
The one compartment modelI. IntroductionThe one feature that the RxKinetics family of pk programs have in common is the ability to edit the default drug models.
"With great power there must also come — great responsibility!" The goal of this tutorial is to help you better understand onecompartment modeling and population kinetics, so that you can create your own drug models and exploit the flexibility that these tools give you. 

II. The one compartment linear modelThe one compartment linear model 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 onecompartment model has, by definition, only one volume term, Vd, which is usually expressed in liters. The second criteria for utilizing a onecompartment linear model is that the drug is eliminated from the body in a firstorder 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. III. 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 a mathematical constant:
Cp_{0} = Dose / Vd where
If we know the dose and we can measure the serum level, then we can calculate the Vd by rearranging the above equation.
Vd = Dose / Cp_{0}
the term describing elimination during infusion has been dropped).
Population model IV. 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:
Kel = (ln Cp_{0}  ln Cp_{t}) / t where
ln Cp_{t} = the natural log of the trough level t = time between levels
Population model Because of this assumption, we can set up a simple proportion equation to derive the components of our Kel equation:
Kel _{Normal} = 0.693 / Halflife _{Normal} Kel _{Nonrenal} = 0.693 / Halflife _{ESRD} Kel _{Renal} = [Kel _{Normal}  Kel _{Nonrenal}] / 120 where
The values from Equation 4 can then be plugged into the equation for a straight line (slopeintercept form):^{3}
Kel = (CrCl * Kel _{Renal}) + Kel _{Nonrenal } The calculation of Kel in this manner is properly referred to as The Wagner Method.^{4} V. Prospective dosingOnce we know the population parameters for a drug, we can plug them into the standard 1compartment dose equations to calculate a dosage regimen for a patient.^{1}
Interval = PeakPredict + Tinf + [(ln Peak  ln Trough) / Kel] where
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 = K_{Nonrenal} + (K_{Renal} X CrCl)
Dose = Kel x Vd x Peak x Tinf x (1  e^{Kel x tau} / 1  e^{Kel x tinf}) where
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 CefepimeI. IntroductionNow that you have a better understanding of how the 1compartment population model works, let's work through creating a model for Cefepime.
"Where can I find model parameters for a drug?" II. Finding pk data in Bennet's tablesProbably 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:
III. Finding pk data in the Package insertThis 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:
IV. Reconciling the literature with the FreeKin Modeler
"The young man knows the rules, but the old man knows the exceptions" Although there are some discrepancies between the two sources, they are close.
A significant problem occurs though when we try to plug these literature values into a model. What looks good on paper does not always 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. The FreeKin Modeler program was developed to help translate the literature data into a practical pk model useful for designing dosing regimens. The basic parameters you will need from the literature are:
The parameters that are created by the modeler are:
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 70 kg patient. Finishing up our Cefepime example using the FreeKin Modeler, our inputs are:
The resulting parameters for Cefepime are:
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 = 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 steadystate 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 (I speak from experience!). A wellmeaning but misguided colleague will tell you just how wrong your Vd value is by quoting some figure from the literature. Just remember this one important fact:
Again, here is the download link for the FreeKin Modeler. It's Free, it's for Kinetics, it's FreeKin awesome! Analyzing your patient populationI. IntroductionThe methods described so far in this tutorial have focused on retrieving data from the published literature. The potential problem with any published data is, it is likely to have been derived from patients who are dissimilar to your own. This is especially true of the package insert pk data which is usually derived from studies of healthly adult males. It is highly unlikely that the majority of your patients are healthy adult males. II. Population analysis with the APK and Kinetics programsOne of the most powerful tools included with the APK and Kinetics programs is population analysis. With this tool you can derive a model which best fits your patient population. Each time you print a consult, the programs save your serum level analysis results. This data is a virtual gold mine of information about your patient population. Below is a screen shot of the population analysis dialog in Kinetics (APK has a similar screen). First select a model to analyze, then select a date range. The other criteria on this screen are optional. You may use these optional criteria to further narrow down the population that you wish to analyze. III. Statistical analysis
Volume of distribution
Kel equation
y = a + bx where
the variable y = Kel the intercept a = K_{Nonrenal} the slope b = K_{Renal} The Kel regression equation is calculated with the linear least squares method. IV. Save your data
"Sometimes I do smart things. Sometimes I do dumb things. Most of the time I don't do anything." We usually only analyze those levels which are "off target". But, if you exclude those patients who do hit target, your population data will be skewed to the outliers. For population analysis to work properly, you must include every single patient. Remember, the trigger to save the data is printing the consult. Always run the numbers, and always print a consult. Please take advantage of this powerful tool to derive pk models which best fit your patient population. ConclusionOnecompartment modeling isn't rocket science. The proven methods described here are simple and reliable for predicting dosage requirements of any drug which:
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. References and recommended reading

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