Skip to main content
SpringerLink
Log in

Linear compartmental systems. I. kinetic analysis and derivation of their optimized symbolic equations

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

The study of many biological systems requires the application of a compartmental analysis, together with the use of isotopic tracers, parameter identification and methods to evaluate the mean parameters. For all this, the kinetic equations of the compartmental system as a function of its parameters are needed. In this paper, we present some considerations on the diagrams of connectivity of linear compartmental systems and obtain new properties from the matrix corresponding to the ordinary first-order linear differential equation systems which describe their kinetic behaviour. Using these properties, symbolic equations are obtained in a simplified form. These equations provide the instantaneous amount of substance in any compartment of the system when zero input is injected into one or more of the system compartments, solely as a function of those parameters of compartmental systems which really have an influence on the sought expression. This is unlike what happens in the other symbolic equations obtained in a previous contribution that included all the fractional transfer coefficients involved in the compartmental system, regardless of whether or not they had an influence on the instantaneous amount of substance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Rowland M., Tozer N.: Clinical Pharmacokinetics: Concepts and Applications, 2nd edn. Lea & Febiger, London (1989)

    Google Scholar 

  2. Riegelman S., Loo J.C., Rowland M.: Shortcomings in pharmacokinetic analysis by conceiving the body to exhibit properties of a single compartment. J. Pharm. Sci. 57(1), 117–123 (1968)

    Article  CAS  Google Scholar 

  3. Watabe H., Ikoma Y., Kimura Y., Naganawa M., Shidahara M.: PET kinetic analysis-compartmental model. Ann. Nucl. Med. 20(9), 583–588 (2006)

    Article  CAS  Google Scholar 

  4. Peletier L.A., Benson N., van der Graaf P.H.: Impact of protein binding on receptor occupancy: a two-compartment model. J. Theor. Biol. 265(4), 657–671 (2010)

    Article  CAS  Google Scholar 

  5. Rescigno A.: Compartmental analysis and its manifold applications to pharmacokinetics. AAPS J. 12(1), 61–72 (2010)

    Article  CAS  Google Scholar 

  6. Rescigno A.: Compartmental analysis revisited. Pharmacol. Res. 39(6), 471–478 (1999)

    Article  CAS  Google Scholar 

  7. Jacquez J.A.: Modelling with Compartmets. Biomedware, Ann Arbor (1999)

    Google Scholar 

  8. R. Varon, Estudios de Sistemas de Compartimentos y su Aplicación a la Fase de Transición de Ecuaciones cinéticas. Doctoral Thesis. (Universidad de Murcia 1979)

  9. Bevan D.R., Weyand E.H.: Compartmental analysis of the disposition of benzo[a]pyrene in rats. Carcinogenesis 9(11), 2027–2032 (1988)

    Article  CAS  Google Scholar 

  10. Garcia-Meseguer M.J., Vidalde Labra J.A., Garcia-Canovas F., Havsteen B.H., Garcia-Moreno M., Varon R.: Time course equations of the amount of substance in a linear compartmental system and their computerized derivation. Biosystems 59(3), 197–220 (2001)

    Article  CAS  Google Scholar 

  11. Galvez J., Varon R.: I. Transient phase kinetics of enzyme reactions. J. Theor. Biol. 89(1), 1–17 (1981)

    Article  CAS  Google Scholar 

  12. Fleishaker J.C., Smith R.B.: Compartmental model analysis in pharmacokinetics. J. Clin. Pharmacol. 27(12), 922–926 (1987)

    CAS  Google Scholar 

  13. Orisakwe O.E., Afonne O.J., Ilondu N.A., Obi E., Ufearo C.S., Agbasi P.U. et al.: Influence of prokinetics on the gastrointestinal transit and residence times of activated charcoal. J. Pak. Med. Assoc. 52(8), 354–356 (2002)

    CAS  Google Scholar 

  14. Endres C.J., De Jesus O.T., Uno H., Doudet D.J., Nickles J.R., Holden J.E.: Time profile of cerebral [18F]6-fluoro-L-DOPA metabolites in nonhuman primate: implications for the kinetics of therapeutic L-DOPA. Front. Biosci. 9, 505–512 (2004)

    Article  CAS  Google Scholar 

  15. Kuypers D.R., Vanrenterghem Y.: Time to reach tacrolimus maximum blood concentration,mean residence time, and acute renal allograft rejection: an open-label, prospective, pharmacokinetic study in adult recipients. Clin. Ther. 26(11), 1834–1844 (2004)

    Article  CAS  Google Scholar 

  16. Sarria B., Dainty J.R., Fox T.E., Fairweather-Tait S.J.: Estimation of iron absorption in humans using compartmental modelling. Eur. J. Clin. Nutr. 59(1), 142–144 (2005)

    Article  CAS  Google Scholar 

  17. Harmsen M.M., van Solt C.B., Fijten H.P., Van Setten M.C.: Prolonged in vivo residence times of llama single-domain antibody fragments in pigs by binding to porcine immunoglobulins. Vaccine 23(41), 4926–4934 (2005)

    Article  CAS  Google Scholar 

  18. Juillet B., Saccomani M.P., Bos C., Gaudichon C., Tome D., Fouillet H.: Conceptual, methodological and computational issues concerning the compartmental modeling of a complex biological system: Postprandial inter-organ metabolism of dietary nitrogen in humans. Math. Biosci. 204(2), 282–309 (2006)

    Article  CAS  Google Scholar 

  19. Varon R., Masia-Perez J., Garcia-Molina F., Garcia-Canovas F., Arias E., Arribas E. et al.: An alternative analysis of enzyme systems based on the whole reaction time. Evaluation of the kinetic parameters and initial enzyme concentration. J. Math. Chem. 42(4), 789–813 (2007)

    Article  CAS  Google Scholar 

  20. de Graaf A.A., Freidig A.P., De R.B., Jamshidi N., Heinemann M., Rullmann J.A. et al.: Nutritional systems biology modeling: from molecular mechanisms to physiology. PLoS Comput. Biol. 5(11), e1000554 (2009)

    Article  Google Scholar 

  21. Fouillet H., Juillet B., Gaudichon C., Mariotti F., Tome D., Bos C.: Absorption kinetics are a key factor regulating postprandial protein metabolism in response to qualitative and quantitative variations in protein intake. Am. J. Physiol. Regul. Integr. Comp. Physiol. 297(6), R1691–R1705 (2009)

    Article  CAS  Google Scholar 

  22. Meier P., Zierler K.L.: On the theory of the indicator-dilution method for measurement of blood flow and volume. J. Appl. Physiol. 6(12), 731–744 (1954)

    CAS  Google Scholar 

  23. Nye J.A., Votaw J.R., Jarkas N., Purselle D., Camp V., Bremner J.D. et al.: Compartmental modeling of 11C-HOMADAM binding to the serotonin transporter in the healthy human brain. J. Nucl. Med. 49(12), 2018–2025 (2008)

    Article  Google Scholar 

  24. Brix G., Zwick S., Kiessling F., Griebel J.: Pharmacokinetic analysis of tissue microcirculation using nested models: multimodel inference and parameter identifiability. Med. Phys. 36(7), 2923–2933 (2009)

    Article  Google Scholar 

  25. Rescigno A.: A contribution to the theory of tracer methods II. Biochim. Biophys. Acta. 21(1), 111–116 (1956)

    Article  CAS  Google Scholar 

  26. Anderson D.H.: Compartmental Modelling and Tracer Kinetics. Springer, Berlin (1983)

    Google Scholar 

  27. Godfrey K.: Compartmental Models and Their Application. Academic Press, London (1983)

    Google Scholar 

  28. Jacquez J.A.: Compartmental Analysis in Biology and Medicine, 2nd edn. Ann Arbor, Michigan (1985)

    Google Scholar 

  29. Rescigno A., Thakur S.: New Trends in Pharmacokinetics. Plenum Press, New York (1991)

    Google Scholar 

  30. Varon R., Garcia-Meseguer M.J., Garcia-Canovas F., Havsteen B.H.: General linear compartment model with zero input: I. Kinetic equations. Biosystems 36(2), 121–133 (1995)

    Article  CAS  Google Scholar 

  31. Varon R., Garcia-Meseguer M.J., Havsteen B.H.: General linear compartment model with zero input: II. The computerized derivation of the kinetic equations. Biosystems 36(2), 135–144 (1995)

    Article  CAS  Google Scholar 

  32. Varon R., Garcia-Meseguer M.J., Valero E., Garcia-Moreno M., Garcia-Canovas F.: General linear compartment model with zero input: III. First passage residence time of enzyme systems. Biosystems 36(2), 145–156 (1995)

    Article  CAS  Google Scholar 

  33. M.J. Garcia-Meseguer, Análisis cinético de los Sistemas Lineales de Compartimentos: Aplicación a la Evaluación de Parámetros Medios. Doctoral Thesis. (Servicio de publicaciones de la UCLM, Cuenca, 1998)

  34. Rescigno A.: Compartmental analysis and its manifold applications to pharmacokinetics. AAPS J. 12(1), 61–72 (2010)

    Article  CAS  Google Scholar 

  35. Jacquez J.A.: Compartmental Analysis in Biology and Medicine, 3rd edn. Thompson-Shore Inc, Dexter (1996)

    Google Scholar 

  36. Holz M., Fahr A.: Compartment modeling. Adv. Drug Deliv. Rev. 48(2–3), 249–264 (2001)

    Article  CAS  Google Scholar 

  37. Rescigno A., Segre G.: On some metric properties of the systems of compartments. Bull. Math. Biophys. 27(3), 315–323 (1965)

    Article  CAS  Google Scholar 

  38. Chou K.C.: Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady-state systems. Biophys. Chem. 35(1), 1–24 (1990)

    Article  CAS  Google Scholar 

  39. Lal R., Anderson D.H.: Calculation and utilization of component matrices in linear bioscience models. Math. Biosci. 99(1), 11–29 (1990)

    Article  CAS  Google Scholar 

  40. Gibaldi M.: Biopharmaceutics and Clinical Pharmacokinetics, 4th edn. Lea and Febiger, Londres (1991)

    Google Scholar 

  41. Cheng H.Y.: A method for calculating the mean residence times of catenary metabolites. Biopharm. Drug Dispos. 12(5), 335–342 (1991)

    Article  CAS  Google Scholar 

  42. Green M.H.: Introduction to modelling. J. Nutr. 122(3 Suppl), 690–694 (1992)

    CAS  Google Scholar 

  43. Galvez J., Varon R., Garcia-Carmona F.: III. Kinetics of enzyme reactions with inactivation steps. J. Theor. Biol. 89(1), 37–44 (1981)

    Article  CAS  Google Scholar 

  44. Hearon J.Z.: Theorems on linear systems. Ann. N. Y. Acad. Sci. 108, 36–38 (1963)

    Article  CAS  Google Scholar 

  45. Garcia-Meseguer M.J., Vidalde Labra J.A., Garcia-Moreno M., Garcia-Canovas F., Havsteen B.H., Varon R.: Mean residence times in linear compartmental systems. Symbolic formulae for their direct evaluation. Bull. Math. Biol. 65(2), 279–308 (2003)

    Article  CAS  Google Scholar 

  46. Taussky O.: On a theorem of Latimer and Macduffee. Can. J. Math. 1, 300–302 (1949)

    Article  Google Scholar 

  47. Jacquez J.A., Simon C.P.: Qualitative theory of compartmental systems with lags. Math. Biosci. 180, 329–362 (2002)

    Article  Google Scholar 

  48. Juillet B., Bos C., Gaudichon C., Tome D., Fouillet H.: Parameter estimation for linear compartmental models-a sensitivity analysis approach. Ann. Biomed. Eng. 37(5), 1028–1042 (2009)

    Article  Google Scholar 

  49. Travis C.C., Haddock G.: On structural identification. Math. Biosci. 56, 157–173 (1981)

    Article  Google Scholar 

  50. Cobelli C., Lefschy A., Jacur R.: Identifiability results on some constrained compartment systems. Math. Biosci. 47, 173–195 (1979)

    Article  Google Scholar 

  51. Rubinow S.I.: Introduction to Mathematical Biology. Wiley, New York (1975)

    Google Scholar 

  52. Chau N.: Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve. Math. Biosci. 76, 185–206 (1985)

    Article  Google Scholar 

  53. Benet L.Z.: General treatment of linear mammillary models with elimination from any compartment as used in pharmacokinetics. J. Pharm. Sci. 61(4), 536–541 (1972)

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Electronics, Automation and Communications, Technical School of Industrial Engineering, University of Castilla-La Mancha, Albacete, Spain

    Francisco Garcia-Sevilla

  2. Department of Physical Chemistry, Technical School of Industrial Engineering, University of Castilla-La Mancha, Albacete, Spain

    Manuela Garcia-Moreno & Ramón Varon

  3. Nursing Department, School of Nursing, University of Castilla-La Mancha, Albacete, Spain

    Milagros Molina-Alarcon & María J. Garcia-Meseguer

  4. Department of Medical Science, Faculty of Medicine, University of Castilla-La Mancha, Albacete, Spain

    José M. Villalba

  5. Applied Physics Department, High School of Informatics Engineering, Albacete, Spain

    Enrique Arribas

  6. Escuela de Ingenieros Industriales, Universidad de Castilla-La Mancha, Av. España s/n, Campus Universitario, 02071, Albacete, Spain

    Ramón Varon

Authors
  1. Francisco Garcia-Sevilla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuela Garcia-Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milagros Molina-Alarcon
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. María J. Garcia-Meseguer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. José M. Villalba
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Enrique Arribas
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ramón Varon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ramón Varon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garcia-Sevilla, F., Garcia-Moreno, M., Molina-Alarcon, M. et al. Linear compartmental systems. I. kinetic analysis and derivation of their optimized symbolic equations. J Math Chem 50, 1598–1624 (2012). https://doi.org/10.1007/s10910-012-9991-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-012-9991-z

Keywords

Search

Navigation