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Multifractal characterization of healing process after bone loss
Biomedical Signal Processing and Control 52 (2019) 179–186
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Multifractal characterization of healing process after bone loss a ˛ Marta Borowska a,∗ , Ewelina Bebas , Janusz Szarmach b , Edward Oczeretko a a Bialystok University of Technology, Faculty of Mechanical Engineering, Department of Materials and Biomedical Engineering, Wiejska 45C, 15-351, Białystok, Poland b Medical University of Bialystok, Department of Oral Surgery, M. Curie-Skłodowskiej 24A, 15-276, Białystok, Poland
a r t i c l e
i n f o
Article history: Received 6 August 2018 Received in revised form 12 March 2019 Accepted 13 April 2019 Available online 24 April 2019 Keywords: Multifractal analysis Fractal dimension Guided bone regeneration Xenografts Radiographic images
a b s t r a c t Objective: In this work we have proposed to use multifractal analysis to evaluate the effectiveness of the healing process in postresectal and postcystal bone loss using the guided bone regeneration (GBR). Methods: The material of the study consists of 19 radiographic images obtained from patients (13 females and 6 males), observed within 1-year-long period, who had undergone bone augmentation with xenogenic material. Using radiographic images (RVG) made with digital radiography set Kodak RVG 6100 on the day of the procedure (Group A) and 12 months later (Group B), the degree of reconstruction of intraosseous defects were compared. A visual subjective evaluation of the healing process based on RVG images may not be sufficient. For this purpose, modern computer image analysis methods can be used. Pre-processing of radiographic images consisted of separating areas of interest (ROI) and binarization. The resulting images were analyzed using multifractals. The properties of biological structures cannot be sufficiently described in the Euclidean space, but they can be characterized by a parameter called the fractal dimension. Results: Significant increase (p < 0.0001) of multifractal features were found in patients from group B. Individual parameters may characterize bone structure, which had undergone bone augmentation with xenogenic material. Finally, we showed that the properly selected multifractal quantificators obtained for those images could be used to assign those images to the appropriate group. Conclusion: The proposed analysis based on RVG images could help to evaluate the healing process after bone loss. The analysis and interpretation of the healing process after bone loss using monofractal and multifractal analysis as a multi-steps process quantify a stable (good) clinical result. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The healing process of the boneless alveolar areas after resection or cyst removal is a rare and complicated reparative operation that leads to complete fill-up of the empty space with new properly build bone tissue. The most common finding is loss of the bone volume, and regeneration seen as a main healing action. Frequently, bone loss exceeds it’s physiological limits, resulting in bone missing areas. Tissue engineering is a helpful tool facilitating reconstruction process of the alveolar bone shape and function. The basic materials used for bone regeneration are osseous materials, replacing bone specimens and barrier membranes. Therefore, in certain cases, the application of guided bone regeneration technique (GBR) with biomaterials and barrier membranes is justified. These materials
∗ Corresponding author. E-mail address: [email protected] (M. Borowska). https://doi.org/10.1016/j.bspc.2019.04.014 1746-8094/© 2019 Elsevier Ltd. All rights reserved.
enable proliferation of the bone tissue to the inner and outer layer of the osseous gap due to its osteoconductive properties. A visual subjective evaluation of the healing process based on RVG images may not be sufficient. For this purpose, modern computer image analysis methods can be used. These methods allow to determine the differences between healthy and diseased tissue, and hence, it s possible to assess the healing process quantitatively and qualitatively. Various studies on the texture of dental images concern the use of methods based on the statistical properties of the surfaces. Through their use, it is possible to detect the areas covered by caries more accurately [1]. If used together with other methods it allows for better recognition of osteoporosis in women [2], finding texture boundaries of the cysts region [3], identifying asymptomatic osteoporotic and osteopenic patients through dental examinations [4], automatizing the segmentation of lower jaw and mandible on panoramic x-rays [5]. It can also be used as a tool for dental implant treatment planning [6]. Modern methods of fractal and multifractal analysis are increasingly used nowadays [7–12]. The analysis of the fractal dimension has been used in the assessment of alveolar bone defects treated
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with various bone substitute materials [13]. The fractal dimensions of individual implants were determined on the basis of X-ray image analysis. The results show that the fractal dimension differed depending on the biomaterial used and reflected the individual nature of bone remodeling during observation. In Pârvu [14], it has been shown that fractal and multifractal analysis of tissue images as a non-invasive technique can be used to measure contrasting morphological changes in human gingival cells and tissues and can provide detailed information for the study of healthy and diseased gums in patients with root disease and generalized chronic periodontitis GCP. In our previous publications [15–17] we assessed the effectiveness of the treatment process of postresectional and postcystal bone loss using the GBR method on the basis of fractal dimensions. Significant changes were determined in the calculated fractal dimensions from RVG images taken immediately after the procedure and 12 months after the surgery. The box and information dimensions were calculated, showing significant differences, on the basis of which the healing process could be positively assessed. We showed fractal characteristics of images and we also wanted to point to multifractal characteristics, which we presented in this work. The tendency of changes in fractal dimensions was the same as in work [17] in which the calculations were performed based on binarization. In publication [15], the fractal dimension was also calculated and the method of calculation was based on differences in the intensity of pixels. Multifractal analysis shows the possibility of predicting an individual patient’s response to neoadjuvant cytotoxic therapy in case of osteosarcoma (over 70% compliance for the Dqmax parameter) based on magnetic resonance images thanks to which, in the future, the implementation of alternative treatment options directed at a specific patient [18] may be allowed. In Gao [19] multifractal spectrum features of micro-CT images in combination with the C4.5 classifier allows for early diagnosis of osteoporosis with accuracy greater than the methods used so far. Multifractal methods are often used for segmentation of anatomical structures or diseased areas [20,21]. The multifractal analysis gave better results than fractal analysis when identifying the primary cancer, in the case of metastatic bone disease diagnosis based on medical images. It also showed a high probability of correct diagnosis [22]. In our study we did not do any research to show bone mineral density (it is impossible to determine it from RVG images), however in the article [22] multifractal analysis was used to predict vertebral damage in CT images. RVG images were useful for specifying parameters that make it possible to describe the bone healing process. There are not many reports on the use of the fractal dimension and multifractal behavior in the evaluation of the healing process after bone augmentation with xenogenic material. Information about such studies can be found in the assessment of bone after the implant insertion. Abdulhameed et al. [23] used fractal dimension in predicting implant stability from intraoral periapical radiographs. Their results confirmed the increase in the fractal dimension with the increase of time after surgery. Fractal analysis was also used for the assessment of bone quantity around the implant [24]. The aim of this paper is to evaluate the effectiveness of the healing process in postresectal and postcystal bone loss using the guided bone regeneration (GBR) based on a multifractal analysis of RVG images.
Medical University in Bialystok from year 2012–2014. The presence of root cysts was examined on the bases of clinical symptoms and radiological images performed. Bone loss in the frontal maxillary and mandibular region among all 19 patients was classified as Class II according to Dietrich and ass. [25]. A few conditions such as root canal treatment of the teeth qualified for the resection and cystectomy; sanitation of the oral cavity, professional hygienisation, were required for acceptance to the surgery. Generally healthy patients, with no medical conditions impairing bone healing, were included in the research. All the procedures were performed by the same operator under local anesthesia with prophylactic antibiotic coverage in a classic surgery protocol manner [26]. Postresectal bone loss was filled with xenograft BioOss® material (spongeous bone granulate grading 0.25–1 mm) and covered with resorbable barrier membrane BioGide® of pork origin. The membrane itself consists of highly purified collagen type I and III, with no organic and/or chemical components. Implanted BioOss® material is a natural specimen of deproteinated beef bone. It shows resemblance to human bone in morphology, crystal structure, chemical composition and internal structure. It’s porosity ranging 60% is similar to that of the human bone. Post-surgery wound was stabilized with single suture knot using synthetic 5-0 thick sutures (Fig. 1). To assess the treatment results, several intraoral radiological images were performed in the straight angle technique, immediately after the procedure (Group A) and 12 months later (Group B). The equipment used for the research was KODAK RVG 6100 set with the real resolution over 14 pl/mm, and a collimator narrowing the radiation beam, with the constant exposure time of 0.08 s; all data were saved as MPG files. During convalescence period, analgesics, 0.12% chlorhexidine gluconate mouth rinse and cold compresses were a part of the therapy. 2.2. Fractal and multifractal analysis Fractal and multifractal methodology is widely applied in many areas of biosciences both in the analysis of medical signals and images [10,27–29]. They can provide valuable information on the geometrical complexity of biological structures [27]. Fractals are geometric structures with specific metric properties such as length, area or volume represented at multiple scales. Benoit Mandelbrot [30] proposed definition of the fractal as a set, which can be described by a parameter called the fractal (Hausdorff) dimension (DH ) with a non-integer value greater than its topological dimension [31]. If a given fractal is embedded in a n dimensional Euclidean space with a topological dimension, then it goes to fill a subset of the whole space. This dimension determines the self-similarity at different scales of the fractal structure. Many methods of fractal dimension calculation are used in the images analysis. The so-called box-counting method is the most commonly used method of estimating the fractal dimension. The geometrical object in the n− dimensional Euclidean space with length L is covered by the boxes with a size r ≤ L. The relationship between fractal dimension DH and number of boxes N(r) covering object with a size r is governed by a power-law: N(r)˜r −DH
Logarithms on both sides of the above relation, determine fractal dimension according to the formula: DH = lim
2. Materials and methods 2.1. Data collection The treatment results of 19 patients (13 females, 6 males; average age of 356) were assessed in Oral Surgery Department of
(1)
r→0
lnN(r) ln 1r
(2)
The next important step in the development of fractal analysis was the introduction of the multifractal concept [32]. Multifractal is an object that combines many fractal subsets with different scaling exponents, each with a different fractal dimension [33]. The description of the internal structure of the multifractal is based on
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Fig. 1. Stages of surgery: a) clinical condition before surgery, b) a dissected periosteal flap with visible external plaque of the alveolar process, c) enucleation of the cyst, d) resection of the apex of the tooth root, e) bone loss after enucleation of the cyst, f) preparation of a xenogenic implant (BioOss) with the patient’s blood, g) a bone defect filled with xenogeneous material, h) coverage of the area augmented by the barrier membrane (BioGide), i) sewing the post-treatment wound with knot stitching.
the consideration of the whole spectrum of dimensions, called the spectrum of generalized dimensions Dq [34]. These spectrum of dimension Dq is shown in the graph as Dq in the function of q or f (˛) in the function ˛. Consider the two dimensional object in n − dimensional Euclidean space with length L covered by boxes of size r ≤ L. The mass probability in the i − th box is given by:
Pi (r) =
Ni (r) N
(3)
where N is the total mass of the object and Ni (r) is the number of pixels in the i − th box. Dependence is quantified for multifractal measure (scaling) of Pi with i − th box size r as:
¨ where ∝i is the Lipschitz-Holder exponent [34]. The number of ¨ boxes N(∝), where the probability Pi (r) has Lipschitz-Holder exponent ∝ is found to scale as: N(∝)˜r f (∝)
where f (∝) provide the fractal dimension of subset with exponent ∝ [34,35]. In order to obtain f (∝) the scaling of the q-th moments of Pi (r) distributions can be used in the following way: ni (r)
(4)
q
Pi (r)˜r q
(6)
i
where q = (q − 1)Dq is the mass exponent of order q. By Legendre transform, the variables f (∝q ) and ∝q are related to q : ∝q =
Pi (r)˜r ˛i
(5)
dq 1 = lim i lnPi (r) dq r→0 lnr i
(7)
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Fig. 2. Description of parameters obtained from multifractal analysis a) example of f (q) and ∝q functions obtained in the range of q values, b) f (∝) - spectra. Structure acronyms are: fractal dimension (D0 ), information dimension (D1 ), fractal spectrum (f (q)), scaling factor (˛ (q)), exponent estimated in the range of q from -1 to 1 (˛q ), the value of minimum exponent ˛ (˛min ), the value of maximum exponent ˛ (˛max ), width of fractal spectrum (˛), the value of the fractal spectrum for minimum exponent ˛ (f (˛min )), the value of the fractal spectrum for maximum exponent ˛ (f (˛max )), range of fractal spectrum (f (˛)), the minimum value of the fractal spectrum for exponent ˛ (f (˛q )min ), the maximum value of the fractal spectrum for exponent ˛ (f (˛q )max ), height of fractal spectrum (f (˛q )).
1 i lni r→0 lnr
f ∝q = q∝q − q = lim q Pi /
(8)
i
q Pi ;
where i = q is the weight; ∝q is the exponent; f (∝q ) represents multifractal spectrum function; q is the mass exponent. Multifractal spectrum is the plot of f (∝q ) vs. ∝q . The generalized entropy Hq (r) and generalized dimension Dq are defined as: Ni (r)
ln
q
Pi
i=1
Hq (r) =
(9)
1−q
Dq = −lim
r→0
Hq (r) q 1 = = [q∝q − f (∝q )] q−1 q−1 lnr
(10)
When q = 0, f (∝q=0 ) represents fractal dimension D0 previously named DH . For q = 1, f (∝q=1 ) represents information dimension (entropy dimension) D1 defined by formula: Ni (r)
ln
Pi (r)lnPi (r)
i=1
D1 = lim
lnr
r→0
(11)
The information dimension D1 is related to the entropy measure of Shannon [36], which gives decrease in information with an increase of the size of the box [37]. For q = 2, f (∝q=2 ) represents correlation dimension D2 defined by formula: Ni (r)
ln D2 = lim
r→0
Pi2 (r)
i=1
lnr
(12)
For q = ±∞, f (∝q=±∞ ) represents D∞ and D−∞ , which are corresponding to the values ∝max and ∝min , respectively. For monofractals, ∝= ∝max − ∝min should be zero. Fig. 2 shows some of the features obtained from multifractal spectrum plot. 3. Results Fig. 3 shows the scheme of RVG image analysis. In the first step, the regions of interest (ROI) were selected for the analysis, in the images taken immediately after the procedure and one year after the surgery. The areas were 128 × 128 pixels each. The extracted fragments of the images were binarized to make the analysis possible (Fig. 3e,f). It is important that all analyzed images were binarized
using the same threshold. It allowed for reliable comparative analysis of the parameters obtained. The optimal binarization thresholds were obtained using the Otsu method [38] implementing MATLAB software: the binarization threshold values were in the range from 79 to 176, and the average threshold value of the determined range was 127.5. Next, multifractal spectrum of a binary image was used to carry out the multifractal analysis. Multifractal analysis was done in Matlab for Windows (MathWorks, Inc., USA), a high-performance language for technical computing. As a result of the analysis, two functions were obtained: the value of the fractal spectrum f (˛) and the value of the scaling exponent ˛ estimated in the range of q from −1 to 1. Additionally, a graph of the fractal spectrum f (˛) was plotted and a graph of theoretical values was fitted to it. Fig. 4 presents the multifractal spectrum f (˛) and the value of the singularity coefficient ˛ depending on the factor q for a patient from group A and group B of the image ROI, respectively. There is a noticeable difference in the distributions of the functions f (q) and ˛(q) in the case of images taken on the day of the surgery and performed one year after the surgery. The graphs depicting the dependence of the fractal spectrum f (q) and the scaling factor ˛ (q) on the coefficient q for each of the images, increase their values by comparing the results obtained for images for patients from group A and group B. In addition, the information dimension D1 , which corresponds to the value f (q) for q = 1 and the fractal dimension D0 in the point q = 0, which is also the maximum of function f , can be read from the obtained results. The results of multifractals analysis are shown in Table 1. Statistical analysis was performed by means of two-sample t-test. Taking the calculated parameters into account it can be concluded that the values of information and fractal dimensions increase after one year after the surgery. The average post-surgery fractal dimension was 1.061, while after one year after the surgery it increased to 1.781. In the case of the information dimension, its mean value after the surgery was 1.045, increasing to 1.7692 one year after it. Statistical analysis of the results allows for a much larger discrepancy in the results obtained for images taken immediately after the procedure than for those performed one year after the surgery. This means that as the healing process progresses, the fractal dimension of the image being examined stabilizes at one level and amounts to about 1.77. The change in the shape of the multifractal spectrum for images taken from group B is clearly visible compared to the spectrum obtained for images taken from group A. The spectrum loses the shape of the bell curve, by shortening its left side, which is caused by
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Fig. 3. The scheme of multifractal analysis: a) post-surgical image with the region of interest, b) post-surgical image with the region of interest one year after, c) grey region of interest after the surgery, d) grey region of interest one year after the surgery, e) binarized regions of interest after the surgery, f) binarized regions of interest one year after the surgery, g) multifractal spectrum f (˛) and the volume of the singularity coefficient ˛ depending on the factor q, for the post-surgical image, h) multifractal spectrum f (˛) and the volume of the singularity coefficient ˛ depending on the factor q, for post-surgical image one year later, i) multifractal spectrum for the image after surgery, j) multifractal spectrum for the image one year after the surgery.
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Fig. 4. Example of f (q) and ∝q function estimated in range of q values for a post-surgery image (a) and one year after the surgery image (b). Multifractal post-surgery spectra f (∝q ) (c) and one year after the surgery (d). Multifractal spectra f (∝q ) with a parabola fit to post-surgery f (∝q ) (e) and one year after the surgery (f). Table 1 The results of the multifractal analysis (mean ± standard deviation). Structure acronyms are: fractal dimension (D0 ), information dimension (D1 ), the value of minimum exponent ˛ (˛min ), the value of maximum exponent ˛ (˛max ), width of fractal spectrum (˛), the value of the fractal spectrum for minimum exponent ˛ (f (˛min )), the value of the fractal spectrum for maximum exponent ˛ (f (˛max )), range of fractal spectrum (f (˛)), the minimum value of the fractal spectrum for exponent ˛ (f (˛q )min ), the maximum value of the fractal spectrum for exponent ˛ (f (˛q )max ), height of fractal spectrum (f (˛q )).
D0 D1 ∝min ∝max ∝ f (∝min ) f (∝max ) f (∝) f (∝q )min f (∝q )max f (∝q )
Group A
Group B
p
1.061 ± 0.385 1.045 ± 0.396 1.028 ± 0.390 1.109 ± 0.379 0.081 ± 0.033 1.047 ± 0.396 1.047 ± 0.383 0.001 ± 0.026 1.034 ± 0.387 1.065 ± 0.386 0.031 ± 0.006
1.780 ± 0.192 1.769 ± 0.196 1.741 ± 0.217 1.916 ± 0.248 0.174 ± 0.082 1.772 ± 0.194 1.721 ± 0.168 −0.052 ± 0.043 0.084 ± 0.112 1.919 ± 0.245 1.835 ± 0.264
<0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.01 <0.0001 <0.0001 <0.0001
cumulating the left-hand points at the top of the curve. In contrast, the spectrum values increase. The spectral width ˛ = ∝min − ∝max , and the difference in the height of the spectral arms f = f (∝)min − f (∝)max can be deter-
mined from the multifractal spectrum. These data are summarized in Table 1. On the basis of the obtained results (Fig. 5), it can be noticed that the multifractal spectrum width increases significantly (p < 0.001) after one year from the operation. On average, the post-surgery spectrum width was 0.081, while after one year from the operation it increased to 0.174. Both after the surgery and after a year from the procedure, the diversity of the spectral width is large. Narrow spectrum can indicate the occurrence of pathological or unnatural conditions, which in this case is the appearance of a fresh implant. The narrow range of ∝ value variability indicates the existence of a homogeneous surface, which is the implant. The increase in the ˛ value variability range indicates a diversified surface structure, which can be bone tissue. The wide multifractal spectrum confirms that the implant has been received and overgrown with the bone tissue of the patient. 4. Discussion The starting point of the described research is the hypothesis that analyzing the geometry of multifractal processes is the right tool for modeling RVG images after applying bone regeneration technique using Bio-Oss xenogenic material (step1) and covering Bio-Oss xenogenic material with restorable membrane
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fractal parameters were examined in the range of moment orders (q) form -1 to 1. The minimum values of multifractal parameters ∝min (scaling exponent) and f (∝)min (multifractal spectrum), corresponded to q = 1, and the maximum values of multifractal parameters ∝max and f (∝)max corresponded to q = −1. A higher values of ∝ indicated the multifractal nature of the image properties obtained 12 months after surgery in contrast to smaller values of ∝ which indicated the monofractal nature of the images obtained immediately after the procedure. Both the monofractal and the multifractal methods give a stable results in our research. However, in the multifractal method, the image is described by a greater number of parameters that can be additional indicators. The obtained results have a cognitive and applicatory value. Literature studies have shown that research into the multifractal nature of textures in medical applications is the subject of very few scientific reports. 5. Conclusion
Fig. 5. Boxplots obtained of multifractal analysis for Group A and Group B. Structure acronyms are: fractal dimension (D0 ), information dimension (D1 ), the value of minimum exponent ˛ (˛min ), the value of maximum exponent ˛ (˛max ), width of fractal spectrum (˛), the value of the fractal spectrum for minimum exponent ˛ (f (˛min )), the value of the fractal spectrum for maximum exponent ˛ (f (˛max )), range of fractal spectrum (f (˛)), the minimum value of the fractal spectrum for exponent ˛ (f (˛q )min ), the maximum value of the fractal spectrum for exponent ˛ (f (˛q )max ), height of fractal spectrum (f (˛q )).
BioGide (Biomateriale Geistlich) (step2). Parameters obtained from fractal geometry provide a powerful tool for characterizing objects in many medical imaging applications. It is often used in image analysis because the objects being imaged are discontinuous, complex and fragmented. The significance and the advantage of this geometry compared with classic signal processing methods, lies in its way of assuming irregularities. We combined fractal and multifractal features in order to have both global and local descriptions of the heterogeneities of the image texture. The finding that the image has the characteristics of fractal or multifractal requires conducting appropriate research. The use of simple techniques allows for characterizing the image using a single scaling factor, for example, counting boxes (box-counting method). The analysis showed an increase in the fractal and information dimension 12 months after the surgery, until the image stabilization at one level was achieved. RVG images can also be characterized not only by a single scaling factor, but by using more complex methods, for example multifractal analysis. Multifractal analysis methods are used to study processes / patterns at various levels of resolution. They allow for the direct determination of the scaling exponent function of probability distributions and statistical moments that characterize multifractal processes. Multifractal analysis of the healing process after bone loss has pointed to various properties characterizing this process. Multi-
The study of processes occurring in living organisms requires more sophisticated image processing methods that improve visual interpretation, measurement and characterization. Multifractal theory explains patterns and processes in biological organisms, their developments both in structures and functions. Fractal and multifractal studies show that the RVG images are of fractal nature, and the intensity values of RVG images indicate the multifractal, cascading nature of the healing process after resection and alveolation. The analysis and interpretation of the healing process after bone lost, using monofractal and multifractal analysis as a multi-steps process, quantify a stable (good) clinical result. Compliance with ethical standards The study protocol was approved by the Local Ethical Committee of Medical University of Bialystok, Poland. Conflicts of interest The authors declare that there are no financial or personal relationships with other people or organizations that could inappropriately influence this study. Acknowledgement The research was performer as a part of the projects S/WM/1/2017 and was financed with the founds for science from the Polish Ministry of Science and Higher Education. References [1] R. Obuchowicz, K. Nurzynska, B. Obuchowicz, A. Urbanik, A. Piórkowski, Caries Detection Enhancement Using Texture Feature Maps of Intraoral Radiographs, Springer, 2018. [2] M.S. Kavitha, S.-Y. An, C.-H. An, K.-H. Huh, W.-J. Yi, M.-S. Heo, S.-S. Lee, S.-C. Choi, Texture analysis of mandibular cortical bone on digital dental panoramic radiographs for the diagnosis of osteoporosis in Korean women, Oral Surg. Oral Med. Oral Pathol. Oral Radiol. 119 (2015) 346–356. [3] A. Jatti, R. Joshi, Characterization of dental pathologies using digital panoramic X-ray images based on texture analysis, Engineering in Medicine and Biology Society (EMBC), 2017 39th Annual International Conference of the IEEE (2017) 592–595. [4] C. Muramatsu, K. Horiba, T. Hayashi, T. Fukui, T. Hara, A. Katsumata, H. Fujita, Quantitative assessment of mandibular cortical erosion on dental panoramic radiographs for screening osteoporosis, Int. J. Comput. Assist. Radiol. Surg. 11 (2016) 2021–2032. [5] A. Naik, S. Tikhe, S. Bhide, K. Kaliyamurthie, T. Saravanan, Automatic segmentation of lower jaw and mandibular bone in digital dental panoramic radiographs, Indian J. Sci. Technol. (2016) 9.
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´ T. Stoˇsic, ´ [22] R. Baravalle, F. Thomsen, C. Delrieux, Y. Lu, J.C. Gómez, B. Stoˇsic, Three-dimensional multifractal analysis of trabecular bone under clinical computed tomography, Med. Phys. 44 (2017) 6404–6412. [23] E.A. Abdulhameed, N.H. Al-Rawi, A.T. Uthman, A.R. Samsudin, Bone texture fractal dimension analysis of ultrasound-treated bone around implant site: a double-blind clinical trial, Int. J. Dent. 2018 (2018). [24] C. Diana, S. Mohanty, Z. Chaudhary, S. Kumari, J. Dabas, R. Bodh, Does platelet-rich fibrin have a role in osseointegration of immediate implants? A randomized, single-blind, controlled clinical trial, Int. J. Oral Maxillofac. Surg. 47 (2018) 1178–1188. [25] T. Dietrich, P. Zunker, D. Dietrich, J.-P. Bernimoulin, Apicomarginal defects in periradicular surgery: classification and diagnostic aspects, Oral Surg. Oral Med. Oral Pathol. Oral Radiol. Endod. 94 (2002) 233–239. [26] G. Pecora, S. Kim, R. Celletti, M. Davarpanah, The guided tissue regeneration principle in endodontic surgery: one-year postoperative results of large periapical lesions, Int. Endod. J. 28 (1995) 41–46. [27] G. Reishofer, F. Studencnik, K. Koschutnig, H. Deutschmann, H. Ahammer, G. Wood, Age is reflected in the fractal dimensionality of MRI diffusion based tractography, Sci. Rep. 8 (2018) 5431. [28] G. Landini, Fractals in microscopy, J. Microsc. 241 (2011) 1–8. [29] P. Waliszewski, F. Wagenlehner, S. Gattenlöhner, W. Weidner, On the relationship between tumor structure and complexity of the spatial distribution of cancer cell nuclei: a fractal geometrical model of prostate carcinoma, Prostate 75 (2015) 399–414. [30] B.B. Mandelbrot, Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands, Proc. Natl. Acad. Sci. 72 (1975) 3825–3828. [31] B.B. Mandelbrot, The Fractal Geometry of Nature, WH freeman, New York, 1983. [32] H. Salat, R. Murcio, E. Arcaute, Multifractal methodology, Phys. A Stat. Mech. Its Appl. 473 (2017) 467–487. [33] J. Feder, Fractals (physics of Solids and Liquids), Plennum, New york, 1988. [34] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A (Coll Park) 33 (1986) 1141. [35] A.B. Chhabra, C. Meneveau, R.V. Jensen, K. Sreenivasan, Direct determination of the f (␣) singularity spectrum and its application to fully developed turbulence, Phys. Rev. A (Coll Park) 40 (1989) 5284. [36] C. Shanon, W. Weaver, The Mathematical Theory of Communication, Wiener N. Cybernetics, Urbana, 1949, 1949. [37] A. Beghdadi, C. Andraud, J. Lafait, J. Peiro, M. Perreau, Entropic and multifractal analysis of disordered morphologies, Fractals 1 (1993) 671–679. [38] N. Otsu, A threshold selection method from gray-level histograms, IEEE Trans. Syst. Man Cybern. 9 (1979) 62–66.
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Biomedical Signal Processing and Control journal homepage: www.elsevier.com/locate/bspc
Multifractal characterization of healing process after bone loss a ˛ Marta Borowska a,∗ , Ewelina Bebas , Janusz Szarmach b , Edward Oczeretko a a Bialystok University of Technology, Faculty of Mechanical Engineering, Department of Materials and Biomedical Engineering, Wiejska 45C, 15-351, Białystok, Poland b Medical University of Bialystok, Department of Oral Surgery, M. Curie-Skłodowskiej 24A, 15-276, Białystok, Poland
a r t i c l e
i n f o
Article history: Received 6 August 2018 Received in revised form 12 March 2019 Accepted 13 April 2019 Available online 24 April 2019 Keywords: Multifractal analysis Fractal dimension Guided bone regeneration Xenografts Radiographic images
a b s t r a c t Objective: In this work we have proposed to use multifractal analysis to evaluate the effectiveness of the healing process in postresectal and postcystal bone loss using the guided bone regeneration (GBR). Methods: The material of the study consists of 19 radiographic images obtained from patients (13 females and 6 males), observed within 1-year-long period, who had undergone bone augmentation with xenogenic material. Using radiographic images (RVG) made with digital radiography set Kodak RVG 6100 on the day of the procedure (Group A) and 12 months later (Group B), the degree of reconstruction of intraosseous defects were compared. A visual subjective evaluation of the healing process based on RVG images may not be sufficient. For this purpose, modern computer image analysis methods can be used. Pre-processing of radiographic images consisted of separating areas of interest (ROI) and binarization. The resulting images were analyzed using multifractals. The properties of biological structures cannot be sufficiently described in the Euclidean space, but they can be characterized by a parameter called the fractal dimension. Results: Significant increase (p < 0.0001) of multifractal features were found in patients from group B. Individual parameters may characterize bone structure, which had undergone bone augmentation with xenogenic material. Finally, we showed that the properly selected multifractal quantificators obtained for those images could be used to assign those images to the appropriate group. Conclusion: The proposed analysis based on RVG images could help to evaluate the healing process after bone loss. The analysis and interpretation of the healing process after bone loss using monofractal and multifractal analysis as a multi-steps process quantify a stable (good) clinical result. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction The healing process of the boneless alveolar areas after resection or cyst removal is a rare and complicated reparative operation that leads to complete fill-up of the empty space with new properly build bone tissue. The most common finding is loss of the bone volume, and regeneration seen as a main healing action. Frequently, bone loss exceeds it’s physiological limits, resulting in bone missing areas. Tissue engineering is a helpful tool facilitating reconstruction process of the alveolar bone shape and function. The basic materials used for bone regeneration are osseous materials, replacing bone specimens and barrier membranes. Therefore, in certain cases, the application of guided bone regeneration technique (GBR) with biomaterials and barrier membranes is justified. These materials
∗ Corresponding author. E-mail address: [email protected] (M. Borowska). https://doi.org/10.1016/j.bspc.2019.04.014 1746-8094/© 2019 Elsevier Ltd. All rights reserved.
enable proliferation of the bone tissue to the inner and outer layer of the osseous gap due to its osteoconductive properties. A visual subjective evaluation of the healing process based on RVG images may not be sufficient. For this purpose, modern computer image analysis methods can be used. These methods allow to determine the differences between healthy and diseased tissue, and hence, it s possible to assess the healing process quantitatively and qualitatively. Various studies on the texture of dental images concern the use of methods based on the statistical properties of the surfaces. Through their use, it is possible to detect the areas covered by caries more accurately [1]. If used together with other methods it allows for better recognition of osteoporosis in women [2], finding texture boundaries of the cysts region [3], identifying asymptomatic osteoporotic and osteopenic patients through dental examinations [4], automatizing the segmentation of lower jaw and mandible on panoramic x-rays [5]. It can also be used as a tool for dental implant treatment planning [6]. Modern methods of fractal and multifractal analysis are increasingly used nowadays [7–12]. The analysis of the fractal dimension has been used in the assessment of alveolar bone defects treated
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with various bone substitute materials [13]. The fractal dimensions of individual implants were determined on the basis of X-ray image analysis. The results show that the fractal dimension differed depending on the biomaterial used and reflected the individual nature of bone remodeling during observation. In Pârvu [14], it has been shown that fractal and multifractal analysis of tissue images as a non-invasive technique can be used to measure contrasting morphological changes in human gingival cells and tissues and can provide detailed information for the study of healthy and diseased gums in patients with root disease and generalized chronic periodontitis GCP. In our previous publications [15–17] we assessed the effectiveness of the treatment process of postresectional and postcystal bone loss using the GBR method on the basis of fractal dimensions. Significant changes were determined in the calculated fractal dimensions from RVG images taken immediately after the procedure and 12 months after the surgery. The box and information dimensions were calculated, showing significant differences, on the basis of which the healing process could be positively assessed. We showed fractal characteristics of images and we also wanted to point to multifractal characteristics, which we presented in this work. The tendency of changes in fractal dimensions was the same as in work [17] in which the calculations were performed based on binarization. In publication [15], the fractal dimension was also calculated and the method of calculation was based on differences in the intensity of pixels. Multifractal analysis shows the possibility of predicting an individual patient’s response to neoadjuvant cytotoxic therapy in case of osteosarcoma (over 70% compliance for the Dqmax parameter) based on magnetic resonance images thanks to which, in the future, the implementation of alternative treatment options directed at a specific patient [18] may be allowed. In Gao [19] multifractal spectrum features of micro-CT images in combination with the C4.5 classifier allows for early diagnosis of osteoporosis with accuracy greater than the methods used so far. Multifractal methods are often used for segmentation of anatomical structures or diseased areas [20,21]. The multifractal analysis gave better results than fractal analysis when identifying the primary cancer, in the case of metastatic bone disease diagnosis based on medical images. It also showed a high probability of correct diagnosis [22]. In our study we did not do any research to show bone mineral density (it is impossible to determine it from RVG images), however in the article [22] multifractal analysis was used to predict vertebral damage in CT images. RVG images were useful for specifying parameters that make it possible to describe the bone healing process. There are not many reports on the use of the fractal dimension and multifractal behavior in the evaluation of the healing process after bone augmentation with xenogenic material. Information about such studies can be found in the assessment of bone after the implant insertion. Abdulhameed et al. [23] used fractal dimension in predicting implant stability from intraoral periapical radiographs. Their results confirmed the increase in the fractal dimension with the increase of time after surgery. Fractal analysis was also used for the assessment of bone quantity around the implant [24]. The aim of this paper is to evaluate the effectiveness of the healing process in postresectal and postcystal bone loss using the guided bone regeneration (GBR) based on a multifractal analysis of RVG images.
Medical University in Bialystok from year 2012–2014. The presence of root cysts was examined on the bases of clinical symptoms and radiological images performed. Bone loss in the frontal maxillary and mandibular region among all 19 patients was classified as Class II according to Dietrich and ass. [25]. A few conditions such as root canal treatment of the teeth qualified for the resection and cystectomy; sanitation of the oral cavity, professional hygienisation, were required for acceptance to the surgery. Generally healthy patients, with no medical conditions impairing bone healing, were included in the research. All the procedures were performed by the same operator under local anesthesia with prophylactic antibiotic coverage in a classic surgery protocol manner [26]. Postresectal bone loss was filled with xenograft BioOss® material (spongeous bone granulate grading 0.25–1 mm) and covered with resorbable barrier membrane BioGide® of pork origin. The membrane itself consists of highly purified collagen type I and III, with no organic and/or chemical components. Implanted BioOss® material is a natural specimen of deproteinated beef bone. It shows resemblance to human bone in morphology, crystal structure, chemical composition and internal structure. It’s porosity ranging 60% is similar to that of the human bone. Post-surgery wound was stabilized with single suture knot using synthetic 5-0 thick sutures (Fig. 1). To assess the treatment results, several intraoral radiological images were performed in the straight angle technique, immediately after the procedure (Group A) and 12 months later (Group B). The equipment used for the research was KODAK RVG 6100 set with the real resolution over 14 pl/mm, and a collimator narrowing the radiation beam, with the constant exposure time of 0.08 s; all data were saved as MPG files. During convalescence period, analgesics, 0.12% chlorhexidine gluconate mouth rinse and cold compresses were a part of the therapy. 2.2. Fractal and multifractal analysis Fractal and multifractal methodology is widely applied in many areas of biosciences both in the analysis of medical signals and images [10,27–29]. They can provide valuable information on the geometrical complexity of biological structures [27]. Fractals are geometric structures with specific metric properties such as length, area or volume represented at multiple scales. Benoit Mandelbrot [30] proposed definition of the fractal as a set, which can be described by a parameter called the fractal (Hausdorff) dimension (DH ) with a non-integer value greater than its topological dimension [31]. If a given fractal is embedded in a n dimensional Euclidean space with a topological dimension, then it goes to fill a subset of the whole space. This dimension determines the self-similarity at different scales of the fractal structure. Many methods of fractal dimension calculation are used in the images analysis. The so-called box-counting method is the most commonly used method of estimating the fractal dimension. The geometrical object in the n− dimensional Euclidean space with length L is covered by the boxes with a size r ≤ L. The relationship between fractal dimension DH and number of boxes N(r) covering object with a size r is governed by a power-law: N(r)˜r −DH
Logarithms on both sides of the above relation, determine fractal dimension according to the formula: DH = lim
2. Materials and methods 2.1. Data collection The treatment results of 19 patients (13 females, 6 males; average age of 356) were assessed in Oral Surgery Department of
(1)
r→0
lnN(r) ln 1r
(2)
The next important step in the development of fractal analysis was the introduction of the multifractal concept [32]. Multifractal is an object that combines many fractal subsets with different scaling exponents, each with a different fractal dimension [33]. The description of the internal structure of the multifractal is based on
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Fig. 1. Stages of surgery: a) clinical condition before surgery, b) a dissected periosteal flap with visible external plaque of the alveolar process, c) enucleation of the cyst, d) resection of the apex of the tooth root, e) bone loss after enucleation of the cyst, f) preparation of a xenogenic implant (BioOss) with the patient’s blood, g) a bone defect filled with xenogeneous material, h) coverage of the area augmented by the barrier membrane (BioGide), i) sewing the post-treatment wound with knot stitching.
the consideration of the whole spectrum of dimensions, called the spectrum of generalized dimensions Dq [34]. These spectrum of dimension Dq is shown in the graph as Dq in the function of q or f (˛) in the function ˛. Consider the two dimensional object in n − dimensional Euclidean space with length L covered by boxes of size r ≤ L. The mass probability in the i − th box is given by:
Pi (r) =
Ni (r) N
(3)
where N is the total mass of the object and Ni (r) is the number of pixels in the i − th box. Dependence is quantified for multifractal measure (scaling) of Pi with i − th box size r as:
¨ where ∝i is the Lipschitz-Holder exponent [34]. The number of ¨ boxes N(∝), where the probability Pi (r) has Lipschitz-Holder exponent ∝ is found to scale as: N(∝)˜r f (∝)
where f (∝) provide the fractal dimension of subset with exponent ∝ [34,35]. In order to obtain f (∝) the scaling of the q-th moments of Pi (r) distributions can be used in the following way: ni (r)
(4)
q
Pi (r)˜r q
(6)
i
where q = (q − 1)Dq is the mass exponent of order q. By Legendre transform, the variables f (∝q ) and ∝q are related to q : ∝q =
Pi (r)˜r ˛i
(5)
dq 1 = lim i lnPi (r) dq r→0 lnr i
(7)
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Fig. 2. Description of parameters obtained from multifractal analysis a) example of f (q) and ∝q functions obtained in the range of q values, b) f (∝) - spectra. Structure acronyms are: fractal dimension (D0 ), information dimension (D1 ), fractal spectrum (f (q)), scaling factor (˛ (q)), exponent estimated in the range of q from -1 to 1 (˛q ), the value of minimum exponent ˛ (˛min ), the value of maximum exponent ˛ (˛max ), width of fractal spectrum (˛), the value of the fractal spectrum for minimum exponent ˛ (f (˛min )), the value of the fractal spectrum for maximum exponent ˛ (f (˛max )), range of fractal spectrum (f (˛)), the minimum value of the fractal spectrum for exponent ˛ (f (˛q )min ), the maximum value of the fractal spectrum for exponent ˛ (f (˛q )max ), height of fractal spectrum (f (˛q )).
1 i lni r→0 lnr
f ∝q = q∝q − q = lim q Pi /
(8)
i
q Pi ;
where i = q is the weight; ∝q is the exponent; f (∝q ) represents multifractal spectrum function; q is the mass exponent. Multifractal spectrum is the plot of f (∝q ) vs. ∝q . The generalized entropy Hq (r) and generalized dimension Dq are defined as: Ni (r)
ln
q
Pi
i=1
Hq (r) =
(9)
1−q
Dq = −lim
r→0
Hq (r) q 1 = = [q∝q − f (∝q )] q−1 q−1 lnr
(10)
When q = 0, f (∝q=0 ) represents fractal dimension D0 previously named DH . For q = 1, f (∝q=1 ) represents information dimension (entropy dimension) D1 defined by formula: Ni (r)
ln
Pi (r)lnPi (r)
i=1
D1 = lim
lnr
r→0
(11)
The information dimension D1 is related to the entropy measure of Shannon [36], which gives decrease in information with an increase of the size of the box [37]. For q = 2, f (∝q=2 ) represents correlation dimension D2 defined by formula: Ni (r)
ln D2 = lim
r→0
Pi2 (r)
i=1
lnr
(12)
For q = ±∞, f (∝q=±∞ ) represents D∞ and D−∞ , which are corresponding to the values ∝max and ∝min , respectively. For monofractals, ∝= ∝max − ∝min should be zero. Fig. 2 shows some of the features obtained from multifractal spectrum plot. 3. Results Fig. 3 shows the scheme of RVG image analysis. In the first step, the regions of interest (ROI) were selected for the analysis, in the images taken immediately after the procedure and one year after the surgery. The areas were 128 × 128 pixels each. The extracted fragments of the images were binarized to make the analysis possible (Fig. 3e,f). It is important that all analyzed images were binarized
using the same threshold. It allowed for reliable comparative analysis of the parameters obtained. The optimal binarization thresholds were obtained using the Otsu method [38] implementing MATLAB software: the binarization threshold values were in the range from 79 to 176, and the average threshold value of the determined range was 127.5. Next, multifractal spectrum of a binary image was used to carry out the multifractal analysis. Multifractal analysis was done in Matlab for Windows (MathWorks, Inc., USA), a high-performance language for technical computing. As a result of the analysis, two functions were obtained: the value of the fractal spectrum f (˛) and the value of the scaling exponent ˛ estimated in the range of q from −1 to 1. Additionally, a graph of the fractal spectrum f (˛) was plotted and a graph of theoretical values was fitted to it. Fig. 4 presents the multifractal spectrum f (˛) and the value of the singularity coefficient ˛ depending on the factor q for a patient from group A and group B of the image ROI, respectively. There is a noticeable difference in the distributions of the functions f (q) and ˛(q) in the case of images taken on the day of the surgery and performed one year after the surgery. The graphs depicting the dependence of the fractal spectrum f (q) and the scaling factor ˛ (q) on the coefficient q for each of the images, increase their values by comparing the results obtained for images for patients from group A and group B. In addition, the information dimension D1 , which corresponds to the value f (q) for q = 1 and the fractal dimension D0 in the point q = 0, which is also the maximum of function f , can be read from the obtained results. The results of multifractals analysis are shown in Table 1. Statistical analysis was performed by means of two-sample t-test. Taking the calculated parameters into account it can be concluded that the values of information and fractal dimensions increase after one year after the surgery. The average post-surgery fractal dimension was 1.061, while after one year after the surgery it increased to 1.781. In the case of the information dimension, its mean value after the surgery was 1.045, increasing to 1.7692 one year after it. Statistical analysis of the results allows for a much larger discrepancy in the results obtained for images taken immediately after the procedure than for those performed one year after the surgery. This means that as the healing process progresses, the fractal dimension of the image being examined stabilizes at one level and amounts to about 1.77. The change in the shape of the multifractal spectrum for images taken from group B is clearly visible compared to the spectrum obtained for images taken from group A. The spectrum loses the shape of the bell curve, by shortening its left side, which is caused by
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Fig. 3. The scheme of multifractal analysis: a) post-surgical image with the region of interest, b) post-surgical image with the region of interest one year after, c) grey region of interest after the surgery, d) grey region of interest one year after the surgery, e) binarized regions of interest after the surgery, f) binarized regions of interest one year after the surgery, g) multifractal spectrum f (˛) and the volume of the singularity coefficient ˛ depending on the factor q, for the post-surgical image, h) multifractal spectrum f (˛) and the volume of the singularity coefficient ˛ depending on the factor q, for post-surgical image one year later, i) multifractal spectrum for the image after surgery, j) multifractal spectrum for the image one year after the surgery.
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Fig. 4. Example of f (q) and ∝q function estimated in range of q values for a post-surgery image (a) and one year after the surgery image (b). Multifractal post-surgery spectra f (∝q ) (c) and one year after the surgery (d). Multifractal spectra f (∝q ) with a parabola fit to post-surgery f (∝q ) (e) and one year after the surgery (f). Table 1 The results of the multifractal analysis (mean ± standard deviation). Structure acronyms are: fractal dimension (D0 ), information dimension (D1 ), the value of minimum exponent ˛ (˛min ), the value of maximum exponent ˛ (˛max ), width of fractal spectrum (˛), the value of the fractal spectrum for minimum exponent ˛ (f (˛min )), the value of the fractal spectrum for maximum exponent ˛ (f (˛max )), range of fractal spectrum (f (˛)), the minimum value of the fractal spectrum for exponent ˛ (f (˛q )min ), the maximum value of the fractal spectrum for exponent ˛ (f (˛q )max ), height of fractal spectrum (f (˛q )).
D0 D1 ∝min ∝max ∝ f (∝min ) f (∝max ) f (∝) f (∝q )min f (∝q )max f (∝q )
Group A
Group B
p
1.061 ± 0.385 1.045 ± 0.396 1.028 ± 0.390 1.109 ± 0.379 0.081 ± 0.033 1.047 ± 0.396 1.047 ± 0.383 0.001 ± 0.026 1.034 ± 0.387 1.065 ± 0.386 0.031 ± 0.006
1.780 ± 0.192 1.769 ± 0.196 1.741 ± 0.217 1.916 ± 0.248 0.174 ± 0.082 1.772 ± 0.194 1.721 ± 0.168 −0.052 ± 0.043 0.084 ± 0.112 1.919 ± 0.245 1.835 ± 0.264
<0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.01 <0.0001 <0.0001 <0.0001
cumulating the left-hand points at the top of the curve. In contrast, the spectrum values increase. The spectral width ˛ = ∝min − ∝max , and the difference in the height of the spectral arms f = f (∝)min − f (∝)max can be deter-
mined from the multifractal spectrum. These data are summarized in Table 1. On the basis of the obtained results (Fig. 5), it can be noticed that the multifractal spectrum width increases significantly (p < 0.001) after one year from the operation. On average, the post-surgery spectrum width was 0.081, while after one year from the operation it increased to 0.174. Both after the surgery and after a year from the procedure, the diversity of the spectral width is large. Narrow spectrum can indicate the occurrence of pathological or unnatural conditions, which in this case is the appearance of a fresh implant. The narrow range of ∝ value variability indicates the existence of a homogeneous surface, which is the implant. The increase in the ˛ value variability range indicates a diversified surface structure, which can be bone tissue. The wide multifractal spectrum confirms that the implant has been received and overgrown with the bone tissue of the patient. 4. Discussion The starting point of the described research is the hypothesis that analyzing the geometry of multifractal processes is the right tool for modeling RVG images after applying bone regeneration technique using Bio-Oss xenogenic material (step1) and covering Bio-Oss xenogenic material with restorable membrane
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fractal parameters were examined in the range of moment orders (q) form -1 to 1. The minimum values of multifractal parameters ∝min (scaling exponent) and f (∝)min (multifractal spectrum), corresponded to q = 1, and the maximum values of multifractal parameters ∝max and f (∝)max corresponded to q = −1. A higher values of ∝ indicated the multifractal nature of the image properties obtained 12 months after surgery in contrast to smaller values of ∝ which indicated the monofractal nature of the images obtained immediately after the procedure. Both the monofractal and the multifractal methods give a stable results in our research. However, in the multifractal method, the image is described by a greater number of parameters that can be additional indicators. The obtained results have a cognitive and applicatory value. Literature studies have shown that research into the multifractal nature of textures in medical applications is the subject of very few scientific reports. 5. Conclusion
Fig. 5. Boxplots obtained of multifractal analysis for Group A and Group B. Structure acronyms are: fractal dimension (D0 ), information dimension (D1 ), the value of minimum exponent ˛ (˛min ), the value of maximum exponent ˛ (˛max ), width of fractal spectrum (˛), the value of the fractal spectrum for minimum exponent ˛ (f (˛min )), the value of the fractal spectrum for maximum exponent ˛ (f (˛max )), range of fractal spectrum (f (˛)), the minimum value of the fractal spectrum for exponent ˛ (f (˛q )min ), the maximum value of the fractal spectrum for exponent ˛ (f (˛q )max ), height of fractal spectrum (f (˛q )).
BioGide (Biomateriale Geistlich) (step2). Parameters obtained from fractal geometry provide a powerful tool for characterizing objects in many medical imaging applications. It is often used in image analysis because the objects being imaged are discontinuous, complex and fragmented. The significance and the advantage of this geometry compared with classic signal processing methods, lies in its way of assuming irregularities. We combined fractal and multifractal features in order to have both global and local descriptions of the heterogeneities of the image texture. The finding that the image has the characteristics of fractal or multifractal requires conducting appropriate research. The use of simple techniques allows for characterizing the image using a single scaling factor, for example, counting boxes (box-counting method). The analysis showed an increase in the fractal and information dimension 12 months after the surgery, until the image stabilization at one level was achieved. RVG images can also be characterized not only by a single scaling factor, but by using more complex methods, for example multifractal analysis. Multifractal analysis methods are used to study processes / patterns at various levels of resolution. They allow for the direct determination of the scaling exponent function of probability distributions and statistical moments that characterize multifractal processes. Multifractal analysis of the healing process after bone loss has pointed to various properties characterizing this process. Multi-
The study of processes occurring in living organisms requires more sophisticated image processing methods that improve visual interpretation, measurement and characterization. Multifractal theory explains patterns and processes in biological organisms, their developments both in structures and functions. Fractal and multifractal studies show that the RVG images are of fractal nature, and the intensity values of RVG images indicate the multifractal, cascading nature of the healing process after resection and alveolation. The analysis and interpretation of the healing process after bone lost, using monofractal and multifractal analysis as a multi-steps process, quantify a stable (good) clinical result. Compliance with ethical standards The study protocol was approved by the Local Ethical Committee of Medical University of Bialystok, Poland. Conflicts of interest The authors declare that there are no financial or personal relationships with other people or organizations that could inappropriately influence this study. Acknowledgement The research was performer as a part of the projects S/WM/1/2017 and was financed with the founds for science from the Polish Ministry of Science and Higher Education. References [1] R. Obuchowicz, K. Nurzynska, B. Obuchowicz, A. Urbanik, A. Piórkowski, Caries Detection Enhancement Using Texture Feature Maps of Intraoral Radiographs, Springer, 2018. [2] M.S. Kavitha, S.-Y. An, C.-H. An, K.-H. Huh, W.-J. Yi, M.-S. Heo, S.-S. Lee, S.-C. Choi, Texture analysis of mandibular cortical bone on digital dental panoramic radiographs for the diagnosis of osteoporosis in Korean women, Oral Surg. Oral Med. Oral Pathol. Oral Radiol. 119 (2015) 346–356. [3] A. Jatti, R. Joshi, Characterization of dental pathologies using digital panoramic X-ray images based on texture analysis, Engineering in Medicine and Biology Society (EMBC), 2017 39th Annual International Conference of the IEEE (2017) 592–595. [4] C. Muramatsu, K. Horiba, T. Hayashi, T. Fukui, T. Hara, A. Katsumata, H. Fujita, Quantitative assessment of mandibular cortical erosion on dental panoramic radiographs for screening osteoporosis, Int. J. Comput. Assist. Radiol. Surg. 11 (2016) 2021–2032. [5] A. Naik, S. Tikhe, S. Bhide, K. Kaliyamurthie, T. Saravanan, Automatic segmentation of lower jaw and mandibular bone in digital dental panoramic radiographs, Indian J. Sci. Technol. (2016) 9.
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