# Biologically effective dose Biologically effective dose Status: LIVE Price: $0.005 USDC `GET /api/calc/bed` HTML: https://magentlab.com/docs/bed Markdown: https://magentlab.com/docs/bed.md Not a medical device. Not a diagnosis, dose, or listing recommendation. ## What it computes Compute Fowler 1989 biologically effective dose from equal-fraction count, dose per fraction in Gy, and a caller-assigned tissue α/β, without a time or repopulation term. ## When to use When an agent already has n, d, and tissue α/β and must not invent a radiotherapy prescription, apply T_k or T_pot, or treat the result as EQD2. ## When not to use - Not a medical device. Not a radiotherapy prescription or treatment plan. magent does not plan radiotherapy. - Fowler 1989 BED without a time or repopulation term. Does not apply overall-time T_k or T_pot (those are later Fowler and other papers). - Does not convert BED to EQD2 or normalized total dose. Caller supplies n, d_gy, and tissue alpha_beta; magent does not assign α/β. - Rejects non-positive n×d or α/β. n 1-50, d_gy 0.5-30, alpha_beta 1-20 are magent clamps, not bands published as input limits in Fowler 1989. ## Formula ``` bed_gy = n * d_gy * (1 + d_gy / alpha_beta) ``` ## Citation [The linear-quadratic formula and progress in fractionated radiotherapy](https://doi.org/10.1259/0007-1285-62-740-679) Fowler JF Br J Radiol. 1989;62(740):679-694 DOI: [10.1259/0007-1285-62-740-679](https://doi.org/10.1259/0007-1285-62-740-679) ## Worked example ### n 30, d 2 Gy, α/β 3 Gy Given: `alpha_beta=3, d_gy=2, n=30` 1. bed_gy = 30 * 2 * (1 + 2 / 3) = 60 * (5 / 3) = 100.0 ### single 2 Gy fraction, α/β 10 Gy Given: `alpha_beta=10, d_gy=2, n=1` 1. bed_gy = 1 * 2 * (1 + 2 / 10) = 2 * 1.2 = 2.4 ## Also searched as - BED - biologically effective dose - linear quadratic formula - Fowler BED - alpha beta ratio - fractionated radiotherapy BED - LQ model BED - dose per fraction BED ## Parameters - `n` (integer, required): Number of equal fractions (integer 1-50, magent clamp; Fowler 1989 does not publish an input band). Caller supplies n. magent does not plan a fraction schedule. - `d_gy` (number, required): Dose per fraction in Gy (0.5-30, magent clamp). Caller supplies d. magent does not prescribe dose. - `alpha_beta` (number, required): Caller-assigned tissue α/β in Gy (1-20, magent clamp). magent does not assign α/β from a tissue table. ## Bulk (POST) POST the same path with a JSON items array. The first item is the GET unit; each additional item is $0.002. Invalid items return HTTP 400 before settlement. Method: `POST /api/calc/bed` Max items: 25 Price: unit_amount + extra_item_amount * (n - 1) (unit 5000 atomic, extra 2000 atomic) Status: PREVIEW - 1 to 25 items. Each item uses the same keys as the GET query parameters. - POST of 1 item costs the same as GET. Lists longer than 25: chunk into multiple POSTs. Extra items still execute; do not pad dummy rows. - The first invalid item fails the whole request (index in the 400 JSON). You are not charged. - HTTP 200 is {count, path, items}. Each items[i] is the GET 200 body for that row. - Catalog misses (empty matches, found=false) stay paid 200s inside that slot. - x402 resource URL is this path plus n and body_sha256. Browsers that prefer text/html get the same paywall as GET; Pay retries POST with the same items. Example request: ```json { "items": [ { "alpha_beta": "3", "d_gy": "2", "n": "30" }, { "alpha_beta": "3", "d_gy": "2", "n": "30" } ] } ``` Human paywall (two-item fixture): https://api.magentlab.com/paywall/bulk/bed Example 200: ```json { "count": 2, "path": "/api/calc/bed", "items": [ { "n": 30, "disclaimer": "Not a medical device. Deterministic published-formula or published-code output for autonomous agents. A licensed clinician remains responsible for patient care and billing submissions.", "citation": { "authors": "Fowler JF", "doi": "10.1259/0007-1285-62-740-679", "id": "fowler-1989", "pmid": "2670032", "source": "Br J Radiol. 1989;62(740):679-694", "title": "The linear-quadratic formula and progress in fractionated radiotherapy" }, "formula": "bed", "formula_expression": "bed_gy = n * d_gy * (1 + d_gy / alpha_beta)", "alpha_beta": 3.0, "bed_gy": 100.0, "d_gy": 2.0 }, { "n": 30, "disclaimer": "Not a medical device. Deterministic published-formula or published-code output for autonomous agents. A licensed clinician remains responsible for patient care and billing submissions.", "citation": { "authors": "Fowler JF", "doi": "10.1259/0007-1285-62-740-679", "id": "fowler-1989", "pmid": "2670032", "source": "Br J Radiol. 1989;62(740):679-694", "title": "The linear-quadratic formula and progress in fractionated radiotherapy" }, "formula": "bed", "formula_expression": "bed_gy = n * d_gy * (1 + d_gy / alpha_beta)", "alpha_beta": 3.0, "bed_gy": 100.0, "d_gy": 2.0 } ] } ``` ## Response fields - `formula` (string): bed - `formula_expression` (string): Exact expression used - `n` (integer): Equal-fraction count used - `d_gy` (number): Dose per fraction used, Gy - `alpha_beta` (number): Caller-assigned tissue α/β used, Gy - `bed_gy` (number): Fowler 1989 biologically effective dose in Gy, 1 decimal - `citation` (object): Fowler 1989 Br J Radiol citation - `disclaimer` (string): Not-a-device notice ## Example JSON Frozen fixture. Not a live lookup. ```json { "n": 30, "disclaimer": "Not a medical device. Deterministic published-formula or published-code output for autonomous agents. A licensed clinician remains responsible for patient care and billing submissions.", "citation": { "authors": "Fowler JF", "doi": "10.1259/0007-1285-62-740-679", "id": "fowler-1989", "pmid": "2670032", "source": "Br J Radiol. 1989;62(740):679-694", "title": "The linear-quadratic formula and progress in fractionated radiotherapy" }, "formula": "bed", "formula_expression": "bed_gy = n * d_gy * (1 + d_gy / alpha_beta)", "alpha_beta": 3.0, "bed_gy": 100.0, "d_gy": 2.0 } ``` ## Errors - HTTP 400 `invalid_bed`: BED inputs must be fraction count, dose per fraction, and alpha/beta as used in Fowler 1989. Returned before settlement; you are not charged. - HTTP 400 `invalid_items`: POST body must be a JSON object with only an items array Returned before settlement; you are not charged. - HTTP 400 `invalid_item_count`: items must be an array of 1 to 25 objects Returned before settlement; you are not charged. - HTTP 400 `invalid_item`: each items entry must be a JSON object Returned before settlement; you are not charged. Shared HTTP 402 and 503: https://magentlab.com/docs/payments ## Related tools - [Morphine milligram equivalents](https://magentlab.com/docs/mme) — Compute morphine milligram equivalents from a CDC 2022 table opioid and daily dose (or transdermal fentanyl mcg/hr). - [Phenytoin correction](https://magentlab.com/docs/phenytoin-correction) — Estimate an albumin-adjusted total-equivalent phenytoin concentration from observed total phenytoin and albumin, with an optional renal binding-factor change. ## Payment Unpaid requests return HTTP 402 even if query parameters or POST items are missing or invalid, so CDP and agents can index the path. After a PAYMENT-SIGNATURE, invalid params return HTTP 400 JSON before settlement. magent does not charge. Fix the params, then request a new 402 quote for that complete URL. Catalog misses (empty matches, found=false) are valid paid 200s. - GET the tool with Accept: application/json (do not send Accept: text/html unless you want the human paywall). - HTTP 402 means you have not paid. Decode the base64 PAYMENT-REQUIRED header. HTTP 503 is not a new quote. - Sign accepts[0] (exact, EIP-3009) with validBefore = now + maxTimeoutSeconds (600 seconds), or accepts[1] (batch-settlement deposit/voucher) on product GET. Ping and POST {items} stay exact. Magent retries transient facilitator 429/5xx before answering. - Retry the same URL with PAYMENT-SIGNATURE (base64 JSON of accepted and payload). Magent attaches canonical extensions.bazaar on CDP verify/settle; echoing 402 bazaar is optional. - Success is HTTP 200 JSON plus PAYMENT-RESPONSE. exact settles before the tool. batch-settlement verifies (and deposit-settles) before the tool and commits the charge after HTTP 2xx. HTTP 503: retry the same PAYMENT-SIGNATURE. Mint a new exact nonce only after a new HTTP 402. - offered `exact` on eip155:8453: Fixed-price USDC on Base. The buyer authorizes the advertised amount (EIP-3009). magent settles before the tool runs (paymentFlow upfront). extra.assetTransferMethod=eip3009 extra.paymentFlow=upfront. - offered `batch-settlement` on eip155:8453: Payment-channel vouchers claimed later in batches. Exact stays accepts[0]. Same-path POST {items} is still exact. extra.assetTransferMethod=eip3009 extra.paymentFlow=authorization. - not offered `upto`: Not offered. Usage-based: buyer authorizes a ceiling, seller settles the actual amount after work. Reserved for a future metered tool. Not for current calculators or code search. Same-path POST {items} is still scheme exact: one EIP-3009 authorization for GET unit plus $0.002 per extra item. That is not x402 batch-settlement (payment channels / vouchers). Shared HTTP 402 and 503 table: https://magentlab.com/docs/payments Stdio MCP (one process per host): https://magentlab.com/docs/mcp ## Guarantees - HTTP 402 is unpaid, including a missing query. After PAYMENT-SIGNATURE, invalid parameters return HTTP 400 before settlement. You are not charged. HTTP 402 never includes the tool JSON. - exact: settlement finishes before the tool (paymentFlow upfront). batch-settlement: verify/deposit before the tool; chargedCumulativeAmount commits after HTTP 2xx. Uncertain or failed settlement fails closed (HTTP 503). Retry the same PAYMENT-SIGNATURE. - HTTP 402 is unpaid only. Do not sign a second authorization because of a 503. - Execution is timeout-bounded and concurrency-capped. - Calculator inputs are stored encrypted for allowlisted operators (Magent Console). They are not in public logs, SSH lookups, catalog, or rollups. Request logs store path without query.