title: Scale-specific multifractal medical image analysis.
abstract: Fractal geometry has been applied widely in the analysis of medical images to characterize the irregular complex tissue structures that do not lend themselves to straightforward analysis with traditional Euclidean geometry. In this study, we treat the nonfractal behaviour of medical images over large-scale ranges by considering their box-counting fractal dimension as a scale-dependent parameter rather than a single number. We describe this approach in the context of the more generalized Rényi entropy, in which we can also compute the information and correlation dimensions of images. In addition, we describe and validate a computational improvement to box-counting fractal analysis. This improvement is based on integral images, which allows the speedup of any box-counting or similar fractal analysis algorithm, including estimation of scale-dependent dimensions. Finally, we applied our technique to images of invasive breast cancer tissue from 157 patients to show a relationship between the fractal analysis of these images over certain scale ranges and pathologic tumour grade (a standard prognosticator for breast cancer). Our approach is general and can be applied to any medical imaging application in which the complexity of pathological image structures may have clinical value.
authors.affiliation: Department of Physics, MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
authors.forename: Boris
authors.initials: B
authors.lastname: Braverman
authors.forename: Mauro
authors.initials: M
authors.lastname: Tambasco
mesh_terms: D000465:Algorithms; D056808:Biostatistics; D001943:Breast Neoplasms / Q000473:pathology; D018270:Carcinoma, Ductal, Breast / Q000473:pathology; D005260:Female; D017709:Fractals; D006801:Humans; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D011379:Prognosis
references.citation: Med Hypotheses. 2000 Mar;54(3):360-6
references.pmid: 10783467
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Mar;65(3 Pt 2B):037201
references.pmid: 11909318
references.citation: Hum Pathol. 2008 Dec;39(12):1859-60; author reply 1860-1
references.pmid: 19010250
references.citation: Med Image Anal. 2009 Aug;13(4):634-49
references.pmid: 19535282
references.citation: Hum Pathol. 2008 May;39(5):740-6
references.pmid: 18439940
references.citation: Med Phys. 1996 Apr;23(4):585-94
references.pmid: 8860906
references.citation: J Transl Med. 2010 Dec 31;8:140
references.pmid: 21194459
references.citation: Cancer Res. 2001 Nov 15;61(22):8347-50
references.pmid: 11719469
references.citation: Nat Med. 1998 Aug;4(8):882-5
references.pmid: 9701230
references.citation: Micron. 2009 Jun;40(4):486-94
references.pmid: 19171487