Kinetics of Drugs
description
Transcript of Kinetics of Drugs
Dandamun, Benbellah Ali Y.
BSCHE-5 Engr. Elaine G. Mission
KINETICS OF DRUGS
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Contents Introduction ...................................................................................................................................... 2
Drug kinetics ..................................................................................................................................... 2
Rate, Order, and Molecularity .................................................................................................. 2
Zero-Order, First-Order, and Second-Order Reactions and Their Rate Equations, Half-
Life and Shelf-Life ......................................................................................................................... 3
Zero-Order Reaction ................................................................................................................ 4
First-Order Reaction ................................................................................................................. 7
Second-Order Reactions ...................................................................................................... 10
Special Cases ......................................................................................................................... 13
Sample Problems ....................................................................................................................... 13
Solutions to Problems ................................................................................................................ 15
References ...................................................................................................................................... 17
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KINETICS OF DRUGS
Introduction For us to be able to predict the shelf-life of a dosage, it is essential to determine the
kinetics of the breakdown of the drug under carefully controlled conditions. Drug kinetics
is essential because it determines the following:
Stability of Drugs (Half-life or t1/2)
Shelf Life (t0.90)
It is even used in the determination of the drugβs expiration date. But the discussion will
only focus on the basic principles of Drug Kinetics.
Drug kinetics Drug kinetics is simply defined as how drug changes with time. Many drugs are not
chemically stable and the principles of chemical kinetics are used to predict the time
span for which a drug (pure or formulation) will maintain its therapeutic effectiveness or
efficacy at a specified temperature.
Rate, Order, and Molecularity The underlying principle on which all of the science of kinetics is built is the law of mass
action (Cairns, 2008). The law of mass action states that, the rate of a reaction is
proportional to the molar concentrations of the reactants each raised to power equal to
the number of molecules undergoing reaction.
ππ΄ + ππ΅ β πππππ’ππ‘π
π ππ‘π = βπ [π΄]π[π΅]π
The Rate of a chemical reaction is the speed of the reaction or simply, how fast the
reaction occurs. It is, in a dilute solution, proportional to the concentrations of the various
reactants each raised to the power of the number of moles of the reactant in the
balanced chemical equation.
Reactions are classified according to number of reacting species whose concentration
determines the rate at which the reaction occurs (Florence and Attwood, 2006). The
Order of the reaction (Overall Order) is the sum of the powers to which these
concentrations are raised. The individual order of the reactant is simply the exponent of
the reactant.
πππππ ππ π‘βπ πππππ‘πππ = ππ’π ππ ππ₯ππππππ‘π
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πππππ ππ π΄ = π
πππππ ππ π΅ = π
ππ£πππππ πππππ = π + π
In practice, the rate of a chemical reaction depends only on a small number of
concentration terms, and the sum of the powers to which these concentrations are raised.
This is because chemical reactions occur in a number of steps, or stages (called a
mechanism) and the rate of the overall reaction is often governed by the rate of the
slowest step (called, not surprisingly, the rate-determining step). Even if every other stage
of a chemical reaction occurs essentially instantaneously, the rate of the reaction as a
whole cannot exceed that of the slowest stage. (Cairns, 2008)
The order of a chemical reaction cannot be predicted from the chemical equation, even
if it has been balanced. The order of a reaction is determined experimentally from
accurate measurements of the rate under different conditions (Cairns, 2008). Typically,
drugs react in the zero, first, and second order. There are also rare cases wherein the
drugs undergo a third order reaction, or even a fractional order of reaction.
The Molecularity is the total number of molecules taking part in the slowest of the
elementary reaction steps. In most chemical reactions, two molecules collide and react;
the molecularity is 2 and the reaction is said to be bimolecular.
e.g.: π΄ + π΅ β πππππ’ππ‘π
Reactions in which only one molecule is involved (unimolecular) are known, but usually
occur in the gas phase.
e.g.: π΄ β πππππ’ππ‘π
Reactions with a molecularity higher than 2 are very rare, since this would require three
or more reactants all encountering each other at the same time.
Zero-Order, First-Order, and Second-Order Reactions and
Their Rate Equations, Half-Life and Shelf-Life Taking an example of a reaction:
π΄ β πππππ’ππ‘
π ππ‘π = βπ [π΄]
Rate can be rewritten as:
ππ₯
ππ‘= βπ[π΄]
Differential rate equations like these are not much use to the practicing chemist, so it is
usual to integrate the differential form of the rate equation, shown above, to obtain more
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useful expressions (Cairn, 2008). The next discussion will be all about the derivation of the
rate reactions for zero, first, and second order reactions. For the following derivations, we
let:
πͺπ = πΆππππππ‘πππ‘πππ ππ π΄
πͺππ= πΌπππ‘πππ πΆππππππ‘πππ‘πππ ππ π΄ (@π‘ = 0)
π = π ππ‘π πΆπππ π‘πππ‘
π‘ = π‘πππ
π‘1/2 = π»πππ β ππππ
π‘0.9 = πβπππ β ππππ
The shelf life of a pharmaceutical product is the length of time the product may safely
be stored on the dispensary shelf before significant decomposition occurs. This is
important since, at best, drugs may decompose to inactive products; in the worst case
the decomposition may yield toxic compounds. The shelf-life is often taken to be the time
for decomposition of 10% of the active drug to occur, leaving 90% of the activity.
The unit for the rate constant (k) can be determined using the formula:
(π‘πππ)β1(πππππππ‘πππ‘πππ)1βπ
Zero-Order Reaction There are some reactions in which the rate of the reaction is independent of the
concentration of the reactants but does depend on some other factor, such as the
amount of catalyst present. These reactions are termed zero-order reactions, and rate
equations can be derived as follows:
π[π΄]
ππ‘= βπ[π΄]0
β« π[π΄] = β β« π ππ‘
β« π[π΄]πͺπ
πͺππ
= βπ β« ππ‘π‘
0
πͺπ β πͺππ= βππ or πͺππ
β πͺπ = ππ
We can also rewrite the equation in terms of the mole fraction of A (Xa).
πͺππ πΏπ = ππ
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Zero-order process takes place at a constant rate independent of the existing
concentration or initial concentration.
Example:
A patient was given 100 mg drug A orally. Assume that the drug absorption follows zero-
order kinetics at a rate of 10 mg/min.
In the example, 10 mg of drug will undergo absorption for every minute. It does not matter
even if the initial concentration administered to the patient is higher or lower than 100
mg. The same rate (10 mg/min) will be followed. Eventually, the process will come to an
end when the amount of drug administered is absorbed by the body completely (for the
example, the process ends in 10 minutes.)
In zero-order reactions the amount of product formed varies with time so that the amount
of product formed after 20 minutes will be twice that formed after 10 minutes. Reactions
that follow zero-order kinetics are quite rare, but they do occur in solid-phase reactions
such as release of drug from a pharmaceutical suspension.
A plot of the amount remaining (as ordinate) against time (as abscissa) is linear with a
slope of k0. Many decomposition reactions in the solid phase or in suspensions apparently
follow zero-order kinetics. (Florence and Attwood, 2006)
FIGURE 1 PLOT OF T VS CA
FIGURE 2 HYDROLYSIS OF A SUSPENSION OF
ACETYLSALICYLIC ACID AT 34Β°C.
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Drugs that follow Zero-order kinetics
Phenytoin
Phenylbutazone
Warfarin
Heparin
Ethanol
Aspirin
Theophylline
Tolbutamide
Salicylates
Half-life for a Zero-Order Reaction
The half-life of a reaction can be derived by letting πΆπ=0.5πΆππ and π‘ = π‘1/2.
πΆππβ πΆπ = ππ‘
πΆππβ 0.5πΆππ
= ππ‘1/2
0.5πΆππ= ππ‘1/2
ππ/π =π. ππͺππ
π
Note that for the half-life of a zero-order reaction, rate constant (k) and half-life (t1/2)
depend on the initial concentration of the reactant (πΆππ).
FIGURE 3 SAMPLE PLOT FOR THE HALF-LIFE OF A ZERO-ORDER REACTION
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Shelf-life for a Zero-Order Reaction
The shelf-life of a reaction can be derived by letting πΆπ=0.9πΆππ and π‘ = π‘0.90.
πΆππβ πΆπ = ππ‘
πΆππβ 0.9πΆππ
= ππ‘0.90
0.1πΆππ= ππ‘0.90
ππ.ππ =π. ππͺππ
π
Same as the half-life, rate constant (k) and shelf-life (t0.90) depend on the initial
concentration of the reactant (πΆππ).
First-Order Reaction This type is the most common in the pharmaceutical industry (e.g.: drug absorption and
drug degradation). The rate of first-order reactions is determined by one concentration
term and may be derived as follows:
π[π΄]
ππ‘= βπ[π΄]
β«π[π΄]
[π΄]= β β« πππ‘
β«π[π΄]
[π΄]
πΆπ
πΆππ
= βπ β« ππ‘π‘
0
ln πΆπ β ln πΆππ= βππ‘
β π₯π§πͺπ
πͺππ
= ππ
In terms of the mole fraction of A (Xa), the equation becomes:
β π₯π§(π β πΏπ) = ππ
First-order process takes place at a constant proportion of the drug concentration
available at that time so the process is depending on the initial concentration.
Example:
A patient was given 100 mg of drug B orally and it was assumed to be following first-order
kinetics, a proportion of 10% per minute of the existing concentration at that time.
For this example, the dose given to the patient is again 100 mg but the proportion being
absorbed is 10% per minute. This means that in the first minute, 10% of the initial drug will
be absorbed (10 mg in the example). For the second minute, again 10% of the drug that
remained for absorption will be absorbed (10% of 90 mg = 9 mg, meaning, after the
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second minute, only 81 mg of the reactant will remain). And this process goes on. This
means that the first-order process never comes to an end.
If a plot of equation is made, with t on the horizontal axis and βlnπΆπ
πΆππ
on the vertical axis,
a straight line passing through the origin will be obtained for a reaction obeying first-order
kinetics. The slope of this straight line will be equal to k, the rate constant for the reaction.
FIGURE 4 PLOT OF T VS LN(CA/CAO)
FIGURE 5 FIRST-ORDER PLOT FOR HYDROLYSIS OF HOMATROPINE IN
HYDROCHLORIC ACID (0.226 MOL DM-03) AT 90Β°C.
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As can be seen in the figure, the main difference between a first-order and a zero-order
reaction is that in zero-order reaction, the process will end eventually while in a first-order
reaction, the process does not end.
Half-life for a First-Order Reaction
The half-life of a reaction can be derived by letting πΆπ=0.5πΆππ and π‘ = π‘1/2.
β lnπΆπ
πΆππ
= ππ‘
β ln0.5πΆππ
πΆππ
= ππ‘1/2
β ln 0.5 = ππ‘1/2
ππ/π = βπ₯π§ π. π
π
Note that for the half-life of a first-order reaction, rate constant (k) and half-life (t1/2) does
not depend on the initial concentration of the reactant (πΆππ).
FIGURE 6 GRAPHS SHOWING THE DIFFERENCE OF FIRST AND ZERO-ORDER KINETICS IN TERMS
OF ELIMINATION OF DRUGS
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Shelf-life for a First-Order Reaction
The shelf-life of a reaction can be derived by letting πΆπ=0.9πΆππ and π‘ = π‘0.90.
β lnπΆπ
πΆππ
= ππ‘
β ln0.9πΆππ
πΆππ
= ππ‘0.90
β ln 0.9 = ππ‘0.90
π‘0.90 = βln 0.9
π
Same as the half-life, rate constant (k) and shelf-life (t0.90) does not depend on the initial
concentration of the reactant (πΆππ).
Second-Order Reactions For reactions of the type:
2π΄ β πππππ’ππ‘π or π΄ + π΅ β πππππ’ππ‘π
The rate of the reaction will be first order with respect to each reactant and hence
second order overall. The rate of a second-order reaction is determined by the
concentrations of two reacting species. The rate equation for a second-order reaction is
derived as follows:
FIGURE 7 SAMPLE PLOT FOR THE HALF-LIFE OF A FIRST-ORDER REACTION
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π[π΄]
ππ‘= βπ[π΄]2
β«π[π΄]
[π΄]2= β β« πππ‘
β«π[π΄]
[π΄]2
πΆπ
πΆππ
= βπ β« ππ‘π‘
0
π
πͺπβ
π
πͺππ
= ππ
In terms of the mole fraction of A (Xa), the equation becomes:
π
πͺππ
πΏπ
π β πΏπ= ππ
The equations derived are only valid for second-order reactions in which the
concentrations of the reactants are equal. In most cases it is possible to arrange for the
concentrations of the reactants to be equal and the equations may be used.
The equation derived is a straight line of the type y = mx + b, so that a plot of 1/Ca against
t yields a straight line of slope k, with an intercept on the vertical axis of 1/Cao.
Half-life for a Second-Order Reaction
The half-life of a reaction can be derived by letting πΆπ=0.5πΆππ and π‘ = π‘1/2.
1
πΆπβ
1
πΆππ
= ππ‘
1
0.5πΆππ
β1
πΆππ
= ππ‘1/2
πΆππβ 0.5πΆππ
0.5πΆππ2 = ππ‘1/2
FIGURE 8 PLOT OF T VS 1/CA
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1
πΆππ
= ππ‘1/2
ππ/π =π
πͺπππ
The half-life for a second-order reaction is inversely proportional to the initial
concentration.
Shelf-life for a Second-Order Reaction
The shelf-life of a reaction can be derived by letting πΆπ=0.9πΆππ and π‘ = π‘0.90.
1
πΆπβ
1
πΆππ
= ππ‘
1
0.9πΆππ
β1
πΆππ
= ππ‘0.90
πΆππβ 0.9πΆππ
0.9πΆππ2 = ππ‘0.90
FIGURE 9 SAMPLE PLOT FOR THE HALF-LIFE OF A SECOND-ORDER REACTION
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0.1
0.9πΆππ
= ππ‘0.90
π0.90 =π. π
π. ππͺπππ
The shelf-life for a second-order reaction is inversely proportional to the initial
concentration.
Special Cases
Apparent Zero-Order of Reaction
In aqueous suspensions of drugs, as the dissolved drug decomposes more drug dissolve
to maintain drug concentration i.e. drug concentration kept constant, once all
undissolved drug is dissolved, rate becomes first order.
Pseudo First-Order
In some second-order reactions the concentration of one of the reactants is many times
more than the concentration of the other, too large in fact as to be considered constant
throughout the reaction. In these cases, the reaction appears to follow first-order kinetics,
even though, strictly speaking, it is still a second-order process.
The majority of decomposition reactions involving drugs fall into this category, either
because the species reacting with the drug is maintained constant by buffering or
because, as in the case of uncatalysed hydrolysis reactions, the water is in such large
excess that any change in its concentration is negligible.
Example:
In a Hydrolysis Reaction catalyzed by [π»+]:
π[π΄]
ππ‘= π[π΄][π»+]
When the solution is buffered at constant pH, [π»+] is constant and we can write the
equation as:
π[π΄]
ππ‘= π[π΄]
Sample Problems 1. A solution of a drug was freshly prepared at a concentration of 300 mg/ml, after
30 days, the drug concentration in the solution was 75 mg/ml.
a. Assuming First-order kinetics. What is the rate constant? When will the drug decline
to one-half of the original concentration?
b. Assuming Zero-order kinetics. What is the rate constant? When will the drug decline
to one-half of the original concentration?
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c. Assuming Second-order kinetics. What is the rate constant? When will the drug
decline to one-half of the original concentration?
2. If the half-life for decomposition of a drug is 12 hr., how long will it take for 125 mg
of the drug to decompose 30%? Assume First-order kinetics and constant
temperature.
3. A pharmacist dissolved a few milligrams of a new antibiotic drug into exactly 100
ml of distilled water and placed the solution in a refrigerator (5 CΒ°). At various time
intervals, the pharmacist removed a 10 ml aliquot from the solution and measured
the amount of drug contained in each aliquot.
a. What is the order of the decomposition process of this antibiotic?
b. What is the rate of decomposition of this antibiotic?
c. How many milligrams of antibiotic were in the original solution prepared by the
pharmacist?
4. Determine the first-order rate constant for the hydrolysis of acetyl-Ξ²-methylcholine
at 85ΛC from the information given below.
5. Hydrogen peroxide solutions are normally stable, but when metal ions are added,
hydrogen peroxide decomposes:
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2π»2π2 β 2π»2π + π2
In a solution containing FeCl3, the concentration of H2O2 varied as follows:
Using these data, determine the order of the reaction with respect to peroxide, and the
value of the rate constant (include appropriate units).
Solutions to Problems 1. πΆπ = 75 ππ/ππ
πΆππ= 300 ππ/ππ
π‘ = 30 πππ¦π
a. β πππΆπ
πΆππ
= ππ‘
β ππ75
300= π(30 πππ¦π )
π = π. ππ Γ ππβπ /π ππ
π‘1/2 = βππ 0.5
π
π‘1/2 = βππ 0.5
4.62 Γ 10β2/πππ¦
ππ/π = ππ π πππ
b. πΆπ β πΆππ= βππ‘
(75 β 300)ππ/ππ = βπ(30 πππ¦π )
π = π. π ππ/(ππ β π ππ)
π‘1/2 =0.5πΆππ
π
π‘1/2 =0.5(300
ππππ
)
7.5 ππ/(ππ β πππ¦)
ππ/π = ππ π πππ
c. 1
πΆπβ
1
πΆππ
= ππ‘
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1
75ππππ
β1
300ππππ
= π(30 πππ¦π )
π = π. ππ Γ ππβπ (π ππ)βπ (ππ
ππ)
βπ
π‘1/2 =1
πΆπππ
π‘1/2 =1
300ππππ
(3.33 Γ 10β4 (πππ¦)β1 (ππππ
)β1
ππ/π = ππ π πππ
2. π‘1/2 = 12 βππ
π‘1/2 = βππ 0.5
π
12 βππ = βππ 0.5
π
π = π. ππ/ππ
β ππ(1 β ππ) = ππ‘
β ππ(1 β 0.30) =0.06
βππ‘
π = π. ππ πππ
3. a. The plot of t vs Ca formed a straight line, therefore the reaction is at zero-
order.
b. The slope of the graph (-k) is -6.55. Therefore, rate constant (k) = 6.55 (mg/ml)
hr
c. The intercept of the graph is 87.30. Therefore, Cao= 87.30 mg/ml
4. The slope (k) of the graph is 0.35. Therefore, rate constant (k) = 0.35/day
5. The plot of t vs ln(Ca/Cao) formed a straight line, therefore the reaction order is
first-order. The slope (k) of the graph is 3.99 x 10-3. The rate constant (k) is 3.99 x
10-3.
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References Attwood, D. and Florence, A. (2006). Physicochemical Principles of Pharmacy, 4th ed.
Retrieved from: http://www.scribd.com
Cairns, D. (2008). Essentials of Pharmaceutical Chemistry, 3rd ed. Retrieved from:
http://www.4shared.com/office/_wd4P3B4/Essentials_of_Pharmaceutical_C.htm
Differences between Zero-order kinetics and First-order kinetics. Retrieved from:
http://www.pharmainfo.net/og/rcp/downloads
Drug Stability and Kinetics. Retrieved from: http://www.scribd.com
Drugs following zero order pharmacokinetics.
http://www.lifehugger.com/moc/84/Drugs_following_zero_order_pharmacokinetics
First and zero order kinetics. Retrieved from:
https://sites.google.com/site/pharmacologyinonesemester/2-drug-distribution-
metabolism-and-elimination/2-5-blood-levels/2-5-3-first-and-zero-order-kinetics
Half-lives. Retrieved from:
http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Reaction_Rates/Half-
lives_and_Pharmacokinetics
Levenspiel, O. Chemical Reaction Engineering, 3rd ed. Retrieved from:
http://www.scribd.com