Lecture notes on pharmacokinetics

2. • QUANTITATIVE ASPECTS OF PHARMACOKINETICS
• Pharmacokinetics is the branch of pharmacology concerned with mathematical
description of the time course of plasma drug concentrations measured after
administration of a dose.
• Specifically, pharmacokinetics is the use of mathematical modeling to describe
how a drug behaves in the body during absorption, distribution, metabolism,
and excretion (together known as ADME)
• Pharmacokinetic parameters are calculated from plasma drug concentration-
versus-time data after a dose of the drug is administered at least via the desired
route and ideally also after IV administration (100% bioavailability).
• The mathematical models developed should enable predictions regarding drug
movement in and elimination from the body, so that optimal dosing regimens
can be designed, and when necessary, drug withdrawal times (for competition,
food safety) can be estimated.

3. • Most pharmacokinetic studies are conducted in healthy animals, yet dosing regimens
should be individualized to adjust for differences in physiology (age, sex, species, and
breed), pharmacology (drug interactions), and pathology (renal or hepatic disease).
• In most instances, tissue drug concentrations parallel plasma drug concentrations,
which is why plasma data can usually be used as a substitute for tissue
concentrations.
• The most relevant pharmacokinetic parameters that describe drug movement and
provide the basis for dosing regimens are the apparent volume of distribution (Vd)
and the plasma clearance (Cl), both of which determine the elimination rate constant
(kel) and elimination half-life (t½).
• Additional parameters may include the distribution rate constant and distribution
half-life and, if the drug is also administered PO, the absorption rate constant and
absorption half-life.

4. • With each drug movement, usually a constant fraction or percentage (rather
than an amount) moves per unit time (ie, first-order kinetics). When a constant
amount moves per unit of time it is called zero order kinetics.
• Assuming linear pharmacokinetics (direct proportionality between dose and
exposure), plasma drug concentrations plotted on a semilogarithmic scale (ie, a
logarithmic scale on the y-axis and a linear scale on the x-axis) can be fit by a
straight line (suggesting the data are described by a one-compartment
pharmacokinetic model) or a biphasic curve (the data are described by a two-
compartment model).

5. • In a two-compartment model, the first portion of the concentration-versus-
time curve (distribution phase), with a steeper slope, represents distribution
and elimination.
• The terminal portion of the curve (elimination phase), with a flatter slope,
represents primarily elimination (ie, distribution of the drug into and out of the
tissues is at equilibrium, and distribution into the tissues no longer contributes
appreciably to the decline in plasma concentrations).
• Macro rate constants can be determined for the distribution and elimination
phases through extrapolation back to the y-axis in a process called curve
stripping (See plot of two-compartment model). The y-intercept of the line that
describes the distribution phase is designated A, and the y-intercept of the line
that describes the elimination phase is designated B.
•

6. • The absolute value of the slope of the elimination phase is the elimination rate
constant (often referred to as beta or kel), and from it is derived the elimination
half-life (t1/2).
• The absolute value of the slope of the line that describes the distribution phase
is called the distribution rate constant (often referred to as alpha or kd). This
rate constant enables calculation of a distribution half-life.
• Once the data are mathematically described by the slopes and y-intercepts of
these lines, the plasma drug concentration at any given time point after the
drug is administered can be predicted by the following equation:
•

7. • where Cp(t) is the plasma concentration as a function of time, A is the y-
intercept of the line that describes the distribution phase, α is the rate constant
for the distribution portion of the plasma concentration-versus-time curve, B is
the y-intercept of the line that describes the elimination phase, β is the rate
constant for the elimination portion of the plasma concentration-versus-time
curve, e is the base of the natural logarithm (mathematical constant
approximately equal to 2.71828), and t is the time since administration.

8. PLOT OF TWO COMPARTMENT MODEL PHARMACOKINETICS

9. • APPARENT VOLUME OF DISTRIBUTION
• The pharmacokinetic parameter used to assess the extent of drug distribution
throughout the body is known as the apparent volume of distribution (Vd). If
both the dose and plasma drug concentration (CP) are known, then Vd can be
calculated as follows:
•
• where Vd is the apparent volume of distribution (in L/kg), D is the dose (in
mg/kg), and Cp is the plasma drug concentration (in mg/L).
• This theoretical volume is the volume into which the drug must be distributed if
the concentration in plasma represents the concentration throughout the body.
The term "apparent" underscores the fact that Vd does not indicate where the
drug is distributed, but only that it goes somewhere.

10. • Vd is useful for several reasons. Perhaps most importantly, it can be used to
calculate a dose if the target Cp is known, by rearrangement of the equation
for Vd. For example, if the steady-state Vd (Vdss) of phenobarbital is 0.6 L/kg, and
the target concentration of phenobarbital in a drug-naive animal is 10 mg/L, the
IV dose is calculated as follows:
An additional reason Vd is useful is that if Cp is known at any time after the dose is administered, then Vdss can be
used to calculate how much drug is left in the body.

11. • Finally, Vdss can be used to predict the relative ability of the drug to move to different
body compartments. If limited to the extracellular compartment (interstitial fluid,
plasma), as is typical of water-soluble drugs, the drug represents 20%–30% of body
weight, and the Vd of such a drug should be < 0.3 L/kg.
• Lipid-soluble drugs are generally able to penetrate cell membranes and thus are
distributed to both extracellular and intracellular fluid, which represents ~60% of the
body weight. Such drugs are generally characterized by Vd > 0.6 L/kg.
• Some drugs are limited to the plasma compartment and do not distribute well. For
example, for drugs very tightly bound to plasma proteins, Vd approximates the size of
the blood compartment, or < 0.1 L/kg. As the drug is freed from the protein,
however, it will leave the plasma compartment and distribute into tissues.
• Many drugs are characterized by a Vd that exceeds the body weight of the animal (ie,
> 1 L/kg). For example, the mean digoxin Vd in dogs is 13 L/kg. This means that
digoxin binds appreciably in other tissues (ie, if the drug leaves the plasma,
regardless of where it goes, Vd will increase).

12. • The Vdss of a drug is usually consistent over a wide dose range for a given species, and
it is the best estimate of Vd for extrapolating across species.
• However, a number of clinically important factors can influence Vd, including age
(Vd is larger in neonates and pediatrics, smaller in geriatrics);
• Functional status of the kidneys (Vd is decreased with dehydration),
• Liver (Vd is increased with edema), and heart; fluid accumulations;
• Concentration of plasma proteins (influencing unbound drug only);
• Acid-base status (particularly if ion trapping causes the drug to accumulate in tissues);
• Inflammatory processes or necrosis (tends to increase distribution); and any other
causes for alteration in the extent of plasma-protein binding.

13. • DRUG CLEARANCE
• As soon as a drug reaches the systemic circulation, it immediately begins to be
cleared from plasma.
• Clearance is the volume of blood from which a drug is irreversibly eliminated, or
cleared.
• Usually plasma is sampled; however, plasma clearance represents the sum clearances
by all organs. If the drug is cleared by only a single organ, then plasma clearance is
the clearance of that organ.
• An alternative definition of clearance is the volume of plasma that would contain the
amount of drug excreted per unit time. This definition demonstrates the link between
volume of distribution (Vd) and clearance. If the elimination rate constant (kel) is
known, it describes the fraction of Vdss cleared, and together, these two values can be
used to calculate clearance:

14. where Cl is clearance (in mL/kg/min), Vdss is the apparent volume of distribution at steady-state (in mL/kg), and kel is the
elimination rate constant (in min−1).
Like Vd, then, Cl directly influences kel, the rate at which drug is eliminated from the body: as Cl increases, kel becomes
steeper.
Clearance is independent of the Vd of a drug and thus of the concentration of drug in the blood; no matter how much
drug is in the blood, the same volume will be cleared per unit time.
The two major organs responsible for clearance are the liver and kidneys. After a drug is metabolized, it is irreversibly
eliminated from the body. Its metabolites, however, must be excreted (usually by the kidneys).
Hepatic clearance is defined as the volume of plasma totally cleared per unit time as blood passes through the liver. The
rate of hepatic clearance depends on drug delivery to the liver—ie, blood flow (Q) and the extraction (E) ratio of the
drug, or fraction of the drug removed as it passes through the liver. Extraction, in turn, is determined by the intrinsic
clearance (metabolic capacity) of the liver.

15. • Drugs cleared by the liver fall into two major categories:
• Flow-limited drugs: are extracted so rapidly that Q becomes the limiting factor
of hepatic clearance. Binding to plasma proteins will not influence the clearance
of such drugs.
• In contrast, the rate-limiting step of capacity-limited drugs is intrinsic clearance,
the metabolic capacity of the liver. For such drugs, binding to serum proteins
will decrease the rate of clearance. Therefore, highly protein-bound drugs are
referred to as "capacity limited, binding sensitive," as opposed to drugs not
highly protein bound and thus "capacity limited, binding insensitive."
•

16. • Hepatic disease differentially impacts flow- and capacity-limited drugs.
• Hepatic clearance of flow-limited drugs markedly decreases with changes in
hepatic blood flow, such as might occur with portosystemic shunting.
• When administered PO, such drugs are normally characterized by high first-
pass metabolism and low oral bioavailability.
• With portosystemic shunting, oral bioavailability can markedly increase, so oral
doses must be decreased in proportion to the extent of shunted blood.
• Changes in hepatic mass and function will affect capacity-limited drugs.
• In general, if liver disease has negatively affected serum albumin and BUN
concentration, the intrinsic metabolic capacity of the liver is also likely to be
negatively affected.
• However, if protein-binding decreases for a highly protein-bound drug such
that more of the drug is unbound, hepatic clearance may not be as negatively
affected.

17. • Renal clearance is defined as the volume of plasma totally cleared of a drug per
unit time (eg, L/min) during passage through the kidneys.
• The renal clearance of drugs depends primarily on renal blood flow;
• It is also affected by urine pH,
• Extent of plasma-protein binding,
• Urine-concentrating ability,
• Concomitant use of certain drugs.
• Serum creatinine concentration or serum creatinine clearance can be used to
assess changes in renal clearance as renal function declines.
• Either the dose or the interval can be proportionately modified.
• For drugs with a short half-life, intervals are more appropriately prolonged
(compared with decreasing dose) as serum creatinine concentration increases;
• for drugs that accumulate because of a long half-life, the dose or interval might
be proportionately decreased or prolonged, respectively.

18. • ELIMINATION RATE CONSTANT
• The elimination half-life is the time that lapses as Cp decreases by 50%.
• The elimination half-life is derived from the elimination rate constant, kel,
which is the slope of the terminal component of the plasma concentration-
versus-time curve
• A hybrid parameter, kel is affected by both Cl and Vd.
• Cl determines the rate of decline in Cp; thus, the greater the volume of drug
cleared, the steeper the slope, or kel.
• The impact of Vd on half-life reflects its effect on Cp: a larger Vd means that less
drug is in the volume of blood cleared by the liver or kidneys.
• Therefore, the rate of elimination declines as Vd increases, resulting in an
inverse relationship.
• The elimination half-life is calculated as follows:

19. where ln 2 ≈ 0.693.
The relationship between kel and t1/2 reflects the fact that t1/2 becomes the run of the slope as concentration decreases by
50% (ie, C1/C2 = 2, where C1 is the concentration at time 1 and C2 is the concentration at time 2). Because t1/2 is inversely
proportional to kel, t1/2 is directly proportional to Vd (larger Vd results in a longer half-life) and inversely proportional to Cl.
t1/2 = Vd x Ln2
CL
Note that Cl and Vd can be profoundly altered, yet t1/2 may not change.
For example, in an animal dehydrated because of renal dysfunction, Cl may be decreased by 50%, thereby doubling t1/2. If
the animal is markedly dehydrated, however, then Vd will decrease because of the contraction of extracellular fluid
volume. Because more drug is in each milliliter of blood cleared by the kidney, the same amount of drug may be
eliminated, and thus kel or t1/2 may not change.

20. • The elimination half-life determines the amount of time to steady state and the
amount of time for a drug to be eliminated from the body after drug
administration is discontinued.
• After a drug is discontinued, 50% of it is eliminated after one half-life, 75%
after the second half-life (half of 50%), 87.5% after the third, and so on. For
practical purposes, most of the drug is eliminated by 3–5 half-lives.

21. • Single-Dose Concentration Curves After Extravascular Administration of Drugs
• When a drug is administered by an extravascular route, plasma drug concentrations
rise until a peak or maximum drug concentration (Cmax) is reached.
• After the drug enters circulation, it is subjected immediately and simultaneously to
distribution, metabolism, and excretion.
• The plasma concentration-time curve after extravascular administration has an
additional y-intercept and slope, and the slope reflects the absorption rate
constant, ka.
• The absorption half-life is the time that elapses as 50% of the drug is absorbed into
the system.
• Absorption generally is sufficiently slow that drug distribution is generally masked by
the absorption phase.
• Therefore, as plasma drug concentrations decline after Cmax is reached, the slope
generally reflects kel.
•

22. • Steady-State Plasma Concentration (Repeated Administration or Constant IV
Infusion) of Drugs
• In some cases, the desired therapeutic effect of a drug is produced with a
single dose.
• To achieve a satisfactory response, however, it is frequently necessary to
maintain drug concentrations in the therapeutic range for a longer time. Rather
than administering large doses, which could result in potentially toxic plasma
drug concentrations, multiple dosing allows regular, safer intervals.
• For drugs with a very short half-life, the drug may be administered through a
catheter as a constant-rate infusion, which is essentially continuous IV delivery.
• The rate of administration depends on the amount of fluctuation in drug
concentration that can occur during a dosing interval, which in turn is
determined by the relationship between t1/2 and the dosing interval, Τ.

23. • If a drug is administered at an interval substantially longer than its half-life, most of the drug
will be eliminated during each dosing interval.
• Therefore, little drug remains when the subsequent dose is administered, and plasma drug
concentrations will fluctuate (from maximum drug concentration [Cmax] to the minimum drug
concentration [Cmin]) during the dosing interval.
• For example, if a drug with a 4-hour half-life is administered every 12 hours, 87.5% of the
drug will be eliminated during each dosing interval. With each dose there is a risk that drug
concentrations will become subtherapeutic; increasing the dose will result in a small increase
in Cmin but may substantially increase Cmax, thus increasing the risk of toxicosis.
• A more appropriate response would be to decrease the dosing interval. However, this may
be necessary only if drug efficacy depends on the presence of the drug. For example, this
amount of fluctuation may be acceptable for a concentration-dependent antimicrobial such
as gentamicin. If the drug is an anticonvulsant, however, the risk of seizures increases just
before the next dose. If the drug is time dependent, drug concentrations may decrease to
below the minimum inhibitory concentration of the infecting microbe.

24. • In contrast to drugs with a short half-life, drugs with a long half-life compared
with the dosing interval will accumulate with each dose because much of the
drug remains in the body when the next dose is administered.
• Drugs with a long half-life begin to accumulate with the first dose and continue
to do so until a steady-state equilibrium is reached such that the amount of
drug eliminated during each dosing interval is equivalent to the amount of drug
administered during that same interval.
• The accumulation ratio describes the magnitude of increase of
either Cmax or Cmin at steady state compared with the first dose. The longer the
half-life is compared with the dosing interval, the greater the accumulation
ratio is.
•

25. • As with drug elimination, for practical purposes, steady state is achieved within
3–5 half-lives, regardless of the drug or dose, provided the preparation and
dosing regimen are the same.
• In such cases, 50% of the plateau or steady-state concentration will be reached
after 1 half-life, 75% after 2 half-lives, 87.5% after 3 half-lives, and 93.6% after 4
half-lives.
• Response to the drug, whether efficacy or toxicosis, cannot be assessed until
steady state is reached. Because the amount of drug in the body is large
compared with each dose, manipulating plasma drug concentrations for such
drugs is difficult, because changes require dosing for 3–5 half-lives at the new
dose.
• If the time to reach steady state, and thus time to therapeutic effect, is
unacceptable, steady-state plasma drug concentrations may be achieved more
rapidly by administration of a loading dose or doses, as follows:

26. where D is the dose (in mg/kg), Vd is the volume of distribution at steady-state (in mL/kg), and Cp is the target plasma
concentration (in mg/mL). If the drug is administered PO, the dose must account for bioavailability:
where F is the bioavailability (in %). However, the drug will not be at steady state, but only at steady-state
concentrations.
If the maintenance dose does not maintain what the loading dose achieved, then as steady state at the maintenance
dose is reached, plasma drug concentrations may increase to cause toxicosis or decrease to a subtherapeutic
concentration.
Drugs with very short half-lives are often administered by constant-rate infusions in animals in critical condition. In such
cases, the interval is infinitely short compared with the half-life, and the drug accumulates until steady state is reached.
The rate of infusion can be calculated as follows:
where I is the rate of infusion (in mcg/kg/min), Cl is clearance (in mL/kg/min), and Cp is the target plasma concentration
(in mcg/mL).
A loading dose should be administered if the time to steady state is unacceptably long.