A model of competing saturable kinetic processes with application to thepharmacokinetics of the anticancer drug paclitaxel

Rebeccah E. Marsh; Jack A. Tuszyński; Michael Sawyer; Kenneth J. E. Vos

doi:10.3934/mbe.2011.8.325

Article Contents

2011, Volume 8, Issue 2: 325-354. Doi: 10.3934/mbe.2011.8.325

This issue Previous Article Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy Next Article Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis

A model of competing saturable kinetic processes with application to the pharmacokinetics of the anticancer drug paclitaxel

1.
Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2J1
2.
Department of Experimental Oncology, Faculty of Medicine and Dentistry, University of Alberta, Edmonton, Alberta, T6G 2J1
3.
Department of Physics and Astronomy, University of Lethbridge, Lethbridge, Alberta, T1K 3M4

Received: February 28, 2010

Revised: July 31, 2010

Published: March 2011

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Abstract

A saturable multi-compartment pharmacokinetic model for the anti-cancer drug paclitaxel is proposed based on a meta-analysis of pharmacokinetic data published over the last two decades. We present and classify the results of time series for the drug concentration in the body to uncover the underlying power laws. Two dominant fractional power law exponents were found to characterize the tails of paclitaxel concentration-time curves. Short infusion times led to a power exponent of $-1.57 \pm 0.14$, while long infusion times resulted in tails with roughly twice the exponent. Curves following intermediate infusion times were characterized by two power laws. An initial segment with the larger slope was followed by a long-time tail characterized by the smaller exponent. The area under the curve and the maximum concentration exhibited a power law dependence on dose, both with compatible fractional power exponents. Computer simulations using the proposed model revealed that a two-compartment model with both saturable distribution and elimination can reproduce both the single and crossover power laws. Also, the nonlinear dose-dependence is correlated with the empirical power law tails. The longer the infusion time the better the drug delivery to the tumor compartment is a clinical recommendation we propose.

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Mathematics Subject Classification: Primary: 92C45, 80A30; Secondary: 34-02.

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References

[1]	R. Advani, G. A. Fisher, B. L. Lum, J. Hausdorff, J. Halsey, M. Litchman and B. I. Sikic, A phase I trial of doxorubicin, paclitaxel, and valspodar (PSC 833), a modulator of multidrug resistance, Clin. Cancer Res., 7 (2001), 1221-1229.
[2]	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 (1963), 287-295.doi: 10.1088/0031-9155/8/3/305.
[3]	J. B. Bassingthwaighte and D. A. Beard, Fractal 15O-labeled water washout from the heart, Circ. Res., 77 (1995), 1212-1221.
[4]	D. A. Beard and J. B. Bassingthwaighte, Power-law kinetics of tracer washout from physiological systems, Ann. Biomed. Eng., 26 (1998), 775-779.doi: 10.1114/1.105.
[5]	H. G. Boxenbaum, Pharmacokinetics tricks and traps: Flip-flop models, J. Pharm. Pharm. Sci., 1 (1998), 90-91.
[6]	T. Brown, K. Havlin, G. Weiss, J. Cagnola, J. Koeller, J. Kuhn, J. Rizzo, J. Craig, J. Phillips and D. Von Hoff, A phase I trial of taxol given by a 6-hour intravenous infusion, J. Clin. Oncol., 9 (1991), 1261-1267.
[7]	P. Chelminiak, R. E. Marsh, J. A. Tuszyński, J. M. Dixon and K. J. E. Vos, Asymptotic time dependence in the fractal pharmacokinetics of a two-compartment model, Phys. Rev. E, 72 (2005), Art. Id 031903, 1-7.
[8]	B. Damascelli, G. Cantu, F. Mattavelli, P. Tamplenizza, P. Bidoli, E. Leo, F. Dosio, A. M. Cerrotta, G. Di Tolla, L. F. Frigerio, F. Garbagnati, R. Lanocita, A. Marchiano, G. Patelli, C. Spreafico, V. Ticha, V. Vespro and F. Zunino, Intraarterial chemotherapy with polyoxyethylated castor oil free paclitaxel, incorporated in albumin nanoparticles (ABI-007): Phase II study of patients with squamous cell carcinoma of the head and neck and anal canal: preliminary evidence of clinical activity, Cancer, 92 (2001), 2592-2602.doi: 10.1002/1097-0142(20011115)92:10<2592::AID-CNCR1612>3.0.CO;2-4.
[9]	A. Dokoumetzidis and P. Macheras, Fractional pharmacokinetics and pharmacodynamics, J. of Pharmacokinetics and Pharmacodynamics, 36 (2009), 165-178.doi: 10.1007/s10928-009-9116-x.
[10]	A. Dokoumetzidis, R. Magin and P. Macheras, A commentary on fractionalization of multi-compartmental models, J. of Pharmacokinetics and Pharmacodynamics, 37 (2010), 203-207.doi: 10.1007/s10928-010-9153-5.
[11]	F. Doz, J. C. Gentet, F. Pein, D. Frappaz, P. Chastagner, S. Moretti, G. Vassal, J. Arditti, O. Van Tellingen, A. Iliadis and J. Catalin, Phase I trial and pharmacological study of a 3-hour paclitaxel infusion in children with refractory solid tumors: A SFOP study, British Journal of Cancer, 84 (2001), 604-610.doi: 10.1054/bjoc.2000.1637.
[12]	J. Fuite, R. Marsh and J. Tuszyński, Fractal pharmacokinetics of the drug mibefradil in the liver, Phys. Rev. E, 66 (2002), Art. Id. 021904, 1-11.
[13]	H. Gelderblom, J. Verweij, D. M. van Zomeren, D. Buijs, L. Ouwens, K. Nooter, G. Stoter and A. Sparreboom, Influence of Cremophor EL on the bioavailability of intraperitoneal Paclitaxel, Clin. Cancer Res., 8 (2002), 1237-1241.
[14]	H. Gelderblom, S. D. Baker, A. Zhao, J. Verwij and A. Sparrreboom, Distribution of paclitaxel in plasma and cerebrospinal fluid, Anti-cancer drugs, 14 (2003), 365-368.doi: 10.1097/00001813-200306000-00007.
[15]	K. Gelmon, E. Eisenhauer, C. Bryce, A. Tolcher, L. Mayer, E. Tomlinson, B. Zee, M. Blackstein, E. Tomiak, J. Yau, G. Batist, B. Fisher and J. Iglesias, Randomized phase II study of high-dose paclitaxel with or without amifostine in patients with metastatic breast cancer, J. Clin. Oncol., 17 (1999), 3038-3047.
[16]	L. Gianni, C. M. Kearns, A. Giani, G. Capri, L. Vigano, A. Lacatelli, G. Bonadonna and M. J. Egorin, Nonlinear pharmacokinetics and metabolism of paclitaxel and its pharmacokinetic/pharmacodynamic relationships in humans, J. Clin. Oncol., 13 (1995), 180-190.
[17]	M. Gibaldi and D. Perrier, "Pharmacokinetics," 2nd edition, M. Dekker, New York, 1982.
[18]	K. Gough, M. Hutchinson, O. Keene, B. Byrom, S. Ellis, L. Lacey and J. McKellar, Assessment of dose proportionality: report from the statisticians in the pharmaceutical industry/pharmacokinetics UK joint working party, Drug Inf. J., 29 (1995), 1039-1048.
[19]	A. Henningsson, M. O. Karlsson, L. Vigano, L. Gianni, J. Verweij and A. Sparreboom, Mechanism-based pharmacokinetic model for paclitaxel, J. Clin. Oncol., 19 (2001), 4065-4073.
[20]	M. T. Huizing, V. H. Misser, R. C. Pieters, W. W. ten Bokkel Huinink, C. H. Veenhof, J. B. Vermorken, H. M. Pinedo and J. H. Beijnen, Taxanes: A new class of antitumor agents, Cancer Invest., 13 (1995), 381-404.doi: 10.3109/07357909509031919.
[21]	J. A. Jacquez, "Compartmental Analysis in Biology and Medicine," BioMedware, Ann Arbor, 1996.
[22]	M. A. Jordan, R. J. Toso, D. Thrower and D. L. Wilson, Mechanism of mitotic block and inhibition of cell proliferation by taxol at low concentrations, Proc. Natl. Acad. Sci. USA, 102 (1993), 9552-9556.doi: 10.1073/pnas.90.20.9552.
[23]	M. A. Jordon and L. Wilson, Taxane Anticancer Agents: Basic Science and Current Status, in "ACS Symp. Series 583" (eds. G. Georg, T. Chen, I. Ojima and D. Vyas), The American Chemical Society, Washington, (1995), 138-153.
[24]	M. A. Jordan, Mechanism of action of antitumor drugs that interact with microtubules and tubulin, Curr. Med. Chem. Anti-Canc. Agents, 2 (2002), 1-17.
[25]	M. O. Karlsson, V. Molnar, A. Freijs, P. Nygren, J. Bergh and R. Larsson, Pharmacokinetic models for the saturable distribution of paclitaxel, Drug Metab. Dispos., 27 (1999), 1220-1223.
[26]	C. M. Kearns, L. Gianni and M. J. Egorin, Paclitaxel pharmacokinetics and pharmaco-dynamics, Semin. Oncol., 22 (1995), 16-23.
[27]	T. Y. Kim, D. W. Kim, J. Y. Chung, S. G. Shin, S. C. Kim, D. S. Heo, N. K. Kim and Y. J. Bang, Phase I and pharmacokinetic study of Genexol-PM, a cremophor-free, polymeric micelle-formulated paclitaxel, in patients with advanced malignancies, Clin. Cancer Res., 10 (2004), 3708-3716.doi: 10.1158/1078-0432.CCR-03-0655.
[28]	K. Kosmidis, V. Karalis, P. Argyrakis and P. Macheras, Michaelis-Menten kinetics under spatially constrained conditions: Application to mibefradil pharmacokinetics, Biophys. J., 87 (2004), 1498-1506.doi: 10.1529/biophysj.104.042143.
[29]	M. Lopez-Quintela and J. Casado, Revision of the methodology in enzyme kinetics: A fractal approach, J. Theor. Biol., 139 (1989), 129-139.doi: 10.1016/S0022-5193(89)80062-1.
[30]	T. M. Ludden, S. L. Beal and L. B. Sheiner, Comparison of the akaike information criterion, the Schwartz criterion and the F-test as guides to model selection, J. Pharmacokinet. Biopharm., 22 (1994), 431-445.doi: 10.1007/BF02353864.
[31]	P. Macheras, A fractal approach to heterogeneous drug distribution: Calcium pharmacokinetics, Pharm. Res., 13 (1996), 663-670.doi: 10.1023/A:1016031129053.
[32]	H. Maier-Lenz, B. Hauns, B. Haering, J. Koetting, K. Mross, C. Unger, T. Bauknecht, A. du Bois, H. G. Meerpohl, N. Hollaender and K. Diergarten, Phase I study of paclitaxel administered as a 1-hour infusion: Toxicity and pharmacokinetics, Semin. Oncol., 24 (1997), Art. Id. S19, 16-19.
[33]	R. E. Marsh and J. A. Tuszyński, Fractal Michaelis-Menten kinetics under steady state conditions: Application to mibefradil, Pharm. Res., 12 (2006), 2760-2767.doi: 10.1007/s11095-006-9090-6.
[34]	R. E. Marsh, J. A. Tuszyński, M. B. Sawyer and K. J. E. Vos, Emergence of power laws in the pharmacokientics of paclitaxel due to competing saturable processes, J. Pharm. Pharmaceut. Sci., 11 (2008), 77-96.
[35]	L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.
[36]	B. Monsarrat, E. Mariel, S. Cros, M. Gares, D. Guenard, F. Gueritte-Voegelein and M. Wright, Taxol metabolism. Isolation and identification of three major metabolites of taxol in rat bile, Drug Metab. Dispos., 18 (1990), 895-901.
[37]	T. Mori, Y. Kinoshita, A. Watanabe, T. Yamaguchi, K. Hosokawa and H. Honjo, Retention of paclitaxel in cancer cells for 1 week in vivo and in vitro, Cancer Chemother. Pharmacol., 58 (2006), 665-672.doi: 10.1007/s00280-006-0209-6.
[38]	K. Mross and N. Hollander and B. Hauns and M. Schumacher and H. Maier-Lenz, The pharmacokinetics of a 1-h paclitaxel infusion, Cancer Chemother. Pharmacol., 45 (2000), 463-470.doi: 10.1007/s002800051020.
[39]	V. R. Nannan Panday, R. de Wit, J. H. Schornagel, M. Schot, H. Rosing, J. Lieverst, W. W. ten Bokkel Huinink, J. H. Schellens and J. H. Beijnen, Pharmacokinetics of paclitaxel administered in combination with cisplatin, etoposide and bleomycin in patients with advanced solid tumours, Cancer Chemother. Pharmacol., 44 (1999), 349-353.doi: 10.1007/s002800050988.
[40]	J. H. Nettles, H. Li, B. Cornett, J. M. Krahn, J. P. Snyder and K. H. Downing, The binding mode of epothilone A on alpha, beta-tubulin by electron crystallography, Science, 305 (2004), 866-869.doi: 10.1126/science.1099190.
[41]	W. P. Norris, S. A. Tyler and A. M. Brues, Retention of radioactive bone-seekers, Science, 128 (1958), 456-462.doi: 10.1126/science.128.3322.456.
[42]	K. H. Norwich and S. Siu, Power functions in physiology and pharmacology, J. Theor. Biol., 95 (1982), 387-398.doi: 10.1016/0022-5193(82)90253-3.
[43]	T. Ohtsu, Y. Sasaki, T. Tamura, Y. Miyata, H. Nakanomyo, Y. Nishiwaki and N. Saijo, Clinical pharmacokinetics and pharmacodynamics of paclitaxel: A 3-hour infusion versus a 24-hour infusion, Clin. Cancer Res., 1 (1995), 599-606.
[44]	K. P. Papadopoulos, M. J. Egorin, M. Huang, A. Troxel, E. Kaufman, C. Balmaceda, L. T. Vahdat and C. S. Hesdorffer, The pharmacokinetics and pharmacodynamics of high-dose paclitaxel monotherapy (825 mg/m2 continuous infusion over 24h) with hematopoietic support in women with metastatic breast cancer, Cancer Chemother. Pharmacol., 47 (2001), 45-50.doi: 10.1007/s002800000193.
[45]	J. Parness and S. B. Horwitz, Taxol binds to polymerized tubulin in vitro, J. Cell Biol., 91 (1981), 479-487.doi: 10.1083/jcb.91.2.479.
[46]	A. Patnaik, E. Warner, M. Michael, M. J. Egorin, M. J. Moore, L. L. Siu, P. M. Fracasso, S. Rivkin, I. Kerr, M. Litchman and A. M. Oza, Phase I dose-finding and pharmacokinetic study of paclitaxel and carboplatin with oral valspodar in patients with advanced solid tumors, J. Clin. Oncol., 18 (2000), 3677-3689.
[47]	F. Pellegrini and D. R. Budman, Review: Tubulin function, action of antitubulin drugs, and new drug development, Cancer Invest., 23 (2005), 264-273.doi: 10.1081/CNV-200055970.
[48]	W. Press, "Numerical Recipes in C: The Art of Scientific Computing," Cambridge University Press, New York, 1992.
[49]	D. M. Robinson and G. M. Keating, Albumin-bound paclitaxel inmetastatic breast cancer drugs, Drugs, 66 (2006), 941-948.doi: 10.2165/00003495-200666070-00007.
[50]	E. K. Rowinsky, P. J. Burke, J. E. Karp, R. W. Tucker, D. S. Ettinger and R. C. Donehower, Phase I and pharmacodynamic study of taxol in refractory acute leukemias, Cancer Res., 49 (1989), 4640-4647.
[51]	E. K. Rowinsky, M. Wright, B. Monsarrat, G. J. Lesser and R. C. Donehower, Taxol: Pharmacology, metabolism and clinical implications, Cancer Surv., 17 (1993), 283-304.
[52]	O. Soepenberg, A. Sparreboom, M. J. de Jonge, A. S. Planting, G. de Heus, W. J. Loos, C. M. Hartman, C. Bowden and J. Verweij, Real-time pharmacokinetics guiding clinical decisions; phase i study of a weekly schedule of liposome encapsulated paclitaxel in patients with solid tumours, Eur. J. Cancer, 40 (2004), 681-688.doi: 10.1016/j.ejca.2003.11.027.
[53]	D. S. Sonnichsen, C. A. Hurwitz, C. B. Pratt, J. J. Shuster and M. V. Relling, Saturable pharmacokinetics and paclitaxel pharmacodynamics in children with solid tumors, J. Clin. Oncol., 12 (1994), 532-538.
[54]	C. Sottani, C. Minoia, M. D'Incalci, M. Paganini and M. Zucchetti, High-performance liquid chromatography tandem mass spectrometry procedure with automated solid phase extraction sample preparation for the quantitative determination of paclitaxel (taxol) in humanplasma, Rapid Commun. Mass Spectrom., 12 (1998), 251-255.doi: 10.1002/(SICI)1097-0231(19980314)12:5<251::AID-RCM145>3.0.CO;2-Z.
[55]	A. Sparreboom, L. van Zuylen, E. Brouwer, W. J. Loos, P. de Bruijn, H. Gelderblom, M. Pillay, K. Nooter, G. Stoter and J. Verweij, Cremophor EL-mediated alteration of paclitaxel distribution in human blood: Clinical pharmacokinetic implications, Cancer Res., 59 (1999), 1454-1457.
[56]	L. van Zuylen, M. O. Karlsson, J. Verweij, E. Brouwer, P. de Bruijn, K. Nooter, G. Stoter and A. Sparreboom, Pharmacokinetic modeling of paclitaxel encapsulation in cremophor EL micelles, Cancer Chemother. Pharmacol., 47 (2001), 309-318.doi: 10.1007/s002800000215.
[57]	L. van Zuylen, J. Verweij and A. Sparreboom, Role of formulation vehicles in taxane pharmacology, Invest. New Drugs, 19 (2001), 125-141.doi: 10.1023/A:1010618632738.
[58]	D. Verotta, Fractional compartmental models and multi-term Mittag-Leffler response functions, Journal of Pharmacokinetics and Pharmacodynamics, 37 (2010), 209-215.doi: 10.1007/s10928-010-9155-3.
[59]	D. Verotta, Fractional dynamics pharmacokinetics-pharmacodynamic models, Journal of Pharmacokinetics and Pharmacodynamics, 37 (2010), 257-276.doi: 10.1007/s10928-010-9159-z.
[60]	J. Verweij, M. Clavel and B. Chevalier, Paclitaxel (taxol) and docetaxel (taxotere): Not simply two of a kind, Annals of Oncology, 5 (1994), 495-505.
[61]	M. Weiss, Use of gamma distributed residence times in pharmacokinetics, Eur. J. Clin. Pharmacol., 25 (1983), 695-702.doi: 10.1007/BF00542361.
[62]	P. H. Wiernik, E. L. Schwartz, J. J. Strauman, J. P. Dutcher, R. B. Lipton and E. Paietta, Phase I clinical and pharmacokinetic study of Taxol, Cancer Res., 47 (1987), 2486-2493.
[63]	P. H. Wiernik, E. L. Schwartz, A. Einzig, J. J. Strauman, R. B. Lipton and J. P. Dutcher, Phase I trial of taxol given as a 24-hour infusion every 21 days: Responses observed in metastatic melanoma, J. Clin. Oncol., 5 (1987), 1232-1239.
[64]	M. E. Wise, S. B. Osborn, J. Anderson and R. W. S. Tomlinson, A stochastic model for turnover of radiocalcium based on the observed power laws, Math. Biosci. l, 2 (1968), 199-224.doi: 10.1016/0025-5564(68)90019-9.
[65]	M. E. Wise, The evidence against compartments, Biometrics, 27 (1971), 262.
[66]	M. E. Wise, Interpreting both short and long-term power laws in physiological clearance curves, Math. Biosci., 20 (1974), 327-337.doi: 10.1016/0025-5564(74)90008-X.
[67]	M. H. Woo, D. Gregornik, P. D. Shearer, W. H. Meyer and M. V. Relling, Pharmacokinetics of paclitaxel in an anephric patient, Cancer Chemother. Pharmacol., 43 (1999), 92-96.doi: 10.1007/s002800050868.