The role of fractional calculus in modeling biological phenomena: a review. (English) Zbl 1467.92050
Summary: This review provides the latest developments and trends in the application of fractional calculus (FC) in biomedicine and biology. Nature has often showed to follow rather simple rules that lead to the emergence of complex phenomena as a result. Of these, the paper addresses the properties in respiratory lung tissue, whose natural solutions arise from the midst of FC in the form of non-integer differ-integral solutions and non-integer parametric models. Diffusion of substances in human body, e.g. drug diffusion, is also a phenomena well known to be captured with such mathematical models. FC has been employed in neuroscience to characterize the generation of action potentials and spiking patters but also in characterizing bio-systems (e.g. vegetable tissues). Despite the natural complexity, biological systems belong as well to this class of systems, where FC has offered parsimonious yet accurate models. This review paper is a collection of results and literature reports who are essential to any versed engineer with multidisciplinary applications and bio-medical in particular.
Keywords:
fractional calculus; fractional order impedance; impedance models; diffusion; respiratory impedance; biomedicine; power-lawReferences:
| [1] | Adolfsson, K.; Enelund, M.; Olsson, M., On the fractional order model of viscoelasticity, Mechanics of Time- Dependent Materials, 9, 15-34 (2005) |
| [2] | Agrawal, O., Symposium on fractional derivatives and applications in engineering and sciences, ASME International Design Engineering Technical Conference, Chicago, 1-385 (2002) |
| [3] | Ala, G.; Di Paola, M.; Francomano, E.; Li, Y.; Pinnola, F., Electrical analogous in viscoelasticity, Commun Nonlinear Sci Numer Simul, 19, 2513-2527 (2014) · Zbl 1455.74022 |
| [4] | Amblard, F.; Maggs, A.; Yurke, B.; Pargellis, A.; Leibler, S., Subdiffusion and anomalous viscoelasticity in actin networks, Phys Rev Lett, 77, 4470-4473 (1996) |
| [5] | Ando, Y.; Maeda, Y.; Mizutani, K.; Wakatsuki, N.; Hagiwara, S.; Nabetani, H., Effect of air-dehydration pretreatment before freezing on the electrical impedance characteristics and texture of carrots, J Food Eng, 169, 114-121 (2016) |
| [6] | Ando, Y.; Mizutani, K.; Wakatsuki, N., Electrical impedance analysis of potato tissues during drying, J Food Eng, 121, 24-31 (2014) |
| [7] | Atanackovic, T.; Pilipovic, S.; Zorica, D., A diffusion wave equation with two fractional derivatives of different order, J Pharmacokinetics Pharmacodynamics, 37, 507-524 (2010) |
| [8] | Babik, B.; T. Asztalo and, F. P.; Deak, Z.; Hantos, Z., Changes in respiratory mechanics during cardiac surgery, Anesth Analg, 96(5), 1280-1287 (2003) |
| [9] | Bagley, R.; Torvik, P., Fractional calculus - a different approach to the analysis of viscoelastically damped structures, AIAA J, 21, 741-748 (1983) · Zbl 0514.73048 |
| [10] | Bagley, R.; Torvik, P., Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J, 23, 918-925 (1985) · Zbl 0562.73071 |
| [11] | Bak, P., How nature works. the science of self-organized criticality, Springer-Verlag, New York, NY, 31-37 (1996) · Zbl 0894.00007 |
| [12] | Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J., Fractional calculus models and numerical methods (2012), World Scientific Publishing Company · Zbl 1248.26011 |
| [13] | Bao, G.; Mitragotri, S.; Tong, S., Multifunctional nanoparticles for drug delivery and molecular imaging, Annu Rev Biomed Eng, 15, 253-282 (2013) |
| [14] | Barabasi, A.; Albert, R., Emergence of scaling in random networks, Science, 286(5439), 509-512 (1999) · Zbl 1226.05223 |
| [15] | Barkai, E.; Klafter, J., Comment on subdiffusion and anomalous viscoelasticity in actin networks, Phys Rev Lett, 81, p.1134 (1998) |
| [16] | Bates, J., A recruitment model of quasi-linear power-law stress adaptation in lung tissue, Ann Biomed Eng, DOI 10.1007/s10439-007-9291-0 (2007) |
| [17] | Bates, J., Lung mechanics, an inverse modelling approach (2009), Cambridge University Press |
| [18] | Battaglia, J.; Cois, O.; Puigsegur, L.; Oustaloup, A., Solving an inverse heat conduction problem using a non-integer identified model, Int J Heat Mass Transfer, 44, 2671-2680 (2001) · Zbl 0981.80007 |
| [19] | Beaulieu, A.; Bosse, D.; Micheau, P.; Avoine, O.; Praud, J.; Walti, H., Measurement of fractional order model parameters of respiratory mechanical impedance in total liquid ventilation, IEEE Trans Biomed Eng, 59(2), 323-331 (2012) |
| [20] | Beierlein, M.; Gibson, J.; Connors, B., Two dynamically distinct inhibitory networks in layer 4 of the neocortex, J Neurophysiol, 90, -, 2987-3000 (2003) |
| [21] | Benchellal, A.; Poinot, T.; Trigeassou, J., Approximation and identification of diffusive interfaces by fractional models, Signal Process, 86, 2712-2727 (2006) · Zbl 1172.93398 |
| [22] | Benson, D.; Tadjeran, C.; Meerschaert, M.; Farnham, I.; Pohll, G., Radial fractional-order dispersion through fractured rock, Water Resour Res, 40, W12416 (2004) |
| [23] | Berg, J., Biochemistry (2002), Freeman and Company: Freeman and Company New York |
| [24] | Berkowitz, B.; Scher, H., Anomalous transport in random fracture networks, Phys Rev Lett, 79, 4038-4041 (1997) |
| [25] | Black, K.; Suki, B.; Madwed, J.; Jackson, A., Airway resistance and tissue elastance from input or transfer impedance in bronchoconstricted monkeys, J Appl Physiol, 90(2), 571-578 (1985) |
| [26] | Borges, E.; Matos, A.; Cardoso, J.; Correia, C.; Vasconcelos, T.; Gomes, N., Early detection and monitoring of plant diseases by bioelectric impedance spectroscopy, IEEE 2nd Portuguese Meeting in Bioengineering (ENBENG), 1-4 (2012), IEEE |
| [27] | Boser, S.; Park, H.; Perry, S.; Menache, M.; Green, F., Fractal geometry of airway remodeling in human asthma, Am J Respir Crit Care Med, 172(7), 817-823 (2005) |
| [28] | Bosse, D.; Beaulieu, A.; Avoine, O.; Micheau, P.; Praud, J.; Walti, H., Neonatal total liquid ventilation: is low frequency forced oscillation technique suitable for respiratory mechanics assessment?, J Appl Physiol, 109, 501-510 (2010) |
| [29] | Böttcher, C.; van Belle, O.; Bordewijk, P.; Rip, A., Theory of electric polarization (1978), Elsevier Science Ltd |
| [30] | Brennan, S.; Hall, G.; Horak, F.; Moeller, A.; Pitrez, P.; Franzmann, A., Correlation of forced oscillation technique in preschool children with cystic fibrosis with pulmonary inflammation, Thorax, 60, 159-163 (2005) |
| [31] | Brenner, N.; Bialek, W.; de Ruyter van Steveninck, R., Adaptive rescaling maximizes information transmission, Neuron, 26, 3, 695-702 (2000) |
| [32] | Cao, Y.; Repo, T.; Silvennoinen, R.; Lehto, T.; Pelkonen, P., Analysis of the willow root system by electrical impedance spectroscopy, J Exp Bot, 62, 1, 351-358 (2011) |
| [33] | Caputo, M., Linear models of dissipation whose q is almost frequency independent, J Geophys, 13, 529-539 (1967) |
| [34] | Carpinteri, A.; Mainardi, F., Fractals and fractional calculus in continuum mechanics (1997), Springer, Wien, New York · Zbl 0917.73004 |
| [35] | Cavalcanti, J.; Lopes, A.; Jansen, J.; Melo, P., Detection of changes in respiratory mechanics due to increasing degrees of airway obstruction in asthma by the forced oscillation technique, Respir Med, 100(12), 2207-2219 (2006) |
| [36] | Chowdhury, A.; Bera, T.; Ghoshal, D.; Chakraborty, B., Studying the electrical impedance variations in banana ripening using electrical impedance spectroscopy (eis), Third International conference on computer, communication, control and information technology (C3IT), 1-4 (2015), IEEE |
| [37] | Claret, L.; Iliadis, A.; Macheras, P., A stochastic model describes the heterogeneous pharmacokinetics of cyclosporin, J Pharmacokinet Pharmacodyn, 28, 5, 445-463 (2001) |
| [38] | Cole, K.; Cole, R., Dispersion and absorption in dielectrics i. alternating current characteristics, J Chem Phys, 9, 4, 341-351 (1941) |
| [39] | da Costa, G.; Faria, A. D.; Mango, A. D.; Lopes, A.; de Melo, P., Respiratory impedance and response to salbutamol in healthy individuals and patients with copd, Respiration, 88, 101-111 (2014) |
| [40] | Craiem, D.; Armentano, R., A fractional derivative model to describe arterial viscoelasticity, Biorheology, 44, 251-263 (2007) |
| [41] | Davidson, D.; Cole, R., Dielectric relaxation in glycerol, propylene glycol, and \(n\)-propanol, J Chem Phys, 19, 12, 1484-1490 (1951) |
| [42] | Deb, K., Multi-objective optimization using evolutionary algorithms, 16 (2001), John Wiley & Sons · Zbl 0970.90091 |
| [43] | Debye, P., Interferenz von röntgenstrahlen und wärmebewegung, Ann Phys, 348, 1, 49-92 (1913) |
| [44] | Debye, P., Polar molecules (1929), Chemical Catalog Company, Incorporated · JFM 55.1179.04 |
| [45] | Dejmek, P.; Miyawaki, O., Relationship between the electrical and rheological properties of potato tuber tissue after various forms of processing, Biosci Biotechnol Biochem, 66, 6, 1218-1223 (2002) |
| [46] | Destexhe, A.; Mainen, Z.; Sejnowski, T., Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism, J Comput Neurosci, 1, 3, 195-230 (1994) |
| [47] | Di Paola, M.; Pinnola, F.; Zingales, M., Fractional differential equations and related exact mechanical models, Computers and mathematics with applications, 66, 608-620 (2013) · Zbl 1381.74038 |
| [48] | Dokoumetzidis, A.; Macheras, P., A population growth model of dissolution, Pharm Res, 14, 1122-1126 (1997) |
| [49] | Dokoumetzidis, A.; Macheras, P., A tube model for transport and dispersion in the circulatory system based on the vascular fractal tree, Ann Biomed Eng, 31, 284-293 (2003) |
| [50] | Dokoumetzidis, A.; Macheras, P., Fractional kinetics in drug absorption and disposition processes, J Pharmacokinet Pharmacodyn, 36, 165-178 (2009) |
| [51] | Dokoumetzidis, A.; Magin, R.; Macheras, P., A commentary on fractionalization of multi-compartmental models, J Pharmacokinet Pharmacodyn, 37, 203-207 (2010) |
| [52] | Dokoumetzidis, A.; Magin, R.; Macheras, P., Fractional kinetics in multi-compartmental systems, J Pharmacokinet Pharmacodyn, 37, 507-524 (2010) |
| [53] | Dorkin, H.; Lucthen, K.; Jackson, A., Human respiratory input impedance from 200 Hz: physiological and modeling consideration, Am Physiol Soc, 823-831 (1998) |
| [54] | Drew, P.; Abbott, L., Models and properties of power-law adaptation in neural systems, J Neurophysiol, 96, 2, 826-833 (2006) |
| [55] | DuBois, A.; Brody, A.; Lewis, D.; Burgess, B., Oscillation mechanics of lungs and chest in man, J Appl Physiol, 8(6), 587-594 (1956) |
| [56] | Dumais, J.; Forterre, Y., Vegetable dynamics: the role of water in plant movements, Ann Rev Fluid Dyn, 44, 453-478 (2012) · Zbl 1347.76066 |
| [57] | Eckert, J.; Ogawa, J., The chemical control of postharvest diseases: deciduous fruits, berries, vegetables and root/tuber crops, Ann Rev Biomed Eng, 26, 433-469 (1988) |
| [58] | Eke, A.; Herman, P.; Kocsis, I.; Kozak, L., Fractal characterization of complexity in temporal physiological signals, Physiol Meas, 23, R1-R38 (2002) |
| [59] | Ellerkmann, R.; Riazanski, V.; Elger, C.; Urban, B.; Beck, H., Slow recovery from inactivation regulates the availability of voltage-dependent na(+) channels in hippocampal granule cells, hilar neurons and basket cells, J Physiol, 532, Pt2, 385-397 (2001) |
| [60] | Ellis, T.; Murray, W.; Kavalieris, L., Electrical capacitance of bean (vicia faba) root systems was related to tissue density-test for the Dalton model, Plant Soil, 366, 1-2, 575-584 (2013) |
| [61] | Emmert, S.; Wolf, M.; Gulich, R.; Krohns, S.; Kastner, S.; Lunkenheimer, P., Electrode polarization effects in broadband dielectric spectroscopy, Eur Phys J B, 83, 2, 157-165 (2011) |
| [62] | Famulare, M.; Fairhall, A., Feature selection in simple neurons: how coding depends on spiking dynamics, Neural Comput, 22, -, 1-18 (2009) |
| [63] | Farre, R.; Peslin, R.; Oostveen, E.; Suki, B.; Duvivier, C.; Navajas, D., Human respiratory impedance from 8 to 256 Hz corrected for upper airway shunt, J Appl Physiol, 72, 427-433 (1992) |
| [64] | Feldman, Y.; Puzenko, A.; Ryabov, Y., Non-debye dielectric relaxation in complex materials, Chem Phys, 284, 1, 139-168 (2002) |
| [65] | Fredberg, J.; Stamenovic, D., On the imperfect elasticity of lung tissue, J Appl Physiol, 67, 2408-2419 (1989) |
| [66] | Freeborn, T., A survey of fractional-order circuit models for biology and biomedicine, IEEE J Emerging Sel Top Circuits Syst, 3, 3, 416-424 (2013) |
| [67] | Freeborn, T.; Maundy, B.; Elwakil, A., Cole impedance extractions from the step-response of a current excited fruit sample, Comput Electron Agric, 98, 100-108 (2013) |
| [68] | Fröhlich, H., Theory of dielectrics: dielectric constant and dielectric loss (1958), Clarendon Press · Zbl 0079.23603 |
| [69] | Fuite, J.; Marsh, R.; Tuszynski, J., Fractal pharmacokinetics of the drug in the liver, Physical Rev, 22 (2002) |
| [70] | Gabano, J.; Poinot, T., Estimation of thermal parameters using fraction al modelling, Signal Process, 91, 938-948 (2011) · Zbl 1217.80150 |
| [71] | Gabano, J.; Poinot, T., Fractional modelling and identification of thermal systems, Signal Process, 91, 531-541 (2011) · Zbl 1203.94029 |
| [72] | Gal, A.; Eytan, D.; Wallach, A.; Sandler, M.; Schiller, J.; Marom, S., Dynamics of excitability over extended timescales in cultured cortical neurons, J Neurosci, 30, 48, 16332-16342 (2010) |
| [73] | Garra, R.; Giusti, A.; Mainardi, F.; Pagnini, G., Fractional relaxation with time-varying coefficient, Fractional Calculus Appl Anal, 17, 2, 424-439 (2014) · Zbl 1305.26018 |
| [74] | Glenny, R., Heterogeneity in the lung, Lung Biology in Health and Disease, 121, 571-609 (1998), Marcel Dekker, USA |
| [75] | Golberg, D., Genetic algorithms in search, optimization, and machine learning, Addion Wesley, 1989 (1989) · Zbl 0721.68056 |
| [76] | Goutelle, S.; Maurin, M.; Rougier, F.; Barbaut, X.; Bourguignon, L.; Ducher, M., The Hill equation: a review of its capabilities in pharmacological modelling, Fundam Clin Pharmacol, 22, 633-648 (2008) |
| [77] | Greenham, G., Bruise and pressure injury in apple fruits, J Exp Bot, 17, 2, 404-409 (1966) |
| [78] | Greenham, G.; Helms, K.; Müller, W., Influence of virus inflections on impedance parameters, J Exp Bot, 29, 4, 867-877 (1978) |
| [79] | Grotberg, J., Respiratory fluid mechanics and transport processes annual review on biomedical engineering, Ann Rev of Biomed Engi, 3, 421-457 (2001) |
| [80] | Gutmaniene, N.; Svirskiene, N.; Svirskis, G., Firing properties of frog tectal neurons in vitro, Brain Res, 981, -, 213-216 (2003) |
| [81] | Habib, R.; Chalker, R.; Jackson, A., Airway geometry and wall mechanical properties estimated from subglottal input impedance in humans, J Appl Physiol, 77(1), 441-451 (1994) |
| [82] | Hall, G.; Hantos, Z.; Sly, P., Altered respiratory tissue mechanics in asymptomatic wheezy infants, Am J Respir Crit Care Med, 164(8), 1387-1391 (2001) |
| [83] | Hantos, Z.; Adamicz, A.; Govaerts, E.; Daroczy, B., Mechanical impedances of lungs and chest wall in the cat, J Appl Physiol, 73(2), 427-433 (1992) |
| [84] | Hantos, Z.; Daroczy, B.; Suki, B.; Galgoczy, G.; Csendes, T., Forced oscillatory impedance of the respiratory system at low frequencies, J Appl Physiol, 60(1), 123-132 (1985) |
| [85] | Hantos, Z.; Daroczy, B.; Suki, B.; Nagy, S.; Fredberg, J., Input impedance and peripheral inhomogeneity of dog lungs, J Appl Physiol, 72, 168-178 (1992) |
| [86] | Havriliak, S.; Negami, S., A complex plane analysis of \(α\)-dispersions in some polymer systems, Journal of polymer science part C: Polymer symposia, 14, 99-117 (1966), Wiley Online Library |
| [87] | Hennion, M.; Hanert, E., How to avoid unbouded drug accumulation with fractional pharmacokinetics, J Pharmacokinet Pharmacodyn, 40, 691-700 (2013) |
| [88] | Henry, B.; Langlands, T.; Wearne, S., Fractional cable models for spiny neuronal dendrites, Phys Rev Lett, 28, 1-4 (2008) |
| [89] | Heymans, N.; Bauwens, J., Fractal rheological models and fractional differential equations for viscoelastic behavior, Rheol Acta, 33, 210-219 (1994) |
| [90] | Higaki, K.; Yamashita, S.; Amidon, G., Time-dependent oral absorption models, J Pharmacokinet Pharmacodyn, 28, 109-128 (2001) |
| [91] | Hildebrandt, J., Comparison of mathematical models for cat lung and viscoelastic balloon derived by Laplace transform methods from pressure-volume data, Bull Math Biophys, 31, 651-667 (1969) · Zbl 0181.47901 |
| [92] | Hildebrandt, J., Pressure-volume data of cat lung interpreted by a plastoelastic, linear viscoelastic model, J Appl Physiol, 28(3), 365-372 (1970) |
| [93] | Hilfer, R., Applications of fractional calculus in physics (2000), World Scientific-Singapore River Edge · Zbl 0998.26002 |
| [94] | Hilfer, R., Analytical representations for relaxation functions of glasses, J Non Cryst Solids, 305, 1, 122-126 (2002) |
| [95] | Hilfer, R., Foundations of fractional dynamics: a short account (2011), World Scientific, Singapore |
| [96] | Hilfer, R.; Metzler, R.; Blumen, A.; Klafter, J., Special issue on strange kinetics, Chem Phys, 284, 1-541 (2002) |
| [97] | Hirai, T.; McKeown, K.; Gomes, R.; Bates, J., Effects of lung volume on lung and chest wall mechanics in rats, J Appl Physiol, 86(1), 16-21 (1985) |
| [98] | Holford, N.; Sheiner, L., Understanding the dose-effect relationship: clinical application of pharmacokinetic-pharmacodynamic models, Clin Pharmacokinet, 6, 429-453 (1999) |
| [99] | Horsfield, K.; Dart, G.; Olson, D.; Cumming, G., Models of the human bronchial tree, J Appl Physiol, 31, 207-217 (1971) |
| [100] | Hou, C.; Gheorghiu, S.; Coppens, M.; Huxley, V.; Pfeifer, P., Gas diffusion through the fractal landscape of the lung, Fractals in Biology and Medicine, IV, 17-30 (2005), Berlin, Birkhauser |
| [101] | Içier, F.; Baysal, T., Dielectrical properties of food materials-1: factors affecting and industrial uses, Crit Rev Food Sci Nutr, 44, 6, 465-471 (2004) |
| [102] | Ionescu, C., Phase constancy in a ladder model of neural dynamics, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 42, 6, 1543-1551 (2012) |
| [103] | Ionescu, C., Emerging tools in engineering: fractional order ladder impedance models for respiratory and neural systems, IEEE Emerging and Selected Topics in Circuits and Systems, 3(3), 425-431 (2013) |
| [104] | Ionescu, C., The human respiratory system: an analysis of the interplay between anatomy, structure, breathing and fractal dynamics (2013), Springer Verlag · Zbl 1273.92016 |
| [105] | Ionescu, C.; Copot, D.; Copot, C.; Keyser, R. D., Bridging the gap between modelling and control of anesthesia: an ambitious ideal, Fractional Differ Appl, 1-6 (2014) |
| [106] | Ionescu, C.; Derom, E.; Keyser, R. D., Modelling respiratory impedance in patients with kyphoscoliosis, Biomed Signal Process Control, 11, 36-41 (2014) |
| [107] | Ionescu, C.; Desager, K.; Keyser, R. D., Fractional order model parameters for the respiratory input impedance in healthy and in asthmatic children, Comput Programs Methods Med, 101(3), 315-323 (2011) |
| [108] | Ionescu, C.; Desager, K.; Vandersteen, G.; Keyser, R. D., Respiratory mechanics in children with cystic fibrosis, Biomed Signal Process Control, 11, 74-79 (2014) |
| [109] | Ionescu, C.; Hernandez, A.; Keyser, R. D., A recurrent parameter model to characterize the high-frequency range of respiratory impedance in healthy subjects, IEEE Trans Biomed Eng, 7(6), 882-892 (2013) |
| [110] | Ionescu, C.; Keyser, R. D., Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease, IEEE Trans Biomed Eng, 56, 978-987 (2009) |
| [111] | Ionescu, C.; Keyser, R. D.; Torrico, B.; Smet, T. D.; Struys, M.; Normey-Rico, J., Robust predictive control strategy applied for propofol dosing using bis as a controlled variable during anesthesia, IEEE Trans Biomed Eng, 55, 2161-2170 (2008) |
| [112] | Ionescu, C.; Machado, J.; Keyser, R. D., Fractional order impulse response of the respiratory system, Comput Math Applications, 62, 845-854 (2011) · Zbl 1228.65251 |
| [113] | Ionescu, C.; Machado, J.; Keyser, R. D., Analysis of the respiratory dynamics during normal breathing by means of pseudophase plots and pressure-volume loops, IEEE Trans Syst, Man Cybern, 43(1), 53-62 (2013) |
| [115] | Jesus, I.; Machado, J.; Cunha, J., Fractional electrical impedances in botanical elements, J Vib Control, 14, 9-10, 1389-1402 (2008) · Zbl 1229.78023 |
| [116] | Kaczka, D.; Lutchen, K.; Hantos, Z., Emergent behavior of regional heterogeneity in the lung and its effects on respiratory impedance, J Appl Physiol, 110, 1473-1481 (2011) |
| [117] | Kaczka, D.; Mitzler, W.; Brown, R., Effects of lung inflation on airway heterogeneity during histaminergic brochoconstriction, J Appl Physiol, 115, 626-633 (2013) |
| [118] | Kaczka, D.; Smallwood, J., Constant-phase descriptions of canine lung, chest wall, and total respiratory system viscoelasticity: effects of distending pressure, Resp Physiol Neurobiol, 183, 75-84 (2012) |
| [119] | Karalis, V.; Dokoumetzidis, A.; Macheras, P., A physiologically based approach for the estimation of recirculatory parameters, J Pharmacol Exp Ther, 308, 198-205 (2004) |
| [120] | Karalis, V.; Macheras, P., Drug disposition viewed in terms of the fractal volume of distribution, Pharm Res, 19, 3287-3291 (2002) |
| [121] | Karalis, V.; Tsantili-Kakoulidou, A.; Macheras, P., Multivariate statistics of disposition pharmacokinetic parameters for structurally unrelated drugs used in therapeutics, Pharm Res, 19, 1827-1834 (2002) |
| [122] | Kertész, A.; Hlaváčová, Z.; Vozáry, E.; Staroňová, L., Relationship between moisture content and electrical impedance of carrot slices during drying, Int Agrophys, 29, 1, 61-66 (2015) |
| [123] | Khaled, D.; Castellano, N.; Gazquez, J.; Salvador, R.; Manzano-Agugliaro, F., Cleaner quality control system using bioimpedance methods: a review for fruits and vegetables, J Clean Prod (2015) |
| [124] | Khaled, D.; Gazquez, J.; Garcia, R.; Manzano-Agugliaro, F., Fruit and vegetable quality assessment via dielectric sensing, Sensors, 15, 7, 15363-15397 (2015) |
| [125] | Klafter, J.; Lim, S.; Metzler, R., Fractional dynamics in physics: recent advances (2011), World Scientific-Singapore |
| [126] | Koch, C., Biophysics of computation: information processing in single neurons (1999), New York: Oxford University Press |
| [127] | Koch, H.; Zacek, H., Fractals also in pharmacokinetics, Pharmazie, 46, H12, 870-871 (1991) |
| [128] | Kopelman, R., Fractal reaction kinetics, J Stat Phys, 42, 1/2, 870-871 (1986) |
| [129] | Kopelman, R., Rate processes on fractals: theory, simulations, and experiments, J Stat Phys, 42, 1/2, 870-871 (1986) |
| [130] | Kopf, M.; Metzler, R.; Haferkamp, O.; Nonnenmacher, T., Nmr studies of anomalous diffusion in biological tissues: experimental observation of Lévy stable processes, Fractals Biol Med, 2, -, 345-364 (1998) |
| [131] | Kosmidis, K.; Argyrakis, P.; Macheras, P., Fractal kinetics in drug release from finite fractal matrices, J Chem Phys, 119, 6373-6377 (1997) |
| [132] | Kosmidis, K.; Karalis, V.; Argyrakis, P.; Macheras, P., Michaelismenten kinetics under spatially constrained conditions: application to mibefradil pharmacokinetics, J Biophys, 87, 1498-1506 (2004) |
| [133] | Kou, X.; Chai, L.; Jiang, L.; Zhao, S.; Yan, S., Modeling of the permittivity of holly leaves in frozen environments, IEEE Trans Geosci Remote Sens, 53, 11, 6048-6057 (2015) |
| [134] | Kuang, W.; Nelson, S., Dielectric relaxation characteristics of fresh fruits and vegetables from 3 to 20 GHz, J Microwave Power Electromagn Energy, 32, 2, 115-123 (1997) |
| [135] | Kufudakis, A., Network with lumped RC parameters as an electro-analog model of diffusion process. simulation of diffusion through membranes, Czech J Phys, 21, 1163-1173 (1971) |
| [136] | Kytariolos, J.; Dokoumetzidis, A.; Macheras, P., Power law ivivc: an application of fractional kinetics for drug release and absorption, Eur J Pharm Sci, 41, 299-304 (2010) |
| [137] | Langlands, T.; Henry, B., Fractional chemotaxis diffusion equations, Phys Rev, 81, 5, 051102 (2010) |
| [138] | Langlands, T.; Henry, B.; Wearne, S., Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions, J Math Biol, 59, 6, 761-808 (2009) · Zbl 1232.92037 |
| [139] | Lansky, P.; Weiss, M., Does the dose-solubility ratio affect the mean dissolution time of drugs?, Pharm Res, 16, 1470-1476 (1999) |
| [140] | Lansky, P.; Weiss, M., Modeling heterogeneity of properties and random effects in drug dissolution, Pharm Res, 18, 1061-1067 (2001) |
| [141] | Laogun, A.; Ajayi, N., Radio-frequency dielectric properties of some tropical african leaf vegetables, Tech. Rep (1985), International Centre for Theoretical Physics, Trieste (Italy) |
| [142] | Larcombe, A.; Zoski, G.; Thamrin, C.; Bozanich, E.; Hantos, Z.; Sly, P., Factors influencing the assessment of lung function in mice with influenza-induced lung disease, Influenza Other Respir Viruses, 7(6), 889-894 (2013) |
| [143] | Laufer, S.; Ivorra, A.; Reuter, V.; Rubinsky, B.; Solomon, S., Electrical impedance characterization of normal and cancerous human hepatic tissue, Physiol Meas, 31, 7, 995 (2010) |
| [144] | Li, S.; Aliyeva, M.; Dphtary, N.; Martin, R.; Poynter, M.; Kostin, S., Antigen-induced mast cell expansion and bronchoconstriction in a mouse model of asthma, Am J Lung Cell Mol Physiol, 306, L196-L206 (2014) |
| [145] | Lijuan, S.; Wenqia, W.; Zhoxima, Y., Finite difference approximations for the fractional advection-diffusion equation, Phys Lett A, 373, 4405-4408 (2009) · Zbl 1234.65034 |
| [146] | Lin, J.; Poinot, T.; Trigeassou, J.; Ouvrard, R., Parameter estimation of fractional systems: application to the modeling of a lead-acid battery, 12th IFAC symposium on system identification (2000) |
| [147] | Lopes, A.; Machado, J., Fractional order models of leaves, J Vib Control, 20, 7, 998-1008 (2014) |
| [148] | Lopes, A.; Machado, J., Modeling vegetable fractals by means of fractional-order equations, J Vib Control (2015) |
| [149] | Lorx, A.; Szabo, B.; Hercsuth, M.; Penzes, I.; Hantos, Z., Low-frequency assessment of airway and tissue mechanics in ventilated copd patients, Journal Appl Physiol, 107(6), 1884-1892 (2009) |
| [150] | Losa, G.; Merlini, D.; Nonnenmacher, T.; Weiber, E., Fractals in biology and medicine, vol. IV (2005), Birkhauser Verlag, Basel |
| [151] | Lundstrom, B.; Fairhall, A.; Maravall, M., Multiple timescale encoding of slowly varying whisker stimulus envelope in cortical and thalamic neurons in vivo, J Neurosci, 30, 14, 50-71 (2010) |
| [152] | Lundstrom, B.; Higgs, M.; Spain, W.; Fairhall, A., Fractional differentiation by neocortical pyramidal neurons, Nat Neurosci, 11, 11, 1335-1342 (2008) |
| [153] | Ma, B.; Breen, B.; Bates, J., Influence of parenchymal heterogeneity on airway-parenchymal interdependence, Respir Physiol Neurobiol, 188(2), 94-101 (2013) |
| [154] | Ma, B.; Sanderson, M.; Bates, J., Airway-parenchymal interdependence in the lung slice, Respir Physiol Neurobiol, 185(2), 211-216 (2013) |
| [155] | Machado, J., Special issue on fractional order calculus and its applications, Nonlinear Dyn, 29, 1-385 (2002) |
| [156] | Machado, J.; Galhano, A., Fractional order inductive phenomena based on the skin effect, Nonlinear Dyn, 68, 1-2, 107-115 (2012) |
| [157] | Machado, J.; Kiryakova, V.; Mainardi, V., Recent history of fractional calculus, Commun Nonlinear Sci Numer Simul, 16(3), 4756-4767 (2011) · Zbl 1236.49030 |
| [158] | Macheas, P., A fractal approach to heterogeneous drug distribution: calcium pharmacokinetics, Pharm Res, 13, 663-670 (1996) |
| [159] | Macheras, P., Carrier-mediated transport can obey fractal kinetics, Pharm Res, 12, 4, 541-548 (1995) |
| [160] | Macheras, P., A fractal approach to heterogeneous drug distribution: calcium pharmacokinetics, Pharm Res, 13, 663-670 (1996) |
| [161] | Macheras, P.; Dokoumetzidis, A., On the heterogeneity of drug dissolution and release, Pharm Res, 17, 108-112 (2000) |
| [162] | Magin, R., Fractional calculus in bioengineering (2006), Begell House Redding |
| [163] | Magin, R., Fractional calculus model of complex dynamics in biological tissues, Computers and mathematics with applications, 59, 1586-1593 (2010) · Zbl 1189.92007 |
| [164] | Magin, R.; Abdullah, O.; Baleanu, D.; Zhou, X. J., Anomalous diffusion expressed through fractional order differential operators in the blochtorrey equation, J Magn Reson, 190, -, 255-270 (2008) |
| [165] | Mainardi, F., Fractional relaxation in anelastic solids, J Alloys Compd, 211, 212, 534-538 (1994) |
| [166] | Mainardi, F., Fractional calculus and waves in linear viscoelasticity (2010), Imperial College Press, World Scientific, London · Zbl 1210.26004 |
| [167] | Maksym, G.; Bates, J., A distributed nonlinear model of lung tissue elasticity, J Appl Physiol, 82(1), 32-41 (1997) |
| [168] | Mancuso, S., Seasonal dynamics of electrical impedance parameters in shoots and leaves related to rooting ability of olive (olea europea) cuttings, Tree Physiol, 19, 2, 95-101 (1999) |
| [169] | Mandelbrot, B., The fractal geometry of nature (1983), NY, Freeman & Co |
| [170] | Marchal, F.; Mazurek, H.; Habib, M.; Duvivier, C.; Derelle, J.; Peslin, R., Input respiratory impedance to estimate airway hyperreactivity in children: standard method versus head generator, Eur Respir J, 7(3), 601-607 (1994) |
| [171] | Maundy, B.; Elwakil, A., Extracting single dispersion Cole-Cole impedance model parameters using an integrator setup, Analog Integr Circuits Signal Process, 71, 1, 107-110 (2012) |
| [172] | Mazurek, H.; Marchal, F.; Derelle, J.; Hatahet, R.; Moneret-Vautrin, D.; Monin, P., Specificity and sensitivity of respiratory impedance in assessing reversibility of airway obstruction in children, Chest, 107(4), 996-1002 (1995) |
| [173] | Meerschaert, M.; Scheffler, H., Limit theorems for continuous-time random walks with infinite mean waiting times, J Appl Probab Trust, 638, 623-638 (2004) · Zbl 1065.60042 |
| [174] | Meerschaert, M.; Straka, P., Fractional dynamics at multiple time, Phys Rev, 149, 5, 578-886 (2012) · Zbl 1263.82040 |
| [175] | Mehauté, A.; Machado, J.; Trigeassou, J.; Sabatier, J., Fractional differentiation and its applications (2005), books Verlag, Diedorf |
| [176] | Mehauté, A. L., Transfer processes in fractal media, J Stat Phys, 36, 5/6, 1632-1647 (1984) |
| [177] | Mercik, S.; Weron, K., Stochastic origins of the long-range correlations of ionic current fluctuations in membrane channels, Phys Rev, 63, 5, 051910 (2001) |
| [178] | de Mesquita Jr, J.; Lopes, A.; Jansen, J.; de Melo, P., Using the forced oscillation technique to evaluate respiratory resistance in individuals with silicosis, Jornal brasileiro de pneumologia, 32(3), 213-220 (2006) |
| [179] | Metzler, R.; Klafter, J., The random’s walk guide to anomalous difussion: a fractional dynamics approach, Phys Rep, 339, 1-77 (2000) · Zbl 0984.82032 |
| [180] | Miller, M.; Okaty, B.; Nelson, S., Region-specific spike-frequency acceleration in layer 5 pyramidal neurons mediated by kv1 subunits, J Neurosci, 28, -, 13716-13726 (2008) |
| [181] | Mizukami, Y.; Yamada, K.; Sawai, Y.; Yamaguchi, Y., Measurement of fresh tea leaf growth using electrical impedance spectroscopy, Agric J, 2, 1, 134-139 (2007) |
| [182] | Mount, L., The ventilation flow-resistance and compliance of rat lungs, J Physiol, 127(1), 157-167 (1955) |
| [183] | Muñoz-Huerta, R.; Ortiz-Melendez, A.; Guevara-Gonzalez, R.; Torres-Pacheco, I.; Herrera-Ruiz, G.; Contreras-Medina, L., An analysis of electrical impedance measurements applied for plant N status estimation in lettuce (lactuca sativa), Sensors, 14, 7, 11492-11503 (2014) |
| [184] | Myers, C.; Fontao, F.; Janos, T.; Boda, K.; Petak, F.; Habre, W., Sevoflurane and desflurane protect cholinergic-induced bronchocostriction of hyperreactive airways in rabbits, Canadian journal of anesthesia, 58, 1007-1015 (2011) |
| [185] | Nelson, S.; Trabelsi, S., Factors influencing the dielectric properties of agricultural and food products, J Microwave Power Electromagn Energy, 46, 2, 93-107 (2012) |
| [186] | Nigmatulin, R.; Baleanu, D.; Al-Zhrani, A.; Alhamed, Y.; Zahid, A.; Youssef, T., Spectral analysis of HIV drugs for acquired immunodeficiency syndrome within modified non-invasive methods, Revista de Chimie - Bucharest (2013) |
| [187] | Nigmatulin, R.; Ionescu, C.; Baleanu, D., Nimrad: novel technique for respiratory data treatment, Signal Image Video Process, 8, 1517-1532 (2014) |
| [188] | Nigmatullin, R., The realization of the generalized transfer in a medium with fractal geometry, Physica Status Solidi B, 133, p.425 (1986) |
| [189] | Nigmatullin, R.; Ionescu, C.; Osokin, S.; Baleanu, D.; Toboev, V., Non-invasive methods applied for complex signals, Rom Rep Phys, 64, 4, 1032-1045 (2012) |
| [190] | Nigmatullin, R.; Nelson, S., Recognition of the “fractional” kinetics in complex systems: dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz, Signal Processing, 86, 10, 2744-2759 (2006) · Zbl 1172.94464 |
| [191] | Nigmatullin, R.; Ryabov, Y., Cole-Davidson dielectric relaxation as a self-similar relaxation process, Phys Solid State, 39, 1, 87-90 (1997) |
| [192] | Novikov, V.; Wojciechowski, K.; Komkova, O.; Thiel, T., Anomalous relaxation in dielectrics. equations with fractional derivatives, Mater Sci, 23, 4, 977 (2005) |
| [193] | Ogihasra, T.; Tamai, I.; Tsuji, A., Application of fractal kinetics for carrier-mediated transport of drugs across intestinal epithelial membrane, Pharm Res, 4, 620-625 (1998) |
| [194] | Ohnishi, S.; Miyawaki, O., Osmotic dehydrofreezing for protection of rheological properties of agricultural products from freezing-injury, Food Sci Technol Res, 11, 1, 52-58 (2005) |
| [195] | Oldham, K.; Spanier, J., The fractional calculus (1974), Academic Press, London · Zbl 0292.26011 |
| [196] | de Oliveira, E.; Mainardi, F.; Vaz, J., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur Phys Journal Special Topics, 193, 1, 161-171 (2011) |
| [197] | Ortigueira, M.; Machado, J., Special section: fractional calculus applications in signals and systems, Signal Processing, 86, 2503-3094 (2006) · Zbl 1172.94301 |
| [198] | Oustaloup, A., Diversity and non-integer differentiation for system dynamics (2014), Wiley · Zbl 1294.93001 |
| [199] | Özarslan, E.; Basser, P.; Shepherd, T.; Thelwall, P.; Vemuri, B.; Blackband, S., Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal, J Magn Reson, 183, -, 315-323 (2006) |
| [200] | Ozier-Lafontaine, H.; Bajazet, T., Analysis of root growth by impedance spectroscopy (EIS), Plant Soil, 277, 1-2, 299-313 (2005) |
| [201] | OShaughnessy, B.; Procaccia, I., Analytical solution for diffusion on fractal objects, Phys Rev Lett, 54, 5, 455-458 (1985) |
| [202] | Peppas, N.; Huang, Y.; Torres-Lugo, M.; Ward, J.; Zhang, J., Physicochemical foundations and structural design of hydrogels in medicine and biology, Ann Rev Biomed Eng, 2, 9-29 (2000) |
| [203] | Petak, F.; Hayden, M.; Hantos, Z.; Sly, P., Volume dependence of respiratory impedance in infants, Am J Respir Crit Care Med, 156, 1172-1177 (1997) |
| [204] | Petras, I.; Magin, R., Simulation of drug uptake in a two compartmental fractional model for a biological system, Communication nonlinear science and numerical simulation, 16, 4588-4595 (2011) · Zbl 1229.92043 |
| [205] | Pham, Q.; Bourgkard, E.; Chau, N.; Willim, G.; Megherbi, S.; Teculescu, D., Forced oscillation technique (fot): a new tool for epidemiology of occupational lung diseases?, Eur Respir J, 8(8), 1307-1313 (1995) |
| [206] | Pliquett, W., Bioimpedance: a review for food processing, Food Eng Rev, 2, 2, 74-94 (2010) |
| [207] | Podlubny, I., Fractional differential equations (1999), Academic Press, San Diego · Zbl 0924.34008 |
| [208] | Poinot, T.; Trigeassou, J., Modelling and simulation of fractional systems, Workshop on fractional differetiation and its applications, 656-663 (2004) |
| [209] | Popovic, J.; Atanackovic, M.; Rapaic, M.; Atanackovic, T., A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac, J Pharmacokinet Pharmacodyn, 37(2), 135-139 (2010) |
| [210] | Popovic, J.; Dolicanin, D.; Rapaic, M.; Popovic, S.; Pilipovic, S.; Atanackovic, T., A nonlinear two compartmental fractional derivative model, Eur J Drug Metab Pharmacokinet, 36, 189-196 (2011) |
| [211] | Popovic, J.; Posa, M.; Popovic, K.; Popovic, D.; Milosevic, N.; Tepavcevic, V. V., Individualizion of a pharmacokinetic model by fractional and nonlinear fit improvement, Eur J Drug Metab Pharmacokinet, 38, 69-76 (2013) |
| [212] | Repo, T.; Pulli, S., Application of impedance spectroscopy for selecting frost hardy varieties of english ryegrass, Ann Bot, 78, 5, 605-609 (1996) |
| [213] | Richardson, L., Atmosphere diffusion shown on a distance-neighbour graph, Proc R Soc Lond A, 110, p.109 (1926) |
| [214] | Rosa, E.; de Oliveira, E., Relaxation equations: fractional models, arXiv preprint arXiv:151001681 (2015) |
| [215] | Sabatier, J.; Agrawal, P.; Machado, J., Advances in fractional calculus: theoretical developments and applications in physics and engineering (2007), Springer, Dordrecht · Zbl 1116.00014 |
| [216] | Sabatier, J.; Aoun, M.; Oustaloup, A.; Gregoire, G.; Ragot, F.; Roy, P., Fractional system identification for lead acid battery state charge estimation, Signal Processing, 10, 2645-2657 (2006) · Zbl 1172.93399 |
| [217] | Salazar, E.; Knowles, J., An analysis of pressure-volume characteristics of the lungs, J Appl Physiol, 19, 97-104 (1964) |
| [218] | Savageau, M., Michaelismenten mechanism reconsidered: implications of fractal kinetics, J Theor Biol, 176, 115-124 (1995) |
| [219] | Schnider, T.; Minto, C.; Gambus, P.; Andresen, C.; Goodale, D.; Youngs, E., The influence of method of administration and covariates on the pharmacokinetics of propofol in adult volunteers, Anesthesiology, 88, 1170-1182 (1998) |
| [220] | Schnider, T.; Minto, C.; Shafer, S.; Gambus, P.; Andresen, C.; Goodale, D., The influence of age on propofol pharmacodynamicss, Anesthesiology, 90, 1502-1516 (1999) |
| [221] | Schwartz, B.; Anafi, R.; Aliyeva, M.; Figueroa, J. T.; Allen, G.; Lundblad, L., Effects of central airway shunting on the mechanical impedance of the mouse lung, Ann Biomed Eng, 39(1), 497-507 (2011) |
| [222] | Sibatov, T.; Uchaikin, D., Fractional relaxation and wave equations for dielectrics characterized by the Havriliak-Negami response function, arXiv preprint arXiv:10083972 (2010) |
| [223] | Siepmann, J.; Siepmann, F., Mathematical modeling of drug delivery, Int J Pharm, 364, 328-343 (2008) |
| [224] | Sierociuk, D.; Dzielinski, A.; Sarwas, G.; Petras, I.; Podlubny, I.; Skovranek, T., Modelling heat transfer in heterogeneous media using fractional calculus, Phisilogical Trans Royal Soc, 371 (2013) · Zbl 1382.80004 |
| [225] | Sierociuk, D.; Skovranek, T.; Macias, M.; Podlubny, I.; Petras, I.; Dzielinski, A., Diffusion process modelling by using fractional order models, Appl Math Comput, 257, 2-11 (2015) |
| [226] | Stanislavsky, A.; Weron, K.; Trzmiel, J., Subordination model of anomalous diffusion leading to the two-power-law relaxation responses, EPL (Europhysics Letters), 91, 4, 40003 (2010) |
| [227] | Suki, B.; Barabasi, A.; Lutchen, K., Lung tissue viscoelasticity: a mathematical framework and its molecular basis, J Appl Physiol, 76(6), 2749-2759 (1994) |
| [228] | Suki, B.; Yuan, H.; Zhang, Q.; Lutchen, K., Partitioning of lung tissue response and inhomogeneous airway constriction at the airway opening, J Appl Physiol, 82(4), 1349-1359 (1997) |
| [229] | Teka, W.; Marinov, T.; Santamaria, F., Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model, PLoS Comput Biol, 10, 3 (2014) |
| [230] | Teka, W.; Stockton, D.; Santamaria, F., Power-law dynamics of membrane conductances increase spiking diversity in a Hodgkin-Huxley model, PLoS Comput Biol, 12, 3, 1-23 (2016) |
| [231] | Teorell, T., Kinetics of distribution of substances administered to the body, Arch Intern Pharmacodyn Ther, 57, 205-225 (1937) |
| [232] | Thamrin, C.; Janosi, T.; Collins, R.; Sly, P.; Hantos, Z., Sensitivity analysis of respiratory parameter estimates in the constant-phase model, Ann Biomed Eng, 32 (6), 815-822 (2004) |
| [233] | Tiitta, M.; Tomppo, L.; Järnström, H.; Löija, M.; Laakso, T.; Harju, A., Spectral and chemical analyses of mould development on scots pine heartwood, Eur J Wood Wood Prod, 67, 2, 151-158 (2009) |
| [234] | Tingay, D.; Polglase, G.; Bhatia, R.; Berry, C.; Kopotic, R.; Kopotic, C., Pressure-limited sustained inflation vs. gradual tidal inflations for resuscitation in preterm lambs, J Appl Physiol, 118, 890-897 (2015) |
| [235] | Tolnai, J.; Szabari, M.; Albu, G.; Maar, B.; Parameswaran, H.; Suki, E., Functional and morphological assessment of early impairment of airway function in a rat model of emphysema, J Appl Physiol, 112, 1932-1939 (2012) |
| [236] | Tomkiewicz, D.; Piskier, T., A plant based sensing method for nutrition stress monitoring, Precis Agric, 13, 3, 370-383 (2012) |
| [237] | Toyoda, K.; Tsenkova, R.; Nakamura, M., Characterization of osmotic dehydration and swelling of apple tissues by bioelectrical impedance spectroscopy, Drying Technol, 19, 8, 1683-1695 (2001) |
| [238] | Uebachs, M.; Schaub, C.; Perez-Reyes, E.; Beck, H., T-Type Ca2+ channels encode prior neuronal activity as modulated recovery rates, J Physiol, 571, Pt3, 519-536 (2006) |
| [239] | Urban, J.; Bequet, R.; Mainiero, R., Assessing the applicability of the earth impedance method for in situ studies of tree root systems, J Exp Bot, 62, 6, 1857-1869 (2011) |
| [240] | Väinölä, A.; Repo, T., Impedance spectroscopy in frost hardiness evaluation of rhododendron leaves, Ann Bot, 86, 4, 799-805 (2000) |
| [241] | Valsami, G.; Dokoumetzidis, A.; Macheras, P., Modeling of supersaturated dissolution data, Int J Pharm, 181, 153-157 (1999) |
| [242] | Verotta, D., Fractional compartmental models and multi-term Mittag-Leffer response functions, J Pharmacokinet Pharmacodyn, 37, 209-215 (2010) |
| [243] | Verotta, D., Fractional dynamics pharmacokinetics-pharmacodynamics models, J Pharmacokinet Pharmacodyn, 37, 257-276 (2010) |
| [244] | Vosika, Z.; Lazarević, M.; Simic-Krstić, J.; Koruga, D., Modeling of bioimpedance for human skin based on fractional distributed-order modified Cole model, FME Trans, 42, 1, 74-81 (2014) |
| [245] | Wada, D.; Ward, D., The hybrid model: a new pharmacokinetic model for computer-controlled infusion pumps, IEEE Trans Biomed Eng, 41, 133-142 (1994) |
| [246] | Wagers, S.; Lundblad, L.; Ekman, M.; Irvin, C.; Bates, J., The allergic mouse model of asthma: normal smooth muscle in an abnormal lung?, J Appl Physiol, 96(6), 2019-2027 (1985) |
| [247] | Wark, B.; Lundstrom, B.; Fairhall, A., Sensory adaptation, Curr Opin Neurobiol, 17, 4, 423-429 (2007) |
| [248] | Watanabe, T.; Orikasa, T.; Shono, H.; Koide, S.; Ando, Y.; Shiina, T., The influence of inhibit avoid water defect responses by heat pretreatment on hot air drying rate of spinach, J Food Eng, 168, 113-118 (2016) |
| [249] | Wawrzkiewicz, A.; Pawelek, K.; Borys, P.; Dworakowska, B.; Grzywna, Z., On the simple random-walk models of ion-channel gate dynamics reflecting long-term memory, Eur Biophys J, 41, 6, 505-526 (2012) |
| [250] | Weibel, E., Morphometry of the human lung (1963), Berlin: Springer |
| [251] | Weinberg, S., Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model, PLoS ONE, 10, 5, e0126629 (2015) |
| [252] | Weiss, J., The hill equation revisited: uses and misuses, FASEB J, 11, 835-841 (1997) |
| [253] | West, B.; Barghava, V.; Goldberger, A., Beyond the principle of similitude: renormalization of the bronchial tree, J Appl Physiol, 60, 1089-1097 (1986) |
| [254] | West, B.; Bologna, M.; Grigolini, P., Physics of Fractal Operators (2003), Springer, New York |
| [255] | West, B.; Schlesinger, M., On the ubiquity of 1/f noise, Int J Mod Phys, 3, 795-819 (1989) |
| [256] | Winkler, T.; Suki, B., Emergent structure-function relations in emphysema and asthma, Crit Rev Biomed Eng, 39(4), 263-280 (2011) |
| [257] | Winkler, T.; Venegas, J., Self-organized patterns of airway narrowing, J Appl Physiol, 110, 1482-1486 (2011) |
| [258] | Wu, L.; Ogawa, Y.; Tagawa, A., Electrical impedance spectroscopy analysis of eggplant pulp and effects of drying and freezing-thawing treatments on its impedance characteristics, J Food Eng, 87, 2, 274-280 (2008) |
| [259] | Wu, L.; Orikasa, T.; Ken, T.; Takeo, S.; Akio, T., Applicability of vacuum-dehydrofreezing technique for the long-term preservation of fresh-cut eggplant: effects of process conditions on the quality attributes of the samples, J Food Eng, 91, 4, 560-565 (2009) |
| [260] | XiaoHong, L.; TingLin, H.; GuoDong, W.; Gang, Z., Effect of salt stress on electrical impedance spectroscopy parameters of wheat (triticum aestivum l.) leaves, J Zhejiang Univ (Agric Life Sci), 35, 5, 564-568 (2009) |
| [261] | Ysasi, A.; Belle, J.; Gibney, B.; Fedulov, A.; Wagner, W.; Tsuda, A., Effect of unilateral diaphragmatic paralysis on postpneumonectomy lung growth, Am J Physiololgy Lung Cell Mol Physiol, 305, L439-L445 (2013) |
| [262] | Zerah, F.; Lorino, A.; Lorino, H.; Harf, A.; Mavier, I. M., Forced oscillation technique vs spirometry to assess bronchodilatation in patients with asthma and copd, Chest, 108(1), 41-47 (1995) |
| [263] | Zhang, M.; Repo, T.; Willison, J.; Sutinen, S., Electrical impedance analysis in plant tissues: on the biological meaning of Cole-Cole \(α\) in scots pine needles, Eur Biophys J, 24, 2, 99-106 (1995) |
| [264] | Zhang, M.; Willison, J., Electrical impedance analysis in plant tissues: the effect of freeze-thaw injury on the electrical properties of potato tuber and carrot root tissues, Can J Plant Sci, 72, 2, 545-553 (1992) |
| [265] | Zhang, M.; Willison, J., Electrical impedance analysis in plant tissues, J Exp Bot, 44, 8, 1369-1375 (1993) |
| [266] | Zhou, X.; Gao, Q.; Abdullah, O.; Magin, R., Studies of anomalous diffusion in the human brain using fractional calculus, Magn Reson Med, 63, -, 562-569 (2010) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
