Drug concentration models
Drug concentration models describe how dosing schedules create changing concentrations through time. The mathematics links dose, timing, absorption and elimination to the concentration profile that the body experiences.
Start with a one-compartment model
Let \(A(t)\) be drug amount in a well-mixed model compartment and \(V\) its apparent volume. Then
\[\boxed{C(t)=\frac{A(t)}{V}.}\]If elimination is first order,
\[\frac{dA}{dt}=-kA,\]so
\[\boxed{C(t)=C_0e^{-kt}.}\]Single intravenous dose
An idealised IV bolus places the dose into the central compartment immediately. The concentration starts at
\[C_0=\frac{D}{V}\]and then falls exponentially:
\[\boxed{C(t)=\frac{D}{V}e^{-kt}.}\]Half-life controls how quickly concentration falls
\[\boxed{t_{1/2}=\frac{\ln2}{k}.}\]A smaller elimination rate constant gives a longer half-life and a slower decline. A larger \(k\) gives faster elimination.
Constant intravenous infusion
Now suppose drug enters continuously at rate \(R\). For concentration,
\[\boxed{\frac{dC}{dt}=\frac{R}{V}-kC.}\]The exact solution from \(C(0)=0\) is
\[\boxed{C(t)=C_{ss}(1-e^{-kt}),}\]where
\[\boxed{C_{ss}=\frac{R}{kV}.}\]At steady state, input and elimination balance.
Why steady state is approached gradually
At first, little drug is present, so elimination is small and concentration rises quickly. As concentration increases, elimination \(kC\) increases. Eventually elimination nearly matches the input rate, so further increase becomes very small.
Stopping an infusion
If the infusion is stopped at time \(T\), concentration afterwards follows exponential elimination from the value reached at \(T\):
\[C(t)=C(T)e^{-k(t-T)},\qquad t\ge T.\]The same elimination law therefore governs the decline after input stops.
Repeated bolus doses
Suppose identical doses are given every \(\tau\) hours. Each dose adds a new exponential contribution to the concentration.
For doses given at times \(t_j\),
\[\boxed{C(t)=\sum_{t_j\le t}\frac{D}{V}e^{-k(t-t_j)}.}\]This is the principle of superposition for a linear one-compartment model.
Why repeated dosing creates peaks and troughs
Immediately after each dose, concentration jumps upward. Between doses, it declines. If the next dose arrives before the previous drug is fully eliminated, the new dose is added on top of residual drug.
Accumulation
With repeated regular dosing, concentration may rise over the first several doses until a repeating pattern is reached. This happens because each dose adds drug before the previous dose has completely disappeared.
In the linear model, the accumulation factor for repeated IV boluses is
\[\boxed{R_{acc}=\frac{1}{1-e^{-k\tau}}.}\]Shorter dosing intervals or slower elimination increase accumulation.
Peak and trough at steady repeating dosing
For repeated IV boluses in a linear one-compartment model, the post-dose peak at steady state is
\[\boxed{C_{max,ss}=\frac{D/V}{1-e^{-k\tau}}.}\]The pre-dose trough is
\[\boxed{C_{min,ss}=C_{max,ss}e^{-k\tau}.}\]These equations show mathematically how dose, dosing interval and elimination determine concentration fluctuation.
Long versus short dosing intervals
If \(\tau\) is long relative to half-life, much of the drug disappears before the next dose, so peak–trough variation is large.
If \(\tau\) is short relative to half-life, the profile is smoother but accumulation is greater.
Oral concentration models
For oral dosing, drug must usually be absorbed before reaching the central compartment:
With first-order absorption and elimination,
\[\boxed{C(t)=\frac{F D k_a}{V(k_a-k_e)}\left(e^{-k_et}-e^{-k_at}\right).}\]Unlike an IV bolus, the concentration begins near zero, rises to a peak, and then falls.
Absorption rate changes the shape
A large \(k_a\) gives rapid absorption and an earlier, sharper peak. A smaller \(k_a\) spreads absorption over more time and typically gives a later, broader peak.
Time of the oral peak
For the simple first-order absorption model,
\[\boxed{t_{max}=\frac{\ln(k_a/k_e)}{k_a-k_e}.}\]The peak occurs where absorption into the central compartment is exactly balanced by elimination from it.
Input rate versus elimination rate
Nearly every simple drug concentration model can be read as a balance:
\[\boxed{\frac{dA}{dt}=\text{input rate}-\text{elimination rate}.}\]For oral dosing, input changes through time because the absorption compartment is being depleted. For infusion, input is constant. For an IV bolus, input is represented by the initial condition.
Loading dose
If a target concentration is required quickly, the simplest one-compartment relation is
\[\boxed{D_L\approx VC_{target}.}\]For non-IV administration, bioavailability may also need to be included.
Maintenance dosing
Maintenance dosing aims to replace drug as it is eliminated. For a constant infusion, the maintenance input at steady state satisfies
\[R=CL\,C_{target}.\]For intermittent dosing, dose and interval together determine the repeating peak–trough profile.
Multi-compartment concentration models
Some drugs do not behave as if they mix instantly throughout one effective volume. A two-compartment model allows exchange between a central and peripheral compartment:
\[\frac{dA_1}{dt}=-(k_{10}+k_{12})A_1+k_{21}A_2+\text{input},\]\[\frac{dA_2}{dt}=k_{12}A_1-k_{21}A_2.\]The measured central concentration is often
\[C_1=\frac{A_1}{V_1}.\]Why distribution can create a fast and slow decline
Immediately after a central dose, drug can leave the central compartment both by elimination and by distribution into peripheral tissue. Later, redistribution from the peripheral compartment can sustain a slower terminal decline.
This is why some concentration curves are better described by sums of exponentials than by a single exponential.
Nonlinear concentration models
If elimination becomes saturated, the rate is no longer proportional to concentration. A Michaelis–Menten elimination model is
\[\boxed{\frac{dC}{dt}=-\frac{V_{max}C}{K_m+C}.}\]Then doubling concentration does not necessarily double the elimination rate, so superposition and simple accumulation formulas no longer apply.
Concentration is not effect
A concentration curve tells us exposure, not automatically the biological response.
The same concentration profile can produce different effects in different biological systems.
Choosing a concentration model
| Situation | Useful starting model |
|---|---|
| single IV bolus | exponential one-compartment decay |
| constant infusion | input minus first-order elimination |
| repeated IV dosing | sum of exponential dose contributions |
| oral dose | absorption compartment plus elimination |
| clear distribution phase | two- or multi-compartment model |
| saturable elimination | nonlinear elimination model |
A practical modelling workflow
Specify the dosing route and timing. Decide whether the state variable is amount or concentration. Write the input process. Add elimination and, if needed, absorption or distribution. Solve the model analytically when possible or numerically otherwise. Plot the resulting concentration–time function. Check peaks, troughs, accumulation and steady state. Finally, compare the model with observed concentration data.