Drug kinetics after multiple dosing
Pharmacokinetics of multiple dose
•
The
purpose of the multiple dosing is to maintain a desired plasma level of drugs
for long periods of time.
•
The plasma
level obtained from each succeeding dose is always higher than those obtained
from the previous dose since some drug remains in the plasma when repeated doses are
administered. • Since the rate of elimination is proportional to the amount of drug in the
body, it increases with the increasing amount of drug until it approaches the
rate of input (rate of input=rate of output). • At
this point the amount reaches a plateau in the body. (Rate of input=rate of
output).
•
The main
difference is the fluctuation in the plasma
concentration due to multiple dosing.
2
•
Dosage regimen:
– size
of the drug dose – frequency of drug administration, i.e. time interval
between doses (τ: tau)
•
Pharmacokinetic parameters
obtained from single dose administration are used to predict drug plasma
levels during a multiple-dosage regimen. • The principle of superposition
assumes that early doses of drug do not affect the pharmacokinetics of the
subsequent doses.
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•
AUC0-∞= AUC0-τ
•
If the drug is administered at a
fixed dose and fixed dosing interval (τ), the amount of the drug in the body
will increase and then plateau to a mean plasma level higher than the peak Cp
obtained from the first dose.
•
This is known as “steady state”, at
which the plasma drug levels fluctuate between Cpssmax and Cpssmin
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• The average Cpssav is calculated by dividing
the AUC for a dosing interval (AUC0-τ) by τ, at steady state
• Regardless of the route of administration:
 Cpssav = F×D0
= F×D0 =
AUCss0−τ =
AUC0−∞
CL×τ
Vd×kd×τ τ τ
• Accumulation factor (R): Cpssmax Cpssmin
 = max = min = C1 C1 1−e
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•
The maximum accumulation of drug
in the body can be given in terms of the administered dose as:
1.44 t
 X/Xo = =
1/2 ,
where:
X: is the maximum drug accumulation
Xo: is the amount after first dose
:
dosing interval
•
Therefore, when the dosing
interval is equal to the half-life of the drug, the maximum accumulation will
be 1.44 times the administered dose.
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•
As the frequency of dosing increases
the accumulation level also increases.
•
For
example:
If the dosing interval is one-fourth
the half-life, the maximum accumulation is 5.76 times the administered
dose.
•
Therefore, a desired plateau level
can be achieved either by:
– Selecting a dosing interval and adjusting the dose or – By
changing the dosing interval to give a fixed dose
•
Drug accumulation is not a phenomenon that depends on the property
of a drug.
•
Accumulation is the result of the frequency of administration relative to half-life.
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• Some
fluctuation is always present and the percentage of fluctuation is given by:
% = ( − )
• This means that the fluctuation in the plasma
concentration is only a
. function of the drug half life ( . =
 ) or the dosing interval.
• Since the half life of drug is constant, the adjustment in the dosing interval is required to
achieve desired fluctuation.
• If a drug is administered more frequently (smaller τ), the
fluctuation decreases and reaches a limit
of (0) percent fluctuation when the dosing interval is zero as in the case of
constant intravenous infusion.
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•
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For
a given drug in multiple-doses, the time required to reach steady state is dependent on the t1/2 of the drug and is independent of the size of the dose,
length of τ, and number of doses.
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•
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For
a drug given at the same dosing rate (e.g. 20 mg/h: 120 mg/6h, 240 mg/12h, or
480 mg/24h), the Cpssav will be the same, but fluctuation between
Cpssmax and Cpssmin will vary.
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•
The rate with which the plateau
level is reached is dependent only on the elimination half-life.
•
It will therefore take one
half-life to reach 50% of the plateau level (Cpss), and 3.3 half-lives to
reach 90% of the maximum level.
× = × (1 −
×)
tss (90%) = ‒ ln(0.1)/ k = 3.3 t1/2
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• At steady state:
Rate in (input) = Rate out
(output)
F×D/τ= CL×Cpss
• Regardless of the route of administration:
 Cpssav =
F×D = F×D =
AUC0−τ =
AUC0−∞
CL×τ
Vd×kd×τ τ τ
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•
Following multiple dosing, the
plasma concentration can be estimated by using the following equation:
 = D 0 1−
e− nkτ
× −kt
Cp
t (
− −kτ )
e
Vd 1 e
, where: n = number of doses given, and t = time after nth
dose
•
As the number of doses (n)
approaches infinity (∞), a plateau (steady state) is reached at which the
plasma concentration at any time t (post dose) can be estimated by the
following equation:
D 1
 Cpss
= 0 ( −
−kτ
)×e−kt Vd
1 e
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D
Cp 0
  Cpssmax= −0kτ
;Cpssmax =
−eVd−kτ
1−e 1
 Cpssmin
= Cp−0kτ ×e−kτ =Cpssmax ×e−kτ
1−e
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• For a continuous i.v. infusion:
Cpt =
 R
(1−e−kt )=  R (1−e−kt ) CL Vd×kd
•
After one or more short i.v.
infusions: D
Cp t =
 (1 −
e−kt ) (modified from above) tinf ×Vd
× kd
• After the infusion is stopped:
Cp t =
C
stop
×e−
kt
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• For multiple oral dose regimen:
– Before reaching the steady state:
  t F×ka×D0 ⎢⎡
1−−ee−−nkakaττ ×e−kat −(11−−ee−−nkkττ)×e−kt ⎤⎥⎦
Cp
= ( )
Vd(k−ka) ⎣
1
– At steady state:
  Cpss=
F×(ka×D)0 ⎡⎢⎣(1−1e−kτ)×e−kt
−(1−1e−kaτ)×e−kat
⎤⎦⎥
Vd
ka−k
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 Cpssmax =
F×D0 ( 1−kτ
)×e−ktp
Vd 1−e
 Cpssmin =
ka×F×D0
( 1−kτ )×e−kτ
Vd(ka−k)
1−e
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• Following a single oral dose:
tmax =
 2.3
×log ka
ka −k k
• At steady state:
 tp(tmax, ss)
= 1 ×ln⎡⎢ka×(1−e−−kakττ)⎤⎥ ka−k
⎢⎣k×(1−e
)⎥⎦
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Loading
dose (Priming dose)
•
It is the dose that provides an
effective concentration in the plasma instantaneously • The
main objective of the loading dose is to achieve the desired (therapeutic)
concentration as quickly as possible.
•
Oral DL can be
calculated as follows:
  DL0 = ka −e k
)
D (1−e )(1
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Loading
dose
•
For extremely rapid absorption or
in the case of IV infusion:
 DL
= 1−kτ
D0 (1−e )
•
As a general rule, the dose ratio
should be equal to 2 if the selected dosage interval τ = t1/2
D
Dose
ratio =
 L
D0
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• A
rapid approximation of DL may be estimated from the following
equation:
= Vd
×Cpssav
Loading dose (DL)
(S)(F)
,where:
S: fraction of the administered
dose which is the active form of the drug F: bioavailability of the drug
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•
After multiple dosing, the maintenance
dose required is based on CL, Cpss, dosing interval (τ):
CL×Cpssav
×τ
Maintenance Dose (D0) =
(S)(F)
,where:
S: fraction of the administered
dose which is the active form of the drug
F: bioavailability of the drug
•
When Cpss & τ are fixed, a
drug with a smaller CL would require a smaller maintenance dose.
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3-
Check your calculations: By applying these equations
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to
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find
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the
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corresponding Cpssmin & Cpssav
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• Factors
that should be considered in designing a dosage regimen:
1. Activity and toxicity:
¾ Minimum therapeutic dose
¾ Toxic dose
¾ Therapeutic index (Therapeutic window)
¾ Side effects
¾ Dose-response relationships
2. Pharmacokinetic processes of drugs which can be influenced
by dosage form:
¾
 Absorption
¾ Distribution
(ADME)
¾ Metabolism
¾ Excretion
3. Clinical state of patients:
¾ Age, Weight, Urine pH, Condition being treated, Existence
of other diseases.
4. Other factors:
¾ Tolerance
¾ dependence
¾ Drug interactions
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Reference
• Applied
Biopharmaceutics & Pharmacokinetics by Leon Shargel, Susanna Wu-Pong (4th
edition): Chapters 15 & 17
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