The 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!"
- - Stan Lee
The goal of this tutorial is to help you better understand one-compartment 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 model
The 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 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.
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:
Equation 1. Concentration1
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 distribution1
Vd = Dose / Cp0
(Note: these equations have been simplified for this discussion, i.e.,
the term describing elimination during infusion has been dropped).
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
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:
Equation 3. Elimination rate constant 1
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
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. This assumption is true for most drugs which are excreted by glomerular filtration.
Because of this assumption, we can set up a simple proportion equation to derive the
components of our Kel equation:
Equation 4. Deriving Kel components2
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 equation for a straight line (slope-intercept form):3
y = mx + b
The equation thus becomes:
Equation 5. Population Kel equation4
Kel = (CrCl * Kel Renal) + Kel Nonrenal
The calculation of Kel in this manner is properly referred to as The Wagner Method.4
V. 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.1
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 Dose1
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?"
This 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, impractical or inconsistent.
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:
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 2. Average Plasma Concentrations in µg/mL of Cefepime
and Derived Pharmacokinetic Parameters (±SD),
500 mg IV
1 g IV
2 g IV
C max , µg/mL
Number of subjects (male)
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.
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.
IV. 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.
Table 3. Bennett, et al vs Package insert.
2.2 / 18
2 / 19
Volume of distribution
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:
Peak level from normal dose
Length of infusion
The parameters that are created by the modeler are:
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 70 kg patient.
Finishing up our Cefepime example using the FreeKin Modeler, our inputs are:
Normal Half-Life = 2.2 hrs
ESRD Half-Life = 18 hrs
Normal dose = 1000 mg
Peak level from normal dose = 78.7 mcg/ml
Peak time = 0 min
Length of infusion = 30 min
The resulting parameters for Cefepime are:
Kel equation = 0.0385 + (0.0026 X CrCl)
Vd (L/kg) = 0.17 L/kg
Target peak = 85 mcg/ml
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 (I speak
from experience!). A well-meaning 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:
The Vd in these equations has no true physiological significance, it is a mathematical constant.
Again, here is the download link for the FreeKin Modeler. It's Free, it's for Kinetics, it's FreeKin awesome!
The 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 programs
One 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
The population analysis routine calculates these basic statistics describing the Vd of your patient population:
If we examine the relationship of Kel vs CrCl for a group of patients, we can
derive a regression equation to explain this relationship.
Equation 8. Linear regression
y = a + bx
the variable x = CrCl
the variable y = Kel
the intercept a = KNonrenal
the slope b = KRenal
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."
- - Hintz
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.
One-compartment modeling isn't rocket science. The proven methods described here are
simple and reliable for predicting dosage requirements of any drug which:
Can be described by a one-compartment linear model.
Is largely excreted by glomerular filtration.
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.