Basics of reaction kinetics
The kinetics of every process influencing the fate of any given pharmacon (absorption, elimination, and transport between the compartments in a multi-compartment model) may be described using the same mathematical tools. Of the processes influencing the pharmacons, elimination can be assessed most easily using experimental tools.
As we previously discussed elimination is everything that (1) causes a change in the pharmacon‘s primary chemical bonds (metabolism) or (2) clears the pharmacon from the living organism (excretion). The first is a chemical reaction; however in the second the transport process may also be viewed as a chemical reaction. The time course of change in the concentration of the reaction materials known as the rate (and closely relating to this the mechanism) of chemical reactions is described by reaction kinetics.
When using an empiric approach the equation expressing the rate of a chemical reaction contains the concentration of every reacting chemical powered to the exponent according to their stoichiometry (stoichiometric exponents), and the goal is to identify a k constant that if multiplied by the product of the powered concentrations will provide the reaction rate obtained experimentally (v):
The (gross) order of a chemical reaction is determined by the sum of the concentrations‘
exponents determining the reaction velocity (a+b+…+x using the above equation as an example). Order of the reaction may be defined for a single constituent as well, in this case only the exponent of the given chemical‘s concentration is taken into consideration (using the above example this would equal to a for chemical 1). The most important consideration from a practical point of view is for k constant, as this has to be invariable within the experimental
In the living organism pharmacons are metabolized in a chemically rich environment generally with the participation of several different molecules (usually present in a watery solution). Although theoretically only the first step is considered as metabolism (in which the structure of the pharmacon changes), metabolism is usually a chain reaction where the subsequent steps influence the rate of the first step, therefore the complete process should be considered.
Regardless the elimination of pharmacons may be approximated relatively simply due to the validity of the following principles:
1. The rate of a chain reaction (at least in the beginning of the process) is determined by the step with the smallest rate constant (rate limiting step). The kinetic order of the complete process is therefore well approximated by the order of the rate limiting step, so it is sufficient to deal with that.
2. If one of the reacting agents is present in such abundance that its concentration remains practically unaltered during the entire process, than it doesn‘t contribute to the order of the reaction that is determined experimentally, therefore it may be omitted from the equation describing the rate of the reaction. This means for example that the concentration of water often present during metabolism may be left out of the calculations similar to the endogenous molecules that are present in high concentration (e.g. ATP, glucuronic acid).
3. The rate limiting step (similar to the other steps) is generally an enzymatic catalyzed reaction. The catalyst is retrieved unchanged at the end of the reaction therefore the enzyme may also be omitted from the equation describing the rate of the equation.
Summarizing, during the elimination of pharmacons (from the mathematical standpoint including their absorption and any other transport) only the rate limiting step of the chain reaction in which the pharmacon attaches to an enzyme (or a carrier) should be considered (while other molecules possibly also participating in the reaction are present in abundance compared to the pharmacon). Using these considerations we can say that the elimination (or absorption or transport) of pharmacons may be viewed as monomolecular process therefore the rate of the process is given by the product of the rate constant and the pharmacon‘s concentration raised to the first power. In other words the elimination (and mostly all other processes) of pharmacons generally shows first order kinetics.
Elimination in the one-compartment open model
Due to the fact that elimination is the most readily investigated process of all processes involving the pharmacon, the principles of kinetics will be illustrated using that as an example. For the sake of simplicity -if not indicated otherwise- elimination will be discussed using a one-compartment open model.
Although the rate of elimination can be generally well estimated by the product of the elimination constant and the first power of the pharmacon concentration, the rate of elimination may be determined more precisely. Elimination (as an enzyme-catalyzed reaction) may be described using Michaelis-Menten kinetics, that is based on the law of mass action (more precisely, on the law of mass action described for equilibrium). Forerunner of the Michaelis-Menten equation is the Hill equation that is well-known in pharmacodynamics. The model of Michaelis and Menten also contains some presumptions, owing to this fact it is mathematically relatively simple. According to it, the rate of elimination (v):
where: vmax is the maximal rate of elimination, c is the concentration of the enzyme‘s substrate in the rate limiting step of the pharmacon‘s elimination (in the simplest case this is the concentration of the pharmacon itself, but this is not necessarily true), KM is the substrate concentration at which the rate limiting enzyme works at half-maximal rate (Michaelis-Menten constant). Using the above equation if rate is plotted as a function of concentration (using linear axes), a hyperbolic or saturation curve is obtained (Fig. 3).
KM 4 KM Substrate concentration
Rate of enzymatic elimination
vmax/2 vmax
Figure 3: Rate of enzymatic elimination as a function of substrate concentration
Although KM is the half-saturating concentration of the substrate for the rate-limiting enzyme, it describes the pharmacon‘s concentration well in our case since the other processes barely contribute to the overall kinetics of complete elimination process.
The pharmacon‘s rate of elimination in the 0 - KM concentration interval may be simplified as follows since KM > c in this case:
here: ke=vmax/KM is the rate constant for elimination. It should also be noticed that in the equation expressing the rate (v=ke·c) the first power of concentration occurs, so the elimination of pharmacons in the 0 - KM concentration interval follows first order kinetics with good approximation. In this case the quantity of substrate is small when compared to the quantity (or more precisely the activity) of the enzyme, therefore substrate molecules (forming from the additionally added pharmacon dose) will readily find free enzymes so the rate of elimination will increase. According to this, Fig. 3 reflects that the rate-concentration function is approximately linear at concentrations below KM; therefore the rate of elimination is roughly proportional to the concentration of the substrate (or the pharmacon in question).
The rate of elimination may be given in a simpler for concentrations higher than 4 KM as well, using the assumption that KM < c:
So for concentration above 4 KM the elimination rate constant equals to vmax, and the rate of elimination is always at maximum. It should be noted that lack of dependence on the concentration means the zero power of the concentration (v=vmax=ke·c0), so elimination of pharmacons at concentrations exceeding 4 KM shows zero order kinetics with good approximation. In this case substrate quantity greatly exceeds the quantity of the enzyme (or its activity), most of the enzyme molecules are occupied therefore it is difficult for the newly introduced (or produced) substrate molecules to find a free enzyme. This is reflected in Figure 3, where the rate-concentration function is approximately linear and runs almost horizontally showing that elimination became independent of the substrate concentration (or pharmacon in question).
It is not worth the effort to mathematically simplify the equation describing the rate of elimination in the concentration interval KM - 4 KM as this would yield a fractional order kinetic reaction (0-1 in this interval). In practice however this concentration interval is handled as either a first- or a zero order kinetic reaction depending on whether the therapeutically or toxicologically significant concentration is in the higher or lower range.
Although the elimination of pharmacons follow at least two kinds of (zero and first order) kinetics, in practice we see that elimination of most pharmacons is kinetically uniform;
generally show first order kinetics, with only a limited scope of pharmacons that follow zero order kinetics (for example ethanol). The reason for this is that the concentration of most pharmacons has no significance in such a wide dosage range that would be lead to a change from first order kinetics to zero order kinetics. If a substance is efficacious in a concentration well below the KM value of the rate limiting enzyme (or transporter), the levels at 4 KM could be toxic or couldn‘t develop (as these are incompatible with life). However if the concentration in interest is above the value of 4 KM, concentrations below KM are generally ineffective, so these are not investigated.
Only a few special pharmacons are used over such wide concentration ranges where the change in kinetics may be seen in practice. For example acetylsalicylic acid is eliminated with first order kinetics when given in the indication of platelet aggregation inhibition (75-325 mg per day), however when given to ameliorate the symptoms of rheumatoid arthritis it may be given in a high dose (2-4 g), and is eliminated with zero order kinetics. For the sake of simplicity we disregard the fractional exponent kinetics in this case as well (although, for example, the dose range for the antipyretic and pain killer indication of acetylsalicylic acid – 0.5-2 g – would be best described using fractional exponents).