Methodology
Model card: how the plotter and calculators work
Short answer
The plotter adds one Bateman curve per dose: A(t) = Σ D·ka/(ka−ke)·(e^(−ke·τ) − e^(−ka·τ)), with ke = ln2 ÷ half-life and ka solved from time to peak. Example: retatrutide (t½ 6 days, tmax 1 day) gives 0.8909 mg-eq one day after 1 mg. Results below are recomputed from the code at each build; as of Oct 19, 2026.
Equations
- Elimination rate: ke = ln 2 ÷ t½ (per day).
- Absorption rate: solve tmax = ln(ka/ke) ÷ (ka − ke) for ka > ke by bisection.
- One dose D at time tᵢ: A(t) = D · ka/(ka − ke) · (e^(−ke(t−tᵢ)) − e^(−ka(t−tᵢ))) for t ≥ tᵢ, else 0. When |ka − ke| < 10⁻⁹ the limit D · ke · τ · e^(−ke·τ) is used.
- Total amount: the sum over all doses. Sampled hourly; computed exactly in O(n) by carrying the two exponential sums forward.
- Accumulation for a constant dose every τ days: R = 1 ÷ (1 − e^(−ke·τ)). Fraction of steady state after n half-lives: 1 − 2^(−n) (96.9% at 5).
- Stable window: the first run of two consecutive week-over-week trough changes both under 5% during dosing.
- Vial math (U-100): C = vial mg ÷ water mL; V = dose ÷ C; units = V × 100, displayed to the nearest 0.5 U with the raw value; draws per vial = floor(vial mg ÷ dose mg).
- Cost: vials = ceil(total mg ÷ vial mg); cost = vials × price; per mg = price ÷ vial mg. Discard-by: start + N days, with N only from a cited label or the user.
- First peak about 24 hours after the first dose (time to peak).
- Trough: the lowest point, just before the next weekly dose.
- Accumulation: each dose adds to what is left, so peaks and troughs climb toward about 1.8× a single dose.
- Stable window: from about day 28, the weekly trough changes by less than 5% from one week to the next.
- Washout: after the last dose the amount halves every 6 days.
Relative amount (mg-equivalents), not a blood level. Model and sources: model card.
Parameters
| Compound | t½ (d) | tmax (h) | Published basis |
|---|---|---|---|
| Retatrutide | 6 | 24 | Phase 1b multiple-ascending-dose study reported a half-life of about 6 days; the Phase 2 protocol summary of single-dose data gives about 5–7 days. Single-dose data summarized in the Phase 2 protocol put peak concentration at about 1–3 days; the model uses 1 day. Source |
| Tirzepatide | 5 | 24 | FDA labels: Zepbound states approximately 5–6 days (pharmacokinetics section) and approximately 5 days (overdosage section); Mounjaro states approximately 5 days. The model uses 5 days. FDA label (Zepbound): median time to maximum concentration 24 hours, range 8 to 72 hours. The model uses 24 hours. Source |
| Semaglutide | 7 | 36 | FDA label: elimination half-life of about 1 week. FDA label: maximum concentration reached 1 to 3 days after a dose; the model uses 36 hours. Source |
Validation vectors
These values are recomputed from the production code each time the site is built and compared with the hand-calculated targets in the build specification (tolerance ±0.0005). The same values are asserted in the unit test suite.
| Compound | ke | ka | R (7 d) | 1 mg, A(7 d) | 2 mg ×4, A(28 d) | Result |
|---|---|---|---|---|---|---|
| Retatrutide | 0.115525expected 0.115525 | 3.5371expected 3.5371 | 1.8033expected 1.8033 | 0.4605expected 0.4605 | 1.5954expected 1.5954 | Matches |
| Tirzepatide | 0.138629expected 0.138629 | 3.3122expected 3.3122 | 1.6101expected 1.6101 | 0.3955expected 0.3955 | 1.2473expected 1.2473 | Matches |
| Semaglutide | 0.099021expected 0.099021 | 2.1514expected 2.1514 | 2expected 2 | 0.5241expected 0.5241 | 1.9655expected 1.9655 | Matches |
| Day | A(t), mg-eq |
|---|---|
| 1 | 0.8909 |
| 3 | 0.731 |
| 7 | 0.4605 |
| 14 | 0.2051 |
| Vial | Water | Amount | Units |
|---|---|---|---|
| 10 mg | 2 mL | 2.5 mg | 50 U |
| 10 mg | 1 mL | 2 mg | 20 U |
| 20 mg | 3 mL | 2 mg | 30 U |
| 30 mg | 3 mL | 2.5 mg | 25 U |
| 5 mg | 2 mL | 0.25 mg | 10 U |
Other vectors, recomputed the same way:
- Cost: 8 mg weekly for 4 weeks with 10 mg vials at $100 = 32 mg, 3.2 vials, 4 whole vials, $400 ($320 fractional), $10 per mg.
- Titration: the Phase 3 retatrutide 12 mg ladder from Jan 5, 2026 reaches its first 12 mg week on 2026-04-27.
- Storage: a 28-day period from Oct 19, 2026 ends 2026-11-16.
- Pen to vial: a Zepbound 5 mg pen dose matched from a 30 mg vial in 3 mL is 50 U.
- Edge cases: a 0 mg dose gives a zero curve; a dose contributes nothing before it is given; a tier with no weekdays, a horizon over 104 weeks, water of 0 mL, a dose larger than the vial and a tmax too long for the half-life are all rejected with a message.
What the model does not do
- It does not estimate blood levels, effects or side effects, and it does not adjust for body weight, kidney function, injection site or missed-dose rules in labels.
- It assumes linear kinetics and a constant half-life across doses, which the labels and Phase 1 data support only within the studied dose range.
- For retatrutide, parameters come from early-phase publications; the Phase 3 program has not published a population PK model.
Frequently asked questions
Is the plotter output a plasma concentration?
- No. It is the modeled relative amount of drug in the body in mg-equivalents, with bioavailability set to 1. Converting to concentration would need a volume of distribution and individual clearance, which vary from person to person. Use the curve to compare schedules.
Why a one-compartment model?
- Published labels and Phase 1 papers report a half-life and a time to peak for these weekly drugs. A one-compartment model with first-order absorption is the simplest model that reproduces both, and its assumptions are easy to state and test.
How is the absorption rate chosen?
- The absorption rate ka is solved by bisection so that the single-dose peak falls at the published time to peak: tmax = ln(ka/ke) ÷ (ka − ke). A tmax so long that ka would not exceed ke is rejected as invalid input.
How are the calculators tested?
- Each engine function has unit tests in Vitest that must reproduce the values on this page to within 0.0005, plus edge cases such as zero doses, doses before time zero, missing weekdays, horizons over 104 weeks and water volumes of zero.
Sources
- LY3437943, a novel triple GIP, GLP-1, and glucagon receptor agonist in people with type 2 diabetes: a phase 1b, multicentre, double-blind, placebo-controlled, randomised, multiple-ascending dose trial. The Lancet 2022;400(10366):1869-1881 (abstract via PubMed PMID 36354040; doi:10.1016/S0140-6736(22)02033-5), Oct 27, 2022. Accessed Oct 19, 2026.
- Study Protocol J1I-MC-GZBF(b) for NCT04881760 (Prot_000.pdf). Eli Lilly and Company, hosted on ClinicalTrials.gov, Aug 29, 2022. Accessed Oct 19, 2026.
- ZEPBOUND (tirzepatide) injection, for subcutaneous use - Full Prescribing Information, Medication Guide and Instructions for Use (DailyMed SPL set ID 487cd7e7-434c-4925-99fa-aa80b1cc776b, version 40). U.S. National Library of Medicine (DailyMed) / Eli Lilly and Company, Aug 28, 2026. Accessed Oct 19, 2026.
- WEGOVY (semaglutide) injection, for subcutaneous use; WEGOVY (semaglutide) tablets, for oral use - Full Prescribing Information (SPL, Revised 6/2026). Novo Nordisk / National Library of Medicine DailyMed, Jun 18, 2026. Accessed Oct 19, 2026.