# Drug Half-Life Calculator

> Applies first-order elimination kinetics to a drug's half-life: elimination rate constant, fraction and amount of a dose remaining after a given time, times to steady state and to 97% elimination, and, for repeated dosing, the accumulation ratio and the approach to steady state.

- Calculator id: `drug-half-life` · Category: Medical & Clinical (`medical`) · Tool name: `calculate_drug_half_life_kinetics`
- Canonical page: https://tttkmbb.com/medical/drug-half-life · This document: https://tttkmbb.com/medical/drug-half-life.md · JSON definition: https://tttkmbb.com/medical/drug-half-life.json

## Purpose

Applies first-order elimination kinetics to a drug's half-life: elimination rate constant, fraction and amount of a dose remaining after a given time, times to steady state and to 97% elimination, and, for repeated dosing, the accumulation ratio and the approach to steady state.

**Use when:** You know a drug's elimination half-life and want how much remains after some hours, how long washout or steady state takes, or how much a regular dosing interval causes accumulation.

**Do not use when:** Elimination is not first-order (saturable kinetics such as phenytoin or ethanol), active metabolites or multi-compartment distribution dominate the time course, or a dosing decision is being made; results are informational and not medical advice. For radioactive or chemical decay use half-life.

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `half_life_hours` | number | h | required | Terminal elimination half-life in hours (minutes ÷ 60, days × 24). (> 0, max 100000) |
| `dose_mg` | number | mg | optional, default 100 | Amount of a single dose in mg (any unit works, outputs scale with it); the default 100 makes amounts read as percentages. (> 0, max 10000000) |
| `hours_elapsed` | number | h | optional | Hours after the dose at which the remaining fraction is evaluated; defaults to one half-life. (min 0, max 1000000) |
| `dosing_interval_hours` | number | h | optional | Interval between repeated equal doses; enables the accumulation ratio and steady-state outputs. (> 0, max 100000) |
| `doses_taken` | integer |  | optional | Number of doses given so far at that interval; reports the fraction of the steady-state level reached (needs dosing_interval_hours). (min 1, max 10000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `elimination_rate_constant_per_hour` | number | 1/h | k = ln 2 / half-life. |
| `half_lives_elapsed` | number |  | hours_elapsed / half_life_hours. |
| `fraction_remaining` | number |  | e^(−k × hours_elapsed) = 0.5^(half-lives elapsed). |
| `amount_remaining_mg` | number | mg | dose_mg × fraction remaining. |
| `percent_eliminated` | number | % | 100 × (1 − fraction remaining). |
| `time_to_steady_state_hours` | number | h | 5 × half-life, the usual rule of thumb (96.9% of the plateau reached). |
| `time_to_eliminate_97_percent_hours` | number | h | ln(100/3) / k ≈ 5.06 half-lives, when 3% of a dose remains. |
| `accumulation_ratio` | number |  | R = 1 / (1 − e^(−k τ)): steady-state level relative to the first dose; only when dosing_interval_hours is given. |
| `fraction_at_steady_state_after_n_doses` | number |  | 1 − e^(−n k τ); only when dosing_interval_hours and doses_taken are given. |
| `schedule` | list |  | Hours, percent remaining and amount remaining at 0–10 half-lives after a single dose. |

## Formula

`k = ln(2) / half_life_hours; fraction_remaining = e^(−k × hours_elapsed); amount_remaining_mg = dose_mg × fraction_remaining; percent_eliminated = 100 × (1 − fraction_remaining); time_to_steady_state = 5 × half_life_hours; time_to_eliminate_97_percent = ln(100 / 3) / k; accumulation_ratio = 1 / (1 − e^(−k × dosing_interval_hours)); fraction_at_steady_state = 1 − e^(−doses_taken × k × dosing_interval_hours)`

One-compartment model with first-order (linear) elimination: each half-life removes half of what remains, so 50 / 75 / 87.5 / 93.75 / 96.9% is eliminated after 1–5 half-lives regardless of dose; the accumulation ratio assumes equal doses at fixed intervals with complete absorption. Results are informational only and not a substitute for clinical judgement or medical advice.

## Data Sources

- Wikipedia – Biological half-life (first-order elimination, fraction remaining after n half-lives) — https://en.wikipedia.org/wiki/Biological_half-life (reference, retrieved 2026-09-24)
- Wikipedia – Pharmacokinetics (accumulation ratio, time to steady state) — https://en.wikipedia.org/wiki/Pharmacokinetics (reference, retrieved 2026-09-24)
- Hallare J, Gerriets V – Half Life, StatPearls (NCBI Bookshelf) — https://www.ncbi.nlm.nih.gov/books/NBK554498/ (textbook, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/drug-half-life?half_life_hours=…`
- `POST https://tttkmbb.com/api/v1/calculate/drug-half-life` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/drug-half-life · OpenAPI operationId `calculate_drug_half_life_kinetics` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "drug-half-life", "inputs": {…}}`

## Example

- Half-life 6 h, 200 mg, 24 h later, every 12 h, 3 doses: inputs `{"half_life_hours":6,"dose_mg":200,"hours_elapsed":24,"dosing_interval_hours":12,"doses_taken":3}` → `{"elimination_rate_constant_per_hour":0.1155,"fraction_remaining":0.0625,"amount_remaining_mg":12.5,"percent_eliminated":93.75,"time_to_steady_state_hours":30,"accumulation_ratio":1.333,"fraction_at_steady_state_after_n_doses":0.9844,"time_to_eliminate_97_percent_hours":30.35}`
- Half-life 12 h, 500 mg, 36 h later: inputs `{"half_life_hours":12,"dose_mg":500,"hours_elapsed":36}` → `{"elimination_rate_constant_per_hour":0.0578,"half_lives_elapsed":3,"amount_remaining_mg":62.5,"percent_eliminated":87.5,"time_to_steady_state_hours":60}`

```
GET https://tttkmbb.com/api/v1/calculate/drug-half-life?half_life_hours=6&dose_mg=200&hours_elapsed=24&dosing_interval_hours=12&doses_taken=3
```

## Limitations

Elimination is not first-order (saturable kinetics such as phenytoin or ethanol), active metabolites or multi-compartment distribution dominate the time course, or a dosing decision is being made; results are informational and not medical advice. For radioactive or chemical decay use half-life. One-compartment model with first-order (linear) elimination: each half-life removes half of what remains, so 50 / 75 / 87.5 / 93.75 / 96.9% is eliminated after 1–5 half-lives regardless of dose; the accumulation ratio assumes equal doses at fixed intervals with complete absorption. Results are informational only and not a substitute for clinical judgement or medical advice. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Is a drug gone after 5 half-lives?**

About 3.1% remains after 5 and 0.1% after 10 half-lives; whether that is negligible depends on the drug's potency and the reason for the washout.

**Does a bigger dose take longer to clear?**

Not with first-order kinetics: the fraction remaining depends only on time, so doubling the dose doubles the amount at every time point but not the half-life.

**What does the accumulation ratio mean?**

The steady-state peak (and trough) is R times the peak (or trough) after the first dose; dosing every half-life gives R = 2, dosing every 2 half-lives gives 1.33.

## Related

- [Half-Life Calculator](https://tttkmbb.com/chemistry/half-life.md) — General half-life and decay calculator (radioactive or chemical).
- [Dose by Weight Calculator](https://tttkmbb.com/medical/dose-by-weight.md) — Weight-based dose that feeds the amount.
- [Creatinine Clearance Calculator](https://tttkmbb.com/medical/creatinine-clearance.md) — Renal function that lengthens the half-life of renally cleared drugs.