 ### The one compartment model

#### I. Introduction

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
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 distribution1
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).

#### 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
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. 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
where
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

Equation 6. Ideal Interval1
Interval = PeakPredict + Tinf + [(ln Peak - ln Trough) / Kel]
where
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)
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

#### I. 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?"
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.

#### II. 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:

Table 1. Cefepime PK data from Bennett, et al.
Drug Percent
excreted unchanged
Half-life
(Normal/ESRD)
Plasma
Protein Binding
Volume of
Distribution
Dose for Normal
Renal Function

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

#### III. 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.

 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.

#### 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.

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

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:

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 70 kg patient.

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 (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!

#### I. Introduction

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:

Kel equation
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
where
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.

"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.

#### 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:

1. Can be described by a one-compartment linear model.
2. Is largely excreted by glomerular filtration.
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.