The one feature that the RxKinetics family of pk programs have in common is the ability
to edit the default drug models. The pk program is no longer a sealed box, we've cracked
it open to let you tune it yourself.
With this freedom comes responsibility. 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.
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
where
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.
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
where
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.
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
where
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.
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)
where
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
Introduction
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.
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
MAXIPIME
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:
Normal Half-Life
ESRD Half-Life
Normal dose
Peak level from normal dose
Length of infusion
The parameters that are created by the modeler are:
Kel equation
Vd (L/kg)
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.
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. 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.
Conclusion
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 renally excreted.
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.
