First-order elimination and what a half-life means

In a first-order model, a constant fraction of what is present is eliminated per unit time. That gives exponential decay, and the half-life is the time for the amount to fall by half regardless of where it started.

The practical consequence is the five half-lives rule. After one half-life 50 percent remains, after two 25, after three 12.5, and after five about three percent, which is close enough to gone for most purposes.

Accumulation, and why the interval matters more than the amount

If a second amount arrives before the first has cleared, the two overlap. Repeat that and the concentration climbs to a plateau where the amount eliminated per interval equals the amount arriving.

The plateau depends on the ratio of interval to half-life, not on the size of each amount. An interval equal to the half-life gives a twofold accumulation. An interval of half the half-life gives about 3.4-fold. An interval of three half-lives gives about 1.14-fold, which is practically none.

• •Interval equal to the half-life: 2-fold accumulation
• •Interval of half the half-life: about 3.4-fold
• •Interval of two half-lives: about 1.33-fold
• •Interval of three half-lives: about 1.14-fold

What a one-compartment model leaves out

It assumes instant appearance, which is wrong for anything absorbed rather than injected intravenously. Real curves rise to a peak over a period set by absorption and then decline.

It assumes the body is one well-mixed compartment. Real distribution into tissues produces a two-phase curve, a fast distribution phase and a slower elimination phase, and a single reported half-life usually refers to the terminal phase only.

Why the trough figure was wrong here before

The plotter used to look for the trough by testing whether a sampled time point coincided exactly with a dosing time. On a floating-point sample grid that is almost never true, so the running minimum stayed at its initial value and the tool reported a trough of zero for most inputs.

Troughs are now evaluated at the dosing times directly, and the steady-state peak and trough are computed in closed form so they do not depend on how long a window happened to be plotted.

How the concentration curve is calculated

Superposition of exponentially decaying inputs, sampled densely enough to draw, with the steady-state values taken from the closed-form expressions rather than read off the sampled curve.

k                = ln(2) / half-life
C(t)             = SUM over past doses of D x e^(-k(t - t_dose))
accumulation     = 1 / (1 - e^(-k x interval))
steady peak      = D x accumulation
steady trough    = D x accumulation x e^(-k x interval)
time to plateau  = about 5 half-lives

1. 1Convert half-life to a rate constant. The natural logarithm of two divided by the half-life. This is the fraction eliminated per unit time in the exponential.
2. 2Superpose the inputs. Each amount decays independently from the moment it arrives, and the concentration at any time is the sum of all of them. Linear kinetics is what makes this valid.
3. 3Sample either side of each input. The curve is sampled just before and just after every dosing time, so the plot shows the vertical step a bolus produces rather than a diagonal ramp into it.
4. 4Read troughs at the dosing times. Evaluated directly at each dose time rather than searched for on the sample grid, which is what the previous implementation attempted and almost always failed to find.
5. 5Compute steady state in closed form. The accumulation ratio and the steady peak and trough come from the analytic expressions, so they are independent of how long a window is plotted.

What this method cannot tell you

• •One compartment with instant input. Real absorption produces a rise to a peak that this model does not show.
• •Single-phase elimination. Most real curves have a distribution phase and an elimination phase with different slopes.
• •Linear kinetics. At saturating concentrations elimination becomes concentration-independent and the model breaks.
• •The concentration axis is in arbitrary units of one amount. It is a shape, not a prediction of any measurable level.