Nonlinear Pharmacokinetics
This concluding unit addresses situations where standard linear (first-order) pharmacokinetics does NOT apply. Nonlinear (dose-dependent) pharmacokinetics occurs when one or more ADME processes become saturated — the rate of the process no longer increases proportionally with drug concentration. The Michaelis-Menten equation is the mathematical framework for describing saturable processes. Phenytoin is the classic clinical example where nonlinear kinetics makes dosing treacherous.
Syllabus & Topics
- 1Linear vs Nonlinear Pharmacokinetics: Linear (first-order) PK: rate of ADME processes is directly proportional to drug concentration → doubling the dose doubles the AUC, Css, etc. Elimination rate = KE × Cp → KE and t½ are constant regardless of dose. Nonlinear (dose-dependent) PK: AUC increases disproportionately with dose. Kinetic parameters (t½, CL, Vd) CHANGE with dose or concentration. Causes: saturation of one or more processes (enzymes, transporters, protein binding). Also called capacity-limited, saturable, or Michaelis-Menten kinetics.
- 2Characteristics of Nonlinear PK: (1) AUC NOT proportional to dose (AUC/dose ↑ with ↑dose). (2) t½ changes with dose (↑t½ at higher doses for saturable metabolism). (3) Steady-state Cp increases disproportionately with dose: small dose increase → large Css increase → DANGER of toxicity. (4) Metabolite ratios change with dose. (5) Composition of excretory products changes. (6) Clearance changes with dose. These characteristics make dosing adjustments for nonlinear drugs clinically challenging and potentially dangerous.
- 3Factors Causing Nonlinearity – Absorption: (1) Saturable active transport: Vitamins (B₁, B₂, B₁₂, C) — absorbed by specific carriers with limited capacity → ↑dose does NOT ↑proportional absorption. (2) Saturable first-pass metabolism: at high doses → hepatic enzymes saturated → ↑bioavailability (Propranolol, Verapamil — oral BA increases at higher doses). (3) Drug dissolution limitation: at high doses, dissolution (not absorption) becomes rate-limiting → incomplete absorption. (4) GI degradation: at high doses → degradation pathways saturated → ↑absorption (certain prodrugs).
- 4Factors Causing Nonlinearity – Distribution: (1) Saturable plasma protein binding: at therapeutic concentrations, many drugs bind linearly. At HIGH concentrations (overdose), binding sites saturate → disproportionate ↑ free drug → ↑Vd, ↑effect/toxicity. Example: Phenytoin — 90% bound at therapeutic levels; at near-toxic levels, binding saturates → ↑free fraction → exaggerated toxicity. (2) Saturable tissue binding: drug accumulation in tissues reaches capacity. (3) Concentration-dependent partitioning into erythrocytes or fat.
- 5Factors Causing Nonlinearity – Metabolism: MOST COMMON CAUSE of nonlinear PK. Saturable enzymatic metabolism: at low drug concentrations → first-order (rate ∝ Cp). At high concentrations → enzyme capacity approached → transition to zero-order (rate = constant, independent of Cp). Michaelis-Menten kinetics describes this transition. Classic example: Phenytoin — CYP2C9/2C19 metabolizes phenytoin → at therapeutic range (10-20 μg/mL), enzymes are near saturation → small dose increases cause LARGE ↑Css. Other examples: Ethanol (ADH saturated → zero-order elimination: ~10 mL/h), Theophylline (at high doses), Salicylates.
- 6Factors Causing Nonlinearity – Excretion: (1) Saturable active tubular secretion: organic acid transporters (OAT) have limited capacity → at high penicillin concentrations, secretion saturates → ↓CLR → ↑t½. (2) Saturable tubular reabsorption: glucose — normally completely reabsorbed (CLR = 0). Above renal threshold (~180 mg/dL) → transport maximum (Tm) exceeded → glucose appears in urine (diabetes). (3) pH-dependent reabsorption changes: if drug itself changes urine pH at high doses → altered reabsorption.
- 7Michaelis-Menten Equation: Rate of metabolism: −dCp/dt = (Vmax × Cp) / (Km + Cp). Where: Vmax = maximum rate of metabolism (mg/h or mg/L/h — when enzyme is fully saturated). Km = Michaelis constant = drug concentration at which rate = ½Vmax (mg/L or μg/mL). Km reflects enzyme affinity — low Km → high affinity. Two limiting cases: (1) When Cp << Km: rate ≈ (Vmax/Km) × Cp → FIRST ORDER (linear PK). (2) When Cp >> Km: rate ≈ Vmax → ZERO ORDER (constant rate elimination).
- 8Michaelis-Menten – Steady State: At steady state with nonlinear PK: Css = (Km × R₀) / (Vmax − R₀). Where R₀ = dosing rate (F·Dose/τ). As R₀ approaches Vmax → Css → ∞ (vertical asymptote!). This means small increases in dosing rate near Vmax → disproportionately large ↑Css → toxicity. For Phenytoin: Vmax typically 6-8 mg/kg/day, Km ≈ 4-6 μg/mL. At therapeutic Css (10-20 μg/mL), the dosing rate is close to Vmax → VERY sensitive to dose changes. A 50 mg/day ↑ in Phenytoin dose can increase Css from 15 → 40 μg/mL → seizures from toxicity.
- 9Parameter Estimation – Graphical Methods: (1) Direct linear plot: each (Dose rate, Css) pair gives a line → intersection point gives Vmax and Km. At least 2 Css values at 2 different doses needed. (2) Lineweaver-Burk (double reciprocal): plot 1/Rate vs 1/Cp → straight line: slope = Km/Vmax, y-intercept = 1/Vmax, x-intercept = −1/Km. (3) Eadie-Hofstee: plot Rate vs Rate/Cp → slope = −Km, y-intercept = Vmax. For clinical Phenytoin dosing: Orbit graph method or direct equations: R₁/Css₁ = Vmax/(Km + Css₁) and R₂/Css₂ = Vmax/(Km + Css₂) → solve two equations for two unknowns (Vmax, Km).
- 10Phenytoin – Clinical Example: Phenytoin (Dilantin): antiepileptic drug with narrow TI (10-20 μg/mL). Exhibits nonlinear PK because CYP2C9 metabolism is saturable within the therapeutic range. Clinical consequences: (1) Apparent t½ changes: 12 h at low Cp → 40+ h at high Cp. (2) Steady-state time unpredictable: may take >> 5 × t½ because t½ itself changes. (3) Dose adjustments MUST be small (25-50 mg increments, not the usual 100 mg). (4) TDM essential — monitor total AND free levels (90% protein bound; in uremia/hypoalbuminemia → free fraction ↑). (5) Genetic polymorphism: CYP2C9 poor metabolizers → lower Vmax → more pronounced nonlinearity → ↑toxicity risk.
- 11Ethanol – Nonlinear Example: Ethanol elimination: ADH (alcohol dehydrogenase) → acetaldehyde → ALDH → acetic acid. ADH has LOW Km (~0.01%) → saturated at very low ethanol concentrations → elimination follows ZERO-ORDER kinetics at virtually all physiological concentrations. Rate ≈ Vmax ≈ constant (7-10 g/h or ~10 mL absolute alcohol/h). BAC declines linearly (not exponentially) with time → implications for legal/forensic toxicology. Disulfiram: inhibits ALDH → acetaldehyde accumulates → unpleasant symptoms (used in alcohol aversion therapy).
- 12Other Nonlinear Examples: Salicylates (Aspirin → Salicylic acid): multiple metabolic pathways — glycine conjugation and glucuronidation saturate at anti-inflammatory doses (3-6 g/day) → nonlinear kinetics → risk of toxicity with small dose increases → tinnitus as early warning. Theophylline: at high concentrations, N-demethylation by CYP1A2 approaches saturation → nonlinear. Worsened by: liver disease, CHF, drug interactions (Cimetidine, Erythromycin inhibit CYP1A2). Carbamazepine: autoinduction — CYP3A4 induced by carbamazepine itself → ↑metabolism over weeks → ↓Css → may need dose increase 2-3 weeks after initiation.
Learning Objectives
Exam Prep Questions
Q1. Why is Phenytoin dosing considered dangerous?
Phenytoin’s metabolizing enzymes (CYP2C9) are near saturation within the THERAPEUTIC range (10–20 μg/mL). The Css vs Dose relationship follows a hyperbolic curve — at therapeutic doses, you’re near the steep part of this curve. A small increase in dose (e.g., 300 → 350 mg/day, just a 17% increase) can cause Css to jump from 15 → 40+ μg/mL → toxicity (nystagmus, ataxia, sedation, seizures paradoxically). This is why dose adjustments must be ≤50 mg at a time, with therapeutic drug monitoring.
Q2. What is the difference between Vmax and Km?
Vmax = the MAXIMUM rate of metabolism (mg/h) — achieved when ALL enzyme molecules are saturated (working at full capacity). Determined by: amount of enzyme and its catalytic efficiency. Km = the drug concentration at which metabolism is at HALF of Vmax (mg/L). It reflects AFFINITY — lower Km means higher affinity (enzyme reaches half-speed at lower drug levels). Together, they characterize how a drug is metabolized: high Vmax + low Km = drug handled very efficiently at low concentrations.
Q3. How does Ethanol show zero-order elimination?
Ethanol is metabolized by alcohol dehydrogenase (ADH), which has a very LOW Km (~0.01% or 0.1 g/L). Since social drinking produces BAC of 0.05–0.2% (50–200× higher than Km), ADH is ALWAYS fully saturated → rate of elimination = Vmax = constant (~7–10 g/h or ~10 mL/h of pure alcohol). Therefore, blood alcohol concentration declines LINEARLY with time (not exponentially as with first-order drugs). This is why “one standard drink per hour” is the commonly cited elimination rate.