What is meant by non-linear pharmacokinetics?
When the dose of a drug is increased, we expect that the concentration at steady state will increase proportionately, i.e. if the dose rate is increased or decreased say two-fold, the plasma drug concentration will also increase or decrease two-fold. However, for some drugs, the plasma drug concentration changes either more or less than would be expected from a change in dose rate. This is known as non-linear pharmacokinetic behaviour and can cause problems when adjusting doses.
What causes non-linear pharmacokinetic behaviour?
In a previous article (Article 1 - `Clearance' Aust Prescr 1988;11:12-3), it was shown that the steady state blood concentration (C_{ss}) is a function of both the dose and the clearance of the drug.
equation 1
C_{ss} = | F x dose rate clearance |
where F is the bioavailability.
In most dosing situations, total clearance (CL) is determined by protein binding and intrinsic clearance (CL_{int}) (Article 4 - `How drugs are cleared by the liver' Aust Prescr 1990;13:88-9).
equation 2
CL = f_{u} x CL_{int}
where f_{u} is the fraction unbound to protein.
Combining equations 1 and 2, the determinants of C_{ss} during chronic dosing are
equation 3
C_{ss} = | F x dose rate f_{u} x CL_{int} |
F, f_{u} and CL_{int} usually do not change with drug concentration so that C_{ss} is directly proportional to dose rate. However, there are some situations where this predictable relationship between dose rate and C_{ss} breaks down due to dose dependency of F, f_{u} and/or CL_{int}.
1. Saturation of elimination mechanisms causes a change in intrinsic clearance
Drug metabolism
The metabolism of drugs is carried out by a variety of enzymes such as cytochrome P450 and N-acetyltransferase. The dependence of the rate of an enzyme reaction on substrate concentration is given by the Michaelis-Menten equation and is illustrated in Fig. 1
equation 4
v = | V_{max} x S K_{m} + S |
where v is the velocity of reaction, S is the substrate concentration, V_{max} is the maximum velocity at very high substrate concentrations and K_{m} is the substrate concentration at half V_{max}. Km is a measure of the affinity of the substrate for the enzyme.
In pharmacokinetic terms, v is equivalent to the rate of elimination (v = C_{u} x CL) and S is equivalent to the unbound drug concentration (C_{u}). Equation 4 can then be rearranged to give a function for intrinsic clearance (see also equation 1).
equation 5
CL_{int} = | V C_{u} | = | V_{max} K_{m} + C_{u} |
where V_{max} is the maximum rate of metabolism at high concentrations of unbound drug and K_{m} is the unbound drug concentration at half V_{max}.
Usually, unbound plasma drug concentration (C_{u}) in the therapeutic range is very small compared to the K_{m} for the metabolising enzyme and equation 5 approximates to
equation 6
CL_{int} = | V_{max} K_{m} |
CL_{int} is then independent of unbound drug concentration which is therefore linear with dose. In some cases, unbound drug concentration is close to or above K_{m} at therapeutic doses, and the kinetics begin to become non-linear (see
Fig. 1). In this situation, CL_{int} decreases as unbound drug concentration increases (see equation 5) and steady state drug concentration increases more than proportionately with dose (equation 3). At high drug concentrations, the maximal rate of metabolism is reached and cannot be exceeded. Under these conditions, a constant amount of drug is eliminated per unit time no matter how much drug is in the body. Zero order kinetics then apply rather than the usual first order kinetics where a constant proportionof the drug in the body is eliminated per unit time. Some examples of drugs which exhibit non-linear kinetic behaviour are phenytoin, ethanol, salicylate and, in some individuals, theophylline.
Phenytoin: Phenytoin exhibits marked saturation of metabolism at concentrations in the therapeutic range (10-20 mg/L) (Fig. 2). Consequently, small increases in dose result in large increases in total and unbound steady state drug concentration. As an example, for a patient with typical K_{m} of 5 mg/L (total drug) and V_{max} of 450 mg/day, steady state concentrations at doses of 300, 360 and 400 mg/day would be 10.0, 20.0 and 40.0 mg/L respectively (Fig. 2). Thus, small dosage adjustments are required to achieve phenytoin concentrations in the therapeutic range of 10-20 mg/L.
A second consequence is that, because clearance decreases, apparent half-life increases from about 12 hours at low phenytoin concentrations to as long as a week or more at high concentrations. This means that
i. the time to reach steady state can be as long as 1-3 weeks at phenytoin concentrations near the top of the therapeutic range
ii. in the therapeutic range, the phenytoin concentration fluctuates little over a 24 hour period allowing once daily dosing and sampling for drug concentration monitoring at any time between doses
iii. if dosing is stopped with concentrations in the toxic range, phenytoin concentration initially falls very slowly and there may be little change over a number of days.
Alcohol: Alcohol is an interesting example of saturable metabolism. The Km for alcohol is about 0.01 g% (100 mg/L) so that concentrations in the range of pharmacological effect are well above the K_{m}. The V_{max} for ethanol metabolism is about 10 g/hour (12.8 mL/hour) and it can be calculated (see legend to Fig. 2) that at the common legal driving limit of 0.05 g%, the rate of alcohol metabolism per hour is 8.3 g/hour. This amount of alcohol is contained in 530 mL light beer, 236 mL standard beer, 88 mL wine or 27 mL spirit. Higher rates of ingestion will result in further accumulation.
Renal excretion
In Article 7 (`Clearance of drugs by the kidneys' Aust Prescr 1992;15:16-9), it was shown that renal drug clearance is the sum of filtration clearance plus secretion clearance minus reabsorption. Clearance by glomerular filtration is a passive process which is not saturable, but secretion involves saturable drug binding to a carrier. Even when secretion is saturated, filtration continues to increase linearly with plasma drug concentration. The extent to which saturation of renal secretion results in non-linear pharmacokinetics depends on the relative importance of secretion and filtration in the drug's elimination. Because of the baseline of filtration clearance, saturation of renal secretion does not usually cause clinically important problems.
2. Saturation of first pass metabolism causing an increase in bioavailability
After oral administration, the drug-metabolising enzymes in the liver are exposed to relatively high drug concentrations in the portal blood. For drugs with high hepatic extraction ratios, e.g. alprenolol, an increased dose can result in saturation of the metabolising enzymes and an increase in bioavailability (F). Steady state drug concentration then increases more than proportionately with dose (equation 3). Other drugs with saturable first pass metabolism are tropisetron and paroxetine.
3. Saturation of protein binding sites causing a change in fraction of drug unbound in plasma
The fraction unbound of a drug in plasma (f_{u}) is given by
equation 7
f_{u} = | 1 1 + K_{a}P_{u} |
where Ka is the affinity constant for binding to a protein such as albumin or a1 acid glycoprotein and P_{u} is the concentration of free (unbound) protein, i.e. protein that does not have drug bound to it. The total concentration of albumin in plasma is about 0.6 mM (40 g/L) and the concentration of a1 acid glycoprotein is about 0.015 mM. Usually drug concentrations are well below those of the binding proteins and unbound protein (P_{u}) approximates to total protein (P_{T}). Then, fu depends only on the affinity constant and the total concentration of protein binding sites, and remains constant with changes in drug concentration. In a few cases (e.g. salicylate, phenylbutazone, diflunisal), therapeutic drug concentrations are high enough to start to saturate albumin binding sites so that unbound protein concentration decreases and f_{u} increases while total drug concentration increases less than proportionately with increases in dose (equation 3). This occurs more commonly for drugs such as disopyramide which bind to a1 acid glycoprotein because of the lower concentration of binding protein.
What are the practical consequences of saturable protein binding? From equation 3, it can be seen that as f_{u} increases, total drug concentration at steady state decreases. However, f_{u} does not affect the steady state concentration of the unbound drug. In other words, unbound concentration will increase linearly with dose, but total drug concentration will increase less than proportionately. This is illustrated in Fig. 3 for the case of disopyramide. This dissociation between total and unbound drug concentration causes difficulties in therapeutic drug monitoring where total drug concentration is nearly always measured. Total drug concentration may appear to plateau despite increasing dose (Fig. 3) leading to further dose increases. However, unbound concentrations and drug effect do increase linearly with dose - if this is not realised, n appropriate dose increases with consequent toxicity can occur.
Self-test questions
The following statements are either true or false.
1. In a drug with non-linear kinetics, doubling the dose will double the concentration.
2. Phenytoin needs to be given twice or 3 times daily because it has a half-life of about 12 hours.
Answers to self-test questions
1. False
2. False