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Aims and learning outcomes
Pharmacokinetics is a fundamental scientific discipline that underpins applied therapeutics. Patients need to be prescribed appropriate medicines for a clinical condition. The medicine is chosen on the basis of an evidence- based approach to clinical practice and assured to be compatible with any other medicines or alternative therapies the patient may be taking. The design of a dosage regimen is dependent on a basic under- standing of the drug use process (DUP). When faced with a patient who shows specific clinical signs and symptoms, pharmacists must always ask a fundamental question: ‘Is this patient suffering from a drug-related problem?’ Once this issue is evaluated and a clinical diagnosis is avail- able, the pharmacist can apply the DUP to ensure that the patient is prescribed an appropriate medication regimen, that the patient under- stands the therapy prescribed, and that an agreed concordance plan is achieved. Pharmacists using the DUP consider:
● (^) Need for a drug ● (^) Choice of a drug ● (^) Goals of therapy ● (^) Design of regimen
- Route
- Dose and frequency
- Duration ● (^) Monitoring and review ● (^) Counselling
Once a particular medicine is chosen, the principles of clinical pharmaco- kinetics are required to ensure the appropriate formulation of drug is chosen for an appropriate route of administration. On the basis of the patient’s drug handling parameters, which require an understanding of
Basic pharmacokinetics
Soraya Dhillon and Kiren Gill
absorption, distribution, metabolism and excretion, the dosage regimen for the medicine in a particular patient can be developed. The pharmacist will then need to ensure that the appropriate regimen is prescribed to achieve optimal efficacy and minimal toxicity. Pharmacists then ensure that the appropriate monitoring is undertaken and that the patient receives the appropriate information to ensure compliance. Clinical pharmaco- kinetics is thus a fundamental knowledge base that pharmacists require to ensure effective practice of pharmaceutical care. The aim of this chapter is to provide the practising clinical pharma- cist with the appropriate knowledge and skills of applied clinical phar- macokinetics, which can be applied in everyday practice. The objectives for this chapter are to enable the reader to:
● (^) State the rationale for using therapeutic drug monitoring (TDM) to optimise drug therapy. ● (^) Identify drugs that should be routinely monitored. ● (^) Define first-order and zero-order kinetics. ● (^) Apply one-compartment pharmacokinetics to single and multiple dosing following the intravenous and oral administration of drugs. ● (^) Apply the basic principles of interpretation of serum drug concen- trations in practice. ● (^) Apply one-compartment pharmacokinetics to describe steady-state serum drug concentrations following oral slow-release dosing. ● (^) Use the method of iteration to derive individualised pharmaco- kinetic parameters from serum drug concentration data. ● (^) Apply nonlinear pharmacokinetics to describe steady-state plasma concentrations following parenteral and/or oral phenytoin therapy.
Introduction
Pharmacokinetics provides a mathematical basis to assess the time course of drugs and their effects in the body. It enables the following processes to be quantified:
A bsorption D istribution M etabolism E xcretion
These pharmacokinetic processes, often referred to as ADME, determine the drug concentration in the body when medicines are prescribed. A fundamental understanding of these parameters is required to design an
2 Basic pharmacokinetics
The rate of a reaction or process is defined as the velocity at which it proceeds and can be described as either zero-order or first-order.
Zero-order reaction
Consider the rate of elimination of drug A from the body. If the amount of the drug, A , is decreasing at a constant rate, then the rate of elimin- ation of A can be described as:
where k * the zero-order rate constant. The reaction proceeds at a constant rate and is independent of the concentration of A present in the body. An example is the elimination of alcohol. Drugs that show this type of elimination will show accumula- tion of plasma levels of the drug and hence nonlinear pharmacokinetics.
First-order reaction
If the amount of drug A is decreasing at a rate that is proportional to A , the amount of drug A remaining in the body, then the rate of elimination of drug A can be described as:
where k the first-order rate constant. The reaction proceeds at a rate that is dependent on the concentration of A present in the body. It is assumed that the processes of ADME fol- low first-order reactions and most drugs are eliminated in this manner. Most drugs used in clinical practice at therapeutic dosages will show first-order rate processes; that is, the rate of elimination of most drugs will be first-order. However, there are notable exceptions, for example phenytoin and high-dose salicylates. In essence, for drugs that show a first-order elimination process one can show that, as the amount of drug administered increases, the body is able to eliminate the drug accordingly and accumulation will not occur. If you double the dose you will double the plasma concentration. However, if you continue to increase the amount of drug administered then all drugs will change from showing a first-order process to a zero-order process, for example in an overdose situation.
d d
A
t
kA
d d
A *
t
k
4 Basic pharmacokinetics
Pharmacokinetic models
Pharmacokinetic models are hypothetical structures that are used to describe the fate of a drug in a biological system following its administration.
One-compartment model
Following drug administration, the body is depicted as a kinetically homo- geneous unit (see Figure 1.1). This assumes that the drug achieves instant- aneous distribution throughout the body and that the drug equilibrates instantaneously between tissues. Thus the drug concentration–time profile shows a monophasic response (i.e. it is monoexponential; Figure 1.2a). It is important to note that this does not imply that the drug concentration in plasma ( C p) is equal to the drug concentration in the tissues. However, changes in the plasma concentration quantitatively reflect changes in the tissues. The relationship described in Figure 1.2a can be plotted on a log C p vs time graph (Figure 1.2b) and will then show a linear relation; this represents a one-compartment model.
Two-compartment model
The two-compartment model resolves the body into a central compart- ment and a peripheral compartment (see Figure 1.3). Although these com- partments have no physiological or anatomical meaning, it is assumed that the central compartment comprises tissues that are highly perfused such as heart, lungs, kidneys, liver and brain. The peripheral compart- ment comprises less well-perfused tissues such as muscle, fat and skin. A two-compartment model assumes that, following drug adminis- tration into the central compartment, the drug distributes between that compartment and the peripheral compartment. However, the drug does not achieve instantaneous distribution, i.e. equilibration, between the two compartments. The drug concentration–time profile shows a curve (Figure 1.4a), but the log drug concentration–time plot shows a biphasic response
Pharmacokinetic models 5
Single component
k a k
Figure 1.1 One-compartment model. ka absorption rate constant (h^1 ), k elimination rate constant (h^1 ).
(Figure 1.4b) and can be used to distinguish whether a drug shows a one- or two-compartment model. Figure 1.4b shows a profile in which initially there is a rapid decline in the drug concentration owing to elimination from the central compart- ment and distribution to the peripheral compartment. Hence during this rapid initial phase the drug concentration will decline rapidly from the central compartment, rise to a maximum in the peripheral compartment, and then decline. After a time interval ( t ), a distribution equilibrium is achieved between the central and peripheral compartments, and elimination of the drug is assumed to occur from the central compartment. As with the one- compartment model, all the rate processes are described by first-order reactions.
Pharmacokinetic models 7
C (^) p
(a) Time
logC (^) p
(b) Time
Figure 1.4 (a) Plasma concentration versus time profile of a drug showing a two- compartment model. (b) Time profile of a two-compartment model showing logCp versus time.
Multicompartment model
In this model the drug distributes into more than one compartment and the concentration–time profile shows more than one exponential (Figure 1.5a). Each exponential on the concentration–time profile describes a compart- ment. For example, gentamicin can be described by a three-compartment model following a single IV dose (see Figure 1.5b).
Pharmacokinetic parameters
This section describes various applications using the one-compartment open model system.
8 Basic pharmacokinetics
Cp
(a) Time
log Cp
(b) Time
Figure 1.5 (a) Plasma concentration versus time profile of a drug showing multicompartment model. (b) Time profile of a multicompartment model showing logCp versus time.
and V d can be used to convert drug amount X to concentration. Since
X X 0 exp( kt )
then
Thus
C p t C p^0 exp( kt )
This formula describes a monoexponential decay (see Figure 1.2), where C p t plasma concentration at any time t. The curve can be converted to a linear form (Figure 1.6) using natural logarithms (ln):
ln C p t ln Cp^0 kt
where the slope k , the elimination rate constant; and the y intercept ln C p^0. Since
then at zero concentration ( C p^0 ), the amount administered is the dose, D , so that
If the drug has a large V d that does not equate to a real volume, e.g. total plasma volume, this suggests that the drug is highly distributed in tis- sues. On the other hand, if the V d is similar to the total plasma volume this will suggest that the total amount of drug is poorly distributed and is mainly in the plasma.
Half-life
The time required to reduce the plasma concentration to one half its initial value is defined as the half-life ( t 1/2 ). Consider
ln C p t ln C p^0 kt
C D p V 0 d
V X
d C p
X V
kt d V d
X 0 exp(^ )
10 Basic pharmacokinetics
Let C p^0 decay to C p^0 /2 and solve for t t 1/2 :
ln( C p^0 /2) ln C p^0 kt 1/
Hence
kt 1/2 ln C p^0 ln( C p^0 /2)
and
This parameter is very useful for estimating how long it will take for levels to be reduced by half the original concentration. It can be used to estimate for how long a drug should be stopped if a patient has toxic drug levels, assuming the drug shows linear one-compartment pharmacokinetics.
Clearance
Drug clearance (CL) is defined as the volume of plasma in the vascular compartment cleared of drug per unit time by the processes of metab- olism and excretion. Clearance for a drug is constant if the drug is eliminated by first-order kinetics. Drug can be cleared by renal excretion or by metabolism or both. With respect to the kidney and liver, etc., clearances are additive, that is:
CLtotal CLrenal CLnonrenal
t k
t k
1 2
1 2
2
0 693
/
/
(ln )
.
Pharmacokinetic parameters 11
ln Cp
Time
*Concentration at time 0
*
Figure 1.6 LnCp versus time profile.
Hence
t 4 days
Multiple doses
Some drugs may be used clinically on a single-dose basis, although most drugs are administered continually over a period of time. When a drug is administered at a regular dosing interval (orally or IV), the drug accumu- lates in the body and the serum concentration will rise until steady-state conditions have been reached, assuming the drug is administered again before all of the previous dose has been eliminated (see Figure 1.7).
Steady state
Steady state occurs when the amount of drug administered (in a given time period) is equal to the amount of drug eliminated in that same period. At steady state the plasma concentrations of the drug ( C ssp ) at any time during any dosing interval, as well as the peak and trough, are similar. The time to reach steady-state concentrations is dependent on the half-life of the drug under consideration.
Effect of dose
The higher the dose, the higher the steady-state levels, but the time to achieve steady-state levels is independent of dose (see Figure 1.8). Note that the fluctuations in C p max and C p min are greatest with higher doses.
Pharmacokinetic applications 13
Time
Cp
Figure 1.7 Time profile of multiple IV doses.
Effect of dosing interval
Consider a drug having a half-life of 3 h. When the dosing interval, , is less than the half-life, t 1/2 , greater accumulation occurs, i.e. higher steady-state levels are higher and there is less fluctuation in C p max and C p min (see Figure 1.9, curve A). When t 1/2, then a lower accumulation occurs with greater fluctuation in C p max and C p min (see Figure 1.9, curve C). If the dosing interval is much greater than the half-life of the drug, then C p min approaches zero. Under these conditions no accumulation will occur and the plasma concentration–time profile will be the result of administration of a series of single doses.
Time to reach steady state
For a drug with one-compartment characteristics, the time to reach steady state is independent of the dose, the number of doses administered, and the dosing interval, but it is directly proportional to the half-life.
Prior to steady state
As an example, estimate the plasma concentration 12 h after therapy commences with drug A given 500 mg three times a day.
14 Basic pharmacokinetics
Time
A = 0.75 g
B = 0.5 g
C = 0.35 g
12
10
8
6
4
2
0
Cp
Figure 1.8 Time profiles of multiple IV doses – reaching steady state using different doses.
Thus, total C p t at 12 h is
C p t C p^0 exp( k 12) C p^0 exp( k 4)
Remember that C p^0 D / V d. This method uses the principle of superposition. The following equation can be used to simplify the process of calculating the value of C p at any time t after the n th dose:
where n number of doses, dosing interval and t time after the n th dose.
At steady state
To describe the plasma concentration ( C p) at any time ( t ) within a dosing interval ( ) at steady state (see Figure 1.11):
Remember that C p^0 D / V d. Alternatively, for some drugs it is important to consider the salt factor ( S ). Hence, if applicable, C p^0 SD / V d and:
C
S D kt p t V (^) d k
[exp ] [1 exp )]
( ) (
C D^ kt p t V (^) d k
[exp ] [1 exp )]
^
( ) (
C D^ kn^ kt p t V d k
exp [1 exp ]
^ ^ ^
[ ( ) [exp( )] ( )
16 Basic pharmacokinetics
Time
C p
C p max
C p min
Figure 1.11 Time profile at steady state and the maximum and minimum plasma concentration within a dosage interval.
To describe the maximum plasma concentration at steady state (i.e. t 0 and exp( kt ) 1):
To describe the minimum plasma concentration at steady state (i.e. t ):
To describe the average steady-state concentration, C ssp (this nota- tion will be used throughout the book):
Since
then
Steady state from first principles At steady state the rate of drug administration is equal to the rate of drug elimination. Mathematically the rate of drug administration can be stated in terms of the dose ( D ) and dosing interval ( ). It is always important to include the salt factor ( S ) and the bioavailability ( F ). The rate of drug elim- ination will be the clearance of the plasma concentration at steady state:
Rate of drug elimination CL C ssp
At steady state:
S F D (^) CL C p
ss
Rate of drug administration
S F D
C
D t p V
ss 1/ d
t
V
1
/
d CL
C
D
C
S D
pss^ ^ CL or^ pss ^ CL
C D^ k p min V (^) d k
[exp )] [1 exp )]
^
( (
C D p max V (^) d [1 exp k )]
1 (
Pharmacokinetic applications 17
Following a continuous infusion, the plasma concentrations will increase with time until the rate of elimination (rate out) equals the rate of infusion (rate in) and will then remain constant. The plateau concen- tration, i.e. C ssp, is the steady-state concentration. Steady state will be achieved in 4–5 times the t 1/2. If one considers the previous equation, which describes the plasma concentration during the infusion prior to steady state, then at steady state,
exp( kt ) 0
As rate in rate out at steady state,
where R D / infusion rate (dose/h). When a constant infusion is stopped, the drug concentrations in the plasma decline in an exponential manner, as illustrated in Figure 1.13. To estimate the plasma concentration, C (^) p at t one must describe the decay of C ssp at time t to C (^) p at time t. Thus, from the above:
To describe the decay of C p from t to t , one uses the single-dose IV bolus equation
C p t C p^0 [exp( kt )]
C
D
p
ss CL
C (^) pss D CL
R CL C pss
Pharmacokinetic applications 19
Cp Cp
Time
ss
Figure 1.12 Time profile after IV infusion.
Since C p^0 is C ssp, then from the above,
Loading dose
The time required to obtain steady-state plasma levels by IV infusion will be long if a drug has a long half-life. It is, therefore, useful in such cases to administer an intravenous loading dose to attain the desired drug concentration immediately and then attempt to maintain this con- centration by a continuous infusion. To estimate the loading dose (LD), where C ssp is the final desired concentration, use
If the patient has already received the drug, then the loading dose should be adjusted accordingly:
or
if the salt of the drug (salt factor S ) is used.
LD d^ p
ss p
initial
V C C S
( )
LD V d (^) ( C (^) pss^ C pinitial)
LD V d (^) C pss
C
D k t t p
exp[ ( CL
)]
20 Basic pharmacokinetics
Cp X
Cp
Time
Cpss
Figure 1.13 Profile following discontinuation of an infusion.
