First Order, Zero Order and Non-linear Elimination Kinetics

Created on Mon, 06/29/2015 - 22:16
Last updated on Tue, 10/31/2017 - 05:00

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This chapter answers parts from Section B(v) of the 2017 CICM Primary Syllabus, which expects the exam candidate to "describe the mechanisms of drug clearance and metabolism". One of the possible interpretations of this broad learning objective is some understanding of the concepts of first and zero-order elimination kinetics, which are fairly fundamental in pharmacokinetics.  In the historical Part I exam papers, the only appearance of these concepts was in Question 5(p.2) from the second paper of 2009. The candidates were expected to "explain the difference and the clinical relevance" between first and zero order elimination. The question was passed by only 22% of the candidates that year. 

In brief :

  • First order elimination kinetics: a constant proportion (eg. a percentage) of drug is eliminated per unit time
  • Zero order elimination kinetics: a constant amount (eg. so many milligrams) of drug is eliminated per unit time
  • First order kinetics is a concentration-dependent process (i.e. the higher the concentration, the faster the clearance), whereas zero order elimination rate is independent of concentration.
  • Michaelis-Menten kinetics  describes enzymatic reactions where a maximum rate of reaction is reached when drug concentration achieves 100% enzyme saturation.
  • Non-linear elimination kinetics is the term which describes drig clearance by Michaelis-Menten processes, where a drug at low concentration is cleared by first-order kinetics and at high concentrations by zero order kinetics (eg. phenytoin or ethanol). 

The official college pharmacology text  (Birkett et al, Pharmacokinetics Made Easy - 2009) discusses first order kinetics briefly as a part of discussing the elimination rate constant (p.30 -31). Zero order and mixed elimination kinetics get a mention around p 80, in the chapter on on-linear pharmacokinetics (Chapter 9).  Generally speaking these resources should be enough to write a competent answer to Question 5(p.2) from the second paper of 2009. as well as to mutter something sensible during a viva station. Realistically, what could they possibly ask about this? Nothing unpredictable. At worst, one could be asked to draw a labelled graph, explain the major differences and give examples.

First order elimination kinetics

A definition of this concept can be borrowed from the college answer to  Question 5(p.2):

"First-order kinetics... is where a constant fraction of drug in the body is eliminated per unit of time"

This is a logarithmic function. All enzymes and clearance mechanisms are working at well below their maximum capacity, and the rate of drug elimination is directly proportional to drug concentration. 

first order elimination kinetics on a linear scale

The drug concentration halves predictably according to fixed time intervals. When you plot this on a semi-logarithmic scale, the relationship of concentration and time is linear. 

first order elimination kinetics on a

If you plot the relationship of concentration vs. elimination rate, the same sort of linear relationship is seen:

first order elimination kinetics: relationship of concentration to clearance rate

The term "first order" actually comes from chemistry, where is has classically been used to describe reaction kinetics. When doubling the concentration of reagents also doubles the reaction rate, the increase in rate is by a factor of 2 (2 to the first power, or 21). That "first power" gives rise to the term "first order". In that fashion, one can have a "second order" reaction where doubling the concentration of reagents quadruples the reaction rate (i.e 2 to the second power, 22). Following this trend in nomenclature to its most absurd extent, third-order fourth-order and fiftieth-order reactions can be conceived of. 

Zero-order elimination kinetics

In chemistry, when doubling the concentration of reagents has no effect on the reaction rate, the increase in rate is by a factor of 0 (i.e. 20). This is zero-order kinetics. The relationship of concentration to reaction rate can therefore be plotted as a boring straight line:

Zero order elimination kinetics - relationship of concentration and clearance rate

In the realm of pharmacokinetics, "reaction rate" is elimination of the drug, by whatever clearance mechanisms (some of which might actually involve reactions). Generally speaking first-order kinetics can describe clearance which is driven by diffusion; diffusion rate is directly proportional to drug concentration. If there is a functionally inexhaustable amount of metabolic enzymes available, the reaction will also be first order (i.e. the more substrate you throw at the system, the harder the system will work). However, if there is some limit on how much enzyme activity there can be, then the system is said to be saturable, i.e. it is possible to saturate the enzymes to a point where increases in concentration can no longer produce increases in enzyme activity. This gives rise to non-linear elimination kinetics, known by the uninformatively eponymous  term "Michaelis-Menten elimination".

Michaelis-Menten elimination kinetics

Named after Leonor Michaelis and Maud Menten, this model of enzyme kinetics describes the relationship between the concentration and the rate of enzyme-mediated reaction. In short, at low concentrations, the more substrate you give the faster the reaction rate. At high concentrations, the rate of reaction remains the same because all the enzyme molecules are "busy", i.e. the system is saturated. 

This concept can be described by the unimaginatively named Michaelis-Menten equation, which relates the rate (velocity, V) of a reaction to the concentration of the substrate (lets call it "drug"). A maximum rate of reaction is reached when drug concentration achieves 100% enzyme saturation. Beyond this concentration, clearance will be zero-order.  The maximum rate of reaction in this instance is called Vmax (i.e. maximum velocity). The concentration required to achieve 50% of this maximum reaction rate is called Km, where K presumably stands for Konzentration because everything in science was named by the Germans.

The equation is:

Michaelis-Menten equation

The graph of the enzymatic reaction rate to drug concentration looks a little like this:

Michaelis-Menten relationship of drug concentration and enzymatic reaction rate

Thus, when the patient is receiving regular doses of the drug, if the concentration is already high then relatively small changes in the dose will produce a disproportionately large change in drug concentration.

Michaelis-Menten kinetcis - relationship of concentration to changes in dose

This has relevance to clinical pharmacology. Drugs which have a therapeutic concentration range in the steep part of this curve are said to have a narrow therapeutic range (i.e. the effective dose is not too far off the toxic dose). If relatively large changes in dose produce relatively small changes in the concentration , toxic levels will be difficult to achieve and the drug is said to have a broad therapeutic index. 

broad and narrow therapeutic index

The college answer to Question 5(p.2) from the second paper of 2009 specifically nominates Km as the magic cut-off for where therapeutic index becomes narrow. In other words, if the drug concentration required to produce a useful effect is above the concentration required for 50% of Vmax, then the drug will have a narrow therapeutic index. It is true (it makes logical sense) but it is not clear where the examiners got this from. 

Obviously not all livers are the same. The liver of a veteran alcoholic will be able to metabolise substantially more whiskey than the liver of an average nun.  In short, there will be variation in the population depending on the Vmax of their clearance organs, and everybody's organs will max out at different concentrations. 

Michaleis mentent clearance with different vmax values

Examples of elimination kinetics relevant to intensive care practice

Gentamicin as a model of first-order elimination kinetics

This drug is not metabolised by any saturable enzymes. Its elimination is nothing fancy: it just ends up in the urine at a rate which is proportional to its concentration. In fact it is one of the few drugs which can Goncalves-Pereira et al (2010) were able to generate some nice classical looking curves in their study:

gentamicin elimination kinetics from Goncalves-Pereira et al (2010)

Ethyl alcohol as a model of zero-order elimination kinetics

Ethanol particularly will be interesting to the intensivist, owing to the extreme fondness for it among the intensive care community.  Rangno et al (1981) were able to plot some excellent zero-order elimination curves for IV-infused ethanol (a) and  oral doses (b) among eight healthy male volunteers whom they lured into the experiment with the promise of booze. The high-dose group got fairly sozzled with a dose of 1.25g/kg, which for a 70kg male would be 87.5g (almost 9 standard drinks, more than a full bottle of wine), reaching a blood alcohol concentration of 0.2% - four times over the Australian legal driving limit.   

Across all normal dose ranges, ethanol elimination appears to be quite linear, and independent of concentration. As seen from the graph above, the gradient of diminishing concentration over time from a high concentration is the same as that from a low concentration. This is because the dose consumed is typically quite high (i.e. it is a low potency drug).  The enzymes responsible for its clearance in the normal person are saturated with the first drink; the Km of alcohol dehydrogenase for ethanol is about 1mmol/L (Cederbaum, 2012).  The Vmax for alcohol ends up being about 7-10g/hr. It would therefore take an average person approximately 28 hours to completely metabolise one whole bottle of vodka, which explains some of the adverse functional effects seen during the subsequent morning. 

Interestingly, the Km of alcohol dehydrogenase for other alcohols is much higher; the metabolism of methanol and ethylene glycol therefore proceed according to first-order kinetics. Of all the many isoforms of alcohol dehydrogenase, none show a preference for ethyl alcohol. It clearly has never been high on the list of metabolic priorities, as far as evolution is concerned. 

Phenytoin as a model of nonlinear elimination kinetics

Phenytoin is the classical poster child for non-linear elimination kinetics, because the enzyme saturation point is reached somewhere in the middle of the therapeutic concentration range. The excellent 1979 article by Alan Richards contains within it a famous graphic representation of the changes in phenytoin concentration among a group of epileptic patients. Incontrast to this author, Richards had permission to reproduce it from Richens & Dunlop (1975).

In summary, it is important only ever make small dose adjustments with this drug, and other drugs like this drug.

 

 

References

Richens, Alan. "Clinical pharmacokinetics of phenytoin." Clinical pharmacokinetics 4.3 (1979): 153-169.

Richens, Alan, and Andrew Dunlop. "Serum-phenytoin levels in management of epilepsy." The Lancet 306.7928 (1975): 247-248.

Barza, Michael, et al. "Predictability of blood levels of gentamicin in man." Journal of Infectious Diseases 132.2 (1975): 165-174.

Goncalves‐Pereira, J., A. Martins, and P. Povoa. "Pharmacokinetics of gentamicin in critically ill patients: pilot study evaluating the first dose." Clinical Microbiology and Infection 16.8 (2010): 1258-1263.

Rangno, R. E., J. H. Kreeft, and D. S. Sitar. "Ethanol ‘dose‐dependent’elimination: Michaelis‐Menten v classical kinetic analysis." British Journal of Clinical Pharmacology 12.5 (1981): 667-673.

Cederbaum, Arthur I. "Alcohol metabolism." Clinics in liver disease 16.4 (2012): 667-685.