Maths IA · Modelling · topic 84 of 97
Medicine in the bloodstream: why are doses spaced out? · IB Maths IA idea: guide, sample graph and step-by-step plan
same dose, same fraction lost: what is the limit?Take a tablet every eight hours and the level in your blood does not just repeat. It climbs, then settles. You're going to find the level it settles at, and the longest gap between doses that is still safe and effective.
- ✓ you like exponential decay and want to see it used for something that matters
- ✓ you enjoy a pattern that settles down to a limit you can calculate
- ✓ you want a model that a pharmacist or a doctor could actually use
The big idea
Most medicines leave the body by first-order elimination: each hour a fixed fraction is removed, so the concentration decays exponentially and the half-life tells you the rate. A dose adds a fixed amount to what is already there, so the peak after dose n is a geometric series. As n grows the series converges, which gives a steady-state peak and trough you can write down. From that, one formula links the half-life, the dose size and the gap between doses to the levels you end up with. The series is the routine part. The marks come from testing the model's assumptions: instant absorption, one compartment, no change in the body over time. Compare dosing schedules, justify a loading dose, and say clearly that this is maths on a simplified model, not medical advice.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Sum the peaks after each dose, take the limit, and compare with a spreadsheet that adds one dose at a time.
→ A finite geometric series becomes an infinite one. Say how many doses it takes to get within 10% of the limit.Use a therapeutic window, a lowest useful level and a highest safe level, to find the range of intervals that keep both peak and trough inside it.
→ Two inequalities in the interval, solved with logarithms. The answer is a range, not a single number.Give a larger first dose so the first peak already equals the steady-state peak.
→ A formula for the loading dose in terms of the accumulation factor. Compare how long each schedule takes to reach the window.Replace tablets with a steady infusion and write a differential equation: input rate minus elimination.
→ A first-order equation with a limit. Compare its steady level with the average level of the tablet schedule.Skip one dose and follow the levels until the schedule recovers.
→ A perturbed series. Measure how long the trough stays below the window and what a double dose would do.Take a published half-life range for a substance and vary it, or fit decay to published concentration readings.
→ A sensitivity study: how much does the steady-state level move for a 20% change in half-life?Your Maths IA research question: run the check
which substance? which number?Write your own question first. Then make sure it ticks every box:
- Names the substance or a clearly labelled made-up one, its half-life source, and the dose size and units.
- Names the quantity you will find: the steady-state level, the longest safe interval, or the time to reach the window.
- Says what the model is tested against: a spreadsheet simulation, published concentration data, or a second model.
- Has maths to derive (a series or a differential equation), not just a table of values.
- States what you leave out: absorption time, body weight, kidney or liver differences, and that this is not dosing advice.
The model
A dose lifts the concentration by a fixed amount and the body removes a fixed fraction every hour. Sketch the levels for the substance and the schedule you choose.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Which substance or made-up drug, where the half-life comes from, and what you expect before calculating: where does the level settle?
Exponential decay from a half-life: derive k, define every symbol and check the half-life numerically.
The peak after each dose as a geometric series, summed, then the limit. A spreadsheet beside the formula.
A lowest and highest level, the inequalities for the interval, solved with logarithms.
A loading dose or continuous infusion: the extra model and a graph comparing it with the first.
Your curve beside published concentration readings or a label's quoted values. A number for the gap and a reason for it.
Back to your aim: the steady level, the safe range of intervals, with numbers and units.
One compartment, instant absorption, fixed half-life. Say which assumption matters most and what a real clinician would also weigh.
Where every half-life and rule came from.
What your Maths IA graph could look like
draw the limit as a line!A made-up drug to show the form of a strong graph, not the answer: half-life 6 hours, a dose adds 5 mg/L, one dose every 8 hours. Yours will look different. example only
- Concentration against time for several doses, so the climb and the settling are both visible.
- Axis labels give units: hours on the time axis, mg/L on the concentration axis.
- Calculate the steady-state level, then draw it as a line. Reading it off the graph is not enough.
- Plot some published or simulated readings as points: agreement with the curve is your evidence.
- Another graph to consider: peak and trough against the gap between doses, with the window shaded.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what half-life means. Before any calculation, guess where the level settles if you double the gap between doses.
- Choose the substancePick a real substance with a published half-life, or a made-up drug you label as made-up, and note the source of every number.
- Derive the decayWrite the concentration after one dose, link the half-life to the decay constant and define every symbol.
- Trial run (1 lesson)Build a spreadsheet that adds one dose at a time for ten doses. Compare it with your formula and fix any mismatch now.
- Sum the seriesWrite the peak after n doses as a geometric series, sum it, and find the limit. Check it against the table.
- Use the windowSet a lowest and highest level, turn them into inequalities for the interval, and solve with logarithms.
- Compare two schedulesAdd a loading dose or an infusion. Put both schedules on one graph and measure how long each takes to reach the window.
- Go further, with a reasonVary the half-life by a realistic amount and measure the effect, or derive the infusion model from a differential equation. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a real patient and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Say it is a model, not advice. Then send your teacher one draft.
The maths you'll use
- Exponential decay and logarithms – Write , link to the half-life by , and solve for times and intervals.
- Geometric series – Write the peak after doses as with and sum it.
- Infinite series and limits – Take to get the steady-state peak and trough, and find how many doses reach 90% of it.
- Inequalities with logarithms – Turn the window limits into bounds and solve each with logarithms.
- A differential equation – Model a continuous infusion by , solve by separating variables and compare its limit with the tablet schedule.
Key equations
Syllabus links
- 1.3Geometric sequences and series
- 1.5Exponents and an introduction to logarithms
- 1.7Rational exponents and laws of logarithms
- 1.8Infinite geometric series
- 2.9Exponential and logarithmic functions
- 5.18First-order differential equations; Euler's methodHL
- 1.3Geometric sequences and series
- 1.5Exponents and an introduction to logarithms
- 1.11Infinite geometric seriesHL
- 2.9Logarithmic, logistic, piecewise and half-life modelsHL
- 5.14Setting up differential equations; separating variablesHL
Read up first
Data and sources to start from
A half-life taken from a drug label or pharmacology reference (cited), plus a spreadsheet simulation of the dosing schedule; made-up values are allowed if clearly labelled.
- U.S. National Library of Medicine: DailyMed (current drug labels, many list half-lives)
- Health Sciences Authority, Singapore: product information
References to cite
- Rowland, M., & Tozer, T. N. (2011). Clinical pharmacokinetics and pharmacodynamics: Concepts and applications (4th ed.). Lippincott Williams & Wilkins.
- Gibaldi, M., & Perrier, D. (1982). Pharmacokinetics (2nd ed.). Marcel Dekker.
- U.S. National Library of Medicine. (2026). DailyMed. https://dailymed.nlm.nih.gov/
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
half-life is not decay constant- Treating the half-life as the time the drug 'works' for. It is the time for the concentration to halve, nothing more.
- Adding the new dose to zero instead of to what is left from earlier doses.
- Using the half-life as if it were the decay constant. They are linked by k = ln 2 / half-life, and the units must agree.
- Quoting a limit without showing the series converges (the ratio must be less than 1 in size).
- Giving a single best interval when the window gives a range, and not saying which end is safer.
- Presenting results as advice for real people. This is maths on a simple model; say so and say what it leaves out.
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.