Abstract
Irregularity in shape and behavior is the main feature of every anatomical system, including human organs, tissues, cells, and sub-cellular entities. It has been shown that this property cannot be quantified by means of the classical Euclidean geometry, which is only able to describe regular geometrical objects. In contrast, fractal geometry has been widely applied in several scientific fields. This rapid growth has also produced substantial insights in the biomedical imaging. Consequently, particular attention has been given to the identification of pathognomonic patterns of “shape” in anatomical entities and their changes from natural to pathological states. Despite the advantages of fractal mathematics and several studies demonstrating its applicability to oncological research, many researchers and clinicians remain unaware of its potential. Therefore, this review aims to summarize the complexity and fractal geometry of nuclear medicine images.
Similar content being viewed by others
References
Kane EA, Higham TE (2015) Complex systems are more than the sum of their parts: using integration to understand performance, biomechanics, and diversity. Integr Comp Biol 55:146–165
Grizzi F, Chiriva-Internati M (2005) The complexity of anatomical systems. Theor Biol Med Model 19:2–26
Simon AH (1962) The architecture of complexity. Proc Am Philos Soc 106:467–482
Di Ieva A, Grizzi F, Jelinek H et al (2014) Fractals in the neurosciences, part I: general principles and basic neurosciences. Neuroscientist 20:403–417
Noble D (2008) Claude Bernard, the first systems biologist, and the future of physiology. Exp Physiol 93:16–26
Sargut G, McGrath RG (2011) Learning to live with complexity. Harv Bus Rev 89(68–76):136
Losa GA (2009) The fractal geometry of life. Riv Biol 102:29–59
Losa GA (2002) Fractal morphometry of cell complexity. Riv Biol 95:239–258
Bianciardi G (2015) Differential diagnosis. Shape and function, fractal tools in the pathology lab. Nonlinear Dynamics Psychol Life Sci 19:437–464
Losa GA, Nonnenmacher TF (1996) Self-similarity and fractal irregularity in pathologic tissues. Mod Pathol 9:174–182
Lennon FE, Cianci GC, Cipriani NA, Hensing TA, Zhang HJ, Chen CT, Murgu SD, Vokes EE, Vannier MW, Salgia R (2015) Lung cancer—a fractal viewpoint. Nat Rev Clin Oncol 12:664–675
Di Ieva A, Esteban FJ, Grizzi F et al (2015) Fractals in the neurosciences, part II: clinical applications and future perspectives. Neuroscientist 21:30–43
Im K, Lee JM, Yoon U, Shin YW, Hong SB, Kim IY, Kwon JS, Kim SI (2006) Fractal dimension in human cortical surface: multiple regression analysis with cortical thickness, sulcal depth, and folding area. Hum Brain Mapp 27:994–1003
Gadde SG, Anegondi N, Bhanushali D et al (2016) Quantification of vessel density in retinal optical coherence tomography angiography images using local fractal dimension. Invest Ophthalmol Vis Sci 57:246–252
Noujaim SF, Berenfeld O, Kalifa J, Cerrone M, Nanthakumar K, Atienza F, Moreno J, Mironov S, Jalife J (2007) Universal scaling law of electrical turbulence in the mammalian heart. Proc Natl Acad Sci U S A 104:20985–20989
Goldberger AL (1991) Is the normal heartbeat chaotic or homeostatic? News Physiol Sci 6:87–91
Mandelbrot B (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156(3775):636–638
Bancaud A, Lavelle C, Huet S, Ellenberg J (2012) A fractal model for nuclear organization: current evidence and biological implications. Nucleic Acids Res 40:8783–8792
Glenny RW, Robertson HT, Yamashiro S, Bassingthwaighte JB (1991) Applications of fractal analysis to physiology. J Appl Physiol 70:2351–2367
Liebovitch LS, Todorov AT (1996) Using fractals and nonlinear dynamics to determine the physical properties of ion channel proteins. Crit Rev Neurobiol 10:169–187
West BJ (2010) Fractal physiology and the fractional calculus: a perspective. Front Physiol 14:1–12
Cross SS (1997) Fractals in pathology. J Pathol 182:1–8
Smith TG Jr, Lange GD, Marks WB (1996) Fractal methods and results in cellular morphology—dimensions, lacunarity and multifractals. J Neurosci Methods 69:123–136
Grizzi F, Russo C, Colombo P et al (2005) Quantitative evaluation and modeling of two-dimensional neovascular network complexity: the surface fractal dimension. BMC Cancer 8:5–14
Havlin S, Buldyrev SV, Goldberger AL, Mantegna RN, Ossadnik SM, Peng CK, Simons M, Stanley HE (1995) Fractals in biology and medicine. Chaos, Solitons Fractals 6:171–201
Grizzi F, Colombo P, Taverna G, Chiriva-Internati M, Cobos E, Graziotti P, Muzzio PC, Dioguardi N (2007) Geometry of human vascular system: is it an obstacle for quantifying antiangiogenic therapies? Appl Immunohistochem Mol Morphol 15:134–139
Captur G, Karperien AL, Li C et al (2015) Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation. J Cardiovasc Magn Reson 7:17–80
Lorthois S, Cassot F (2010) Fractal analysis of vascular networks: insights from morphogenesis. J Theor Biol 262:614–633
Moal F, Chappard D, Wang J et al (2002) Fractal dimension can distinguish models and pharmacologic changes in liver fibrosis in rats. Hepatology 36:840–849
Rajkovic K, Bacic G, Ristanovic D, Milosevic NT (2014) Mathematical model of neuronal morphology: prenatal development of the human dentate nucleus. Biomed Res Int 2014:812351
Ristanovic D, Stefanovic BD, Puskas N (2014) Fractal analysis of dendrite morphology using modified box-counting method. Neurosci Res 84:64–67
Ristanovic D, Stefanovic BD, Puskas N (2014) Fractal analysis of dendrite morphology of rotated neuronal pictures: the modified box counting method. Theor Biol Forum 107:109–121
Tambasco M, Magliocco AM (2008) Relationship between tumor grade and computed architectural complexity in breast cancer specimens. Hum Pathol 39:740–746
Liebovitch LS, Toth TI (1990) Fractal activity in cell membrane ion channels. Ann N Y Acad Sci 591:375–391
Dioguardi N, Grizzi F, Fiamengo B, Russo C (2008) Metrically measuring liver biopsy: a chronic hepatitis B and C computer-aided morphologic description. World J Gastroenterol 14:7335–7344
Grizzi F, Russo C, Franceschini B, di Rocco M, Torri V, Morenghi E, Fassati LR, Dioguardi N (2006) Sampling variability of computer-aided fractal-corrected measures of liver fibrosis in needle biopsy specimens. World J Gastroenterol 12:7660–7665
Dioguardi N, Grizzi F, Franceschini B, Bossi P, Russo C (2006) Liver fibrosis and tissue architectural change measurement using fractal-rectified metrics and Hurst’s exponent. World J Gastroenterol 12:2187–2194
Dioguardi N, Franceschini B, Aletti G, Russo C, Grizzi F (2003) Fractal dimension rectified meter for quantification of liver fibrosis and other irregular microscopic objects. Anal Quant Cytol Histol 25:312–320
Dioguardi N, Grizzi F, Bossi P, Roncalli M (1999) Fractal and spectral dimension analysis of liver fibrosis in needle biopsy specimens. Anal Quant Cytol Histol 21:262–266
Sedivy R (1996) Fractal tumours: their real and virtual images. Wien Klin Wochenschr 108:547–551
Kikuchi A, Kozuma S, Yasugi T, Taketani Y (2004) Fractal analysis of surface growth patterns in endometrioid endometrial adenocarcinoma. Gynecol Obstet Investig 58:61–67
Lee LH, Tambasco M, Otsuka S, Wright A, Klimowicz A, Petrillo S, Morris D, Magliocco A, Bebb DG (2014) Digital differentiation of non-small cell carcinomas of the lung by the fractal dimension of their epithelial architecture. Micron 67:125–131
Vasiljevic J, Rel** B, Sopta J, Mijucic V, Tulic G, Rel** I (2012) Application of multifractal analysis on microscopic images in the classification of metastatic bone disease. Biomed Microdevices 14:541–548
Di Ieva A, Grizzi F, Tschabitscher M et al (2010) Correlation of microvascular fractal dimension with positron emission tomography [11C]-methionine uptake in glioblastoma multiforme: preliminary findings. Microvasc Res 80:267–273
Metze K (2013) Fractal dimension of chromatin: potential molecular diagnostic applications for cancer prognosis. Expert Rev Mol Diagn 13:719–735
Fudenberg G, Getz G, Meyerson M, Mirny LA (2011) High order chromatin architecture shapes the landscape of chromosomal alterations in cancer. Nat Biotechnol 29:1109–1113
Misteli T (2010) Higher-order genome organization in human disease. Cold Spring Harb Perspect Biol 2(8):a000794
Irinopoulou T, Rigaut JP, Benson MC (1993) Toward objective prognostic grading of prostatic carcinoma using image analysis. Anal Quant Cytol Histol 15:341–344
Streba CT, Pirici D, Vere CC, Mogoantă L, Comănescu V, Rogoveanu I (2011) Fractal analysis differentiation of nuclear and vascular patterns in hepatocellular carcinomas and hepatic metastasis. Romanian J Morphol Embryol 52:845–854
Strauss LG, Conti PS (1991) The applications of PET in clinical oncology. J Nucl Med 32:623–648
Barrington SF, Mikhaeel NG, Kostakoglu L, Meignan M, Hutchings M, Müeller SP, Schwartz LH, Zucca E, Fisher RI, Trotman J, Hoekstra OS, Hicks RJ, O'Doherty MJ, Hustinx R, Biggi A, Cheson BD (2014) Role of imaging in the staging and response assessment of lymphoma: consensus of the International Conference on Malignant Lymphomas Imaging Working Group. J Clin Oncol 32:3048–3058
Dimitrakopoulou-Strauss A (2015) PET-based molecular imaging in personalized oncology: potential of the assessment of therapeutic outcome. Future Oncol 11:1083–1091
Mijnhout GS, Hoekstra OS, van Tulder MW, Teule GJJ, Devill WLJM (2001) Systematic review of the diagnostic accuracy of (18)F-fluorodeoxyglucose positron emission tomography in melanoma patients. Cancer 91:1530–1542
Kessler LG, Barnhart HX, Buckler AJ, Choudhury KR, Kondratovich MV, Toledano A, Guimaraes AR, Filice R, Zhang Z, Sullivan DC, QIBA Terminology Working Group (2015) The emerging science of quantitative imaging biomarkers terminology and definitions for scientific studies and regulatory submissions. Stat Methods Med Res 24:9–26
Michallek F, Dewey M (2014) Fractal analysis in radiological and nuclear medicine perfusion imaging: a systematic review. Eur Radiol 24:60–69
Miwa K, Inubushi M, Wagatsuma K, Nagao M, Murata T, Koyama M, Koizumi M, Sasaki M (2014) FDG uptake heterogeneity evaluated by fractal analysis improves the differential diagnosis of pulmonary nodules. Eur J Radiol 83:715–719
Dimitrakopoulou-Strauss A, Georgoulias V, Eisenhut M, Herth F, Koukouraki S, Mäcke HR, Haberkorn U, Strauss LG (2006) Quantitative assessment of SSTR2 expression in patients with non-small cell lung cancer using 68Ga-DOTATOC PET and comparison with 18F-FDG PET. Eur J Nucl Med Mol Imaging 33:823–830
Dimitrakopoulou-Strauss A, Strauss LG, Burger C (2001) Quantitative PET studies in pretreated melanoma patients: a comparison of 6-[18F]fluoro-L-dopa with 18F-FDG and 15O-water using compartment and noncompartment analysis. J Nucl Med 42:248–256
Breki CM, Dimitrakopoulou-Strauss A, Hassel J, Theoharis T, Sachpekidis C, Pan L, Provata A (2016) Fractal and multifractal analysis of PET/CT images of metastatic melanoma before and after treatment with ipilimumab. EJNMMI Res 6:61
Dimitrakopoulou-Strauss A, Strauss LG, Schwarzbach M, Burger C, Heichel T, Willeke F, Mechtersheimer G, Lehnert T (2001) Dynamic PET 18F-FDG studies in patients with primary and recurrent soft-tissue sarcomas: impact on diagnosis and correlation with grading. J Nucl Med 42:713–720
Okazumi S, Dimitrakopoulou-Strauss A, Schwarzbach MH, Strauss LG (2009) Quantitative, dynamic 18F-FDG-PET for the evaluation of soft tissue sarcomas: relation to differential diagnosis, tumor grading and prediction of prognosis. Hell J Nucl Med 12:223–228
Dimitrakopoulou-Strauss A, Strauss LG, Burger C, Rühl A, Irngartinger G, Stremmel W, Rudi J (2004) Prognostic aspects of 18F-FDG PET kinetics in patients with metastatic colorectal carcinoma receiving FOLFOX chemotherapy. J Nucl Med 45:1480–1487
Koukouraki S, Strauss LG, Georgoulias V, Eisenhut M, Haberkorn U, Dimitrakopoulou-Strauss A (2006) Comparison of the pharmacokinetics of 68Ga-DOTATOC and [18F]FDG in patients with metastatic neuroendocrine tumours scheduled for 90Y-DOTATOC therapy. Eur J Nucl Med Mol Imaging 33:1115–1122
Sachpekidis C, Kopka K, Eder M et al (2016) 68Ga-PSMA-11 dynamic PET/CT imaging in primary prostate cancer. Clin Nucl Med 41:e473–e479
Sachpekidis C, Baumer P, Kopka K et al (2018) 68Ga-PSMA PET/CT in the evaluation of bone metastases in prostate cancer. Eur J Nucl Med Mol Imaging 45:904–912. https://doi.org/10.1007/s00259-018-3936-0
Sachpekidis C, Hillengass J, Goldschmidt H, Wagner B, Haberkorn U, Kopka K, Dimitrakopoulou-Strauss A (2017) Treatment response evaluation with 18F-FDG PET/CT and 18F-NaF PET/CT in multiple myeloma patients undergoing high-dose chemotherapy and autologous stem cell transplantation. Eur J Nucl Med Mol Imaging 44:50–62
Dimitrakopoulou-Strauss A, Hoffmann M, Bergner R, Uppenkamp M, Haberkorn U, Strauss LG (2009) Prediction of progression-free survival in patients with multiple myeloma following anthracycline-based chemotherapy based on dynamic FDG-PET. Clin Nucl Med 34:576–584
Ben Bouallegue F, Tabaa YA, Kafrouni M et al (2017) Association between textural and morphological tumor indices on baseline PET-CT and early metabolic response on interim PET-CT in bulky malignant lymphomas. Med Phys 44:4608–4619
Tochigi T, Shuto K, Kono T, Ohira G, Tohma T, Gunji H, Hayano K, Narushima K, Fujishiro T, Hanaoka T, Akutsu Y, Okazumi S, Matsubara H (2017) Heterogeneity of glucose metabolism in esophageal cancer measured by fractal analysis of fluorodeoxyglucose positron emission tomography image: correlation between metabolic heterogeneity and survival. Dig Surg 34:186–191
Lopci E, Grizzi F, Russo C, Toschi L, Grassi I, Cicoria G, Lodi F, Mattioli S, Fanti S (2017) Early and delayed evaluation of solid tumours with 64Cu-ATSM PET/CT: a pilot study on semiquantitative and computer-aided fractal geometry analysis. Nucl Med Commun 38:340–346
Kuikka JT, Tiihonen J, Karhu J et al (1997) Fractal analysis of striatal dopamine re-uptake sites. Eur J Nucl Med Mol Imaging 24:1085–1090
Kuikka JT, Hartikainen P (2000) Heterogeneity of cerebral blood flow: a fractal approach. Nuklearmedizin 39:37–42
Lee JS, Lee DS, Park KS, Chung JK, Lee MC (2004) Changes in the heterogeneity of cerebral glucose metabolism with healthy aging: quantitative assessment by fractal analysis. J Neuroimaging 14:350–356
Venegas JG, Galletti GG (2000) Low-pass filtering, a new method of fractal analysis: application to PET images of pulmonary blood flow. J Appl Physiol 88:1365–1373
Kalliokoski KK, Kuusela TA, Nuutila P, Tolvanen T, Oikonen V, Teräs M, Takala TES, Knuuti J (2001) Perfusion heterogeneity in human skeletal muscle: fractal analysis of PET data. Eur J Nucl Med Mol Imaging 28:450–456
Kalliokoski KK, Kuusela TA, Laaksonen MS, Knuuti J, Nuutila P (2003) Muscle fractal vascular branching pattern and microvascular perfusion heterogeneity in endurance-trained and untrained men. J Physiol 546:529–535
Acknowledgments
The “Michele Rodriguez” Foundation is acknowledged for the scientific support.
Funding
The Italian Association for Research on Cancer (AIRC—Associazione Italiana per la Ricerca sul Cancro) provided financial support for the research with the grant no. 18923.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Grizzi, F., Castello, A., Qehajaj, D. et al. The Complexity and Fractal Geometry of Nuclear Medicine Images. Mol Imaging Biol 21, 401–409 (2019). https://doi.org/10.1007/s11307-018-1236-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11307-018-1236-5