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Section 2 - Applied Pharmacokinetics

Creating a one-compartment model


The one compartment linear model

    The one compartment model linear assumes that the drug in question is evenly distributed throughout the body into a single compartment. This model is only appropriate for drugs which rapidly and readily distribute between the plasma and other body tissues. The one-compartment model has, by definition, only one volume term, Vd, which is usually expressed in liters.

    The second criteria for utilizing a one-compartment linear model is that the drug is eliminated from the body in a first-order fashion. That is, the rate of elimination is proportional to the amount of drug in the body. The proportionality constant which relates the rate and amount is the first order elimination rate constant, Kel, which has units of reciprocal time (usually 1/hours). In this model, the Kel is a constant, it does not change when different doses or multiple doses are given.

Volume of distribution (Vd)

    We evaluate serum levels because it is more convenient to measure the concentration of a drug in the body rather than the amount. The volume of distribution is the term that relates the amount of drug to its observed concentration. Vd has no true physiological significance, it is merely a mathematical constant:
      Equation 1. Concentration
      Cp0 = Dose / Vd
        Cp0 = the peak concentration

    If we know the dose and we can measure the serum level, then we can calculate the Vd by rearranging the above equation.

      Equation 2. Volume of distribution
      Vd = Dose / Cp0
        (Note: these equations have been simplified for this discussion, i.e.,
        the term describing elimination during infusion has been dropped).

    Population model
    If we calculate Vd for a group of patients, we can then derive an average Vd for our patient group. A population average Vd is usually expressed in Liters per kilogram (L/kg).

    Figure 1. Vd frequency distribution.
    Figure 1. Vd frequency distribution

    Elimination rate constant (Kel)

      The elimination rate constant is calculated from the serum level decay curve. By measuring two (or more) serum levels we can calculate the Kel by this equation:
        Equation 3. Elimination rate constant
        Kel = (ln Cp0 - ln Cpt) / t
          ln Cp0 = the natural log of the peak level
          ln Cpt = the natural log of the trough level
          t = time between levels

      Population model
      If we calculate Kel for a group of patients, we can then derive an average for our patient group. If the drug is largely excreted unchanged in the urine, it is customary in the literature to list average Half-Life for those patients with normal renal function, and average Half-Life for those with ESRD (End Stage Renal Disease). In order to extrapolate these two points to all patients, we assume a linear relationship between CrCl and half-life:

      Figure 2. CrCl vs Half-Life.
      Figure 2. CrCl vs Half-Life

      Because of this assumption, we can set up a simple proportion equation to derive the components of our Kel equation:

        Equation 4. Deriving Kel components
        Kel Normal = 0.693 / Half-life Normal
        Kel Nonrenal = 0.693 / Half-life ESRD
        Kel Renal = [Kel Normal - Kel Nonrenal] / 120
          120 is the CrCl at 100% renal function.

      The values from Equation 4 can then be plugged into the following equation to estimate Kel for any patient with a known CrCl:

        Equation 5. Population Kel equation
        Kel = Kel Nonrenal + (CrCl * Kel Renal)

    Prospective dosing

      Once we know the population parameters for a drug, we can plug them into the standard 1-compartment dose equations to calculate a dosage regimen for a patient. The population model is used for prospective dosing, prior to obtaining serum level data.
        Equation 6. Ideal Interval
        Interval = PeakPredict + Tinf + [(ln Peak - ln Trough) / Kel]
          PeakPredict = Peak prediction time (usually zero)
          Tinf = Length of the infusion (piggyback)
          ln Peak = the natural log of your target peak level
          ln Trough = the natural log of your target trough level
          Kel = KNonrenal + (KRenal X CrCl)
        Equation 7. Ideal Dose
        Dose = Kel x Vd x Peak x Tinf x (1 - e-Kel x tau / 1 - e-Kel x tinf)
          Kel = KNonrenal + (KRenal X CrCl)
          Vd = Vd (L/kg) X patient weight
          Peak = your target peak level (or extrapolated peak if peak prediction time > 0)
          Tinf = Length of the infusion (piggyback)
          tau = interval

    Creating a prospective model for Cefepime


      Now that you have a better understanding of how the 1-compartment population model works, let's work through creating a model for Cefepime.

      "Where can I find model parameters for a drug", is one of our most frequently asked questions. Unfortunately, this information is not easily found in one reference. Furthermore, what little pk data you may find is often either incomplete or impractical.

    Finding pk data in Bennet's tables

      Probably the single best reference is Drug Prescribing in Renal Failure : Dosing Guidelines for Adults by William M. Bennett, George R. Aronoff, Jeffrey S. Berns, et al.

      Here is their data on Cefepime:

      Drug Percent
      excreted unchanged
      Protein Binding
      Volume of
      Dose for Normal
      Renal Function

      % hrs % L/kg
      Cefepime 85 2.2 / 18 16 0.3 250-2000 mg q8h

    Finding pk data in the Package insert

      This may be surprising to some, but the FDA package insert of newer drugs usually has an excellent pharmacokinetics section. Here is that section from the package insert for Cefepime:
      Pharmacokinetics:   The average plasma concentrations of cefepime observed in healthy adult male volunteers (n=9) at various times following single 30-minute infusions (IV) of cefepime 500 mg, 1 g, and 2 g are summarized in Table 1. Elimination of cefepime is principally via renal excretion with an average (± SD) half-life of 2.0 (±0.3) hours and total body clearance of 120.0 (± 8.0) mL/min in healthy volunteers. Cefepime pharmacokinetics are linear over the range 250 mg to 2 g. There is no evidence of accumulation in healthy adult male volunteers (n=7) receiving clinically relevant doses for a period of 9 days.

      Absorption:   The average plasma concentrations of cefepime and its derived pharmacokinetic parameters after intravenous administration are portrayed in Table 1.

      TABLE 1
      Average Plasma Concentrations in µg/mL of Cefepime
      and Derived Pharmacokinetic Parameters (±SD),
      Intravenous Administration
      Parameter 500 mg IV 1 g IV 2 g IV
      0.5 hr 38.2 78.7 163.1
      1.0 hr 21.6 44.5 85.8
      2.0 hr 11.6 24.3 44.8
      4.0 hr 5.0 10.5 19.2
      8.0 hr 1.4 2.4 3.9
      12.0 hr 0.2 0.6 1.1
      C max , µg/mL 39.1 (3.5) 81.7 (5.1) 163.9 (25.3)
      AUC, hr·µg/mL 70.8 (6.7) 148.5 (15.1) 284.8 (30.6)
      Number of subjects (male) 9 9 9
      Distribution:   The average steady state volume of distribution of cefepime is 18.0 (± 2.0)L. The serum protein binding of cefepime is approximately 20% and is independent of its concentration in serum.

      Renal Insufficiency:   Cefepime pharmacokinetics have been investigated in patients with various degrees of renal insufficiency (n=30). The average half-life in patients requiring hemodialysis was 13.5 ( 2.7) hours and in patients requiring continuous peritoneal dialysis was 19.0 ( 2.0) hours. Cefepime total body clearance decreased proportionally with creatinine clearance in patients with abnormal renal function, which serves as the basis for dosage adjustment recommendations in this group of patients.

    Reconciling the literature with the FreeKin Modeler

      "The young man knows the rules, but the old man knows the exceptions"
      - - Oliver Wendell Holmes

      Although there are some discrepancies between the two sources, they are close.

      Bennett's tables Package insert
      Half-life (Normal/ESRD) 2.2 / 18 2 / 19
      Volume of distribution 0.3 L/kg 0.26 L/kg

      A huge problem occurs though when we try to plug these literature values into a model. What looks good on paper does not translate to a practical dose. If we were to use these numbers in our dose prediction equations, Equation 6 and Equation 7, we get results which are 2 or 3 times the recommended dose!

      Because of this difficulty in translating literature data into a practical dosing model, I created the FreeKin Modeler program.

      The basic parameters you will need from the literature are:

      1. Normal Half-Life
      2. ESRD Half-Life
      3. Normal dose
      4. Peak level from normal dose
      5. Length of infusion

      The parameters that are created by the modeler are:

      1. Kel equation
      2. Vd (L/kg)
      3. Target peak and trough

      Below is a screen shot of FreeKin. The program breaks down the process of creating a model into 3 steps: Kel, Vd, and target levels. After you have created your model, you can then test it with various creatinine clearances. The assumption made in testing is that you are dosing the average 70kg patient.

      freekin modeler

      Finishing up our Cefepime example using the FreeKin Modeler, our inputs are:

      1. Normal Half-Life = 2.2 hrs
      2. ESRD Half-Life = 18 hrs
      3. Normal dose = 1000 mg
      4. Peak level from normal dose = 78.7 mcg/ml
      5. Peak time = 0 min
      6. Length of infusion = 30 min

      The resulting parameters for Cefepime are:

      1. Kel equation = 0.0385 + (0.0026 X CrCl)
      2. Vd (L/kg) = 0.17 L/kg
      3. Target peak = 85 mcg/ml
      4. Target trough = 6 mcg/ml

      The dosage recommendations from this model compare favorably with the published guidelines for dosage adjustment in renal failure. Notice that the parameter which differed most from that cited in the literature is the Volume of distribution. You can double check the results of FreeKin for yourself using Equation 2 for a rough estimate of Vd:

        Vd = Dose / Cp0
        Vd = 1000 mg / 78.7 mcg/ml
        Vd = 12.7 L
        Vd = 0.18 L/kg for our average 70kg patient

      The package insert states that the average steady-state Vd is 18 liters, and multiplying out Bennett's 0.3 L/kg gives us 21 liters for the average 70 kg patient. If you plug either of these Vd's into Equation 7, you will get an ideal dose which is 2 or 3 times the recommended dose. Of course, this makes absolutely no sense. And this leads to one of the problems you will run into when creating a practical model. Some misguided colleague will tell you just how wrong your Vd value is by quoting some figure from the literature. Just remember this one important, irrefutable fact:

        The Vd in these equations has no true physiological significance, it is merely a mathematical constant.


      One-compartment modeling isn't rocket science. The proven methods described here are simple and reliable for predicting dosage requirements of any drug which:
      1. Can be described by a one-compartment linear model.
      2. Is largely renally excreted.
      3. Is administered by intravenous infusion (piggyback).

      The RxKinetics family of pk programs are tools. And just like any other tool, you need to understand how they work before you use them. It is my hope that this tutorial has given you some insight and practical information about our pk tools that will allow you to provide better care for your patients.

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    Section 2 - Applied Pharmacokinetics

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