On Single Compartment Pharmacokinetic Model Systems that Obey Michaelis-Menten Kinetics and Systems that Obey Krebs Cycle Kinetics

Author(s)
Kal Renganathan Sharma

ABSTRACT

The integration of Michaelis-Menten kinetics results in a trancedental equation. The results are not in a form that is readily usable. A more usable form of the model solutions is developed. This was accomplished by using Taylor series expansion of dimensionless concentration u in terms of its derivatives. The infinite series expression for dimensionless concentration is given. It can be seen that for times t < , the Taylor series expression evaluated near the origin up to the third derivative is a reasonable representation of the integrated solution. More terms in the Taylor series expression can be added to suit the application. It can vary with the apparent volume, dosage, enzyme concentration, Michaelis constant and the desired accuracy level needed. The single compartment model solution was obtained by the method of Laplace transform. It can be seen from Figure 2 that the dimensionless drug concentration in the compartment goes through a maxima. The curve is convex throughout the absorption and elimination processes. The drug gets completely depleted after a said time. The curve is asymmetrical with a right skew. The systems under absorption with elimination that obey the kinetics that can be represented by a set of reactions in circle were considered. A system of simple reactions in circle was taken into account. The concentration profile of the reactants were obtained by the method of Laplace transforms. The conditions when subcritical damped oscillations can be expected are derived. A model was developed for cases when absorption kinetics exhibit subcritical damped oscillations. The solution was developed by the method of Laplace transforms. The solution for dimensionless concentration of the drug in single compartment for different values of rate constants and dimensionless frequency are shown in Figures 6-9. The drug profile reaches a maximum and drops to zero concen-tration after a said time. The fluctuations in concentration depends on the dimensionless frequency resulting from the subcritical damped oscillations during absorption. At low frequencies the fluctuations are absent. As the frequency is increased the fluctuations in concentration are pronounced. The fre-quency of fluctuations were found to increase with increase in frequency of oscillations during ab-sorption.

The integration of Michaelis-Menten kinetics results in a trancedental equation. The results are not in a form that is readily usable. A more usable form of the model solutions is developed. This was accomplished by using Taylor series expansion of dimensionless concentration u in terms of its derivatives. The infinite series expression for dimensionless concentration is given. It can be seen that for times t < , the Taylor series expression evaluated near the origin up to the third derivative is a reasonable representation of the integrated solution. More terms in the Taylor series expression can be added to suit the application. It can vary with the apparent volume, dosage, enzyme concentration, Michaelis constant and the desired accuracy level needed. The single compartment model solution was obtained by the method of Laplace transform. It can be seen from Figure 2 that the dimensionless drug concentration in the compartment goes through a maxima. The curve is convex throughout the absorption and elimination processes. The drug gets completely depleted after a said time. The curve is asymmetrical with a right skew. The systems under absorption with elimination that obey the kinetics that can be represented by a set of reactions in circle were considered. A system of simple reactions in circle was taken into account. The concentration profile of the reactants were obtained by the method of Laplace transforms. The conditions when subcritical damped oscillations can be expected are derived. A model was developed for cases when absorption kinetics exhibit subcritical damped oscillations. The solution was developed by the method of Laplace transforms. The solution for dimensionless concentration of the drug in single compartment for different values of rate constants and dimensionless frequency are shown in Figures 6-9. The drug profile reaches a maximum and drops to zero concen-tration after a said time. The fluctuations in concentration depends on the dimensionless frequency resulting from the subcritical damped oscillations during absorption. At low frequencies the fluctuations are absent. As the frequency is increased the fluctuations in concentration are pronounced. The fre-quency of fluctuations were found to increase with increase in frequency of oscillations during ab-sorption.

KEYWORDS

Single Compartment Models, Michaelis and Menten Kinetics, Reactions in Circle, Subcritical Damped Oscillations, Pharmacokinetics

Single Compartment Models, Michaelis and Menten Kinetics, Reactions in Circle, Subcritical Damped Oscillations, Pharmacokinetics

Cite this paper

nullK. Sharma, "On Single Compartment Pharmacokinetic Model Systems that Obey Michaelis-Menten Kinetics and Systems that Obey Krebs Cycle Kinetics,"*Journal of Encapsulation and Adsorption Sciences*, Vol. 1 No. 3, 2011, pp. 43-50. doi: 10.4236/jeas.2011.13007.

nullK. Sharma, "On Single Compartment Pharmacokinetic Model Systems that Obey Michaelis-Menten Kinetics and Systems that Obey Krebs Cycle Kinetics,"

References

[1] K. R. Sharma, “Transport Phenomena in Biomedical Engineering: Artificial Organ Design and Development and Tissue Engineering,” McGraw Hill, New York, 2010.

[2] R. L. Fournier, “Basic Transport Phenomena in Biomedical Engineering,” Taylor & Francis, Philadelphia, 1999.

[3] S. M. Skinner, R. E. Clark, N. Baker, R. A. Shipley and Am. J. Physiolo, “Complete Solution of the Three Compartment Model in Steady State After Single Injection of Radioactive Tracer,” Institute of Physics in Association with the American, Vol. 196, 1959, pp. 238-244.

[4] F. Lundquist and H. Wolthers, “The Kinetics of Alcohol Elimination in Man,” Acta Pharmacol Toxicol, Vol. 14, 1958, pp. 265-289. doi:10.1111/j.1600-0773.1958.tb01164.x

[5] S. L. Beal, “Computation of the Explicit Solution to the Michaelis-Menten Equation,” Journal of Pharmacokinetics and Biopharmaceutics, Vol. 11, 1983, pp. 641-657. doi:10.1007/BF01059062

[6] K. R. Godfrey and W. R. Fitch, “On the Identification of Michaelis-Menten Elimination Parameters from a Single Dose-Response Curve,” Journal of Pharmacokinetics and Biopharmaceutics, Vol. 12, 1984, pp. 193-221. doi:10.1007/BF01059062

[7] S. Tang and Y. Xiao, “One-Compartment Model with Michaelis-Menten Elimination Kinetics and Therapeutic Window: An Analytical Approach,” Journal of Pharmacokinetics and Pharmacodynamics, Vol. 34, 2007, pp. 807-827. doi:10.1007/BF01059062

[8] O. Levenspiel, “Chemical Reaction Engineering,” John Wiley & Sons, New York, 1999.

[9] L. Michaelis and M. L. Menten, “Die Kinetik der Intertinwerkung,” Biochemische Zeitschrift, Vol. 49, 1913, pp. 333-369.

[10] H. S. Mickley, T. K. Sherwood and C. E. Reed, “Applied Mathematics in Chemical Engineering,” McGraw Hill Book Company, New York, 1957.

[11] Sir H. A. Krebs, “Citric Acid Cycle,” Nobel Prize Lecture, 1953. http://nobel.se

[12] K. R. Sharma, “Subcritical Damped Oscillatory Kinetics of Simple Reactions in Circle,” chemcon conferences, Bubanewar, India, December, 2003.

[13] Ans. Magna, “Solution to the Cubic Equation,” Renaissance Mathematics, pp. 1501-1576.

[14] A. Varma and M. Morbidelli, “Mathematical Methods in Chemical Engineering,” Oxford University Press, Oxford, UK, 1997.

[1] K. R. Sharma, “Transport Phenomena in Biomedical Engineering: Artificial Organ Design and Development and Tissue Engineering,” McGraw Hill, New York, 2010.

[2] R. L. Fournier, “Basic Transport Phenomena in Biomedical Engineering,” Taylor & Francis, Philadelphia, 1999.

[3] S. M. Skinner, R. E. Clark, N. Baker, R. A. Shipley and Am. J. Physiolo, “Complete Solution of the Three Compartment Model in Steady State After Single Injection of Radioactive Tracer,” Institute of Physics in Association with the American, Vol. 196, 1959, pp. 238-244.

[4] F. Lundquist and H. Wolthers, “The Kinetics of Alcohol Elimination in Man,” Acta Pharmacol Toxicol, Vol. 14, 1958, pp. 265-289. doi:10.1111/j.1600-0773.1958.tb01164.x

[5] S. L. Beal, “Computation of the Explicit Solution to the Michaelis-Menten Equation,” Journal of Pharmacokinetics and Biopharmaceutics, Vol. 11, 1983, pp. 641-657. doi:10.1007/BF01059062

[6] K. R. Godfrey and W. R. Fitch, “On the Identification of Michaelis-Menten Elimination Parameters from a Single Dose-Response Curve,” Journal of Pharmacokinetics and Biopharmaceutics, Vol. 12, 1984, pp. 193-221. doi:10.1007/BF01059062

[7] S. Tang and Y. Xiao, “One-Compartment Model with Michaelis-Menten Elimination Kinetics and Therapeutic Window: An Analytical Approach,” Journal of Pharmacokinetics and Pharmacodynamics, Vol. 34, 2007, pp. 807-827. doi:10.1007/BF01059062

[8] O. Levenspiel, “Chemical Reaction Engineering,” John Wiley & Sons, New York, 1999.

[9] L. Michaelis and M. L. Menten, “Die Kinetik der Intertinwerkung,” Biochemische Zeitschrift, Vol. 49, 1913, pp. 333-369.

[10] H. S. Mickley, T. K. Sherwood and C. E. Reed, “Applied Mathematics in Chemical Engineering,” McGraw Hill Book Company, New York, 1957.

[11] Sir H. A. Krebs, “Citric Acid Cycle,” Nobel Prize Lecture, 1953. http://nobel.se

[12] K. R. Sharma, “Subcritical Damped Oscillatory Kinetics of Simple Reactions in Circle,” chemcon conferences, Bubanewar, India, December, 2003.

[13] Ans. Magna, “Solution to the Cubic Equation,” Renaissance Mathematics, pp. 1501-1576.

[14] A. Varma and M. Morbidelli, “Mathematical Methods in Chemical Engineering,” Oxford University Press, Oxford, UK, 1997.