Fractal pharmacokinetics

Computational and Mathematical Methods in Medicine

Volumn 11, Issue 2, 2010, Pages 161-184

  Pereira, Luis M ^a  

^a BOSTON CHILDREN S HOSPITAL   (United States)

Author keywords

Anomalous diffusion; Fractal microenvironments; Fractional calculus; Time dependent rate constants

Indexed keywords

CALCULATIONS; DIFFUSION; PHARMACOKINETICS; RATE CONSTANTS;

ANOMALOUS DIFFUSION; DRUG CONCENTRATION; FRACTAL PHARMACOKINETICS; FRACTIONAL CALCULUS; HOMOGENEOUS KINETICS; MICROENVIRONMENTS; TIME DEPENDENT; TWO-COMPARTMENT MODELS;

FRACTAL DIMENSION;

AMIODARONE; CYCLOSPORIN;

ARTICLE; COMPUTER SIMULATION; CONTROLLED STUDY; DIFFUSION; DRUG ACCUMULATION; DRUG DISTRIBUTION; FRACTAL ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; PARAMETRIC TEST; PARSIMONY ANALYSIS;

EID: 77952500845     PISSN: 1748670X     EISSN: 17486718     Source Type: Journal    
DOI: 10.1080/17486700903029280     Document Type: Article

References (85)

1. S. Alexander and R. Orbach, Density of states on fractals: 'fractons', J. Phys. Lett. 43 (1982), pp. L625-L631.
2. F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler, Subdiffusion and anomalous viscoelasticity in actin networks, Phys. Rev. Lett. 77 (1996), pp. 4470-4473.
3. L.W. Anacker and R. Kopelman, Fractal chemical kinetics: Simulations, and experiments, J. Chem. Phys. 81 (1984), pp. 6402-6403.
4. L.W. Anacker, R. Kopelman, and J.S. Newhouse, Fractal chemical kinetics: Reacting random walkers, J. Stat. Phys. 36(5/6) (1984), pp. 591-602.
5. J. Anderson and R.W.S. Tomlinson, The distribution of calcium-47 in the rat, J. Physiol. 182 (1966), pp. 664-670.
6. J. Anderson, S.B. Osborn, R.W. Tomlinson, and M.A. Weinbren, Some applications of power law analysis to radioisotope studies in man, Phys. Med. Biol. 8(3) (1963), pp. 287-295.
7. F.I. Ataullakhanov, S.V. Komarova, and V.M. Vitvitsky, A possible role of adenylate metabolism in human erythrocytes: simple mathematical model, J. Theor. Biol. 179 (1996), pp. 75-86.
8. E. Barkai and J. Klafter, Comment on 'Subdiffusion and anomalous viscoelasticity in actin networks', Phys. Rev. Lett. 81 (1998), p. 1134.
9. A.H. Beckett and G.T. Tucker, Problems in the in vivo evaluation of drug preparations and the interpretation of in vivo data, J. Mond. Pharm. 3 (1967), pp. 181-202.
10. D. ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge, 2000.
11. F. Brouers and O. Sotolongo-Costa, Generalized fractal kinetics in complex systems. Applications to biophysics and biotechnology, Phys. A: Stat. Mech. Appl. 368(1) (2006), pp. 165-175.
12. F. Brouers, O. Sotolongo-Costa, and K. Weron, Burr, Lévy, Tsallis, Phys. A: Stat. Mech. Appl. 344(3-4) (2004), p. 409.
13. F.X. Caroli-Bosc, A. Iliadis, L. Salmon, P. Macheras, A.M. Montet, A. Bourgeon, R. Garrafo, J.P. Delmont, and J.C. Montet, Ursodeoxycholic acid modulates cyclosporin A oral absorption in liver transplant recipients, Fundam. Clin. Pharmacol. 14 (2000), pp. 601-609.
14. P. Chelminiak, R. Marsh, J. Tuszynski, J.M. Dixon, and K.J.E. Vos, Asymptotic time dependence in the fractal pharmacokinetics of a two-compartment model, Phys. Rev. E 72(031903) (2005), pp. 1-7.
15. B.S. Chen, Y.C. Wang, W.S. Wu, and W.H. Li, A new measure of the robustness of biochemical networks, Bioinformatics 21 (2005), pp. 2698-2705.
16. L. Claret, A. Iliadis, and P. Macheras, A stochastic model describes the heterogeneous pharmacokinetics of cyclosporin, J. Pharmacokinet. Pharmacodyn. 28(5) (2001), pp. 445-463.
17. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
18. R. Curto, E.O. Voit, A. Sorribas, and M. Cascante, A power-law model of purine metabolism: comparison of alternative modelling strategies, Math. Biosci. 151 (1998), pp. 1-49.
19. Y. Dassas and P. Duby, Diffusion toward fractal interfaces, J. Electrochem. Soc. 142 (1995), p. 4175.
20. A. Dokoumetzidis and P. Macheras, A population growth model of dissolution, Pharm. Res. 14 (1997), pp. 1122-1126.
21. A. Dokoumetzidis and P. Macheras, A tube model for transport and dispersion in the circulatory system based on the vascular fractal tree, Ann. Biomed. Eng. 31 (2003), pp. 284-293.
22. J. Fuite, R. Marsh, and J. Tuszynski, Fractal pharmacokinetics of the drug mibefradil in the liver, Phys. Rev. E 66(021904) (2002), pp. 1-11.
23. R. Grima and S. Schnell, How reaction kinetics with time-dependent rate coefficients differs from generalized mass action, Chem. Phys. Chem. 7 (2006), pp. 1422-1424.
24. R. Heinrich and S. Schuster, The Regulation of Cellular Systems, Chapman and Hall, New York, 1996.
25. K. Higaki, S. Yamashita, and G.L. Amidon, Time-dependent oral absorption models, J. Pharmacokinet. Pharmacodyn. 28 (2001), pp. 109-128.
26. T.H. Hughes and J.H. Matis, An irreversible two-compartment model with age-dependent turnover rates, Biometrics 40 (1984), pp. 501-505.
27. B.D. Hughes, M.F. Schlesinger, and E.W. Montroll, Random walks in self similar clusters, Proc. Natl Acad. Sci. USA 78 (1981), pp. 3287-3291.
28. V. Karalis and P. Macheras, Drug disposition viewed in terms of the fractal volume of distribution, Pharm. Res. 19 (2002), pp. 696-703.
29. V. Karalis, A. Dokoumetzidis, and P. Macheras, A physiologically based approach for the estimation of recirculatory parameters, J. Pharmacol. Exp. Ther. 308 (2004), pp. 198-205.
30. V. Karalis, A. Tsantili-Kakoulidou, and P. Macheras, Multivariate statistics of disposition pharmacokinetic parameters for structurally unrelated drugs used in therapeutics, Pharm. Res. 19 (2002), pp. 1827-1834.
31. V. Karalis, A. Tsantili-Kakoulidou, and P. Macheras, Quantitative structure pharmacokinetic relationships for disposition parameters of cephalosporins, Eur. J. Pharm. Sci. 20 (2003), pp. 115-123.
32. V. Karalis, L. Claret, A. Iliadis, and P. Macheras, Fractal volume of drug distribution: it scales proportionally to body mass, Pharm. Res. 18 (2001), pp. 1056-1060.
33. H.P. Koch and H. Zacek, Fractals also in pharmacokinetics, Pharmazie 46(H12) (1991), pp. 870-871.
34. R. Kopelman, Rate processes on fractals: theory, simulations, and experiments, J. Stat. Phys. 42(1/2) (1986), pp. 185-200.
35. R. Kopelman, Fractal reaction kinetics, Science 241 (1988), pp. 1620-1626.
36. K. Kosmidis, P. Argyrakis, and P. Macheras, Fractal kinetics in drug release from finite fractal matrices, J. Chem. Phys. 119 (2003), pp. 6373-6377.
37. M. Kotulski and K. Weron, Random walk approach to relaxation in disordered systems, in Athens Conference on Applied Probability and Time Series Analysis Volume I: Applied Probability, C.C. Heyde, Y.V. Prohorov, R. Pyke, and S.T. Rachev, eds., Lecture Notes in Statistics 114, Springer, New York, 1996, pp. 379-388.
38. P. Lansky and M. Weiss, Does the dose-solubility ratio affect the mean dissolution time of drugs?, Pharm. Res. 16 (1999), pp. 1470-1476.
39. P. Lansky and M. Weiss, Modeling heterogeneity of properties and random effects in drug dissolution, Pharm. Res. 18 (2001), pp. 1061-1067.
40. P. Lansky and M. Weiss, Classification of dissolution profiles in terms of fractional dissolution rate and a novel measure of heterogeneity, J. Pharm. Sci. 92 (2003), pp. 1632-1647.
41. A. Le Mehaute, Transfer processes in fractal media, J. Stat. Phys. 36(5/6) (1984), pp. 665-676.
42. A. Le Mehaute and G. Crepy, Introduction to transfer and motion in fractal media: The geometry of kinetics, Solid State Ionics 9(10) (1983), p. 17.
43. X.-F. Li, Approximate solution of linear ordinary differential equations with variable coefficients, Math. Comput. Simulation 75(3-4) (2007), pp. 113-125.
44. P. Macheras, Carrier-mediated transport can obey fractal kinetics, Pharm. Res. 12(4) (1995), pp. 541-548.
45. P. Macheras, A fractal approach to heterogeneous drug distribution: calcium pharmacokinetics, Pharm. Res. 13 (1996), pp. 663-670.
46. P. Macheras and A. Argyrakis, Gastrointestinal drug absorption: Is it time to consider heterogeneity as well as homogeneity, Pharm. Res. 14 (1997), pp. 842-847.
47. P. Macheras and A. Dokoumetzidis, On the heterogeneity of drug dissolution and release, Pharm. Res. 17 (2000), pp. 108-112.
48. F. Mainardi, Fractional relaxation in anelastic solids, J. Alloys Comp. 211(212) (1994), pp. 534-538.
49. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, San Francisco, CA, 1983.
50. R. Marsh and J.A. Tuszynski, Fractal Michaelis-Menten kinetics under steady-state conditions: Application to mibefradil, Pharm. Res. 23(12) (2006), pp. 2760-2767.
51. R.E. Marsh, T.A. Riauka, and S.A. McQuarrie, Use of a simulated annealing algorithm to fit compartmental models with an application to fractal pharmacokinetics, J. Pharm. Pharmaceut. Sci. 10(2) (2007), pp. 168-179.
52. G.H. Marshall, Calcium pools and the power function, in Compartments, Pools and Spaces in Medical Physiology, P.E. Bergner and C.C. Lushbaugh, eds., Proceedings of AEC Symposium Series no.11, Oak Ridge, 1966, pp. 451-468.
53. H.P. McKean, Hausdorff-Besicovitch dimension of Brownian motion paths, Duke Math. J. 22(2) (1955), pp. 229-234.
54. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
55. R.R. Nigmatullin, On the theory of relaxation with 'remnant' memory, Phys. Status Solidi B 124 (1984), pp. 389-393.
56. R.R. Nigmatullin, The realization of the generalized transfer in a medium with fractal geometry, Phys. Status Solidi B 133 (1986), p. 425.
57. W.P. Norris, S.A. Tyler, and A.M. Brues, Retention of radioactive bone-seekers: calculations based on the power function cast doubt on the present concept of biological half-times, Science 128(3322) (1958), pp. 456-462.
58. T. Ogihasra, I. Tamai, and A. Tsuji, Application of fractal kinetics for carrier-mediated transport of drugs across intestinal epithelial membrane, Pharm. Res. 4 (1998), pp. 620-625.
59. B. O'Shaughnessy and I.I. Procaccia, Analytical solution for diffusion on fractal objects, Phys. Rev. Lett. 54(5) (1985), pp. 455-458.
60. L.M. Pereira, Comment on 'Asymptotic time dependence in the fractal pharmacokinetics of a two-compartment model', Phys. Rev. E 78 (2008), p. 5.
61. A. Rescigno, The rise and fall of compartmental analysis, Pharmacol. Res. 44 (2001), pp. 337-342.
62. L.F. Richardson, Atmosphere diffusion shown on a distance-neighbour graph, Proc. R. Soc. Lond. A. 110 (1926), p. 709.
63. J. Rudnik and G. Gaspari, The shapes of random walks, Science 237(4813) (1987), pp. 384-389.
64. A.I. Saichev and G.M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos 7(4) (1997), pp. 753-764.
65. M.A. Savageau, Biochemical systems analysis: i. Some mathematical properties of the rate law for component enzymatic enzymatic reactions, J. Theor. Biol. 25 (1969), pp. 365-369.
66. M.A. Savageau, Michaelis-Menten mechanism reconsidered: Implications of fractal kinetics, J. Theor. Biol. 176 (1995), pp. 115-124.
67. M.A. Savageau, Development of fractal kinetic theory for enzyme-catalysed reactions and implications for the design of biochemical pathways, Biosystems 47 (1998), pp. 9-36.
68. S. Schnell and T.E. Turner, Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws, Prog. Biophys. Mol. Biol. 85 (2004), pp. 235-260.
69. V. Seshadri and B.J. West, Fractal dimensionality of Lévy processes, Proc. Natl Acad. Sci. USA 79 (1982), pp. 4501-4505.
70. M.V. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Losungen, Z. Phys. Chem. 92 (1917), pp. 129-168.
71. D. Stauffer, Introduction to Percolation Theory, Taylor & Francis, London, 1985.
72. T. Teorell, Kinetics of distribution of substances administered to the body, Arch. Intern. Pharmacodyn. Ther. 57 (1937), pp. 205-225.
73. C. Tsallis, Nonextensive statistical mechanics and thermodynamics. Historical background and present status, in Nonextensive Statistical Mechanics and its Applications, S. Abe and Y. Okamoto, eds., Series Lecture Notes in Physics, Springer, Berlin, 2001, pp. 3-98.
74. C. Tsallis, Dynamical scenario for nonextensive statistical mechanics, Phys. A: Stat. Mech. Appl. 340(1-3) (2004), pp. 1-10.
75. G.T. Tucker, Compartmental vs. empirical vs. physiological models, in Topics in Pharmaceutical Sciences, D.D. Breimer and P. Speiser, eds., Elsevier, Amsterdam, 1982, pp. 33-48.
76. G.T. Tucker, P.R. Jackson, and G.C.A. Storey, Amiodarone disposition: poliexponential, power and gamma functions, Eur. J. Clin. Pharmacol. 26 (1984), pp. 655-656.
77. G. Valsami, A. Dokoumetzidis, and P. Macheras, Modeling of supersaturated dissolution data, Int. J. Pharm. 181 (1999), pp. 153-157.
78. S. Vasilach, Algebraic method for solving systems of linear differential equations with variable coefficients, SIAM J. Math. Anal. 12(1) (1981), pp. 23-39.
79. M. Weiss, The anomalous pharmacokinetics of amiodarone explained by nonexponential tissue trapping, J. Pharmacokinet. Biopharm. 27 (1999), pp. 383-396.
80. M.E. Wise, The evidence against compartments, Biometrics 27 (1971), p. 262.
81. M.E. Wise, Interpreting both short- and long-term power laws in physiological clearance curves, Math. Biosci. 20 (1974), pp. 327-337.
82. M.E. Wise, Negative power functions of time in pharmacokinetics and their implications, J. Pharmacok. Pharmacodyn. 13(3) (1985), pp. 309-346.
83. M.E. Wise and J.M. Borsboom, Two exceptional sets of physiological clearance curves and their mathematical form: Test cases?, Bull. Math. Biol. 51 (1989), pp. 579-596.
84. M. Yamamoto, On the homogeneous linear systems of differential equations with variable coefficients, Proc. Japan Acad. Ser. A 53(1) (1977), pp. 17-20.
85. G.M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), pp. 461-580.