Medicinal Chemistry Applet

ln Cp vs time - iv bolus

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Introduction

Plotting the natural logarithm of the plasma concentration (ln Cp) against time for a drug administered by an iv bolus provides a line (Equation 1).  In this equation, ln Cpo refers to the natural logarithm of the plasma concentration at t = 0 and corresponds to the y-intercept.  The elimination of the drug is a first-order process.  The slope of the resulting line is equivalent to the negative of the elimination rate constant (kel).  This type of plot is the most common method for determining kel for a drug.  The kel can be easily converted to half-life (t½) by the relationship t½ = 0.693/kel.  Typically, the rate of drug elimination is reported as t½ instead of kel.  Also, the y-intercept can be used to determine ln Cp at t = 0.  Knowledge of Cpo can be used with the dosage (D) to estimate the distribution volume of a drug (Vd), commonly listed on a per kg patient weight basis (L/kg) (Equation 2).

     (1)

     (2)

For a very closely related discussion, see the applet on Cp vs time plots.

Applet

This applet plots data for up to three different drugs with arbitrary initial concentrations and kel values.  Data points are generated and plotted from t = 0 through five half-lives of the slowest eliminating drug.  The y-axis is unitless as a logarithmic function, but common units for Cpo would be µg/mL.  The x-axis would typically be hours - the same units as t½ and the inverse of the units of kel.

Drug
(Color)		 ln Cpo
 		 kel
 
Drug 1 (red)
Drug 2 (blue)
Drug 3 (green)
calculation may be slow

Problem information - also see the Cp vs time applet

Advances within a class of drugs, such as cephalosporins, provide nice examples for this applet.  Since their discovery in the late 1940s, cephalosporins have undergone a series of improvements.  Examples from three generations of cephalosporins are shown below.  The pharmacokinetic data for each drug were taken from Goodman and Gilman's The Pharmacological Basis of Therapeutics, a standard reference on drugs and their activity.  The kel values were derived from reported half-lifes.  ln Cpo values were determined based on a single-compartment model with the formula Cpo = Do/Vd.  The true ln Cpo values are likely considerably higher than shown in the table since distribution from the plasma is not instantaneous.  The Vd values are adjusted to a 70 kg patient.

cephalosporin t½
(h)		 Vd
(L)		 dose
(g)		 Cpo
(µg/mL)		 ln Cpo
(unitless)		 kel
(h-1)
cephalothin (1) 0.6 18 2 110 4.7 1.2
cefotetan (2) 3.6 10 2 200 5.3 0.19
ceftriaxone (3) 7.3 14 2 140 4.9 0.095


Problems

The treatment in these questions is virtually identical to that of the Cp vs time applet.

  1. Enter the cephalosporin data into the applet fields and graph all three compounds.  Be sure to use the bolded data from the table, and check that the rate constant and concentration data are placed in the proper text boxes.

  2. If all three cephalosporins are active at concentrations greater than 25 µg/mL (ln 25 = 3.22), how long will each drug be effective after a single iv injection?

  3. Having a long half-life is generally considered to be advantageous for a drug.  For what types of drugs would it be particularly beneficial to have a long half-life?

  4. A very long half-life is not necessarily good.  What type of drug activity would be an extended half-life be unfavorable?

Reference

Goodman and Gilman's The Pharmacological Basis of Therapeutics, 10th ed.; Hardman, J. G., Limbird, L. E., Eds.; McGraw-Hill: New York, 2001.

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