# PSA density (Benson 1992)

PSA density

Status: LIVE
Price: $0.005 USDC
`GET /api/calc/psad`
HTML: https://magentlab.com/docs/psad
Markdown: https://magentlab.com/docs/psad.md

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

## What it computes

Compute Benson 1992 prostate-specific antigen density as serum PSA (ng/ml) divided by prostate volume (ml). Volume is length × width × depth × 0.5, or a caller-supplied volume.

## When to use

When an agent already has PSA and either a measured prostate volume or three orthogonal dimensions and must not invent a π/6 ellipsoid, diagnose cancer, or treat PSAD as a biopsy order.

## When not to use

- Not a medical device. magent does not measure the gland or diagnose prostate cancer.
- Benson 1992 volume is length × width × depth × 0.5 (D1×D2×D3×1/2). This is not the π/6 ellipsoid.
- Provide psa_ng_ml with prostate_volume_ml (1-500 ml) or length_cm, width_cm, and depth_cm (each 1-20 cm), not both volume styles.
- Retrospective PSAD categories ≤0.05, >0.05 and <0.10, and ≥0.10 are Benson 1992 only. Not a biopsy order.

## Formula

```
volume_ml = length_cm × width_cm × depth_cm × 0.5 (Benson: D1×D2×D3×1/2, not π/6); psad = psa_ng_ml / volume_ml
```

## Citation

[Prostate specific antigen density: a means of distinguishing benign prostatic hypertrophy and prostate cancer](https://doi.org/10.1016/s0022-5347(17)37393-7)
Benson MC, Whang IS, Pantuck A, Ring K, Kaplan SA, Olsson CA, Cooner WH
J Urol. 1992;147(3 Pt 2):815-816
DOI: [10.1016/s0022-5347(17)37393-7](https://doi.org/10.1016/s0022-5347(17)37393-7)

## Worked example

### PSA 4 ng/ml, 4 × 4 × 4 cm gland

Given: `depth_cm=4, length_cm=4, psa_ng_ml=4, width_cm=4`

1. volume_ml = 4 × 4 × 4 × 0.5 = 32
2. psad = 4 / 32 = 0.125
3. 0.125 is ≥0.10 (Benson abnormal)


### PSA 2 ng/ml, volume 40 ml

Given: `prostate_volume_ml=40, psa_ng_ml=2`

1. psad = 2 / 40 = 0.050
2. 0.050 is ≤0.05 (Benson normal)


## Also searched as

- PSA density
- prostate tumor volume
- PSAD
- prostate-specific antigen density
- Benson PSAD
- PSA divided by prostate volume
- BPH versus prostate cancer density

## Parameters

- `psa_ng_ml` (number, required): Serum prostate-specific antigen in ng/ml (0.01-10000, magent clamp). magent does not assay PSA.
- `prostate_volume_ml` (number, optional): Caller-supplied prostate volume in ml (1-500). Provide this or length_cm, width_cm, and depth_cm, not both. magent does not measure the gland.
- `length_cm` (number, optional): Prostate length (D1) in cm (1-20). Required with width_cm and depth_cm when prostate_volume_ml is omitted. Benson volume = L×W×D×0.5.
- `width_cm` (number, optional): Prostate width (D2) in cm (1-20). Required with length_cm and depth_cm when prostate_volume_ml is omitted.
- `depth_cm` (number, optional): Prostate depth (D3) in cm (1-20). Required with length_cm and width_cm when prostate_volume_ml is omitted.

## 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/psad`
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": [
    {
      "depth_cm": "4",
      "length_cm": "4",
      "psa_ng_ml": "4",
      "width_cm": "4"
    },
    {
      "depth_cm": "4",
      "length_cm": "4",
      "psa_ng_ml": "4",
      "width_cm": "4"
    }
  ]
}
```

Human paywall (two-item fixture): https://api.magentlab.com/paywall/bulk/psad

Example 200:

```json
{
  "count": 2,
  "path": "/api/calc/psad",
  "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": "Benson MC, Whang IS, Pantuck A, Ring K, Kaplan SA, Olsson CA, Cooner WH",
        "doi": "10.1016/s0022-5347(17)37393-7",
        "id": "benson-1992-psad",
        "pmid": "1371554",
        "source": "J Urol. 1992;147(3 Pt 2):815-816",
        "title": "Prostate specific antigen density: a means of distinguishing benign prostatic hypertrophy and prostate cancer"
      },
      "formula": "psad",
      "formula_expression": "volume_ml = length_cm × width_cm × depth_cm × 0.5 (Benson: D1×D2×D3×1/2, not π/6); psad = psa_ng_ml / volume_ml",
      "psa_ng_ml": 4.0,
      "benson_band": ">=0.10",
      "prostate_volume_ml": 32.0,
      "psad": 0.125,
      "depth_cm": 4.0,
      "length_cm": 4.0,
      "width_cm": 4.0
    },
    {
      "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": "Benson MC, Whang IS, Pantuck A, Ring K, Kaplan SA, Olsson CA, Cooner WH",
        "doi": "10.1016/s0022-5347(17)37393-7",
        "id": "benson-1992-psad",
        "pmid": "1371554",
        "source": "J Urol. 1992;147(3 Pt 2):815-816",
        "title": "Prostate specific antigen density: a means of distinguishing benign prostatic hypertrophy and prostate cancer"
      },
      "formula": "psad",
      "formula_expression": "volume_ml = length_cm × width_cm × depth_cm × 0.5 (Benson: D1×D2×D3×1/2, not π/6); psad = psa_ng_ml / volume_ml",
      "psa_ng_ml": 4.0,
      "benson_band": ">=0.10",
      "prostate_volume_ml": 32.0,
      "psad": 0.125,
      "depth_cm": 4.0,
      "length_cm": 4.0,
      "width_cm": 4.0
    }
  ]
}
```

## Response fields

- `formula` (string): psad
- `formula_expression` (string): Exact Benson 1992 expressions used
- `psa_ng_ml` (number): Serum PSA used, ng/ml
- `prostate_volume_ml` (number): Prostate volume used, ml (caller-supplied or length × width × depth × 0.5), 3 decimals when computed
- `length_cm` (number): Length D1 used, cm. Omitted on the prostate_volume_ml path.
- `width_cm` (number): Width D2 used, cm. Omitted on the prostate_volume_ml path.
- `depth_cm` (number): Depth D3 used, cm. Omitted on the prostate_volume_ml path.
- `psad` (number): PSA density (ng/ml per ml), dimensionless, 3 decimals
- `benson_band` (string): Benson 1992 retrospective category: <=0.05, 0.05-0.10 (open interval), or >=0.10
- `citation` (object): Benson 1992 citation
- `disclaimer` (string): Not-a-device notice

## Example JSON

Frozen fixture. Not a live lookup.

```json
{
  "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": "Benson MC, Whang IS, Pantuck A, Ring K, Kaplan SA, Olsson CA, Cooner WH",
    "doi": "10.1016/s0022-5347(17)37393-7",
    "id": "benson-1992-psad",
    "pmid": "1371554",
    "source": "J Urol. 1992;147(3 Pt 2):815-816",
    "title": "Prostate specific antigen density: a means of distinguishing benign prostatic hypertrophy and prostate cancer"
  },
  "formula": "psad",
  "formula_expression": "volume_ml = length_cm × width_cm × depth_cm × 0.5 (Benson: D1×D2×D3×1/2, not π/6); psad = psa_ng_ml / volume_ml",
  "psa_ng_ml": 4.0,
  "benson_band": ">=0.10",
  "prostate_volume_ml": 32.0,
  "psad": 0.125,
  "depth_cm": 4.0,
  "length_cm": 4.0,
  "width_cm": 4.0
}
```

## Errors

- HTTP 400 `invalid_psad`: PSA density needs PSA (ng/ml) and either prostate_volume_ml or three dimensions as used in Benson 1992 (volume = L×W×D×0.5). Returned before settlement; you are not charged.
- HTTP 400 `invalid_psa`: psa_ng_ml must be a number from 0.01 to 10000. Returned before settlement; you are not charged.
- HTTP 400 `invalid_prostate_dimension`: Prostate length, width, and depth in cm must each be a number from 1 to 20, or provide prostate_volume_ml (1-500). 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

- [Gleason score](https://magentlab.com/docs/gleason) — Sum caller-assigned Gleason 1974 primary and secondary histologic patterns (each 1-5) and return score 2-10. magent does not read slides. Not ISUP/Epstein 2016 Grade Groups.
- [D'Amico](https://magentlab.com/docs/damico) — Apply the D'Amico 1998 JAMA clinically localized three-group rule to caller-assigned clinical T (t1c, t2a, t2b, t2c), Gleason sum 2-10, and PSA ng/mL, and return risk low, intermediate, or high. Does not assay PSA, assign Gleason, or compute CAPRA or NCCN groups.
- [CAPRA](https://magentlab.com/docs/capra) — Apply Cooperberg 2005 Table 1 CAPRA points from caller-assigned preoperative PSA, Gleason primary/secondary, clinical T (t1/t2/t3a), percent positive biopsies, and age, and return the 0-10 score, Table 4 Kaplan-Meier band, and Table 4 5-year recurrence-free survival point estimate. Not a diagnosis.

## 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.