Continuous BMD

Invert a supplied continuous mean model μ(d)=μ0·exp(kd) at a positive relative mean-response change; no dichotomization, parameter fitting or lower confidence bound is invented.

Description

Calculate the dose giving a specified relative increase in a supplied continuous exponential mean response.

Continuous endpoints measure quantities such as an assay response or organ weight rather than affected-versus-unaffected counts. A continuous benchmark response can be defined as a relative change from a modeled background mean. The endpoint and change criterion need to be stated explicitly.^[1]

This calculator assumes an increasing mean curve μ(d) = μ0 × exp(k × d). It accepts the background mean, positive exponential rate, and positive relative increase, and returns the corresponding dose and target mean. Enter 0.2 for a 20% increase; this input is not a probability.

When to use Continuous BMD

Use it to inspect a chosen increasing exponential fit after its parameters and benchmark criterion have been estimated elsewhere.

How to use Continuous BMD and understand its results

Enter positive backgroundMean in the response unit and positive exponentialRatePerDose in reciprocal dose units.
Enter positive benchmarkRelativeChange as a fraction of background, then run the calculation.
Read benchmarkDose in the dose unit implied by the rate and targetMeanResponse in the response unit. Retain the model and endpoint definition with the result.

target mean = μ0 × (1 + r); benchmark dose = ln(1 + r)/k

The implementation uses log1p for the logarithm. Background controls the target mean but cancels from the relative-change dose.

json

{
  "input": {
    "backgroundMean": 10,
    "exponentialRatePerDose": 0.1,
    "benchmarkRelativeChange": 0.2
  },
  "output": {
    "benchmarkDose": 1.823215567939546,
    "targetMeanResponse": 12
  }
}

A relative change greater than one is allowed: 1 means a 100% increase. Zero or negative background, rate, or change is rejected. Doubling background leaves the dose unchanged while doubling the target mean.

Limitations and assumptions

No observations are fitted and no response variance, lower confidence bound, residual check, or model comparison is calculated.

The model supports increasing positive means only. It does not cover a decreasing endpoint, an additive change, a standard-deviation benchmark, or a quantal extra-risk criterion.

Alternative or Complementary methods

A probit model serves a different binary-response question. A margin calculation can use an independently justified point of departure and compatible exposure after reviewing the model evidence.

References

Benchmark Dose Technical Guidance (2012) — U.S. Environmental Protection Agency

Similar or alternative tools

Benchmark Dose Probit
Calculate an extra-risk benchmark for a supplied binary-response probit model.
Margin of Exposure
Compare an assessed point of departure with a supplied exposure.