메뉴 건너뛰기




Volumn 126, Issue 4, 2009, Pages 2072-2081

Fractal ladder models and power law wave equations

Author keywords

[No Author keywords available]

Indexed keywords

ADIABATIC EQUATIONS; ATTENUATION COEFFICIENT; DASHPOTS; EQUIVALENT LUMPED PARAMETERS; FRACTAL NETWORK; FRACTIONAL DERIVATIVES; FREQUENCY-DEPENDENT; KRAMERS-KRONIG RELATIONS; LADDER MODEL; LOW FREQUENCY; LUMPED PARAMETER; MAMMALIAN TISSUES; POWER LAW; POWER-LAW BEHAVIOR; SELF-SIMILAR; SOFT TISSUE; STRESS-STRAIN; ULTRASONIC ATTENUATION;

EID: 70350503592     PISSN: 00014966     EISSN: None     Source Type: Journal    
DOI: 10.1121/1.3204304     Document Type: Article
Times cited : (56)

References (58)
  • 1
    • 0018572279 scopus 로고
    • Ultrasonic absorption and attenuation in mammalian tissues
    • DOI 10.1016/0301-5629(79)90086-3
    • S. A. Goss, L. A. Frizzell, and F. Dunn, " Ultrasonic absorption and attenuation in mammalian tissues.," Ultrasound Med. Biol. 5, 181-186 (1979). 10.1016/0301-5629(79)90086-3 (Pubitemid 10197116)
    • (1979) Ultrasound in Medicine and Biology , vol.5 , Issue.2 , pp. 181-186
    • Goss, S.A.1    Frizzell, L.A.2    Dunn, F.3
  • 2
    • 0022800407 scopus 로고
    • Frequency dependence of ultrasound attenuation and backscatter in breast tissue
    • 10.1016/0301-5629(86)90077-3
    • F. T. D'Astous and F. S. Foster, " Frequency dependence of ultrasound attenuation and backscatter in breast tissue.," Ultrasound Med. Biol. 12, 795-808 (1986). 10.1016/0301-5629(86)90077-3
    • (1986) Ultrasound Med. Biol. , vol.12 , pp. 795-808
    • D'Astous, F.T.1    Foster, F.S.2
  • 3
    • 0032662508 scopus 로고    scopus 로고
    • Experimental verification of models for determining dispersion from attenuation
    • 10.1109/58.764857
    • P. He, " Experimental verification of models for determining dispersion from attenuation.," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 706-714 (1999). 10.1109/58.764857
    • (1999) IEEE Trans. Ultrason. Ferroelectr. Freq. Control , vol.46 , pp. 706-714
    • He, P.1
  • 5
    • 0020849422 scopus 로고
    • On the frequency dependence of attenuation in normal and fatty liver
    • P. A. Narayana and J. Ophir, " On the frequency dependence of attenuation in normal and fatty liver.," IEEE Trans. Sonics Ultrason. SU-30, 379-383 (1983).
    • (1983) IEEE Trans. Sonics Ultrason. , vol.30 , pp. 379-383
    • Narayana, P.A.1    Ophir, J.2
  • 6
    • 0023430894 scopus 로고
    • Frequency-dependent ultrasonic differentiation of normal and diffusely diseased liver
    • 10.1121/1.395303
    • T. Lin, J. Ophir, and G. Potter, " Frequency-dependent ultrasonic differentiation of normal and diffusely diseased liver.," J. Acoust. Soc. Am. 82, 1131-1138 (1987). 10.1121/1.395303
    • (1987) J. Acoust. Soc. Am. , vol.82 , pp. 1131-1138
    • Lin, T.1    Ophir, J.2    Potter, G.3
  • 7
    • 0003983549 scopus 로고
    • (Princeton University Pres, Princeton, NJ)
    • P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Pres, Princeton, NJ, 1968), pp. 270-300.
    • (1968) Theoretical Acoustics , pp. 270-300
    • Morse, P.M.1    Ingard, K.U.2
  • 8
    • 77956725039 scopus 로고
    • Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range
    • 10.1121/1.1908239
    • M. A. Biot, " Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequency range.," J. Acoust. Soc. Am. 28, 168-178 (1956). 10.1121/1.1908239
    • (1956) J. Acoust. Soc. Am. , vol.28 , pp. 168-178
    • Biot, M.A.1
  • 9
    • 0025036253 scopus 로고
    • An equation for acoustic propagation in inhomogeneous media with relaxation losses
    • 10.1121/1.400317
    • A. I. Nachman, J. F. Smith, and R. C. Waag, " An equation for acoustic propagation in inhomogeneous media with relaxation losses.," J. Acoust. Soc. Am. 88, 1584-1595 (1990). 10.1121/1.400317
    • (1990) J. Acoust. Soc. Am. , vol.88 , pp. 1584-1595
    • Nachman, A.I.1    Smith, J.F.2    Waag, R.C.3
  • 10
    • 0017905041 scopus 로고
    • Signal-processing of broad-band pulsed ultrasound-Measurement of attenuation of soft biological tissues
    • 10.1109/TBME.1978.326259
    • A. C. Kak and K. A. Dines, " Signal-processing of broad-band pulsed ultrasound-Measurement of attenuation of soft biological tissues.," IEEE Trans. Biomed. Eng. 25, 321-344 (1978). 10.1109/TBME.1978.326259
    • (1978) IEEE Trans. Biomed. Eng. , vol.25 , pp. 321-344
    • Kak, A.C.1    Dines, K.A.2
  • 11
    • 0020353767 scopus 로고
    • A dispersive model for the propagation of ultrasound in soft tissue
    • DOI 10.1016/0161-7346(82)90019-0
    • K. V. Gurumurthy and R. M. Arthur, " A dispersive model for the propagation of ultrasound in soft tissue.," Ultrason. Imaging 4, 355-377 (1982). 10.1016/0161-7346(82)90019-0 (Pubitemid 13169105)
    • (1982) Ultrasonic Imaging , vol.4 , Issue.4 , pp. 355-377
    • Gurumurthy, K.V.1    Arthur, R.M.2
  • 13
    • 0031698487 scopus 로고    scopus 로고
    • Simulation of ultrasound pulse propagation in lossy media obeying a frequency power law
    • 10.1109/58.646916
    • P. He, " Simulation of ultrasound pulse propagation in lossy media obeying a frequency power law.," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 114-125 (1998). 10.1109/58.646916
    • (1998) IEEE Trans. Ultrason. Ferroelectr. Freq. Control , vol.45 , pp. 114-125
    • He, P.1
  • 14
    • 0029409111 scopus 로고
    • An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation
    • 10.1109/58.476548
    • M. G. Wismer and R. Ludwig, " An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation.," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 1040-1049 (1995). 10.1109/58.476548
    • (1995) IEEE Trans. Ultrason. Ferroelectr. Freq. Control , vol.42 , pp. 1040-1049
    • Wismer, M.G.1    Ludwig, R.2
  • 16
    • 0018894763 scopus 로고
    • Ultrasound pulse propagation in dispersive media
    • S. Leeman, " Ultrasound pulse propagation in dispersive media.," Ultrasound Med. Biol. 25, 481-488 (1980).
    • (1980) Ultrasound Med. Biol. , vol.25 , pp. 481-488
    • Leeman, S.1
  • 17
    • 0020944687 scopus 로고
    • Green's functions for acoustic fields in dispersive media
    • 10.1088/0022-3727/16/12/001
    • J. M. Blackledge and S. Leeman, " Green's functions for acoustic fields in dispersive media.," J. Phys. D: Appl. Phys. 16, L247-L250 (1983). 10.1088/0022-3727/16/12/001
    • (1983) J. Phys. D: Appl. Phys. , vol.16
    • Blackledge, J.M.1    Leeman, S.2
  • 18
    • 0028292355 scopus 로고
    • Time domain wave equations for lossy media obeying a frequency power law
    • DOI 10.1121/1.410434
    • T. L. Szabo, " Time-domain wave-equations for lossy media obeying a frequency power-law.," J. Acoust. Soc. Am. 96, 491-500 (1994). 10.1121/1.410434 (Pubitemid 24209147)
    • (1994) Journal of the Acoustical Society of America , vol.96 , Issue.1 , pp. 491-500
    • Szabo, T.L.1
  • 19
    • 0346305000 scopus 로고    scopus 로고
    • Power-law attenuation in acoustic and isotropic anelastic media
    • DOI 10.1111/j.1365-246X.2003.02086.x
    • A. Hanyga and M. Seredyńska, " Power-law attenuation in acoustic and isotropic anelastic media.," Geophys. J. Int. 155, 830-838 (2003). 10.1111/j.1365-246X.2003.02086.x (Pubitemid 38039118)
    • (2003) Geophysical Journal International , vol.155 , Issue.3 , pp. 830-838
    • Hanyga, A.1    Seredynska, M.2
  • 20
    • 1642641759 scopus 로고    scopus 로고
    • Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency
    • DOI 10.1121/1.1646399
    • W. Chen and S. Holm, " Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency.," J. Acoust. Soc. Am. 115, 1424-1430 (2004). 10.1121/1.1646399 (Pubitemid 38451295)
    • (2004) Journal of the Acoustical Society of America , vol.115 , Issue.4 , pp. 1424-1430
    • Chen, W.1    Holm, S.2
  • 21
    • 0035130739 scopus 로고    scopus 로고
    • Eulerian derivation of the fractional advection-dispersion equation
    • DOI 10.1016/S0169-7722(00)00170-4, PII S0169772200001704
    • R. Schumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, " Eulerian derivation of the fractional advection-dispersion equation.," J. Contam. Hydrol. 48, 69-88 (2001). 10.1016/S0169-7722(00)00170-4 (Pubitemid 32141059)
    • (2001) Journal of Contaminant Hydrology , vol.48 , Issue.1-2 , pp. 69-88
    • Schumer, R.1    Benson, D.A.2    Meerschaert, M.M.3    Wheatcraft, S.W.4
  • 22
    • 84977255207 scopus 로고
    • Linear models of dissipation whose Q is almost frequency independent II
    • M. Caputo, " Linear models of dissipation whose Q is almost frequency independent II.," Geophys. J. R. Astron. Soc. 13, 529-539 (1967).
    • (1967) Geophys. J. R. Astron. Soc. , vol.13 , pp. 529-539
    • Caputo, M.1
  • 23
    • 33845373149 scopus 로고    scopus 로고
    • Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation
    • DOI 10.1121/1.2354032
    • M. G. Wismer, " Finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation.," J. Acoust. Soc. Am. 120, 3493-3502 (2006). 10.1121/1.2354032 (Pubitemid 44888049)
    • (2006) Journal of the Acoustical Society of America , vol.120 , Issue.6 , pp. 3493-3502
    • Wismer, M.G.1
  • 24
    • 0027146124 scopus 로고
    • Representation of the absorption of nonlinear waves by fractional derivatives
    • DOI 10.1121/1.407192
    • M. Ochmann and S. Makarov, " Representation of the absorption of nonlinear waves by fractional derivatives.," J. Acoust. Soc. Am. 94, 3392-3399 (1993). 10.1121/1.407192 (Pubitemid 24000848)
    • (1993) Journal of the Acoustical Society of America , vol.94 , Issue.6 , pp. 3392-3399
    • Ochmann, M.1    Makarov, S.2
  • 25
    • 0033950415 scopus 로고    scopus 로고
    • Transient acoustic wave propagation in rigid porous media: A time-domain approach
    • DOI 10.1121/1.428250
    • Z. E. A. Fellah and C. Depollier, " Transient acoustic wave propagation in rigid porous media: A time-domain approach.," J. Acoust. Soc. Am. 107, 683-688 (2000). 10.1121/1.428250 (Pubitemid 30082794)
    • (2000) Journal of the Acoustical Society of America , vol.107 , Issue.2 , pp. 683-688
    • Fellah, Z.E.A.1    Depollier, C.2
  • 26
    • 0034071987 scopus 로고    scopus 로고
    • A model for longitudinal and shear wave propagation in viscoelastic media
    • DOI 10.1121/1.428630
    • T. L. Szabo and J. Wu, " A model for longitudinal and shear wave propagation in viscoelastic media.," J. Acoust. Soc. Am. 107, 2437-2446 (2000). 10.1121/1.428630 (Pubitemid 30304890)
    • (2000) Journal of the Acoustical Society of America , vol.107 , Issue.5 I , pp. 2437-2446
    • Szabo, T.L.1    Wu, J.2
  • 27
    • 36149036329 scopus 로고
    • Fractional diffusion equation on fractals: One-dimensional case and asymptotic behavior
    • 10.1088/0305-4470/25/8/023
    • M. Giona and H. E. Roman, " Fractional diffusion equation on fractals: One-dimensional case and asymptotic behavior.," J. Phys. A 25, 2093-2105 (1992). 10.1088/0305-4470/25/8/023
    • (1992) J. Phys. A , vol.25 , pp. 2093-2105
    • Giona, M.1    Roman, H.E.2
  • 28
    • 43949160695 scopus 로고
    • Fractional model equation for anomalous diffusion
    • 10.1016/0378-4371(94)90064-7
    • R. Metzler, W. G. Glöckle, and T. F. Nonnenmacher, " Fractional model equation for anomalous diffusion.," Physica A 211, 13-24 (1994). 10.1016/0378-4371(94)90064-7
    • (1994) Physica A , vol.211 , pp. 13-24
    • Metzler, R.1    Glöckle, W.G.2    Nonnenmacher, T.F.3
  • 29
    • 11544266356 scopus 로고
    • Fox function representation of non-Debye relaxation processes
    • 10.1007/BF01058445
    • W. G. Glöckle and T. F. Nonnenmacher, " Fox function representation of non-Debye relaxation processes.," J. Stat. Phys. 71, 741-756 (1993). 10.1007/BF01058445
    • (1993) J. Stat. Phys. , vol.71 , pp. 741-756
    • Glöckle, W.G.1    Nonnenmacher, T.F.2
  • 30
    • 0000902727 scopus 로고
    • Generalized viscoelastic models: Their fractional equations with solutions
    • 10.1088/0305-4470/28/23/012
    • H. Schiessel, R. Metzler, A. Blumen, and T. F. Nonnenmacher, " Generalized viscoelastic models: Their fractional equations with solutions.," J. Phys. A 28, 6567-6584 (1995). 10.1088/0305-4470/28/23/012
    • (1995) J. Phys. A , vol.28 , pp. 6567-6584
    • Schiessel, H.1    Metzler, R.2    Blumen, A.3    Nonnenmacher, T.F.4
  • 31
    • 0020765202 scopus 로고
    • THEORETICAL BASIS FOR THE APPLICATION OF FRACTIONAL CALCULUS TO VISCOELASTICITY.
    • DOI 10.1122/1.549724
    • R. L. Bagley and P. J. Torvik, " A theoretical basis for the application of fractional calculus to viscoelasticity.," J. Rheol. 27, 201-210 (1983). 10.1122/1.549724 (Pubitemid 13540406)
    • (1983) Journal of Rheology , vol.27 , Issue.3 , pp. 201-210
    • Bagley, R.L.1    Torvik, P.J.2
  • 32
    • 1042276663 scopus 로고    scopus 로고
    • Wave equations, dispersion relations, and van Hove singularities for applications of doublet mechanics to ultrasound propagation in bio- and nanomaterials
    • DOI 10.1121/1.1642620
    • J. Wu and C. Layman, " Wave equations, dispersion relations, and van Hove singularities for applications of doublet mechanics to ultrasound propagation in bio- and nanomaterials.," J. Acoust. Soc. Am. 115, 893-900 (2004). 10.1121/1.1642620 (Pubitemid 38200675)
    • (2004) Journal of the Acoustical Society of America , vol.115 , Issue.2 , pp. 893-900
    • Wu, J.1    Layman, C.2    Liu, J.3
  • 34
    • 28444433394 scopus 로고    scopus 로고
    • Mechanical models for living cells - A review
    • DOI 10.1016/j.jbiomech.2004.12.008, PII S0021929004005949
    • C. T. Lim, E. H. Zhou, and S. T. Quek, " Mechanical models for living cells A review.," J. Biomech. 39, 195-216 (2006). 10.1016/j.jbiomech.2004.12.008 (Pubitemid 41728994)
    • (2006) Journal of Biomechanics , vol.39 , Issue.2 , pp. 195-216
    • Lim, C.T.1    Zhou, E.H.2    Quek, S.T.3
  • 35
    • 0024706974 scopus 로고
    • Cortical shell-liquid core model for passive flow of liquid-like spherical cells into micropipets
    • 10.1016/S0006-3495(89)82659-1
    • A. Yeung and E. Evans, " Cortical shell-liquid core model for passive flow of liquid-like spherical cells into micropipets.," Biophys. J. 56, 139-149 (1989). 10.1016/S0006-3495(89)82659-1
    • (1989) Biophys. J. , vol.56 , pp. 139-149
    • Yeung, A.1    Evans, E.2
  • 36
    • 38149080489 scopus 로고    scopus 로고
    • Viscoelastic properties of human mesenchymally-derived stem cells and primary osteoblasts, chondrocytes, and adipocytes
    • 10.1016/j.jbiomech.2007.06.019
    • E. M. Darling, M. Topel, S. Zauscher, T. P. Vail, and F. Guilak, " Viscoelastic properties of human mesenchymally-derived stem cells and primary osteoblasts, chondrocytes, and adipocytes.," J. Biomech. 41, 454-464 (2008). 10.1016/j.jbiomech.2007.06.019
    • (2008) J. Biomech. , vol.41 , pp. 454-464
    • Darling, E.M.1    Topel, M.2    Zauscher, S.3    Vail, T.P.4    Guilak, F.5
  • 38
    • 2542494985 scopus 로고    scopus 로고
    • Rheological properties of living materials. From cells to tissues
    • DOI 10.1080/10273360410001678083
    • C. Verdier, " Rheological properties of living materials. From cells to tissues.," J. Theoretical Medicine 5, 67-91 (2003). 10.1080/10273360410001678083 (Pubitemid 38678943)
    • (2003) Journal of Theoretical Medicine , vol.5 , Issue.2 , pp. 67-91
    • Verdier, C.1
  • 39
    • 70350494099 scopus 로고
    • Ladder structures for representation of viscoelastic systems
    • 10.1002/pol.1956.120199104
    • B. Gross and R. M. Fuoss, " Ladder structures for representation of viscoelastic systems.," J. Polym. Sci. 19, 39-50 (1956). 10.1002/pol.1956.120199104
    • (1956) J. Polym. Sci. , vol.19 , pp. 39-50
    • Gross, B.1    Fuoss, R.M.2
  • 40
    • 36149037022 scopus 로고
    • Hierarchical analogues to fractional relaxation equations
    • 10.1088/0305-4470/26/19/034
    • H. Schiessel and A. Blumen, " Hierarchical analogues to fractional relaxation equations.," J. Phys. A 26, 5057-5069 (1993). 10.1088/0305-4470/26/19/034
    • (1993) J. Phys. A , vol.26 , pp. 5057-5069
    • Schiessel, H.1    Blumen, A.2
  • 41
    • 0029304889 scopus 로고
    • Mesoscopic pictures of the sol-gel transition: Ladder models and fractal networks
    • 10.1021/ma00115a038
    • H. Schiessel and A. Blumen, " Mesoscopic pictures of the sol-gel transition: Ladder models and fractal networks.," Macromolecules 28, 4013-4019 (1995). 10.1021/ma00115a038
    • (1995) Macromolecules , vol.28 , pp. 4013-4019
    • Schiessel, H.1    Blumen, A.2
  • 42
    • 0001488637 scopus 로고
    • Fractal rheological models and fractional differential equations for viscoelastic behavior
    • 10.1007/BF00437306
    • N. Heymans and J.-C. Bauwens, " Fractal rheological models and fractional differential equations for viscoelastic behavior.," Rheol. Acta 33, 210-219 (1994). 10.1007/BF00437306
    • (1994) Rheol. Acta , vol.33 , pp. 210-219
    • Heymans, N.1    Bauwens, J.-C.2
  • 44
    • 0345698053 scopus 로고
    • Ladder structures for representation of viscoelastic systems. II
    • B. Gross, " Ladder structures for representation of viscoelastic systems. II.," J. Polym. Sci. 20, 123-131 (1956).
    • (1956) J. Polym. Sci. , vol.20 , pp. 123-131
    • Gross, B.1
  • 45
    • 0021440218 scopus 로고
    • Applications of fractional calculus to the theory of viscoelasticity
    • 10.1115/1.3167616
    • R. C. Koeller, " Applications of fractional calculus to the theory of viscoelasticity.," J. Appl. Mech. 51, 299-307 (1984). 10.1115/1.3167616
    • (1984) J. Appl. Mech. , vol.51 , pp. 299-307
    • Koeller, R.C.1
  • 46
    • 0344252790 scopus 로고
    • A theory of linear viscoelastic properties of dilute solutions of coiling polymers
    • 10.1063/1.1699180
    • P. E. Rouse, " A theory of linear viscoelastic properties of dilute solutions of coiling polymers.," J. Chem. Phys. 21, 1272-1280 (1953). 10.1063/1.1699180
    • (1953) J. Chem. Phys. , vol.21 , pp. 1272-1280
    • Rouse, P.E.1
  • 47
    • 0000958814 scopus 로고
    • Viscoelastic pulse propagation and stable probability distributions
    • A. Kreis and A. C. Pipkin, " Viscoelastic pulse propagation and stable probability distributions.," Q. Appl. Math. 44, 353-360 (1986).
    • (1986) Q. Appl. Math. , vol.44 , pp. 353-360
    • Kreis, A.1    Pipkin, A.C.2
  • 48
    • 27244437249 scopus 로고    scopus 로고
    • Causality, Stokes' wave equation, and acoustic pulse propagation in a viscous fluid
    • DOI 10.1103/PhysRevE.72.026610, 026610
    • M. J. Buckingham, " Causality, Stokes' wave equation, and acoustic pulse propagation in a viscous fluid.," Phys. Rev. E 72, 026610 (2005). 10.1103/PhysRevE.72.026610 (Pubitemid 41520883)
    • (2005) Physical Review E - Statistical, Nonlinear, and Soft Matter Physics , vol.72 , Issue.2 , pp. 1-9
    • Buckingham, M.J.1
  • 49
    • 0028976328 scopus 로고
    • Causal theories and data for acoustic attenuation obeying a frequency power-law
    • 10.1121/1.412332
    • T. L. Szabo, " Causal theories and data for acoustic attenuation obeying a frequency power-law.," J. Acoust. Soc. Am. 97, 14-24 (1995). 10.1121/1.412332
    • (1995) J. Acoust. Soc. Am. , vol.97 , pp. 14-24
    • Szabo, T.L.1
  • 50
    • 56749132453 scopus 로고    scopus 로고
    • Analytical time-domain Green's functions for power-law media
    • 10.1121/1.2977669
    • J. F. Kelly, M. M. Meerschaert, and R. J. McGough, " Analytical time-domain Green's functions for power-law media.," J. Acoust. Soc. Am. 124, 2861-2872 (2008). 10.1121/1.2977669
    • (2008) J. Acoust. Soc. Am. , vol.124 , pp. 2861-2872
    • Kelly, J.F.1    Meerschaert, M.M.2    McGough, R.J.3
  • 52
    • 0033889221 scopus 로고    scopus 로고
    • On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law
    • DOI 10.1121/1.429586, PII S0001496600031076
    • K. R. Waters, M. S. Hughes, J. Mobley, G. H. Brandenburger, and J. G. Miller, " On the applicability of Kramers-Krönig relations for ultrasonic attenuation obeying a frequency power law.," J. Acoust. Soc. Am. 108, 556-563 (2000). 10.1121/1.429586 (Pubitemid 30625878)
    • (2000) Journal of the Acoustical Society of America , vol.108 , Issue.2 , pp. 556-563
    • Waters, K.R.1    Hughes, M.S.2    Mobley, J.3    Brandenburger, G.H.4    Miller, J.G.5
  • 53
    • 0026348707 scopus 로고
    • Fractal geometry: A design principle for living organisms
    • E. R. Weibel, " Fractal geometry: A design principle for living organisms.," Am. J. Physiol. Lung Cell. Mol. Physiol. 261, L361-L369 (1991).
    • (1991) Am. J. Physiol. Lung Cell. Mol. Physiol. , vol.261
    • Weibel, E.R.1
  • 54
    • 0034660890 scopus 로고    scopus 로고
    • Fractals and cancer
    • J. W. Baish and R. K. Jain, " Fractals and cancer.," Cancer Res. 60, 3683-3688 (2000).
    • (2000) Cancer Res. , vol.60 , pp. 3683-3688
    • Baish, J.W.1    Jain, R.K.2
  • 55
    • 36749120008 scopus 로고
    • Lattices of effectively nonintegral dimensionality
    • 10.1063/1.523316
    • D. Dhar, " Lattices of effectively nonintegral dimensionality., " J. Math. Phys. 18, 577-585 (1977). 10.1063/1.523316
    • (1977) J. Math. Phys. , vol.18 , pp. 577-585
    • Dhar, D.1
  • 56
    • 0020186194 scopus 로고
    • Density of states on fractals: Fractons
    • S. Alexander and R. Orbach, " Density of states on fractals: Fractons.," J. Phys. (Paris) 43, L625-L631 (1982).
    • (1982) J. Phys. (Paris) , vol.43
    • Alexander, S.1    Orbach, R.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.