Skip to main content
SpringerLink
Log in

Two-compartment dispersion model for analysis of organ perfusion system of drugs by fast inverse laplace transform (FILT)

  • Published:
Journal of Pharmacokinetics and Biopharmaceutics Aims and scope Submit manuscript

Abstract

A dispersion model developed in Chromatographic theory is applied to the analysis of the elution profile in the liver perfusion system of experimental animals. The equation for the dispersion model with the linear nonequilibrium partition between the perfusate and an organ tissue is derived in the Laplace-transformed form, and the fast inverse Laplace transform (FILT) is introduced to the pharmacokinetic field for the manipulation of the transformed equation. By the analysis of the nonlinear least squares method associated with FILT, this model (two-compartment dispersion model) is compared to the model with equilibrium partition between the perfusate and the liver tissue (one-compartment dispersion model) for the outflow curves of ampicillin and oxacillin from the rat liver. The model estimation by Akaike's information criterion (AIC) suggests that the two-compartment dispersion model is more proper than the one-compartment dispersion model to mathematically describe the local disposition of these drugs in the perfusion system. The blood space in the liver, VB, and the dispersion number DN are estimated at 1.30 ml (±0.23 SD) and 0.051 (±0.023 SD), respectively, both of which are independent of the drugs. The efficiency number, RN, of ampicillin is 0.044 (±0.049 SD) which is significantly smaller than 0.704 (±0.101 SD) of oxacillin. The parameters in the two-compartment dispersion model are correlated to the recovery ratio, FH, mean transit time, ¯tH, and the relative variance, σ2/¯tH 2, of the elution profile of drugs from the rat liver.

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

Abbreviations

A :

Cross-sectional area of the blood space

C(t, z) :

Concentration of drug (one-compartment dispersion model)

∼C(s, z) :

Laplace transform of C(t, z)

C 1(t, z):

Concentration of drug in blood space (two-compartment dispersion model)

C 2(t, z):

Concentration of drug in the liver tissue (two-compartment dispersion model)

∼C 1 (s, z):

Laplace transform ofC 1(t, z)

D :

Axial or longitudinal dispersion coefficient

D c(=D· A 2):

Corrected dispersion coefficient

D N :

Dispersion number

f I(t):

Input function with respect tot

fI(z):

Input function with respect toz

FI(s):

Laplace transform of fI(t)

fs(t):

System weight function with respect tot

fs(z):

System weight function with respect to z

FH :

Recovery ratio

k′ :

Partition ratio (distribution ratio)

k12, k21 :

Forward and backward partition rate constant in the central elimination two-compartment dispersion model

k p12 ,k p21 :

Forward and backward partition rate constant in the peripheral elimination two-compartment dispersion model

ke :

Elimination (or irreversible transfer) rate constant

k pe :

Elimination rate constant in peripheral elimination model

KH :

Distribution constant

L :

Length of blood space in liver

M :

Amount of drug injected

m :

Coefficient related to the injected amount

ph :

Mass transfer coefficient from perfusate to hepatic tissue

Q :

Flow rate of perfusate

RN :

Efficiency number

s :

Laplace variable

t :

Time

¯ tH :

Mean transit time

υ :

Linear flow velocity of the perfusate

V B(= L·A):

Blood volume (sum of the sinusoid volume and the space of Disse)

vh :

Apparent volume of distribution

V ′H :

Anatomical volume of liver tissue

z :

Axial coordinate in the liver

δ(t):

Delta function

ɛ :

Volume ratio of the anatomical liver tissue to the blood space

δ2 :

Variance of transit time

δ2/¯t 2H :

Relative dispersion to transit time

∂ :

Partial derivatives

References

  1. K. S. Pang and M. Rowland. Hepatic clearance of drugs: 1. Theoretical considerations of a “well-stirred” model and a “parallel tube” model. Influence of hepatic blood flow, plasma and blood cell binding, and the hepatocellular enzymatic activity on hepatic drug clearance.J. Pharmacokin. Biopharm. 5:625–653 (1977).

    Article  CAS  Google Scholar 

  2. G. R. Wilkinson and D. G. Shand. Commentary. A physiological approach to hepatic drug clearance.Clin. Pharmacol. Ther. 18:377–390 (1975).

    CAS  PubMed  Google Scholar 

  3. M. Rowland, L. Z. Benet, and G. G. Graham. Clearance concepts in pharmacokinetics.J. Pharmacokin. Biopharm. 1:123–136 (1977).

    Article  Google Scholar 

  4. L. Bass, S. Keiding, K. Winkler, and N. Tygstrup. Enzymatic elimination of substrates flowing through the intact liver.J. Theor. Biol. 61:393–409 (1976).

    Article  CAS  PubMed  Google Scholar 

  5. R. W. Brauer, G. F. Leong, P. F. McElory, Jr., and R. J. Holloway. Circulatory pathways in the rat liver as revealed by32P chromic phosphate colloid uptake in the isolated perfused liver preparation.Am. J. Physiol. 184:593–598 (1956).

    CAS  PubMed  Google Scholar 

  6. M. Weiss. Moment of physiological transit time distribution and the time course of drug disposition in the body.J. Math. Biol. 15:305–318 (1982).

    Article  CAS  PubMed  Google Scholar 

  7. L. Bass, P. Robinson, and A. J. Bracken. Hepatic elimination of flowing substrates: the distributed model.J. Theor. Biol. 72:161–184 (1978).

    Article  CAS  PubMed  Google Scholar 

  8. E. L. Forker and B. Luxton. Hepatic transport kinetic and plasma disappearance curves: Distributed modeling versus conventional approach.Am. J. Physiol. 235:E648-E660 (1978).

    CAS  PubMed  Google Scholar 

  9. C. A. Goresky, W. H. Ziegler, and G. G. Bach. Capillary exchange modeling. Barrier-limited and flow limited distribution.Circ. Res. 27:739–764 (1970).

    Article  CAS  PubMed  Google Scholar 

  10. M. S. Roberts and M. Rowland. Hepatic elimination: dispersion model.J. Pharm. Sci. 74:585–587 (1985).

    Article  CAS  PubMed  Google Scholar 

  11. M. S. Roberts and M. Rowland. A dispersion model of hepatic elimination: 1. Formulation of the model and bolus considerations.J. Pharmacokin. Biopharm. 14:227–260 (1986).

    Article  CAS  Google Scholar 

  12. L. Lapidus and N. R. Amundson. Mathematics of adsorption in beds. VI. The effect of longitudinal diffusion in ion exchange and Chromatographic columns.J. Phys. Chem. 56:984–988 (1952).

    Article  CAS  Google Scholar 

  13. K. Yamaoka, and T. Nakagawa. Moment analysis for reaction chromatography.J. Chromatog. 117:1–10 (1976).

    Article  CAS  Google Scholar 

  14. T. Hosono. Numerical inversion of Laplace transform and some applications to wave optics. Radio Sci.16:1015–1019 (1981).

    Article  Google Scholar 

  15. K. Yamaoka, Y. Tanigawara, T. Nakagawa, and T. Uno. A pharmacokinetic analysis program (MULTI) for microcomputer.J. Pharmacobio-Dyn. 4:879–885 (1981).

    Article  CAS  PubMed  Google Scholar 

  16. A. B. Littlewood.Gas Chromatography, Academic Press, New York, 1970, pp. 164–232.

    Book  Google Scholar 

  17. O. Levenspiel.Chemical Reaction Engineering, Wiley, New York, 1972, pp. 235–315.

    Google Scholar 

  18. C. A. Goresky and G. G. Bach. Membrane transport and the hepatic circulation.Ann. N.Y. Acad. Sci. 170 (1):18–47 (1970).

    Article  CAS  Google Scholar 

  19. K. Yamaoka, T. Nakagawa, and T. Uno. Application of Akaike's information criterion (AIC) in the evaluation of linear pharmacokinetic equations.J. Pharmacokin. Biopharm. 6:165–175 (1978).

    Article  CAS  Google Scholar 

  20. M. Barza and L. Weinstein. Pharmacokinetics of the penicillins in man.Clin. Pharmacokin. 1:297–308 (1976).

    Article  CAS  Google Scholar 

  21. Y. Murai, T. Nakagawa, K. Yamaoka, and T. Uno. High performance liquid Chromatographic analysis and pharmacokinetic investigation of oxacillin and its metabolites in man.Chem. Pharm. Bull. 29:3290–3297 (1981).

    Article  CAS  PubMed  Google Scholar 

  22. C. A. Goresky. A linear method for determining liver sinusoidal and extravascular volumes.Am. J. Physiol. 204:626–640 (1963).

    CAS  PubMed  Google Scholar 

  23. W. Perl and F. P. Chinard. A convection-diffusion model of indicator transport through an organ.Circ. Res. 22:273–298 (1967).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Pharmaceutical Sciences, Kyoto University, Sakyo-ku, 606, Kyoto, Japan

    Yoshitaka Yano, Kiyoshi Yamaoka, Yoshihiro Aoyama & Hisashi Tanaka

Authors
  1. Yoshitaka Yano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kiyoshi Yamaoka
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoshihiro Aoyama
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hisashi Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yano, Y., Yamaoka, K., Aoyama, Y. et al. Two-compartment dispersion model for analysis of organ perfusion system of drugs by fast inverse laplace transform (FILT). Journal of Pharmacokinetics and Biopharmaceutics 17, 179–202 (1989). https://doi.org/10.1007/BF01059027

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01059027

Key words

Search

Navigation