Why Relate Washout to Half-Life?
In a crossover study, a washout interval separates treatment periods. The interval gives the drug from the preceding period time to decline before the next period begins. A useful first design calculation asks: under a specified elimination model, what fraction of the drug concentration present at the start of washout is expected to remain at its end?
The answer is expressed in half-lives. One half-life leaves one-half of the starting concentration; two leave one-quarter; three leave one-eighth. This calculation turns a proposed washout time into an interpretable residual fraction. It is a design aid, not by itself a finding that carryover is absent: the calculation concerns the modeled drug concentration, not necessarily every pharmacologic effect or the resulting study comparison.
Exponential Elimination and the Half-Life Formula
The calculation assumes first-order elimination over the interval of interest: the rate of decline in concentration is proportional to the concentration present. For a single exponential elimination phase, let \(C_0\) be the concentration at the start of washout, \(C(w)\) the concentration \(w\) time units later, and \(k\) the positive elimination rate constant, in inverse time units. The model is \(C(w)=C_0e^{-kw}\). It assumes no additional dosing during the interval and that this exponential model is an appropriate description of the relevant concentration decline.
The half-life is defined by \(C(t_{1/2})=C_0/2\). Substitution into the model gives \(C_0/2=C_0e^{-kt_{1/2}}\). Dividing by \(C_0\) and taking natural logarithms yields \(k t_{1/2}=\ln 2\), so \(k=\ln 2/t_{1/2}\). The residual fraction follows by dividing the concentration at the end by the concentration at the start:
The ratio \(w/t_{1/2}\) is the number of half-lives elapsed. The fraction is dimensionless; multiplying it by 100 gives the percentage remaining. If \(C_0\) is known, multiplying \(C_0\) by the fraction gives the model-predicted concentration at the end of washout. Use consistent time units for \(w\) and \(t_{1/2}\), such as both in hours.
Worked Example: A 52-Hour Washout With a 12-Hour Half-Life
Suppose, hypothetically, that a drug has a relevant elimination half-life of 12 hours and a proposed washout lasts 52 hours. Assume the concentration follows single-exponential first-order elimination throughout the washout, with no further dose. Calculate the fraction and percentage expected to remain. If the concentration at the start is 160 mg/L, also calculate the predicted concentration at the end.
First calculate the number of half-lives elapsed: \(52/12=4.3333\). Then apply the fraction formula:
As a percentage, this is \(100(0.0496062829\ldots)\%=4.96062829\ldots\%\), or approximately 4.961%. The rounding is to five decimal places for the fraction and three decimal places for the percentage.
The predicted concentration is the starting concentration multiplied by the fraction:
Thus, under the stated model, about 4.961% of the starting concentration remains, corresponding to about 7.937 mg/L from a starting concentration of 160 mg/L. This is a model-based estimate, not evidence that any pharmacologic effect has declined by the same percentage.
Choosing a Duration From a Residual-Fraction Goal
A design team can also work backward. If it specifies a maximum model-predicted residual fraction \(q\), with \(0<q<1\), it can find the washout duration that reaches that fraction or less. Starting with \(2^{-w/t_{1/2}}\le q\), take logarithms and solve for \(w\):
The inequality reverses when multiplying by \(-1\). For a target of 1%, \(q=0.01\), so the required duration is about \(6.644\) half-lives. A target is a planning choice for this calculation, not a universal threshold that proves the absence of carryover.
Worked Example: Converting a 1% Target Into Hours
For a hypothetical drug with a 9-hour elimination half-life, calculate the duration corresponding to a target residual fraction of 1%. Assume the same first-order, single-phase model applies during washout and that the half-life is expressed in hours.
Here \(q=0.01\). The number of half-lives is \(\log_2(1/0.01)=\log_2(100)=\ln(100)/\ln(2)=6.643856\ldots\). Therefore:
At the unrounded duration, the model predicts a residual fraction of 0.01, or 1%. If a practical schedule uses a 60-hour interval, check that rounded-up duration directly: \(60/9=6.6667\) half-lives, giving \(2^{-60/9}\approx0.009843\), or approximately 0.9843%. Thus the rounded-up interval is just over the chosen 1% target under the assumed model.
Worked Example: A Washout Specified as Five Half-Lives
A hypothetical protocol proposes a washout of five elimination half-lives. The relevant half-life is estimated to be 14.4 hours. Assume single-exponential first-order elimination applies and no dosing occurs during the interval. Find the washout duration in hours and the predicted fraction remaining.
The duration is \(5\times14.4\ \mathrm{h}=72.0\ \mathrm{h}\). Since exactly five half-lives elapse, the residual fraction is:
The calculation predicts 3.125% of the starting concentration remains at 72 hours. It does not predict that 3.125% of a clinical effect remains: concentration and effect need not decline in parallel.
Checking the Calculation and Its Interpretation
Before using the result in design discussions, check the inputs and the scope of the model. The half-life must describe the relevant drug concentration decline, the washout time must start from the intended point, and the units must match. For example, a duration in hours cannot be divided by a half-life in days without conversion. The assumed elimination phase must also be appropriate across the interval being calculated.
A useful diagnostic is whether the result behaves sensibly as washout duration changes. At \(w=0\), the formula gives \(f_{\mathrm{rem}}=1\), or 100%. At one half-life, it gives \(1/2\); at two, \(1/4\). For positive \(w\), the predicted fraction must lie between zero and one and decrease as \(w\) increases. A value above one for a positive washout, or a value that increases with duration, usually signals a sign, unit, or substitution error.
The half-life formula is convenient when the half-life is known or is the design input. An equivalent calculation uses the elimination rate constant directly, \(e^{-kw}\), when \(k\) is available; the two expressions agree because \(k=\ln(2)/t_{1/2}\). A fixed number of half-lives is another way to express the same calculation, while choosing a residual target makes the design criterion explicit. These are alternative ways to calculate a model-predicted concentration fraction, not separate methods for demonstrating that carryover is absent.
| Design input | Calculation | Interpretation |
|---|---|---|
| Washout duration \(w\) and half-life \(t_{1/2}\) | \(2^{-w/t_{1/2}}\) | Predicted fraction remaining at the proposed interval |
| Number of half-lives \(m\) | \(2^{-m}\) | Predicted fraction after \(m\) half-lives |
| Target fraction \(q\) and half-life \(t_{1/2}\) | \(t_{1/2}\log_2(1/q)\) | Duration corresponding to the target fraction |
Why the Calculation Matters—and a Common Pitfall
A washout calculation makes a design decision more transparent: it shows how a proposed interval relates to the assumed elimination half-life and what residual concentration fraction that model predicts. It also lets a team compare candidate intervals consistently. The number is most useful when reported with its assumptions, time units, and interpretation rather than as an unexplained rule of thumb.
Common mistake: “Five half-lives have passed, so carryover is impossible.” Five half-lives leave \(2^{-5}=3.125\%\) of the starting concentration under the model, not zero. More importantly, a concentration fraction is not itself a measure of pharmacodynamic persistence or proof that a subsequent treatment comparison is unaffected. The correction is to state what the calculation actually predicts, alongside the assumed half-life and washout duration, and not to treat the result as a guarantee about carryover.
The estimate is only as applicable as its inputs and elimination model. A half-life should not be treated as a universal constant detached from the concentration phase and context it describes. If the model does not adequately represent the relevant decline, the calculated fraction may not represent the actual concentration at the next period. This tutorial uses the calculation to inform a design; it does not substitute a numerical residual threshold for evidence about the study’s pharmacologic or statistical consequences.
Check Your Understanding
Try each problem, then compare your reasoning with the solutions.
- A hypothetical drug has a 10-hour half-life. Under the single-exponential model, what fraction and percentage remain after 30 hours?
- A proposed washout is 48 hours and the assumed half-life is 16 hours. How many half-lives pass, and what fraction remains?
- For a drug with a 6-hour half-life, how long does the model require to reach a 12.5% residual fraction?
- Why does a calculated residual concentration fraction not establish that pharmacodynamic carryover is absent?
Solutions. (1) \(30/10=3\) half-lives, so \(2^{-3}=0.125\), or 12.5%. (2) \(48/16=3\) half-lives, so the predicted fraction is \(2^{-3}=0.125\), or 12.5%. (3) Since 12.5% is \(0.125=1/8=2^{-3}\), three half-lives are required; \(3\times6=18\) hours. (4) The calculation predicts a concentration ratio under a specified elimination model. Drug effect may not decline in proportion to concentration, and the fraction alone does not test whether a later-period comparison is affected.