{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"drug-half-life"},"result":{"entity_type":"calculator","id":"drug-half-life","calculator_id":"drug-half-life","canonical_url":"https://tttkmbb.com/medical/drug-half-life","name":"Drug Half-Life Calculator","title":"Drug Half-Life Calculator – Elimination Rate, Fraction Remaining, Accumulation Ratio and Time to Steady State","category":"medical","category_name":"Medical & Clinical","tool_name":"calculate_drug_half_life_kinetics","featured_mcp_tool":false,"description":"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.","inputs":[{"name":"half_life_hours","label":"Elimination half-life","type":"number","unit":"h","required":true,"max":100000,"exclusive_min":0,"description":"Terminal elimination half-life in hours (minutes ÷ 60, days × 24).","example":6},{"name":"dose_mg","label":"Dose","type":"number","unit":"mg","required":false,"default":100,"max":10000000,"exclusive_min":0,"description":"Amount of a single dose in mg (any unit works, outputs scale with it); the default 100 makes amounts read as percentages.","example":200},{"name":"hours_elapsed","label":"Time since the dose","type":"number","unit":"h","required":false,"min":0,"max":1000000,"description":"Hours after the dose at which the remaining fraction is evaluated; defaults to one half-life.","example":24},{"name":"dosing_interval_hours","label":"Dosing interval τ","type":"number","unit":"h","required":false,"max":100000,"exclusive_min":0,"description":"Interval between repeated equal doses; enables the accumulation ratio and steady-state outputs.","example":12},{"name":"doses_taken","label":"Doses taken","type":"integer","required":false,"min":1,"max":10000,"description":"Number of doses given so far at that interval; reports the fraction of the steady-state level reached (needs dosing_interval_hours).","example":3}],"outputs":[{"name":"elimination_rate_constant_per_hour","label":"Elimination rate constant k","type":"number","unit":"1/h","decimals":4,"description":"k = ln 2 / half-life."},{"name":"half_lives_elapsed","label":"Half-lives elapsed","type":"number","decimals":3,"description":"hours_elapsed / half_life_hours."},{"name":"fraction_remaining","label":"Fraction remaining","type":"number","decimals":4,"description":"e^(−k × hours_elapsed) = 0.5^(half-lives elapsed)."},{"name":"amount_remaining_mg","label":"Amount remaining","type":"number","unit":"mg","decimals":3,"description":"dose_mg × fraction remaining."},{"name":"percent_eliminated","label":"Eliminated","type":"number","unit":"%","decimals":2,"description":"100 × (1 − fraction remaining)."},{"name":"time_to_steady_state_hours","label":"Time to steady state (5 half-lives)","type":"number","unit":"h","decimals":2,"description":"5 × half-life, the usual rule of thumb (96.9% of the plateau reached)."},{"name":"time_to_eliminate_97_percent_hours","label":"Time to eliminate 97%","type":"number","unit":"h","decimals":2,"description":"ln(100/3) / k ≈ 5.06 half-lives, when 3% of a dose remains."},{"name":"accumulation_ratio","label":"Accumulation ratio","type":"number","decimals":3,"description":"R = 1 / (1 − e^(−k τ)): steady-state level relative to the first dose; only when dosing_interval_hours is given."},{"name":"fraction_at_steady_state_after_n_doses","label":"Fraction of steady state after n doses","type":"number","decimals":4,"description":"1 − e^(−n k τ); only when dosing_interval_hours and doses_taken are given."},{"name":"schedule","label":"Decay schedule","type":"list","decimals":3,"description":"Hours, percent remaining and amount remaining at 0–10 half-lives after a single dose."}],"input_schema":{"type":"object","properties":{"half_life_hours":{"description":"Terminal elimination half-life in hours (minutes ÷ 60, days × 24). Unit: h.","type":"number","maximum":100000,"exclusiveMinimum":0,"examples":[6],"x-unit":"h"},"dose_mg":{"description":"Amount of a single dose in mg (any unit works, outputs scale with it); the default 100 makes amounts read as percentages. Unit: mg.","type":"number","maximum":10000000,"exclusiveMinimum":0,"default":100,"examples":[200],"x-unit":"mg"},"hours_elapsed":{"description":"Hours after the dose at which the remaining fraction is evaluated; defaults to one half-life. Unit: h.","type":"number","minimum":0,"maximum":1000000,"examples":[24],"x-unit":"h"},"dosing_interval_hours":{"description":"Interval between repeated equal doses; enables the accumulation ratio and steady-state outputs. Unit: h.","type":"number","maximum":100000,"exclusiveMinimum":0,"examples":[12],"x-unit":"h"},"doses_taken":{"description":"Number of doses given so far at that interval; reports the fraction of the steady-state level reached (needs dosing_interval_hours).","type":"integer","minimum":1,"maximum":10000,"examples":[3]}},"additionalProperties":false,"required":["half_life_hours"]},"output_schema":{"type":"object","properties":{"elimination_rate_constant_per_hour":{"description":"k = ln 2 / half-life. Unit: 1/h.","type":"number","x-unit":"1/h"},"half_lives_elapsed":{"description":"hours_elapsed / half_life_hours.","type":"number"},"fraction_remaining":{"description":"e^(−k × hours_elapsed) = 0.5^(half-lives elapsed).","type":"number"},"amount_remaining_mg":{"description":"dose_mg × fraction remaining. Unit: mg.","type":"number","x-unit":"mg"},"percent_eliminated":{"description":"100 × (1 − fraction remaining). Unit: %.","type":"number","x-unit":"%"},"time_to_steady_state_hours":{"description":"5 × half-life, the usual rule of thumb (96.9% of the plateau reached). Unit: h.","type":"number","x-unit":"h"},"time_to_eliminate_97_percent_hours":{"description":"ln(100/3) / k ≈ 5.06 half-lives, when 3% of a dose remains. Unit: h.","type":"number","x-unit":"h"},"accumulation_ratio":{"description":"R = 1 / (1 − e^(−k τ)): steady-state level relative to the first dose; only when dosing_interval_hours is given.","type":"number"},"fraction_at_steady_state_after_n_doses":{"description":"1 − e^(−n k τ); only when dosing_interval_hours and doses_taken are given.","type":"number"},"schedule":{"description":"Hours, percent remaining and amount remaining at 0–10 half-lives after a single dose.","type":"array"}}},"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)","method":"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.","sources":[{"name":"Wikipedia – Biological half-life (first-order elimination, fraction remaining after n half-lives)","url":"https://en.wikipedia.org/wiki/Biological_half-life","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Pharmacokinetics (accumulation ratio, time to steady state)","url":"https://en.wikipedia.org/wiki/Pharmacokinetics","type":"reference","retrieved_at":"2026-09-24"},{"name":"Hallare J, Gerriets V – Half Life, StatPearls (NCBI Bookshelf)","url":"https://www.ncbi.nlm.nih.gov/books/NBK554498/","type":"textbook","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"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},"expected":{"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},"url":"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"},{"name":"Half-life 12 h, 500 mg, 36 h later","inputs":{"half_life_hours":12,"dose_mg":500,"hours_elapsed":36},"expected":{"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},"url":"https://tttkmbb.com/api/v1/calculate/drug-half-life?half_life_hours=12&dose_mg=500&hours_elapsed=36"}],"faq":[{"q":"Is a drug gone after 5 half-lives?","a":"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."},{"q":"Does a bigger dose take longer to clear?","a":"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."},{"q":"What does the accumulation ratio mean?","a":"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."}],"tags":["drug half-life","elimination rate constant","steady state","accumulation ratio","pharmacokinetics","washout"],"related":[{"calculator_id":"half-life","reason":"General half-life and decay calculator (radioactive or chemical)."},{"calculator_id":"dose-by-weight","reason":"Weight-based dose that feeds the amount."},{"calculator_id":"creatinine-clearance","reason":"Renal function that lengthens the half-life of renally cleared drugs."}],"links":{"html":"https://tttkmbb.com/medical/drug-half-life","markdown":"https://tttkmbb.com/medical/drug-half-life.md","json":"https://tttkmbb.com/medical/drug-half-life.json","api":"https://tttkmbb.com/api/v1/calculate/drug-half-life","schema":"https://tttkmbb.com/api/v1/calculators/drug-half-life","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:51:44Z"}