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Pharmacokinetics

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Statement

Absorption, distribution, metabolism and excretion of drugs.

Why it matters

receptor-pharmacology and dose-response (siblings in this unit) describe pharmacodynamics — what a drug does to the body once bound to its target. Pharmacokinetics is the complementary half of the question: what the body does to the drug, from the moment it is administered until it is fully cleared. Together, the two determine whether a given dose actually reaches and sustains an effective concentration at the target for a clinically useful length of time; a drug with excellent receptor pharmacology is useless if it is cleared before it can act, or accumulates to toxic levels with repeated dosing.

genomic-medicine, also in this unit, depends directly on pharmacokinetics: individual variation in drug-metabolising enzymes (most commonly the cytochrome P450 family) changes a person's elimination rate constant, and hence their effective dose requirement, which is exactly the quantitative framework this result establishes.

Hypotheses
Elimination is first-order: the rate of drug removal from plasma is proportional to the current plasma concentration.This holds whenever the metabolising enzymes or excretory transporters responsible for clearing the drug are operating well below their maximum throughput (saturation), so the removal rate simply tracks however much drug is currently present. The drug behaves as though distributed instantaneously and uniformly through a single effective volume (a one-compartment model).This is a simplification: real distribution into tissue takes finite time, and for many drugs a distinct early "distribution phase" is clinically important. It is nonetheless the standard first approximation, and is adequate for drugs that equilibrate with tissue quickly relative to their elimination half-life. Organ function (hepatic and renal) is assumed constant across the dosing period; impaired liver or kidney function lowers clearance and requires dose adjustment, since the same formulas otherwise predict drug accumulation to higher-than-intended steady-state levels.
Proof
1
\[ \dfrac{dC}{dt} = -kC \]
First-order elimination (Hypotheses): the instantaneous rate of decline in plasma concentration \(C\) is proportional to \(C\) itself, with proportionality constant \(k\), the elimination rate constant, reflecting that metabolic and excretory pathways operating below saturation clear drug at a rate scaling linearly with how much is present. A
2
\[ C(t) = C_0\,e^{-kt} \]
Integrating Step 1's differential equation with initial condition \(C(0)=C_0\) gives exponential decay; plotted on a semi-logarithmic axis, this appears as a straight line, the standard diagnostic that a drug's elimination is genuinely first-order over the plotted range. A
3
\[ t_{1/2} = \dfrac{\ln 2}{k} \]
Setting \(C(t_{1/2})=C_0/2\) in Step 2 and solving gives a half-life that is independent of \(C_0\) — a defining feature of first-order kinetics: it takes the same amount of time to halve the concentration regardless of the starting concentration. A
4
\[ CL = k \cdot V_d, \qquad V_d = \dfrac{\text{Dose}}{C_0} \]
Clearance \(CL\), the volume of plasma effectively cleared of drug per unit time, links the rate constant \(k\) to the apparent volume of distribution \(V_d\), the hypothetical volume the drug would need to occupy, at the observed plasma concentration, to account for the entire administered dose. B
5
\[ C_{ss,\text{avg}} = \dfrac{F\cdot\text{Dose}}{CL\cdot\tau}; \qquad \text{steady state reached after} \approx 4\text{-}5\ t_{1/2} \]
With repeated dosing at interval \(\tau\), plasma concentration accumulates cycle to cycle until the amount eliminated between doses equals the amount administered, reaching a steady oscillating (or, for continuous infusion, constant) average level; this equilibrium is approached asymptotically, reaching roughly \(90\text{-}97\%\) of its final value after about four to five half-lives, regardless of the dose or interval chosen. A
Result
\[ C(t) = C_0\,e^{-kt}, \qquad t_{1/2}=\dfrac{\ln2}{k} \]

Reading. Plasma drug concentration falls exponentially once absorption and distribution are complete, with a fixed half-life that does not depend on the starting dose — the quantitative backbone used to design dosing intervals and predict how long a drug remains present, or pharmacologically active, in the body.

Scope. Valid for first-order elimination in a one-compartment approximation (Hypotheses); breaks down at saturating doses, where elimination becomes zero-order (Fails without), and is only a first approximation for drugs with a pharmacologically significant distribution phase, which require a two-compartment model instead.

Corollaries & converses
  • dose-response sets the concentration range a drug must reach and sustain to be effective; pharmacokinetics determines when, and for how long, plasma concentration will actually sit within that effective range for a given dosing regimen.
  • A loading dose (\(\text{Dose}_{\text{load}}=V_d\cdot C_{\text{target}}\)) rapidly achieves a target concentration by filling the whole distribution volume at once, while a maintenance dose rate (\(\text{Rate}=CL\cdot C_{\text{target}}\)) simply replaces what clearance removes — two direct, practical applications of Steps 3 and 4's parameters.
  • Converse: a semi-logarithmic plasma-concentration-versus-time plot that is not a single straight line, but shows two distinct slopes, is itself evidence that the drug does not obey simple one-compartment, first-order kinetics (Hypotheses), and instead has a separate distribution phase.
Fails without
  • Drop first-order elimination at high, saturating dose (Hypotheses): once the responsible metabolising enzyme is working at its maximum rate, elimination becomes zero-order (a constant rate, independent of concentration); half-life is then no longer well-defined as a fixed constant, and a modest further dose increase can cause a disproportionately large rise in steady-state concentration — the mechanism behind the narrow, dangerous therapeutic window of drugs like phenytoin at typical clinical doses.
  • Drop the one-compartment assumption for a drug with a genuine distribution phase: plasma concentration can fall rapidly at first, as drug moves out of plasma into peripheral tissue, even while pharmacologically active tissue concentrations remain high; using the single-exponential \(C_0e^{-kt}\) to predict how quickly the drug's effect ends would then significantly underestimate the true duration of action.
Common errors
  • Treating half-life as the duration of a drug's pharmacological effect, rather than the time for plasma concentration to fall by half; effect duration also depends on where the therapeutic threshold sits relative to \(C_0\).
  • Assuming that doubling the dose doubles the half-life. For first-order kinetics, \(t_{1/2}\) depends only on \(k\) (Step 3); doubling the dose changes \(C_0\), not \(k\), so half-life is unchanged.
  • Confusing clearance \(CL\) (a volume cleared per unit time) with the elimination rate constant \(k\) (a fractional rate); \(CL=k\cdot V_d\) relates them, but they carry different units and describe different things.
  • Assuming steady state is reached after a single dosing interval, rather than after roughly four to five half-lives (Step 5) — a frequent source of premature, and therefore misleading, dose adjustments made before a drug has actually equilibrated.
Discussion

Compartmental pharmacokinetic modelling has its roots in Torsten Teorell's 1937 mathematical treatment of drug distribution and elimination, decades before the underlying enzymology (notably the cytochrome P450 system responsible for most phase I drug metabolism) was well characterised. Oral drugs are additionally subject to the first-pass effect: metabolism by the liver (and, for some drugs, the gut wall) before the drug ever reaches the systemic circulation, which is why oral bioavailability \(F\) is often well below \(1\) even for drugs that are completely absorbed from the gut.

Many clinically important drugs are better described by a two-compartment model, with a fast initial distribution phase (drug equilibrating into well-perfused tissue) followed by a slower true elimination phase; and a smaller number, including ethanol and phenytoin at ordinary doses, follow Michaelis-Menten-type saturable elimination kinetics rather than simple first-order decay, requiring the more careful treatment noted in Fails without.

Common misconception: that the formulas above implicitly give the concentration actually delivered by an oral dose without qualification. Most of this page's algebra assumes bioavailability \(F=1\) (all administered drug reaches systemic circulation), true by definition for intravenous dosing but rarely exactly true for oral dosing, where incomplete absorption and first-pass metabolism reduce \(F\) below \(1\).

Worked examples
1
\[ \text{IV bolus, } t_{1/2}=6\,\text{h}, \ C_0=80\,\text{mg/L}: \text{ find } C(12\,\text{h})\text{ and } C(24\,\text{h}) \]
\(k=\ln2/6\approx0.1155\,\text{h}^{-1}\). Twelve hours is exactly two half-lives, so \(C(12)=80/4=20\,\text{mg/L}\); twenty-four hours is four half-lives, so \(C(24)=80/16=5\,\text{mg/L}\), directly from repeated halving (Step 3) without needing the exponential formula explicitly. A
2
\[ \text{Dose}=500\,\text{mg produces }C_0=80\,\text{mg/L}: \text{ find } V_d \text{ and } CL \]
\(V_d=\text{Dose}/C_0=500/80=6.25\,\text{L}\); \(CL=k\cdot V_d=0.1155\times6.25\approx0.72\,\text{L/h}\). These two derived parameters, together with \(k\), fully characterise this drug's disposition for the purposes of designing a maintenance dosing regimen (Corollaries). A
\[ C(t)=80\,e^{-0.1155t}\,\text{mg/L}, \qquad t_{1/2}=6\,\text{h}, \qquad V_d=6.25\,\text{L}, \qquad CL\approx0.72\,\text{L/h} \]

Reading. A single measured half-life and initial concentration are sufficient to derive every other standard pharmacokinetic parameter for a one-compartment, first-order drug.

Scope. The identical procedure applies to any drug obeying the Hypotheses; departures from a clean single-exponential decline signal that a more complex model (Fails without, second bullet) is needed instead.

Further example. A patient with renal impairment requires a drug that is eliminated by first-order kinetics. The drug has a volume of distribution \(V_d = 40\,\text{L}\) and a total clearance \(CL = 4\,\text{L/h}\). The therapeutic window is \(10\) to \(20\,\text{mg/L}\). Determine the elimination rate constant \(k\), the half-life \(t_{1/2}\), and the single intravenous bolus dose required to achieve an initial concentration \(C_0\) of \(15\,\text{mg/L}\) (the midpoint of the therapeutic range).

1
\[ k = \frac{CL}{V_d} = \frac{4\,\text{L/h}}{40\,\text{L}} = 0.1\,\text{h}^{-1} \]
The elimination rate constant is derived from the fundamental relationship between clearance and volume of distribution for a one-compartment model. A
2
\[ t_{1/2} = \frac{\ln 2}{k} = \frac{0.6931}{0.1\,\text{h}^{-1}} \approx 6.93\,\text{h} \]
The half-life is calculated using the natural logarithm of 2 divided by the rate constant. This indicates the time required for the plasma concentration to decrease by 50%. A
3
\[ \text{Dose} = C_0 \times V_d = 15\,\text{mg/L} \times 40\,\text{L} = 600\,\text{mg} \]
For an instantaneous intravenous bolus, the dose is the product of the desired initial concentration and the volume of distribution. This ensures the drug is distributed throughout the body to reach the target concentration immediately. A
\[ k=0.1\,\text{h}^{-1}, \quad t_{1/2}\approx6.93\,\text{h}, \quad \text{Dose}=600\,\text{mg} \]

Reading. A dose of 600 mg will immediately place the patient at the center of the therapeutic window. Given the half-life of approximately 7 hours, the concentration will drop to 7.5 mg/L (below the therapeutic window) after one half-life, necessitating a dosing interval shorter than 6.93 hours to maintain efficacy.

Scope. This calculation assumes linear pharmacokinetics, where clearance and volume of distribution are constant and independent of concentration. It applies to drugs that follow first-order elimination kinetics and do not exhibit saturation effects.

Problems
  1. A drug has \(CL=5\,\text{L/h}\) and \(V_d=50\,\text{L}\). Find \(k\) and \(t_{1/2}\).
    SolutionFrom Step 4, \(k=CL/V_d=5/50=0.1\,\text{h}^{-1}\). From Step 3, \(t_{1/2}=\ln2/0.1\approx6.93\,\text{h}\).
  2. Explain, referencing Fails without, why exceeding the saturation capacity of a drug's metabolising enzyme converts its elimination kinetics from first-order to zero-order, and why this is clinically dangerous for a narrow-therapeutic-index drug like phenytoin.
    SolutionBelow saturation, elimination rate scales with concentration (first-order, Step 1); once the enzyme is working at its maximum capacity (\(V_{\max}\)), elimination proceeds at a roughly constant rate regardless of concentration (zero-order). Near this saturation point, a small increase in dose can no longer be matched by a proportional increase in clearance, so steady-state concentration rises disproportionately — a small, seemingly safe dose increase can push a narrow-therapeutic-index drug from a therapeutic to a toxic concentration.
  3. If a drug is dosed once every half-life (\(\tau=t_{1/2}\)), approximately how many doses are needed to reach about \(94\%\) of steady state?
    SolutionSince \(\tau=t_{1/2}\), each dosing interval corresponds to one half-life, so the "four to five half-lives" rule (Step 5) translates directly into four to five doses. After four half-lives, the fraction remaining to reach steady state is \(2^{-4}=6.25\%\), i.e. about \(93.75\%\) of steady state is achieved — approximately four doses.
  4. A drug is administered via a constant-rate intravenous infusion. The drug has a clearance \(CL = 12\,\text{L/h}\) and a volume of distribution \(V_d = 30\,\text{L}\). The target steady-state concentration \(C_{ss}\) is \(8\,\text{mg/L}\). Calculate the required infusion rate \(R_0\) in mg/h and the time required to reach 90% of the steady-state concentration.
    SolutionThe steady-state concentration during a constant infusion is given by \(C_{ss} = R_0 / CL\). Rearranging for the infusion rate: \(R_0 = C_{ss} \times CL = 8\,\text{mg/L} \times 12\,\text{L/h} = 96\,\text{mg/h}\). The time to reach a certain fraction of steady state depends on the elimination rate constant \(k\). First, calculate \(k = CL / V_d = 12 / 30 = 0.4\,\text{h}^{-1}\). The fraction of steady state reached at time \(t\) is \(1 - e^{-kt}\). Setting this to 0.90: \(0.90 = 1 - e^{-0.4t}\), which simplifies to \(e^{-0.4t} = 0.10\). Taking the natural logarithm: \(-0.4t = \ln(0.10) \approx -2.3026\). Solving for \(t\): \(t = 2.3026 / 0.4 \approx 5.76\,\text{h}\).
  5. Explain why the area under the curve (AUC) of a drug's plasma concentration-time profile is inversely proportional to its clearance, and describe how this relationship is used to determine bioavailability (\(F\)) when comparing an oral dose to an intravenous dose.
    SolutionClearance (\(CL\)) is defined as the volume of plasma completely cleared of the drug per unit time. The AUC represents the total systemic exposure to the drug. By definition, \(CL = \text{Dose} / \text{AUC}\) for an intravenous bolus dose where bioavailability \(F=1\). Therefore, \(\text{AUC} = \text{Dose} / CL\). This inverse relationship means that a drug with low clearance will have a high AUC (long exposure) for a given dose, while a drug with high clearance will have a low AUC. When comparing an oral dose (\(D_{oral}\)) to an IV dose (\(D_{iv}\)), bioavailability is calculated as \(F = (\text{AUC}_{oral} \times D_{iv}) / (\text{AUC}_{iv} \times D_{oral})\). If the doses are equal, \(F = \text{AUC}_{oral} / \text{AUC}_{iv}\). This ratio quantifies the fraction of the administered dose that reaches the systemic circulation unchanged.