Fractional calculus in viscoelasticity: an experimental study. (English) Zbl 1221.74012

Summary: Viscoelastic properties of soft biological tissues provide information that may be useful in medical diagnosis. Noninvasive elasticity imaging techniques, such as Magnetic Resonance Elastography (MRE), reconstruct viscoelastic material properties from dynamic displacement images. The reconstruction algorithms employed in these techniques assume a certain viscoelastic material model and the results are sensitive to the model chosen. Developing a better model for the viscoelasticity of soft tissue-like materials could improve the diagnostic capability of MRE. The well known “integer derivative” viscoelastic models of Voigt and Kelvin, and variations of them, cannot represent the more complicated rate dependency of material behavior of biological tissues over a broad spectral range. Recently the “fractional derivative” models have been investigated by a number of researchers. Fractional order models approximate the viscoelastic material behavior of materials through the corresponding fractional differential equations. This paper focuses on the tissue mimicking materials CF-11 and gelatin, and compares fractional and integer order models to describe their behavior under harmonic mechanical loading. Specifically, Rayleigh (surface) waves on CF-11 and gelatin phantoms are studied, experimentally and theoretically, in order to develop an independent test bed for assessing viscoelastic material models that will ultimately be used in MRE reconstruction algorithms.


74D05 Linear constitutive equations for materials with memory
74-05 Experimental work for problems pertaining to mechanics of deformable solids
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
74L15 Biomechanical solid mechanics
Full Text: DOI


[1] Fung, Y.C., Biomechanics: mechanical properties of living tissues, (1993), Springer-Verlag New York
[2] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, () · Zbl 0917.73004
[3] Schiessel, H.; Friedrich, C.; Blumen, A., Applications to problems in polymer physics and rheology, () · Zbl 1046.82039
[4] Magin, R.L., Fractional calculus in bioengineering, Crit rev biomed eng, 32, 1, 1-104, (2004)
[5] Hilfer, R., Fractional calculus in bioengineering, (2000), World Scientific River Edge, NJ · Zbl 1046.82009
[6] Grimnes, S.; Martinsen, O.G., Bioimpedance and bioelectricity basics, (2000), Academic Press San Diego, CA
[7] Lakes, R.S., Viscoelastic solids, (1999), CRC Press Boca Raton, FL
[8] Bard, A.J.; Faulkner, L.R., Electrochemical methods: fundamentals and applications, (2001), John Wiley & Sons New York, NY
[9] Muthupillai, R.; Lomas, D.; Rossman, P.; Greenleaf, J.; Manduca, A.; Ehman, R., Magnetic resonance elastography by direct visualization of propagating acoustic strain waves, Science, 269, 5232, 1854-1857, (1995)
[10] Muthupillai, R.; Rossman, P.; Lomas, D.; Greenleaf, J.; Riederer, S.; Ehman, R., Magnetic resonance imaging of transverse acoustic strain waves, Magn reson med, 36, 2, 266-274, (1996)
[11] Manduca, A.; Oliphant, T.E.; Dresner, M.A.; Mahowald, J.L.; Kruse, S.A.; Amromin, E., Magnetic resonance elastography: non-invasive mapping of tissue elasticity, Med image anal, 5, 4, 237-254, (2001)
[12] Romano, A.J.; Shirron, J.; Bucaro, J., On the non-invasive deformation of material parameters from a knowledge of elastic displacement: theory & simulation, Trans ultrason ferroelectr freq control, 45, 3, 751-759, (1998)
[13] Romano, A.J.; Bucaro, J.; Ehman, R.; Shirron, J., Evaluation of a material parameter extraction algorithm using mri-based displacement measurements, Trans ultrason ferroelectr freq control, 47, 6, 1575-1581, (2000)
[14] Oliphant, T.E.; Manduca, A.; Ehman, R.L.; Greenleaf, J.F., Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation, Magn reson med, 45, 2, 299-310, (2001)
[15] VanHouten, E.E.; Doyley, M.M.; Kennedy, F.E.; Weaver, J.B.; Paulsen, K.D., Initial in vivo experience with steady-state subzone-based mr elastography of the human breast, J magn reson imaging, 17, 1, 72-85, (2003)
[16] VanHouten, E.E.; Doyley, M.M.; Kennedy, F.E.; Paulsen, K.D.; Weaver, J.B., A three-parameter mechanical property reconstruction method for mr-based elastic property imaging, IEEE trans med imaging, 24, 3, 311-324, (2005)
[17] Chen, Q.; Suki, B.; An, K.-N., Dynamic mechanical properties of agarose gels modeled by a fractional derivative model, J biomech eng, 126, 5, 666-671, (2004)
[18] Craiem, D.; Armentano, R., A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44, 251263, (2007)
[19] Kiss, M.Z.; Varghese, T.; Hall, T.J., Viscoelastic characterization of in vitro canine tissue, Phys med biol, 49, 18, 4207-4218, (2004)
[20] Klatt, D.; Hamhaber, U.; Asbach, P.; Braun, J.; Sack, I., Noninvasive assessment of the rheological behavior of human organs using multifrequency mr elastography: a study of brain and liver viscoelasticity, Phys med biol, 52, 24, 7281-7294, (2007)
[21] Sinkus, R.; Siegmann, K.; Xydeas, T.; Tanter, M.; Claussen, C.; Fink, M., Mr elastography of breast lesions: understanding the solid/liquid duality can improve the specificity of contrast-enhanced mr mammography, Magn reson med, 58, 6, 1135-1144, (2007)
[22] Royston, T.J.; Mansy, H.A.; Sandler, R.H., Excitation and propagation of surface waves on a viscoelastic half-space with application to medical diagnosis, J acoust soc am, 106, 6, 3678-3686, (1999)
[23] Oestreicher, H.L., Field and impedance of an oscillating sphere in a viscoelastic medium with an application to biophysics, J acoust soc am, 23, 707-714, (1951)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.