Tratamiento ofrecido a los temas en los debates

handles a drug from absorption, distribution and metabolism to excretion (ADME). The inter-action of these processes determines both the plasma concentration and how long a drug persists in the blood. Clinical pharmacokinetics is the application of this area of pharmacology and is used to construct dosage regimens and, in the context of therapeutics, it takes account of the variations between individual patients.

Basic pharmacokinetic principles

Absorption and bioavailability

Drugs are administered by various routes, includ-ing orally, intravenously, intramuscularly, sub-cutaneously, by inhalation and rectally. Clearly, the oral route is the most convenient and pre-ferred by patients but it can be less predictable compared with the intravenous route, because absorption can be affected by the presence of food in the stomach, acidic pH, gastric emptying and ‘first-pass’ metabolism. This last complica-tion is due to absorbed drugs passing straight to the liver where they may be extensively metabol-ized and their entrance to the systemic circu-lation is limited. Common examples of drugs undergoing first-pass metabolism include mor-phine, propranolol and glyceryl trinitrate (GTN);

indeed GTN is given as a sublingual spray to overcome this problem. Interestingly, in the case of statins, first-pass metabolism is an advantage because it limits the actions of statins to the liver (see Chapter 12).

The above processes limit the amount of drug administered reaching the systemic circulation.

To account for this, the bioavailability of the drug is considered. This is the fraction of drug administered that reaches the systemic circu-lation, i.e. the fraction absorbed. Bioavailability (F) is calculated from the ratio of the area under the curve (AUC) of the oral dose to the AUC of an intravenous dose:

AUC oral F = –––––––––

AUC i.v.

For example, the F for digoxin tablets is 0.70 (i.e. 70% is absorbed) and so if 250 micrograms are given orally then 70% of the dose (0.70 ⫻ 250

= 175 micrograms) enters the circulation. This can then be applied to determine the amount of a drug required. So:

Amount needed Dose given = –––––––––––––––

F

Intravenous

Oral Time

Log plasma concentration

Figure 6.1 A comparison of the log plasma concentra-tion–time plot for a drug via the intravenous and the oral routes. The oral route shows a rising phase as the drug is absorbed. As the drug has a bioavailability <1, the area under the curve (AUC) under the oral plot is proportionally less than under the intravenous plot.

Salt factor

In addition to bioavailability, some medicines are made up of active drug and inert salts. The best example is aminophylline, which is the salt of theophylline, so 80% of the amount of amino-phylline is theoamino-phylline. This fraction is termed the ‘salt factor’ or S. So:

Amount = S ⫻ Dose

Amount of drug administered

To take account of F and S the following is applied:

Amount = Dose ⫻ F ⫻ S

Volume of distribution

The volume of distribution (V_d) of a drug is the apparent volume in which a drug is dissolved in the body to give the plasma concentration measured:

Amount in body Dose Vd= –––––––––––––––– = ––––––––––––––

Concentration Concentration at t = 0 where t is time.

The volume of distribution and (therefore) plasma concentration are determined by where the drug goes in the bodily fluids. A widely dis-tributed drug (e.g. one that moves into fat or other tissues) or one that has extensive protein binding has a high V_d and one that is largely retained in the plasma or has low binding has a low value. A commonly encountered drug with a high Vd is digoxin and this reflects wide tissue distribution. In the case of digoxin, a population average V_dis 7.3 L/kg body mass, so for a 70-kg person this would equate to a V_dof 511 L. This value clearly greatly exceeds body volume and is due to the plasma concentration being lower as a result of wide sequestration in the tissues. The Vdis not merely of scientific interest because it is applied clinically in determining loading doses (see later).

Elimination

Elimination is the removal of the active drug from the body and is largely made up of hepatic metabolism and renal excretion. Generally speaking, it is the sum of these two components, the contributions of which vary greatly. Indeed, knowledge of the routes of elimination is essen-tial for drug choice in renal and hepatic impair-ment, e.g. digoxin is cleared predominantly (about two-thirds) by the renal route and so doses of digoxin are determined in relation to renal function. The dihydropyridine drug amlodipine is cleared via the hepatic route and is therefore a suitable antihypertensive for patients with renal disease because its elimination is not impaired.

Clearance

Clearance (CL) is a measure of the rate of elimi-nation and is largely made up of renal and hepatic clearance. It is defined as the volume of plasma cleared of a drug in unit time and usually expressed in litres per hour or day. The clearance of drug determines the amount of drug required in a maintenance regimen (see later).

Kinetics of elimination

Most drugs illustrate first-order kinetics and the plasma concentration undergoes an exponential decay (analogous to radioactive decay), so there is a logarithmic relationship between plasma concentration and time (Figure 6.2).

Due to first-order kinetics, the rate of elimina-tion is proporelimina-tional to the concentraelimina-tion of drug:

Plasma concentration

Time

Figure 6.2 The plasma concentration–time plot for a drug illustrating exponential decay due to first-order kinetics.

C_t= C₀e^–kt

where C_t is the concentration at time t, C₀ the concentration at t = 0 and k the rate constant (/min or /h).

This equation is applied to predict plasma concentrations when the concentration at time 0 is known; it may also be used to estimate the time to reach a safe concentration from a toxic level, e.g. if the plasma concentration of a drug such as digoxin is too high, it enables the time for with-holding doses to be calculated to reach safe levels.

Rate constant

First-order kinetics involves a rate constant (k) and this is simply the fraction of drug eliminated per unit time, when k is 0.1/day, 10% is elimi-nated per day. This can be applied mathemati-cally as the clearance per unit time is the fraction of V_dthat is cleared per unit time. So:

k = –––CL V_d or

CL = k⫻ Vd

Half-life

This is simply the time for the plasma concen-tration to decrease by 50%. Therefore at the half-life (t1/2):

C_t= 0.5 C₀ So:

0.5 C₀= C₀e^–kt ln 0.5 = –kt –0.693 = –kt 0.693 So t1/2 = –––––

k

Regimens

Pharmacokinetics is applied in the construction of dosing regimens, the purpose of which is to maintain the plasma concentration within the

therapeutic window, which is the desired range between toxicity and sub-therapeutic levels. The aim is also to limit the differences between peaks and troughs (Figure 6.3).

In order to achieve a steady state within the therapeutic window regimens are generally based on:

• once daily

• twice daily

• three times daily, often based around ‘break-fast, lunch and dinner’

• four times daily.

A once-daily interval is favoured as the most convenient and is likely to achieve the best com-pliance. However, the above applies to a stable regimen, and at the start of therapy it will take time to achieve steady-state concentration (C_ss).

A general rule is that it takes 5 half-lives to reach steady state (Figure 6.4).

The explanation to this rule of thumb is provided in Table 6.1, in which the amount

Time

Plasma concentration

Therapeutic window

Figure 6.3 The plasma–concentration–time plot for a drug regimen to illustrate the usual approach of maintaining the peaks and troughs within the therapeutic window, to main-tain clinical effectiveness and prevent toxicity.

Plasma concentration

One

half-life Time

Steady state

Figure 6.4 The plasma–concentration–time plot for a drug at the start of an oral regimen, when it takes approximately 5 half-lives to reach steady-state concentration.

present after each half-life is added cumulatively until the amount approaches 100%.

Loading doses

This rule is clinically significant because drugs with a short half-life will soon reach Css. How-ever, for drugs with long half-lives the time to attain C_ss will be prolonged, e.g. digoxin has a half-life of 40 h and will therefore take 200 h or over 8 days to reach C_ss. Digoxin is widely used to control ventricular rate in atrial fibrillation and the therapeutic effect is required urgently.

Therefore, to overcome the delay, drugs with long half-lives or drugs that require immediate action (e.g. lidocaine in ventricular ectopic beats) are often given as a loading dose to take the plasma concentrations quickly up to a desired level (Figure 6.5).

To calculate a loading dose, the amount required is determined by the desired (safe)

plasma concentration multiplied by the volume of distribution:

Amount required = target concentration ⫻ Vd. The dose required must then take account of F and S as follows:

Amount required Loading dose = ––––––––––––––––

F ⫻ S

Maintenance doses

Many therapies involve a maintenance dose to maintain the C_sswithin the therapeutic window.

The maintenance dose equals the amount of drug eliminated per dosage interval and main-tains the steady state. Therefore the rate of input equals the rate of output. To consider mainten-ance dosing, the pharmacokinetics are analogous to a continuous infusion, which involves an increase in plasma concentration followed by a plateau at the steady state where input equals output (Figure 6.6).

So the maintenance dose equals the amount removed. Therefore, we have the relationship:

Infusion rate = CL ⫻ Css

This can then be applied to the oral dosing and the ‘infusion’ rate equates to the dose per dosage interval (tau or ␶). Therefore for an oral regimen there is the important equation:

[CL ⫻ Css⫻ ␶]

Dose = –––––––––––––

F

For drugs that display first-order kinetics, the relationship between dose and plasma concen-tration is clearly linear, so doubling the dose will Ta b l e 6 . 1 Demonstration that steady state is reached after approximately 5 half-lives

First 50 50

Second 50 + 25 75

Third 50 + 37.5 87.5

Fourth 50 + 43.8 93.8

Fifth 50 + 46.9 96.9

Dose Amount present in the body Percentage of Css

after one t_1/2from dose (%)

Maintenance regimen

Plasma concentration

Time Loading

dose

Figure 6.5 The plasma–concentration–time plot for a drug following a loading dose to raise the plasma concentration to therapeutic levels, before the maintenance regimen.

double the concentration, whereas halving the dose will halve the concentration.

Peak and trough concentrations

The steady-state plasma concentration is the time-averaged concentration i.e. the mean plasma concentration. For drugs with a wide therapeutic window Css is appropriate but, for drugs with a narrow therapeutic window, it is important to consider the maximum (C_max) or peak and minimum (C_min) or trough concentra-tions (Figure 6.7). In many instances C_max and C_min should be maintained within the thera-peutic window, usually by altering the dose or the dosage interval.

The following equations are used to determine peak and trough plasma concentrations (Winter 2004):

F⫻ Dose Peak C_ss= ––––––––––

V_d(1 – e^–k␶) F⫻ Dose Trough Css= ––––––––––– ⫻ e^–k␶

Vd(1 – e^–k␶)

Dosage interval (tau)

As indicated above, the dosage interval deter-mines the peaks and troughs. For drugs with a long half-life the dosage interval is 24 h and the dose required equals the amount eliminated in 24 h. However, for drugs with a relatively short half-life and a narrow therapeutic window, the dosage interval may be determined from the half-life in relation to the magnitude of the thera-peutic window, e.g. for theophylline, the C_maxis 20 mg/L and the C_minis 10 mg/L, so the width of A patient requires digoxin to manage atrial fibrillation. For this patient, his Vd is 511 L, the thera-peutic range is 0.8–2 micrograms/L and the F value for digoxin tablets is 0.7.

Solution: the target plasma concentration is chosen as 1.5 micrograms/L, therefore the amount required is 1.5 micrograms ⫻ 511 = 767 micrograms. To take account of F, 767 is divided by 0.7, so the dose needed is 1.096 mg. To take account of the tablets available (62.5, 125, 250 micrograms), the patient might receive 1062.5 micrograms in divided doses over 24 h. This would give a Cssof 1.46 micrograms/L which is acceptable.

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Plateau: input = output

Plasma concentration

Time Start

infusion

Figure 6.6 The plasma–concentration–time plot for a drug administered by intravenous infusion. At plateau the rate of input equals the rate of output and the steady-state plasma concentration is achieved.

Toxic levels

C_{ss max}

C_{ss min}

Css av

Concentration

Subtherapeutic levels

Figure 6.7 This illustrates the time-averaged steady-state concentration in relation to peak and trough concentrations.

the therapeutic window is 10 mg/L and it would take one half-life for the plasma concentration to drop from Cmax to Cmin. Therefore, the dosage interval chosen is less than one half-life and this will ensure that the plasma concentration remains within the safe area (Figure 6.8).

An exception to this approach is the amino-glycosides, such as gentamicin, where, although the peak plasma concentration is within the therapeutic window, the trough concentration is chosen to be subtherapeutic to avoid toxicity, e.g. the trough concentration for gentamicin is

<2 mg/L and the peak is in the range 5–10 mg/L.

Once again the dosage interval is determined by the half-life, e.g. if the peak plasma concentra-tion after an intravenous dose were 8 mg/L then the time to decrease to 2 mg/L would be two half-lives, so the dosage interval would be chosen to be over two half-lives (Figure 6.9).

Real life clinical pharmacokinetics

In spite of the precise science surrounding pharmacokinetic models, patients do not always conform to mathematical equations and this is a result of individual variations that may alter pharmacokinetics, including:

To take account of individual variation, in terms of renal function and weight there are empirical population-based equations that enable one to calculate a reasonable estimate of volume of distribution and clearance for a drug. The popu-lation-based equations are specific to a drug and are useful for recommending a regimen for a patient, the V_d(Example 2) helping to determine the loading dose and CL (Example 3) the main-tenance dose.

Applications of population-based equations Population-based equations are used to predict pharmacokinetics and may be used to recom-mend dosages, e.g. given the patient below, (Example 2), we now know that his Vdis 415 L.

Therefore to determine his loading dose, we can carry out the following:

Css⫻ 415

Loading dose = 889 ␮g and using available tablets one might recommend the loading dose as 7 ⫻ 125 ␮g tablets and a 62.5 ␮g tablet = 937.5 ␮g.

Therapeutic window

Plasma concentration Time 20 10

Figure 6.8 A plasma concentration–time plot to illustrate the therapeutic window in relation to dosage intervals. In the case of theophylline, the maximum concentration is 20 mg/L and the minimum 10 mg/L. The time to drop from 20 mg/L to 10 mg/L is logically one half-life. Therefore, to maintain a plasma concentration within the therapeutic window, the dosage interval should be less than one half-life.

10

2

Concetration of gentamicin (mg/L)

Time C_min

Cmax

Figure 6.9 The plasma concentration–time plot for the aminoglycoside gentamicin, where the peak (Cmax) is usually between 5 and 10 mg/L for clinical effects, but the trough (Cmin) is subtherapeutic to prevent adverse effects. If the peak is 8 mg/L, a dosage interval of two times the half-life will mean that the trough is 2 mg/L or less.

The Vdfor digoxin is given by the equation (Winter 2004):

Vd (L) = (3.8) (weight in kg) + (3.1) (CLcrin mL/min)

As digoxin is largely eliminated by the renal route, an estimate of renal function, i.e. creatinine clearance (CLcr) is used to estimate CL. This is obtained from the Cockcroft–Gault equation:

[1.23(140 – Age) ⫻ Weight]

CLcr= ––––––––––––––––––––––––––

[Serum creatinine]

In females, 1.04 replaces 1.23.

So for a 70-year-old man, who weighs 70 kg and has a plasma creatinine = 125 ␮mol/L, his Vdfor digoxin is given by:

Vd(L) = (3.8) (weight in kg) + (3.1) (CLcrin mL/min) (Note that, although the CLcris given in mL/min, the calculated Vdis in litres).

Vd(L) = (3.8 ⫻ 70) + 3.1 (CLcr) [1.23(140 – 70) ⫻ 70]

CLcr= ––––––––––––––––––––– = 48.2 mL/min 125

Therefore Vd= 266 + 3.1(48.2) = 415 L.

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For digoxin the clearance (CL) is given by the equation (Winter 2004):

CL (mL/min) = (0.8 mL/kg per min) (weight in kg) + CLcr(mL/min)

So for a 70-year-old man, who weighs 70 kg and has a plasma creatinine = 125 ␮mol/L, his CL for digoxin is given:

CLdig= (0.8) ⫻ (70) + CLcr

[1.23(140 – 70) ⫻ 70]

CLdig= 56 + –––––––––––––––––––––

125 CLdig= 104.2 mL/min = 6.25 L/h

Population-based equations may also take account of disease states and drug interactions, e.g. in chronic heart failure the clearance for digoxin is reduced and the population-derived equation is:

CL (L/h) = (0.33 ⫻ Weight in kg) + 0.9 CLcr(mL/min)

In hyperthyroidism the CL for digoxin is increased by 1.3 fold, and in hypothyroidism it is reduced by 0.7 (Winter 2004).

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Similarly, using the CL one can calculate the maintenance dose given:

Dose rate = C_ss⫻ CL

From Example 3, dose rate = 1.5 ⫻ 6.25 (L/h)

= 9.375 ␮g/h. As digoxin is given daily the patient requires 24 ⫻ 9.375 ␮g/day = 225 ␮g/day.

To take account of F the oral dose must be 225/0.7 = 321␮g, and to account for available tablets he should receive 250 mg and 62.5␮g tablets, as a dose of 312.5 ␮g/day.

Revising a regimen

Once a patient has been prescribed a regimen based on population data, he or she may actually handle the drug in a different manner and this will result in a different plasma concentration from predicted. The difference is due to the clear-ance being at a different rate. If clearclear-ance is greater than predicted then C_sswill be lower and if it is less than predicted then the C_ss will be higher. An actual measurement of the C_ss for this

patient can be used to determine the revised dosage. If the Cssis measured then, as we know the dose rate (Example 4, above), we can calcu-late the patient’s actual clearance. This actual (as opposed to population-based) value enables the new maintenance dose to be determined.

Zero-order kinetics

Most drugs show first-order kinetics with the rate of elimination being proportional to the concen-tration of the drug. However, for some drugs, notably the antiepileptic drug phenytoin and ethanol, the enzymes responsible for their elimination become saturated, so the rate of elim-ination is no longer proportional to the drug’s concentration. This means that these drugs exhibit zero-order kinetics with the rate of elimin-ation not being proportional to the drug concen-tration, so small changes in their dosage leads to disproportionate increases in plasma concen-trations. Controlling the dose of phenytoin is The patient in Examples 2 and 3 was receiving 312.5␮g digoxin/day and his plasma digoxin concentration was found to be 2.5␮g/L. In practice the digoxin would be stopped until therapeutic levels (<2 ␮g/L) returned but the data allow for a dosage alteration. Given:

Dose rate = Css⫻ CL.

Then 312.5 ⫻ F = Css⫻ CL (per day) 312.5 ⫻ 0.7 = 2.5 ⫻ CL

218.8

Therefore CL = –––––– = 87.5 L/day 2.5

Therefore the new daily dose to attain a Cssof 1.5 should be:

Daily dose ⫻ F = Css⫻ CL [1.5 ⫻ 87.5]

Daily dose = –––––––––––– = 187.5 ␮g/day 0.7

and so he could be given a new regimen of 125␮g plus 62.5 ␮g tablets, which equals 187.5μg/day.

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therefore difficult to achieve. The pharmaco-kinetics of phenytoin are described by the Michaelis–Menten enzyme kinetics (Figure 6.10), where V_max relates to the maximum rate of elimination and K_mis analogous to the affinity of the metabolizing enzymes for phenytoin.

The consequence of the zero-order kinetics is a dose–plasma concentration profile that varies between individuals due to variability in V_max (Figure 6.11).

In constructing regimens for phenytoin the loading dose is calculated in the usual way by (C_ss ⫻ Vd), because this is independent of metabolism. However, the maintenance dose is calculated from a modified form of the Michaelis–Menten equation:

[S ⫻ F ⫻ Dose] [Vmax⫻ Css] –––––––––––––– = –––––––––––

␶ K_m+ C_ss

where V_maxand K_mare obtained from population averages. When adjusting regimens (i.e. once the plasma concentration is known in relation to the dosage), one can calculate the patient’s individ-ual V_max, assuming that this is more variable and the population K_mis fixed.

Therapeutic drug monitoring (Appendix 2) The major clinical application of pharmaco-kinetics is in therapeutic drug monitoring (TDM), which involves taking a blood sample and measuring the concentration of a specific drug. This enables a dosage regimen to be altered in relation to a patient’s individual clearance and also detects toxic dosing, subtherapeutic dosing and non-compliance. The sample is taken at a specific time point post-dose (to allow the drug

Therapeutic drug monitoring (Appendix 2) The major clinical application of pharmaco-kinetics is in therapeutic drug monitoring (TDM), which involves taking a blood sample and measuring the concentration of a specific drug. This enables a dosage regimen to be altered in relation to a patient’s individual clearance and also detects toxic dosing, subtherapeutic dosing and non-compliance. The sample is taken at a specific time point post-dose (to allow the drug

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