{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"cell-doubling-time"},"result":{"entity_type":"calculator","id":"cell-doubling-time","calculator_id":"cell-doubling-time","canonical_url":"https://tttkmbb.com/science/cell-doubling-time","name":"Cell Doubling Time Calculator","title":"Cell Doubling Time Calculator – Population Doubling Time, Growth Rate and Generations from Two Counts","category":"science","category_name":"Biology, Earth & Space","tool_name":"calculate_cell_doubling_time","featured_mcp_tool":false,"description":"Computes the doubling time of an exponentially growing cell culture (or any population) from an initial count, a final count and the elapsed time with Td = t × ln 2 / ln(Nf / N0), plus the number of doublings, the specific growth rate and an optional projected count.","use_when":"You have cell counts at two time points (e.g. seeding and harvest) and need the doubling time, the number of population doublings, or the expected count after a further period.","do_not_use_when":"The culture has left exponential growth (lag or plateau phase), the final count is not larger than the initial one, or you want to project a population with a known rate (use population-growth).","inputs":[{"name":"initial_count","label":"Initial count N0","type":"number","required":true,"max":1000000000000000,"exclusive_min":0,"description":"Cells (or population) at the start, e.g. cells seeded.","example":100000},{"name":"final_count","label":"Final count Nf","type":"number","required":true,"max":1000000000000000,"exclusive_min":0,"description":"Cells at the end of the interval; must exceed initial_count.","example":800000},{"name":"elapsed_hours","label":"Elapsed time","type":"number","unit":"h","required":true,"max":1000000,"exclusive_min":0,"description":"Time between the two counts in hours.","example":72},{"name":"project_hours","label":"Project to time","type":"number","unit":"h","required":false,"min":0,"max":1000000,"description":"Optional time after the initial count at which to project the count, assuming the same exponential growth.","example":96}],"outputs":[{"name":"doubling_time_hours","label":"Doubling time","type":"number","unit":"h","decimals":2,"description":"Td = elapsed_hours × ln 2 / ln(final_count / initial_count)."},{"name":"doubling_time_days","label":"Doubling time (days)","type":"number","unit":"days","decimals":3,"description":"Td / 24."},{"name":"number_of_doublings","label":"Population doublings","type":"number","decimals":3,"description":"log2(final_count / initial_count)."},{"name":"growth_rate_per_hour","label":"Specific growth rate µ","type":"number","unit":"1/h","decimals":5,"description":"ln(final_count / initial_count) / elapsed_hours."},{"name":"generations_per_day","label":"Generations per day","type":"number","unit":"1/day","decimals":3,"description":"24 / Td."},{"name":"projected_count","label":"Projected count at project_hours","type":"number","decimals":0,"description":"initial_count × 2^(project_hours / Td) (when project_hours is given)."}],"input_schema":{"type":"object","properties":{"initial_count":{"description":"Cells (or population) at the start, e.g. cells seeded.","type":"number","maximum":1000000000000000,"exclusiveMinimum":0,"examples":[100000]},"final_count":{"description":"Cells at the end of the interval; must exceed initial_count.","type":"number","maximum":1000000000000000,"exclusiveMinimum":0,"examples":[800000]},"elapsed_hours":{"description":"Time between the two counts in hours. Unit: h.","type":"number","maximum":1000000,"exclusiveMinimum":0,"examples":[72],"x-unit":"h"},"project_hours":{"description":"Optional time after the initial count at which to project the count, assuming the same exponential growth. Unit: h.","type":"number","minimum":0,"maximum":1000000,"examples":[96],"x-unit":"h"}},"additionalProperties":false,"required":["initial_count","final_count","elapsed_hours"]},"output_schema":{"type":"object","properties":{"doubling_time_hours":{"description":"Td = elapsed_hours × ln 2 / ln(final_count / initial_count). Unit: h.","type":"number","x-unit":"h"},"doubling_time_days":{"description":"Td / 24. Unit: days.","type":"number","x-unit":"days"},"number_of_doublings":{"description":"log2(final_count / initial_count).","type":"number"},"growth_rate_per_hour":{"description":"ln(final_count / initial_count) / elapsed_hours. Unit: 1/h.","type":"number","x-unit":"1/h"},"generations_per_day":{"description":"24 / Td. Unit: 1/day.","type":"number","x-unit":"1/day"},"projected_count":{"description":"initial_count × 2^(project_hours / Td) (when project_hours is given).","type":"number"}}},"formula":"Td = t × ln 2 / ln(Nf / N0); doublings = log2(Nf / N0); µ = ln(Nf / N0) / t; generations per day = 24 / Td; N(t') = N0 × 2^(t' / Td)","method":"Assumes exponential (log-phase) growth with a constant specific growth rate and no cell death; counts taken during lag or plateau phase overestimate the doubling time. Population doubling level (PDL) accumulates the doublings across passages.","sources":[{"name":"Wikipedia – Doubling time","url":"https://en.wikipedia.org/wiki/Doubling_time","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Exponential growth","url":"https://en.wikipedia.org/wiki/Exponential_growth","type":"reference","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":"1 × 10⁵ to 8 × 10⁵ cells in 72 h","inputs":{"initial_count":100000,"final_count":800000,"elapsed_hours":72,"project_hours":96},"expected":{"doubling_time_hours":24,"doubling_time_days":1,"number_of_doublings":3,"growth_rate_per_hour":0.02888,"generations_per_day":1,"projected_count":1600000},"url":"https://tttkmbb.com/api/v1/calculate/cell-doubling-time?initial_count=100000&final_count=800000&elapsed_hours=72&project_hours=96"},{"name":"2 × 10⁵ to 1.2 × 10⁶ cells in 48 h","inputs":{"initial_count":200000,"final_count":1200000,"elapsed_hours":48},"expected":{"doubling_time_hours":18.57,"number_of_doublings":2.585,"growth_rate_per_hour":0.03733,"generations_per_day":1.292},"url":"https://tttkmbb.com/api/v1/calculate/cell-doubling-time?initial_count=200000&final_count=1200000&elapsed_hours=48"}],"faq":[{"q":"What is the difference between doubling time and generation time?","a":"For a population growing exponentially they are the same: the time for the number of cells to double. Individual cell-cycle times vary around it."},{"q":"Why must the final count be larger?","a":"The formula divides by ln(Nf / N0), which is zero or negative when the culture has not grown; a shrinking culture has a half-life, not a doubling time."},{"q":"Can I use bacteria OD readings instead of counts?","a":"Yes, while OD is proportional to cell number (roughly OD600 < 0.5); enter the two OD values as the counts."}],"tags":["doubling time","cell growth","population doubling","generation time","growth rate","cell culture"],"related":[{"calculator_id":"population-growth","reason":"Project a population forward with a known rate or a carrying capacity."},{"calculator_id":"hemocytometer-count","reason":"Get the two cell counts from hemocytometer readings."},{"calculator_id":"half-life","reason":"The equivalent calculation for a decreasing quantity."}],"links":{"html":"https://tttkmbb.com/science/cell-doubling-time","markdown":"https://tttkmbb.com/science/cell-doubling-time.md","json":"https://tttkmbb.com/science/cell-doubling-time.json","api":"https://tttkmbb.com/api/v1/calculate/cell-doubling-time","schema":"https://tttkmbb.com/api/v1/calculators/cell-doubling-time","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:46:19Z","next_actions":[{"tool":"run_calculator","calculator_id":"cell-doubling-time","reason":"Run Cell Doubling Time Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/science/cell-doubling-time.md"}}