical activity in 1988.
In addition to his hospital-based  work,  Jean Paul was  consultant doctor for several public nurseries (crêches) - where he encouraged the mothers to offer tasty food (Roquefort cheese, for example) to their young children - and, in collaboration with the local social services,  provided free consultations in the huge shanty-town then existing at Saint-Denis.   He was one of the few officials accepted in the shanty-town, where he was welcomed as the "doctor in sandals".  It was clearly  an important  formative experience in his full and varied life.
From 1975 onwards, while preparing his PhD, Jean Paul also taught biological science at the University Paris North.  At this time, his scientific research was unrelated to medicine; it dealt with the izoenzymes of … fish!  Jean Paul loved to tell how he scoured fishmongers' stalls in search of different species of fish  for his experiments (later in life, he became expert in cooking … fish!).  After obtaining his PhD in 1978, he was offered a position at the national institute for fisheries research (IFREMER) at Roscoff in Brittany.   As a lover of Brittany, he must have been very tempted, but finally - in October 1980 -  preferred to accept a full-time post as lecturer at Paris North University.  He thus essentially ceased all clinical work, with the exception of a much reduced activity in neonatology at the hospital of Aulnay-sous-Bois.
His research continued in the Laboratory of Developmental Biology and Differentiation in the Biomedical faculty of the University at Bobigny, directed by Mme M-T Chalumeau.  He thus abandoned his work on fish in favour of more biomedical research, in particular in carcinogenesis.  His close colleagues included Michel Kraëmer,  Jean Foucrier and Jany Vassy and they formed a small closely knit  team.  It was, for him,  a happy time with plenty of hard work but also lots of fun and laughter.  Always fascinated by models, whether biological or mathematical, and in response to the pressing needs of the laboratory,  he interested himself increasingly in  quantitative methods: firstly stereology, becoming a member of ISS (International Society for Stereology) in 1979 , and then - very rapidly - image analysis.  He was among the first to work on the estimation of stereological parameters by automated image analysis (Caryometric determination of ploidy by automated image analysis (Leitz TAS) on liver sections, Mikroscopie 1980).  Following his early experience with the TAS of Leitz, he was able to acquire (not without difficulties in importing it from Germany), the first IBAS (Kontron) in France.  It was around this time, and partly to help pay for the IBAS, that Jean Paul accepted a contract with the European Commission to prepare an official report on the toxic effects of nickel.  As usual with Jean Paul, there were no half measures and his final report, completed in 1983, contained a mammoth 1009 pages!  As expert consultant for the Commission, he participated actively in the WHO (Copenhagen) Working Group for Directives on Air Quality for Inorganic Carcinogenic Pollutants (1984-1987) and in the International Committee on the Evaluation of the Epidemiology of Nickel, chaired by Sir Richard Doll (1984-1990).  In Bobigny, his innovative teaching of epidemiology inspired at least one of his students to follow a successful career  in risk evaluation; she tells how his course was the only one in which new knowledge was acquired in the course of the final exam, and that she now tries to follow this example with her own students in the University of Montreal.
The arrival of the IBAS in Bobigny posed serious space problems, resolved (very temporarily) by Mme Chalumeau moving her desk into the corridor to make way for the IBAS in her office!  This resulted in a rapid unblocking of the situation and Jean Paul set to work with his legendary enthusiasm, developing computer programmes which he shared generously with collaborators in several countries of Europe. It was a highly productive period, rich in collaborations, notably with research laboratories in Algeria, Norway and Sweden.  Several of these collaborations continued for many years and resulted in deep and enduring friendships.
In 1984, Jean Paul obtained his third and final doctorate ("Doctorat d'Etat ès-Sciences Naturelles",  a higher degree no longer in existence) for his collection of published work on morphometric models and their use in computer-automated biometry. Leaving Bobigny, he moved to the biomathematics and biostatistics unit (URBB: director Alain-jacques  Valleron) of the French national institute of health and medical research (INSERM), where he was joined a few years later by Jany Vassy.  At URBB, he directed the image analysis group and taught image analysis and stereology to biomathematics students at the University Paris 7.   Remaining at URBB until 1995, he supervised doctoral students working on subjects such as the application of stereological methods to the interpretation of DNA histograms from sectioned cell nuclei, computer simulation of tissue architecture and prognostic grading in histopathology.   Dieter König,  well-known specialist of point processes from Berlin, spent a sabbatical year in Paris and co-directed a thesis applying these methods to the analysis of tissue architecture in cell pathology.  Jean Paul himself became more and more interested in fractals (J. Microsc 1984) and participated regularly  - until March 2004 - in the series of fractal symposia organised by Gabriele Losa at Ascona in Switzerland.
One day, Jean Paul returned very excited from a meeting in Cambridge where he had seen the first demonstration of the new confocal microscope.  It represented a revolution and he had to have one - immediately!  Luckily, it proved possible to redirect a sum of money ear-marked for other equipment, and the first confocal microscope (Biorad) arrived very soon in the lab.  In close collaboration with Jany Vassy (still in Bobigny, but soon to join URBB) for specimen preparation, work on confocal microscopy took off (development of new quantitative methods and applications of all sorts).
In 1995, despite serious health problems (his brain tumour was diagnosed in August 1994), Jean Paul achieved his ambition to found his own laboratory of image analysis in cellular pathology (AIPC), set up within the University Institute of Haematology (IUH, University Paris 7) situated at the hospital Saint-Louis in Paris.   Thanks to the creation of a small biological laboratory (non-existent at URBB) and the proximity in the Institute of other biologists, the collaborations based on the confocal microscope increased and diversified, while the arrival of Damien Schoëvaërt in the Laboratory opened the way to research on image analysis of living cells.  In  histopathology, research became mainly concentrated on prognostic markers for breast and prostate cancer.  With new students, Jean Paul set out to attack the problems posed by tumour heterogeneity, participating in three of four specialised workshops held at Kananaskis, Canada.  He had the idea to use geostatistical methods, developed at the Ecole des Mines, Fontainebleau, to quantitate the degree of intra-tumour heterogeneity from single histologic sections of tumour biopsies. His last student, Vénus Sharifi-Salamatian, obtained her PhD on this topic in November 2002 and their joint article was published in 2004 (J. Microsc).
During his 25 years of scientific research, Jean Paul published or co-published a total of 75 articles in peer-reviewed journals (30 as first author) and contributed around 20 chapters to multi-author books.  His preferred journals were Journal of Microscopy, for the more methodological papers (11 articles, 6 as first author), with Analytical Cellular Pathology (ACP), Analytical Quantitative Cytology and Histology (AQCH) and Cytometry, for applications in cellular pathology (14 articles, mainly  as co-author with one of his students).  He also contributed extensively to Acta Stereologica.  At various times, he belonged to the editorial boards of all of these journals and his scientific rigour and pertinent comments made him a much-solicited referee for these and other journals.  He was an active and well-known member of a number of societies, notably the International Society for Stereology (ISS), the Royal Microscopical Society (RMS), the International Society for Analytical Cellular Pathology (ISAC), the European Society for Analytical Cellular Pathology (ESACP), the International Society for Diagnostic Quantitative Pathology (ISDQP), l'Association Fran����������������������������������?§aise de Cytométrie (AFC, honorary member since 1998) et La Société Fran����������������������������������?§aise de Biologie Théorique (SFBT).  He travelled a great deal and presented or co-presented a large number of communications in national and international congresses.  His former students are dispersed far and wide, not only in France and elsewhere in Europe but also in the USA and Canada.  They will surely miss him, as will his many friends and colleagues in Europe and elsewhere.
On a more personal note, I feel privileged to have shared the last 25 years with Jean Paul.  We met through ISS congresses, firstly in Ljubljana in 1981, and then -  somewhat explosively - in  Gainesville, Florida in 1983, and were thus a "stereological" couple.  I owe him a great deal, both professionally and personally.  I miss him, and will continue to miss him, enormously, but I also retain and cherish so many unforgettable memories.  Thank you, Jean Paul, for everything.


Gabriel Landini, Oral Pathology Unit, School of Dentistry, University of Birmingham, UK.

Jean Paul Rigaut's contribution to Fractal Geometry in Biology.

One of the most inspiring contributions of Jean Paul Rigaut to the field quantitative image analysis is his Asymptotic Fractal model published in 1984 [1]. Professor Rigaut proposed
an empirical law derived from enzyme kinetics that described the relationships between boundary length to resolution in 'nonideally fractal' objects.
While in the physical sciences fractal structures and processes seem to span over several orders of magnitude of scale, in Biology it is relatively uncommon to find fractal regimes spanning more than 2 orders of magnitude; many times this is due to the effect of measuring perimeter lengths and surface areas at resolutions that are close to the size of the elemental constituents (e.g. cells in a tissue). These 'asymptotic fractals' or 'semi-fractals' exhibit fractal behaviour at coarse resolutions but appear to become Euclidean at high resolution. The scaling of boundary lengths in those biological specimensoften result in curved log-log plots rather than linear. Instead of characterising these using a single dimensional value, Rigaut's asymptotic fractal model provides a formal way describing both, bounded and continuously - changing fractional dimensions.
Selected references
[1] Rigaut JP. An empirical formulation relating boundary lengths to resolution in specimens showing'non-ideally fractal' dimensions. J Microsc 1984, 133(1):41-54.
[2] Rigaut JP, Breggren P, Robertson B. Resolution dependence of stereological estimations:
interpretation, with a new fractal concept, of automated image analyser-obtained results on lung sections. Acta Stereol 1983, 2/suppl I: 121-124.
[3] Rigaut JP. Fractals, semi-fractals et biométrie. In: Cherbit G. and Kahane J-P. Fractals: dimensions non entières et applications. Masson, Paris; New York, 1987.
[4] Rigaut JP, Lantuejoul C, Deverly F. Relationship between variance of area density and quadrat area. Interpretation by fractal and random models. Acta Stereol 1987; 6/1:107-113.
[5] Rigaut JP, Berggren P, Robertson B. Stereology, fractals and semi-fractals. The lung alveolar structure studied through a new model. Acta Stereol 1987, 6/1: 63-67
[6] Rigaut JP, Robertson B. Modèles fractals en biologie. J Microsc Spectrosc Electron 1987,
12: 163-167.
[7] Rigaut JP. Natural objects show fractal grey tone functions. A novel approach to automated image segmentation using mathematical morphology. Acta Stereol 1987, 6/3: 799-802.
[8] Rigaut JP. Automated image segmentation by mathematical morphology and fractal geometry. J Microsc 1988, 150 Pt 1: 21-30.
[9] Rigaut JP. Automated image segmentation by fractal grey tone functions. Gegenbaurs morphol Jahrb 1989, 135 (1,S): 77-82.
[10] Rigaut JP. Fractals in biological image analysis and vision. In: Gli oggetti frattali in Astrofisica, Biologia, Fisica e Matematica. Losa GA, Merlini D, Eds. Edizioni Cerfim, Locarno.1989: 111-145.
[11] Landini G, Rigaut JP. A method for estimating the dimension of asymptotic fractal sets. Bioimaging 1997, 5(2): 65-70.
[12] Rigaut JP, Schoëvaërt-Brossault D, Downs AM, Landini G,. Asymptotic fractals in the context of greyscale images. J Microsc 1998,189 (1): 57-63.
[13] Sharifi-Salamatian V, de Roquancourt A, Rigaut JP. Breast carcinoma, intratumour heterogeneityand histological grading, using geostatistics. Anal Cell Path 2000, 20: 83-91.


Gabriel Landini and David A. Randell.

Oral Pathology Unit, School of Dentistry, The University of Birmingham, U. K.

The complexity of cellular neighbourhoods.

Aims . Our aim was to investigate quantitative architectural information from Haematoxylin and
Eosin histological
sections of stratified oral epithelia (normal, dysplastic and neoplastic) and
their possible application to tissue characterisation.
Methods. An algorithmic partitioning of the epithelial tissue compartment was used to create'virtual' cells (v-cells).
These 2D discrete objects were determined from the spatial localisation
of the epithelial cell nuclei. The partitioning was
done with a greyscale watershed algorithm
applied to a Haematoxylin-only image (after H&E dye separation using colour deconvolution). For an arbitrary v-cell x, we defined a neighbourhood membership of level e = 1 as the set of
v-cells adjacent to x and so forth for e > 1. 'Complete neighbourhoods' were identified basedon their topology (i.e. forming
a set of nested (complete) rings of v-cells centred on x). Since
the partition into v-cells on the plane results in a non-regular
lattice with variable number of
neighbours, the scaling (D) of the number of neighbours N(e) on the neighbourhood level e (a
measure of their complexity) was found to be an non-integer number (unlike the case forregular lattices where D = 1).
Only complete neighbourhoods with e = 3 were considered.

Results. The methodology was applied to a variety of oral epithelial tissues as well assurrogate (randomised) sets.
A large spread of D at low e was observed. The average D (over
all sizes) was marginally higher in neoplastic epithelium
(1.194) than normal (1.167) and
dysplastic (1.181) epithelia. The maximum D (considering neighbourhood sizes) was also
higher in neoplastic than in normal and dysplastic tissue.
Conclusions. The arrangement of the v-cells in 2D space approached an irregular hexagonallattice. The scaling of neighbours
 as a function of neighbourhood level had an exponent > 1 and it was marginally larger in carcinomas than in dysplasti
 and normal tissues. Furthermore,the scaling in carcinoma neighbourhoods was found to be the closest to that observed in
surrogate randomised sets. We hypothesise that this may be an expression of the increase intissue disorder and loss of normal architecture classically (and subjectively) described in neoplastic epithelia. Although discrimination based on scaling exponents alone may not behigh enough for reliable diagnostic purposes, it seems to be a useful measure of architecturaltissue make up.
Since the geometry of cellular neighbourhoods does not exactly correspond tothe intuitive concept of distances, it is perhaps
relevant to cell signalling pathways and tissue maintenance.
Acknowledgement: This research has been supported in part by the Science and Technology
Facilities Council, UK, grant ST/F003404/1.   


V. Sharifi-Salamatian,  J. Simony-Lafontaine (*)
(*)Departments of Pathology, Montpellier Cancer Institute, Montpellier cedex, France

On stereological estimation in the presence of spatial heterogeneity

Spatial correlation and heterogeneity are general concerns in stereological. In practice we have to determine the size of observation that is needed to reach a particular confidence interval size in a stereological study. We aim toward answering this fundamental question and describing the effect of heterogeneity on estimation of stereological parameters. A heterogeneity index will be used to develop a methodology for estimating confidence interval size around stereological estimates. This will give answers to the important question of observation size needed to reach a predefined confidence interval size.
We will first present a spatial heterogeneity index for a spatial random field and will assess its interest in image analysis in biology.  Let  be a random field defined on a d-dimensional support set . Some regularity and reproducibility conditions on the random field are required to enable statistical inference based on an observed realization  of the random field. Most of stereological methods are based on spatial means; the spatial mean over a support  is defined by , where is the size of the support v; it is an unbiased estimator for the mean m of the function. The variance of this estimator is related to the autocorrelation function of the random field as well as the size of the support: .

The observation scale is introduced into the analysis of spatial random processes by the integral range [1]. This parameter is defined as  and is interpreted for finite value of A, as the natural scale of the random field, i.e. samples of size in the order of A appear as uncorrelated, and classical spatial statistics and stereological theory as described in [2] can be applied to them. Moreover random process with finite integral range appears to be homogeneous at observation scales larger than integral range. For making reliable stereological estimation the observation size should be at least in the order of several times the integral range; smaller observation size results in large variance and bias in estimation.

The definition of integral range gives the asymptotic behavior of the variance of the spatial mean for random processes with finite integral range as . However, all random processes have not finite integral range. Examples of infinite integral range random process that do not have any natural scale have also been observed. Despite being stationary, these processes appear heterogeneous at all observation scales. Generally fractal fields exhibit such infinite integral range. For such processes, an exponent governs the speed of convergence of  towards zero, . In [3] we proposed the value of  as a heterogeneity index. A value of  close to 1 means homogeneity and a value close to 0 is representative of large heterogeneity.

This property is central for the evaluating the quality of stereological estimators. The size of observation needed to achieve an error smaller than some given bound depends on β. This size is much larger for more heterogeneous field (β close to 0) than for homogeneous field. One has to evaluate the heterogeneity of a field before doing any estimation based on sample means. As confidence interval size obtained at observation scale , (), will scale down as standard deviation one would expect to see an asymptotical decrease of the confidence interval length as . This asymptotic behaviour could be used to predict the size of the confidence interval size as a function of observation size. We have therefore for observation size  and  where the asymptotic behaviour holds the following relation:

A simple two-step process could determine the size of observation needed to reach a confidence interval. In first step heterogeneity parameter β is derived and a basic observation size  is chosen, in the second step the size of the confidence interval  for this observation scale is derived and is used to infer the value of the confidence interval for larger observation size. This enables us to find the suitable size for reaching the needed confidence size target. The value β and the basic observation size could be estimated using the wavelet-based approach described in [3]. The scale where the asymptote holds could be seen over the log-scale diagram as the scale where the linear alignment occurs. Let’s call  the smallest observation scale where linear alignment occurs in log-scale diagram. In the second step we have to calculate the confidence interval of the density at this observation scale. Using this value and the asymptotic relation of spatial means variance, we could easily derive the needed observation scale to reach a target confidence interval size. The confidence interval at scale , i.e.  , could  be obtained empirically by calculating the spatial mean at several spots of size over the whole sample. If we target a confidence interval of , the needed observation size should be equal to  .

To validate the analysis we apply it to three samples coming from 2D cuts obtained over breast cancer tumours and labelled as D1, D2 and D3. First we obtain for each sample the heterogeneity index using the wavelet-based estimator described in [3]. It could be seen that for these data the convergence observation scales larger than 105 μm2. The heterogeneity index is estimated to be equal to 0.15. This value could be used to predict confidence intervals for observation scales larger than 105 μm2. This prediction is shown in Fig. We have also plotted for the sample D1 the 95% Confidence Interval as predicted by the homogeneous assumption that is frequently used in stereology textbooks. As it could be seen using the homogeneous assumption results in the large underestimation of confidence interval and might results in incorrect estimation of stereological parameters

95%-Confidence Interval measured for different observation scales.. For sample D1 the Confidence Interval predicted by homogeneous hypothesis is also plotted.


Yaglom, A.M. 1986 Correlation Theory of Stationary Related Random Functions. Springer Series in      Statistics, New York.
[2].Howard, C. V. & Reed, M. G. 1998 Unbiased Stereology: Three-dimensional Measurement in Microscopy,
bios Scientific Publishers.
[3].Sharifi-Salamatian, V., Pesquet-Popescu, B., Simony-Lafontaine, J., Rigaut, JP., 2004 Index for spatial heterogeneity in breast cancer, J. Micros.,  216 (2) : 110-122.


Architectural Changes Associated with Ageing of the Normal Oral Buccal Mucosa

Rasha Abu Eid, BDS, PhD
Assistant Professor in Oral Pathology, Faculty of Dentistry, The University of Jordan, Amman, Jordan Faleh Sawair, BDS, FDS RCS(Eng.), PhD Associate Professor in Oral Pathology, Faculty of Dentistry, The University of Jordan, Amman, Jordan
Gabriel Landini, DrOdont, PhD Reader in Oral Pathology, School of Dentistry, The University of Birmingham, Birmingham, United Kingdom
Takashi Saku, DDS, PhD Professor of Oral Pathology, Graduate School of Medical and Dental Science, Niigata University, Niigata, Japan

Background and Objective:
With the advances in medicine and the extended average human life expectancy, there is a tendency for the number of
geriatric individuals to increase. It is therefore important to understand the aging process of human tissues to aid in the care of that portion of the population.
The lining of the oral cavity provides protection from the forces of mastication and other potentially noxious effects.
The aging of the oral mucosa has so far been characterised mostly in relation to changes in the oral epithelium such as less prominent rete-ridges, decreased mean thickness (Shklar 1966, Scott et al 1983, Williams and Cruchley 1994), decreased cell density (Hill 1988), decreased mitotic activity and consequently a slow down in tissue regeneration and
healing rates (Barakat et al 1969, Karring and Loe 1973, Hill et al 1994).
Unfortunately, few studies have concentrated on quantitative morphological changes associated with the aging of the oral mucosa. Therefore, the aim of this work is to quantitatively study the architecture and morphology of histological sections representing normal oral mucosa (from biopsies with no mucosal pathology) from different age
groups, to elucidate any associated age changes. Studying the architectural changes associated with aging may help to understand the mechanisms leading to these tissue alterations and help in their monitoring and prevention.

Forty two digital images of normal oral mucosa samples representing different age groups (each group represented a
decade in life ranging from the first to the ninth decade) were captured at x40 magnification (resolution 1.61μm) and 26 images representing the same age groups were captured at x20 magnification (resolution 0.32μm).
The images were analyzed at two levels:
A) The tissue level: The x40 images were analyzed for the irregularity of the epithelial connective tissue interface (ECTI) to estimate changes in rete-pegs prominence using the box counting method to estimate the global fractal dimension.
B) The cellular level : morphometric properties of the epithelial cells were quantitatively assessed and compared across different age groups in the x20 images.
The epithelial cell borders were determined automatically by localization of nuclei based on the optical density of the haematoxylin stain after applying a colour deconvolution algorithm (Ruifrok and Johnston 2001). The nuclei are used as
nuclear “seeds” using greyscale reconstruction (Landini and Othman 2003) and a watershed transform (Vincent and Soille 1991) is then applied to the result of the reconstruction to divide the epithelial compartment into areas of influence unique to each nuclear seed. An example of such analysis is shown in Figure 1.
Cellular morphologic properties were extracted from the segmented images, these included: cell perimeter, area, radius of the inscribed circle centered at the centre of the mass, radius of the enclosing circle centered at the centre of the mass, largest axis length (feret), breadth, convex hull, area of the convex hull polygon, radius of the minimal bounding circle,
aspect ratio, roundness, area equivilent diameter, perimeter equivalent diameter, equivalent ellipse area, compactness, solidity, concavity, convexity, shape, Rfactor, modification ratio, sphericity, feret length, breadth and rectangularity.

The mean box fractal dimension for each age group is shown in (Table 1). No significant trends in the ECTI complexity values with age group were found (One Way ANOVA p>0.05).
Figure 1: A) x20 H&E image of the epithelium B) Haematoxylin component of A C) Segmented epithelial compartment after applying the watershed transform. D) Segmented epithelial compartment after performing the logical AND operation between A and C.
Table 1. Mean Box fractal dimension and the number of cases for different age groups.
Age Range (years) Mean Box Fractal Dimension Number of cases
0-10 1.1069 + 0.03267 2
11-20 1.1117 + 0.08987 2
21-30 1.1414 + 0.07095 4
31-40 1.1290 + 0.03822 4
41-50 1.0933 + 0.05661 5
51-60 1.1373 + 0.05657 8
61-70 1.0992 + 0.04456 9
71-80 1.1260 + 0.04751 5
81-90 1.0903 + 0.01675 3
A total of 52507 “cells” from a selection of cases were analyzed for various morphological parameters. Despite some of the parameters analyzed (case-wise) not showing statistically significant differences across the different age groups (One Way ANOVA p>0.05), cluster analysis showed that two main clusters exist with different average ages.
Cell-wise analysis of the different morphological parameters showed statistically significant differences between the cells of different age ranges (One Way ANOVA p<0.001). The homogenous subsets in the data (post hoc Tukey’s honestly significant difference test) based on different parameters clearly suggested that morphological differences were
present between 3 main age ranges; the first included cases from the first 2 decades of life (0-20 years), the second included cases between 21-50 years of age and the third included cases over 50 years of age.

Various age-related changes have been described in the literature, but preliminary quantitative results obtained in this study indicate that ageing of the oral mucosa does not seem to affect significantly the irregularity of the epithelial connective tissue interface in the range of scales investigated. Mean (case-wise) morphological cellular features
extracted from theoretical cell constructs, also showed no tendency to change with the ageing process when consideredalone, but when submitted to a clustering algorithm, natural groups with different average ages emerged.
When morphological features were compared cell-wise, statistically significant differences were found to exist between
3 main age ranges; 0-20 year, 21-50 year and 51-90 years.
Further studies are needed to look into association of these variables and further morphological features of epithelia in different age groups preferably using large sample sizes. Animal models might also provide further insights into the problem.

1. Abu Eid, R. and G. Landini, Quantification of the global and local complexity of the epithelial-connective tissue interface of normal,
dysplastic, and neoplastic oral mucosae using digital imaging, Pathology Research and Practice. 199: 7 (2003) 475-82.
2. Abu Eid, R. and G. Landini, Morphometry of pseudoepitheliomatous hyperplasia: objective comparison to normal and dysplastic oral mucosae, Analytical and Quantitative Cytology and Histology. 27:4 (2005) 232-40.
3. Abu Eid, R. and G. Landini, Morphometrical differences between pseudo-epitheliomatous hyperplasia in granular cell tumours and squamous cell carcinomas, Histopathology. 48:4 (2006) 407-416.
4. Anderson, D., Cause and prevention of lip canccer, Journal of the Canadian Dental Association. 4 (1971) 138-142.
5. Barakat, N., P. Toto ans N Choukas, Aging and cell renewal of oral epithelium, Journal of Periodontology. 40:10 (1969) 599-602.
6. Cawson, R., Oral ulceration-Clinical aspects, Oral Surgery, Oral Medicine and Oral Pathology. 33:6 (1972) 912-21.
7. Hill, M., Influence of age on the morphology and transit time of murine stratified squamous epithelia, Archives of Oral Biology. 33:4 (1988) 221-9 .
8. Hill, M., J. Karthigasan, J. Berg and C. Squier, Influence of age on the response of oral; mucosa to injury. In: Squier, C. and M. Hill (Eds)
The effect of aging in oral mucosa and skin. (1994) 129-142. CRC Press, Boca Raton.
9. Johnson, N., Orofacial neoplasms: global epidemiology, risk factors and recommendations for research, International Dental Journal. 41 (1991) 365-375.
10. Karring, T. ans H. Loe, The effect of age on mitotic activity in rat oral epithelium, Journal of Periodontal Research. 8:3 (1973) 164-70.
11. Krolls, S. and S. Hoffman, Squamous cell carcinoma of the oral tissues: a statistical analysis of 14,253 cases by age, sex and race of patients, Journal of the American Dental Association. 92 (1967) 571-574.
12. Landini, G. and I. E. Othman, Estimation of tissue layer level by sequential morphological reconstruction, Journal of Microscopy. 209: Pt 2 (2003) 118-25.
13. Ruifrok, A. and D. Johnston, Quantification of histochemical staining by colour deconvolution, Analytical and Quantitative Cytology and Histology. 23: (2001) 291-299.
14. Scott, J., J. Valentine, C. St. Hill and B. Balasooriya, A quantitative histological analysis of the effects of age and sex on human lingual epithelium, Journal de Biologie Buccale. 11:4 (1983) 303-15.
15. Shklar, G. The effects of aging upon oral mucosa, The Journal of Investigative Dermatology. 47:2 (1966) 115-20.
16. Vincent, L. and P. Soille, Watersheds in digital spaces: an efficient algorithm based on immersion simulations, IEEE Transactions on Pattern Analysis and Machine Intelligence. 13 (1991) 583–598.
17. Williams, D. and A. Cruchley, Structural aspects of aging in the oral mucosa. In: Squier, C. and M. Hill (Eds) The effect of aging in oral mucosa and skin. (1994) 65-74. CRC Press, Boca Raton.


Marielda Cataldi (3), Francesca Carella (1), Gionata De Vico (1), Paola Maiolino (4),
Stefano Beltraminelli (2),  Gabriele A. Losa (2)
(1) Department of Biological Sciences, Faculty of  MM. FF. NN. Sciences, Naples University  Federico II, 80138 Naples,  Italy.
(2) Institute of  Scientific Interdisciplinary Studies,  6600 Locarno,  Switzerland.
(3) Department of Veterinary Public Health, Faculty of Veterinary Medicine, Polo Universitario Annunziata, viale Annunziata, 98168  Messina, Italy.
(4) Department of Pathology & Animal Health, Faculty of Veterinary Medicine, Naples University  Federico II,  80138 Naples,  Italy.

The expression of ß-catenin in relation to the fractal organization of canine trichoblastoma tissues.

 Aims : The aim of  this study was  to  assess the  role  of   ß-catenin - a downstream effector of Wnt signalling pathway - in relation to the fractal organization of canine trichoblastoma tissues. Canine trichoblastoma are benign tumours arising from the reduplication of the hair follicle, constituted by a close interaction between epithelial (hair germ) and mesenchymal components (dermal papilla). They are classified into different histological subtypes, namely ribbon type, trabecular type, granular cell type and spindle cell type.

Methods: Formalin fixed, paraffin embedded, haematoxilin and eosin stained histological sections from canine tricoblastoma tissues were analysed by fractal and conventional morphometry. The Fractal Dimension (FD) of different canine tricoblastoma subtypes was determined from  the slope of the regression line describing the fractal region within the bi-asymptotic curve experimentally established by means of the FANAL++ software, which includes subroutines for applying box counting method, data fitting and visualization procedures. ß-catenin expression was evidenced  by  immunohistochemistry.

Rsults: The FD obtained from masks and outlines after grey threshold segmentation of tumour epithelial components showed self-similar fractal properties. Masks but not outlines of  canine Trichoblastoma subtypes showed significant different FD values ranging from 1.75 to 1.85, thus enabling a complete discrimination of different histological types. Trichoblastoma ribbon subtype with the higher amount of mesenchymal stroma displayed an epithelial component with the lowest FD value. ß-catenin immunostaining was found either in the cytoplasm and/or the nucleus of both mesenchymal and epithelial neoplastic cells. Furthermore, the expression pattern of ß-catenin (nucleus, cytoplasm, or both) was closely related with tumour growth pattern morphology.

Conclusions : The FD data suggest that an iterative morphogenetic process, involving both the hair germ and the associated dermal papilla, may be responsible for the tumour architecture and emphasizes the advantages of fractal analysis in the objective characterization of tumour growth. Among  the cellular elements involved in the above morphogenetic processes we investigated  the expression of ß-catenin, a Wnt signalling pathway molecule which has been suggested to play a relevant role in the development of human and  animal hair follicles tumours. In our study, ß-catenin displayed different patterns which could consubstantiate with  the morphogenetic unit, represented by dermal papilla and hair germ, along with its own iterative behaviour and so far assuming an heuristic value in understanding carcinogenesis.


Mauro Tambasco,1,2 Alexei Kouznetsov,2
Anthony M. Magliocco1,3

Department of Oncology,1 Department of Physics & Astronomy,2 Department of Pathology & Laboratory Medicine3 University of Calgary & Tom Baker Cancer Centre, 1331-29th Street NW, Calgary, Alberta, Canada T2N 4N2


Quantifying Local Variations in the Architectural Complexity of Histology Specimens

 Background: Tumor grade (a measure of the degree of cellular differentiation) is one of the factors typically used for the diagnosis and prognosis of cancer. High grade (poorly differentiated) tumors are more abnormal and aggressive than low grade (well differentiated) tumors. However, one of the difficulties with grading tumors is that they often consist of a heterogeneous mixture of cells that exhibit variations in their degree of differentiation and consequently in their ability to drive and maintain tumor growth.  In general, tumor structures derived from poorly differentiated tumor cells possess greater complexity as characterized by   a higher degree of irregularity of the histological structures.1 Hence, fractal dimension, a useful parameter for characterizing complex irregular structures, may provide an objective method for quantifying local variations in cellular differentiation that may exist in histology specimens. Such a tool could aid pathologists in grading heterogeneous histology specimens by drawing attention to the poorly differentiated regions of the tissue and displaying the spatial extent of such regions. It could also be useful for assessing the spatial extent of sub-clinical spread of cancer in tissue obtained from surgical resection.

Objectives: In this paper we describe a technique for estimating spatial variations in the fractal dimension of histological structures. As a proof of concept, we apply this technique to breast and prostate histology specimens containing spatial variations in assigned tumor grade and investigate the correspondence between spatial variations in fractal dimension and tumor grade (an index for the degree of cellular differentiation).

: In previous work,1,2 we developed a fractal based method to segment epithelial structures and compute the fractal dimension of these structures using the box-counting method.  In this work we followed the same approach for estimating the fractal dimension; however, to compute spatial variations in fractal dimension, we applied a user specified rectangular grid to the images and computed the fractal dimension for each grid element. We validated the algorithm using outline structures derived from horizontal planar sections of Takagi surfaces of spatially mixed fractal dimension (Fig. 1).  To test our algorithm on real histology specimens, we selected breast and prostate specimens ranging from normal to high grade (as assigned by breast and prostate cancer pathologists, respectively) from the Calgary Lab Services. We stained the specimens with pan-keratin to highlight the pathologically relevant structures (epithelial cells), and acquired images of the specimens using an optical microscope (Zeiss Axiosope) and digital camera (Zeiss AxioCam) system at a magnification of 10x objective. We joined together histology images consisting of a mixed level of cellular differentiation by joining together normal, and grades 1 to 3 breast specimen images (Fig. 2a), and normal and high grade prostate specimen images. Finally, we applied our method to the constructed mixed histology images to examine the spatial correlation between fractal dimension and tumor grade.

Results: Fig. 1 illustrates the excellent agreement between local variations in fractal dimension and spatial variations in the complexity of the Takagi outline structures.  Fig. 2 shows that spatial variations in tumor grade correspond to variations in the local fractal dimension in the mixed breast histology image. We obtained similar results (not shown) for prostate specimens.


Conclusions: Local variations in fractal dimension correspond to spatial changes in the architectural complexity of breast and prostate histology specimens.  Since architectural complexity has been shown to correlate with tumor grade,1,2 we conclude that our local fractal dimension method is a potentially useful tool for identifying spatial variations in tumor grade.


Fig.1: (a) 5x5 array of outlines of Takagi surfaces of different fractal dimensions that have been joined together. (b) Fractal dimension map showing spatial variations in fractal dimension (FD).


Fig. 2: (a) Mixed image formed by joining four breast histology specimens: (i) normal, (ii) grade 1, (iii) grade 2, and (iv) grade 3. (b) Fractal map of image (a) showing the correlation between fractal dimension (FD) and specimen grade.




[1] Tambasco, M., and Magliocco, A. M., 2007, “Relationship Between Tumor Grade and Computed Architectural Complexity in Breast Cancer Specimens,” Human Pathology, In Press.

[2] Tambasco, M., Costello, B. M., Kouznetsov, A, and Magliocco, A. M., 2007, “Estimating the fractal Dimension of Microscopic Images of Histology specimens,” Journal of Microscopy, Submitted.


G. Bianciardi [1], C. Traversi [2], C. De Felice [3], G.M. Tosi [2], G. Latini [4]

[1] Dpt.of Human Pathology and Oncology , University of Siena, Siena, Italy
[2] Dpt. of Ophthalmology and Neurosurgery, University of Siena, Siena, Italy

[3] Neonatal Intensive Care Unit, Azienda Ospedaliera Universitaria Senese, Siena, Italy
[4]  Division of Neonatology, Perrino Hospital, Brindisi, Italy

Phase transition of vascular network architecture in human pathologies


We have investigated the microvascular pattern in acquired or genetic diseases in humans.
The lower gengival and vestibular oral mucosa, as well as the optic nerve head, was chosen to characterize the vascular pattern complexity due to the simple accessibility and visibility.
Local fractal dimensions, fractal dimension of the minimum path and Lempel-Ziv complexity have been used as operational numerical tools to characterize the microvascular networks.
In the normal healthy subjects microvascular networks show nonlinear values corresponding to the complexity of a diffusion limited aggregation (DLA) model, while in several acquired or genetic diseases, they are approaching the ones of an invasion percolation model. 
Our findings provide new markers to perform  diagnosis, or  differential diagnosis, in human pathology.


Herbert F. Jelinek1, Nebojša T. Miloševic 2 and Dušan Ristanovic 2
1 School of Community Health, Charles Sturt University, Albury, 2640, Australia.
2Department of Biophysics, School of Medicine, University of Belgrade, 11000 Belgrade, Serbia.

Fractal dimension as a tool for classification of rat retinal ganglion cells

Aims. To present a new classification of rat retinal ganglion cells using fractal analysis. To show that the fractal dimension is a robust tool for neuronal classification.

Introduction. Any biological form may have an infinite number of features that differ in significance between groups. Therefore the relationship between morphological features and their functional role is often impossible to ascertain. Retinal ganglion cells (RGCs) provide visual information to the brain and classification of retinal ganglion cells is an essential step in understanding visual function. Using eccentricity information combined with cell body size, dendritic tree size and level of stratification of the dendritic tree within the inner plexiform layer (IPL) of the retina, led to the cat retinal ganglion cells being divided into the a, ß and ? cells (Boycott and Wässle [1974]). Rat ganglion cell classification is, in contrast to the cat, not as straight forward. Several researchers have suggested a number of classification systems and this paper aims, by using a known set of cells that have been classified (Peichl [1989]), to provide a set of novel and robust feature parameters for cell classification. Due to the small centropheripheral gradient of ganglion cell densities that reduces the cell size variability we measured the soma area, dendritic field area and applied fractal analysis to the inner and outer rat ganglion a and d cells of Peichl in an attempt to answer whether these cells can be subdivided based on soma and dendritic field size, and dendritic morphology.

Method. Bearing in mind the great diversity in RGCs classification paradigms, we would like to present the elementary classification method as a theoretically well-established technique to estimate if a measurable feature can or cannot be used for a classification. Let a large set of cells be available and let a variable (a measurable characteristics) of the cell be selected. In order to decide if this variable can be used as a classification property of these cells, we measure this variable over all cells and draw a histogram representing the frequency distribution of the obtained data. If this distribution is multimodal having at least two maxima, there are grounds for believing that this set of cells can be classified into as many classes, as the histogram shows maxima. This assertion is due to the phenomenon known as a superposition: two or more events may be superimposed upon each other to give a single complex event there being no mutual interaction.
101 binary skeletonised
a and d cells were scanned into the computer. The area of both the soma and dendritic field, and the fractal dimension of each cell were measured. The box counting method was used for the fractal analysis as it has been previously shown to be robust for neurons (Fernández and Jelinek [2001]; Miloševic and Ristanovic [2006]). Early work showed that the cell eccentricity has an important influence on the morphology of RGCs (Boycott and Wässle [1974]; Perry [1979]; Peichl [1989]; Huxlin and Goodchild [1997]). As for the RGCs in the rat the dendritic field area increases and dendritic branching complexity decreases with eccentricity (Peichl [1989]; Huxlin and Goodchild [1997]). Changes in cell morphology are more prominent for eccentricities exceeding 2 mm from the area centralis. Therefore, for this work, we restricted inclusion of cells to less than 2 mm distant from the area centralis. The RGCs in the rat are classified after analysing the frequency distributions of the following parameters: the fractal dimension (D), soma area (AS) and dendritic field area (ADF). The distributions of the AS and ADF are normal. On the contrary, the distribution of D is bimodal indicating that there are two independent samples with normal distributions (D1 and D2). To prove this assumption, we tested the mean values of distributions D1 and D2 and found that they are significantly different (p < 0.01). Statistical estimations of these means are shown in the table.

Results. The RGCs could be divided into two groups according to their Ds: simple cellsD from 1.331 to 1.399, and complex cells – D from 1.382 to 1.480. Each group had two subgroups based on their stratification so that four classes of cells are obtained: 1) simple inner, 2) simple outer, 3) complex inner and 4) complex outer cells.

The simple group differed in its soma area only between inner and outer cells (p < 0.05). For the complex group the soma area and fractal dimension were significantly different between inner and outer cells (p < 0.001 and p < 0.01, respectively) suggesting possible differences in functional attributes. Similarly simple and complex cell types differed significantly in fractal dimension for both the inner and outer layer (p < 0.001 and p < 0.01, respectively). Complex inner and outer cell types were significantly different for dendritic field area (p < 0.05) as well. Therefore, it would be expected that these cell groups perform different functions.






m ± CL

95% interval

Dendritic branching pattern




1.348 – 1.381

1.331 – 1.399




1.411 – 1.451

1.382 – 1.480

Cell body

AS (µm2)



264 - 348

188 – 423

Dendritic field

ADF (mm2)



0.060 – 0.099

0.021 – 0.138

CL is 95% confidence limit for the m. ** - p <0.01


The t-test was used to assess whether there was a significant difference between means among the four cell classes. For the simple group, significant difference between inner and outer cells was only obtained for the AS (p < 0.05). The inner and outer complex cells differed significantly in D (p < 0.01) and in AS (p < 0.001). Also for the simple and complex groups the inner cells differed significantly in D (p < 0.001). Outer cells differed in D (p < 0.01) and in ADF (p < 0.05).

Discussion. In the present study we analyzed a large set of the rat RGCs for fractal dimension, cell body size and dendritic field size. In the later two cases we obtained no multimodal frequency distribution of data but for the fractal dimension a bimodal histogram was obtained. Thus we classified our cells according to their fractal dimension into simple and complex cells. We call this method “elementary” since we used only one feature for classification. Using two or more corresponding cell properties, the number of cell classes could be greater. In the present study, we adopted in fact two features: a morphometric one - the fractal dimension, and a morphologic one - the stratification level of the dendritic tree within the IPL.  The distribution of the parameters bring about the conclusion that the sample of RGCs could only be classified using the fractal dimension of their dendritic branching complexity.

Conclusion. The fractal dimension proved to be a robust feature for the rat RGC classification. We identified 2 subgroups of a and d cells, the simple and complex group, with members of these found both in the inner and outer sublamina of the IPL. The results of this study extend those of previous reports in that the large cells can be classified into complex and simple cell types that are located in both the inner and outer IPL. Thus the visual processing of these cells is more complex then has been previously suggested. Moreover, this study supports previously observed differences in the rat RGCs morphology. Unlike previous studies, where eccentricity-dependent changes were qualitatively observed from the corresponding plots, in this study all conclusions stand, since they were drawn after mathematically and statistically correct quantitative analysis.
Boycott, B.B. and H. Wässle [1974], The Morphological Types of Ganglion Cell of the Domestic Cat's Retina. Journal of Physiology 240: 397-419.
Fernández, E. and H.F. Jelinek [2001], Use of Fractal Theory in Neuroscience: Methods, Advantages, and Potential Problems. Methods 24: 309-321.
Huxlin, K.R/ and A.K. Goodchild [1997], Retinal Ganglion Cells in the Albino Rat: Revised Morphological Classification, The Journal of Comparative Neurology 385: 309-313.
Miloševic, N.T. and D. Ristanovic [2006], Fractality of Dendritic Arborization of Spinal Cord Neurons. Neuroscience Letters 396: 172-176.
Peichl, L. [1989], Alpha and Delta Ganglion Cells in the Rat Retina. The Journal of Comparative Neurology 286: 120-139.
Perry, V.H. [1979], The Ganglion Cell Layer of the Retina of the Rat: a Golgi Study. Proc. R. Soc. London B  204: 363-375.


Nebojša T. Milošević, Dušan Ristanović
Department of biophysics, School of Medicine, University of Belgrade, Serbia

The box-counting method as an efficient tool for 2D fractal analysis of neuronal dendritic arbor

The aim of the present report is to show that using the box-counting method as most popular method of fractal analysis the interval of self-similarity of 2D image of neuronal dendritic patterns can be extended over three orders of magnitude. In addition, this report presents our main results relating to applications of fractal geometry to quantitative study of neuronal dendritic branching patterns.

The drawings of neurons were converted into 2D digitized images by scanner and transformed into skeletonized images using Image J software ( Skeletonized images were analyzed by the box-counting method using the same software and public domain Fractop software (, developed by Herbert Jelinek (1996). The box-counting method covers the image with sets of squares (Fig. 1A). The fractal dimension D of the image is determined from the slope of the log-log relationship between numbers of squares covering the image border (N) and the size of the square edge (r) (Fig. 1B). In performing the box-counting method, the box sizes were taken as a power of 2, that is, from 2 to 2048 pixels. Such methodology enables highly significant linear relationship between log N and log r on three decade of the range (Fig. 1B, dotted lines).



Figure 1. An example of the box-counting analysis of the skeleton image (A) and corresponding log-log plot of a relationship between N and r (B). The fractal dimension is negative value of the slope of fitted line. R is the correlation coefficient of fitted line and p is the significance level. An interval of self-similarity is shown with two adjacent dotted lines.


In performing box-counting method to arborization patterns of stalked and islet neurons in substantia gelatinosa of four species (Miloševic et al., 2007a) we showed that its Ds were significantly different in the human species only. Analysis of the stalked and islet dendritic patterns among different species showed that the Ds of both patterns are significantly different between all pairs of species considered. Applying box-counting method on the images of human and rat neurons (Miloševic et al., 2005), we showed that the D discriminated the populations of neurons among different laminae. Finally, by applying the same method of fractal analysis to the dendritic arbors of six types of neurons in monkey dentate nucleus we proved that only three cell types can be clearly discriminated (Miloševic et al., 2007b).

By using statistical assessment of correlation coefficients of a straight line fit, we show in the present report that interval of self-similarity of neuronal dendritic patterns can be extended over two decades of range. In addition, the fractal dimension calculated appears to be the unique characteristics of all the data points. We assume that a simple object, such as 2D image of neuronal dendritic pattern, could be completely described by a single fractal dimension. Obtained fractal dimensions characterize the global dimension i.e., a statistical characteristics of the whole object representing a measure of its global complexity.
Also, this report shows that the fractal analysis is a suitable tool for the morphological analysis of dendro-architectonics of 2D neuronal images. The fractal dimension represents a parameter which could be of relevance for intra- and/or inter-species comparisons of neuronal populations. However, the question remains open whether the fractal dimension can establish a link between neuronal structure and function.

Jelinek HF, Steinke A, Bowdren P. [1996] A biological application of fractal analysis on the World Wide Web. Complex systems: From local interactions to global phenomena. IOS Press: Amsterdam; 24-33.
Milošević N.T., Ristanović D., Stanković J.B., Gudović R. [2007a] Fractal analysis of dendritic arborisation patterns of stalked and islet neurons in substantia gelatinosa of different species. Fractals 15(1): 1-7.
Milošević N.T., Ristanović D., Stanković J.B. [2005] Fractal analysis of the laminar organization of spinal cord neurons. J. Neurosci. Methods 146(2): 198-204.
Milošević N.T., Ristanović D., Gudović R, Rajković K., Marić D. [2007b] Aplication of fractal analysis to neuronal dendritic arborisation patterns of the monkey dentate nucleus. Neurosci. Lett. 425: 23-27.



Rangaraj M. Rangayyan, PhD, PEng,
Department of Electrical and Computer Engineering
Schulich School of Engineering University of Calgary,
Calgary, Alberta, Canada

Detection of architectural distorsion in prior mammograms using Gabor filters, phase portraits,  fractal analysis, and texture analysis.

Architectural distortion is a subtle sign of early breast cancer that is commonly missed in screening mammography. We propose techniques for the detection of architectural distortion in mammograms based on the analysis of oriented texture using Gabor filters and phase portraits. We show that constraining the shape of the general phase portrait model can reduce the false-positive rate in the detection of architectural distortion. A sensitivity of 84% with 4.5 false positives per image was obtained in tests using a database of 19 cases of architectural distortion and 41 normal mammograms. Furthermore, using additional methods of fractal analysis and texture analysis, a sensitivity of 79% at 8.4 false positives per image was achieved in the detection of sites of architectural distortion in 14 “prior mammograms” acquired two years (on the average) before the corresponding screening mammograms on which breast cancer was detected. The results indicate that the methods developed can help in the detection of breast cancer at earlier stages than possible by manual interpretation.


Konradin Metze(1), Randall Luis Adam(2), Maria Luiza de Castro Ramos Valladão(3), Karime Cury Scarpelli(4), Neucimar Jerônimo  Leite(2),  Irene G.H.Lorand-Metze(5)

1) Department of Pathology, Faculty of Medicine, State University of Campinas (UNICAMP), Brazil; 2) Institute of Computing, State University of Campinas,Brazil;
3) Postgraduate Course in Medical Pathophysiology, Faculty of Medicine, State University of Campinas (UNICAMP), Brazil; 4) Postgraduate Course in Medicine, Faculty of Medicine, State University of Campinas (UNICAMP), Campinas, Brazil; 5) Department of Medicine, Faculty of Medicine, State University of Campinas (UNICAMP), Brazil


 Introduction: Examination of nuclei in histological or cytological preparations reveals important information on cell physiology and, furthermore, is of great diagnostic and prognostic importance. Malignant neoplastic growth induces important modifications not only of the genome, but also of the composition and distribution of the histone and non-histone nuclear proteins thus provoking alterations of the nuclear architecture. Usually, neoplasias with a higher number of genetic or epigenetic changes show a more aggressive behaviour. In other words, accentuated remodeling of the nuclear architecture could be of prognostic relevance and indicate a bad prognosis. Recently it has been postulated that under physiologic conditions the DNA arrangement in nuclei is of fractal nature and that this might be a common feature of all cell nuclei. Therefore one might hypothesize that changes of the nuclear architecture in malignant cells could be accompanied by modifications of the fractality.

Aim: to investigate whether alterations of the fractal dimension in nuclei of malignant neoplastic cells in routinely stained cytological smears could be of prognostic relevance.

Material and Methods: We studied two different biological models:  1. Survival of patients with acute B-precursor lymphoblastic leukemia (B-ALL). 2. Therapy response of dogs with canine transmissible venereal tumor treated (cTVT) with vincristine until complete clinical remission. Random pictures from at least 100 nuclei per case of routinely May-Grünwald-Giemsa stained bone marrow slides (model 1) or HE stained tumor imprints were captured and nuclei were segmented interactively. The images were converted to grayscale and viewed as “irregular surfaces”, with the z coordinate represented by the gray level of each pixel. The fractal dimension (FD) of these surfaces was calculated according to Minkowski-Bouligand method extended to three dimensions. We estimated the goodness-of- fit with the help of the R2 value between the real and the theoretically estimated values in the linear regression. The distribution of the residuals was evaluated with the Kolmogorov-Smirnov test. We calculated the percentage of cells (N) with normally distributed residuals. The prognostic relevance of FD and R2 was analyzed in univariate and multivariate Cox regressions comparing them in each model with established prognostic factors. The internal stability of the final models was tested using bootstrap resampling.

Results: The Minkowski FD ranged between 2.238 and 2.285. for the nuclei of the ALL patients and between 2.106 and 2.199  for the nuclei of the cTVT. The R2 values were found between 0.945 and 0.986 for the all nuclei and between 0.873 and 0.937. for the canine tumor. The N values varied considerably with a minimum of 2.0% in ALL blasts and 12% in cTVT cells and a maximum of 100% in both neoplasias. In both models R2 was  indirectly correlated with FD and directly with N. R2 entered the multivariate Cox regression as an independent statistically significant prognostic variable in model 1, but this was not the case for the Minkowski fractal dimension. These results were confirmed by bootstrap resampling studies. In model 2, again in a multivariate Cox regression, the R2 values were an independent prognostic factor for therapy response (together with the clinical prognostic factor “season” of treatment) but not the Minkowski fractal dimension.

Discussion: In the log-log diagram of the Minkowski fractal dimension the dots form a more or less pronounced concavity, due to the presence of local maxima and minima of the image. Nuclei with a high number of local maxima and minima show a higher FD but simultaneously lower goodness-of-fit, since more maxima and minima increase both the complexity (FD) and the expression of the concavity in the log-log plot. High R2 values, together with normally distributed, randomly scattered residuals, are only found in nuclei with few local maxima and minima. Therefore the goodness-of-fit may be interpreted as a measure of roughness of the surface of the pseudo 3D image of the nucleus.

The main question, whether a given structure should be considered as fractal, is linked to its scaling characteristics, following power laws, expressed as linear regressions in log-log plots. A high R2 value, equivalent to the goodness-of fit of these regressions, as well as a normal distribution of the residuals are also important. Since in our study many cells did not fulfill these criteria, we should interpret  FDs with great caution.

Conclusion: In summary, we suggest that the goodness-of-fit of the Minkowski-Bouligand dimension of routinely stained nuclei in cytologic preparations is a measure of the remodeling of the nuclear architecture with prognostic relevance.

Literature :

Dubuc, B, Quiniou, J F, Roques-Carmes C, Tricot C, Zucker S W. Evaluating the fractal dimension of profiles.  Physical Review A 1989, 39:1500-1513.
Minkowski H. Über die Begriffe der Länge, Oberfläche und Volumen.  Jahresbericht der Deutschen Mathematischen Gesellschaft 1901,9:115-121.


Przemysław Waliszewski
Department of Urology, Klinikum Fichtelgebirge Schillerhainstrasse 1-8, 95615 Marktredwitz, Germany,


Gompertzian dynamics is considered to be universal dynamics of cellular growth. It is shown that this dynamics emerges in the Markovian systems already. Gompertzian dynamics emerges according to the principle of fractal-stochastic dualism. This occurs owing to the complex coupling of probabilities of stochastic processes and the existence of fractal structure of time and space; the essence of life.

A Markovian model of growth
Let us consider a small cellular colony with less than 106 cancer cells growing within a normal tissue environment. First, let those cells possess broad autonomy. Let metabolic exchange with normal surounding cells be very weak or does not exist. Second, there is no blood vessels in the colony. Feeding of cells occurs by diffusion. Third, cancer cells continue to proliferate spontaneously owing to a large number of molecular defects. Fourth, there is a minimal reaction of the external tissue systems, such as the neuroimmunohumoral system or the internal mechanisms, such as apoptosis. Finally, cancer cells belong to a single clone. Cells do not undergo differentiation or do not express multiple transitional phenotypes. For the purpose of this study, space is defined by a system of the geometrical co-ordinates. Those co-ordinates build up a volume, in which the nonlinear dynamic process occurs. Time is a parameter, which takes the sense of the evolutional co-ordinate

There is a difference between the time-scales of molecular signaling, i.e., femtoseconds to miliseconds, cellular growth, i.e., hours and cellular proliferation, i.e., days. Single cells in the colony integrate molecular signals much faster than the colony expands in space-time. There is no memory of the state at previous timepoints in that tissue object. Hence, it is possible to describe a growth trajectory under those assumptions as a Markov chain of transitions for each timepoint by Eq. 1:





in which is a probability that the growing cellular colony is at the positions x0,…, xn at the timepoints 0,…n; P is a conditional probability that between timepoints n-1 and n the growth succeeds from the position xn-1 to xn.

Since cellular growth and proliferation into a tissue structure occurs simulataneously in space and in time, it is particularly interesting to introduce the probability P as a function of geometrical spatial variable x and scalar time t. A speed of both processes is usually not large. So, the spatial expansion of cellular system  in the small time step  will also not be large. A change of the probability P in the infinitezimal intervall of time can be described by differential Eq. (2), in which such the change results from a difference between the probabilities of the jump from the position k to x and the probabilities of return:






This leads to Eq. (3), which has a well-known form of the Fokker-Planck equation:






in which both V(x) and D(x) represent a potential function and a diffusion coefficient in the classical physics, respectively.

Eq. (3) can be transformed to the Langevin equation given by Eq. (4)





in which v(t) is the dynamical variable, i.e., the velocity of the division process, l is the dissipation parameter, and dM(t) stands for the fluctuations, which compose a stationary differential Markov process.

The latter process is specified by the probability distribution P(M(t), t) given by Eq. (5):







in which t stands for scalar time, k is the Fourier variable, a, b>0 are real, constant factors.

The above-defined Langevin equation, Eq. (4) possesses a characteristic function f of the following form, Eq. (6):







It is worth to notice that there a relationship between such the conditional probability density P(v, t), and the Gompertz function f(t). Indeed, the conditional probability density P(v,t) can be expressed in the form of the Fourier transform taken with respect to the variable (v-v0e-lt) containing the Gompertz function f(t), equation (7) [1], [2]:






in which the Gompertz function f(t) is defined by Eq. 8:






The gompertz function given by eq. (8) possesses the following derivative,






This derivative reflects a speed of tumor growth. it produces an asymmetric bell-shaped curve. this feature of eq. (9) suggests that at least two different dynamic processes determine dynamics of tumor growth.

The Gompertz function is a solution of the mathematical model of tumor growth [3]. The best fit of experimental data reflecting growth of normal cells or tissue structures as well as transformed cells or tumor tissues can also be done with that function. Owing to the high accuracy of the fit, the Gompertzian model rather than the logistic model, the von Bertalanffy model, or the Bertalanffy-Richards model has been used to investigate details of growth. Although the Markovian model of growth does not apply to more complex conditions of cellular growth in a tissue, its solution points out the existence of an important principle underlying growth of biological structures. A large number of parallel, dynamic processes at the microscopic scale, such as molecular biochemical reactions gives rise to Gompertzian dynamics of growth and self-organization at the macroscopic scale. Those microscopic processes are integrated into mean macroscopic dynamics according to the principle of fractal-stochastic dualism [4]. This occurs owing to the complex coupling of probabilities of stochastic processes and the existence of fractal structure of time and space [5]; the essence of life.

[1]  B. J. West, M. Bologna ands P. Grigolini: Physics of Fractal Operators, Springer, New York, 2003.
[2]  P. Waliszewski and J. Konarski: The Complex Couplings and Gompertzian Dynamics, Complexus Mundi Emergent Patterns in Nature Proceedings of the International Conference Fractals 2006, Vienna 2006, pp. 343–344.

A.K.Laird: Dynamics of tumour growth. Br J Cancer 18 (1964), 490-502.
P. Waliszewski: A Principle of Fractal-Stochastic Dualism and Gompertzian Dynamics of Growth and Self-Organization. Biosystems 82(1) (2005), 61-73
[5]  P. Waliszewski and J. Konarski:
On time-space of nonlinear phenomena with Gompertzian dynamics. Biosystems 80 (2005), 91-97.



Rosario Santoro 1), Giorgio Turchetti 2), Ugo Albisinni 3), Fiorenzo  Marinelli 1)
1) Molecular Genetic Institute (IGM-CNR) c/o IOR Via di Barbiano 1/10 40136 Bologna
2) Dept. of Physics - University of Bologna, Italy
3) Servizio di radiologia e diagnostica per immagini - IOR Bologna, Itala


Introduction - We considered here a set of radiographic images of the hip, for which we compute the  spectrum. The spectrum consists of two asymmetric parabolas. The aim of this study was to evaluate the differences of bone structure in patients with and without osteoporosis. Bone fragility at the femoral neck is characterized not only by low mineral density but also by the changes of the organization of the trabecular architecture. Early diagnosis based on multifractal analysis of bone structure deterioration represents a suitable tool in the prevention of osteoporotic fracture. Analysing the architecture of the femoral neck one observe that a progressive deterioration of the trabecular structure induces  lacunae and random defects in the network of the cancellous bone, with high risk of fracture.

Results –The multifractal spectrum is narrow for normal bone,  whereas the lacunae and defects, present in the osteoporotic bone, provide a broad spectrum.  The spectral analysis relies on the identification of two geometrical parameters related to the width and asymmetry of the function . The results show that non-invasive methods to measure bone architectural changes can be effective in order to identify the patients at risk of fracture.



A. Ben Abdallah (1), H. Akkari (1), I. Bhouri (2),  P. Dubois (3) and M.H.Bedoui (1)

1 Laboratoire de Biophysique, TIM, Facult¶e de M¶edecine de Monastir,Tunisie
2 Unit¶e de recherche Multifractals et Ondelettes, Facult¶e des Sciences de Monastir, Tunisie
3 INSERM, U 703, ITM, CHRU Lille, France

Lacunarity estimation method for analysis of 3D images of trabecular bone

1 Aims
The main objective is to better undestand osteopeny, which is an intermediate bone disease
between healthy and osteoporotic pathologic state. We present a methodology using the whole
information contained in the considered volume. A new approach is investigated: 3D hetero-
geneity measurement lacunarity [2]. This model is elected both for its mathematical properties
and physical properties. The aim of this work is to adopt this 3D method to study tabecular
volumes obtained from the clinical images.
2 Methods
Proposed method : The method that we propose is a combination of the idea of RDBC
method (Relative Di®erential Box Counting) [4] and gliding box algorithm [1] for calculation of
lacunarity using the following function :
L(r) = r® + °: (2.1)
The parameter ® represents the order of convergence of L(r), ¯ estimates the concavity of the
hyperbola and ° is a translation term. Those parameters, noted ®P, ¯P and °P, are obtained
from the lacunarity curves and are computed as the solution of a least squares problem.
Theoretical method of generalised self-a±ne sets : The relation 2.1 correlates with
the theoretical behavior function ¤ for multifractal sets. In [3]; the authors have proved that for these sets the lacunarity is given by ¤(r) = ¸rD¡3: Where D is the correlation dimension D(2) ([5]) and the prefactor ¸ is the parameter related to the lacunarity and also called the lacunarity parameter. The ®TH and ¯TH are deduced from this function.
3 Results
Application to models : Twenty 3D models are constructed to test our method. A ¯rst
group of models called healthy representing bones in a good state. The moderately a®ected
model refers to a bone starting to lose density. The seriously a®ected model comprises bones
a®ected by osteoporosis. Table 1 shows the results (®P, ¯P) of the application of our proposed method and results of ®TH, ¯TH obtained by applying the theoretical method to calculate lacunarity for the three groups of models. The parameter P undergoes the most signi¯cant variation. Application to medical imaging : CT scan wrist images were obtained in 8
patients. For each of the 8 patients, a Volume Of Interest (VOI) was extracted strictly covering
the trabecular area, as shown in Figure 1. We represent the variation of the parameter ¯P of
3D volume bone in the studied population (1).
1 proposed method   theoretical method
Models ®P ¯P ®TH ¯TH
Healthy 2,345����������������������������������������������������������������?§0,045 4,404����������������������������������������������������������������?§0,22 1,188����������������������������������������������������������������?§0,19 0,0095����������������������������������������������������������������?§0,0028
Moderately a®ected 2,27����������������������������������������������������������������?§0,03 5,48����������������������������������������������������������������?§0,4 1,776����������������������������������������������������������������?§0,22 0,06����������������������������������������������������������������?§0,04
Seriously a®ected 2,10����������������������������������������������������������������?§0,08 7,80����������������������������������������������������������������?§1,52 2,437����������������������������������������������������������������?§0,32 0,54����������������������������������������������������������������?§0,76
Table 1: Parameters obtained by the application of our proposed method and theoretical
Figure 1: The Volume Of Interest of trabecular bone and parameters BetaP of Lacunarity for
CT scan images

4 Conclusion
In this study, we extend the 2D lacunarity estimation method based on gliding box to a 3D
measure of lacunarity and the technique of relative di®erential box counting was included. We
prove that the parameter ¯ obtained from the lacunarity curve increases with the degradation
of structure of models. It has allowed us to classify models and to detect the moderatly a®ected case. In the clinical fields, this preliminary study has shown that lacunarity, specifically theparameter ¯, is appreciable for trabucular bone state classi¯cation.

[1] A.Zaia, R.Eleonori, P.Maponi, R.Rossi, and R.Murri, "MR Imaging and osteoporosis: Fractal lacunarity
analysis of trabecular bone," IEEE Transactions On Information Technology in Biomedecine, vol 10(3),.pp.
484-489, 2006.
[2] B.B.Mandelbrot. "The fractal geometry of nature", San Francisco, CA: freeman, 1982.
[3] C.Allain, and M.Cloitre, "Characterising the lacunarity of random and deterministic fractal sets," Physical
review A, vol 44(6), pp. 3552-3558, 1991.
[4] X.C.Jin, S.H.Ong, and Jayasooriah, "A practical method for estimation fractal dimension," Pattern Recog-
nition letters, vol 16, pp. 457-464, May 1995.
[5] L.Olsen, "Self-a±ne multifractal Sierpenski Sponges in Rd; ".Bac Math, vol 183, N±1, pp. 143-199, 1998.


Marino Viganò

Speculating about Leonardo

The “Rivellino” of Locarno’s castle    (1507)

 The medieval castle of Locarno, eventually built in the 9th century, enlarged and reinforced in the 14th and 15th centuries, has been a frontier bulwark of Lombardy up to 1513, when it was conquered by the Swiss cantons opposing Louis XII, king of France, duke of Milan from 1499 to 1513. Almost totally destroyed by the new lords, the Swiss, in 1532 the castle of Locarno has been then turned into a palace, from where the swiss governor of the land (Landvogt) ruled the village and the surrounding country.

Few remains remember the powerful castle’s and harbour’s fortifications: a round tower, walls with machicoulation, moats; and a still standing “ravelin” which - even if partially hided by houses - reveals its unusual character of military architecture from Tuscany in central Italy. The “ravelin” shows in fact a pentagonal plan, with two faces, two flanks, a gorge. The two faces meet with an angle of 90°, the walls are divided by a cordon in an escarpment some 9 to 10 meters and a parapet some 1,5 meters high. The original highness has been calculated in 17 to 18 metres, the lower part of the building and the embankment on which it stands laying now underground no less than 7 metres.

Four casemates are opened into the bastion, two in the eastern and two in the northern face, one of them have been widened in the 20th century to become a “door” through which one can now visit the internal galleries. There the visitor realizes that another branch of gallery linked the castle to the “ravelin” only indoor, with no possibility to enter it from outdoor. An ancient tower, eventually from the 15th century, stands in the very center of the building, acting as the galleries’ vaults supporting pillar. Above every casemate a muzzle chimney rises to the upper platform and let remove the artillery black powder’s smoke and light and air come in. A fifth opening, a slit in the central pillar’s upper half, is probably a pit to descend ammunitions from the platform into the galleries.

An investigation through archival documents has given elements to establish when, by which government, perhaphs by which engineer this bastion has been added to the castle. Began in summer 1507, under the french rule in Lombardy, the “ravelin” has been ordered by Charles II d’Amboise, seigneur of Chaumont, baron of Charenton, general governor for Louis XII of Valois-Orléans and is a simple “ring” of a chain of fortifications built to prevent the Imperial and Swiss armies threaten the lower plains of the duchy of Milan, to stop a foreseen campaign of the emperor Maximilian I of Habsburg wishing to become duke of Lombardy pulling the king of France out from northern Italy.

But whilst the other fortifications have been built with round-tower or horse-shoe plan, the “ravelin” of Locarno’s castle is a bastion. Johann Rudolf Rahn, an historian of the monuments of the Canton Ticino, has written in 1894: “This building remembers a drawing by Leonardo da Vinci in a manuscript of the “Bibliothèque de l’Institut””. May circumstantial prooves seems to confirm his speculation. Leonardo was then again in Milan, in the service of the king of France as “painter and ordinary engineer”, he knew the typologies from central Italy, he had built two similar works on the front and rear of Milan’s castle... and no other expert seems to have had the same knowledge.


Giovanni Dietler

Laboratoire de Physique de la Matière Vivante, Ecole Polytechnique Fédérale de Lausanne(EPFL), 1015 Lausanne, Switzerland 
Polymer Physics versus DNA Topology

DNA appears in linear, circular and knotted forms, depending on the situation and conditions. DNA is therefore an excellent playground for testing statistical mechanics theories. In the present lecture, I will focus on the statistical properties of linear and knotted DNA and show that their critical exponents or their fractal dimensions describing the divergence of the end-to-end distance can be determined from Atomic Force Microscopy (AFM) images of DNA. Additional quantities like the distribution function of the end-to-end distances as a function of the DNA length can also be extracted from the AFM data. Then, the localization of the knot crossings for simple and complex DNA knots will be reported, confirming theoretical predictions. This result points to a possible mechanism, used by enzymes (Topoisomerases) that control the DNA topology inside the cell, to recognize essential crossings on DNA.




DNA knots of different complexities deposited on mica and imaged by Atomic Force Microscopy.
Top row: simple knots (number of essential crossings < 10)
Bottom row: complex knots (number of essential crossings >= 10). DNA length 3.5 µm. Scale bar 250 nm.


Darja Kanduc and Giovanni M. Capone

Department of Biochemistry and Molecular Biology, University of Bari, Italy


The fractal concept offers a new dimension for analyzing the structure-function relationship of
a protein in a wide proteomic context. At present, protein functional analysis is carried out
protein-by-protein, through the definition of specific motifs, peptide features and activity
domains for each protein. Given a length of five amino acid residues as a minimal functional
peptide unit in cell biology and immunology (1), the 36,103 proteins that collectively form the
human proteome ( result into 15,771,565 occurrences of
2,388,563 unique 5-mers. In statistical terms, this means that each unique pentapeptide is
repeated a mean average of 6.6 times throughout the human proteome. However, this
mathematical pentapeptide dissection does not offer a/the key to understand the link betweenspecific 5-mers and cell functions. Moreover the available analyses of protein sequences are based on heterogeneous parameters such as hydro-phobicity(philicity), protrusion degree, periodicity of amphipathic structures, patterns of binding motifs etc. (see, thus, again, offering no common platform to a comprehensive
understanding of functions specifically residing in specific peptide units.
We define the pentapeptide profiling of human proteome based on the assumption that high
specificity is intrinsic to high biological efficiency. Every protein corresponds to a specific
function(s), still the specificities collected in the human proteome must find their definition into common basic principles. We advance the similarity hypothesis as a first platform able to
collect the functional protein motifs into a common peptidome platform having no-similarity to
the host proteome as a minimum common denominator (2). The similarity hypothesis
presupposes that the protein world has evolved according to a scheme defined by two
peptidome sets: the set of unique pentapeptide units, each of which carries specific catalytic,
reactive, interactive or recognitive function, and the set of repeated high-similarity peptapeptide units with no coded specific functional properties.
The similarity hypothesis of the human proteome appears experimentally defined by a fractal
dimension. Indeed, analyzing human proteins for pentapeptide similarity to the human
proteome produces regular similarity profiles where peptide areas repeatedly present in the
human proteome alternate to peptide areas with a low match number or no matches at all, thus representing unique molecular signatures of the analyzed protein (3). Here we analyzed the global proteomic distribution of the similarity quality by using an ampler sample of human
proteins. Exactly, we scanned a sample of 30 human proteins for similarity and obtained the
scanning illustrated in the histogram of Fig 1, reporting the density of each 5-mer from the
human sub-proteome sample along the entire human proteome. It can be seen that protein walks with high level of similarity alternate to protein regions with no counterpart in the human
proteome. The similarity profile described in Fig. 1 implies a repetitive, self-affine, fractal
property in the human proteome landscape.
Fig. 1 -Pentapeptide similarity profile to the human proteome of: A) a sub-proteomic human
sample formed by 30 protein protein walk and amounting to 16,646 pentamers overlapping by
4 residues; B) magnification of the solid box in A) corresponding to 1665 pentamers; C)
magnification of the solid box in B) corresponding to 167 pentamers. The profile of A, B and
C plots is consistent with the existence of a scale-free or fractal phenomenon known as a fractal landscape (5).
The finding is clearly visualized in the progressive magnification of segments of the proteomic
walks: panels A, B and C statistically resemble a shared overall pattern. This overall pattern of
sharing appears to be a basic property of the human proteome as a whole (4). The similarity
quantity as a fractal dimension provides a new, qualitative method to characterize short linear
peptide motifs critically involved in cell biology and immunology. De facto, the short
functional motifs that mediate protein–protein interaction in cell compartment targeting,
enzymatic catalysis and immune recognition are represented by rare, unique motifs specifically
owned by specific proteins.
1. Lucchese G, Stufano A, Trost B, Kusalik A, Kanduc D. Peptidology: short amino acid
modules in cell biology and immunology. Amino Acids. 2007;33:703-7.
2. Kanduc, D. Immunogenicity in peptide-immunotherapy: from self/nonself to
similar/dissimilar sequences. In: Multichain Immune Recognition Receptor Signaling:
From Spatiotemporal Organization to Human Disease, A. Sigalov ed. Landes Biosciences,
Austin, TX, 2008.
3. Capone G, De Marinis A, Simone S, Kusalik A, Kanduc D. Mapping the human proteome
for non-redundant peptide islands. Amino Acids. 2007, in press.
4. Dummer R, Mittelman A, Fanizzi FP, Lucchese G, Willers J, Kanduc D. Non-selfdiscrimination
as a driving concept in the identification of an immunodominant HMWMAA
epitopic peptide sequence by autoantibodies from melanoma cancer patients. Int J
Cancer. 2004;111:720-6.
5. Buldyrev SV, Goldberger AL, Havlin S, Peng CK, Stanley HE, Stanley MH, Simons M.
Fractal landscapes and molecular evolution: modeling the myosin heavy chain gene
family. Biophys J 1993;65:2673-9.


Loretta Ichim  (1,2) Radu Dobrescu (1)
Faculty of Control and Computers, Politehnica University of Bucharest, 313 Splaiul Independentei, Romania,  (2) ”Stefan S. Nicolau” Institute of Virology, Romanian Academy,  285 Mihai Bravu, Bucharest, Romania

Fractal analysis usage to determinate HeLa cell nuclei size distribution in spectrophotometric images.

The paper aims to use fractal analysis techniques for evaluating viral infection effects on cellular substructures and to compute a quantitative measure of cell nucleisizedistribution.
To reach our goal it was started using an improved method compared to that proposed by Perelman et al [1]. This method was used to establish the amount of cellular nuclei size distribution based on Light-Scattering Spectroscopy.
The Mie scattering spectra are collected between 350 – 619 nm wavelengths, in the visible spectral range with an experimental spectrophotometer device which is subject of a patent [2]. The authors consider that the fractal analysis of cell nuclei size distribution using Mie spectrophotometric evaluation is for the first time described in this paper. Because a variety of methods are available to compute the fractal dimension, Df, representing the measure of the nucleus shape irregularity, data were treated using three of the most prominent methods of fractal analysis: Petrosian, Hurst exponent and Height-height correlation (HH). To estimate changes in cell nuclei size distribution using fractal analysis method they were implemented an experimental model for in vitro infection with Herpes Simplex Virus (HSV) virus in human cervix epithelial carcinoma cell line (HeLa CLL#2). The reference used was represented by non-infected HeLa culture.For non-infected HeLa cells it was registered spectra from 10 different areas from culture flask, microscopic chosen, bur for HSV infected HeLa cells it was collected spectra for 60 different areas from the flask’s HSV infected HeLa cells, because of a high level of heterogeneity due to cellular nuclei diameter variation, function of infection degree.Fractal dimension for interest area of Mie scattering spectra was computed using a software package, which can store and process recorded data. Based on experimental data were computed average and standard deviation for fractal dimensions related to non-infected HeLa cellular nuclei size distribution, for a group of 10 investigated areas using all methods. So, fractal dimensions for HSV infected HeLa cellular nuclei size distribution computing using all methods needed first of all an ascendant average for values, followed by 5 value groups dividing. After that they were computed the average and standard deviation for each group.

Analyzing this data it was concluded that ascendant arranged averages for computed fractal dimensions are not normal statistics distributions. Based on computed averages for fractal dimension groups,
they were plotted graphics for non-infected and HSV

infected cells using the three methods (fig. 1) to identify the distribution way of obtained values.It comes out that fractal dimension averages for HSV infected cells have a type of a logistic equation. The equation has non-linear type and has a different parameter variation for the three methods involved.

To evidence the correlation between the three methods using HSV infected cells they were plotted the graphics from using linear regression for fractal dimension by groups. It comes out that there is a positive correlation between the three methods involved, related to those 12 groups of fractal dimensions averages values. The signification of this correlation’s: 99.2% between Petrosian and HH methods and HH and Hurst methods and the 98.2% between Hurst and Petrosian methods.

For each method it was possible to establish a threshold from which all HeLa cells groups having higher fractal dimension averages were HSV infected. The value of this threshold was 1.0061 when using Petrosian method, 1.422 with Hurst method and 1.376 with HH method.


Konradin Metze(1); Randall Luis Adam(2); Neucimar J.Leite(2); Irene G.H.Lorand-Metze (3)

1) Department of Pathology, Faculty of Medicine, State University of Campinas (UNICAMP), Brazil; 2) Institute of Computing, State University of Campinas, Brazil; 3) Department of Medicine, Faculty of Medicine, State University of Campinas (UNICAMP), Brazil.



 Introduction: The fractal analysis of chromatin is becoming more important since it has been postulated that under physiologic conditions the DNA arrangement in nuclei is of fractal nature. Physiologic or pathologic cell changes, which affect the nuclear architecture, might modify these charateristics.  The fractal dimension (FD) of nuclear chromatin can be determined in different ways. 

Aim:The aim of our study was to compare the FDs obtained by different techniques in different models: a) “in silico”, using synthetic surface images applying the method of Develi  and Babadagli  b) in hematoxylin-stained cytologic preparations of cardiomyocytes during different stages of development of 89 Wistar rats and c) in routinely May-Grünwald-Giemsa stained blasts of patients with acute B-precursor lymphoblastic leukemia.

Material and Methods: Random pictures from at least 100 nuclei per case of HE stained cardiomyocytes (model b) or routinely May –Grünwald – Giemsa stained bone marrow slides (model c) were captured and the nuclei segmented interactively. The images were converted to grayscale format. We viewed the images as “irregular surfaces”, where the z coordinate is defined by the gray level of each pixel. The fractal dimensions (FDs), as a measure of self-affinity of these surfaces, were calculated applying the following methods:

According to Sarkar et al. we used three -dimensional cubes and counted the number of intersections with the surface. In a variation of this method (Sarkar variation) columns instead of cubes were used. Another way of estimating the fractal dimensions is to calculate the volume differences between three-dimensional dilations and erosions of the image surface applying various morphological elements with size epsilon. The regression line is calculated between log(volume) and log(epsilon). According to Minkowski this element is spherical. Einstein described a circular disc. Furthermore, a diamond with connexity of 4 (plane-4) or a square with connexity 8 (plane 8) may be used (modified method of Dubuc et al.). Other structural elements are a pyramid with connexity 4 (blanket 4) or connexity 8 (blanket 8) according to the suggestion of Peleg  extended to three dimensions


Fiorenzo Marinelli 1), Giorgio Turchetti 2), Nicoletta  Zini 1), Rosario Santoro 1)
1) Molecular Genetic Inst.
(IGM-CNR) c/o IOR Via di Barbiano 1/10 40136 Bologna
2) Dept. of Physics - University of Bologna, Italy


Introduction – We consider a set of electron microscope images of chromatin during apoptosis induced by APO-1/FAS treatment of Jurkat cultured cells. Apoptosis is a controlled physiological process that occurs in all tissues. By using multifractal analysis it is possible to identify a progressive change of chromatin arrangement during the different steps of apoptosis. In the apoptotic process we have considered three different phases of the ultrastructural rearrangement of chromatin texture, to which correspond different multifractal spectra. In the first phase the chromatin appears to be marginated and exhibits high fluctuation of the density, which decreases in the cap-shaped phase to disappear in the micronuclei.

Results - The width of multifractal spectrum progressively decreases as for a fractal measure which becomes progressively uniform. This technique has been used to measure specific characteristics associated with morphological aspects of apoptosis. The characteristics of this approach suggest it can be used to study the abnormalities of the nuclear chromatin arrangement in various types of healthy, pathological, and tumor cells. It has already been reported that neoplastic nuclei show changes in chromatin arrangement, as assessed by textural features.


Włodzimierz KLONOWSKI
Lab. of Biosignal Analysis Fundamentals, Inst. of Biocybernetics and Biomedical Engineering
Polish Academy of Sciences, Warsaw,
and GBAF, Medical Research Center, Polish Academy of Sciences, Warsaw,

Simple fractal method of assessment of histological texture for medical diagnostics

Fractal and symbolic methods of image and signal analysis can be very useful for assessment of material properties, especially surface properties, based on analysis of experimental data such as microscopic images. Our philosophy is that to be applicable a method should preferably be really simple and easily understandable to non-specialists in the field. Presented methods are very simple and they both draw from multiple disciplines and have multidisciplinary applications.
T.Mattfeldt applied nonlinear deterministic methods from chaos theory to pattern analysis of tumor cells. He compared histological texture in 20 cases of mastopathy with 20 cases of mammary cancer. Epithelial texture plays a central role in histopathological diagnosis and grading of malignancy. T.Mattfeldt pre-processed microscopic 2-D images of tumor cells’ epithelium into 1-D ‘signals’ (so called ‘landscapes’) and then by embedding these signals in a phase space using ‘time-delay’ method; he found that correlation dimension differs considerably between benign and malignant mammary gland tumors [1]. We have proposed to use a similar simple method for pre-processing of the surface’s 2-D image to construct 1-D landscapes, but in the second step we use much simpler and more appropriate in this case Higuchi’s fractal dimension method (cf. .[2] - [3]) for analysis of obtained landscapes.
Fractal dimension of landscapes obtained from surface images does change with the surface properties. The smoother is a surface, i.e. the smaller are its unevenness at any particular scale, the greater is fractal dimension of any landscape obtained from an image of this surface at given magnification. If a surface shows anisotropic roughness properties (texture) then fractal dimensions of its horizontal and vertical landscapes differ from one another ([4] [5]).
Our method draws from multiple disciplines and may find multidisciplinary applications. The same data-processing methods based on fractal and symbolic computational methods may be used for extraction, fusion, and visualization of multi-modal information from nanosensors, for representing and managing signal complexity; these methods are computationally effective and may be applied in real-time.
This work was partially supported by the Polish Ministry of Science and Higher Education grant COST/190/2006 and by FP6. Integrated Project SENSATION (IST 507231).
1.T.Mattfeldt: Spatial Pattern Analysis using Chaos Theory: A Nonlinear Deterministic Approach to the Histological Texture of Tumours; in: Losa G.A., Merlini D., Nonnenmacher T.F., Weibel E.R. (Eds.): Fractals in Biology and Medicine. Vol. II, 1997, pp 50-72, Birkhäuser, Basel, Boston, Berlin.
2.W.Klonowski: From conformons to human brains: an informal overview of nonlinear dynamics and its applications in biomedicine, Nonlinear Biomedical Physics, 1, 5 , 2007, BioMed Central, London, .
3.W.Klonowski: Signal and Image Analysis Using Chaos Theory and Fractal Geometry. Machine Graphics & Vision 2004; 9: 403-431 .
4.W.Klonowski, E.Olejarczyk, R.Stepien: A new simple fractal method for nanomaterials science and nanosensors, Materials Science-Poland, Vol. 23, No.3, 2005, pp.607-612
5.W.Klonowski, E.Olejarczyk, R.Stepien: SEM Image Analysis for Roughness Assessment of Implant Materials, M.Kurzynski, M.Wozniak, E.Puchala, A.Zolnierek, Eds. Springer Verlag, Berlin, Heidelberg, 2005, Advances in Soft Computing Series, pp. 553-560 .



M. Bouguerra, M. A. Alimi
REGIM, ENIS, BP W, 3038 Sfax, University of Sfax, Tunisia

Fractal Analysis Limitation to Describe and Classify (normal and pathological cases) of Human Retinal blood Vessels

The pattern of retinal blood vessels in the retinal circulation of the human eyes is a wonderful and interesting branching network. In this abstract, our interpretation on the dissatisfaction to apply a fractal analysis for the human retinal blood vessels with present’s a complexity biological. A number of retinal images (20 of details "ah" and 20 of details "vk") from the STARE PROJECT database [1] are analyzed, corresponding to both normal and pathological states of the retina.

Over the past decade, there have been several attempts [2]-[3] in the direction of employing the fractal dimension as a measure for quantifying the state of human retinal vessel structures (considered as geometrical objects), with the expectation that such analysis may contribute to automatic detection of pathological cases, and, therefore, to computerization of the diagnostic process. While this is certainly a valid question with possibly high impact on real world diagnostic issues, there are some issues that should be addressed before such investigations may prove useful for the standard clinical practice. First, the fact that retinal vessels represent “finite size” realizations of a fractal growth process, imposes questions about how exactly should one go about measuring the fractal dimension of a particular instance [4]. Then, some of these works [5], address the point that the retinal vessels may have different properties in different regions, and do indeed find different characteristics depending on the locale of measurement, although the procedures adopted in these works are only remotely related to established concepts of multifractality, and the corresponding commonly accepted procedures for its measurement [6]–[7].

We demonstrate that vascular structures of the human retina represent geometrical multifractals, characterized by a hierarchy of exponents rather then a single fractal dimension [8]. In contrast to monofractals, multifractals are characterized by a hierarchy of exponents, rather then a single fractal dimension. Then, multifractals may be viewed as a union of intervowne monofractals, embedded into each other. Therefore, the multifractality test may be performed by computing the fractal box counting dimension [9] for each image. If this turn out to be superior to DLA fractal dimension estimate (Dq = 1.71) [10] and the complexity biological of theses blood vessels, we have to pass for multifractal approach.  

 In the same way, we confirm the excellence of multifractal analysis to classify the blood vessels for normal and pathological cases. We also observe a tendency of images corresponding to the pathological states of the retina to have lower generalized dimensions and a shifted spectrum range, in comparison with the normal cases.



The STructured Analysis of the Retina (STARE) Project [Online]. Available: Last seen on, Oct. 8, 2005.


F. Family, B. R. Masters, and D. E. Platt, "Fractal pattern formation in human retinal vessels", Physica D, vol. 38, pp. 98–103, 1989.


B. R. Masters, "Fractal analysis of the vascular tree in the human retina", Ann. Rev. Bio. Eng., vol. 6, pp. 427–452, 2004.


M. A. Mainster, "The fractal properties of retinal vessels: Embryological and clinical implications" Eye, vol. 4, pp. 235–241, 1990.


A. Avakian, R. E. Kalina, H. E. Sage, A. H. Rambhia, K. E. Elliott, E. L. Chuang, J. I. Clark, J.-N. Hwang, and P. Parsons-Wingerter, "Fractal analysis of region-based vascular change in the normal and non-proliferative diabetic retina", Curr. Eye. Res., vol. 24, pp. 274–280, 2002.


J. Feder, "Fractals". New York: Plenum, 1988


A. B. Chhabra, C. Meneveau, R. V. Jensen, and K. R. Sreenivasan, "Direct determination of the f(alpha) singularity spectrum and its application to fully developed turbulence" Phys. Rev. A, vol. 40, pp. 5284–5294, 1989.


Tatijana Stosic and Borko D. Stosic, "Multifractal Analysis of Human Retinal Vessels" IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 25, no. 8, August 2006.


B.R. Masters, "Noninvasive Diagnostic Techniques in Ophthalmology", B.R.  Masters, Ed, Springer-Verlag, New York, 515, 1990.


T. Vicsek, F. Family, and P. Meakin, "Multifractal geometry of diffusion-limited aggregates", Europhys. Lett., vol. 12, pp. 217–222, 1990.  



Catalin-Iulian Chiurciu (2), Ion Dina (3), Camelia-Doina Vrabie (3), Marius Cornitescu (1),
(1) "Stefan S. Nicolau" Institute of Virology, Bucharest; (2) "Dr. C. Chiurciu" Medical Praxis, Bucharest; (3) "St. John" Emergency Hospital, Bucharest, Romania.

Structure recognition in human optical microscopy specimens by using the fractal analysis of the spatial colour density distribution.

he microscopy analysis of histology preparations is an important element in the diagnosis in a wide range of pathological states. Although the staining of the preparations itself is generally not difficult to perform, it is the analysis step that requires the activity of very skilled personnel. With the higher availability of increasingly performant computing machines, efforts are being made to automate the microscopy image analysis in order to speed up the diagnostic process by rapidly identifying pathological specimens.
Our work is subscribed to the idea of designing image analysis algorithms that would help machines rapidly identify abnormal structures in histology preparations.
We present a method of automatic recognition of the size and distribution of structures in a microscopy image, based on the fractal analysis of the spatial colour distribution in the slide. The method involves the analysis of the image at different simulated resolutions, the identification of colour structure inflexion points in the image, equivalent to contour elements and the fractal evaluation of the contour elements in relation to the unit size used for the generation of the simulated resolution images. The method permits the discrimination and
localization of structures in the image by their size and colour composition, without the intervention of the human operator, leaving the computer learning process of naming structures particular to specific staining methods and tissue types to a later stage in the image analysis.
Figure 1: Human colonic polip preparation, hematoxylin-eosin stain, 10x magnification:
A) original image (24 bits per pixel bit depth, 240x240 pixels image size);
B) contours image generated using the described algorithm.
A) B)


Julia Maria Kröpfl a, Roland Sedivy b, Helmut Ahammer a
Insitute of Biophysics, Medical University of Graz, Harrachgasse 21/V, A-8010 Graz, Austria b Landesklinikum St. Pölten, Propst-Führer-Straße 4, A-3100 St. Pölten, Austria

The importance of the Fast Fourier dimension and texture statistics for a medical classification problem

Aim of the study
The analysis of digital images of Anal Intraepithelial Neoplasia (AIN), a precancerous condition, is used to identify the difference between healthy (Control) and pathological tissues (AIN1, AIN2, AIN3) with the help of image processing and statistical methods like data mining (Fig. 1).  The Fast Fourier dimension as well as first and second order texture statistics were used as attributes in the analysis. Basic exploratory statistics, Principal Component Analysis (PCA) and decision tree analysis were performed in WEKA and R in order to evaluate the worth of the attributes in use and to search for the most effective way of separating the four different classes. From the medical point of view, this is of great importance because to address these questions only subjective visual investigation of Regions of Interest (ROIs) has been possible so far.



Fig. 1: Histological images showing a Control group and three different groups of Anal Intraepithelial Neoplasia (AIN). Visual classification was done by a pathologist. The first row shows with haematoxylin and eosin stained histological slices of AIN. In the second row, the according grey- level images are represented. The increasing level of disease can be recognized by increasing cell growth and the different morphology of the cell nuclei 1.

b = slope

IDL TIFF file  


Fig.2: Calculating the Fast Fourier dimension;

The first image shows the grey- level conversion of a histological image of AIN. A 2D Fast Fourier analysis is being conducted and leads to the power spectrum shown in the middle. Since the power spectrum represents a plane in 3D space, one has to come up with the 2D projection (image to the right) in order to be able to calculate its slope b and furthermore the Fast Fourier dimension according to the following formula:

 Df =                                                                


The analysis is based on the Fast Fourier dimension and first/ second order grey- level texture statistics that were extracted from the ROIs with IQM (Interactive Quantitative Morphology), an IDL based program. The original histological slices were digitized with 768x589 pixels and an original magnification of 400x and converted into grey- level images with NTSC standard for the luminance.
The Fast Fourier dimension is computed by using the power spectrum of a digital signal. The problem of computing the fractal dimension is reduced to that of finding a ‘best fit’ of the data to the curve |k |-d, where, in this case, k equals the spatial frequency in cycles per meter 2 (Fig. 2). First order texture statistics are derived from the original image values. They are global statistical parameters and measure the probability of observing a certain grey value at a randomly chosen location in an image. Second order texture statistics are local statistical parameters and consider the relationship between neighbouring pixels. They are based on the grey- level co- occurrence matrix. Single attribute as well as attribute subset evaluation is used as investigation technique in order to evaluate the most important attributes in the classification process. Different decision tree algorithms are chosen to search for the best class separation.


Different single attribute evaluators and attribute subset evaluators ranked the Fast Fourier dimension in the front among five other attributes such as Variance, Skewness, Kurtosis, Fourth Moment and Correlation (Table 1). Summarizing several weighted variables to principal components significantly improved the correct classification rate of the original dataset (Fig. 3).




average merit      average rank       attribute

0.149 +- 0.075     3.4 +- 2.5         Df

0.048 +- 0.073     5.1 +- 0.94        Variance

0.099 +- 0.152     4.4 +- 2.24        Skewness

0.154 +- 0.052     3.2 +- 0.75        Kurtosis

0     +- 0         4.5 +- 1.2         FirstEnergy

0     +- 0         7.5 +- 2.54        FirstEntropy

0.016 +- 0.049     9.6 +- 2.37        FourthMoment

0     +- 0        10.1 +- 0.83        SecondEntropy

0     +- 0         8.1 +- 1.37        Contrast

0     +- 0         8   +- 1.1         Homogeneity

0.177 +- 0.06      2.1 +- 2.02        Correlation


number of folds (%)  attribute

    8( 80 %          Df

    2( 20 %)         Variance

    3( 30 %)         Skewness

    7( 70 %)         Kurtosis

    0(  0 %)         FirstEnergy

    0(  0 %)         FirstEntropy

    1( 10 %)         FourthMoment

    0(  0 %)         SecondEntropy

    0(  0 %)         Contrast

    0(  0 %)         Homogeneity

    9( 90 %)         Correlation


Table 1: Results of a single attribute and attribute subset evaluator; InfoGain attribute evaluator decides about the worth of an attribute by measuring the information gain with respect to the class. CfsSubset evaluator searches for a subset of attributes by considering the individual predictive ability of each feature along with the degree of redundancy between them 3. InfoGain evaluator selects Df as the third most important attribute, whereas CfsSubset evaluator chooses the Fast Fourier dimension in 8 of 10 folds using 10- fold cross- validation for evaluation.





Fig. 3: Comparison of the correct classification rate for different datasets; All of the decision trees performed best when using the PCA scores dataset. The highest achieved classification accuracy was 74.6%.



The Fast Fourier dimension turned out to be a very useful parameter in multivariate analysis. It might serve as additional information to extracted first and second order image statistical parameters in order to classify histological images of Anal Intraepithelial Neoplasia according to their grade of disease. Different single- attribute and attribute subset evaluators were used for analysis and they all confirmed this hypothesis. The fractal dimension was always ranked in the front among five other attributes such as Variance, Skewness, Kurtosis, Fourth Moment and Correlation. Ranking according to the mean decrease in the Gini Index in a Random Forest analysis also showed the predictive value of the Fast Fourier dimension. It was put on the fourth position in the ranking process. To conclude, the fractal dimension is a very valuable attribute. Its application as an additional parameter to support difficult diagnosis is definitely a great perspective for the future.

1. Ahammer, H.; Image statistics and Data Mining of Anal Intraepithelial Neoplasia; submitted
2.Turner, M.J. et al., Fractal Geometry in Digital Imaging. 1998: Academic Press.
3.Witten, I.H. and E. Frank, Data Mining- Practical Machine Learning Tools and Techniques. 2005, San Francisco: Elsevier Inc. 525


Dimitri Voronine (1), Elizabeth Wendt (3), Carol A. Heckman(3), Sonal O. Uppal (2)
Institut für Physikalische Chemie, Universität Würzburg, Würzburg  97074 Germany
(2) Lerner Research Institute, Cleveland Clinic, OH 44106 USA
(3) Department of Biological Sciences, Bowling Green State University, OH 43403 USA

Morphological fractal analysis of cell shape in cancer chemotherapy

Aims: The current prognostic parameters, including tumor volume, biochemistry, or immunohistochemistry are not sufficient for a satisfactory quantitative diagnosis. Our focus is to evaluate the effects of a combination of MT-polymerizing Taxol® and MT-depolymerizing colchicine on IAR20 PC1 liver cells by measuring the surface fractal dimension as a descriptor of two-dimensional vascular geometrical complexity. Furthermore, we show correlations of fractal dimensions of cell contours with the latent factors from our previously employed cell shape analysis.

Methods: IAR20 PC1 line derived from the liver of in-bred BD-VI was treated with the combination of MT- polymerizing Taxol® and MT- depolymerizing colchicine. After 2-5 hours of incubation, samples were recovered, fixed in 3% glutaraldehyde, and air dried. Information about the shape features was derived from optical interference images. Cells were imaged in a Zeiss universal microscope, and the outlines of their three lowermost interference contours were obtained for each cell as described in [1, 2]. Fractal analysis of the outer, middle, and inner contours of the control and the transformed cells was carried out using the conventional box-counting method [3].

Results: Application of a combination of drugs induced characteristic morphological changes in the contours of the transformed cells. These changes are shown in Fig. 1, where we compare the effects of MT transformations on the cell contour shapes of the treated (A) and the control (B) cells with the immunolocalization of b-tubulin [4]. The MTs in (A) are more perpendicular to the cell edge and form bundles.  In (B) their align along the edge, forming wavy patterns. Fig. 2 shows typical images of a transformed S35 (a) and a control (b) cell. The values of the fractal dimension obtained from these plots allow to distinguish these contours quantitatively. Generally, the transformed cells were more irregular and had higher fractal dimensions. The results of the two-sample T-test performed in this study were similar to our previous analysis using 102 variables and latent factors [4, 5]. This provides a support to the proposed method of fractal analysis, and reveals advantages of using the fractal geometry. To evaluate global differences between the two cell lines we present the results of fractal analysis with fractal histograms and plots of cumulative probabilities, and demonstrate that the degree of fractality is higher in the transformed cells. Cumulative probabilities indicate the total number of cells with the values of fractal dimension less than Df as a function Df. These plots reveal the increased degree of roughness or fractality in the S35 cells. Fractality may be regarded as a quantitative measure of the irregularity of cell contours. The fractal dimension, therefore, serves as a sensitive descriptor of cell shape. Our results indicate that the inner and outer contours undergo the most drastic changes in the distribution of fractal dimensions.

Conclusions: The success of the surface fractal dimension in quantifying the highly heterogeneous shape, size and distribution of vessels completely eliminates the need for biased histopathological interpretations. We conclude that factor variables are related to fractality of IAR20 PC1 cells. Fractal analysis allows to follow the complex response of cells to a combination of MT-interacting agents, and, therefore, provides a practical method of examining complex morphological transformations. Further application of fractal analysis will include investigation of the formation of tumor cell microfoci and small metastases in breast cancer, as a diagnostic histopathology, and identification of tumors at very early stages.



Fig. 1                                                                          Fig. 2


Fig. 1   Typical images of the cells immunolocalized for b-tubulin reveal the roles of microtubules in the shapes of cell contours: (A) cells treated with 2 mM each colchicine and paclitaxel; (B) control cell.


Fig. 2  (A) typical transformed IAR20 PC1 liver cell (S35), (B) control cell.



1. Heckman CA, Plummer III HK, Runyeon CS. Persistent effects of phorbol 12myristate 13-acetate: possible implication of vesicle traffic. J Cell Physiol 1996;166:217-30.
2. Heckman CA, Plummer III HK, Mukherjee R. Enhancement of the transformed shape phenotype by microtubule inhibitors and reversal by an inhibitor combination. Int J Oncology 2000; 16:700-23.
3. Feder J. Fractals. New York: Plenum Press; 1988.
4. Uppal SO, Li Y, Wendt E, Cayer ML, Barnes J, Conway D, Boudreau N, Heckman CA. Pattern analysis of microtubule-polymerizing and-depolymerizing agent combinations as cancer chemotherapies. Int. J. Oncology, 2007; 31
5. Uppal SO. Studies of Microtubule inhibitor combinations on cytoskeleton architecture. PhD Dissertation; 2006.


1  Ralf Metzler (1,2) and Michael A. Lomholt (3)
1 Physics Department, Technical University of Munich, D-85747 Garching, Germany. E-mail:
2 Center for NanoScience, Ludwig Maximilians Universit¨at, Geschwister-Scholl-Platz 1, D-80539 Munich, Germany
3 Department of Physics and Chemistry, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. E-mail:


Aims. We show that non-Brownian search dynamics arise naturally in connection with gene regulation in vitro and in vivo, however, with different effects: While in vitro under dilute conditions proteins searching for their specific targets include L´evy flights to minimise the search time, in vivo molecular crowding causes subdiffusion in the volume that changes the standard picture of gene regulation. We also discuss general aspects of search and their relevance to animal foraging.
We use a continuous time random walk approach, combined with fractional Fokker-Planck equations[1]. L´evy walks are modelled in terms of generalised master equations. Via Fourier-Laplace techniques we obtain expressions for the search times. Our analytical results are supplemented by numerical analysis and Langevin dynamics simulations.
1. In vitro gene regulation. In gene regulation (see Fig. 1) while non-specifically bound [3], proteins diffusively move along the DNA backbone, as verified by single molecule techniques [4]. Proteins also detach to the volume and, after a bulk excursion, reattach successively before reaching the target. In addition, mediated by DNA-looping [5] proteins may jump from one segment of the DNA to another, which is close in the embedding space but remote in
chemical distance measured along the DNA (see Fig. 1). In an intersegmental jump a protein covers a distance ℓ in the chemical coordinate of the DNA that is distributed like p(ℓ) ≃ ℓ−c, where c is the contact exponent for a polymer.
In good solvent under dilute conditions (i.e., for typical in vitro experiments), c ≈ 2.1, while for a Gaussian chain, c = 1.5. Both cases give rise to a diverging variance of the associated jump length distribution, i.e., a L´evy flight [7]. It will be shown that the dynamics of the transcription factor can be modelled in terms of a fractional Fokker-Planck equation for the line density n(x, t) of transcription factors on the DNA. The L´evy flights improve the search and at optimum reduce the time the transcription factor spends in the cell volume, fully detached from the DNA [7].
Subdiffusion in the cell. In vivo the abundance of a large multitude of proteins and other biopolymers present in the cytoplasm of a cell have been shown to cause a state of molecular crowding: The extremely dense concentration of large molecules hinder each other’s motion, causing a subdiffusive motion of an individual protein [8–10], such that the mean squared displacement acquires the scaling form hr2(t)i = 6K t /��(1+α), with 0 < α < 1,. Here K is the
anomalous diffusion coefficient. Under crowding conditions, one typically finds α ≈ 0.75 [9, 10]. An important question for such types of processes is, how a subdiffusive particle interacts with a reactive boundary.
We present the reactive boundary condition and show how, together with the long-tailed returning probability to the boundary from the bulk, a weak ergodicity breaking arises. Accordingly, the protein either does not leave the DNA or does not return to the DNA for time spans that are of the order of the process time. It will be argued how this weak ergodicity breaking may assist the precise regulation of genes by a small number of transcription factors [11]. General aspects of optimised search, and animal foraging. Of general importance of a search process is its efficiency. Brownian search in one and two dimensions involves frequent returns to an area, leading to oversampling. From theoretical and data analysis L´evy strategies, in which the searching agent performs excursions whose length is drawn from heavy-tailed distributions λ(x) ∼ |x|−1−μ (1)
for 0 < α < 2, were shown to be advantageous: occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling. While pure L´evy flights overshoot a target and are not sufficient to
2 k Bulk excursion DL Sliding Intersegmental transfer j(t) nbulk koff
Target finding DB on O
Figure 1: Left: Gene regulation by DNA-binding proteins CI repressor and Cro in the example of the bacteriophage  switch [2]:
Depending on the binding of CI or Cro, the two genes cI and cro diverging from the operator sites OR1 to OR3 are transcribed by polymerase. Both genes are autoregulative, i.e., the encoded gene is the same as the transcription factor activating those
genes. Right: Schematic of the search mechanisms in the Berg-von Hippel model [6].
quickly localise it [12], we will demonstrate how an additional local search mode leads to highly optimised search.
At very low target densities, the Cauchy distribution with μ = 1 corresponds to optimal search. We also show that searching agents following L´evy search strategies are much more robust to changes in the environment than agents following strategies bound by the central limit theorem [13].

Conclusions. Search processes for rare targets are optimised by stochastic processes that leave the spell of the central limit theorem. Namely, trajectories following non-Gaussian statistics are more efficient in localising the target. This can be achieved in different ways, depending on the constraints put on the system. One the one hand, L´evy flight
strategies effect a more efficient scanning. On the other hand, subdiffusion may optimise chemical reactions or gene regulation when either the number of searchers is to be minimised, or a reaction to be completed during first encounter.
[1] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000); J. Phys. A 37, R161 (2004).
[2] M. Ptashne, A Genetic Switch: Phage  and Higher Organisms, 2nd ed. (Cell Press & Blackwell, Cambridge, MA, 1992).
[3] A. Bakk and R. Metzler, FEBS Lett. 563, 66 (2004); J. Theor. Biol. 231, 525 (2004).
[4] I. M. Sokolov, R. Metzler, K. Pant, and M. C. Williams, Biophys. J. 89, 895 (2005); Y. M. Wang, R. H. Austin, and E. C. Cox, Phys. Rev. Lett. 97, 048302 (2006).
[5] A. Hanke and R. Metzler, Biophys. J. 85, 167 (2003); J. M. G. Vilar and L. Saiz, Curr. Opin. Genetics & Developm. 15, 136 (2005).
[6] P. H. von Hippel and O. G. Berg, J. Biol. Chem. 264, 675 (1989).
[7] M. A. Lomholt, T. Ambj¨ornsson, and R. Metzler, Phys. Rev. Lett. 95, 260603 (2005).
[8] D. S. Banks and C. Fradin, Biophys J. 89, 2960 (2005).
[9] I. Golding and E.C. Cox, Phys. Rev. Lett. 96, 098102 (2006).
[10] M. Weiss, M. Elsner, F. Kartberg, and T. Nilsson, Biophys. J. 87, 3518 (2004).
[11] M. A. Lomholt, I. M. Zaid, and R. Metzler, Phys. Rev. Lett. 98, 200603 (2007).
[12] T. Koren, M. A. Lomholt, A. V. Chechkin, J. Klafter, and R. Metzler, Phys. Rev. Lett. 99, 160602 (2007).
[13] M. A. Lomholt, T. Koren, R. Metzler, and J. Klafter, The advantage of L´evy strategies in intermittent search processes, unpublished.


Vincenzo Capasso  -  Alessandra Micheletti  -  Davide  Morale
ADAMSS and Department of Mathematics, University of Milan 
Via Saldini 50, 20133 Milano, Italy
Stochastic geometry  and related statistical  issues in  the 
mathematical modelling of  tumour-induced angiogenesis

Tumour driven   angiogenesis  is an extremely complex  process,  subject to random fluctuations in the geometry  of the vessel  network.
A  major  difficulty in the mathematical modelling of this process derives  from the strong coupling of   the kinetic parameters of the relevant  branching-and-growth  process to the stochastic geometry  of  the capillary network,   via  suitable   underlying  fields;  indeed the morphology of   the vessel network is characterized  by   random closed sets  of different Hausdorff  dimensions.

Methods for reducing  complexity include homogenization at  the macroscale  of these fields,  while keeping stochasticity  at the  microscale of the individual cells or vessels.  This kind of models are known as hybrid models.

For  diagnosis and  validation of  simulation tools,  an original  statistical  method to estimate the intensity, or  mean length density, of a fibre process has been developed and is here presented , based on a representation of  mean geometric densities of random closed sets, as expectations of suitably  defined  random distributions ´a la Dirac-Schwartz.

Application of this method both to simulated processes and to real test cases shows that the method can provide a useful tool for the quantitative description and comparison of the morphology of different networks of vessels.



Agnieszka Kitlas1, Marta Borowska1, Edward Oczeretko1, Marek Kowalewski2, Mirosława Urban2

1Institute of Computer Science, University of Bialystok, Sosnowa 64,15-887 Bialystok, Poland,2 2-nd Department of Children Disease, Medical University of Bialystok, Waszyngtona 17,15-274 Bialystok, Poland

 Lempel-Ziv complexity measure and approximate entropy in the analysis of the heart rate variability in children

The analysis of heart rate variability is based mainly on analysis of RR intervals, i.e. the series of time intervals between the heartbeats in ECG. We can observe RR intervals in electrocardiogram, which is a representation of the electrical forces produced by the heart.


The purpose of this study is the quantitative assessment of heart rate variability by means of Lempel-Ziv complexity measure and approximate entropy. These methods characterize the degree of disorder in finite nonlinear time series, for example in medical signals. Estimation of complexity is of great interest in nonlinear signal analysis.

Materials and methods

Lempel-Ziv complexity measure (L-Z complexity measure) was defined by A. Lempel and J. Ziv in 1976. The complexity counter c(n) counts all distinct patterns in a sequence. When a new pattern is found the complexity counter c(n) increases by one. After normalization L-Z complexity measure has range form 0 to 1, where 0 is adequate to order and 1 is adequate to random pattern.

Approximate entropy (ApEn) is a nonlinear measure that quantifies the amount of regularity. This method was proposed by Steven Pincus in 1991. It describes the complexity and irregularity of the signal and is low in regular time series and high in complex irregular ones.

We have analyzed two groups of patients: test group A – children with diabetes type 1 and microalbuminuria, test group C - healthy children. Diabetes type 1 is characterized by loss of the insulin-producing cells in the pancreas and microalbuminuria, i.e. small amounts of albumin in the urine, is a complication in diabetes.

ECG records were divided into two segments: day activity (from 6.00 a.m. to 10.00 p.m.) and night activity (from 10.00 p.m. to 6.00 a.m.).

Programs for computing Lempel-Ziv complexity measure and approximate entropy were written in Matlab (MathWorks Inc., USA), a high performance language for technical computing and in C++. Statistical analysis was performed by means of nonparametric Mann-Whitney test.


In table 1 we present normalized Lempel-Ziv complexity measure in the studied groups (mean values ± standard deviation).


Normalized L-Z

complexity measure









* statistically significant differences (p<0.05)


The values of Lempel-Ziv complexity measure for healthy children are higher than for unhealthy patients.


In table 2 we present approximate entropy in the studied groups (mean values ± standard deviation).











* statistically significant differences (p<0.05)


The values of approximate entropy were lower in unhealthy patients than in healthy children.



We concluded that using Lempel-Ziv complexity measure and approximate entropy we could quantitatively study the heart rate variability in healthy and diabetic children. For each group the mean value of Lempel-Ziv complexity measure and approximate entropy was higher at night which indicates higher nighttime activity. Our results showed that the values of Lempel-Ziv complexity measure and approximate entropy indicated more deterministic signal form unhealthy patients and more regular heart rate in diabetic children with microalbuminuria.

Diabetes type 1 with microalbuminuria, which is a syndrome characterized by disordered metabolism, has an effect on cardiovascular system and in consequence on heart rate variability. Lempel-Ziv complexity measure and approximate entropy may be very helpful tools in analyzing nonlinear signals.


[1] De Felice C., Bianciardi G., DiLeo L., Latini G., Parrini F. Abnormal oral vascular network geometric complexity in Ehlers-Danlos syndrome. Oral Surg Oral Med Oral Path Oral Radiol Endod., 98, (2004) 429-434.
Fusheng Y., Bo H., Qingyu T. Approximate entropy and its application in biosignal analysis. In: Nonlinear biomedical signal processing. Vol II: Dynamic analysis and modeling (ed. Metin Akay), New York: IEEE Press and John Wiley & Sons, Inc.; (2001) 72-91.
Huang L., Yu P., Ju F, Cheng J. Prediction of response to incision using the mutual information of electroencephalograms during anaesthesia. Medical Engineering & Physics, 25, (2003) 321-327.
Kitlas A., Borowska M., Oczeretko E., Kowalewski M., Urban M. Lempel-Ziv complexity measure in the analysis of heart rate variability. In: Leszek Rutkowski (ed.) XIV National Conference „Biocybernetics and Biomedical Engineering”, Częstochowa, Poland, 21-23.09.2005, Vol. 2, (2005) 829 – 833.
Kitlas A., Oczeretko E., Kowalewski M, Borowska M., Półjanowicz W., Urban M. Approximate entropy in analysis of the heart rate variability. International Conference “Advanced Information and Telemedicine Technologies for Health”, AITTH’ 2005, 8-10.11.2005, Mińsk, Bielarus, Proceedings, 1, (2005) 52-56.
Lempel A., Ziv J. On the complexity of finite sequences. IEEE Trans Inform Theory, 22(1), (1976) 75-81.
Nagarajan R., Szczepanski J., Wajnryb E. Interpreting non-random signatures in biomedical signals with Lempel-Ziv complexity. Physica D, 237, (2008) 259-364.
Pincus S. Approximate entropy (ApEn) as a complexity measure. Chaos, 5(1), (1995) 110-117.
Pincus S.M. Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA, 88, (1991) 2297-2301.
Szczepanski J., Amigo J.M., Wajnryb E., Sanches-Vives M.V. Characterizing spike trains with Lempel-Ziv complexity. Neurocomputing, 58(60), (2004) 84-97.
Torres M. E., Gamero L. G. Relative complexity changes in time series using information measures. Physica A, 286, (2000) 457-473.



Alexander Delides MD1, PhD,  Alexandros Viskos MD, PhD2

1Otolaryngology, Head & Neck Surgery Department 2 Pulmonary Medicine Dept., Sleep Laboratory Athens Medical Center, Marousi, Athens, Greece

 Fractals and Sleep: Fractal Quantitative Endoscopic Analysis of the upper airway as a predictor for Obstructive Sleep Apnea Syndrome


Obstructive Sleep Apnea Syndrome (OSAS) is the syndrome associated with the cease of airflow in the upper respiratory tract during sleep that leads to oxygen desaturation and increase of arterial CO2 and causes, among others, long term permanent changes to cardiac and pulmonary functions. The gold standard method for the diagnosis of the syndrome is Polysomnographic (PSG) sleep testing, an overnight sleep monitoring of basic functions that calculates the apnea-hypopnea index (AHI) a parameter which quantifies the syndrome  and guides treatment.

Clinical evaluation of the OSAS patient includes, apart from thorough history taking, a complete ENT examination accompanied by fiberoptic naso-pharyngo-laryngeal endoscopic examination with and without Muller's maneuver, the latest being described as a forceful inspiration with nose and mouth closed while the endoscope is in the airway. Fiberoptic Naso-pharyngoscopy with Muller's maneuver (FNPMM) is the method of choice for evaluating the collapsibility of the airway caliber and for deciding the site of obstruction. For years FNPMM has being evaluated subjectively by an “eyeball” rule, but recently a number of studies have shown that computer-assisted quantification of the findings are important for performing accurate observations. In order to obtain objective and accurate area measurements of endoscopic images one has to overcome a practical problem: Distances in images captured through an endoscope are not accurately comparable because measurements are related to the distance of the tip of the scope to the measured characteristic. The relative distances are altered when approaching too close  due to the magnifying lens mounted in the tip. To overcome such a problem authors have used different methods but such processes even though appear to solve the problem are impractical to perform and have thus not gained further applications.


A total of 42 subjects (9 females, 33 males) who underwent PSG overnight monitoring of nine parameters were included in the study.  They all underwent complete ENT examination accompanied by  FNPMM in the erect posture on the day of admittance.

The total Apnea Hypopnea Index (AHI) was calculated and subjects with AHI more than 5 and maximal oxygen desaturation less than 90% were considered positive for OSAS and included as patients. Subjects with AHI less than 5 or/and maximum oxygen desaturation above 90% were regarded as controls. Images of FNPMM were documented through a single-chip camera, including images of the resting airway and of maximum collapsibility. The airway area in the images was noted with an image editing software and Box-Counting Fractal Dimension (FD) was calculated for the resting and collapsed airway area. Absolute and percentage of the airway Fractal collapsibility was calculated. Results were related to findings of PSG.


From the 42 subjects, 25 had positive findings for OSAS (patients) and 17 were regarded as controls. A 9.5% cutoff point (ROC curve) was set for FD-area collapsibility to differentiate between a positive (above 9.5% airway collapse) and negative (below 9.5% airway collapse) FD Muller test. Doing such, 23 subjects had a positive test and 16 negative. Prevalence was  59.5%, sensitivity 92%, specificity 82.4%, with a positive predictive value of 88.5% and a negative predictive value of 87.5 % (p<0.0001).

Differences of FD of the airway caliber during relaxed inspiration between patients and controls were found to be non-significant but differences of FD during Muller maneuver as well as FD-area collapsibility were highly significant (p<0.0001)


Calculating Box-counting Fractal Dimension of endoscopic images provides an easy and effective way to quantify findings of FNPMM and diagnose OSAS. The method can be used to screen ENT patients and differential diagnose those from habitual snorers, assisting in selection of patients that should undergo a complete sleep evaluation.


1.   Stuck, B.A. and J.T. Maurer, Airway evaluation in obstructive sleep apnea. Sleep Med Rev, 2007.

2.   Sher, A.E., et al., Predictive value of Muller maneuver in selection of patients for uvulopalatopharyngoplasty. Laryngoscope, 1985. 95(12): p. 1483-7.

3.   Doghramji, K., et al., Predictors of outcome for uvulopalatopharyngoplasty. Laryngoscope, 1995. 105(3 Pt 1): p. 311-4.

4.   Ritter, C.T., et al., Quantitative evaluation of the upper airway during nasopharyngoscopy with the Muller maneuver. Laryngoscope, 1999. 109(6): p. 954-63.

5.   Petri, N., et al., Predictive value of Muller maneuver, cephalometry and clinical features for the outcome of uvulopalatopharyngoplasty. Evaluation of predictive factors using discriminant analysis in 30 sleep apnea patients. Acta Otolaryngol, 1994. 114(5): p. 565-71.

6.   Katsantonis, G.P., C.S. Maas, and J.K. Walsh, The predictive efficacy of the Muller maneuver in uvulopalatopharyngoplasty. Laryngoscope, 1989. 99(7 Pt 1): p. 677-80.

7.   Hsu, P.P., et al., Clinical predictors in obstructive sleep apnea patients with computer-assisted quantitative videoendoscopic upper airway analysis. Laryngoscope, 2004. 114(5): p. 791-9.

8.   Hsu, P.P., et al., Quantitative computer-assisted digital-imaging upper airway analysis for obstructive sleep apnoea. Clin Otolaryngol Allied Sci, 2004. 29(5): p. 522-9.

9.   Ye, J., et al., Computer-assisted fiberoptic pharyngoscopy in obstructive sleep apnea syndrome. ORL J Otorhinolaryngol Relat Spec, 2007. 69(3): p. 153-8.


Marta Borowska1, Edward Oczeretko1, Piotr Sobaniec2, Tadeusz Laudanski3

(1) Department of Medical Informatics, Institute of Computer Science, University of Bialystok, Sosnowa 64, 15-887 Bialystok, Poland  (2) Department of Electrical Engineering, Technical University of Bialystok, Wiejska 45D, 15-351 Bialystok, Poland; Neurology Students' Society, Department of Children Neurology, Medical University of Bialystok, Waszyngtona 17, 15-274 Bialystok, Poland;  (3)Department of Pathophysiology of Pregnancy, Medical University of Bialystok, M. Skłodowskiej-Curie 24a, 15-273, Bialystok, Poland


Synchronization in multivariate biomedical time series – nonlinear dynamics methods


In physiological research, we often study multivariate data sets, containing two or more simultaneously recorded time series. It is important to examine synchronization in these kinds of signals. Here the notion of synchronization will be used in a loose sense as the synonym of correlation, the similarity of the signals or the similarity of their dynamics.

The aims of this study are: (i) to present some of the nonlinear dynamics methods of synchronization: the mutual correlation dimension (D2m), the cross-approximate entropy (cross-ApEn), the mutual information function (MI), and (ii) to apply these measures to two kinds of time series: uterine contractions and EEG signals. As the reference method, the linear cross-correlation (xcorr) function was used.
Materials and methods
Spontaneous uterine activity was recorded directly by a dual micro-tip catheter (Millar Instruments, Inc. USA). The distance between ultra miniature pressure sensors was 30 mm, which after introducing the distal tip near to the uterine fundus, allowed collecting signals both from uterine isthmus and fundus. We study two groups of patients: 5 patients with primary dysmenorrhea and 5 patients with fibromyomas. The EEG signals were referenced to the electrodes placed at the ear lobes. One patient with normal brain activity and one patient with abnormal activity were analyzed.
For the analysis of the recorded signals we used programs written in MATLAB and C++.
The cross-correlation function and the mutual information function are sensitive to the phase differences and can be used to identify time delays between two signals. These measures can be computed in a moving window with a width corresponding to approximately two or three oscillations. As a result, the running synchronization functions were obtained. The running synchronization functions visualize changes in the synchronization of the two simultaneously recorded signals. From running function the propagation% parameter was assessed. This parameter allows quantitative description of synchronization in time series. Cross-approximate entropy measures the synchronicity of the time series, but on the contrary of the cross-correlation and mutual information, this parameter does not indicate the time of the synchronization. The mutual correlation dimension may be interpreted as the amount of degrees of freedom shared between two dynamical systems.

In table 1 we present the results obtained by means of different methods in the studied groups representing uterine contractions signals (mean ± standard deviation).

                        Table 1.













propagation% (MI)






propagation% (xcorr)




Table 2 shows the results obtained by means of different methods in the EEG signals.








propagation% (MI)


propagation% (xcorr)

Patient A














Patient B















The results obtained by different methods (mutual correlation dimension, cross-approximate entropy, mutual information and cross correlation function) are different but qualitatively similar. The running synchronization functions visualize changes in the propagation of the two simultaneously recorded signals, the qualitative and quantitative studies may be performed from the plots.


Mircea V. Rusu
Department of Atomic and Nuclear Physics, Faculty of Physics, University of Bucharest, Bucharest, Romania

Aspects of pattern formation outside of equilibrium

Abstract We present here a view of complex systems research which focuses on pattern formation outside of equilibrium. The concepts and methods are largely inspired from the physics of disordered systems, but the basic ideas are related to structural stability and the physics of phase transitions as classes of universality. Since Mandelbrot developed the basic concept of fractal many researchers have applied fractal theory to describe phenomena in physics, chemistry, biology, medicine and so on. Topographic images obtained with atomic force microscopy (AFM) are usually used to compute fractal dimension of physical surfaces. A film of dimyristoylphosphatidylethanol-amine (DMPE), made by Langmuir-Blodgett (LB) technique, was used to study the conditions and topology of coexistence of both liquid-expanded (LE) and liquid-condensed (LC) phases. Adhesion AFM is used to measure and study the patterns. Other examples described here are: detachment pattern formation experiments using viscous liquids, the swarm pattern formation, and cellular automata modelling of pattern formation. From these and other experiments we reinforce the idea that emergence of order and stability of pattern is a typical result of the nonlinear complex systems even they do not evolve under long range forces that could be sometimes    responsible   for stability of the  system.


Complex systems can vary in composition. Sometimes they can vary in the chemical and even cellular composition but we consider them as the same system, with identical global properties. Maintaining this identity is much more than maintaining the invariance of some average over the properties of the constituents. A central issue in complex system research is the functional organization. So we can ask some fundamental questions:

a) Under what conditions are complex systems organised? b) How does this organisation emergence? c) Organisation at the global level relates to a discretisation of the space of what is possible and recognisable. We found the organisation of living organisms in a hierarchical classification with species, families, genera, etc. Why do we observe these well separated classes rather than a continuum of living organisms?

The concept of morphogenesis is related to the process by which living things develop organised structures. It is now widely used in a variety of fields, like physics, chemistry etc. From physics of complex systems, morphogenesis corresponds to a particular dynamics of system that creates and preserves, under some circumstances, his topology in space and time. This is related to interactions between parts of the system but sometimes such kind of interactions is not seen. We consider these cases as emergence of order from chaos. So, in following examples we will discuss aspects of pattern formation. Organization under the influence of long range order could be mentioned like galaxy or planetary systems subject to gravitational forces, or liquid crystals under electric forces. In the case of short range forces, the pattern formation is a result of coherent structure formation like in processes of fingering, fluid flow or diffusion, mixing, or filament formation. In these cases short rang forces favour adhesion, coalescence, close packing, like in dendrites formations. But in some cases we witness order (pattern) formation and coherent structures in complex systems which do not exhibit long or short range forces. These are typical behaviour for swarms (bees, fishes), or populations, city topology, market, and so on. We conjecture that even brain activity and memory, consciousness, self awareness, logic thinking, are such cases of pattern formation, selected during evolution and giving fundamental advantage to the system. To reach this point we started studying some cases.

Patterns at micro scale

Since Mandelbrot developed the basic concept of fractal [1] many researchers have applied fractal theory to describe phenomena in physics, chemistry, biology, medicine and so on. Topographic images obtained from atomic force microscopy (AFM) are usually used to measure fractal dimension of physical surfaces [2]. A film of DMPE on mica, made by Langmuir-Blodgett (LB) technique, was used to study the conditions and topology of coexistence of both liquid-expanded (LE) and liquid-condensed (LC) phases. Adhesion AFM is used to measure and study the patterns. In the AFM techniques of surface imaging, there are forces that can give signal by bending the cantilever. The shape of an isolated domain results from the competition between line tension and electrostatic repulsion, and is shown as different grey scale on AFM image (figure 1).  If we imagine that lipid molecules (like DMPE) are interacting as linear dipoles, the excess dipole density of the liquid-condensed phase relative to the liquid-expanded phase acts to form non-compact, needle-like domains (figure 2). On the other hand, there is a surface tension which acts to minimize the perimeter of the domain and form circular shapes. This surface tension arises from the excess free energy associated with the formation of the interface between the LC domain and the surrounding LE phase. Mayer and Venderlick [3] derived two simplified, yet rigorous formulations of the electrostatic energy: one being more computationally efficient for circular domains, the other for non-circular domains. Free energy is given by the sum of interfacial and electrostatic contributions, .

The interfacial free energy may be expressed as FL=lP where l is the line tension and P is the perimeter of the domain. The electrostatic energy is given by:

where m is the average excess dipole density and g(r) the pair distribution function.


Figure 1. Adhesion AFM images at three different magnitudes, showing statistical self-similar pattern [2]



Figure 2. a) DMPE molecules on surface, b) Islands: random and circular

Figure 3. Free energy vs. fractal dimension for three values of line tension l.


Measured and computed, using box counting method, the LC domains formed by DMPE molecules on mica are fractal with dimension D = 1.3. The question is why are these domains fractals? A simple free-energy calculation was done in order to obtain at least a qualitative answer to the question above. Also we computed the free energy of hypothetical LC domain with a given fractal dimension that was generated using the real part of Weierstrass-Mandelbrot function. We chose this function of generating fractal shapes because it allows us to tune the fractal dimension D to any desired value. From simulations we saw that interactions between molecules lead to the formation of rough domain frontiers. The interaction between the LC phase and LE phase - which is represented in terms of interfacial energy - acts to minimize the domain perimeter, that thus minimize the interaction between the two phases. The competition between the two effects gives rise to the particular geometry of LC domains. A strong interaction between the LC and LE phases will lead to circular domains. On the contrary, a strong dipolar interaction in LC phase will lead to fractal domains (figure 3).

I. Pattern formation in detachment phenomena
Real fluids that flow under different forces, exhibit pattern formation. Classical case study pinch off and reconnection in binary fluid flow in a Hele-Shaw cell as different viscosity fluids are forced to mix, or dynamics of granular stratification [4]. Another known situation that is governed by intermolecular forces in fluids is the whole domain of bonds. If two solid bodies are in contact having glue or other viscous fluids in between, the two bodes exhibit attractive forces that means that it is necessary to use a force in order to detach them (figure 4). The dynamics of detachment is typical (phenomenological) for non-rigid solids (plastic flows). The phenomena are controlled also by the speed the detachment is conducted. But if we focus the attention on the pattern formed after detachment we see formation of fractal patterns with fractal dimension between 1.72 and 1.79. The filaments are formed (figure 5) as a competition between the mechanical detachment force and surface tension of the viscous fluid, in order to minimise free energy. These types of mechanisms are present also in bubble soup films or other equivalent processes. Cell membrane exhibits other specific phenomena of attachment – detachment  at  molecular  level  that  are involved  in all cell’s  activity.

II. The swarm theory

"Ants aren't smart," but "Ant colonies are" [5]. A colony can solve problems unthinkable for individual ants, such as finding the shortest path to the best food source, allocating workers to different tasks, or defending a territory from neighbours. How do hundreds of honeybees make a critical decision about their hive if many of them disagree? The collective abilities of such animals—none of which grasps the big picture, but each of which contributes to the group's success—seem miraculous. One key to an ant colony, for example, is that no one's in charge. Even with half a million ants, a colony functions just fine with no management at all—at least none that we would recognize. It relies instead upon countless interactions between individual ants, each of which is following simple rules of thumb.


a)  b)


Figure 4. a) Detachment of two plates, b) experimental device

Figure 5. Different pattern formation under the process of detachment. Oil paints or water paints are used.

Figure 6. Patter formation starting from a point like seed (a 24x24 pixel box).


This could be described as a self-organizing system. No ant sees the big picture. No ant tells any other ant what to do. The finding is that simple creatures following simple rules, each one acting on local information could reach a highly organised system. It is a great example of self-organization without long range forces in the system.

III. Cellular automata modelling of pattern formation
Simple rules can mimic evolution of complex systems as Wolfram pointed out [6]. Our computer experiments with different rules show pattern formation, and the usual chaotic changes, periodic changes or a final stabile pattern formation. It is interesting to remark that not all kinds of rules give stability of the pattern (figure 6). Also if we want to assure an evolution in our box, we have to introduce asymmetry in the system. Otherwise the system performs in most cases, after a period of functioning, a periodic return of a given pattern. To ensure a possible evolution in the system it is necessary to have asymmetries, perturbations, in the boundary or inside the box.

Pattern formation could be controlled by long or short range forces, but also we specifically draw attention to short range order that could be “transferred” at high distance by a specific organisation of system parts, that form patterns. These phenomena are found at all scales. We conjecture that also the specific brain activity could be a direct result of such “point-to-point” interactions that create ideas and self-consciousness, and appear as an emergent pattern of brain activity. Further information and studies could be found in [7 – 11]

[1]. B.B.Mandelbrot "Fractals: Form, Chance and Dimension", Freeman, San Francisco, 1977; "The Fractal Geometry of Nature", Freeman, San Francisco, 1983.
[2] Gianiana Dobrescu, Camelia Obreja, M.Rusu, Adhesion AFM Applied to Lipid Minelayers. A fractal Analysis. In Fractals: Theory and Applications in Engineering, Eds: M.Dekking, J.L. Véhel, Evelyne Lutton, C.Tricot, Springer 1999
[3]. M.A.Mayer, T.K.Vanderlick, Langmuir, 1992, 8, 3131.
[4] Makse, H.A., Ball, R.C., Stanley, H.E.,Warr, S., 1998. Dynamics of granular stratification. Physical Review E 58, 3357–3368;
[5] D. M. Gordon.1999. Ants at Work: how an insect society is organized. Free Press, Simon and Schuster. 2000 paperback, W. W. Norton., 2007 Gordon, D. M. Control without hierarchy. Nature 4468:143.
Wolfram S. (2002), A New Kind of Science, Wolfram Media, Inc., 2002
[7] D’Arcy Thomson, On Grows and Form, Cambridge University Press, 2000
[8] Monod, J.
Chance and Necessity. London: Collins. 1972.
[9] Nicolis, G., and I. Prigogine.
Exploring Complexity,  New York: Freeman. 1989
[10] S. Kauffman,
The Origins of Order: Self-organization and Selection in Evolution, Oxford University Press (1990); S. Kauffman, Complexity and Genetic Networks,
[11] R. Thom (1975),
Structural Stability and Morphogenesis Benjamin-Cummings Publishing, Reading, Massachusetts, 1975


Fabio Grizzi
Laboratories of Quantitative Medicine, Istituto Clinico Humanitas IRCCS, Rozzano, Milan, Italy.

The fractal design of human anatomy  in the era of system biology

The conception of anatomical entities as a hierarchy of graduated forms and the increase in the number of observed anatomical sub-entities and structural variables has generated a growing complexity, thus highlighting new properties of organised biological matter.
It is now accepted that complexity is so pervasive in the anatomical world that it has come to be considered as a primary characteristic of human anatomical systems. The need to tackle system complexity has become even more evident since completion of the various genome projects. The still unsolved central question is how to transform molecular knowledge into an understanding of complex phenomena in cells, tissues, organs and organisms. Almost all the anatomical entities display hierarchical forms: their component sub-entities at different spatial scales or their process at different time scales are related to each other. One of the pre-eminent characteristics of the entire living world is its tendency to form multi-level structures of "systems within systems", each of which forms a Whole in relation to its parts and is simultaneously part of a larger Whole. Anatomical entities, when viewed at microscopic as well as macroscopic level of observation, show a different degree of complexity. In addition, complexity can reside in the structural organization of the anatomical system or in its behaviour, and often complexity in structure and behaviour go together.
The need to find a new way of classifying anatomical entities, and objectively quantifying their different structural changes, prompted us to investigate the Fractal geometry and the theories of complexity, and to apply their concepts to human anatomy. This attempt has led us to reflect upon the complex significance of the shape of an observed anatomical entity. Its changes have been defined in relation to variations in its status: from a normal to a pathological state introducing the concepts of kinematics and dynamics of anatomical forms, speed of their changes, and that of scale of their observation.



Rangaraj M. Rangayyan, PhD, P Eng

Department of Electrical and Computer Engineering
Schulich School of Engineering University of Calgary,
Calgary, Alberta, Canada

Fractal analysis of breast masses in mammograms

Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. Breast masses present shape and gray-scale characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses. The fractal dimension of the contour of a mass may be computed either directly from the two-dimensional (2D) contour or from a one-dimensional (1D) signature derived from the contour.
We present a study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box-counting method applied to 1D and 2D representations of the contours. The methods were applied to a dataset of 111 contours of breast masses. Receiver operating characteristics (ROC) analysis was performed to assess and compare the performance of fractal dimension and four previously developed shape factors in the classification of breast masses as benign or malignant. Fractal dimension was observed to complement the other shape factors, in particular fractional concavity, in the representation of the complexity of the contours. The combination of fractal dimension (computed using the 2D ruler method) with fractional concavity yielded the highest area under the ROC curve (AUC) of 0.93 with the Bayesian classifier; the two measures, on their own, resulted in AUC values of 0.91 and 0.89, respectively.


Andras Eke
Institute of Human Physiology and Clinical Experimental Research, Semmelweis University,
Budapest, Hungary

Fractal physiological time series analysis: from methods to applications in assessment of perfusion dynamics in the human brain cortex from the standpoint of age, gender and localization.

The momentum of the newly emerging paradigm of fractal geometry [1] and fractal physiology [2] has created an upsurge in the 90’s of the last century in the application of fractal tools well before their thorough evaluation has been completed. In the course of carrying out the task of applying the fractal concept to physiological time series, such as perfusion fluctuation in the brain cortex [3], we realized that each fractal analysis tool had different performance, prerequisite conditions, and limitations, and each needed thorough evaluation in order to avoid bias or misinterpretation of the derived fractal parameters, especially when applied to physiological signals which may be contaminated with noise [4,5].

This lecture will review the cardinal problems in physiological time series analysis, and the ways to overcome them by elaborating approaches that made the analysis conscious, transparent and thus more reliable. Having laid down the grounds of the basics in terminology and mathematical formalism, the concept and methods of monofractal time series analysis will be reviewed; an area far less understood than fractal analysis of structures. Finally, our experimental findings will be presented where the fractal time series analysis revealed scale-free dynamics in spontaneous fluctuations of human cerebral blood volume with reference to age, gender and localization in the brain.

Characterization of methods used in fractal time series analysis: Limitation and precision of various fractal tools such as autocorrelation (AC), power spectral density (PSD), dispersional (Disp), scaled windowed variance (SWV), and detrended fluctuation (DFA) analyses, were assessed by using results of numerical experiments on ideal monofractal signals. A method (signal summation conversion, SSC) was developed [4, 5] to enhance the precision of signal classification based on the dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) [6]. The Hurst exponent, H, was estimated either directly (Disp, SWV, AC) or indirectly (PSD, DFA) and its mean squared error (MSE) and bias were computed for a set of time series with known H. A flowchart for a reliable fractal
analysis was established based on the compatibility of the methods with the fGn/fBm model and their performance as captured in bias and MSE.
Non-invasive monitoring of cerebral blood volume (CBV) in human subjects: The total hemoglobin concentration change (HbT) was recorded non-invasively by near-infrared spectroscopy (NIRS) at a rate of 2 Hz [7]. As recommended by Eke et al. [4, 5], an extended record of 214 (N = 16384) samples was collected for each subject in a session of approximately 2.5 h. The source and detector fibers were secured in a rubber pad. The optodes (a pair of optical fibers connected to the light source and the detector) were mounted just under the hairline over the forehead (for studies on the influenceof age and gender) or in a more lateral position across the frontal and temporal lobes using an array of optodes
(for studies on the influence of localization). The optode array was placed such that the optical signals are sampled from the frontal lobe supplied by the anterior cerebral artery and the temporal lobe supplied by the medial cerebral artery (MCA). This arrangement allowed for mapping across the borderline zones in between adjacent lobes known to be susceptible to hypotensive provocation, and ischemia, both having severe clinical consequences. The cranium was shielded from ambient light by a black cloth. Because the average tissue hematocrit in the region of interest is constant in a steady state, HbT is proportional to CBV [7].
Fractal analysis of human CBV time series: Spontaneous CBV fluctuations were analyzed by our improved power spectral density method [4] by estimating the spectral index, β, as the negative of the power slope in the CBV spectra [7]. Random and fractal patterns were distinguished based on β.

Performance of fractal tools applied in time series analyses: Estimates of H proved reliable by Disp and bridge detreded SWV, but only for fGn and fBm, respectively. These methods can thus be applied only if signal class is determined by SCC prior to analysis. When signal class remains uncertain or both signal class is present in the signal set, a tool that can be applied to both signal categories should be used, such as our improved PSD method [4] or DFA [5] of comparable performance. Impact of age and gender on the fractal properties of CBV time series: The motivation behind these studies was to demonstrate whether age related stiffening of the cerebral vessels would have any significant influence on the fluctuation pattern thought to be impacted by vasomotion and/or flowmotion in the arterial tree. An important finding of this study was that a gradual shift in this pattern as characterized by the spectral index began to develop in the young adult male, while this tendency was found absent in the age-matched female groups of the premenopausal female, but emerged
powerfully once the female hormonal protection of the cerebral vasculature got removed by menopause [7, 8]. Impact of localization on the fractal properties of CBV time series: The spectral index, β, of the CBV fluctuations is a measure of the fractal correlation in the signal. A smaller β indicates the presence of a less correlated temporal fractal in any particular region. We found a significantly lower β within the borderline MCA territory than in the surrounding areas of the brain cortex [8]. This regional difference in temporal complexity is interpreted as a manifestation of an altered vasomotion activity of the small arteries within the borderline MCA territory which is less synchronized than in the neighboring regions. This results in a less efficient “arterial pump”, which otherwise could aid local perfusion when perfusion pressure drops to critically low levels. A lack of this function can lead to susceptibility of these borderline zones to ischemic insults; a well known clinical scenario [8].

We argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. We advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) of Mandelbrot and van Ness [6, 4]. We demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates unless the chosen method is not class-specific (such as PSD or DFA) [4, 5].
Our approach of utilizing a combination of noninvasive NIRS monitoring and fractal analysis of a long-term correlation pattern in spontaneous CBV fluctuations may prove a useful tool to identify a potentially risky cerebrovascular condition before any manifest symptoms of ischemic cerebrovascular disease or hemorrhagic stroke develop: if a much narrowed range of fractal fluctuation amplitudes develop in the elderly, it may lead to an impaired adaptation to suddenly developing under- or overperfusion tendencies in the peripheral cerebrovascular bed that cannot be effectively handled by the persisting large vessel reactions and their neuronal control mechanisms, both being much too distant from the peripheral vascular events [7].
(Supported by OTKA Grants T016953, T34122, NIH Grants TW00442 and RR1243 and High
Performance Computing of the Hungarian National Information Infrastructure Development Program.)

1. Mandelbrot B (1982) The Fractal Geometry of Nature (New York: W H Freeman)
2. Bassingthwaighte J, Liebovitch L and West B (1994) Fractal Physiology (New York: Oxford University Press)
3. Eke A, Hermán P, Bassingthwaighte J B, Raymond G M, Cannon M, Balla I and Ikrényi C (1997) Temporal fluctuations in regional red blood cell flux in the rat brain cortex is a fractal process Adv Exp Med Biol 428 703–709
4. Eke A, Hermán P, Bassingthwaighte J B, Raymond G M, Percival D B, Cannon M, Balla I and Ikrényi C (2000) Physiological time series: distinguishing fractal noises from motions Pflügers Arch.—Eur. J. Physiol. 439 403–415
5. Eke A, Herman P, Kocsis L, Kozak LR (2002) Fractal characterization of complexity in temporal physiological signals. Physiol Meas 23:R1–38
6. Mandelbrot BB, van Ness JW (1968) Fractional brownian motions, fractional noises and applications. SIAM Rev 10:422– 437
7. Eke, A., Herman, P. and Hajnal, M. (2006) Fractal and noisy cbv dynamics in humans: Influence of age and gender. J Cereb Blood Flow Metab 26, 891-898
8. A. Eke, P. Herman, L. Kocsis (2008) Fractal characterization of complexity in dynamic signals: application to cerebral hemodynamics. Methods Mol Med (under publication).