1
Pharmacokinetics
is the study of drug and/or
metabolite kinetics in the body.
It deals with a mathematical
description of the rates of drug
movement into, within and exit
from the body. The body is a
complex system and a drug
undergoes many steps being
absorbed, distributed through
the body, metabolized or
excreted. The drug also
interacts with receptors and
causes therapeutic and/or toxic
responses.
Concentration of drug
in blood or at the
receptor is variable of
interest.
Site of
Action
Dosage Effects
Plasma
Concen.
Pharmacokinetics Pharmacodynamics
Compartmental Approach Three-compartment model
+D
k2
k23
k32
C2(t) C3(t)
C1(t)
k12
332223
3
3322232112
2
1
112
1
)(
)(
CkCk
dt
dC
CkCkkCk
dt
dC
ttDCk
dt
dC n
i
i
−=
+⋅+−=
−+−= ∑=
δ
Stochastic three-compartment model
33332223
3
223322232112
2
11
1
112
1
)(
)(
ξσ
ξσ
ξσδ
+−=
++⋅+−=
+−+−= ∑=
CkCk
dt
dC
CkCkkCk
dt
dC
ttDCk
dt
dC n
i
i
White (or other) noise can be added to any of the
concentrations
In drug compliance studies – variable is ti
Single-compartment model with
absorption
1
1
0
0
)exp()(
)0(,
CkDk
dt
dC
tkDtD
DDDk
dt
dD
ea
a
a
−=
−=
=−= D
C1(t)
ka
↓ke
2
Plasma concentration vs. time profile of a
single dose of a drug ingested orally
0
2
4
6
8
10
12
14
0 5 10 15 20
TIME (hours)
PlasmaConcentration
0
2
4
6
8
10
12
14
0 5 10 15 20
TIME (hours)
Plasmaconcentration
Influence of Variations in Relative Rates of
Absorption and Elimination on Plasma
Concentration of an Orally Administered Drug
Ka/Ke=10
Ka/Ke=0.1
Ka/Ke=0.01
Ka/Ke=1
Multiple dosing
• On continuous steady administration of a drug,
plasma concentration will rise fast at first then
more slowly and reach a plateau, where:
rate of administration = rate of
elimination i.e. steady state is reached.
Changes are due to timing or dosing, then the
steady state is disturbed and consequently
the effect of therapy is modified. This is called
Non-compliance or non-adherence
Steady state
As successive doses are administered, drug begins to accumulate in the body.
With first order elimination, at a certain point in therapy, the amount of drug administered
during a dosing interval exactly replaces the amount of drug excreted
(rate in = rate out). When this equilibrium occurs, the peak and trough drug
concentrations are the same for each additional dose given, steady-state is reached.
The time required to reach steady-state is approximately 4 to 5 half-lives.
“In HIV therapy, the biggest obstacle to successful treatment is adherence to
medications. The method by which antiretroviral medications suppress the
HIV virus necessitates a very strict regimen of medication. Drugs must be
taken exactly as prescribed without missing doses. With any type of
medication regimen, whether it is to treat HIV, diabetes, or high blood
pressure, there are several reasons why people have difficulty adhering to
their prescribed medications. Several studies have been done to identify
these reasons:”
40% of people said they simply forgot
37% slept through a dose
34% were away from home
27% had made a change in their therapy routine
22% were too busy to take their meds
13% were too sick
10% were experiencing side effects
9% were suffering from depression
Motivation and Basic Terms
• Patient compliance with medication, both in timing and in dosing, is
an important issue in evaluation the success of therapy.
• Noncompliance: - intentional (long term) and non-intentional (short
term).
• The basic one-compartment model is investigated analytically under
the steady-state conditions. It is assumed that the errors in the drug
administration are mutually independent and that a new error in drug
administration occurs always at the steady state. We concentrate on
the effect of short-term noncompliance. ANTIBIOTICS
• Complex model is investigated by computer simulation
• The most frequent type of noncompliance is beside occasional
omission (delay) of a dose a failure to take several consecutive
doses.
3
Therapeutic Window
concentration in plasma yields optimal benefit at a minimal risk of toxicity
AUC – area under the curve, reflects therapeutic effect
Area under the threshold
not only the time spent outside
the therapeutic window but the depth
of the curve is important
Basic Model - Fast absorption rate
Time
Concentration
Cmax
Cmin
a
Cr
t0
Basic Model - Fast absorption rate
Drug concentration influenced by noncompliance
Distribution of max and min
t0 is replaced by a random variable T
Regular drug application
Transient state Steady state
Cmin
C1
C2
C3
Concentration
Time
4
ConcentrationC3
Time
Noncompliance measures
T1
S1
Steady
state
1. Number
2. Time
3.Area
4. Optimality
(f – gaussian half)
∑ iT
∑ iS
∑
∫
∆≈
≈
)(
))(( 3
f
dttCf
f
x
Optimum
Types of dose omission
Complete omission
of a dose
Gauss distribution
Exponential
distribution
Gamma distribution
Delayed dose
Regular dosing





 −
2
2
2 2
x
exp
2
1
σπσ





 −
µµ
x
exp
1
( )





 −
Γ
−
b
x
expx
ba
1 1a
a
x
Types of Fluctuation
MODELING THE INFLUENCE OF NON-ADHERENCE ON
ANTIBIOTIC EFFICACY: APPLICATION TO
CIPROFLOXACIN
• Multiple dosing, 250 mg every 12 hours or
500 mg every 24 hours, of orally applied
ciprofloxacin over ten days.
• Mortality rate of sensitive bacterial
population (sigmoidal)
S
S
SS
zMIC
tC
zMIC
tC
t
max
min
minmax
)(
)(
)(
)(
Ψ
Ψ
−
Ψ−Ψ
=µ
Bacterial population under
treatment
• a new bacterial cell arises, the probability,
f, that it will be resistant
( ) )()()()()1(
)(
maxmax tXttXtXf
dt
tdX STTSS
S
µ−Ψ+Ψ−=
( ))()(
)(
maxmax tXtXf
dt
tdX TTSS
T
Ψ+Ψ=
Decrease (left) in bacterial density during ciprofloxacin
therapy 250mg/12h for regular dosing. Effect of innate
capability to contribute to bacterial kill (f=0.0001, innate killing
rate mT=0.01 [1/h]) . The line gives the density of resistant
population, the saw-like curve is the density of sensitive
bacterial population
5
Bacterial density during ciprofloxacin therapy for 500 mg/24 h regular
dosing (higher maxima), regular dosing 250mg/12h (lower maxima).
Discontinuation of the treatment at time 100 is illustrated
Drug – receptor interaction
• Most drugs combine with specific sites on
macromolecules (e.g. cell membrane
components) by precise physiochemical
interactions between specific chemical
groups of the drug. These sites are
termed receptors.
• Pharmacodynamics
Theory and assumptions
of drug-receptor interaction
• Combination or binding to receptor causes some event
which leads to the response.
• Response to a drug is graded or dose-dependent.
Drug receptor interaction follows simple mass-action
relationships, i.e., only one drug molecule occupies each
receptor site and binding is reversible.
• For a given drug, the magnitude of response is directly
proportional to the fraction of total receptor sites occupied
by drug molecules (i.e. the occupancy assumption).
• The number of drug molecules is assumed to be much
greater than the number of receptor sites.
Drug-receptor
• Combination of drug with a receptor produces a
specific response. "lock and key".
• Drug-receptor interactions are analogous to
enzyme-substrate interactions. Most of the same
principles apply.
• Drug-receptor interactions with characteristics
outlined above can be treated with an equation
analogous to the Michaelis Menten equation utilized
for enzyme-substrate interactions
The Log Dose-Response Curve
• Advantages of expression as log versus response
– Dose-response relationship expressed as a nearly
straight line over a large range of drug doses.
– Wide range of doses can be plotted on a single
graph, allowing easy comparison of different drugs.
– Use of log dose-response curves to compare
different drugs which produce the same response
Typical log dose-response curve