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Winters' formula

Compute expected arterial PCO2 in metabolic acidosis from plasma bicarbonate using the Albert–Winters 1967 formula.

LIVE $0.005 USDC GET /api/calc/winters

Markdown: /docs/winters.md · Canonical: https://magentlab.com/docs/winters

Not a medical device. Not a diagnosis, dose, or listing recommendation.

What it computes

Compute expected arterial PCO2 in metabolic acidosis from plasma bicarbonate using the Albert–Winters 1967 formula.

When to use

When an agent needs published Winters expected PCO2 and must not invent a metabolic acidosis compensation band.

When not to use

• Not a treatment target. Applies to metabolic acidosis compensation as published; the ±2 mm Hg band is the published Winters range.
• Does not compute metabolic alkalosis expected PCO2 formulas.

Formula

expected_pco2_mm_hg = 1.5 * bicarbonate_mmol_l + 8

Citation

Title

Quantitative displacement of acid-base equilibrium in metabolic acidosis

Authors

Albert MS, Dell RB, Winters RW

Source

Ann Intern Med. 1967;66(2):312-322

DOI

10.7326/0003-4819-66-2-312

Worked example

Try

Opens the live endpoint. Unpaid browser requests show the paywall; agents should send Accept: application/json.

/api/calc/winters?bicarbonate_mmol_l=12

Full URL: https://api.magentlab.com/api/calc/winters?bicarbonate_mmol_l=12

x402 V2 curl

Expect HTTP 402 and a PAYMENT-REQUIRED header until you retry with PAYMENT-SIGNATURE. Incomplete query params return HTTP 400 before any quote or charge. HTTP 503 after a signature: retry the same PAYMENT-SIGNATURE — that is not a new 402.

curl -i -H "Accept: application/json" "https://api.magentlab.com/api/calc/winters?bicarbonate_mmol_l=12"

Cursor mcp.json

Intended config for @magent/mcp. This is not npm @x402/fetch (a payment fetch wrapper).

{
  "mcpServers": {
    "magent": {
      "args": [
        "-y",
        "@magent/mcp"
      ],
      "command": "npx",
      "env": {
        "X402_PRIVATE_KEY": "0xYOUR_SPENDING_KEY"
      }
    }
  }
}

Parameters

Bulk (POST)

POST the same path with a JSON items array. Price is n times the GET unit. Invalid items return HTTP 400 before settlement.

POST /api/calc/winters Max 25 unit_amount * n PREVIEW

• 1 to 25 items. Each item uses the same keys as the GET query parameters.
• 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 (fixture)

{
  "items": [
    {
      "bicarbonate_mmol_l": "12"
    },
    {
      "bicarbonate_mmol_l": "12"
    }
  ]
}

Open bulk paywall →

Opens a two-item fixture on the API host. Unpaid browsers get the same MetaMask paywall as GET; Pay retries POST with those items.

Example 200 (fixture)

{
  "count": 2,
  "path": "/api/calc/winters",
  "items": [
    {
      "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": "Albert MS, Dell RB, Winters RW",
        "doi": "10.7326/0003-4819-66-2-312",
        "id": "albert-winters-1967",
        "source": "Ann Intern Med. 1967;66(2):312-322",
        "title": "Quantitative displacement of acid-base equilibrium in metabolic acidosis"
      },
      "formula": "winters",
      "formula_expression": "expected_pco2_mm_hg = 1.5 * bicarbonate_mmol_l + 8",
      "bicarbonate_mmol_l": 12.0,
      "expected_pco2_high_mm_hg": 28.0,
      "expected_pco2_low_mm_hg": 24.0,
      "expected_pco2_mm_hg": 26.0
    },
    {
      "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": "Albert MS, Dell RB, Winters RW",
        "doi": "10.7326/0003-4819-66-2-312",
        "id": "albert-winters-1967",
        "source": "Ann Intern Med. 1967;66(2):312-322",
        "title": "Quantitative displacement of acid-base equilibrium in metabolic acidosis"
      },
      "formula": "winters",
      "formula_expression": "expected_pco2_mm_hg = 1.5 * bicarbonate_mmol_l + 8",
      "bicarbonate_mmol_l": 12.0,
      "expected_pco2_high_mm_hg": 28.0,
      "expected_pco2_low_mm_hg": 24.0,
      "expected_pco2_mm_hg": 26.0
    }
  ]
}

Response

Example JSON

Machine form of the same example. Frozen fixture. Not a live lookup.

{
  "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": "Albert MS, Dell RB, Winters RW",
    "doi": "10.7326/0003-4819-66-2-312",
    "id": "albert-winters-1967",
    "source": "Ann Intern Med. 1967;66(2):312-322",
    "title": "Quantitative displacement of acid-base equilibrium in metabolic acidosis"
  },
  "formula": "winters",
  "formula_expression": "expected_pco2_mm_hg = 1.5 * bicarbonate_mmol_l + 8",
  "bicarbonate_mmol_l": 12.0,
  "expected_pco2_high_mm_hg": 28.0,
  "expected_pco2_low_mm_hg": 24.0,
  "expected_pco2_mm_hg": 26.0
}

Errors

Related tools

• Bicarbonate deficit — Estimate bicarbonate deficit from weight and measured plasma HCO3 using a 50 percent body-weight apparent space.
• Serum osmolality — Compute Worthley calculated serum osmolality from sodium, glucose, and BUN, plus an optional osmolar gap when a measured osm is supplied.

Also searched as

• Winters formula
• expected PCO2
• metabolic acidosis compensation
• Winters PCO2
• expected PaCO2
• HCO3 expected CO2
• Albert Winters formula

Payment

Required query parameters must be present and valid. 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. Unpaid valid requests return HTTP 402 with a base64 PAYMENT-REQUIRED header. HTTP 503 is not a new quote: retry the same PAYMENT-SIGNATURE. Full handshake and shared payment errors: Payments.

Guarantees

• Invalid query parameters return HTTP 400 before settlement. You are not charged.
• Settlement finishes before the tool executes (paymentFlow upfront). 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.