Zero-Order vs First-Order Elimination: Graphs & PEA Drugs

First-order elimination removes a constant fraction of the drug per unit time: the rate is proportional to concentration, the half-life is constant, and the concentration falls along a curve. Zero-order elimination removes a constant amount per unit time: the rate is fixed at the enzyme's maximum, there is no constant half-life, and the concentration falls in a straight line. Most drugs are first-order at therapeutic doses. The classic zero-order exceptions are phenytoin, ethanol and high-dose aspirin (PEA), whose metabolizing enzymes saturate. Phenytoin follows Michaelis-Menten (saturation) kinetics: its liver metabolism saturates at high levels, so once levels reach the upper therapeutic range a small dose increase can produce a large jump in its level.

The visual hack: ruler vs ski slope

Picture two graphs of plasma concentration against time.

• Zero-order = ruler. On ordinary (linear) axes the concentration falls along a straight line, like a ruler laid on a slope: the same number of milligrams disappears every hour.
• First-order = ski slope. On linear axes the curve is steep at first and flattens out: the same percentage disappears every hour, so less and less is removed as the level falls.
• Flip to a semilog plot (log of concentration on the y-axis) and the pictures swap: first-order becomes the straight line, and zero-order curves downward ever more steeply.

The one-line rule: straight on normal paper = zero-order; straight on log paper = first-order.

Merck's example makes it concrete. Start with 500 mg in the body:

Side-by-side comparison

The equations (and why half-life behaves differently)

First-order: the rate depends on how much drug is left.

$\frac{dC}{dt} = -kC \quad\Longrightarrow\quad C_t = C_0\,e^{-kt} \quad\Longrightarrow\quad \ln C_t = \ln C_0 - kt$

The last form is why first-order data make a straight line on a semilog plot. Setting $C_t = C_0/2$ gives a half-life that does not depend on concentration:

$t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}, \qquad k = \frac{CL}{V_d} \;\Longrightarrow\; t_{1/2} = \frac{0.693 \times V_d}{CL}$

After $n$ half-lives the fraction left is $(1/2)^n$: about 6% after 4 half-lives and 3% after 5. That is the math behind the "4 to 5 half-lives" rule for washout and for reaching steady state, covered in the half-life and steady state rule.

Zero-order: the rate is a constant, so concentration falls linearly.

$\frac{dC}{dt} = -k_0 \quad\Longrightarrow\quad C_t = C_0 - k_0 t \quad\Longrightarrow\quad t_{1/2} = \frac{C_0}{2k_0}$

The "half-life" is proportional to the starting concentration: the higher the level, the longer it takes to halve. That is why a zero-order drug has no single half-life, and why the Merck formula $t_{1/2} = 0.693/k$ applies to first-order elimination only.

Why saturation switches the order: Michaelis-Menten kinetics

Enzymatic elimination follows the Michaelis-Menten equation:

$v = \frac{V_{max}\, C}{K_m + C}$

• When $C \ll K_m$: $v \approx (V_{max}/K_m)\,C$. The rate is proportional to concentration, so elimination is first-order. At therapeutic levels of most drugs only a small fraction of enzyme sites is occupied, which is why most drugs sit here.
• When $C \gg K_m$: $v \approx V_{max}$. The enzyme is saturated and the rate stops rising, so elimination is zero-order.
• $K_m$ is the concentration at which the rate is half of $V_{max}$.

So a "zero-order drug" is one whose usual concentrations are high relative to its $K_m$, so the enzyme runs near $V_{max}$. As a drug's concentration rises, its metabolism shifts from first-order toward zero-order. The same $V_{max}$ and $K_m$ define enzyme inhibition graphs in the Lineweaver-Burk plot guide.

PEA: the classic zero-order drugs

Mnemonic: PEA, because a pea is round like the 0 in zero-order.

Phenytoin is also a classic CYP450 inducer; see the CYP450 inducers and inhibitors list.

Phenytoin: saturation kinetics with numbers

At steady state, the daily dose ($R$) equals the elimination rate, so rearranging the Michaelis-Menten equation gives:

$C_{ss} = \frac{K_m \cdot R}{V_{max} - R}$

As the dose approaches $V_{max}$, the denominator shrinks toward zero and the level climbs steeply. If the dose reaches $V_{max}$, there is no steady state: the level keeps rising. Illustrative numbers for a hypothetical patient (not reference values; bioavailability and salt form ignored) with $V_{max}$ = 500 mg/day and $K_m$ = 4 mg/L (4 mcg/mL):

A first-order drug would have risen in proportion to the dose (6 → 8 → 9). This is why the Dilantin label warns that "small incremental doses may increase the half-life and produce very substantial increases in serum levels, when these are in the upper range," that a dose increase of 10% or more can disproportionately raise the steady-state level, and that dose changes should be at least 7–10 days apart.

Exam traps

• "Constant half-life" always means first-order. A zero-order drug has no single half-life.
• The direction of the switch: zero-order at high concentrations, first-order at low concentrations, never the reverse.
• Aspirin is first-order at usual doses; it becomes zero-order only when its pathways saturate at high doses.
• Zero-order does not mean "no elimination." It means a fixed amount per hour, however high the level.
• Clearance is not constant in zero-order kinetics: with a fixed rate, clearance (rate ÷ concentration) falls as the level rises.

Practice question

A 43-year-old man with alcohol use disorder is brought to the emergency department after drinking heavily at a party. He is drowsy but protecting his airway, and his vital signs are stable. Serum ethanol levels are drawn every 2 hours while he is observed. Each level is lower than the last by the same number of mg/dL, and a plot of concentration against time on standard (linear) axes is a straight line. Which of the following best describes the elimination of ethanol in this patient?

• A. Clearance is constant across all plasma concentrations
• B. A constant fraction of the ethanol is eliminated per unit time
• C. A constant amount is eliminated per unit time because alcohol dehydrogenase is saturated
• D. The half-life lengthens as the ethanol concentration falls
• E. Elimination is first-order at these concentrations and becomes zero-order as levels fall

Answer: C. A straight-line fall on linear axes means the same amount is removed each hour: zero-order elimination. Alcohol dehydrogenase has a low $K_m$ for ethanol, so it is saturated at intoxicating concentrations and works at $V_{max}$ regardless of how much alcohol is present.

Why the others are wrong:

• A: Constant clearance is a first-order property. When the rate is fixed, clearance (rate ÷ concentration) is lowest at the highest levels and rises as the level falls.
• B: A constant fraction per unit time is the definition of first-order elimination, which gives a curve on linear axes and a straight line only on a semilog plot.
• D: This is backwards. With a fixed amount removed per hour, the time needed to halve the level ($C_0/(2k_0)$) gets shorter as the level falls; the half-life lengthens as levels rise, as with phenytoin.
• E: The switch runs the other way: saturable drugs are zero-order at high concentrations and approach first-order once levels fall below saturation.

Key takeaways

• First-order: constant fraction per unit time, rate proportional to concentration, constant half-life ($t_{1/2} = 0.693/k$), curved on linear axes and straight on semilog axes. Most drugs.
• Zero-order: constant amount per unit time, fixed rate at $V_{max}$, no constant half-life, straight line on linear axes.
• Michaelis-Menten: first-order when $C \ll K_m$, zero-order when $C \gg K_m$; saturation drives the switch.
• PEA = phenytoin, ethanol, high-dose aspirin.
• Phenytoin: small dose increases can cause disproportionate rises in level; watch for nystagmus and ataxia, and change doses no more often than every 7–10 days.
• Ethanol: alcohol dehydrogenase saturates at low concentrations, so alcohol is removed at a roughly constant rate.

Sources

• Drug Metabolism (Rate: first-order vs zero-order kinetics) — Merck Manual Professional Edition
• Overview of Pharmacokinetics (Formulas Defining Basic Pharmacokinetic Parameters) — Merck Manual Professional Edition
• 12.4 Integrated Rate Laws — OpenStax Chemistry 2e
• Michaelis-Menten Kinetics — Chemistry LibreTexts
• Dilantin (extended phenytoin sodium capsules) Prescribing Information — DailyMed, NLM (2023)
• Cederbaum AI. Alcohol Metabolism — Clinics in Liver Disease (2012)
• Runde TJ, Nappe TM. Salicylates Toxicity — StatPearls, NCBI Bookshelf (2023)
• Aspirin and Other Salicylate Poisoning — Merck Manual Professional Edition