Abstract
A complete analysis has been performed of the mean residence times in linear compartmental systems, closed or open, with or without traps and with zero input. This analysis allows the derivation of explicit and simple general symbolic formulae to obtain the mean residence time in any compartment of any linear compartmental system, closed or open, with or without traps, as well as formulae to evaluate the mean residence time in the entire system like the above situations. The formulae are given as functions of the fractional transfer coefficients between the compartments and, in the case of open systems, they also include the excretion coefficients to the environment from the different compartments. The relationship between the formulae derived and the particular connection properties of the compartments is discussed. Finally, some examples have been solved.
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References
Anderson, D. H. (1983). Compartmental modeling and tracer kinetics, Berlin: Springer.
Aufderheide, A. C. and L. E. Wittmers (1992). Selected aspects of the spatial distribution of lead in bone. Neurotoxins 13, 809–819.
Cheng, H. (1991). A method for calculating the mean residence times of catenary matabolites. Biopharm. Drug Dispos. 12, 335–342.
Cheng, H. and W. J. Jusko (1990). Mean residence times of multicompartmental drugs undergoing reversible metabolism. Pharm. Res. 7, 103–107.
Cremonesi, M. et al. (1999). Biokinetics and dosimetry in patients administered with In-111-DOTA-Tyr(3)-octeotride: implications for internal radiotherapy with Y-90-DOTATOC. Eur. J. Nucl. Med. 26, 877–886.
Cutler, D. J. (1987). Definitions of mean residence times in Pharmacokinetics. Biopharm. Drug Dispos. 8, 87–97.
Gálvez, J. and R. Varón (1981). Transient phase kinetics of enzyme reactions. J. Theor. Biol. 89, 1–17.
García-Meseguer, M. J., J. A. Vidal de Labra, F. García-Cánovas, B. H. Havsteen, M. García-Moreno and R. Varón (2001). Time course equations of the amount of substance in a linear compartmental system and their computerised derivation. Biosystems 59, 197–220.
Gibaldi, M. (1991). Biopharmaceutics and Clinical Pharmacokinetics, 4th edn, London: Lea & Febiger.
Green, M. H. (1992). Introduction to modeling. J. Nutr. 122, 690–694.
Hearon, J. Z. (1963). Theorems on linear systems. Ann. Ny. Acad. Sci. 108, 36–38.
Hearon, J. Z. (1972). Residence time in compartmental system and the moments of a certain distribution. Math. Biosci. 15, 69–77.
Isoherranen, N., E. Lavy and S. Soback (2000). Pharmacokinetics of gentamicine C-1, C-1a, and C-2 in beagles after a single intravenous dose. Antimicrob. Agents. Chemother. 44, 1443–1447.
Jacquez, J. A. (1985). Compartmental Analysis in Biology and Medicine, 2nd edn, Michigan: The University of Michigan Press.
Kong, A. and W. Jusko (1988). Definitions and applications of mean transit and residence times in reference to the two-compartment mammillary plasma clearance model. J. Pharm. Sci. 77, 157–165.
Moolvakar, S. G. and A. Luebeck (1990). Two-events models for carcinogenesis: biological, mathematical and statistical considerations. Risk Anal. 10, 323–340.
Rescigno, A. (1956). A contribution to the theory of tracer methods. Part II. Biochem. Biophys., Acta. 21, 111–116.
Rescigno, A. (1999). Compartmental analysis revisited. Pharm. Res. 36, 471–478.
Rescigno, A. and E. Gurpide (1973). Estimation of average times of residence, recycle and interconversion of blood-borne compounds. JCE & M. 36, 263–276.
Schuster, S. and R. Heinrich (1987). Time hierarchy in enzymatic reaction chains resulting from optimality principles. J. Theor. Biol. 129, 189–209.
Sines, J. J. and D. Hackney (1987). A residence times analysis of enzyme kinetics. Biochem. J. 247, 159–164.
Tozer, E. C. and T. E. Carew (1997). Residence time of low-density lipoprotein in the normal and atherosclerotic rabit aorta. Cos. Res. 80, 208–218.
Varón, R. (1979). Estudio de sistemas de compartimientos y su aplicación a la fase de transición de reacciones enzimáticas, Doctoral Thesis, University of Murcia, Spain.
Varón, R., M. J. García-Meseguer, F. García-Cánovas and B. H. Havsteen (1995). General model compartmental system with zero input I: kinetic equations. Biosystems 36, 121–133.
Veng-Pedersen, P. (1989). Mean time parameters in pharmacokinetics. Definition, computation and clinical implications. Part II. Clin. Pharmacokinet. 17, 424–440.
Wastney, M. E., P. A. Angelus, R. M. Barnes and K. N. S. Subramanian (2000). Kinetic of zinc metabolism: variation with diet, genetics and disease. J. Nutr. 130(2 Suppl. S), 1355S–1359S.
Wastney, M. E., W. A. House, R. M. Barnes and K. N. S. Subramanian (1999). Zinc absorption, distribution, excretion and retention by healthy preterm infants. Ped. Res. 45, 191–196.
Wilson, P. D. G. and J. R. Dainty (1999). Modelling in nutrition: an introduction. Proc. Nutr. Soc. 58, 133–138.
Yamaoka, K., T. Nakagawa and T. Uno (1978). Statistical moments in pharmacokinetics. J. Pharmacokinet. Biopharm. 6, 547–557.
Zimmermann, T., H. Laufen, H. Yeates, F. Scharpf, K. D. Riedel and T. Schumacher (1999). The pharmacokinetics of extended-release formulations of calcium antagonists and amlodipine in subjects with differents gastrointestinal transit times. J. Clin. Pharm. 39, 1021–1031.
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García-Meseguer, M.J., Vidal de Labra, J.A., García-Moreno, M. et al. Mean residence times in linear compartmental systems. Symbolic formulae for their direct evaluation. Bull. Math. Biol. 65, 279–308 (2003). https://doi.org/10.1016/S0092-8240(02)00096-4
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DOI: https://doi.org/10.1016/S0092-8240(02)00096-4