An analysis of errors arising from the direct use of mass balance principles to describe regional drug uptake and elution

  • Richard N. Upton


Errors occurring during the direct application of mass balance principles to describe the uptake and elution of a drug in an organ during and after a constant rate infusion were analyzed. The uptake of lignocaine in the hindquarters of sheep was used as an example—the net mass of lignocaine was calculated from the arterial and inferior vena cava blood lignocaine concentrations and hindquarter blood flow using an integrated form of the Fick equation. The general strategy was to generate a continuous time course of arterial and inferior vena cava drug concentrations that closely resembled the data obtained fromin vivo experiments (the “true” blood concentrations). These were used to calculate the time course of the “true” net mass of lignocaine in the hindquarters by numerical integration with a small step size. The true blood concentrations were then used to generate data sets that simulated different blood sample intervals and random, normally distributed errors added to the blood concentration and blood flow measurements. Simulated data sets were also used to compare different numerical integration methods. There were significant absolute errors in the calculated net mass in the period after the start and end of the constant rate infusion due to numerical integration, but the error resulting from the latter to some extent canceled the error from the former. These errors did not greatly change the time course of the calculated net mass. Decreasing the interval between regular blood samples from 30 to 10 min reduced this absolute error, but greater reductions in error were achieved by optimizing the time interval between blood samples to give an approximate constant error due to numerical integration. There was no advantage in using numerical integration methods other than the linear trapezoidal method. Random noise added to the blood concentration and blood flow terms of the net mass equation added a small bias to the mean value of the calculated net mass. More important, such noise rapidly increased the number of studies required to characterize the calculated mean net mass to a given level of accuracy. It is concluded best results are obtained by minimizing the variability of blood concentration and blood flow measurements, and by using an optimized blood sampling regimen. The direct mass balance calculations and an analysis of their errors are simple enough to be performed using a spreadsheet program on a personal computer.

Key Words

mass balance principles error pharmacokinetics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. L. Price, J. W. Dundee, and E. H. Conner. Rates of uptake and release of thiopental by the human brain; Relation to kinetics of thiopental anaesthesia.Anesthesiology 18:171 (1957).CrossRefGoogle Scholar
  2. 2.
    R. N. Upton, L. E. Mather, W. B. Runciman, C. Nancarrow, and R. J. Carapetis. The use of mass balance principles to describe regional drug distribution and elimination.J. Pharmacokin. Biopharm. 16:13–29 (1988).CrossRefGoogle Scholar
  3. 3.
    S. Björkman, J. Åkeson, F. Nilsson, K. Messeter, and B. Roth. Ketamine and midazolam decrease cerebral blood flow and consequently their own rate of transport into the brain: An application of mass balance pharmacokinetics with a changing regional blood flow.J. Pharmacokin. Biopharm. 20:637–652 (1992).CrossRefGoogle Scholar
  4. 4.
    Y. F. Huang, R. N. Upton, W. B. Runciman. Intravenous bolus administration of subconvulsive dose of lignocaine to conscious sheep. III. Relationships between myocardial pharmacokinetics and pharmacodynamics.Br. J. Anaesth. 70:556–561 (1993).PubMedCrossRefGoogle Scholar
  5. 5.
    R. D. Purvis. Optimum numerical integration methods for estimation of area under the curve (AUC) and area under the moment curve (AUMC).J. Pharmacokin. Biopharm. 20:211–226 (1992).CrossRefGoogle Scholar
  6. 6.
    R. N. Upton, W. B. Runciman, L. E. Mather, R. J. Carapetis, and C. F. McLean. The uptake and elution of lignocaine and procainamide in the hindquarters of the sheep described using mass balance principles.J. Pharmacokin. Biopharm. 16:31–40 (1988).CrossRefGoogle Scholar
  7. 7.
    R. N. Upton, C. Nancarrow, C. R. McLean, L. E. Mather, and W. B. Runciman. The in vivo blood, fat and muscle concentrations of lignocaine and bupivacaine in the hindquarters of sheep.Xenobiotica 21:13–22 (1991).PubMedCrossRefGoogle Scholar
  8. 8.
    J. M. Smith.Scientific Analysis on the Pocket Calculator, Wiley, New York, 1975, pp. 154–179.Google Scholar
  9. 9.
    B. Schmeiser. Approximations to the inverse cumulative normal function for use on hand calculators.Appl. Statist. 28:175–176 (1979).CrossRefGoogle Scholar
  10. 10.
    D. S. Kalonia and A. P. Simonelli. Analysis of consecutive pseudo-first-order reactions. II: Calculation of the rate constants from the co-product or co-reactant data.J. Pharm. Sci. 78:78–84 (1989).PubMedCrossRefGoogle Scholar
  11. 11.
    W. W. Piegorsch and A. J. Bailer. Optimal design allocations for estimating area under curves for studies employing destructive sampling.J. Pharmacokinet. Biopharm. 17:493–507 (1989).PubMedCrossRefGoogle Scholar
  12. 12.
    D. Katz and D. Z. D'Argenio. Experimental design for estimating integrals by numerical quadrature, with applications to pharmacokinetic studies.Biometrics 39:621–628 (1983).PubMedCrossRefGoogle Scholar
  13. 13.
    R. R. Kennedy and A. B. Baker. Analysis of uncertainty in theoretical methods of cardiac output measurement using the “Monte Carlo” technique.Br. J. Anaesth. 71:403–409 (1993).PubMedCrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Richard N. Upton
    • 1
  1. 1.Department of Anaesthesia and Intensive Care, Royal Adelaide HospitalUniversity of AdelaideAdelaideAustralia

Personalised recommendations