Medicinal Chemistry Applet

Cp vs time - iv infusion

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Introduction

Maintaining a steady-state plasma concentration (C_ss) is vital for a proper drug regimen.  The most direct method to achieving a desired C_ss is through iv infusion of a drug solution.  In this situation, the change in plasma concentration (dC_p/dt) is affected by both the rate of infusion, k_in, and the rate of elimination, k_el (Equation 1).  [Note: Some discussions of this topic use different forms of k_in.  In this treatment, the units on k_in are mass/time.]  Integration of Equation 1 gives an expression that allows determination of C_p at any time during the infusion (eq. 2).

     (1)

     (2)

By Equation 2, as time approaches infinity, the exponential term approaches 0.  C_p approaches a constant value called the steady-state concentration, C_ss (eq. 3).  The same relationship may be derived from Equation 1 by setting dC_p/dt to 0.  At this point C_p = C_ss.

     (3)

In a strict sense, C_p never reaches C_ss.  In practice, infusion of a drug for a period of four half-lives will achieve between 90 and 95% of the theoretical C_ss.  Since building C_p to near C_ss can take considerable time, an iv bolus may be used as a loading dose to "jump start" C_p.  Regardless of the status of the infusion or loading dose, once the infusion is halted, C_p will immediately begin to fall in standard, first-order fashion as determined by k_el (Equation 4).

     (4)

Applet

This applet plots data for up to two different drugs.  For each drug, k_in, k_el, V_d, the length of time of the infusion, and the length of time to monitor C_p must be provided.  Data points for C_p vs time are plotted for the specified time period, and all varibles with time units should have the same unit of time measure.  If the units of k_in are mg/h and V_d is in liters, then the units of C_p will be µg/mL. 

Problem information

Vancomycin is an antibiotic that is only used to fight infections that show resistance to standard antibiotics.  These resistant bacteria are often called methicillin-resistant Staphylococcus aureus (MRSA) or "superbugs."  While the term MSRA did initially refer to methicillin-resistant strains, it now is generally used to describe any bacteria that show broad antibiotic resistance.  These bacteria are often found and transmitted in hospitals.  Vancomycin is not orally available and is frequently administered by intravenous infusion.

The pharmacokinetic parameters of vancomycin are shown below.

• t_1/2 = 5.6 h
• k_el = 0.12 h^-1
• V_d = 0.39 L/kg (27 L for a 70 kg patient)

Problems

1. Goodman and Gilman's states that the peak concentration of vancomycin after a one hour infusion at a rate of 7.5 mg/kg/h or 1000 mg/h is 18.5 µg/mL.  Calculate the amount of drug infused at a rate of 7.5 mg/kg/h for a 70 kg patient in 1.0 h.  Use the applet to determine the which of the two infusion rates, if either, will give a peak concentration of 18.5 µg/mL after 1.0 h.
2. Vancomycin can show ototoxicity, affecting hearing or inner ear balance, at concentrations of around 40 µg/mL.  Does either of the one-hour infusions at the rates in the first question lead to toxic levels of vancomycin?
3. What rate of infusion of vancomycin would be required to achieve a C_ss of 20 µg/mL?  (Try different infusion rates in the applet to see the rate necessary to reach the desired concentration.)
4. Although a steady-state concentration of vancomycin can eventually be achieved, how long does it take to reach 95% of the desired concentration of 20 µg/mL in question 3?  The problem of taking too long to reach a desired C_ss can be addressed through a "loading dose" in the form of an intravenous bolus.

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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