The issue of speckle statistics from ultrasound images of soft tissues such as the liver has a long and rich history. A number of theoretical distributions, some related to random scatterers or fades in optics and radar, have been formulated for pulse-echo interference patterns. This work proposes an alternative framework in which the dominant echoes are presumed to result from Born scattering from fluid-filled vessels that permeate the tissue parenchyma. These are modeled as a branching, fractal, self-similar, multiscale collection of cylindrical scatterers governed by a power law distribution relating to the number of branches at each radius. A deterministic accounting of the echo envelopes across the scales from small to large is undertaken, leading to a closed form theoretical formula for the histogram of the envelope of the echoes. The normalized histogram is found to be related to the classical Burr distribution, with the key power law parameter directly related to that of the number density of vessels vs diameter, frequently reported in the range of 2 to 4. Examples are given from liver scans to demonstrate the applicability of the theory.
References
1.
Abramowitz
, M.
, and Stegun
, I. A.
(1964
). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
(U.S. Government Printing Office
, Washington, DC
).2.
Albini
, F. A.
, and Nagelberg
, E. R.
(1962
). “Scattering of a plane wave by an infinite inhomogeneous, dielectric cylinder—an application of the Born approximation
,” J. Appl. Phys.
33
, 1706
–1713
.3.
Bamber
, J. C.
, and Dickinson
, R. J.
(1980
). “Ultrasonic B-scanning: A computer simulation
,” Phys. Med. Biol.
25
, 463
–479
.4.
Bracewell
, R. N.
(1965a
). The Fourier Transform and its Applications
(McGraw-Hill
, New York
), Chap. 8.5.
Bracewell
, R. N.
(1965b
). The Fourier Transform and its Applications
(McGraw-Hill
, New York
), Chap. 12.6.
Burckhardt
, C. B.
(1978
). “Speckle in ultrasound B-mode scans
,” IEEE Trans. Sonics Ultrason.
25
, 1
–6
.7.
Burr
, I. W.
(1942
). “Cumulative frequency functions
,” Ann. Math. Stat.
13
, 215
–232
.8.
Campbell
, J. A.
, and Waag
, R. C.
(1984a
). “Measurements of calf liver ultrasonic differential and total scattering cross sections
,” J. Acoust. Soc. Am.
75
, 603
–611
.9.
Campbell
, J. A.
, and Waag
, R. C.
(1984b
). “Ultrasonic scattering properties of three random media with implications for tissue characterization
,” J. Acoust. Soc. Am.
75
, 1879
–1886
.10.
Chivers
, R. C.
(1977
). “The scattering of ultrasound by human tissues-some theoretical models
,” Ultrasound Med. Biol.
3
, 1
–13
.11.
Cramblitt
, R. M.
, and Parker
, K. J.
(1999
). “Generation of non-Rayleigh speckle distributions using marked regularity models
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
46
, 867
–874
.12.
Feller
, W.
(1948
). “On the Kolmogorov-Smirnov limit theorems for empirical distributions
,” Ann. Math. Statist.
19
, 177
–189
.13.
Fontaine
, I.
, Savery
, D.
, and Cloutier
, G.
(2002
). “Simulation of ultrasound backscattering by red cell aggregates: Effect of shear rate and anisotropy
,” Biophys. J.
82
, 1696
–1710
.14.
Gore
, J. C.
, and Leeman
, S.
(1977
). “Ultrasonic backscattering from human tissue: A realistic model
,” Phys. Med. Biol.
22
, 317
–326
.15.
Insana
, M. F.
, and Brown
, D. G.
(1993
). “Acoustic scattering theory applied to soft biological tissues
,” in Ultrasonic Scattering in Biological Tissues
, edited by K. K.
Shung
and G. A.
Thieme
(CRC Press
, Boca Raton, FL
), pp. 75
–124
.16.
Insana
, M. F.
, Wagner
, R. F.
, Brown
, D. G.
, and Hall
, T. J.
(1990
). “Describing small-scale structure in random media using pulse-echo ultrasound
,” J. Acoust. Soc. Am.
87
, 179
–192
.17.
Kutay
, M. A.
, Petropulu
, A. P.
, and Piccoli
, C. W.
(2001
). “On modeling biomedical ultrasound RF echoes using a power-law shot-noise model
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
48
, 953
–968
.18.
Kutay
, M. A.
, Petropulu
, A. P.
, and Piccoli
, C. W.
(2003
). “Breast tissue characterization based on modeling of ultrasonic echoes using the power-law shot noise model
,” Pattern Recogn. Lett.
24
, 741
–756
.19.
Landini
, L.
, and Verrazzani
, L.
(1990
). “Spectral characterization of tissues microstructure by ultrasounds: A stochastic approach
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
37
, 448
–456
.20.
Laporte
, C.
, Clark
, J. J.
, and Arbel
, T.
(2009
). “Generalized Poisson 3-D scatterer distributions
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
56
, 410
–414
.21.
Lizzi
, F. L.
, Greenebaum
, M.
, Feleppa
, E. J.
, Elbaum
, M.
, and Coleman
, D. J.
(1983
). “Theoretical framework for spectrum analysis in ultrasonic tissue characterization
,” J. Acoust. Soc. Am.
73
, 1366
–1373
.22.
23.
Mamou
, J.
, and Oelze
, M. L.
(2013
). Quantitative Ultrasound in Soft Tissues
(Springer
Netherlands, Dordrecht
).24.
Morse
, P. M.
, and Ingard
, K. U.
(1987
). Theoretical Acoustics
(Princeton University Press
, Princeton, NJ
), Chap. 8.25.
Papoulis
, A.
(1987
). The Fourier Integral and its Applications
(McGraw-Hill
, New York
).26.
Parker
, K. J.
(2016
). “The H-scan format for classification of ultrasound scattering
,” J. OMICS Radiol.
5
, 1000236
.27.
Parker
, K. J.
(2019
). “Shapes and distributions of soft tissue scatterers
,” Phys. Med. Biol.
64
(17
), 175022
.28.
Parker
, K. J.
, Carroll-Nellenback
, J. J.
, and Wood
, R. W.
(2019
). “The 3D spatial autocorrelation of the branching fractal vasculature
,” Acoust.
1
, 369
–381
.29.
Poularikas
, A. D.
(2010
). “Transforms and applications handbook, Chapter 6
,” in The Electrical Engineering Handbook Series
(CRC Press
, Boca Raton, FL)
, p. 7.21
.30.
Prince
, J. L.
, and Links
, J. M.
(2015
). Medical Imaging Signals and Systems
(Pearson
, Boston, MA
).31.
Rayleigh
, L.
(1897
). “XXXVII. On the passage of waves through apertures in plane screens, and allied problems
,” Philos. Mag.
43
, 259
–272
.32.
Rayleigh
, L.
(1918a
). “On the scattering of light spherical shells, and by complete spheres of periodic structure, when the refractivity is small
,” Proc. Royal Soc. London Series A
94
, 296
–300
.33.
Rayleigh
, L.
(1918b
). “XLI. The dispersal of light by a dielectric cylinder
,” London, Edinburgh, Dublin Philos. Mag. J. Sci.
36
, 365
–376
.34.
Rodriguez
, R. N.
(1977
). “A guide to the Burr type XII distributions
,” Biometrika
64
, 129
–134
.35.
Savery
, D.
, and Cloutier
, G.
(2001
). “A point process approach to assess the frequency dependence of ultrasound backscattering by aggregating red blood cells
,” J. Acoust. Soc. Am.
110
, 3252
–3262
.36.
Savery
, D.
, and Cloutier
, G.
(2005
). “Effect of red cell clustering and anisotropy on ultrasound blood backscatter: A Monte Carlo study
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
52
, 94
–103
.37.
Shung
, K. K.
, and Thieme
, G. A.
(1993
). Ultrasonic Scattering in Biological Tissues
(CRC Press
, Boca Raton, FL
).38.
Sleefe
, G. E.
, and Lele
, P. P.
(1988
). “Tissue characterization based on scatterer number density estimation
,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control
35
, 749
–757
.39.
Thijssen
, J. M.
(2003
). “Ultrasonic speckle formation, analysis and processing applied to tissue characterization
,” Pattern Recogn. Lett.
24
, 659
–675
.40.
Tuthill
, T. A.
, Sperry
, R. H.
, and Parker
, K. J.
(1988
). “Deviations from Rayleigh statistics in ultrasonic speckle
,” Ultrason. Imag.
10
, 81
–89
.41.
Waag
, R. C.
, Dalecki
, D.
, and Christopher
, P. E.
(1989a
). “Spectral power determinations of compressibility and density variations in model media and calf liver using ultrasound
,” J. Acoust. Soc. Am.
85
, 423
–431
.42.
Waag
, R. C.
, Dalecki
, D.
, and Smith
, W. A.
(1989b
). “Estimates of wave front distortion from measurements of scattering by model random media and calf liver
,” J. Acoust. Soc. Am.
85
, 406
–415
.43.
Waag
, R. C.
, Lee
, P. P. K.
, Persson
, H. W.
, Schenk
, E. A.
, and Gramiak
, R.
(1982
). “Frequency-dependent angle scattering of ultrasound by liver
,” J. Acoust. Soc. Am.
72
, 343
–352
.44.
Wear
, K. A.
, Wagner
, R. F.
, Brown
, D. G.
, and Insana
, M. F.
(1997
). “Statistical properties of estimates of signal-to-noise ratio and number of scatterers per resolution cell
,” J. Acoust. Soc. Am.
102
, 635
–641
.45.
Zagzebski
, J. A.
, Lu
, Z. F.
, and Yao
, L. X.
(1993
). “Quantitative ultrasound imaging: In vivo results in normal liver
,” Ultrason. Imag.
15
, 335
–351
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