Pharmacokinetics
Pharmacokinetics describes how the amount and concentration of a drug change in the body through time. It studies processes such as absorption, distribution and elimination.
Amount and concentration are different
Let \(A(t)\) be the amount of drug in the body or in a model compartment. If that amount is distributed through an apparent volume \(V\), the concentration is
\[\boxed{C(t)=\frac{A(t)}{V}.}\]Amount may be measured in mg, while concentration may be measured in mg/L.
The simplest model: intravenous bolus
Suppose a dose enters the central compartment immediately and elimination is proportional to the amount present:
\[\boxed{\frac{dA}{dt}=-kA.}\]Here \(k\) is the first-order elimination rate constant.
The exact solution is
\[\boxed{A(t)=A_0e^{-kt}.}\]Dividing by \(V\),
\[\boxed{C(t)=C_0e^{-kt},\qquad C_0=\frac{A_0}{V}.}\]Why does first-order elimination give exponential decay?
The removal rate is \(kA\). When much drug is present, more is removed per unit time. As the amount falls, the absolute removal rate also falls.
The same fraction, rather than the same amount, is removed over equal time intervals.
Concentration after an IV bolus
Half-life
The half-life is the time required for concentration to fall to half its current value:
\[\boxed{t_{1/2}=\frac{\ln2}{k}.}\]For \(k=0.2\) per hour,
\[t_{1/2}\approx3.47\text{ hours}.\]After another half-life, the concentration halves again.
Clearance
Clearance measures the body's ability to eliminate drug. In the simple one-compartment model,
\[\boxed{CL=kV.}\]Therefore
\[k=\frac{CL}{V}.\]The elimination equation for concentration can be written
\[\frac{dC}{dt}=-\frac{CL}{V}C.\]How clearance and volume affect half-life
Since \(k=CL/V\),
\[\boxed{t_{1/2}=\frac{V\ln2}{CL}.}\]A larger apparent volume tends to lengthen half-life if clearance is unchanged. A larger clearance tends to shorten it if volume is unchanged.
Oral dosing requires absorption
An oral dose does not usually enter the central compartment instantaneously. We can introduce an absorption compartment:
Let \(A_g\) be drug amount at the absorption site and \(A_c\) the central amount:
\[\frac{dA_g}{dt}=-k_aA_g,\]\[\boxed{\frac{dA_c}{dt}=F k_aA_g-k_eA_c.}\]Here \(F\) is bioavailability, the fraction of the administered dose reaching the systemic circulation in the model.
Oral concentration–time curve
For first-order absorption and elimination, with \(k_a\ne k_e\),
\[\boxed{C(t)=\frac{F D k_a}{V(k_a-k_e)}\left(e^{-k_et}-e^{-k_at}\right).}\]The concentration initially rises because absorption exceeds elimination. Later it falls because absorption weakens and elimination dominates.
The peak concentration and its time
The maximum concentration is often denoted \(C_{max}\), and the time at which it occurs is \(t_{max}\).
For the simple first-order model,
\[\boxed{t_{max}=\frac{\ln(k_a/k_e)}{k_a-k_e}.}\]The peak occurs where the instantaneous rate of change satisfies
\[\frac{dC}{dt}=0.\]Bioavailability
Bioavailability \(F\) represents the fraction of the administered dose that reaches systemic circulation according to the model. For an idealised IV bolus, \(F=1\).
For non-IV routes, incomplete absorption and presystemic loss can make \(F<1\).
Area under the curve
The area under the plasma concentration–time curve is abbreviated AUC:
\[\boxed{AUC=\int_0^\infty C(t)\,dt.}\]For linear pharmacokinetics after an IV dose,
\[\boxed{AUC=\frac{D}{CL}.}\]For an extravascular dose with bioavailability \(F\),
\[AUC=\frac{FD}{CL}.\]Constant intravenous infusion
If drug enters continuously at rate \(R_0\),
\[\boxed{\frac{dA}{dt}=R_0-kA.}\]Starting from zero,
\[A(t)=\frac{R_0}{k}\left(1-e^{-kt}\right).\]Therefore
\[\boxed{C(t)=\frac{R_0}{CL}\left(1-e^{-kt}\right).}\]Steady state during infusion
As \(t\to\infty\),
\[\boxed{C_{ss}=\frac{R_0}{CL}.}\]At steady state, the drug input rate equals the elimination rate.
Time to steady state
The fraction of steady state reached after time \(t\) is
\[1-e^{-kt}.\]After one half-life it is 50%; after two, 75%; after three, 87.5%; after four, 93.75%; and after five, about 96.9%.
Thus the approach to steady state is controlled primarily by the elimination half-life in this linear model.
Repeated doses
If doses are given repeatedly before the previous dose has been fully eliminated, drug accumulates.
For a linear model, the total concentration can be constructed by adding the contribution from every previous dose:
\[C(t)=\sum_j C_{dose}(t-t_j),\qquad t\ge t_j.\]Eventually a repeating peak–trough pattern can develop under regular dosing.
Loading and maintenance
A loading dose is used when a target concentration is required more quickly than ordinary accumulation would provide. In the simplest model, a target amount is approximately
\[A_{target}=VC_{target}.\]Maintenance input then compensates for ongoing elimination.
These equations describe modelling principles; clinical dosing decisions require drug-specific and patient-specific information.
Two-compartment models
A one-compartment model assumes rapid effective mixing. Some drugs show a fast distribution phase followed by slower elimination. Then a two-compartment model may be more appropriate:
One amount-based representation is
\[\frac{dA_1}{dt}=-(k_{10}+k_{12})A_1+k_{21}A_2,\]\[\frac{dA_2}{dt}=k_{12}A_1-k_{21}A_2.\]The central compartment often represents plasma and rapidly equilibrating tissues, while the peripheral compartment represents more slowly equilibrating tissues in the model.
Why a two-compartment curve can fall in two phases
Immediately after an IV dose, concentration may fall quickly as drug both distributes away from the central compartment and is eliminated. Later, redistribution and elimination produce a slower terminal decline.
This can lead to a biexponential concentration curve rather than a single exponential.
Nonlinear elimination
First-order elimination assumes elimination rate is proportional to concentration. Some elimination pathways can saturate.
A common model is Michaelis–Menten elimination:
\[\boxed{\frac{dC}{dt}=-\frac{V_{max}C}{K_m+C}.}\]When \(C\ll K_m\), elimination is approximately first order. When \(C\gg K_m\), the elimination rate approaches \(V_{max}\).
Pharmacokinetics and pharmacodynamics
Pharmacokinetics describes the drug concentration through time. Pharmacodynamics describes how concentration is related to biological or clinical effect.
Keeping these two modelling questions separate is important.
Parameter estimation
In practice, concentrations are measured at selected times. Parameters such as clearance, volume, absorption rate and inter-compartmental transfer are estimated by fitting the model to those observations.
The fitted model can then be checked against data and compared with alternative model structures.
Population pharmacokinetics
Different individuals can have different pharmacokinetic parameters. Population models describe both typical parameter values and between-person variability.
Covariates such as body size, age, organ function or other measured characteristics may explain part of that variability when scientifically justified.
Stochasticity and measurement error
Observed concentration data vary because of biological differences, model limitations and measurement error. Statistical models are therefore combined with the underlying differential equations during parameter estimation.
A practical modelling workflow
Choose the quantity to model: amount or concentration. Identify the dosing route and input process. Decide how many compartments are required. Write mass-balance equations for movement and elimination. Solve analytically when possible or numerically otherwise. Estimate parameters from concentration data. Examine half-life, AUC, peaks, troughs and steady state as appropriate. Finally, test whether the model adequately explains the observed data.