When clearance stops being constant,
and how a pump compensates.

Left panel. x: drug concentration (mg·L⁻¹). y: rate of elimination (mg·h⁻¹). The curve is a rectangular hyperbola: nearly linear at low concentration, then flattening asymptotically towards V_max. Construction lines show that it reaches V_max/2 exactly at C = K_m.

The three shaded zones are the whole point of the diagram. Zero-order and first-order kinetics are not two different mechanisms — they are the same curve, read at different concentrations. Where you sit on the x-axis relative to K_mdecides which description applies.

Teal · first order, C < K_m/2. The denominator (K_m + C) is dominated by K_m, so the curve is effectively a straight line through the origin and rate is proportional to concentration. Clearance is effectively constant, half-life is constant, and every equation from lesson 1 applies. At the boundary the rate is exactly V_max/3 — substitute C = K_m/2 and the denominator becomes 1.5 K_m.

Gold · mixed order, K_m/2 to 2K_m. Neither simplification is safe. Clearance is measurably falling but the rate has not yet plateaued, so behaviour is genuinely intermediate and cannot be predicted by either set of rules. Note that K_m itself sits in the middle of this zone, at exactly half V_max. This is where phenytoin sits at therapeutic concentrations, which is precisely why it is difficult to dose.

Coral · zero order, C > 2K_m. The denominator is now dominated by C, the C terms very nearly cancel, and rate approaches the constant V_max — at the boundary it is already 2 V_max/3. A fixed amount is removed per unit time regardless of how much is present. Clearance is falling steeply and half-life has ceased to be a useful concept.

Move the K_m slider and watch the zones slide. A low K_m (high affinity) pushes the whole first-order zone into a narrow strip at the left, so the drug is saturated across almost the entire concentration range — that is ethanol. A high K_m pushes the zero-order zone off the right of the plot entirely, so the drug behaves first-order at every clinically achievable concentration — that is most anaesthetic drugs. The drug has not changed; only where K_m falls relative to the concentrations you actually use.

Why linearise at all? V_max is an asymptote. It cannot be read accurately from a hyperbola because the curve never reaches it. Taking reciprocals of the whole equation converts the hyperbola into a straight line whose intercepts give both constants exactly. That was Lineweaver and Burk’s entire motivation in 1934.

The algebra. Invert both sides: 1/rate = (K_m + C) ÷ (V_max·C). Split the fraction: 1/rate = K_m/(V_max·C) + C/(V_max·C). Simplify the second term: 1/rate = (K_m/V_max)·(1/C) + 1/V_max, which has the form y = mx + c.

Right panel. x: 1/C. y: 1/rate. Gradient = K_m/V_max; y-intercept = 1/V_max; x-intercept = −1/K_m — note the negative sign, obtained by setting 1/rate = 0. Move the V_max slider and the line pivots about its x-intercept; move K_m and it pivots about its y-intercept. That behaviour is exactly how competitive inhibition (which raises apparent K_m only) is distinguished experimentally from non-competitive inhibition (which lowers V_max only).

The violet operating point appears on both plots. Slide it from left to right and watch the clearance figure in the readout: constant while C is well below K_m, then falling progressively. That falling number is the whole of phenytoin toxicity.

Axes. x: maintenance dosing rate (mg·day⁻¹). y: steady-state plasma concentration (mg·L⁻¹). The shaded band marks a representative phenytoin therapeutic range of 10–20 mg·L⁻¹.

What to notice. The coral dashed line is what would happen if clearance were constant — a straight line through the origin, so doubling the dose doubles the concentration. The teal curve is the truth for a saturable drug: indistinguishable from linear at low doses, then diverging upward and rising asymptotically as R approaches V_max. Beyond V_max no steady state exists at all.

Try this. With V_max = 400 mg/day and K_m = 6 mg/L, move the dose from 300 to 330 mg/day — a 10% increase — and read the concentration change. This is the arithmetic behind the clinical rule that phenytoin doses are adjusted in small increments with therapeutic drug monitoring. Note too that V_max and K_m vary between individuals, so the position of the vertical asymptote cannot be predicted from population data — which is precisely why monitoring is required rather than a nomogram.

Axes. x: time (hours). y: plasma concentration, switchable between linear and logarithmic. The pair is the point. A first-order decay is a curve on one and a straight line on the other, and it is the straight line that shows the rate constant is constant — neither plot states that on its own.

The four-panel logic to reproduce. First-order elimination gives a curve on linear axes and a straight line on logarithmic axes of gradient −k. Zero-order gives a straight line on linear axes and a curve that becomes progressively steeper on logarithmic axes. If you remember only one thing, remember that a straight line on a semilogarithmic plot is the signature of a first-order process.

Half-life. On the first-order curve the intervals between successive halvings are equal — that is what a constant half-life looks like. On the zero-order line, the readout shows the time to halve from C₀ and from C₀/2: they are different, because a fixed amount is removed each hour from an ever-smaller quantity. Half-life is not a useful concept under zero-order kinetics.

The violet Michaelis–Menten curve is the real behaviour, computed by numerically integrating dC/dt = −V_max·C/(K_m + C). Note that it lies close to the zero-order line while concentration is far above K_m, then bends and develops an exponential tail once concentration falls through K_m.

The shaded bands are the same three zones as Figure 6, turned on their side. There, concentration was the horizontal axis and the zones were vertical strips; here concentration is the vertical axis, so they become horizontal bands. The boundaries are identical: first order below K_m/2, zero order above 2K_m, mixed between.

Now watch what the drug does. A falling concentration does not stay in one zone — it descends through all three. The Michaelis–Menten curve begins in the coral band running almost straight, exactly like the zero-order line beside it; passes through the gold band where it visibly bends; and finishes in the teal band as a true exponential tail parallel to the first-order curve. This is the transition itself — a zero-order process reverting to first-order as substrate falls — and here it is a single continuous curve crossing two boundaries rather than a fact to be recalled.

Raise the K_m slider and the bands rise with it: the transition happens earlier and at a higher concentration, and the curve spends most of its life first-order. Lower K_m and the coral band swallows almost the whole plot, so the drug stays zero-order nearly to exhaustion — the behaviour of ethanol, whose alcohol dehydrogenase has a K_m far below any concentration reached by drinking.

Reading the diagrams. Each box is a volume of distribution; each arrow is a first-order rate constant describing the proportion of drug transferred per unit time. Drug enters the central compartment and is eliminated only from the central compartment — this is true of all three models and is the most commonly mislabelled feature.

Subscript convention. k₁₂ means transfer from compartment 1 to compartment 2; k₂₁ is the reverse. k₁₀ means transfer from compartment 1 to “compartment zero”, meaning out of the body — elimination. The effect site is drawn with a dashed outline because it has a rate constant but no volume, and is therefore excluded from V_ss.

Note what increases with each model. One compartment yields one exponential and a straight line on log axes. Two compartments yield two exponentials, hence a distribution phase followed by an elimination phase. Three compartments yield three exponentials — rapid distribution, slow distribution, and terminal elimination. The number of exponentials always equals the number of compartments.

Axes. x: time after bolus (minutes) — use the time-window buttons to expand the early portion where the α and β phases occur. y: plasma concentration as a percentage of C₀ on a logarithmic scale, which is essential because it renders each exponential component as a straight line.

Curve stripping — how a model is obtained from data. Take the terminalportion of the observed curve, which is straight because only the slowest exponential still contributes: its slope gives γ and its back-extrapolation to time zero gives the coefficient C. Subtract that line from the observed data; the residual is again a straight line, giving β and B. Subtract once more to obtain α and A. Three straight lines, extracted in reverse order.

Compare the models. The one-compartment curve is a single straight line with no distribution phase at all, which is why it cannot describe an induction agent. Adding compartments adds early curvature while barely altering the terminal slope. This is the visual reason why terminal half-life tells you almost nothing about how quickly a patient wakes: waking occurs during the α and β phases, which the terminal slope does not describe.

Separation of the exponents. Widen the gap between α, β and γ and the phases become cleanly distinguishable; bring them together and the curve straightens into something a simpler model would fit adequately. Whether a drug “needs” three compartments is therefore not a property of the drug alone — it depends on whether the exponents are separable given the sampling schedule used.

Axes. x: time (hours). y: plasma concentration expressed as a multiple of the steady-state concentration achieved at rate X.

Compare the teal and coral curves. Doubling the infusion rate doubles the plateau but does not shorten the time taken to reach it — both are at 50% of their own plateau at one half-life, 75% at two, 87.5% at three. Drag the half-life slider and both curves stretch together. Time to steady state is governed by half-life alone.

Press “Show loading dose + infusion”. The violet trace is the solution to the problem: a bolus fills the volume of distribution immediately, and the infusion then replaces only what is being cleared. This is the two-part logic — LD = V_d × C_target, then rate = Cl × C_target — which, extended to three compartments, becomes the BET scheme.

Axes. x: time after bolus (minutes). y: concentration as a percentage of the peak plasma concentration.

Three things to read from it. First, C_e always peaks laterand lower than C_p. Second, the curves cross exactly at the moment C_e peaks — necessarily so, because dC_e/dt = 0 requires C_p = C_e. That crossing is the time to peak effect. Third, after the crossing C_e exceeds C_p: the brain concentration is now higher than the plasma concentration, which is why effect persists while plasma levels are falling.

Drag k_e0 to the left for slow equilibration, as with fentanyl: the effect-site curve becomes flatter, lower and later, and hysteresis is large. Drag it right for rapid equilibration, as with alfentanil: C_e tracks C_p closely, hysteresis is minimal, and both onset and offset are rapid. This single parameter explains why alfentanil is chosen when a rapid, titratable opioid effect is needed for a brief intense stimulus.

Axes. x: duration of infusion (hours). y: context-sensitive half-time (minutes). The grey dashed line is the model’s own terminal elimination half-life.

This is a genuine computation. The figure numerically integrates a three-compartment model, running an infusion whose rate is continuously adjusted to hold the plasma concentration constant, then stopping it and measuring the time for concentration to fall by half. Every point is that measurement repeated for a different infusion duration. It is a generic model illustrating the mechanism, not the parameter set of any real drug — published values for real drugs are tabulated below.

Slider 1 sets how high the curve goes. At a low clearance ratio the curve is nearly flat: elimination dominates and peripheral loading is irrelevant — remifentanil-like behaviour. Raise the ratio and CSHT rises steeply.

Slider 2 sets how long it keeps climbing. The readout shows the deep compartment time constant τ₃ and the point at which the curve plateaus. Make V₃ small and the curve flattens early; make it large and it climbs across the whole 8 hours — which is why fentanyl’s CSHT is still rising at 8 hours while propofol’s has long since levelled off.

Note the grey line. Terminal elimination half-life sits far above the CSHT and does not move when the infusion duration changes, because it is a property of the terminal exponential alone. That divergence is the entire clinical argument for using CSHT instead.

Axes. x: time from starting the infusion (minutes). y: propofol concentration (µg·mL⁻¹). The grey dashed line is the selected target.

What is being simulated. A three-compartment model with an effect site, integrated numerically. The model presets set only the two parameters that differ most between the published algorithms — V₁ and k_e0 — with the remaining compartments held constant, so the figure isolates the effect of those two variables. Real published models differ in other parameters too, and this simulation is illustrative of the principle rather than a reproduction of any commercial pump.

Compare Marsh and Schnider in plasma-targeting mode. Marsh has a V₁ of about 15.9 L in a 70 kg adult; Schnider fixes V₁ at 4.27 L. Since the induction bolus is essentially V₁ × target, Marsh delivers roughly three to four times the initial dose.

Compare Marsh and Modified Marsh. The pharmacokinetics are identical; only k_e0 differs, 0.26 against 1.21 min⁻¹. In plasma targeting the dose delivered is identical and the drug behaves identically — only the pump’s calculated effect-site concentration rises faster. This is the clearest possible demonstration that the model changes the calculation, not the patient.

Now switch to effect-site targeting. A larger bolus is given, the pump pauses, and plasma concentration overshoots the target before falling to meet the rising effect-site curve exactly at target. Note the size of the plasma overshoot with Schnider: a small V₁ combined with a moderate k_e0 drives the calculated plasma concentration to high levels. With Eleveld, the very low k_e0 of 0.146 min⁻¹ means a large initial dose is required, only partly offset by its modest V₁ of 6.28 L.