What pharmacokinetics actually measures
Pharmacokinetics answers a simple question: where does the drug go, and how fast? After you administer a compound, it gets absorbed, distributed through tissues, metabolized, and eliminated. PK quantifies each of these processes with measurable parameters derived from concentration-time data.
You collect blood samples at fixed time points after dosing, measure drug concentration in plasma, and fit a mathematical model to the resulting curve. That curve tells you everything about how the body handles the compound.
The one-compartment model explained visually
The simplest PK model treats the body as a single well-mixed container. The drug enters at rate ka (absorption) and leaves at rate ke (elimination). The plasma concentration at any time follows:
# One-compartment model with first-order absorption
# C(t) = (F * Dose * ka) / (Vd * (ka - ke)) * (exp(-ke*t) - exp(-ka*t))
library(mrgsolve)
mod <- mread("pk1cmt") %>%
param(CL = 5, V = 30, KA = 1.2) %>%
ev(amt = 100, route = "extravascular")
out <- mrgsim(mod, end = 24, delta = 0.1)
plot(out, CP~time)The curve rises as absorption outpaces elimination, peaks at Tmax, then falls as elimination dominates. The shape of that curve is entirely determined by three parameters: clearance, volume of distribution, and absorption rate.
Half-life and what it predicts
The elimination half-life (t½) is the time it takes for plasma concentration to halve. It is derived directly from clearance (CL) and volume of distribution (Vd): t½ = 0.693 × Vd / CL.
Half-life predicts two critical things for drug development:
- Time to steady state: approximately 4-5 half-lives after starting a repeated dosing regimen, the drug reaches steady-state concentration.
- Dosing interval: drugs with short half-lives require frequent dosing or controlled-release formulations to maintain therapeutic concentrations.
AUC: the area that summarizes everything
The Area Under the Curve (AUC) is the integral of the concentration-time profile. It measures total drug exposure: the cumulative amount of drug the body was exposed to over the observation period.
AUC is computed by the trapezoidal method in non-compartmental analysis (NCA): sum up the areas of trapezoids formed by consecutive concentration measurements. In R:
library(PKNCA)
conc_obj <- PKNCAconc(pk_data, conc ~ time | subject)
dose_obj <- PKNCAdose(dose_data, dose ~ time | subject)
data_obj <- PKNCAdata(conc_obj, dose_obj)
results <- pk.nca(data_obj)
# AUC from 0 to last measurable concentration
summary(results)[["auclast"]]AUC is used for bioequivalence studies, dose proportionality assessment, and drug-drug interaction quantification. It is arguably the most important single PK metric.
From PK to PD: linking exposure to effect
Pharmacodynamics (PD) describes what the drug does to the body. The PK/PD link is the relationship between drug concentration and effect. For most compounds, effect increases with concentration but saturates at high doses.
The key insight is that the relevant exposure metric for PD is not necessarily Cmax or AUC in isolation, but the concentration at the target site at the time of the effect measurement. This is why PK/PD modeling is more informative than looking at PK and PD endpoints separately.
The Hill equation and Emax models
The most widely used PD model is the Emax model, derived from the Hill equation:
# Emax model: E(C) = Emax * C^gamma / (EC50^gamma + C^gamma)
# gamma = Hill coefficient (steepness of the dose-response curve)
# EC50 = concentration producing 50% of maximum effect
emax_model <- function(C, Emax, EC50, gamma = 1) {
(Emax * C^gamma) / (EC50^gamma + C^gamma)
}
# Simulate effect over a range of concentrations
conc_range <- seq(0, 20, 0.1)
effect <- emax_model(conc_range, Emax = 100, EC50 = 5, gamma = 1.5)
plot(conc_range, effect, type = "l", xlab = "Concentration (mg/L)", ylab = "Effect (%)")The Hill coefficient controls the steepness of the sigmoid. A value of 1 gives a gradual response. Values above 2 create a switch-like relationship, the drug either works or it does not.
What to do with this in practice
For non-statisticians working in drug development, the practical takeaway is this: when you see a concentration-time plot, you should be able to read off Tmax, estimate Cmax visually, and understand that the declining portion reflects elimination. When you see a dose-response curve, you should recognize whether the relationship is linear, saturating, or switch-like.
These intuitions guide dose selection, dosing frequency decisions, and the interpretation of safety signals. You do not need to fit the models yourself, but understanding what they represent makes you a far more effective collaborator with the pharmacometrician on your team.
Key takeaway
PK/PD is fundamentally about understanding how the body and drug interact over time. The math formalizes intuitions that you can develop by looking at enough concentration-time curves. Start with the one-compartment model, understand half-life and AUC, then build toward PK/PD integration.