- Email: [email protected]
A fractal analysis of skin pigmented lesions using the novel tool of the variogram technique
Chaos, Solitons and Fractals 28 (2006) 1119–1135 www.elsevier.com/locate/chaos
A fractal analysis of skin pigmented lesions using the novel tool of the variogram technique Mario Mastrolonardo
a,*
, Elio Conte
a,b
, Joseph P. Zbilut
c
a
Department of Medical and Occupational Sciences, Unit of Dermatology, Azienda Ospedaliero-Universitaria ‘‘Ospedali Riuniti’’ di Foggia, Italy b Department of Pharmacology and Human Physiology, TIRES-Center for Innovative Technology for Signal Detection and Processing, Bari University, 70100 Bari, Italy c Department of Molecular Biophysics and Physiology, Rush University, Chicago, IL 60612, USA Accepted 5 August 2005
Abstract The incidence of the cutaneous malignant melanoma is increasing rapidly in the world [Ferlay J, Bray F, Pisani P, et al. GLOBOCAN 2000: Cancer incidence, mortality and prevalence worldwide, Version 1.0 IARC Cancer Base no. 5. Lyon: IARC Press, 2001]. The therapeutic address requires a method having high sensitivity and capability to diagnose such disease at an early stage. We introduce a new diagnostic method based on non-linear methodologies. In detail we suggest that fractal as well as noise and chaos dynamics are the most important components responsible for genetic instability of melanocytes. As consequence we introduce the new technique of the variogram and of fractal analysis extended to the whole regions of interest of skin in order to obtain parameters able to identify the malignant lesion. In a preliminary analysis, satisfactory results are reached. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Cutaneous malignant melanoma (MM) is a tumor deriving from a lineage of pigmented cells called melanocytes, and its incidence has increased rapidly in the last five decades [1]. Also, MM responds poorly to currently available therapies, and is still responsible for more than 75% of all skin cancer deaths [2]. The exact discrimination between MM and other non-malignant skin pigmented lesions (called Ômelanocytic naeviÕ), and, even more significantly, the earliest detection of thin MMs, are of the outmost importance for surviving the disease. In current practice, diagnosis of MM is guided by the so called ÔABCD ruleÕ, based on evaluation of four morphological features of lesions, namely, Asymmetry, Border irregularity, Color variegation, and Diameter of more than 5 mm. The sensitivity of this clinical diagnosis of MM is only 65–80%. It does not recognize small MMs (less than 5 mm in diameter) and does not account for very early MMs that may have a regular shape and homogeneous color distribution [3]. In the last decades, the research activity has been prompted by the need to improve diagnostic accuracy. *
Corresponding author. E-mail address: [email protected] (M. Mastrolonardo).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.106
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Epiluminescence microscopy (ELM) [4] has entered daily clinical practice. ELM is an in vivo, non-invasive technique that has widened our possibilities of evaluating a number of morphological features normally invisible to the naked eye. It improves diagnostic accuracy of benign versus malignant skin pigmented lesions. Based on the possibility of analyzing structural elements normally hidden to the naked eye (such as, pigment network, black dots, globules, streaks/radial streaming, blue-white veils, pseudopods, pseudonetwork, horn cysts and structureless areas), a number of diagnostic algorithms have been developed to serve as guidelines to correct diagnosis, such as the Ôpattern analysisÕ [5,6], the ÔABCD rule of dermatoscopyÕ [7], the ÔMenziesÕ methodÕ [8], the Ô7-point check listÕ [9], and the modified ÔABC-point listÕ [10]. However, diagnostic accuracy of such analogical methodologies is to date far from the desirable, ultimate Ô100% thresholdÕ. The best diagnostic performances reported in the literature show in fact sensitivity rates invariably beyond 90% [3,11]. A limit of dermoscopy as well as of all the clinical diagnostic is that they are based on the human observation of the color and of shape and thus the evaluation is subjected to the experience and to the expertise of the individual clinician. In order to overcome such limits, computer-based dermatoscopic systems and electro-optical technologies have been developed in recent years. The aim has been to identify some common features and parameters in analysis of skin lesions, and the ultimate objective is to arrange a fully automated diagnostic device having the capability of diagnosing pigmented lesions without the intervention of an human expert [12]. Usually, clinical and dermoscopic analysis images of lesions are acquired for the analysis. Some groups use also the so called charged-coupled device, CCD. A calibrated three CCD video cameras enables to directly acquire dermoscopic images of pigmented lesions. Melafind and SIAscope [13] use instead an imaging acquisition technique of particular interest since they utilize multispectral narrow bands from 400 to 1000 nm to image the lesion, employing the physical process that light of different wavelength, penetrates the skin to different paths. The technique of the segmentation is utilized in computerized images to separate the lesion from the normal surrounding tissue. Segmented lesions that have the potentiality to be differentiated as benign respect to pigmented cutaneous lesions, are finally identified. The following stage may be called of the objective conversion. Mathematical procedures are used to characterize the lesion morphology as the asymmetry, the blotchiness, the color variation, and the dermoscopic structure. A process of conversion is performed in which the elaboration goes from the qualitative interpretation of dermoscopic structures and patterns to final quantitative parametric features that consequently are retained to be objective expression of the lesion. Finally, a classifier, based on a neural network, is employed. By incorporation of a selected number of features into such classifier, it becomes feasible to decide automatically whether the lesions should or should not biopsied. For a current update of such technique as well as of the other standard techniques, read the excellent papers given in Ref. [14]. The computerized image analysis systems are an interesting and promising field of research. In perspective, such automatized instruments of Ômachine visionÕ will play an important role as adjuvant to clinical and dermoscopic examinations. There are, however, three additional features that must be taken in consideration in the attempt to arrange a valuable method for diagnosis of melanoma. The introduced methods for computer analysis evidence [14] that the number of morphometric variables that must be used for computer analysis, and, generally speaking, the number of significant features to be considered with aim to realize a method discriminating between melanoma and non-melanoma, is very high. This basic feature induces the first important conclusion that, when introducing a new method for diagnosis of melanoma respect to non-melanomas, one must account for the high variability of data characterizing the lesion. The second important feature follows, in some sense, the present clinical tendency. It tends to attribute more importance to the investigation on the color of the lesion, attributing instead less consideration to the dimensions of the lesion itself. Consequently, a new method should account for accurate analysis of color and of variability of such data. Let us examine the third feature. Studying irregularities of a lesion border, various authors arrived to the result to characterize it by a fractal dimension within a multiscale method [15]. This seems to be a very important conclusion since it clears that a way to proceed along the line of an actual investigation and of an actual classification of lesions, must necessarily account for non-linear methodologies which include, in particular, the analysis of noise and chaos dynamics. This last conclusion seems to be strongly supported from two basic indications. The first one is that in melanoma lesions we have, first of all, an altered metabolism states of cancer cells. In addition, there is an important result that was evidenced rather recently from some authors [16]. They outlined that the irregularity of a lesion has a strong correlation with the genetic instability of melanocytes. It then follows that it is unthinkable to aim to analyze the dynamics of the advent of melanoma and to proceed to a lesion classification criteria on the basis of a method unable to make direct and robust use of the currently non-linear theories and methodologies here including in particular the analysis of noise and chaos dynamics. A method that simultaneously should account for analysis of color, for analysis of the high variability in the characteristic of the lesions, and of fractal nature of the border, not only, but possibly, of the whole examined region of interest, must necessarily use non-linear methodologies of investigation. To this purpose we arranged the present work. In the present paper we introduce a new technique that we have elaborated with regard to normal skin tissue, nevi and melanoma. We have called it the variogram analysis of melanocytes lesions. In the subsequent sections, we introduce the variogram analysis, and a proper mathematical model to discriminate the results that
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may be obtained analyzing images of normal skin tissue, nevi, and melanoma. In this second step of our analysis, we also introduce an approach based on the calculation of fractal measure and generalized fractal dimension no more confined to an investigation of the border of the region of interest (lesion) but to all the region itself. Finally, in the third part of the paper, we apply directly the introduced methodology to the analysis of about ten cases of melanomas and nevi.
2. Materials and methods 2.1. Patient selection The Unit of Dermatology at the Azienda Ospedaliero-Universitaria ÔOspedali RiunitiÕ di Foggia, South Italy, was involved in case selection. The patients were examined by one of us (MM), specializing in skin cancer, for the presence of pigmented lesions that were possibly malignant melanomas (MM). All suspect lesions underwent surgical excision subsequent to imaging, and hematoxylin-eosin-stained sections were used to establish diagnosis of MM versus non-melanoma pigmented lesions. 2.2. Lesion types Dermoscopic images were used to test our methodology. Six were MMs, and six non-malignant skin pigmented lesions. Images taken of normal (unaffected) skin from each patient were also submitted to analysis, as were 14 clinical (not dermascopic) images (seven MMs and seven non-malignant skin pigmented lesions) extracted from an image-database available in the Internet (http://www.dermis.net/doia/mainmenu.asp?zugr=d&lang=e). 2.3. Image acquisition The equipment for image acquisition, before surgical excision, consisted of the following components: • A trade digital camera, with the following technical characteristics: 1/1.8 Ôhigh-densityÕ charge-coupled device, CCD; total number of pixels, 3.34 million; range of image size, from 640 · 480 to 2.048 · 1.536. • A lens of a trade hand-held dermatoscope (10· magnification power), attached to the camera lens via a proper photo adapter. A drop of immersion oil was applied to the lesions under examination before applying the glass plate of the dermatoscope onto the skin surface. 2.4. Image processing Images of 180 · 150 pixels were selected as regions of interest (ROI) for analysis. ROI were then converted and digitized by a red/green/blue (RGB) conversion software using MatLab color converter. In detail, each colored map with no hairs was decomposed on the basis of the three fundamental colors (RGB analysis) and each map was analyzed realizing a grid of 180 · 150 pixels (1 6 x 6 180; 1 6 y 6 150) giving 27,000 data points and three basic matrices. In brief, assigning a proper coordinate frame to the map, three basic functions Zi(p) were obtained, being Zi(p) the function of the intensity for colors (i = red, green, and blue, respectively) and being p the pixel location with pair (x, y) of coordinates. A Variogram Analysis of Melanocytic Lesions was performed. Let us explain the essence of the method. The technique of the variogram has been extensively applied in geology and, in particular, in geochemistry studies [17]. Let us examine in detail as it was conceived for our studies. It is well known that the probability models of images are widely used and they are based on Markov random fields models where an image field is retained to be a collection of random variables denoting pixel values on an uniformly spaced grid in the image plan. The central concept of spatial statistics is represented from the regionalized variables z(x) with x 2 D Rd which are treated as realizations of an underlying random function Z(x). This random function Z(x) is called second-order stationary if it fulfills the following conditions: E½Zðx þ hÞ ¼ E½ðZðxÞÞ; cov½Zðx þ hÞ; ZðxÞ ¼ CðhÞ;
ð1Þ d
h2R .
ð2Þ
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This is to say that Z(x) has a constant mean value, E[Z(x)] = m, and its covariance function C(h) is translation invariant according to CðhÞ ¼ E½ZðxÞ Zðx þ hÞ m2 .
ð3Þ
A weaker form of stationarity may be also introduced. It is the intrinsic stationarity that is defined by the following conditions: E½Zðx þ hÞ ZðxÞ ¼ mðhÞ ¼ 0; var½Zðx þ hÞ ZðxÞ ¼ 2cðhÞ;
ð4Þ ð5Þ
1 cðhÞ ¼ E½ðZðx þ hÞ ZðxÞÞ2 . 2
ð6Þ
where
Eq. (6), takes the name of variogram [17]. This is precisely the statistical tool that we will use in the present analysis. In our method Z(x) will represent a second-order stationary random function.
Fig. 1. Normal skin tissue.
Fig. 2. Benign naevus.
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Fig. 3. Melanoma.
Fig. 4. Normal skin tissue. Behavior of Z(x) [see text and (1)] in blue color analysis, as given at each stage of the pixel elaboration.
Further, we will write that cðhÞ ¼ ChD ;
h > 0 and 0 < D < 2;
ð7Þ
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and we will call it the fractal variance function where the constant C will represent the fractal measure while D will represent the generalized fractal dimension [17].
3. Further comments on variogram analysis of skin lesions As previously said, we utilized the variogram to quantify the spatial correlation of a set of measurements of Zi(p), (color intensity or hue for i = red, green, blue) in the case of normal, benign and cancer lesions of skin tissue. In substance, we analyzed the variability of pairs of measurements of Zi in terms of a separation or lag distance. The so called semivariogram is defined in (6) where c(h), the experimental variogram, may be calculated at different separations or lag distances h. As previously said, Zi(p) is a second-order stationary random function whose value is given at the point p in the selected region of interest of the skin tissue. Zi(p + h) represents instead the value of such function at the distance h from p in the same region of interest. The E in (6) represents instead the expectation value of the computed expression. In conclusion, we analyze skin regions of interest where a small value of the variogram indicates that pairs of measurements of Zi(p) are similar for a particular separation distance h. This is to say that they have low variability. Instead, high values of the variogram indicate that the values of the measurements in the lag pairs are very dissimilar on the average, and this is to say that the corresponding data show high variability. We observe here that, introducing the method of variogram in analysis of skin lesions, based on the analysis of the three different colors (red, blue and green), we aim to have satisfied two of the three requirements that we fixed in the previous sections in order to realize a valuable method for diagnosis of melanoma. We mentioned, in fact, as basic requirements for an analysis of skin lesions, the analysis of the color and an examination of the high variability of the data in the characteristics of the lesions. Our method, based on variogram analysis of ROI converted to their basic colors red, blue and green, seems to satisfy such basic requirements since the variogram accounts specifically for the variability of data at each lag step. In order to characterize the essential features of our method, we may add here that, calculated the experimental variogram for each ROI of interest, the introduction of a proper mathematical model con-
Fig. 5. Benign naevus. Behavior of Z(x) in blue color analysis, as given at each stage of the pixel elaboration.
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nected to the calculated experimental variogram, enables us to express and to calculate by a fitting procedure some parameters that may be helpful to discriminate normal skin tissue from benign and cancer lesions. Finally, one has to consider the basic foundation of our approach under the profile of the previously mentioned non-linear features in skin lesions. In our approach, chaos and noise dynamic mechanisms in the distribution of the colors (intensity or hue for RGB) should have an essential role in the uneven distributions and in the measured variability of experimental data of Zi. Our aim is to evaluate the fractal structure of whole lesions and benign nevi with regard to their spatial distribution. Fractal structure should have a strong correlation with the altered metabolism states of cancer cells and with the genetic unstability of melanocytes [15,16]. Consequently, we may consider the equation given in (7) where, so written, the c(h) represents now the fractal variance function, that is in substance a power semivariogram [17]. In this manner, our elaboration of variogram analysis of skin lesions is reduced to a non-linear model in which the basic parameters C (fractal measure) and D (generalized fractal dimension) may be estimated in the cases of interest. In particular, such fractal parameters are able to control the variation of Zi. In brief, the greater the fractal parameters, the greater the variation of Zi should result and it should be correlated in some manner with the altered metabolism of the cancer cells and with the genetic instability of melanocytes.
4. Preliminary results of the analysis. The case of nevi exhibiting linear variogram In Figs. 1–3 we have, respectively, the images of a normal skin tissue, a benign nevi, and a melanoma from our experimental data. In Figs. 4–6 we report instead the values of Zi(p), i = R, G, B, as they were obtained by us step by step at each stage of the pixel elaboration by the previous mentioned MatLab color conversion software. For brevity, only the values of ZB(p) (blue color) are reported. Let us comment, first of all, the range of variability that was obtained for Zi(p), i = R, G, B, in some cases of normal tissue, benign nevi, and melanomas. For normal skin tissue, Zi(x, y) (i = B, R, G) gave the following results:
Mean Mean Mean Mean
value value value value
of of of of
minimum maximum mean variance
ZB(x, y)
ZR(x, y)
ZG(x, y)
146.00 ± 6.63 184.40 ± 8.26 169.47 ± 6.39 20.08 ± 4.59
159.20 ± 7.12 187.00 ± 5.78 174.50 ± 5.60 7.03 ± 1.46
154.00 ± 7.31 185.00 ± 8.27 172.62 ± 6.99 10.83 ± 3.10
Therefore Zi(x, y) showed a very restricted range of variation for colors blue, red, and green, respectively. In the case of the nevi, the results for Zi(x, y) changed profoundly:
Mean Mean Mean Mean
value value value value
of of of of
minimum maximum mean variance
ZB(x, y)
ZR(x, y)
ZG(x, y)
10.40 ± 3.71 90.40 ± 22.07 39.99 ± 12.91 173.14 ± 159.71
29.00 ± 23.76 134.25 ± 35.50 80.18 ± 47.17 308.82 ± 244.28
13.80 ± 4.20 91.80 ± 20.14 45.35 ± 13.61 173.61 ± 153.65
ZB(x, y)
ZR(x, y)
ZG(x, y)
31.66 ± 23.86 139.00 ± 25.51 90.24 ± 38.16 239.44 ± 50.55
124.33 ± 19.65 189.00 ± 15.09 166.59 ± 18.16 71.27 ± 39.80
48.66 ± 28.93 143.66 ± 24.41 106.13 ± 35.30 168.89 ± 72.69
Finally, we calculated the values for the melanomas:
Mean Mean Mean Mean
value value value value
of of of of
minimum maximum mean variance
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Fig. 6. Melanoma. Behavior of Z(x) in blue color analysis, as given at each stage of the pixel elaboration.
As a preliminary indication we may conclude that a very large range of variability was obtained for the data in the case of nevi as well as of melanomas respect to normal skin tissues. We had no significant differences between the two kinds of lesions. Also the calculated mean values with their standard deviations did not result often to have significant differences. However, it must be outlined here that, in spite of such no conclusive statistical results of the data, profound differences could be exhibited from the data when analyzed under the profile of their spatial distribution, and this was, in substance, the reason because we introduced the method of the variogram for such kind of analysis. Therefore, let us look to the results that we obtained calculating the variograms. Each variogram was computed using (6) with a number of lags of 80, and a lag tolerance of 10%. Generally speaking, one expects that there are infinitely many possible behaviors for the obtained experimental variograms, ci(h), i = R, G, B. However, in our preliminary analysis, we accounted for three possible models, that were also identified in other studies [17], the linear, the exponential, and spherical models. Examples of these three possible behaviors are given in Fig. 7(a). In detail, we considered that, if the experimental function ci(h) never levels out, then the use of a linear model should be appropriate. If, otherwise, ci(h) levels out but it is curving all the way, the use of an exponential model should be made. Finally, if ci(h) starts as a straight line and then levels out, the use of a spherical model should result to be more appropriate. It is easily seen that, in such cases, some parameters of interest may be also connected to the considered mathematical models. If the experimental ci(h) shows to have a non-zero intercept on the vertical axis, according to [17], we may consider it what is called a Nugget Effect Component. One may calculate it as a parameter of interest. The second parameter one may use is the Scale: the Scale will be the height on the vertical axis at which the function ci(h) levels off. Finally, the Length will represent the lag distance at which the function ci(h) levels off. We retain that a preliminary use of these parameters [17], should enable to express a satisfactory parametric characterization of the variogram approach that we used. Each time a fitting program of the experimental data of ci(h), based on the least squares method, will be able to select the best model to use (linear, exponential or spherical model), giving each time the values of the calculated parameters (intercept component value), slope (linear case), scale and length values (exponential or spherical models). The results of the experimental variograms that we calculated, are
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Fig. 7. Variogram behavior in normal skin tissue, naevus and melanoma.
reported in Figs. 8–10 for normal skin tissue in one subject. In Figs. 11–13 we have the results of the experimental variograms in a subject with a benign nevi, and, finally, in Figs. 14–16 we report the results of experimental variograms for a melanoma. Also by a simple visual inspection, the first emerging conclusion is that, for nevi exhibiting a linear variogram, we have introduced a very interesting method that is able to discriminate with 100% of sensitivity and specificity a diagnosis of melanoma respect to benign nevi and respect to normal skin tissue. Calculating the experimental variograms, and by simple visual inspection and examination, we are in the condition to differentiate normal skin tissue respect to benign
Fig. 8. Normal skin tissue (ordinate: variogram value; abscissa: max distance)—blue.
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Fig. 9. Normal skin tissue (ordinate: variogram value; abscissa: max distance)—red.
Fig. 10. Normal skin tissue (ordinate: variogram value; abscissa: max distance)—green.
Fig. 11. Benign naevus (ordinate: variogram value; abscissa: max distance)—blue.
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Fig. 12. Benign naevus (ordinate: variogram value; abscissa: max distance)—red.
Fig. 13. Benign naevus (ordinate: variogram value; abscissa: max distance)—green.
Fig. 14. Melanoma (ordinate: variogram value; abscissa: max distance)—blue.
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Fig. 15. Melanoma (ordinate: variogram value; abscissa: max distance)—red.
Fig. 16. Melanoma (ordinate: variogram value; abscissa: max distance)—green.
nevi and melanomas. In fact, in the case of normal skin tissue it is seen that the mathematical model of the variogram is represented by an exponential model of the kind cðhÞ ¼ C 0 þ C 1 ð1 qÞ with q ¼ expðlhÞ
ð8Þ
with low values for the variogram and for such parameters. Instead, in the case of benign nevi, we have that the experimental variogram follows a rather robust linear behavior with higher values of the variogram respect to the case of the normal skin tissue. Finally, the experimental variogram connected to melanomas again exhibits an exponential behavior with values of the variogram and of the connected parameters, that result to be very different respect to the case of normal skin tissue. Fig. 7(b) gives us a direct indication of the status of matter picturing the results of our analysis. To conclude this first part of our analysis, we have only to specify that all the expressed values of the parameters that we calculated by fitting using the very simple mathematical models of variograms that we introduced in Fig. 7(a), furnished optimal numerical indexes of control for the fitting mathematical procedure giving, in particular, a very low sum of residuals (SRE) and an R2 ffi 1. In all the cases as weighting factors the number of pairs/standard deviation were used. The results for the parameters are reported in Table 1 and, obviously, they result very discriminating. Note that our expert clinician expressed doubt of about 70% on the visually examined cases of nevi and melanoma. In conclusion, he had an high level of uncertainty. Our method instead discriminated in this case with 100% of sensitivity and 100% of specificity owing to the basic differences exhibited from the variograms in the two cases of interest of melanoma versus nevi.
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Table 1 Subject
Blue Nugget
Normal skin tissue Sta-pol 6.96 Mit.-pol 4.21 Pac. 0 m.v. 3.72 s.d. 3.51 Naevus DÕAr. Mit. Les. m.v. s.d. Melanoma Lor4. Lor6. Lor7. m.v. s.d.
0 18 7 8.33 9.07 0 5 5 3.33 2.89
Red Scale
Length
Nugget
17.38 23.79 13.26 18.14 5.31
9.49 65.2 28.59 34.43 28.31
0.27 4.53 0 1.6 2.54
0.43 3.23 0.63 1.43 1.56
110 580 80 256.67 280.42 33.83 48 30 37.28 9.48
322.9 490 520 444.3 106.2
Green Scale
Length
Nugget
7.53 3.93 5.66 5.71 1.8
3.66 30.46 5.78 13.3 14.9
2.47 1.17 3 2.21 0.94
8.65 11.61 18 12.75 4.78
19.35 8024 80 59.86 35.09
0 18.1 3 7.03 9.7
5.01 4.35 6,9 5.42 1.32
900 530 800 743.33 191.4
0 3.87 5 2.96 2.62
1.15 5.02 1.4 2.52 2.17
210 580 160 316.67 229.42
12 0 5 5.67 6.03
115 167 250 177.33 68.09
45 52 40 45.67 6.03
20 6 2 9.33 9.45
260 340 400 333.33 70.24
36 30 26 30.67 5.03
Fig. 17. Variogram of naevus.
Fig. 18. Variogram of melanoma.
Scale
Length
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Table 2 Blue melanoma
Blue naevus
C0
I-Lor4 I-Lor6 I-Lor7 m.v. s.d. II-Uma1 II-Uma2 m.v. s.d.
I-Mit I-Les. I-DÕAr
0.00 6.72 0.40 2.37 3.77 15.00 13.00 14.00 1.41
II-Lop II-Nev
C1
2.39 4.64 8.75 5.26 3.23 0.00 0.66 0.3300 0.4667
323.40 410.60 530.70 421.57 104.08 890.40 903.40 896.90 9.19
A1
9610.70 1186.60 8000.00 6265.77 4471.80 325.80 485.50 405.65 112.92
24.08 36.22 34.43 31.58 6.55 32.00 35.00 33.50 2.12
Fractal measure
a
2654.20 3000.00 3575.30 3076.50 465.29 216.40 110.00 163.20 75.24
1.05 1.75 1.39 1.40 0.35 0.28 0.22 0.25 0.04
0.89 0.85 1.28 1.01 0.24 0.51 0.75 0.63 0.17
76.48 42.66 35.92 51.69 21.73 356.15 404.00 380.08 33.84
Generalized fractal dimension 7.29 2.16 0.66 3.37 3.48 24.33 30.63 27.48 4.45
0.30 0.49 0.61 0.47 0.16 0.13 0.10 0.12 0.02
0.92 0.75 1.06 0.91 0.16 0.40 0.47 0.44 0.05
Statistical analysis: (I Mel.–I Naev.) significant differences for parameters: A1 (p = 0.003), FM (p = 0.019), GFD (p = 0.025). (II Mel.–II Naev.) significant differences for parameters: C0 (p = 0.006), C1 (p = 0.025), FM (p = 0.004), GFD (p = 0.013). (I Mel.–II Naev.) significant differences for parameters: A1 (p = 0.046). (II Mel.–I Naev.) significant differences for parameters: C0 (p = 0.04), A1 (p = 0.003), a (p = 0.024), FM (p = 0.0002), GFD (p = 0.0064). Red melanoma
Red naevus
C0
I-Lor4 I-Lor6 I-Lor7 m.v. s.d. II-Uma1 II-Uma2 m.v. s.d.
I-Mit I-Les. I-DÕAr
0.00 5.48 0.00 1.83 3.16 10.00 22.00 16.00 8.49
II-Lop II-Nev
C1
0.50 0.15 6.87 2.51 3.78 0 0 0.00 0.00
123.70 142.90 239.20 168.60 61.89 516.20 590.00 553.10 52.18
A1
1413.70 1622.20 2768.60 1934.83 729.55 156.5 90.18 123.34 46.90
27.08 27.37 34.83 29.76 4.39 30.00 39.00 34.50 6.36
Fractal measure
a
300.00 160.80 223.10 227.97 69.73 187.3 239 213.15 36.56
0.92 1.97 1.91 1.60 0.59 0.74 0.83 0.79 0.06
0.91 1.16 1.44 1.17 0.27 0.61 0.43 0.52 0.13
22.44 28.82 12.80 21.35 8.07 76.55 83.97 80.26 5.25
Generalized fractal dimension 6.04 8.69 2.39 5.71 3.16 8.9 10.59 9.75 1.20
0.36 0.35 0.66 0.46 0.18 0.42 0.41 0.42 0.01
0.94 0.95 1.24 1.04 0.17 0.46 0.3 0.38 0.11
Statistical analysis: (I Mel.–I Naev.) significant differences for parameters: C1 (p = 0.013), A1 (p = 0.008), FM (p = 0.035), GFD (p = 0.014). (II Mel.–II Naev.) significant differences for parameters: C1 (p = 0.013), A1 (p = 0.02), FM (p = 0.0029). (I Mel.–II Naev.) significant differences for parameters: A1 (p = 0.0026). (II Mel.–I Naev.) significant differences for parameters: A1 (p = 0.034), FM (p = 0.0003), GFD (p = 0.016). Green melanoma
Green naevus
C0
I-Lor4 I-Lor6 I-Lor7 m.v. s.d. II-Uma1 II-Uma2 m.v. s.d.
I-Mit I-Les. I-DÕAr
0.00 5.21 0.00 1.74 3.01 17.00 20.00 18.50 2.12
II-Lop II-Nev
C1
7.98 2.61 5.90 5.50 2.71 0.00 0.01 0.00 0.0035
273.00 315.00 397.20 328.40 63.17 714.00 739.40 726.70 17.96
A1
811.90 387.30 590.80 596.67 212.36 345.40 448.00 396.70 72.55
29.70 34.30 29.91 31.30 2.60 31.00 33.00 32.00 1.41
Fractal measure
a
119.00 209.90 210.00 179.63 52.51 433.90 400.00 416.95 23.97
1.01 1.96 1.53 1.50 0.48 0.57 0.48 0.53 0.06
1.20 1.07 1.50 1.26 0.22 0.59 0.64 0.62 0.04
45.08 31.07 27.54 34.56 9.28 157.69 198.21 177.95 28.65
Generalized fractal dimension 2.53 5.03 0.47 2.68 2.28 12.57 16.14 14.36 2.52
0.38 0.51 0.62 0.50 0.12 0.31 0.25 0.28 0.04
0.87 1.00 1.27 1.05 0.20 0.48 0.47 0.48 0.01
Statistical analysis: (I Mel.–I Naev.) significant differences for parameters: A1 (p = 0.008), FM (p = 0.004), GFD (p = 0.016). (II Mel.–II Naev.) significant differences for parameters: C0 (p = 0.006), C1 (p = 0.024), A1 (p = 0.0019), FM (p = 0.015), GFD (p = 0.02). (I Mel.–II Naev.) significant differences for parameters: A1 (p = 0.0001). (II Mel.–I Naev.) significant differences for parameters: C0 (p = 0.011), A1 (p = 0.032), a (p = 0.022), FM (p = 0.0014), GFD (p = 0.015).
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5. The complete variogram analysis of the data Not all the examined nevi exhibited linear variogram. There are cases in which their behavior tends to manifest an exponential behavior. In Figs. 17 and 18 we report the case of variograms of a naevus and of a melanoma both exhibiting non-linear behaviors. The mathematical model to be used in these cases is the following:
Table 3 Blue melanoma
Blue naevus
C0
M16 M11 M12 M14 M15 M17
N17 N11 m.v. s.d. N12 N13 Nc
70.00 80.00 60.00 110.00 130.00 30.00
m.v. s.d.
80.00 35.78
C1
10.00 34.00 22.00 16.97 21.00 0.00 0.00 7.00 12.12
745.70 1413.50 1005.60 1177.20 1070.30 1105.10 1086.23 218.38
A1
8947.60 2500.00 5723.80 4559.14 578.00 196.00 276.20 350.07 201.43
40.00 83.22 80.00 82.00 80.32 40.00 67.59 21.40
Fractal measure
a
3826.00 158.20 1992.10 2593.53 140.00 9.05 500.00 216.35 254.22
0.51 0.83 0.81 0.77 0.17 0.68 0.63 0.25
0.74 0.92 0.83 0.13 0.80 0.21 0.23 0.41 0.34
159.72 128.74 120.00 125.72 501.85 147.54 197.26 149.96
Generalized fractal dimension 12.91 47.39 30.15 24.38 24.68 105.36 77.95 69.33 41.02
0.33 0.44 0.48 0.43 0.11 0.42 0.37 0.14
0.85 0.71 0.78 0.10 0.41 0.08 0.20 0.23 0.17
Statistical analysis: (Mel.–I Naev.) significant differences for parameters: C1 (p = 0.008), GFD (p = 0.0083). (Mel.–II Naev.) significant differences for parameters: C0 (p = 0.012), C1 (p = 0.0018). Red melanoma
Red naevus
C0
M16 M11 M12 M14 M15 M17
N17 N11 m.v. s.d. N12 N13 Nc
40.00 152.50 20.00 289.00 120.00 70.00
m.v. s.d.
115.25 98.37
C1
7.00 50.00 28.50 30.41 20.00 0.00 0.00 6.67 11.55
1105.00 1389.70 929.80 2068.10 1605.60 1682.00 1463.37 412.86
A1
2800.00 1297.00 2048.50 1062.78 578.50 177.60 228.30 328.13 218.30
50.00 66.03 22.03 69.00 65.83 90.00 60.48 22.77
Fractal measure
a
163.40 170.00 166.70 4.67 140.00 3.98 5.18 49.72 78.19
0.95 1.22 1.40 1.63 0.67 0.93 1.13 0.35
1.53 1.12 1.33 0.29 0.80 0.32 0.48 0.53 0.24
49.28 94.94 164.08 42.57 252.66 55.04 109.76 83.38
Generalized fractal dimension 0.84 27.52 14.18 18.87 16.18 108.63 111.12 78.64 54.11
0.69 0.56 0.41 0.85 0.35 0.68 0.59 0.19
1.58 0.64 1.11 0.66 0.68 0.09 0.17 0.31 0.32
Statistical analysis: (Mel.–I Naev.) significant differences for parameters: A1 (p = 0.0008). (Mel.–II Naev.) significant differences for parameters: C1 (p = 0.0033), a (p = 0.034). Green melanoma
Green naevus
C0
M16 M11 M12 M14 M15 M17
N17 N11 m.v. s.d. N12 N13 Nc
70.00 60.00 50.00 152.00 170.00 50.00
m.v. s.d.
92.00 54.26
C1
8.00 40.00 24.00 22.63 20.00 0.00 0.00 6.67 11.55
882.20 1082.00 991.00 1576.00 1237.00 1413.20 1196.90 263.34
A1
2352.00 2344.50 2348.25 5.30 400.00 169.90 230.20 266.70 119.31
70.00 48.78 42.30 81.00 40.00 60.00 57.01 16.29
Fractal measure
a
280.00 200.60 240.30 56.14 337.80 2.81 9.34 116.65 191.55
0.58 0.90 1.03 1.05 0.85 0.74 0.86 0.18
1.03 0.97 1.00 0.04 0.59 0.34 0.27 0.40 0.17
114.03 149.51 70.41 70.04 406.24 127.53 156.29 126.45
Statistical analysis: (Mel.–I Naev.) significant differences for parameters: C1 (p = 0.0011), A1 (p = 0.0002), GFD (p = 0.035). (Mel.–II Naev.) significant differences for parameters: C0 (p = 0.035), C1 (p = 0.0007), a (p = 0.0078).
Generalized fractal dimension 4.49 38.59 21.54 24.11 26.62 110.38 95.08 77.36 44.60
0.41 0.40 0.59 0.64 0.12 0.49 0.44 0.18
1.12 0.69 0.91 0.30 0.41 0.08 0.17 0.22 0.17
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a h cðhÞ ¼ C 0 þ C 1 ð1 qÞ with q ¼ exp . A1
ð9Þ
The fitting procedure determines the parameters C0, C1, A1, a that are sufficient to discriminate melanomas respect to nevi. The results are reported in Tables 2 and 3 for all the cases that we examined. In such tables we have also the results of fractal analysis that we performed. For each case the values of the fractal measure and of generalized fractal dimension were estimated. In our opinion such last results represent the most relevant side of the present paper. For the first time we show that RGB color maps of skin of normal tissue or of nevi or of melanomas exhibit a fractal distribution in their entirety. We find also that in such tissular regions two basic parameters as fractal measure and generalized fractal dimension are able to characterize and to delineate the basic features of such tissular regions itself. It seems that it is confirmed the basic motivation that aimed the present work. The basic features to understand the melanocytic dynamics and to differentiate diagnosis between nevi and melanoma must be reached by application of non-linear methodologies, fractal, noise and chaos dynamics. As obtained in the present paper, it is not the matter to explore fractal dimension in relation to the border of a melanoma lesion but to investigate in detail fractality and thus chaos and noise dynamics to the whole level of tissular regions and, in particular, to the whole region characterizing the lesion itself. The added statistical analysis confirms that variogram analysis with added investigation of fractal dynamics of the whole examined skin regions of interest, enabled us to discriminate benign lesions as nevi respect to melanocytic lesions as melanomas in a satisfactory manner also in the framework of a preliminary analysis.
References [1] Ferlay J, Bray F, Pisani P, et al. GLOBOCAN 2000: Cancer incidence, mortality and prevalence worldwide, Version 1.0 IARC Cancer Base no. 5. Lyon: IARC Press; 2001. [2] Bianchi Scarra` G, Testori A, Mihm M. What is new in melanoma research: genetics and epidemiology of melanoma? Review of a workshop held in Milan in May 2003. Melanoma Res 2004;14:77–84. [3] Ruocco E, Argenziano G, Pellacani G, Seidenari S. Noninvasive imaging of skin tumors. Dermatol Surg 2004;30:301–10. [4] Carli P, De Giorni V, Crocetti E, et al. Improvement of malignant/benign ratio in excised melanocytic lesions in the Ôdermascopy eraÕ: a retrospective study 1997–2001. Br J Dermatol 2004;150:687–92. [5] Pehamberger H, Steiner A, Wolff K. In vivo epiluminescence microscopy of pigmented skin lesions. I. Pattern analysis of pigmented skin lesions. J Am Acad Dermatol 1987;17:571–83. [6] Steiner A, Pehamberger H, Wolff K. In vivo epiluminescence microscopy of pigmented skin lesions. II. Diagnosis of small pigmented skin lesions and early detection of malignant melanoma. J Am Acad Dermatol 1987;17:584–91. [7] Stolz W, Riemann A, Cognetta AB, et al. ABCD rule of dermatoscopy: a new practical method for early recognition of malignant melanoma. Eur J Dermatol 1994;4:521–7. [8] Menzies S, Crotty K, Ingvar C, McCarthy W. An atlas of surface microscopy of pigmented skin lesions. New York: McGraw-Hill Books; 1996. [9] Argenziano G, Fabbrocini G, Carli P, et al. Epiluminescence microscopi for the diagnosis of doubtful melanocytic skin lesions. Comparison of the ABCD rule of dermatoscopy and a new 7-point checklist based on pattern analysis. Arch Dermatol 1998;134:1563–70. [10] Blum A, Rassner G, Garbe C. Modified ABC-point list of dermoscopy: a simplified and highly accurate dermoscopic algorithm for the diagnosis of cutaneous melanocytic lesions. J Am Acad Dermatol 2003;48:672–8. [11] Argenziano G, Soyer HP, Chimenti S, et al. Dermascopy of pigmented skin lesions: results of a consensus meeting via the Internet. J Am Acad Dermatol 2003;48:679–93. [12] Green A, Martin N, McKenzie G, et al. Computer image analysis of pigmented skin lesions. J Am Acad Dermatol 1994;31:958–64; Cascinelli N, Ferrario M, Bufalino R, et al. Results obtained by using as an aid to diagnosis of cutaneous melanoma. Melanoma Res 1992;2:163–70; Seidenari S, Pellicani G, Giannetti A. Digital videomicroscopy and image analysis with automatic classification for detection of thin melanomas. Melanoma Res 1999;9:163–71; Kahofer P, Hofmann-Wellenhof R, Smolle J. Tissue counter analysis of dermatoscopic images of melanocytic skin tumors: preliminary findings. Melanoma Res 2002;12:71–5, and references therein. [13] Elbaum M, Kopf AW, Rabinovitz HS, et al. Automatic differentiation of melanoma from melanocytic nevi with multispectral digital dermoscopy. J Am Acad Dermatol 2001;44:207–18; Moncrieff M, Cotton S, Claridge E, Hall P. Spectrophotometer intracutaneous analysis. A new technique for imaging pigmented skin lesions. Br J Dermatol 2002;146:448–57. [14] Holger V, Classen R. Computer vision and digital imaging technology in melanoma detection. Sem Oncol 2002;29:308–27; Wang SQ, Rabinovitz HS, Kopf AW, Oliviero M. Current technologies for the in vivo diagnosis of cutaneous melanoma. Clin Dermatol 2004;22:217–22;
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See also Marchesini R, Bono A, Bartoli C, Lualdi M, Tomatis S, Cascinelli N. Optical imaging and automated detection: questions and answers. Melanoma Res 2002;12:279–86; Tomatis S, Bono A, Bartoli C, Carrara M, Lualdi M, Tragni G, et al. Automated melanoma detection: multispectral imaging and neural network approach for classification. Med Phys 2003;30(2):212–21; Ruocco E, Argenziano G, Pellicani G, Seidenari S. Noninvasive imaging of skin tumors. Dermatol Surg 2004;30:301–10; Bono A, Bartoli C, Cascinelli N, Lualdi M, Maurichi A, Moglia D, et al. Melanoma Detection: 2002;205:362–6. [15] Claridge E, Hall PN, Kecte M. Shape analysis for classification of malignant melanoma. J Biomed Eng 1992;14:229–34. [16] Maize JC, Ackerman A. Pigmented lesions of skin, Philadelphia: 1987. [17] Wei S, Pengda Z. Multidimensional self-affine distribution with application in geochemistry. Math Geol 2002;34(2):109–23, and references therein.
A fractal analysis of skin pigmented lesions using the novel tool of the variogram technique Mario Mastrolonardo
a,*
, Elio Conte
a,b
, Joseph P. Zbilut
c
a
Department of Medical and Occupational Sciences, Unit of Dermatology, Azienda Ospedaliero-Universitaria ‘‘Ospedali Riuniti’’ di Foggia, Italy b Department of Pharmacology and Human Physiology, TIRES-Center for Innovative Technology for Signal Detection and Processing, Bari University, 70100 Bari, Italy c Department of Molecular Biophysics and Physiology, Rush University, Chicago, IL 60612, USA Accepted 5 August 2005
Abstract The incidence of the cutaneous malignant melanoma is increasing rapidly in the world [Ferlay J, Bray F, Pisani P, et al. GLOBOCAN 2000: Cancer incidence, mortality and prevalence worldwide, Version 1.0 IARC Cancer Base no. 5. Lyon: IARC Press, 2001]. The therapeutic address requires a method having high sensitivity and capability to diagnose such disease at an early stage. We introduce a new diagnostic method based on non-linear methodologies. In detail we suggest that fractal as well as noise and chaos dynamics are the most important components responsible for genetic instability of melanocytes. As consequence we introduce the new technique of the variogram and of fractal analysis extended to the whole regions of interest of skin in order to obtain parameters able to identify the malignant lesion. In a preliminary analysis, satisfactory results are reached. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Cutaneous malignant melanoma (MM) is a tumor deriving from a lineage of pigmented cells called melanocytes, and its incidence has increased rapidly in the last five decades [1]. Also, MM responds poorly to currently available therapies, and is still responsible for more than 75% of all skin cancer deaths [2]. The exact discrimination between MM and other non-malignant skin pigmented lesions (called Ômelanocytic naeviÕ), and, even more significantly, the earliest detection of thin MMs, are of the outmost importance for surviving the disease. In current practice, diagnosis of MM is guided by the so called ÔABCD ruleÕ, based on evaluation of four morphological features of lesions, namely, Asymmetry, Border irregularity, Color variegation, and Diameter of more than 5 mm. The sensitivity of this clinical diagnosis of MM is only 65–80%. It does not recognize small MMs (less than 5 mm in diameter) and does not account for very early MMs that may have a regular shape and homogeneous color distribution [3]. In the last decades, the research activity has been prompted by the need to improve diagnostic accuracy. *
Corresponding author. E-mail address: [email protected] (M. Mastrolonardo).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.106
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Epiluminescence microscopy (ELM) [4] has entered daily clinical practice. ELM is an in vivo, non-invasive technique that has widened our possibilities of evaluating a number of morphological features normally invisible to the naked eye. It improves diagnostic accuracy of benign versus malignant skin pigmented lesions. Based on the possibility of analyzing structural elements normally hidden to the naked eye (such as, pigment network, black dots, globules, streaks/radial streaming, blue-white veils, pseudopods, pseudonetwork, horn cysts and structureless areas), a number of diagnostic algorithms have been developed to serve as guidelines to correct diagnosis, such as the Ôpattern analysisÕ [5,6], the ÔABCD rule of dermatoscopyÕ [7], the ÔMenziesÕ methodÕ [8], the Ô7-point check listÕ [9], and the modified ÔABC-point listÕ [10]. However, diagnostic accuracy of such analogical methodologies is to date far from the desirable, ultimate Ô100% thresholdÕ. The best diagnostic performances reported in the literature show in fact sensitivity rates invariably beyond 90% [3,11]. A limit of dermoscopy as well as of all the clinical diagnostic is that they are based on the human observation of the color and of shape and thus the evaluation is subjected to the experience and to the expertise of the individual clinician. In order to overcome such limits, computer-based dermatoscopic systems and electro-optical technologies have been developed in recent years. The aim has been to identify some common features and parameters in analysis of skin lesions, and the ultimate objective is to arrange a fully automated diagnostic device having the capability of diagnosing pigmented lesions without the intervention of an human expert [12]. Usually, clinical and dermoscopic analysis images of lesions are acquired for the analysis. Some groups use also the so called charged-coupled device, CCD. A calibrated three CCD video cameras enables to directly acquire dermoscopic images of pigmented lesions. Melafind and SIAscope [13] use instead an imaging acquisition technique of particular interest since they utilize multispectral narrow bands from 400 to 1000 nm to image the lesion, employing the physical process that light of different wavelength, penetrates the skin to different paths. The technique of the segmentation is utilized in computerized images to separate the lesion from the normal surrounding tissue. Segmented lesions that have the potentiality to be differentiated as benign respect to pigmented cutaneous lesions, are finally identified. The following stage may be called of the objective conversion. Mathematical procedures are used to characterize the lesion morphology as the asymmetry, the blotchiness, the color variation, and the dermoscopic structure. A process of conversion is performed in which the elaboration goes from the qualitative interpretation of dermoscopic structures and patterns to final quantitative parametric features that consequently are retained to be objective expression of the lesion. Finally, a classifier, based on a neural network, is employed. By incorporation of a selected number of features into such classifier, it becomes feasible to decide automatically whether the lesions should or should not biopsied. For a current update of such technique as well as of the other standard techniques, read the excellent papers given in Ref. [14]. The computerized image analysis systems are an interesting and promising field of research. In perspective, such automatized instruments of Ômachine visionÕ will play an important role as adjuvant to clinical and dermoscopic examinations. There are, however, three additional features that must be taken in consideration in the attempt to arrange a valuable method for diagnosis of melanoma. The introduced methods for computer analysis evidence [14] that the number of morphometric variables that must be used for computer analysis, and, generally speaking, the number of significant features to be considered with aim to realize a method discriminating between melanoma and non-melanoma, is very high. This basic feature induces the first important conclusion that, when introducing a new method for diagnosis of melanoma respect to non-melanomas, one must account for the high variability of data characterizing the lesion. The second important feature follows, in some sense, the present clinical tendency. It tends to attribute more importance to the investigation on the color of the lesion, attributing instead less consideration to the dimensions of the lesion itself. Consequently, a new method should account for accurate analysis of color and of variability of such data. Let us examine the third feature. Studying irregularities of a lesion border, various authors arrived to the result to characterize it by a fractal dimension within a multiscale method [15]. This seems to be a very important conclusion since it clears that a way to proceed along the line of an actual investigation and of an actual classification of lesions, must necessarily account for non-linear methodologies which include, in particular, the analysis of noise and chaos dynamics. This last conclusion seems to be strongly supported from two basic indications. The first one is that in melanoma lesions we have, first of all, an altered metabolism states of cancer cells. In addition, there is an important result that was evidenced rather recently from some authors [16]. They outlined that the irregularity of a lesion has a strong correlation with the genetic instability of melanocytes. It then follows that it is unthinkable to aim to analyze the dynamics of the advent of melanoma and to proceed to a lesion classification criteria on the basis of a method unable to make direct and robust use of the currently non-linear theories and methodologies here including in particular the analysis of noise and chaos dynamics. A method that simultaneously should account for analysis of color, for analysis of the high variability in the characteristic of the lesions, and of fractal nature of the border, not only, but possibly, of the whole examined region of interest, must necessarily use non-linear methodologies of investigation. To this purpose we arranged the present work. In the present paper we introduce a new technique that we have elaborated with regard to normal skin tissue, nevi and melanoma. We have called it the variogram analysis of melanocytes lesions. In the subsequent sections, we introduce the variogram analysis, and a proper mathematical model to discriminate the results that
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may be obtained analyzing images of normal skin tissue, nevi, and melanoma. In this second step of our analysis, we also introduce an approach based on the calculation of fractal measure and generalized fractal dimension no more confined to an investigation of the border of the region of interest (lesion) but to all the region itself. Finally, in the third part of the paper, we apply directly the introduced methodology to the analysis of about ten cases of melanomas and nevi.
2. Materials and methods 2.1. Patient selection The Unit of Dermatology at the Azienda Ospedaliero-Universitaria ÔOspedali RiunitiÕ di Foggia, South Italy, was involved in case selection. The patients were examined by one of us (MM), specializing in skin cancer, for the presence of pigmented lesions that were possibly malignant melanomas (MM). All suspect lesions underwent surgical excision subsequent to imaging, and hematoxylin-eosin-stained sections were used to establish diagnosis of MM versus non-melanoma pigmented lesions. 2.2. Lesion types Dermoscopic images were used to test our methodology. Six were MMs, and six non-malignant skin pigmented lesions. Images taken of normal (unaffected) skin from each patient were also submitted to analysis, as were 14 clinical (not dermascopic) images (seven MMs and seven non-malignant skin pigmented lesions) extracted from an image-database available in the Internet (http://www.dermis.net/doia/mainmenu.asp?zugr=d&lang=e). 2.3. Image acquisition The equipment for image acquisition, before surgical excision, consisted of the following components: • A trade digital camera, with the following technical characteristics: 1/1.8 Ôhigh-densityÕ charge-coupled device, CCD; total number of pixels, 3.34 million; range of image size, from 640 · 480 to 2.048 · 1.536. • A lens of a trade hand-held dermatoscope (10· magnification power), attached to the camera lens via a proper photo adapter. A drop of immersion oil was applied to the lesions under examination before applying the glass plate of the dermatoscope onto the skin surface. 2.4. Image processing Images of 180 · 150 pixels were selected as regions of interest (ROI) for analysis. ROI were then converted and digitized by a red/green/blue (RGB) conversion software using MatLab color converter. In detail, each colored map with no hairs was decomposed on the basis of the three fundamental colors (RGB analysis) and each map was analyzed realizing a grid of 180 · 150 pixels (1 6 x 6 180; 1 6 y 6 150) giving 27,000 data points and three basic matrices. In brief, assigning a proper coordinate frame to the map, three basic functions Zi(p) were obtained, being Zi(p) the function of the intensity for colors (i = red, green, and blue, respectively) and being p the pixel location with pair (x, y) of coordinates. A Variogram Analysis of Melanocytic Lesions was performed. Let us explain the essence of the method. The technique of the variogram has been extensively applied in geology and, in particular, in geochemistry studies [17]. Let us examine in detail as it was conceived for our studies. It is well known that the probability models of images are widely used and they are based on Markov random fields models where an image field is retained to be a collection of random variables denoting pixel values on an uniformly spaced grid in the image plan. The central concept of spatial statistics is represented from the regionalized variables z(x) with x 2 D Rd which are treated as realizations of an underlying random function Z(x). This random function Z(x) is called second-order stationary if it fulfills the following conditions: E½Zðx þ hÞ ¼ E½ðZðxÞÞ; cov½Zðx þ hÞ; ZðxÞ ¼ CðhÞ;
ð1Þ d
h2R .
ð2Þ
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This is to say that Z(x) has a constant mean value, E[Z(x)] = m, and its covariance function C(h) is translation invariant according to CðhÞ ¼ E½ZðxÞ Zðx þ hÞ m2 .
ð3Þ
A weaker form of stationarity may be also introduced. It is the intrinsic stationarity that is defined by the following conditions: E½Zðx þ hÞ ZðxÞ ¼ mðhÞ ¼ 0; var½Zðx þ hÞ ZðxÞ ¼ 2cðhÞ;
ð4Þ ð5Þ
1 cðhÞ ¼ E½ðZðx þ hÞ ZðxÞÞ2 . 2
ð6Þ
where
Eq. (6), takes the name of variogram [17]. This is precisely the statistical tool that we will use in the present analysis. In our method Z(x) will represent a second-order stationary random function.
Fig. 1. Normal skin tissue.
Fig. 2. Benign naevus.
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Fig. 3. Melanoma.
Fig. 4. Normal skin tissue. Behavior of Z(x) [see text and (1)] in blue color analysis, as given at each stage of the pixel elaboration.
Further, we will write that cðhÞ ¼ ChD ;
h > 0 and 0 < D < 2;
ð7Þ
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and we will call it the fractal variance function where the constant C will represent the fractal measure while D will represent the generalized fractal dimension [17].
3. Further comments on variogram analysis of skin lesions As previously said, we utilized the variogram to quantify the spatial correlation of a set of measurements of Zi(p), (color intensity or hue for i = red, green, blue) in the case of normal, benign and cancer lesions of skin tissue. In substance, we analyzed the variability of pairs of measurements of Zi in terms of a separation or lag distance. The so called semivariogram is defined in (6) where c(h), the experimental variogram, may be calculated at different separations or lag distances h. As previously said, Zi(p) is a second-order stationary random function whose value is given at the point p in the selected region of interest of the skin tissue. Zi(p + h) represents instead the value of such function at the distance h from p in the same region of interest. The E in (6) represents instead the expectation value of the computed expression. In conclusion, we analyze skin regions of interest where a small value of the variogram indicates that pairs of measurements of Zi(p) are similar for a particular separation distance h. This is to say that they have low variability. Instead, high values of the variogram indicate that the values of the measurements in the lag pairs are very dissimilar on the average, and this is to say that the corresponding data show high variability. We observe here that, introducing the method of variogram in analysis of skin lesions, based on the analysis of the three different colors (red, blue and green), we aim to have satisfied two of the three requirements that we fixed in the previous sections in order to realize a valuable method for diagnosis of melanoma. We mentioned, in fact, as basic requirements for an analysis of skin lesions, the analysis of the color and an examination of the high variability of the data in the characteristics of the lesions. Our method, based on variogram analysis of ROI converted to their basic colors red, blue and green, seems to satisfy such basic requirements since the variogram accounts specifically for the variability of data at each lag step. In order to characterize the essential features of our method, we may add here that, calculated the experimental variogram for each ROI of interest, the introduction of a proper mathematical model con-
Fig. 5. Benign naevus. Behavior of Z(x) in blue color analysis, as given at each stage of the pixel elaboration.
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nected to the calculated experimental variogram, enables us to express and to calculate by a fitting procedure some parameters that may be helpful to discriminate normal skin tissue from benign and cancer lesions. Finally, one has to consider the basic foundation of our approach under the profile of the previously mentioned non-linear features in skin lesions. In our approach, chaos and noise dynamic mechanisms in the distribution of the colors (intensity or hue for RGB) should have an essential role in the uneven distributions and in the measured variability of experimental data of Zi. Our aim is to evaluate the fractal structure of whole lesions and benign nevi with regard to their spatial distribution. Fractal structure should have a strong correlation with the altered metabolism states of cancer cells and with the genetic unstability of melanocytes [15,16]. Consequently, we may consider the equation given in (7) where, so written, the c(h) represents now the fractal variance function, that is in substance a power semivariogram [17]. In this manner, our elaboration of variogram analysis of skin lesions is reduced to a non-linear model in which the basic parameters C (fractal measure) and D (generalized fractal dimension) may be estimated in the cases of interest. In particular, such fractal parameters are able to control the variation of Zi. In brief, the greater the fractal parameters, the greater the variation of Zi should result and it should be correlated in some manner with the altered metabolism of the cancer cells and with the genetic instability of melanocytes.
4. Preliminary results of the analysis. The case of nevi exhibiting linear variogram In Figs. 1–3 we have, respectively, the images of a normal skin tissue, a benign nevi, and a melanoma from our experimental data. In Figs. 4–6 we report instead the values of Zi(p), i = R, G, B, as they were obtained by us step by step at each stage of the pixel elaboration by the previous mentioned MatLab color conversion software. For brevity, only the values of ZB(p) (blue color) are reported. Let us comment, first of all, the range of variability that was obtained for Zi(p), i = R, G, B, in some cases of normal tissue, benign nevi, and melanomas. For normal skin tissue, Zi(x, y) (i = B, R, G) gave the following results:
Mean Mean Mean Mean
value value value value
of of of of
minimum maximum mean variance
ZB(x, y)
ZR(x, y)
ZG(x, y)
146.00 ± 6.63 184.40 ± 8.26 169.47 ± 6.39 20.08 ± 4.59
159.20 ± 7.12 187.00 ± 5.78 174.50 ± 5.60 7.03 ± 1.46
154.00 ± 7.31 185.00 ± 8.27 172.62 ± 6.99 10.83 ± 3.10
Therefore Zi(x, y) showed a very restricted range of variation for colors blue, red, and green, respectively. In the case of the nevi, the results for Zi(x, y) changed profoundly:
Mean Mean Mean Mean
value value value value
of of of of
minimum maximum mean variance
ZB(x, y)
ZR(x, y)
ZG(x, y)
10.40 ± 3.71 90.40 ± 22.07 39.99 ± 12.91 173.14 ± 159.71
29.00 ± 23.76 134.25 ± 35.50 80.18 ± 47.17 308.82 ± 244.28
13.80 ± 4.20 91.80 ± 20.14 45.35 ± 13.61 173.61 ± 153.65
ZB(x, y)
ZR(x, y)
ZG(x, y)
31.66 ± 23.86 139.00 ± 25.51 90.24 ± 38.16 239.44 ± 50.55
124.33 ± 19.65 189.00 ± 15.09 166.59 ± 18.16 71.27 ± 39.80
48.66 ± 28.93 143.66 ± 24.41 106.13 ± 35.30 168.89 ± 72.69
Finally, we calculated the values for the melanomas:
Mean Mean Mean Mean
value value value value
of of of of
minimum maximum mean variance
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Fig. 6. Melanoma. Behavior of Z(x) in blue color analysis, as given at each stage of the pixel elaboration.
As a preliminary indication we may conclude that a very large range of variability was obtained for the data in the case of nevi as well as of melanomas respect to normal skin tissues. We had no significant differences between the two kinds of lesions. Also the calculated mean values with their standard deviations did not result often to have significant differences. However, it must be outlined here that, in spite of such no conclusive statistical results of the data, profound differences could be exhibited from the data when analyzed under the profile of their spatial distribution, and this was, in substance, the reason because we introduced the method of the variogram for such kind of analysis. Therefore, let us look to the results that we obtained calculating the variograms. Each variogram was computed using (6) with a number of lags of 80, and a lag tolerance of 10%. Generally speaking, one expects that there are infinitely many possible behaviors for the obtained experimental variograms, ci(h), i = R, G, B. However, in our preliminary analysis, we accounted for three possible models, that were also identified in other studies [17], the linear, the exponential, and spherical models. Examples of these three possible behaviors are given in Fig. 7(a). In detail, we considered that, if the experimental function ci(h) never levels out, then the use of a linear model should be appropriate. If, otherwise, ci(h) levels out but it is curving all the way, the use of an exponential model should be made. Finally, if ci(h) starts as a straight line and then levels out, the use of a spherical model should result to be more appropriate. It is easily seen that, in such cases, some parameters of interest may be also connected to the considered mathematical models. If the experimental ci(h) shows to have a non-zero intercept on the vertical axis, according to [17], we may consider it what is called a Nugget Effect Component. One may calculate it as a parameter of interest. The second parameter one may use is the Scale: the Scale will be the height on the vertical axis at which the function ci(h) levels off. Finally, the Length will represent the lag distance at which the function ci(h) levels off. We retain that a preliminary use of these parameters [17], should enable to express a satisfactory parametric characterization of the variogram approach that we used. Each time a fitting program of the experimental data of ci(h), based on the least squares method, will be able to select the best model to use (linear, exponential or spherical model), giving each time the values of the calculated parameters (intercept component value), slope (linear case), scale and length values (exponential or spherical models). The results of the experimental variograms that we calculated, are
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Fig. 7. Variogram behavior in normal skin tissue, naevus and melanoma.
reported in Figs. 8–10 for normal skin tissue in one subject. In Figs. 11–13 we have the results of the experimental variograms in a subject with a benign nevi, and, finally, in Figs. 14–16 we report the results of experimental variograms for a melanoma. Also by a simple visual inspection, the first emerging conclusion is that, for nevi exhibiting a linear variogram, we have introduced a very interesting method that is able to discriminate with 100% of sensitivity and specificity a diagnosis of melanoma respect to benign nevi and respect to normal skin tissue. Calculating the experimental variograms, and by simple visual inspection and examination, we are in the condition to differentiate normal skin tissue respect to benign
Fig. 8. Normal skin tissue (ordinate: variogram value; abscissa: max distance)—blue.
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Fig. 9. Normal skin tissue (ordinate: variogram value; abscissa: max distance)—red.
Fig. 10. Normal skin tissue (ordinate: variogram value; abscissa: max distance)—green.
Fig. 11. Benign naevus (ordinate: variogram value; abscissa: max distance)—blue.
M. Mastrolonardo et al. / Chaos, Solitons and Fractals 28 (2006) 1119–1135
Fig. 12. Benign naevus (ordinate: variogram value; abscissa: max distance)—red.
Fig. 13. Benign naevus (ordinate: variogram value; abscissa: max distance)—green.
Fig. 14. Melanoma (ordinate: variogram value; abscissa: max distance)—blue.
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Fig. 15. Melanoma (ordinate: variogram value; abscissa: max distance)—red.
Fig. 16. Melanoma (ordinate: variogram value; abscissa: max distance)—green.
nevi and melanomas. In fact, in the case of normal skin tissue it is seen that the mathematical model of the variogram is represented by an exponential model of the kind cðhÞ ¼ C 0 þ C 1 ð1 qÞ with q ¼ expðlhÞ
ð8Þ
with low values for the variogram and for such parameters. Instead, in the case of benign nevi, we have that the experimental variogram follows a rather robust linear behavior with higher values of the variogram respect to the case of the normal skin tissue. Finally, the experimental variogram connected to melanomas again exhibits an exponential behavior with values of the variogram and of the connected parameters, that result to be very different respect to the case of normal skin tissue. Fig. 7(b) gives us a direct indication of the status of matter picturing the results of our analysis. To conclude this first part of our analysis, we have only to specify that all the expressed values of the parameters that we calculated by fitting using the very simple mathematical models of variograms that we introduced in Fig. 7(a), furnished optimal numerical indexes of control for the fitting mathematical procedure giving, in particular, a very low sum of residuals (SRE) and an R2 ffi 1. In all the cases as weighting factors the number of pairs/standard deviation were used. The results for the parameters are reported in Table 1 and, obviously, they result very discriminating. Note that our expert clinician expressed doubt of about 70% on the visually examined cases of nevi and melanoma. In conclusion, he had an high level of uncertainty. Our method instead discriminated in this case with 100% of sensitivity and 100% of specificity owing to the basic differences exhibited from the variograms in the two cases of interest of melanoma versus nevi.
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Table 1 Subject
Blue Nugget
Normal skin tissue Sta-pol 6.96 Mit.-pol 4.21 Pac. 0 m.v. 3.72 s.d. 3.51 Naevus DÕAr. Mit. Les. m.v. s.d. Melanoma Lor4. Lor6. Lor7. m.v. s.d.
0 18 7 8.33 9.07 0 5 5 3.33 2.89
Red Scale
Length
Nugget
17.38 23.79 13.26 18.14 5.31
9.49 65.2 28.59 34.43 28.31
0.27 4.53 0 1.6 2.54
0.43 3.23 0.63 1.43 1.56
110 580 80 256.67 280.42 33.83 48 30 37.28 9.48
322.9 490 520 444.3 106.2
Green Scale
Length
Nugget
7.53 3.93 5.66 5.71 1.8
3.66 30.46 5.78 13.3 14.9
2.47 1.17 3 2.21 0.94
8.65 11.61 18 12.75 4.78
19.35 8024 80 59.86 35.09
0 18.1 3 7.03 9.7
5.01 4.35 6,9 5.42 1.32
900 530 800 743.33 191.4
0 3.87 5 2.96 2.62
1.15 5.02 1.4 2.52 2.17
210 580 160 316.67 229.42
12 0 5 5.67 6.03
115 167 250 177.33 68.09
45 52 40 45.67 6.03
20 6 2 9.33 9.45
260 340 400 333.33 70.24
36 30 26 30.67 5.03
Fig. 17. Variogram of naevus.
Fig. 18. Variogram of melanoma.
Scale
Length
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Table 2 Blue melanoma
Blue naevus
C0
I-Lor4 I-Lor6 I-Lor7 m.v. s.d. II-Uma1 II-Uma2 m.v. s.d.
I-Mit I-Les. I-DÕAr
0.00 6.72 0.40 2.37 3.77 15.00 13.00 14.00 1.41
II-Lop II-Nev
C1
2.39 4.64 8.75 5.26 3.23 0.00 0.66 0.3300 0.4667
323.40 410.60 530.70 421.57 104.08 890.40 903.40 896.90 9.19
A1
9610.70 1186.60 8000.00 6265.77 4471.80 325.80 485.50 405.65 112.92
24.08 36.22 34.43 31.58 6.55 32.00 35.00 33.50 2.12
Fractal measure
a
2654.20 3000.00 3575.30 3076.50 465.29 216.40 110.00 163.20 75.24
1.05 1.75 1.39 1.40 0.35 0.28 0.22 0.25 0.04
0.89 0.85 1.28 1.01 0.24 0.51 0.75 0.63 0.17
76.48 42.66 35.92 51.69 21.73 356.15 404.00 380.08 33.84
Generalized fractal dimension 7.29 2.16 0.66 3.37 3.48 24.33 30.63 27.48 4.45
0.30 0.49 0.61 0.47 0.16 0.13 0.10 0.12 0.02
0.92 0.75 1.06 0.91 0.16 0.40 0.47 0.44 0.05
Statistical analysis: (I Mel.–I Naev.) significant differences for parameters: A1 (p = 0.003), FM (p = 0.019), GFD (p = 0.025). (II Mel.–II Naev.) significant differences for parameters: C0 (p = 0.006), C1 (p = 0.025), FM (p = 0.004), GFD (p = 0.013). (I Mel.–II Naev.) significant differences for parameters: A1 (p = 0.046). (II Mel.–I Naev.) significant differences for parameters: C0 (p = 0.04), A1 (p = 0.003), a (p = 0.024), FM (p = 0.0002), GFD (p = 0.0064). Red melanoma
Red naevus
C0
I-Lor4 I-Lor6 I-Lor7 m.v. s.d. II-Uma1 II-Uma2 m.v. s.d.
I-Mit I-Les. I-DÕAr
0.00 5.48 0.00 1.83 3.16 10.00 22.00 16.00 8.49
II-Lop II-Nev
C1
0.50 0.15 6.87 2.51 3.78 0 0 0.00 0.00
123.70 142.90 239.20 168.60 61.89 516.20 590.00 553.10 52.18
A1
1413.70 1622.20 2768.60 1934.83 729.55 156.5 90.18 123.34 46.90
27.08 27.37 34.83 29.76 4.39 30.00 39.00 34.50 6.36
Fractal measure
a
300.00 160.80 223.10 227.97 69.73 187.3 239 213.15 36.56
0.92 1.97 1.91 1.60 0.59 0.74 0.83 0.79 0.06
0.91 1.16 1.44 1.17 0.27 0.61 0.43 0.52 0.13
22.44 28.82 12.80 21.35 8.07 76.55 83.97 80.26 5.25
Generalized fractal dimension 6.04 8.69 2.39 5.71 3.16 8.9 10.59 9.75 1.20
0.36 0.35 0.66 0.46 0.18 0.42 0.41 0.42 0.01
0.94 0.95 1.24 1.04 0.17 0.46 0.3 0.38 0.11
Statistical analysis: (I Mel.–I Naev.) significant differences for parameters: C1 (p = 0.013), A1 (p = 0.008), FM (p = 0.035), GFD (p = 0.014). (II Mel.–II Naev.) significant differences for parameters: C1 (p = 0.013), A1 (p = 0.02), FM (p = 0.0029). (I Mel.–II Naev.) significant differences for parameters: A1 (p = 0.0026). (II Mel.–I Naev.) significant differences for parameters: A1 (p = 0.034), FM (p = 0.0003), GFD (p = 0.016). Green melanoma
Green naevus
C0
I-Lor4 I-Lor6 I-Lor7 m.v. s.d. II-Uma1 II-Uma2 m.v. s.d.
I-Mit I-Les. I-DÕAr
0.00 5.21 0.00 1.74 3.01 17.00 20.00 18.50 2.12
II-Lop II-Nev
C1
7.98 2.61 5.90 5.50 2.71 0.00 0.01 0.00 0.0035
273.00 315.00 397.20 328.40 63.17 714.00 739.40 726.70 17.96
A1
811.90 387.30 590.80 596.67 212.36 345.40 448.00 396.70 72.55
29.70 34.30 29.91 31.30 2.60 31.00 33.00 32.00 1.41
Fractal measure
a
119.00 209.90 210.00 179.63 52.51 433.90 400.00 416.95 23.97
1.01 1.96 1.53 1.50 0.48 0.57 0.48 0.53 0.06
1.20 1.07 1.50 1.26 0.22 0.59 0.64 0.62 0.04
45.08 31.07 27.54 34.56 9.28 157.69 198.21 177.95 28.65
Generalized fractal dimension 2.53 5.03 0.47 2.68 2.28 12.57 16.14 14.36 2.52
0.38 0.51 0.62 0.50 0.12 0.31 0.25 0.28 0.04
0.87 1.00 1.27 1.05 0.20 0.48 0.47 0.48 0.01
Statistical analysis: (I Mel.–I Naev.) significant differences for parameters: A1 (p = 0.008), FM (p = 0.004), GFD (p = 0.016). (II Mel.–II Naev.) significant differences for parameters: C0 (p = 0.006), C1 (p = 0.024), A1 (p = 0.0019), FM (p = 0.015), GFD (p = 0.02). (I Mel.–II Naev.) significant differences for parameters: A1 (p = 0.0001). (II Mel.–I Naev.) significant differences for parameters: C0 (p = 0.011), A1 (p = 0.032), a (p = 0.022), FM (p = 0.0014), GFD (p = 0.015).
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5. The complete variogram analysis of the data Not all the examined nevi exhibited linear variogram. There are cases in which their behavior tends to manifest an exponential behavior. In Figs. 17 and 18 we report the case of variograms of a naevus and of a melanoma both exhibiting non-linear behaviors. The mathematical model to be used in these cases is the following:
Table 3 Blue melanoma
Blue naevus
C0
M16 M11 M12 M14 M15 M17
N17 N11 m.v. s.d. N12 N13 Nc
70.00 80.00 60.00 110.00 130.00 30.00
m.v. s.d.
80.00 35.78
C1
10.00 34.00 22.00 16.97 21.00 0.00 0.00 7.00 12.12
745.70 1413.50 1005.60 1177.20 1070.30 1105.10 1086.23 218.38
A1
8947.60 2500.00 5723.80 4559.14 578.00 196.00 276.20 350.07 201.43
40.00 83.22 80.00 82.00 80.32 40.00 67.59 21.40
Fractal measure
a
3826.00 158.20 1992.10 2593.53 140.00 9.05 500.00 216.35 254.22
0.51 0.83 0.81 0.77 0.17 0.68 0.63 0.25
0.74 0.92 0.83 0.13 0.80 0.21 0.23 0.41 0.34
159.72 128.74 120.00 125.72 501.85 147.54 197.26 149.96
Generalized fractal dimension 12.91 47.39 30.15 24.38 24.68 105.36 77.95 69.33 41.02
0.33 0.44 0.48 0.43 0.11 0.42 0.37 0.14
0.85 0.71 0.78 0.10 0.41 0.08 0.20 0.23 0.17
Statistical analysis: (Mel.–I Naev.) significant differences for parameters: C1 (p = 0.008), GFD (p = 0.0083). (Mel.–II Naev.) significant differences for parameters: C0 (p = 0.012), C1 (p = 0.0018). Red melanoma
Red naevus
C0
M16 M11 M12 M14 M15 M17
N17 N11 m.v. s.d. N12 N13 Nc
40.00 152.50 20.00 289.00 120.00 70.00
m.v. s.d.
115.25 98.37
C1
7.00 50.00 28.50 30.41 20.00 0.00 0.00 6.67 11.55
1105.00 1389.70 929.80 2068.10 1605.60 1682.00 1463.37 412.86
A1
2800.00 1297.00 2048.50 1062.78 578.50 177.60 228.30 328.13 218.30
50.00 66.03 22.03 69.00 65.83 90.00 60.48 22.77
Fractal measure
a
163.40 170.00 166.70 4.67 140.00 3.98 5.18 49.72 78.19
0.95 1.22 1.40 1.63 0.67 0.93 1.13 0.35
1.53 1.12 1.33 0.29 0.80 0.32 0.48 0.53 0.24
49.28 94.94 164.08 42.57 252.66 55.04 109.76 83.38
Generalized fractal dimension 0.84 27.52 14.18 18.87 16.18 108.63 111.12 78.64 54.11
0.69 0.56 0.41 0.85 0.35 0.68 0.59 0.19
1.58 0.64 1.11 0.66 0.68 0.09 0.17 0.31 0.32
Statistical analysis: (Mel.–I Naev.) significant differences for parameters: A1 (p = 0.0008). (Mel.–II Naev.) significant differences for parameters: C1 (p = 0.0033), a (p = 0.034). Green melanoma
Green naevus
C0
M16 M11 M12 M14 M15 M17
N17 N11 m.v. s.d. N12 N13 Nc
70.00 60.00 50.00 152.00 170.00 50.00
m.v. s.d.
92.00 54.26
C1
8.00 40.00 24.00 22.63 20.00 0.00 0.00 6.67 11.55
882.20 1082.00 991.00 1576.00 1237.00 1413.20 1196.90 263.34
A1
2352.00 2344.50 2348.25 5.30 400.00 169.90 230.20 266.70 119.31
70.00 48.78 42.30 81.00 40.00 60.00 57.01 16.29
Fractal measure
a
280.00 200.60 240.30 56.14 337.80 2.81 9.34 116.65 191.55
0.58 0.90 1.03 1.05 0.85 0.74 0.86 0.18
1.03 0.97 1.00 0.04 0.59 0.34 0.27 0.40 0.17
114.03 149.51 70.41 70.04 406.24 127.53 156.29 126.45
Statistical analysis: (Mel.–I Naev.) significant differences for parameters: C1 (p = 0.0011), A1 (p = 0.0002), GFD (p = 0.035). (Mel.–II Naev.) significant differences for parameters: C0 (p = 0.035), C1 (p = 0.0007), a (p = 0.0078).
Generalized fractal dimension 4.49 38.59 21.54 24.11 26.62 110.38 95.08 77.36 44.60
0.41 0.40 0.59 0.64 0.12 0.49 0.44 0.18
1.12 0.69 0.91 0.30 0.41 0.08 0.17 0.22 0.17
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a h cðhÞ ¼ C 0 þ C 1 ð1 qÞ with q ¼ exp . A1
ð9Þ
The fitting procedure determines the parameters C0, C1, A1, a that are sufficient to discriminate melanomas respect to nevi. The results are reported in Tables 2 and 3 for all the cases that we examined. In such tables we have also the results of fractal analysis that we performed. For each case the values of the fractal measure and of generalized fractal dimension were estimated. In our opinion such last results represent the most relevant side of the present paper. For the first time we show that RGB color maps of skin of normal tissue or of nevi or of melanomas exhibit a fractal distribution in their entirety. We find also that in such tissular regions two basic parameters as fractal measure and generalized fractal dimension are able to characterize and to delineate the basic features of such tissular regions itself. It seems that it is confirmed the basic motivation that aimed the present work. The basic features to understand the melanocytic dynamics and to differentiate diagnosis between nevi and melanoma must be reached by application of non-linear methodologies, fractal, noise and chaos dynamics. As obtained in the present paper, it is not the matter to explore fractal dimension in relation to the border of a melanoma lesion but to investigate in detail fractality and thus chaos and noise dynamics to the whole level of tissular regions and, in particular, to the whole region characterizing the lesion itself. The added statistical analysis confirms that variogram analysis with added investigation of fractal dynamics of the whole examined skin regions of interest, enabled us to discriminate benign lesions as nevi respect to melanocytic lesions as melanomas in a satisfactory manner also in the framework of a preliminary analysis.
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