Pharmacokinetics
Mathematical models of drug absorption, distribution, and elimination
A Brief History
What Is Pharmacokinetics?
Pharmacokinetics is the study of how the body processes drugs: how they enter the body, spread through tissues, undergo chemical transformation, and are eventually eliminated. The acronym ADME captures the four fundamental processes: Absorption, Distribution, Metabolism, and Excretion.
It is crucial to distinguish pharmacokinetics from pharmacodynamics. While pharmacokinetics asks "What does the body do to the drug?", pharmacodynamics asks "What does the drug do to the body?" In other words, pharmacokinetics describes the time course of drug concentration in the body, whereas pharmacodynamics describes the relationship between drug concentration and biological response.
The mathematical framework of pharmacokinetics was established by Torsten Teorell in 1937, who proposed the first compartmental models to describe drug kinetics. This pioneering work laid the foundation for all modern quantitative approaches to drug dosing and therapeutic drug monitoring.
The One-Compartment Model
The simplest pharmacokinetic model treats the entire body as a single, well-mixed compartment. This is a gross simplification — the body is obviously not homogeneous — but for many drugs and time scales, this model provides remarkable predictive power.
Consider an intravenous (IV) bolus injection: a dose $D$ is instantaneously delivered into the bloodstream. Assuming first-order kinetics (the rate of elimination is proportional to the concentration $C$), the drug concentration follows the differential equation:
$$ \frac{dC}{dt} = -k_e C $$where $k_e$ is the elimination rate constant (units: time$^{-1}$). This is a simple exponential decay:
$$ C(t) = C_0 e^{-k_e t} $$where $C_0$ is the initial concentration immediately after bolus injection.
Half-life: The time taken for the drug concentration to fall to half its initial value: $$t_{1/2} = \frac{\ln 2}{k_e} \approx \frac{0.693}{k_e}$$
Volume of Distribution: A theoretical volume that relates dose to initial concentration: $$V_d = \frac{D}{C_0}$$ This parameter reflects how widely the drug distributes through the body; a large $V_d$ indicates extensive tissue binding or distribution.
Clearance: The volume of plasma completely cleared of drug per unit time: $$\text{CL} = k_e V_d = \frac{D}{AUC}$$ where AUC is the area under the concentration-time curve. This is perhaps the most clinically important parameter, as it determines the rate at which the body eliminates drug.
The product $k_e V_d$ (clearance) is independent of dose and represents an intrinsic property of the drug–body system. Knowing clearance and target concentration, one can calculate the maintenance dose required to sustain therapeutic levels.
Oral Administration
Most drugs are taken orally, not intravenously. Oral absorption is not instantaneous; instead, drug gradually enters the bloodstream from the gastrointestinal tract. This is often modeled as first-order absorption with rate constant $k_a$.
The one-compartment model with first-order absorption and elimination gives:
$$ \frac{dC}{dt} = \frac{k_a F D}{V_d} e^{-k_a t} - k_e C $$where $F$ is the bioavailability — the fraction of the dose that reaches the systemic circulation. For IV administration, $F = 1$ by definition; for oral dosing, $F < 1$ due to incomplete absorption and first-pass hepatic metabolism.
The solution to this linear ODE is the Bateman equation:
$$ C(t) = \frac{k_a F D}{V_d (k_a - k_e)} \left( e^{-k_e t} - e^{-k_a t} \right) $$This shows biphasic kinetics: an absorption phase (rising concentration) followed by an elimination phase (falling concentration).
The maximum concentration $C_{\max}$ occurs when absorption and elimination rates are equal, i.e., when $\frac{dC}{dt} = 0$. This happens at: $$t_{\max} = \frac{\ln(k_a / k_e)}{k_a - k_e}$$
Note: $t_{\max}$ depends only on the rate constants, not on the dose. The peak concentration is: $$C_{\max} = \frac{k_a F D}{V_d (k_a - k_e)} \left( e^{-k_e t_{\max}} - e^{-k_a t_{\max}} \right)$$
Bioavailability $F$ is estimated from the ratio of oral to IV areas under the curve (AUC): $$F = \frac{\text{AUC}_{\text{oral}}}{\text{AUC}_{\text{IV}}} \times \frac{D_{\text{IV}}}{D_{\text{oral}}}$$
Oral dosing is more practical and safer than IV injection, but also more variable: food, pH, and GI motility affect absorption. The interindividual variation in bioavailability is a major source of dosing uncertainty in clinical practice.
Repeated Dosing
Chronic disease management requires repeated doses. If doses are given at regular intervals (the dosing interval $\tau$), concentrations accumulate over successive doses. The key principle is superposition: the concentration at any time is the sum of contributions from each prior dose.
For a first-order elimination process, if the first dose produces a concentration profile $C(t) = C_0 e^{-k_e t}$, then the second dose (given at time $t = \tau$) contributes $C_0 e^{-k_e(t - \tau)}$ for $t > \tau$. The total concentration after $n$ doses is:
$$ C(t) = C_0 \left[ e^{-k_e t} + e^{-k_e(t - \tau)} + e^{-k_e(t - 2\tau)} + \cdots \right] $$At steady state (as $n \to \infty$), the trough concentration (just before a dose) and peak concentration (just after a dose) are:
$$ C_{\text{peak}}^{\infty} = \frac{C_0}{1 - e^{-k_e \tau}}, \quad C_{\text{trough}}^{\infty} = \frac{C_0 \, e^{-k_e \tau}}{1 - e^{-k_e \tau}} $$The ratio of steady-state concentration to single-dose concentration is the accumulation factor: $$R = \frac{C_{\text{ss}}}{C_0} = \frac{1}{1 - e^{-k_e \tau}}$$
When $k_e \tau \gg 1$ (short half-life relative to dosing interval), $R \approx 1$ and little accumulation occurs. When $k_e \tau \approx \ln 2$ (half-life equals interval), $R \approx 2$. When $k_e \tau \ll 1$ (long half-life relative to interval), $R$ becomes large, potentially causing toxicity if not managed carefully.
Steady state is reached after approximately 5 half-lives, at which point the amount absorbed per dosing interval equals the amount eliminated.
A critical clinical concern is maintaining drug concentration within the therapeutic window: above the minimum effective concentration (MEC) for efficacy, but below the minimum toxic concentration (MTC) to avoid adverse effects. The width of this window and the accumulation factor determine optimal dosing regimens.
Explore: Drug Concentration Profiles
Two-Compartment Model
The one-compartment model assumes instantaneous equilibration throughout the body. However, many drugs exhibit a distinctly slower distribution phase. For example, a drug may rapidly enter the blood and central organs, but then gradually distribute into peripheral tissues over hours or days.
The two-compartment model divides the body into a central compartment (blood, heart, lungs, liver, kidneys — highly perfused organs) and a peripheral compartment (muscle, fat, other tissues — slower to equilibrate). Drug moves between compartments, and only the central compartment is eliminated.
The governing equations are:
$$ \begin{aligned} \frac{dC_c}{dt} &= -k_{cp} C_c - k_e C_c + k_{pc} C_p \\ \frac{dC_p}{dt} &= k_{cp} C_c - k_{pc} C_p \end{aligned} $$where $C_c$ and $C_p$ are central and peripheral concentrations, $k_{cp}$ and $k_{pc}$ are intercompartmental rate constants, and $k_e$ is the elimination rate from the central compartment.
The solution for the central compartment concentration is bi-exponential:
$$ C_c(t) = A e^{-\alpha t} + B e^{-\beta t} $$where $\alpha$ and $\beta$ are the slopes of the fast (distribution) and slow (elimination) phases, respectively, and $A$ and $B$ are their amplitudes. The fast phase represents redistribution from the central to peripheral compartment; the slow phase represents overall body elimination.
The two-compartment model is essential for drugs with significant tissue binding or slow distribution, such as digoxin (a cardiac glycoside), aminoglycosides, and many anticancer agents. When is one compartment enough? The one-compartment model is adequate when distribution equilibrates quickly relative to the timescale of elimination, or when we are interested only in the late, post-distributive phase.
Nonlinear Pharmacokinetics
All models discussed so far assume first-order kinetics: the rate of elimination (and absorption) is proportional to concentration. This is a valid approximation only when drug concentrations are well below the saturation threshold of the metabolic enzymes or transport proteins involved.
At higher concentrations, metabolic capacity is saturated, and the system exhibits Michaelis–Menten kinetics:
$$ \frac{dC}{dt} = -\frac{V_{\max} C}{K_m + C} $$where $V_{\max}$ is the maximum elimination rate (capacity limit) and $K_m$ is the Michaelis constant (concentration at which elimination rate is half $V_{\max}$).
Two limiting cases reveal the importance of this nonlinearity:
Low concentration ($C \ll K_m$): The denominator is dominated by $K_m$, so the rate is approximately $-\frac{V_{\max}}{K_m} C$, giving first-order kinetics with an effective rate constant $k_e = V_{\max}/K_m$.
High concentration ($C \gg K_m$): The denominator is dominated by $C$, so the rate is approximately $-V_{\max}$, a constant — zero-order kinetics. The drug concentration falls linearly with time, independent of how much drug is present.
Critically, nonlinear kinetics creates dose-dependent half-lives. A small dose is eliminated quickly (first-order); a large dose accumulates because the metabolic machinery becomes saturated. Classic examples include ethanol (alcohol), phenytoin (an antiepileptic), and salicylates (aspirin). This phenomenon is clinically dangerous: a modest increase in dose can cause disproportionate, sometimes toxic, increases in steady-state concentration.
When elimination follows Michaelis–Menten kinetics, the clearance is no longer constant but depends on concentration: $$\text{CL}(C) = \frac{V_{\max}}{K_m + C}$$
At low $C$, $\text{CL} \approx V_{\max}/K_m$ is high and fixed. As $C$ increases, clearance decreases — a counterintuitive result that has serious consequences for dosing. The therapeutic index (ratio of toxic to therapeutic dose) narrows considerably for such drugs.
Exercises
Conceptual Questions
- Explain the difference between $V_d$ (volume of distribution) and the actual volume of body fluid. Why might a drug have $V_d > 60$ L even though total body water is ~40 L?
- What is the relationship between clearance $\text{Cl}$, half-life $t_{1/2}$, and volume of distribution? Show that $t_{1/2} = 0.693 V_d / \text{Cl}$.
- In a one-compartment model after IV bolus, plasma concentration follows $C(t) = (D/V_d)e^{-kt}$. Why is the curve exponential, and what does the slope of a semi-log plot reveal?
- Explain why steady-state concentration during continuous infusion is $C_{ss} = R / \text{Cl}$ regardless of the clearance mechanism (hepatic, renal, etc.). How does doubling the infusion rate $R$ affect $C_{ss}$?
- In nonlinear (Michaelis-Menten) pharmacokinetics, why does doubling the dose not double the peak concentration? What happens to the elimination rate as concentration increases?
Computer Problems
- One-Compartment IV Bolus Simulation. Implement $\frac{dC}{dt} = -k C$ with bolus dose $D = 100$ mg and volume $V_d = 50$ L. Use $k = 0.2$ hr$^{-1}$ (half-life $\approx 3.5$ hours). Plot $C(t)$ over 24 hours, compute half-life from the curve, and overlay a semi-log plot to verify linearity.
- Two-Compartment Model with Absorption. Implement first-order absorption and two-compartment elimination: $\frac{dA}{dt} = -k_a A$, $\frac{dC_c}{dt} = (k_a A - k_{10} C_c - k_{12} C_c + k_{21} C_p) / V_c$, and $\frac{dC_p}{dt} = (k_{12} C_c - k_{21} C_p) / V_p$. Use realistic parameters (e.g., $k_a = 0.5$, $k_{10} = 0.1$, $k_{12} = 0.2$, $k_{21} = 0.15$) and plot central vs peripheral concentration over time.
- Therapeutic Range and Dosing Intervals. For a drug with $V_d = 50$ L, $\text{Cl} = 10$ L/hr, therapeutic window [5, 20] mg/L, design a dosing schedule. Compute the loading dose, maintenance dose, and dosing interval to achieve $C_{min} = 5$ and $C_{max} = 20$. Simulate plasma concentration over multiple dose intervals.
- Nonlinear Pharmacokinetics (Michaelis-Menten). Implement $\frac{dC}{dt} = -\frac{V_{\max} C}{K_m + C}$ with $V_{\max} = 100$ mg/hr, $K_m = 50$ mg/L. Plot elimination rate versus concentration for low (10 mg/L), medium (50 mg/L), and high (500 mg/L) doses. Show the transition from first-order to zero-order kinetics and compute the time to reach steady state.
- Multiple-Dose Accumulation. Simulate repeated oral doses (100 mg every 6 hours for 7 days) with absorption half-life 1 hour and elimination $\text{Cl} = 10$ L/hr, $V_d = 50$ L. Plot plasma concentration including all doses and transients. Compute the accumulation factor and steady-state $C_{min}$ and $C_{max}$.
References
- Teorell, T. (1937). Kinetics of distribution of substances administered to the body, I: The extravascular modes of administration. Arch. Int. Pharmacodyn. Ther., 57, 205–225.
- Rowland, M. & Tozer, T. N. (2011). Clinical Pharmacokinetics and Pharmacodynamics: Concepts and Applications (4th ed.). Wolters Kluwer.
- Gibaldi, M. & Perrier, D. (1982). Pharmacokinetics (2nd ed.). Marcel Dekker.
- Shargel, L., Yu, A. B., & Pabla, K. S. (2016). Applied Biopharmaceutics & Pharmacokinetics (6th ed.). McGraw-Hill.
- Wagner, J. G. (1975). Fundamentals of Clinical Pharmacokinetics. Drug Intelligence Publications.
- Bateman, H. (1910). Solution of a system of differential equations occurring in the theory of radioactive transformations. Proc. Cambridge Philos. Soc., 15, 423–427.
- Michaelis, L. & Menten, M. L. (1913). Die Kinetik der Invertinwirkung. Biochem. Z., 49, 333–369.
- Pang, K. S. & Rowland, M. (1977). Hepatic clearance of drugs. I. Theoretical considerations of a “well-stirred” model and a “parallel tube” model. Influence of hepatic blood flow, plasma and blood cell binding, and the hepatocellular enzymatic activity on hepatic drug clearance. J. Pharmacokinet. Biopharm., 5, 625–653.
- Brunton, L. L., Knollmann, B. C., & Hilal-Dandan, R. (Eds.). (2018). Goodman & Gilman's The Pharmacological Basis of Therapeutics (13th ed.). McGraw-Hill.
- Atkinson, A. J., Abernethy, D. R., Daniels, C. E., Dedrick, R. L., & Markey, S. P. (Eds.). (2007). Principles of Clinical Pharmacology (2nd ed.). Academic Press.