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Applied Biopharmaceutics & Pharmacokinetics, 7e >
Chapter 4
Chapter 4: One­Compartment Open Model: Intravenous
Bolus Administration David S.H. Lee
CHAPTER OBJECTIVES
Describe a one­compartment model, IV bolus injection.
Provide the pharmacokinetic terms that describe a one­compartment model, IV bolus injection,
and the underlying assumptions.
Explain how drugs follow one­compartment kinetics using drug examples that follow one­
compartment kinetics.
Calculate pharmacokinetic parameters from drug concentration–time data using a one­
compartment model.
Simulate one­compartment plasma drug level graphically using the one­compartment model
equation.
Calculate the IV bolus dose of a drug using the one­compartment model equation.
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Relate the relevance of the magnitude of the volume of distribution and clearance of various
drugs to underlying processes in the body.
Derive model parameters from slope and intercept of the appropriate graphs.
While the oral route of drug administration is the most convenient, intravenous (IV) administration is the
most desirable for critical care when reaching desirable drug concentrations quickly is needed.
Examples of when IV administration is desirable include antibiotic administration during septic infections
or administration of antiarrhythmic drugs during a myocardial infarction. Because pharmacokinetics is
the science of the kinetics of drug absorption, distribution, and elimination, IV administration is
desirable in understanding these processes since it simplifies drug absorption, essentially making it
complete and instantaneous. This leaves only the processes of drug distribution and elimination left to
study. This chapter will introduce the concepts of drug distribution and elimination in the simplest
model, the one­compartment open model.
The one­compartment open model assumes that the body can be described as a single, uniform
compartment (ie, one compartment), and that drugs can enter and leave the body (ie, open model).
The simplest drug administration is when the entire drug is given in a rapid IV injection, also known as
an IV bolus. Thus, the one­compartment open model with IV bolus administration is the simplest
pharmacokinetic model. It assumes that the drug is administered instantly into the body, it is
instantaneously and rapidly distributed throughout the body, and drug elimination occurs immediately
upon entering the body. This model is a simplistic representation of the processes in the body that
determine drug disposition, but nonetheless, it can be useful to describe and predict drug disposition.
In reality, when a drug is administered intravenously, the drug travels through the bloodstream and
distributes throughout the bloodstream in the body. While this process is not truly instantaneous, it is
relatively rapid enough that we can make this assumption for most drugs. Through the bloodstream,
the drug is distributed to the various tissue organs in the body. The rate and extent of distribution to the
tissue organs depends on several processes and properties. Tissues in the body are presented the
drug at various rates, depending on the blood flow to that organ, and the drug may have different
abilities to cross from the vasculature to the organ depending on the molecular weight of the drug.
Tissues also have different affinity for the drug, depending on lipophilicity and drug binding. Finally,
large organs may have a large capacity for drugs to distribute to.
While drug distribution is complex, if these processes are rapid enough, we can simplify our
conceptualization as if the drug uniformly distributes into a single (one) compartment of fluid. The
volume of this single compartment is termed the apparent volume of distribution, VD. The apparent
volume of distribution is not an actual volume in the body, but is a theoretical volume that the drug
uniformly distributes to immediately after being injected into the body. This uniform and instantaneous
distribution is termed a well­stirred one­compartment model. The apparent volume of distribution is a
proportion between the dose and the concentration of the drug in plasma, , at that time immediately
after being injected.
Most drugs are eliminated from the body by liver metabolism and/or renal excretion. All of the
processes of drug elimination can be described by the elimination rate constant, k. The elimination rate
constant is the proportion between the rate of drug elimination and the amount of drug in the body.
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Because the amount of drug in the body changes over time, the rate of drug elimination changes, but
the elimination rate constant remains constant for first­order elimination. This makes it convenient to
summarize drug elimination from the body independent of time or the amount of drug in the body.
However, because it’s difficult to measure the amount of drug in the body, DB, pharmacokineticists and
pharmacists also prefer to convert drug amounts to drug concentrations in the body using the apparent
volume of distribution. Thus, the elimination rate constant also describes the proportion between the
rate of change of drug concentration and drug concentration in the compartment.
The one­compartment open model with IV bolus dosing describes the distribution and elimination after
an IV bolus administration and is shown in Fig. 4­1. The fluid that the drug is directly injected into is the
blood, and generally, drug concentrations are measured in plasma since it is accessible. Therefore,
this model predicts the concentrations in the plasma, but does not predict the concentrations in tissues.
However, using this model, which assumes distribution to tissues is rapid, we can assume the declines
in drug concentration in the plasma and tissues will be proportional. For these reasons, the one­
compartment open model is useful for predicting concentrations in the plasma, and declines in plasma
concentrations will be proportional to declines in tissue concentrations.
FIGURE 4­1
Pharmacokinetic model for a drug administered by rapid intravenous injection. DB = drug in body; VD =
apparent volume of distribution; k = elimination rate constant.
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ELIMINATION RATE CONSTANT
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APPARENT VOLUME OF DISTRIBUTION
CLEARANCE
CLINICAL APPLICATION
CALCULATION OF κ FROM URINARY EXCRETION
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DATA
PRACTICE PROBLEM
PRACTICE PROBLEM
CLINICAL APPLICATION
CHAPTER SUMMARY
LEARNING QUESTIONS
1. A 70­kg volunteer is given an intravenous dose of an antibiotic, and serum drug concentrations were
determined at 2 hours and 5 hours after administration. The drug concentrations were 1.2 and 0.3
μg/mL, respectively. What is the biologic half­life for this drug, assuming first­order elimination kinetics?
2. A 50­kg woman was given a single IV dose of an antibacterial drug at a dose level of 6 mg/kg. Blood
samples were taken at various time intervals. The concentration of the drug (Cp) was determined in the
plasma fraction of each blood sample and the following data were obtained:
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t (hours)
Cp (µg/mL)
0.25
8.21
0.50
7.87
1.00
7.23
3.00
5.15
6.00
3.09
12.0
1.11
18.0
0.40
a. What are the values for VD, k, and t1/2 for this drug?
b. This antibacterial agent is not effective at a plasma concentration of less than 2 μg/mL. What is
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the duration of activity for this drug?
c. How long would it take for 99.9% of this drug to be eliminated?
d. If the dose of the antibiotic was doubled exactly, what would be the increase in duration of
activity?
3. A new drug was given in a single intravenous dose of 200 mg to an 80­kg adult male patient. After 6
hours, the plasma drug concentration of drug was 1.5 mg/100 mL of plasma. Assuming that the
apparent VD is 10% of body weight, compute the total amount of drug in the body fluids after 6 hours.
What is the half­life of this drug?
4. A new antibiotic drug was given in a single intravenous bolus of 4 mg/kg to 5 healthy male adults
ranging in age from 23 to 38 years (average weight 75 kg). The pharmacokinetics of the plasma drug
concentration–time curve for this drug fits a one­compartment model. The equation of the curve that
best fits the data is
Determine the following (assume units of μg/mL for Cp and hours for t):
a. What is the t1/2?
b. What is the VD?
c. What is the plasma level of the drug after 4 hours?
d. How much drug is left in the body after 4 hours?
e. Predict what body water compartment this drug might occupy and explain why you made this
prediction.
f. Assuming the drug is no longer effective when levels decline to less than 2 μg/mL, when should
you administer the next dose?
5. Define the term apparent volume of distribution. What criteria are necessary for the measurement of
the apparent volume of distribution to be useful in pharmacokinetic calculations?
6. A drug has an elimination t1/2 of 6 hours and follows first­order kinetics. If a single 200­mg dose is
given to an adult male patient (68 kg) by IV bolus injection, what percent of the dose is lost in 24
hours?
7. A rather intoxicated young man (75 kg, age 21 years) was admitted to a rehabilitation center. His
blood alcohol content was found to be 210 mg%. Assuming the average elimination rate of alcohol is
10 mL of ethanol per hour, how long would it take for his blood alcohol concentration to decline to less
than the legal blood alcohol concentration of 100 mg%? (Hint: Alcohol is eliminated by zero­order
kinetics.) The specific gravity of alcohol is 0.8. The apparent volume of distribution for alcohol is 60% of
body weight.
8. A single IV bolus injection containing 500 mg of cefamandole nafate (Mandol, Lilly) is given to an
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adult female patient (63 years, 55 kg) for a septicemic infection. The apparent volume of distribution is
0.1 L/kg and the elimination half­life is 0.75 hour. Assuming the drug is eliminated by first­order kinetics
and may be described by a one­compartment model, calculate the following:
a. The C0p
b. The amount of drug in the body 4 hours after the dose is given
c. The time for the drug to decline to 0.5 μg/mL, the minimum inhibitory concentration for
streptococci
9. If the amount of drug in the body declines from 100% of the dose (IV bolus injection) to 25% of the
dose in 8 hours, what is the elimination half­life for this drug? (Assume first­order kinetics.)
10. A drug has an elimination half­life of 8 hours and follows first­order elimination kinetics. If a single
600­mg dose is given to an adult female patient (62 kg) by rapid IV injection, what percent of the dose
is eliminated (lost) in 24 hours assuming the apparent VD is 400 mL/kg? What is the expected plasma
drug concentration (Cp) at 24 hours postdose?
11. For drugs that follow the kinetics of a one­compartment open model, must the tissues and plasma
have the same drug concentration? Why?
12. An adult male patient (age 35 years, weight 72 kg) with a urinary tract infection was given a single
intravenous bolus of an antibiotic (dose = 300 mg). The patient was instructed to empty his bladder
prior to being medicated. After dose administration, the patient saved his urine specimens for drug
analysis. The urine specimens were analyzed for both drug content and sterility (lack of bacteriuria).
The drug assays gave the following results:
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t (hours)
Amount of Drug in Urine (mg)
0
0
4
100
8
26
a. Assuming first­order elimination, calculate the elimination half­life for the antibiotic in this patient.
b. What are the practical problems in obtaining valid urinary drug excretion data for the
determination of the drug elimination half­life?
ANSWERS
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Frequently Asked Questions
What is the difference between a rate and a rate constant?
A rate represents the change in amount or concentration of drug in the body per time unit. For
example, a rate equal to –5 mg/h means the amount of drug is decreasing at 5 mg/h. A positive
or negative sign indicates that the rate is increasing or decreasing, respectively. Rates may be
zero order, first order, or higher orders. For a first­order rate, the rate of change of drug in the
body is determined by the product of the elimination rate constant, k, and the amount of drug
remaining in the body, that is, rate = –kDB, where k represents “the fraction” of the amount of
drug in the body that is eliminated per hour. If k = 0.1 h–1 and DB = 10 mg, then the rate = 0.1 h–
1 × 10 mg = 1 mg/h. The rate constant in this example shows that one­tenth of the drug is
eliminated per hour, whatever amount of drug is present in the body. For a first­order rate, the
rate states the absolute amount eliminated per unit time (which changes with the amount of
drug in the body), whereas the first­order rate constant, k, gives a constant fraction of drug that
is eliminated per unit time (which does not change with the amount of drug in the body).
Why does k always have the unit 1/time (eg, h–1), regardless of what concentration unit is plotted?
The first­order rate constant k has no concentration or mass units. In the calculation of the
slope, k, the unit for mass or concentration is canceled when taking the log of the number:
If a drug is distributed in the one­compartment model, does it mean that there is no drug in the tissue?
The one­compartment model uses a single homogeneous compartment to represent the fluid
and the vascular tissues. This model ignores the heterogeneity of the tissues in the body, so
there is no merit in predicting precise tissue drug levels. However, the model provides useful
insight into the mass balance of drug distribution in and out of the plasma fluid in the body. If VD
is larger than the physiologic vascular volume, the conclusion is that there is some drug outside
the vascular pool, that is, in the tissues. If VD is small, then there is little extravascular tissue
drug storage, except perhaps in the lung, liver, kidney, and heart. With some knowledge about
the lipophilicity of the drug and an understanding of blood flow and perfusion within the body, a
postulation may be made as to which organs are involved in storing the extravascular drug. The
concentration of a biopsy sample may support or refute the postulation.
How is clearance related to the volume of distribution and k?
Clearance is the volume of plasma fluid that is cleared of drug per unit time. Clearance may also
be derived for the physiologic model as the fraction of drug that is eliminated by an organ as
blood flows through it. The former definition is equivalent to Cl = kVD and is readily adapted to
dosing since VD is the volume of distribution. If the drug is eliminated solely by metabolism in the
liver, then ClH = Cl. ClH is usually estimated by the difference between Cl and ClR. ClH is directly
estimated by the product of the hepatic blood flow and the extraction ratio.
If we use a physiologic model, are we dealing with actual volumes of blood and tissues? Why do
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volumes of distribution emerge for drugs that often are greater than the real physical volume?
Since mass balance (ie, relating dose to plasma drug concentration) is based on volume of
distribution rather than blood volume, the compartment model is used in determining dose.
Generally, the total blood concentrations of most drugs are not known, since only the plasma or
serum concentration is assayed. Some drugs have an RBC/plasma drug ratio much greater
than 1, making the application of the physiologic model difficult without knowing the apparent
volume of distribution.
Learning Questions
1. The Cp decreased from 1.2 to 0.3 μg/mL in 3 hours.
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t (hours)
Cp (µg/mL)
2
1.2
5
0.3
These data may also be plotted on a semilog graph and t1/2 obtained from the graph.
2. Dose (IV bolus) = 6 mg/kg × 50 kg = 300 mg
a. (1) Plot the data on semilog graph paper and use two points from the line of best fit.
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t (hours)
Cp (µg/mL)
2
6
6
3
(2) t1/2 (from graph) = 4 hours
b. Alternatively, time t may be found from a graph of Cp versus t.
c. Time required for 99.9% of the drug to be eliminated:
(1) Approximately 10 t1/2
(2) With 0.1% of drug remaining,
d. If the dose is doubled, then will also double. However, the elimination half­life or first­
order rate constant will remain the same. Therefore,
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Notice that doubling the dose does not double the duration of activity.
3. At 6 hours:
4. Cp = 78e–0.46t (the equation is in the form )
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b. c. (1) (2) d. At 4 hours:
e. VD = 3846 mL
The apparent VD approximates the plasma volume.
f. Cp = 2 μg/mL
Find t.
Alternate Method
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6. For first­order elimination kinetics, one­half of the initial quantity is lost each t1/2. The following
table may be developed:
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Time
Number of
Amount of Drug in Body
Percent of Drug in
Percent of Drug
(hours)
t 1/2
(mg)
Body
Lost
0
0
200
100
0
6
1
100
50
50
12
2
50
25
75
12
2
50
25
75
18
3
25
12.5
87.5
24
4
12.5
6.25
93.75
Method 1
From the above table the percent of drug remaining in the body after each t1/2 is equal to 100%
times (1/2)n, where n is the number of half­lives, as shown below:
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Number of t 1/2
Percent of Drug in Body
0
100
1
50
100 × 1/2
2
25
100 × 1/2 × 1/2
3
12.5
100 × 1/2 × 1/2 × 1/2
N
Percent of drug remaining Percent of Drug Remaining in Body after n t 1/2
100 × (1/2) n
, where n = number of t1/2
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Percent of drug lost At 24 hours, n = 4, since t1/2 = 6 hours.
Percent of drug lost Method 2
The equation for a first­order elimination after IV bolus injection is
where
DB = amount of drug remaining in the body
D0 = dose = 200 mg
k = elimination rate constant
7. The zero­order rate constant for alcohol is 10 mL/h. Since the specific gravity for alcohol is
0.8,
Therefore, the zero­order rate constant, k0, is 8 g/h.
Drug in body at t = 0:
Drug in body at time t:
For a zero­order reaction:
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8.
a. b. c. 9. 10. 11. The total drug concentration in the plasma is not usually equal to the total drug
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concentration in the tissues. A one­compartment model implies that the drug is rapidly
equilibrated in the body (in plasma and tissues). At equilibrium, the drug concentration in the
tissues may differ from the drug concentration in the body because of drug protein binding,
partitioning of drug into fat, differences in pH in different regions of the body causing a different
degree of ionization for a weakly dissociated electrolyte drug, an active tissue uptake process,
etc.
12. Set up the following table:
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Time (hours)
Du (mg)
dDu/t
mg/h
t*
0
0
4
100
100/4
25
2
8
26
26/4
6.5
6
The elimination half­life may be obtained graphically after plotting mg/h versus t*. The t1/2
obtained graphically is approximately 2 hours.
REFERENCE
BIBLIOGRAPHY
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