zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Fractional calculus models of complex dynamics in biological tissues. (English) Zbl 1189.92007
Summary: Fractional (non-integer order) calculus can provide a concise model for the description of the dynamic events that occur in biological tissues. Such a description is important for gaining an understanding of the underlying multiscale processes that occur when, for example, tissues are electrically stimulated or mechanically stressed. The mathematics of fractional calculus has been applied successfully in physics, chemistry, and material sciences to describe dielectrics, electrodes and viscoelastic materials over extended ranges of time and frequency. In heat and mass transfer, for example, the half-order fractional integral is the natural mathematical connection between thermal or material gradients and the diffusion of heat or ions. Since the material properties of tissues arise from the nanoscale and microscale architecture of subcellular, cellular, and extracellular networks, the challenge for bioengineers is to develop new dynamic models that predict macroscale behavior from microscale observations and measurements. We describe three areas of bioengineering research (bioelectrodes, biomechanics, bioimaging) where fractional calculus is being applied to build these new mathematical models.

92C37Cell biology
92C30Physiology (general)
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[1] Keener, J.; Sneyd, J.: Mathematical physiology, (2004) · Zbl 0913.92009
[2] Bar-Yam, Y.: Dynamics of complex systems, (1997) · Zbl 1074.37041
[3] Shelhamer, M.: Nonlinear dynamics in physiology: A state space approach, (2007) · Zbl 1182.92019
[4] Bruce, E. N.: Biomedical signal processing and signal modeling, (2001)
[5] West, B. J.; Bologna, M.; Grigolini, P.: Physics of fractal operators, (2003)
[6] Magin, R. L.: Fractional calculus in bioengineering, (2006)
[7] Hilfer, R.: Applications of fractional calculus in physics, (2000) · Zbl 0998.26002
[8] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[9] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[10] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[11] , Fractals and fractional calculus in continuum mechanics CISM courses and lectures (1997)
[12] Anastasio, T. J.: The fractional-order dynamics of brainstem vestibulo-ocular neurons, Biological cybernetics 72, 69-79 (1994)
[13] Anastosio, T. J.: Nonuniformity in the linear network model of the oculomotor integrator produces approximately fractional-order dynamics and more realistic neuron behavior, Biol cybern. 79, 377-391 (1998) · Zbl 0918.92003 · doi:10.1007/s004220050487
[14] Thorson, J.; Biederman-Thorson, M.: Distributed relaxation processes in sensory adaptation, Science 183, 161-172 (1974)
[15] Magin, R. L.; Ovadia, M.: Modeling the cardiac tissue electrode interface using fractional calculus, Journal of vibration and control 14, 1431-1442 (2008) · Zbl 1229.92018 · doi:10.1177/1077546307087439
[16] Grimnes, S.; Martinsen, O. G.: Bioimpedance and bioelectricity basics, (2000)
[17] Ovadia, M.; Zavitz, D. H.: The electrode--tissue interface in living heart: equivalent circuit as a function of surface area, Electroanalysis 10, 262-272 (1998)
[18] Greatbatch, W.; Chardack, W. M.: Myocardial and endocardial electrodes for chronic implantation, Annals of the New York Academy of sciences 148, 235-251 (1968)
[19] Lakes, R. S.: Viscoelastic solids, (1999) · Zbl 1098.74013
[20] Craiem, D.; Armentano, R. L.: A fractional derivative model to describe arterial viscoelasticity, Biorheology 44, 251-263 (2007)
[21] Craiem, D.; Rojo, F. J.; Atienza, J. M.; Armentano, R. L.; Guinea, G. V.: Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Physics in medicine and biology 53, 4543-4554 (2008)
[22] Puig-De-Morales-Marinkovic, M.; Turner, K. T.; Butler, J. P.; Fredberg, J. J.; Suresh, S.: Viscoelasticity of the human red blood cell, American journal of physiology -- cell physiology 293, 597-605 (2007)
[23] 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, Magnetic resonance in medicine 58, 1135-1144 (2007)
[24] Heymans, N.: Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state, Journal of vibration and control 14, 1587-1596 (2008) · Zbl 1229.74016 · doi:10.1177/1077546307087428
[25] Bates, J. H. T.: A recruitment model of quasi-linear power-law stress adaptation in lung tissue, Annals of biomedical engineering 35, 1165-1174 (2007)
[26] Bergson, H.: Creative evolution, (1998)