Concepts of Pharmacology
- Half Life Calculation C. M. Prada, MD
July 12, 2006
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• Pharmacokinetics = availability
- dosage and rate of administration
- modes of transport – across biologic membranes; bound to
proteins from plasma and tissues
- blood flow to the site of action
- extent and speed of the metabolic process of the drug
- rate of the removal of the drug (and its metabolic products)
• Pharmacodynamics = pharmacologic effect
(in relation with the plasma drug concentration)
- cellular mechanisms of drug action
- clinical evaluation of drug effects
- biologic variability
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Definition: quantitative study of absorption,
distribution, metabolism, and elimination of
chemicals in the body, as well as the time
course of these effects.
Summary:
- absorption
- distribution
- metabolism
- elimination
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• Concentration of a drug at its site of action is a
fundamental determinant of its pharmacologic
effects.
• Drugs are transported to and from their sites of
action in the blood – because of that: the
concentration at the active site is a function of
the concentration in the blood.
• The change in drug concentration over time in
the blood, at the site of action, and in other
tissues is a result of complex interactions of
multiple biologic factors with the
physicochemical characteristics of the drug.
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medicine
movement
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Pharmacokinetic Concepts:
Rate Constants and Half-Lives
• Disposition of most drugs follows first-order kinetics –
a constant fraction of the drug is removed during a
finite period of time.
• This fraction is equivalent to the rate constant of the
process.
• Rate constants: k; min-1 or h-1
• The absolute amount of drug removed is proportional to
the concentration of the drug
• In first-order kinetics, the rate of change of the
concentration at any given time is proportional to the
concentration present at that time.
• When the concentration is high, it will fall faster than
when it is low.
• First-order kinetics apply not only to elimination, but
also to absorption and distribution.
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Half-Lives
•
•
•
The rapidity of pharmacokinetic processes is often
described with half-lives
Half-Life = the time required for the concentration to
change by a factor of 2.
Half-Life = the period of time required for the
concentration or amount of drug in the body to be
reduced to exactly one-half of a given concentration or
amount.
Half-Life = the time required for half the quantity of a drug
or other substance deposited in a living organism to
be metabolized or eliminated by normal biological
processes. Also called biological half-life.
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Zero-Order Elimination
approximately constant rate of elimination
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First-Order Elimination
Ct is concentration after time t
C0 is the initial concentration (t=0)
k is the elimination rate constant
- first-order logarithmic process
- that is, a constant proportion of the agent is eliminated per unit time
(Birkett, 2002)
Birkett DJ (2002). Pharmacokinetics Made Easy (Revised Edition). Sydney: McGraw-Hill Australia. ISBN 0-07-471072-9.
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First-Order Elimination (cont.)
dC
1). ------ = kC
dt
dC
2). ------ = k dt
C
3). ln C = - kt + D
4). C = eD e-kt
5). At time t = 0: C = eD
6). C0 = eD
7). Ct = C0 e-kt
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At the time t = t1/2:
C(1/2) = C0 x 1/2
Ct = C0 e-kt
C0 x 1/2 = C0 e-kt1/2
e-kt1/2 = 1/2
- kt1/2 = ln 1/2 = - ln 2
ln 2
t1/2 = ------k
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The reduction of the quantity
in terms of the number of
half-lives elapsed :
ln 2
t1/2 = ------k
Number of
half-lives
elapsed
Fraction
remaining
As
power
of 2
0
1/1
1 / 20
1
1/2
1 / 21
2
1/4
1 / 22
3
1/8
1 / 23
4
1/16
1 / 24
5
1/32
1 / 25
6
1/64
1 / 26
7
1/128
1 / 27
...
...
N
1 / 2N
1 / 2N
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First-Order Elimination (cont.)
Relationship between the elimination rate constant and half-life:
ln(Cpeak) - ln(Ctrough)
kelim = --------------------------------t interval
t½ = 0.693 / kelim
Half-life is determined by clearance (CL) and volume of
distribution (VD):
Only for IV
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Half-Life Calculation
• Directly from the corresponding rate
constants:
(ln 2)
0.693
t1/2 = ---------- = ----------k
k
Ex.: rate constant of 0.1 min-1 translates into
a half life of
6.93 min.
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Half – Life (cont.)
• Half-Life of any first-order kinetic process can
be calculated – ex.: drug absorption,
distribution, elimination
• First order processes – asymptotically
approach completion, because a constant
fraction of the drug is removed per unit of time
(not an absolute amount).
• The process will be almost complete after
5 (97%)
…half-lives:
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Half-Life Elimination (cont.)
• Repeated equal doses of a drug more
frequently than 5 elimination half-times:
result in drug being administered at a rate
greater than its plasma clearance –
accumulation in plasma.
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Equations for Half Lives
For a zero order reaction A products , rate = k:
t½ = [Ao] / 2k
For a first order reaction A products , rate = k[A]:
t½ = 0.693 / k
For a second order reaction 2A products or A + B products
(when [A] = [B]), rate = k[A]2:
t½ = 1 / k [Ao]
http://www.chem.purdue.edu/gchelp/howtosolveit/Kinetics/Halflife.html
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For a zero order reaction
A products ,
rate = k:
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For a first order reaction
A products , rate = k[A]:
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For a second order reaction 2A products or A + B products
(when [A] = [B]), rate = k[A]2:
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The Elimination Half-Time Limits
• Only in single-compartment models does it actually
represent the time required for a drug to reach half of its
initial concentration after administration
• This is because in a single-compartment model
elimination is the only process that can alter drug
concentration
• Intercompartmental distribution cannot occur because
there are no other compartments for the drug to be
distributed to and from
• Most drugs in Anesthesia: lipophilic – therefore are
more suited to multicompartmental model
• In multicompartmental models, the metabolism and
excretion of some intravenous anesthetic drugs may
have only minor contribution to changes in plasma
concentration when compared with the effects of
intercompartmental distribution
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Drug Elimination
• Elimination = all the various processes that
terminate the presence of a drug in the body.
• Processes:
- metabolism
- renal excretion
- hepato-biliary excretion
- pulmonary excretion (inhaled anesthetics
mainly)
- other: saliva, sweat, breast milk, tears.
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Renal Excretion
• Both metabolically changed and unchanged drugs
• LMW substances: filtered from blood through the
Bowman membrane of the capsule
• Some: actively secreted
• Reabsorption in the tubule: depending on the lipid
solubility, degree of ionization, molecular shape, carrier
mechanism (for some).
• Weak acid: best reabsorbed from an acidic urine.
• Important to know if the drug is dependent on renal
function or excretion.
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Hepatobiliary Excretion
• Drugs metabolites – excreted in the
intestinal tract with the bile.
• Majority: reabsorbed into the blood and
excreted through urine. (enterohepatic
cycle).
• Poorly lipid-soluble organic compounds –
at least three active transport mechanisms
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Pulmonary Excretion
• Volatile anesthetics and anesthetic gases:
in large part eliminated unchanged
through the lung
• The factors that determine uptake operate
in reverse manner
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Multicompartment Pharmacokinetics
• Instead of a single exponential process with one
half-time, the pharmacokinetics are described by
2 or more exponential processes – can
calculate a half-time for each process: l1, l2,
l3, etc. (referred to as: a, b, g).
• The time for the concentration to decrease by
50% is dependent on the preceding dosing
history and can vary with the duration of drug
administration.
• The time to decrease plasma concentration by
half is not equal with the time to remove half of
the drug from the body - terminal half-life.
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Multicompartment Models
• 3 compartments:
– Central = vascular bed
– 2nd = rapidly equilibrating, high perfusion (muscle)
– 3rd = large compartment – slow equilibrating, low
perfusion (fat).
• 5 compartments: (isoflurane and sevoflurane
measurements)
–
–
–
–
–
Central
Vessel-rich
Muscle group
4th compartment
The fat group
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Context-Sensitive Half-Time
• Improved our understanding of anesthetic drug
disposition; is clinically applicable.
• Effect of distribution on plasma drug conc.
varies in magnitude and direction over time depends on the drug concentration gradients
between various compartments.
- ex.: early part of the infusion of a
lipophilic drug, the distributive factor decrease its
plasma conc. as the drug is transported to the
unsaturated peripheral tissues – later, after the
infusion is discontinued: drug will re-enter in the
central circulation
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Context-Sensitive Half-Time (cont.)
• Def.: context-sensitive half-time
describes the time required for the plasma
drug concentration to decline by 50% after
terminating an infusion of a particular drug.
• Calculated by using computer simulation
of multicompartmental models of drug
disposition:
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Context-Sensitive Half-Time (cont.)
• Reflects the combined effects of distribution
and metabolism on drug disposition
• The data confirm the clinical impression that as
the infusion duration increases, the context
sensitive half-time of all drugs increases. This
phenomenon is not described by the
elimination half-life.
• No relationship with the Half-Life.
• N.B.: For a one-compartment model: contextsensitive half-life = elimination half-life.
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Context-Sensitive Half-Time (cont.)
• Compare fentanyl (half life 462 min.) and
sufentanil (half-life 577 min.) – storage and
release of fentanyl from the peripheral binding
sites: delay the declines of plasma
concentration.
• Compare propofol and thiopental –
comparable c-s h-t following a brief infusion only.
• Because of : 1). high metabolic clearance of propofol and
2). relatively slow rate of return to plasma of propofol from
peripheral compartments.
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Context-Sensitive Half-Time (cont.)
• Alfentanil – studied for ambulatory techniques
• Elimination half – life: 111 min.
• Small distribution volume – not significant in plasma decay
after infusion
• Sufentanil – elimin. half – life: 577 min.
• Less context-sensitive half-time (for infusions up to 8 hours);
large volume of distribution for sufentail
• After termination of its infusion, the decay in plasma drug
conc. is accelerated not only by elimination, but also by
continuous redistribution into the peripheral compartments.
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Context-Sensitive Half-Time and
Time to Recovery
• Time to recovery depends on other
additional factors:
– Plasma concentration below which
awakening can be expected
– Awakening from anesthesia is a function of
effect-site concentration decay
– Effect-site equilibration – half-time of
equilibration between drug concentration in
blood and the drug effect can be determined:
t1/2 ke0
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1). Zero order pharmacokinetics
If there is one thing that separates pharmacology from other medical
subjects, it is zero order pharmacokinetics!
Salicylic acid is an example of a drug that behaves this way.
What is drug elimination according to zero order kinetics?
A constant amount of drug is removed per unit of time.
This makes the rate of metabolism saturable, so that small changes in dose,
can give dramatic changes in plasma concentration
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2). A drug was given intravenously at a dose of 200mg.
The initial concencentration in plasma (Co) was 10 µg/ml,
and the Kel (elimination constant) was 0.02/h.
Determine the plasma clearance (Clpl) and t1/2 for this drug.
Distribution volume: Vd = dose/ Co = 200 mg / 10 mg/L = 20 L
Cl pl = Vd x Kel = 20 L x 0.02/h = 0.4 L/h
T1/2 = ln 2 / kel = 0.693 x 0.02 = 35 hours
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The kinetic order of a reaction is determined
by the exponent of the rate equation, n, in dc/dt = K Cn:
Kinetic
order
Equation
Dependency
on C
2
dc/dt = K C2
Exponential
1
dc/dt = K C1 or dc/dt = Linear
KC
0
dc/dt = K C0 or dc/dt = None
K
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Saturable kinetics usually follow Michaelis-Menten equation:
dC/dt = [(dC/dt)m.C] / (Km + C)
where (dc/dt)m is the maximum rate that a reaction can reach
and Km is the Michaelis-Menton constant that corresponds
to a concentartion at which the rate is 1/2 of the maximum.
When C is very small it can be ignored from denominator
and reduces the above equation to:
dC/dt = [(dC/dt)m.C] / Km
or
dC/dt = (dC/dt)m/Km.C.
Hence the rate becomes dependent upon a constant,
[(dC/dt)m / Km], and C, and the kinetics change to a first order.
On the other hand during the time when C is very high,
Km becomes negligible and the equation reduces to:
dC/dt = (dC/dt)m.C / C
Now C can be cancelled from both nominator and denominator to give:
dC/dt = (dC/dt)m
i.e., the rate depends only upon a constant but not upon concentration.
From then on the reaction proceeds according to zero order kinetics.
Between these two extremes the order of the reaction is a mixture of
first and zero kinetics.
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Concentration dependency of the kinetic order of saturable reaction.
www.pharmacy.ualberta.ca/pharm415/orderof.htm
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Michaelis Menten Equation
The Michaelis Menten process is somewhat more complicated with
a maximum rate (velocity, Vm)
and a Michaelis constant (Km) and the amount or concentration remaining.
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Half-Lives for Some Common
Drugs
• Narcan (Naloxone): plasma half-life: - adult:
64±12 min; neonate: 3.1 ± 0.5 hours
• Fentanyl: half-life: 2 – 9 hours
• Morphine: terminal half-life: 1.5 – 4.5 hours
• Midazolam: elimination half-life: 1.8 – 6.4 hours
• Flumazenil (Romazicon): distribution half-life: 7
– 15 min.; terminal half–life: 41 – 79 min
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Drug
Duration
Half-life
Route
Equianalgesic
Dosage
Codeine
4–6 h
3 h
IM/IV/SC
PO
120 mg
200 mg
Fentanyl
Hydrocodone
Hydromorphone
1–2 h
4–8 h
4–5 h
1.5–6 h
3.3–4.5 h
2–3 h
0.1 - 0.2 mg
Levorphanol
6–8 h
12–16 h
Meperidineæ
2–4 h
3–4 h
IM/IV
PO
IM/IV/SC
PO
IM/IV/SC
PO
IM/IV/SC
PO
Methadone
4–6 h
15–30 h
Morphine
3–6 h
1.5–3 h
Oxycodone
Oxymorphone
4–6 h
3–6 h
NA
NA
IM/IV/SC
PO
IM/IV/SC
PO
PO
IM/IV/SC
20-30 mg
1.3–1.5 mg
7.5 mg
2 mg
4 mg
75 mg
300 mg
1-10 mg§ Medline
Short term: 5-10mg
Chronic use: 1-4
mg
(2 mg)
2 - 20 mg§
Medline Short
term use: 20 mg
Chronic dosing: 24 mg
(3mg)
10 mg
30–60 mg#
15-30 mg (20 mg)
1 mg
Important Update:
Opana™ and Opana ER™
(oxymorphone immediate
release and oxymorphone
extended release tablets)
have been approved by
the FDA.
PO
10 mg
Propoxyphene
4–6 h
6–12 h
PO
130-200 mg *
Propoxyphene HCL: 130mg; Napsylate: 200mg. Not recommended for chronic pain management and therefore not
available in program above.
*
#
Acute dosing (opiate naive): 60mg.
Chronic dosing: 30 mg.
§:
Many equianalgesic tables underestimate methadone potency - more studies are needed. Parenteral: Program
utilizes 10mg for short-term dosing and 2 mg for chronic dosing. Oral: Program utilizes 20mg for short-term dosing
and 3 mg for chronic dosing.
æ
Meperidine should be used for acute dosing only and not used for chronic pain management (meperidine has a
short half-life and a toxic metabolite: normeperidine). Its use should also be avoided in patients with renal
insufficiency, CHF, hepatic insufficiency, and the elderly because of the potential for toxicity due to accumulation
of the metabolite normeperidine. Seizures, confusion, tremors, or mood alterations may be seen.
http://www.globalrph.com/narcoticonv.htm
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e = 1 + 1/1! + 1/2! + 1/3! +……
e is the limit of (1 + 1/n)n as n tends to infinity
e = 2.718281828459045235
Euler
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Disclaimer
• This is a compilation intended only for the
personal use of the MHMC residents, and
not for publication
• For the bibliographic sources, please send
an e-mail to: cprada@metrohealth.org
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