Pharmacology for nursing · Lesson 3 of 14

Dose, interval and half-life

Why a drug given every eight hours builds up for two days, why the level swings between a peak and a trough, and how a nurse checks that both sit inside the safe range.

An example lesson built by Chalkwork Labs from public sources, to show the shapes. Your course is drafted from your material and checked by you.

Invented drug. The drug on this page, didactin, does not exist. Its numbers are invented for teaching: a volume of distribution of 0.5 L per kg, a half-life of 8 hours in a healthy adult, and a window of 10 to 20 mg/L. The doses, interval, half-life and range are not any real drug's. They are here to teach the shapes. They are not a prescribing reference.

1. What the body does to a drug

A drug only works while enough of it is in the blood. Three processes decide how much is there at any moment.

Absorption is how the drug gets from where you gave it into the blood. A tablet must dissolve and cross the gut wall. A dose into a vein skips this step.
Distribution is how the drug spreads from the blood into the tissues. The more it spreads, the lower the blood level a dose gives.
Elimination is how the body removes the drug. The liver changes it, and the kidneys pass it out in urine.

Absorption pushes the level up. Elimination pulls it down. Distribution sets how much a dose raises it. The chart of blood level against time is the result of that tug of war.

Key idea

Every dosing decision comes back to one picture: the blood level over time, drawn against the range that works. The rest of this lesson builds that picture one piece at a time.

2. One dose, and the half-life

Start with one dose into a vein. The whole dose is in the blood at once. The level jumps, then falls as the body eliminates the drug.

The jump depends on the volume of distribution: the volume the dose seems to be diluted into. Divide the dose by it and you get the level. Here the patient weighs kg, the volume is L per kg, so L in total. A mg dose gives ÷ = mg/L.

The fall follows a simple rule. The body removes a fixed fraction of the drug each hour, not a fixed amount. This is first-order elimination. Because the fraction is fixed, the time it takes for the level to halve is always the same. That time is the half-life.

One IV dose of mg with a half-life of hours. The dots mark each half-life: the level halves every time.

Two rules of thumb follow from this and nurses use them every day:

After about 5 half-lives, a drug is almost gone (about 3% is left).
With repeated doses, after about 5 half-lives the drug has almost finished building up. The next section shows why the same number works both ways.

3. Repeated doses, the window and steady state

Now give the dose again and again. The first dose has not fully left when the second arrives, so the second starts from a higher floor. The level climbs with each dose. But as it climbs, the body removes more per interval, because it removes a fixed fraction of a bigger amount. When it removes one whole dose per interval, the climb stops. That is steady state.

At steady state the level still swings. It is highest just after a dose (the peak) and lowest just before the next one (the trough). The therapeutic window is the range of levels that works without harm. Both the peak and the trough must sit inside it.

Move the sliders. The shaded band is the window, to mg/L. Watch the trough first: it is usually the number that fails.

Try it

Dose: mg

Interval: every hours

Half-life: hours

Route

–

time to steady state (5 half-lives)

–

peak at steady state, mg/L

–

trough at steady state, mg/L

–

against the window

Blood level over time for repeated doses. The dashed lines mark the steady-state peak and trough. The chart runs until the level has settled.

Three things to try:

Halve the half-life to 4 hours (a smoker clears this drug faster). The level settles in hours instead of , but the trough falls to mg/L. The drug stops working before each dose.
Double the half-life to 16 hours (liver disease or heart failure slows elimination). Now the peak reaches and even the trough is mg/L. The whole curve sits above the window. The dose has not changed; the patient has.
Switch to IV bolus. The peak gets sharper and higher, the trough lower. Slow absorption smooths the swing.

Key idea

The dose sets how high the curve sits. The half-life sets how long it takes to get there and how far it swings. A bigger dose does not reach steady state sooner. It reaches a higher steady state in the same 5 half-lives.

4. Why the interval matters as much as the dose

Go back to the 4-hour half-life with the 8-hour interval. The trough is too low. Two fixes come to mind.

Give it more often. Every 4 hours, same mg: peak , trough mg/L. Both inside the window.
Give a bigger dose. To lift the trough to 10 mg/L at the 8-hour interval you need about mg, and the peak then reaches mg/L. Toxic.

The dose scales the whole curve up or down. It cannot change the ratio of peak to trough. That ratio is set by the interval and the half-life. With an IV bolus every 8 hours and a 4-hour half-life, the level halves twice between doses, so the peak is times the trough. By mouth the swing is softer, about times. The window only allows times. No dose can fit a swing of into a window of 2. Only a shorter interval can.

Key idea

Rule. When the interval is much longer than the half-life, the swing is too big for a narrow window. Shorten the interval or use a slow-release form. Do not just raise the dose.

5. Loading dose and the missed dose

Two situations come up on every ward. First, the patient needs the drug to work now, not in 40 hours. A loading dose is a bigger first dose that fills the body to the target level at once. Its size is target level × volume of distribution: mg/L × L = mg. Then the normal dose holds the level there.

Second, the patient misses a dose. What happens next depends on what they do about it. Try all three.

Try it

Loading dose (first dose): mg

Missed dose

What the patient does

–

hours until the level first reaches the window

–

lowest level after 40 h, mg/L

–

highest level after 40 h, mg/L

mg by mouth every hours, half-life hours, for 96 hours. The grey line is the regimen with no loading dose and no missed dose.

What the numbers say, at the default settings:

Without a loading dose the level first reaches the window after hours. With a mg loading dose it gets there in hours.
Skip dose 6 and the trough drops to mg/L, far below the window, and it takes several doses to climb back.
Take it 4 hours late and the dip is milder, mg/L, and the curve recovers at the next dose.
Double the next dose and the dip is the same as skipping, mg/L, and then the peak overshoots to mg/L. Both ends of the window are broken by one decision.

Common mistake

Doubling up to make up a missed dose is the common mistake. It does not undo the low trough that already happened. It adds a high peak. Follow the medicine's own missed-dose rule and tell the prescriber.

6. Worked example

Beyond the syllabus. This example works in clearance, litres of blood cleared per hour. That is pharmacy, not pre-registration nursing. Read it for the picture. It is not examined.

A 70 kg adult takes mg of didactin by mouth every hours. Half-life hours, volume of distribution L, everything absorbed. What average level will they settle at, and when?

Step 1. Elimination rate. The fraction removed per hour is 0.693 ÷ half-life = 0.693 ÷ = per hour. (0.693 is the natural log of 2; it turns a half-life into a rate.)

Step 2. Clearance. Clearance is the volume of blood fully cleared of drug each hour: rate × volume = × = L per hour.

Step 3. Average level at steady state. At steady state the body removes exactly one dose per interval. So the average level = dose ÷ (clearance × interval) = ÷ ( × ) = mg/L. Inside the 10 to 20 window.

Step 4. When. 5 half-lives = 5 × = hours, which is doses. The chart in section 3 settles at the same point.

Step 5. Check the swing. The chart gives a peak of and a trough of mg/L, either side of the average. Both inside the window, so the regimen holds.

The daily dose is mg. The same daily dose given as an IV bolus every 8 hours has the same average, but swings from down to mg/L: exactly on both edges of the window.

Quick check

1. A drug has a half-life of 6 hours. Straight after an IV dose the level is 40 mg/L. No more doses are given. What is the level 18 hours later?

Quick check

2. A child weighs 18 kg. The prescription is 5 mg/kg. The stock solution is 50 mg in 2 mL. What volume do you draw up?

Quick check

3. A prescriber wants the level inside the window within an hour, not after two days. Which change does that?

7. Read the code

Show the working in code

Optional. The same calculation as Python you can step through.

Beyond the syllabus. This section reads Python code. It is for learners who want to see how the chart is built. It is not examined, and you can skip it.

The chart in section 3 is a sum of curves. This block does the same job in a simpler way: it steps the level forward one dose at a time, for an IV bolus. Read the steps, then press Run. You will not write or edit any of it.

Block 1: Step the level forward, dose by dose

Watch the peak and trough columns. Find the dose number where they stop changing.

Python's maths helpers: log and exp.
The regimen: dose, interval, half-life, and the volume the dose is diluted into.
Turn the half-life into a decay factor: the fraction of the drug still there after one interval. Here the interval equals the half-life, so the factor is 0.5.
The loop. Each dose adds dose ÷ volume to the level (the peak). Then the level decays until the next dose (the trough). Print both.

import math

dose = 350        # mg, IV bolus
interval = 8      # hours between doses
half_life = 8     # hours
volume = 35       # L, volume of distribution

k = math.log(2) / half_life       # per hour
decay = math.exp(-k * interval)   # fraction left

level = 0.0
for n in range(1, 13):
    level = level + dose / volume   # the dose lands
    peak = level
    level = level * decay           # decays until the next
    print(f"dose {n:2d}  peak {peak:4.1f}  trough {level:4.1f}")

Quick check

From dose 9 onward the peak stays at 20.0 and the trough at 10.0. Why do they stop changing?

What to remember

The half-life is the time for the level to halve. It belongs to the drug in that patient, not to the dose.
About 5 half-lives to wash out. About 5 half-lives to reach steady state. The same number both ways.
The dose sets how high the curve sits. The interval and half-life set how far it swings. A bigger dose cannot fix a swing that is too wide.
Check the trough as well as the peak. Both must sit inside the window at steady state.
A loading dose is target level × volume of distribution. It reaches the target now; it does not change the maintenance dose.
A missed dose makes a low trough. Doubling the next dose adds a high peak on top. Follow the medicine's missed-dose rule.
Dose calculations: mg = mg/kg × kg; write the stock as mg per 1 mL; volume = mg ÷ (mg per mL); sense-check the answer.

Sources

Open RN, Nursing Pharmacology, Chippewa Valley Technical College, 2020. Chapter 1, sections 1.2 to 1.6 (Pharmacokinetics, Absorption, Distribution, Metabolism, Excretion). wtcs.pressbooks.pub/pharmacology. Licence: CC BY 4.0. Source of the plain-language definitions of absorption, distribution, elimination and half-life.
Le, J., "Overview of Pharmacokinetics", MSD Manual Professional Edition. msdmanuals.com. Used as a reference for half-life, clearance and volume of distribution; no text is reproduced.
Lapham, R., Drug Calculations for Nurses: A Step-by-Step Approach, 5th edition, Routledge, 2021. Used as a reference for the dose calculation method; no text is reproduced.
The one-compartment model with first-order elimination, superposition of doses, and the formulas for steady state, clearance and loading dose are standard pharmacokinetics. The drug, didactin, is invented, and its parameters are not any real drug's. Every number on this page is computed by export_05.py from those parameters.

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