The Fractal (BSf) Kinetics Equation and Its Approximations
Author(s)
F. Brouers*
ABSTRACT
We discuss the Brouers-Sotolongo fractal (BSf) kinetics model. This formalism interpolates between the first and second order kinetics. But more importantly, it introduces not only a fractional order n but also a fractal time parameter a which characterizes the time variation of the rate constant. This exponent appears in non-exponential relaxation and complex reaction models as demonstrated by the extended use of the Weibull and Hill kinetics which are the two most popular approximations of the BSf (n, a) kinetic equation as well in non-Debye relaxation formulas. We show that the use of nonlinear programs allows an easy and precise fitting of the data yielding the BSf parameters which have simple physical interpretations.
We discuss the Brouers-Sotolongo fractal (BSf) kinetics model. This formalism interpolates between the first and second order kinetics. But more importantly, it introduces not only a fractional order n but also a fractal time parameter a which characterizes the time variation of the rate constant. This exponent appears in non-exponential relaxation and complex reaction models as demonstrated by the extended use of the Weibull and Hill kinetics which are the two most popular approximations of the BSf (n, a) kinetic equation as well in non-Debye relaxation formulas. We show that the use of nonlinear programs allows an easy and precise fitting of the data yielding the BSf parameters which have simple physical interpretations.
KEYWORDS
Fractal Kinetics, Farmacokinetics, Cancer Research, Water Treatment, Adsorption, Porous Materials, Activated Carbons
Fractal Kinetics, Farmacokinetics, Cancer Research, Water Treatment, Adsorption, Porous Materials, Activated Carbons
Cite this paper
Brouers, F. (2014) The Fractal (BSf) Kinetics Equation and Its Approximations. Journal of Modern Physics, 5, 1594-1601. doi: 10.4236/jmp.2014.516160.
Brouers, F. (2014) The Fractal (BSf) Kinetics Equation and Its Approximations. Journal of Modern Physics, 5, 1594-1601. doi: 10.4236/jmp.2014.516160.
References
[1] Brouers, F. and Sotolongo-Costa, O. (2006) Physica A: Statistical Mechanics and Its Applications, 368, 165-175.
http://dx.doi.org/10.1016/j.physa.2005.12.062 [2] Cho, D.H., Chu, K.H. and Kim, E.Y. (2014) International Journal of Environmental Science and Technology, May, 1-10. [3] Snopok, B.A. (2014) Theoretical and Experimental Chemistry, 50, 67-95.
http://dx.doi.org/10.1007/s11237-014-9351-0 [4] Gaspard, S. and Ncibi, M.C. (Eds.) (2013) Royal Society of Chemistry, 25.
http://dx.doi.org/10.1039/9781849737142 [5] Marczewski, A.W., Derylo-Marczewska, A. and Slota, A. (2013) Adsorption, 19, 391-406.
http://dx.doi.org/10.1007/s10450-012-9462-7 [6] Ellis, K.J. (2013) Neutron and Muon Studies of Spin Dynamics in Magnetic Systems. Diss. University of Huddersfield, Huddersfield. [7] Pereira, L.M. (2010) Computational and Mathematical Methods in Medicine, 11, 161-184.
http://dx.doi.org/10.1080/17486700903029280 [8] Jones, L.B., Secomb, T.W., Dewhirst, M.W. and El-Kareh, A.W. (2014) Journal of Theoretical Biology, 357, 10-20.
http://dx.doi.org/10.1016/j.jtbi.2014.04.032 [9] Sobrevals, L., Mato-Berciano, A., Urtasun, N., Mazo, A. and Fillat, C. (2014) Stem Cell Research, 12, 1-10.
http://dx.doi.org/10.1016/j.scr.2013.09.008 [10] Ho, M.L., Judd, J., Kuypers, B.E., Yamagami, M., Wong, F.F. and Suh, J. (2014) Cellular and Molecular Bioengineering, 7, 334-343. [11] Swietach, P., Vaughan-Jones, R.D., Harris, A.L. and Hulikova, A. (2014) Philosophical Transactions of the Royal Society B: Biological Sciences, 369, Article ID: 20130099.
http://dx.doi.org/10.1098/rstb.2013.0099 [12] Alvarez-Berdugo, D., Jiménez, M., Clavé, P. and Rofes, L. (2014) The Scientific World Journal, 2014, Article ID: 184526. [13] Werner, B., Gallagher, R.E., Paietta, E., Litzow, M., Tallman, M.S., Wiernik, P.H. and Dingli, D. (2014) Cancer Research, canres-1210. [14] Tonkin, J.A., Shamsudeen, S., Brown, M.R., Serda, R.E., Rees, P. and Summers, H.D. (2014) IET Optoelectronics, 8, 113-116.
http://dx.doi.org/10.1049/iet-opt.2013.0080 [15] Kjøniksen, A.L., Calejo, M.T., Zhu, K.Z., Cardoso, A.M.S., Pedroso de Lima, M.C., Jurado, A.S., Nystrom, B. and Sande, S.A. (2014) Journal of Pharmaceutical Sciences, 103, 227-234. [16] Wang, L., Luo, Q., Lin, T., Li, R., Zhu, T., Zhou, K., Ji, Z., Song, J., Jia, B., Zhang, C., Chen, W. and Zhu, G. (2014) Drug Development and Industrial Pharmacy, 10, 1-9. [17] Sotolongo-Grau, O., Rodriguez-Perez, D., Antoranz, J.C. and Sotolongo-Costa, O. (2010) Physical Review Letters, 105, Article ID: 158105.
http://dx.doi.org/10.1103/PhysRevLett.105.158105 [18] Sotolongo-Grau, O., Rodriguez-Perez, D., Sotolongo-Costa, O. and Antoranz, J.C. (2013) Physica A: Statistical Mechanics and Its Applications, 392, 2007-2015. [19] Stanislavsky, A. and Weron, K. (2013) Physical Chemistry Chemical Physics, 15, 15595-15601.
http://dx.doi.org/10.1039/c3cp52272e [20] Jurlewicz, A. and Weron, K. (2002) Journal of Non-Crystalline Solids, 305, 112-121.
http://dx.doi.org/10.1016/S0022-3093(02)01087-6 [21] Kopelman, R. (1988) Science, 241, 1620-1626.
http://dx.doi.org/10.1126/science.241.4873.1620 [22] Savageau, M.A. (1995) Journal of Theoretical Biology, 176, 115-124.
http://dx.doi.org/10.1006/jtbi.1995.0181 [23] Jose, K.K. and Raik, N.S. (2009) Communications in Statistics—Theory and Methods, 38, 912-926.
http://dx.doi.org/10.1080/03610920802322474 [24] Brouers, F., Sotolongo-Costa, O. and Weron, K. (2004) Physica A, 344, 409-416.
http://dx.doi.org/10.1016/j.physa.2004.06.008 [25] Burr, I.W. (1942) The Annals of Mathematical Statistics, 13, 215-232.
http://dx.doi.org/10.1214/aoms/1177731607 [26] Hamissa, A.B., Brouers, F., Ncibi, M.C. and Seffen, M. (2013) Separation Science and Technology, 48, 2834-2842.
http://dx.doi.org/10.1080/01496395.2013.809104 [27] Gaspard, S., Altenor, S., Passe-Coutrin, N., Ouensanga, A. and Brouers, F. (2006) Water Research, 40, 3467-3477.
http://dx.doi.org/10.1016/j.watres.2006.07.018 [28] Renugadevi, N., Sangeetha, R. and Lalitha, P. (2011) Archives of Applied Science Research, 3, 492-498.
[1] Brouers, F. and Sotolongo-Costa, O. (2006) Physica A: Statistical Mechanics and Its Applications, 368, 165-175.
http://dx.doi.org/10.1016/j.physa.2005.12.062 [2] Cho, D.H., Chu, K.H. and Kim, E.Y. (2014) International Journal of Environmental Science and Technology, May, 1-10. [3] Snopok, B.A. (2014) Theoretical and Experimental Chemistry, 50, 67-95.
http://dx.doi.org/10.1007/s11237-014-9351-0 [4] Gaspard, S. and Ncibi, M.C. (Eds.) (2013) Royal Society of Chemistry, 25.
http://dx.doi.org/10.1039/9781849737142 [5] Marczewski, A.W., Derylo-Marczewska, A. and Slota, A. (2013) Adsorption, 19, 391-406.
http://dx.doi.org/10.1007/s10450-012-9462-7 [6] Ellis, K.J. (2013) Neutron and Muon Studies of Spin Dynamics in Magnetic Systems. Diss. University of Huddersfield, Huddersfield. [7] Pereira, L.M. (2010) Computational and Mathematical Methods in Medicine, 11, 161-184.
http://dx.doi.org/10.1080/17486700903029280 [8] Jones, L.B., Secomb, T.W., Dewhirst, M.W. and El-Kareh, A.W. (2014) Journal of Theoretical Biology, 357, 10-20.
http://dx.doi.org/10.1016/j.jtbi.2014.04.032 [9] Sobrevals, L., Mato-Berciano, A., Urtasun, N., Mazo, A. and Fillat, C. (2014) Stem Cell Research, 12, 1-10.
http://dx.doi.org/10.1016/j.scr.2013.09.008 [10] Ho, M.L., Judd, J., Kuypers, B.E., Yamagami, M., Wong, F.F. and Suh, J. (2014) Cellular and Molecular Bioengineering, 7, 334-343. [11] Swietach, P., Vaughan-Jones, R.D., Harris, A.L. and Hulikova, A. (2014) Philosophical Transactions of the Royal Society B: Biological Sciences, 369, Article ID: 20130099.
http://dx.doi.org/10.1098/rstb.2013.0099 [12] Alvarez-Berdugo, D., Jiménez, M., Clavé, P. and Rofes, L. (2014) The Scientific World Journal, 2014, Article ID: 184526. [13] Werner, B., Gallagher, R.E., Paietta, E., Litzow, M., Tallman, M.S., Wiernik, P.H. and Dingli, D. (2014) Cancer Research, canres-1210. [14] Tonkin, J.A., Shamsudeen, S., Brown, M.R., Serda, R.E., Rees, P. and Summers, H.D. (2014) IET Optoelectronics, 8, 113-116.
http://dx.doi.org/10.1049/iet-opt.2013.0080 [15] Kjøniksen, A.L., Calejo, M.T., Zhu, K.Z., Cardoso, A.M.S., Pedroso de Lima, M.C., Jurado, A.S., Nystrom, B. and Sande, S.A. (2014) Journal of Pharmaceutical Sciences, 103, 227-234. [16] Wang, L., Luo, Q., Lin, T., Li, R., Zhu, T., Zhou, K., Ji, Z., Song, J., Jia, B., Zhang, C., Chen, W. and Zhu, G. (2014) Drug Development and Industrial Pharmacy, 10, 1-9. [17] Sotolongo-Grau, O., Rodriguez-Perez, D., Antoranz, J.C. and Sotolongo-Costa, O. (2010) Physical Review Letters, 105, Article ID: 158105.
http://dx.doi.org/10.1103/PhysRevLett.105.158105 [18] Sotolongo-Grau, O., Rodriguez-Perez, D., Sotolongo-Costa, O. and Antoranz, J.C. (2013) Physica A: Statistical Mechanics and Its Applications, 392, 2007-2015. [19] Stanislavsky, A. and Weron, K. (2013) Physical Chemistry Chemical Physics, 15, 15595-15601.
http://dx.doi.org/10.1039/c3cp52272e [20] Jurlewicz, A. and Weron, K. (2002) Journal of Non-Crystalline Solids, 305, 112-121.
http://dx.doi.org/10.1016/S0022-3093(02)01087-6 [21] Kopelman, R. (1988) Science, 241, 1620-1626.
http://dx.doi.org/10.1126/science.241.4873.1620 [22] Savageau, M.A. (1995) Journal of Theoretical Biology, 176, 115-124.
http://dx.doi.org/10.1006/jtbi.1995.0181 [23] Jose, K.K. and Raik, N.S. (2009) Communications in Statistics—Theory and Methods, 38, 912-926.
http://dx.doi.org/10.1080/03610920802322474 [24] Brouers, F., Sotolongo-Costa, O. and Weron, K. (2004) Physica A, 344, 409-416.
http://dx.doi.org/10.1016/j.physa.2004.06.008 [25] Burr, I.W. (1942) The Annals of Mathematical Statistics, 13, 215-232.
http://dx.doi.org/10.1214/aoms/1177731607 [26] Hamissa, A.B., Brouers, F., Ncibi, M.C. and Seffen, M. (2013) Separation Science and Technology, 48, 2834-2842.
http://dx.doi.org/10.1080/01496395.2013.809104 [27] Gaspard, S., Altenor, S., Passe-Coutrin, N., Ouensanga, A. and Brouers, F. (2006) Water Research, 40, 3467-3477.
http://dx.doi.org/10.1016/j.watres.2006.07.018 [28] Renugadevi, N., Sangeetha, R. and Lalitha, P. (2011) Archives of Applied Science Research, 3, 492-498.