A storage model with random release rate for modeling exposure to food contaminants

Mathematical Biosciences and Engineering

Volumn 5, Issue 1, 2008, Pages 35-60

  Bertail, Patrice ^a   Clémençon, Stéphan ^b   Tressou, Jessica ^c  

^a CREST   (France)

^b INSTITUT TELECOM   (France)

^c HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

Author keywords

Dietary contamination; Food safety; Pharmacokinetics model; Piecewise deterministic Markov process; Simulation method; Stability analysis

Indexed keywords

EID: 45849136009     PISSN: 15471063     EISSN: 15510018     Source Type: Journal    
DOI: 10.3934/mbe.2008.5.35     Document Type: Article

References (63)

1. B. Aberg, L. Ekman, R. Falk, U. Greitz, G. Persson, and J. O. Snihs, Metabolism of methyl mercury compounds in man, Arch. Environ. Health 19 (1969), no. 4, 478-484.
2. L. S. Allen, An introduction to stochastic processes with biology applications, Pearson/Prentice Hall, Upper Saddle River, N.J., 2003.
3. S. Asmussen, Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities, Adv. Appl. Probab. 8 (1998), no. 2, 354-374.
4. Applied probability and queues, Springer-Verlag, New York, 2003, Second Edition.
5. P. Bertail and S. Clémençon, Edgeworth expansions for suitably normalized sample mean statistics of atomic Markov chains, Prob. Th. Rel. Fields 130 (2004), no. 3, 388-414.
6. Regenerative-block bootstrap for Markov chains, Bernoulli 12 (2005), no. 4, 689-712.
7. P. Bertail, S. Clémençon, and J. Tressou, Statistical analysis of a dynamic model for food contaminant exposure with applications to dietary methylmercury contamination, 2007, http://metarisk.inapg.iura.fr/.
8. P. Bertail, M. Feinberg, J. Tressou, and P. Verger (Coordinateurs), Analyse des risques alimentaires, TEC&DOC, Lavoisier, Paris, 2006.
9. P. Bertail and J. Tressou, Incomplete generalized U-Statistics for food risk assessment, Biometrics 62 (2006), no. 1, 66-74.
10. P. J. Bickel and D. A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196-1217.
11. P. Bougerol and N. Picard, Strict stationarity of generalized autoregressive processes, Ann. Probab. 20 (1992), no. 4, 1714-1730.
12. M. Brett, H. J. Weimann, W. Cawello, H. Zimmermann, G. Pabst, B. Sierakowski, R. Giesche, and A. Baumann, Parameters for compartment-free pharmacokinetics: Standardisation of study design, data analysis and reporting, Shaker Verlag, Aachen, Germany, 1999.
13. P. J. Brockwell, S. I. Resnick, and R. L. Tweedie, Storage processes with general release rule and additive inputs, Adv. Appl. Probab. 14 (1982), no. 2, 392-433.
14. S. Browne and K. Sigman, Work-modulated queues with applications to storage processes, J. Appl. Probab. 29 (1992), no. 3, 699-712.
15. E. Budtz-Jrgensen, N. Keiding, P. Grandjean, P. Weihe, and R. F. White, Statistical methods for the evaluation of health effects of prenatal mercury exposure, Environmetrics 13 (2003), 105-120.
16. F. Cérou, P. Del Moral, F. Le Gland, and P. Lezaud, Genetic genealogical models in rare event analysis, Alea 1 (2006), 183-196.
17. O. L. V. Costa, Stationary distributions for piecewise-deterministic Markov processes, J. Appl. Probab. 27 (1990), no. 1, 60-73.
18. C. Cox, T. W. Clarkson, D. O. Marsh, L. Amin-Zaki, S. Tikriti, and G. G. Myers, Dose-response analysis of infants prenatally exposed to methylmercury: an application of a single compartment model to stringle-stranded hair analysis, Environ. Res. 49 (1989), 318-332.
19. CREDOC-AFSSA-DGAL, Enquête INCA (individuelle et nationale sur les consommations alimentaires), TEC&DOC, Lavoisier, Paris, 1999, (Coordinateur: J.L. Volatier).
20. A. Crépet, J. Tressou, P. Verger, and J. Ch. Leblanc, Management options to reduce exposure to methyl mercury through the consumption of fish and fishery products by the French population, Regul. Toxicol. Pharmacol. 42 (2005), 179-189.
21. D. J. Daley and D. Vere-Jones, An introduction to the theory of Point Processes: I, Springer-Verlag, New York, 2003, Second Edition.
22. M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models, J. R. Statist. Soc. 46 (1984), no. 3, 353-388.
23. M. H. A. Davis and R. J. Elliott, Applied stochastic analysis, Stochastics Monographs, vol. 5, Gordon and Breach, New York, 1991.
24. L. Edler, K. Poirier, M. Dourson, J. Kleiner, B. Mileson, H. Nordmann, A. Renwick, W. Slob, K. Walton, and G. Würtzen, Mathematical modelling and quantitative methods, Food Chem. Toxicol. 40 (2002), 283-326.
25. P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling extremal events for insurance and finance, Applications of Mathematics, Springer-Verlag, Berlin, 1997.
26. R. Falk, J. O. Snihs, L. Ekman, U. Greitz, and B. Aberg, Whole-body measurements on the distribution of mercury-203 in humans after oral-intake of methylradiomercury nitrate, Acta Radiol. 9 (1970), 55-72.
27. FAO/WHO, Evaluation of certain food additives and contaminants, Fifty-seventh report of the Joint FAO/WHO Expert Committee on Food Additives 909, 2002, June 2001.
28. Evaluation of certain food additives and contaminants, Sixty-first report of the Joint FAO/WHO Expert Committee on Food Additives, Technical Report Series 52, WHO, Geneva, Switzerland, 2004, June 2003.
29. M. Gibaldi and D. Perrier, Pharmacokinetics, Drugs and the Pharmaceutical Sciences: a Series of Textbooks and Monographs, Marcel Dekker, New York, 1982, Second Edition.
30. M. J. Gibney and H. van der Voet, Introduction to the Monte Carlo project and the approach to the validation of probabilistic models of dietary exposure to selected food chemicals, Food Addit. Contam. 20 (2003), no. Suppl. 1, S1-S7.
31. P. Grandjean, P. Weihe, R. White, F. Debes, S. Araki, K. Yokoyama, K. Murata, N. Sorensen, R. Dahl, and P. Jorgensen, Cognitive deficit in 7-year-old children with prenatal exposure to methylmercury, Neurotoxicology Teratology 19 (1997), 417-428.
32. D. R. Grey, Regular variation in the tail behaviour of solutions of random difference equations, Adv. Appl. Probab. 4 (1994), no. 1, 169-183.
33. D. R. Helsel, Nondetects and data analysis: Statistics for censored environmental data, Statistics in Practice, Wiley, New York, 2004.
34. IFREMER, Résultat du réseau national d'observation de la qualité du milieu marin pour les mollusques (RNO), 1994-1998.
35. IPCS, Environmental health criteria 70: Principles for the safety assessment of food, Tech. report, WHO, Geneva, 1987, International Programme on Chemical Safety, available at www.who.int/pcs/.
36. J. Jacod and A. N. Shiryaev, Limit theorems for stochastic processes, Springer, New York, 1987.
37. H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math. 131 (1973), 207-248.
38. T. Konstantopoulos and G. Last, On the use of Lyapunov function methods in renewal theory, Stoch. Proc. App. 79 (1999), 165-178.
39. R. Lund, S. P. Meyn, and R. L. Tweedie, Rates of convergence of stochastically monotone Markov processes, Ann. App. Prob. 6 (1996), 218-237.
40. MAAPAR, Résultats des plans de surveillance pour les produits de la mer, Ministère de l'Agriculture, de l'Alimentation, de la Pêche et des Affaires Rurales, 1998-2002.
41. S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous time processes, Adv. Appl. Probab. 25 (1993), no. 3, 518-548.
42. Computable bounds for geometric convergence rates of Markov chains, Ann. Appl. Probab. 4 (1994), no. 4, 981-1011.
43. A survey of Foster-Lyapunov conditions for general state space Markov processes, Proceedings of the Workshop on Stochastic Stability and Stochastic Stabilization (Metz, France), Springer-Verlag, 1994.
44. Markov chains and stochastic stability, Springer-Verlag, London, 1996.
45. J. K. Miettienen, Mercury, mercurials, and mercaptans, M.W. Miller & T.W. Clarkson ed., ch. Absorption and elimination of dietary mercury and methyl mercury in man, pp. 223-240, C.C. Thomas, Springfield, IL., 1973.
46. S. T. Rachev, Probability metrics and the stability of stochastic models, Wiley, New York, 1991.
47. S. T. Rachev and G. Rüschendorf, Mass transportation problems I: Theory, Springer, New York, 1998.
48. S. T. Rachev and G. Samorodnitsky, Limit laws for a stochastic process and random recursion arising in probabilistic modelling, Adv. Appl. Probab. 27 (1995), no. 1, 185-202.
49. T. Rahola, R. K. Aaron, and J. K. Miettienen, Half time studies of mercury and cadmium by whole body counting, Assessment of Radioactive Contamination in Man, Proceedings IAEA-SM-150-13, 1972, pp. 553-562.
50. A. G. Renwick, S. M. Barlow, I. Hertz-Picciotto, A. R. Boobis, E. Dybing, L. Edler, G. Eisenbrand, J. B. Greig, J. Kleiner, J. Lambe, D. J. G. Mueller, S. Tuijtelaars, P. A. Van den Brandt, R. Walker, and R. Kroes, Risk characterisation of chemicals in food and diet, Food Chem. Toxicol. 41 (2003), no. 9, 1211-1271.
51. G. O. Roberts and R. L. Tweedie, Bounds on regeneration times and convergence rates of Markov chains, Stoch. Proc. Appl. 80 (1999), 211-229.
52. Rates of convergence of stochastically monotone and continuous time Markov models, J. Appl. Probab. 37 (2000), no. 2, 359-373.
53. J. S. Rosenthal, Minorization conditions and convergence rates for Markov chain Monte Carlo, J. Amer. Stat. Assoc. 90 (1995), 558-566.
54. M. Rowland and T. N. Tozer, Clinical pharmacokinetics: Concepts and applications, 1995.
55. J. C. Smith, P. V. Allen, M. D. Turner, B. Most, H. L. Fisher, and L. L. Hall, The kinetics of intravenously administered methyl mercury in man, Toxicol. Appl. Pharmacol. 128 (1994), 251-256.
56. J. C. Smith and F. F. Farris, Methyl mercury pharmacokinetics in man: A reevaluation, Toxicol. Appl. Pharmacol. 137 (1996), 245-252.
57. J. Tressou, Méthodes statistiques pour l'évaluation du risque alimentaire, Ph.D. thesis, Université Paris X, 2005, http://tel.archives-ouvertes.fr/tel-00139909.
58. Non parametric modelling of the left censorship of analytical data in food risk exposure assessment, J. Amer. Stat. Assoc. 101 (2006), no. 476, 1377-1386.
59. J. Tressou, A. Crépet, P. Bertail, M. H. Feinberg, and J. C. Leblanc, Probabilistic exposure assessment to food chemicals based on extreme value theory, application to heavy metals from fish and sea products, Food Chem. Toxicol. 42 (2004), no. 8, 1349-1358.
60. T. Tsubaki and K. Irukayama, Minamata disease: Methyl-mercury poisoning in Minamata and Niigata, Japan., ElsevierScientific, New York, 1977.
61. NRC (National Research Council), Committee on the toxicological effects of methylmercury: Toxicological effects of methylmercury, Tech. report, Washington DC, 2000.
62. P. Verger, J. Tressou, and S. Clémençon, Integration of time as a description parameter in risk characterisation: application to methylmercury, Regul. Toxicol. Pharmacol. 49 (2007), no. 1, 25-30.
63. W. Whitt, Stochastic-process limits, an introduction to stochastic-process limits and their application to queues, Springer, New York, 2002.