# Presentation

This software was designed by Ineris to evaluate lethality thresholds of chemicals. The software models binary data (such as lethality; irreversible toxicity: presence or absence of congenital abnormalities, blindness, severe burns, etc.; reversible effects: apparition or not of a transient unconsciousness, etc.) obtained with either single or multiple exposure durations, for several doses or concentrations. The fitted model is then used to calculate the BenchMark Dose (BMD) and the lower limit of the one-sided 95% confidence limit for the BMD (BMDL).

## 1. THE MODEL

### 1.1 Model for one exposure duration

At a given dose, the number of animals with an adverse response is assumed to be binomially distributed.

The mathematical model used is the logprobit model, which was selected in a comparative study by Péry et al. (2010) for its ability to describe distribution of sensitivity in a population.

The model developed in this software assumes that the incidence in the control groups is zero. The probability of an adverse response can be expressed as α+β ×log(C), where α and β are the parameters (β is called the shape or slope parameter) and C is the concentration or dose. The probability of an adverse response follows a cumulative standard normal distribution.

### 1.2 Model for several duration exposures

The model for one exposure duration has been adapted to model data collected in experiments involving different exposure durations.

As in Péry et al. (2010) and the beta version in BMDS 2.1.2 (Davis et al., 2011), the logprobit model was extended using the generalizations of Haber’s law (ten Berge et al., 1986), stating that the effect for exposure to concentration C during a period of time t is a function of C^{n}t, named fixed effect level (Jarabek, 1995), where n is called the Haber constant.

The probability of an adverse response can thus be expressed as α+β ×log(C^{n}× t) and follows a cumulative standard normal distribution.

## 2. BMD AND BMDL CALCULATIONS

### 2.1 ESTIMATION METHOD

The model parameters α and β, and optionally n, are estimated with the maximum likelihood method.

The BMD is computed for a given exposure duration as a function of the parameters of the model. It is the dose (or concentration) that would have an incidence of x%, the BenchMark Response (BMR).

The lower limit of the 95% (1-α) confidence-level (BMDL) is based on the log-likelihood ratio statistic and is computed using constrained optimization, by finding the smallest value BMD such that the log-likelihood is equal to the maximum likelihood minus 1.35 (Χ(1-2α, 1)2) (one-sided confidence interval) as in BMDS software version 2.1.2 (US EPA (Environmental Protection Agency), 2008).

### 2.2 EXTRAPOLATION TO EXPOSURE DURATIONS NOT COVERED BY THE DATA

#### 2.2.1 Model for one exposure duration

BMD and BMDL values can be calculated for several exposure durations, using the generalization on Haber’s law, with n=1 for longer exposure durations than the one experimentally tested, and n=3 for shorter exposure duration than the one tested.

#### 2.2.2 Model for several duration exposures

BMD and BMDL can be estimated for duration exposures which were not experimentally tested, but need to be used with caution when they are out of the range tested.

## 3. DATA REQUIREMENTS

To run the model, the following data are necessary:

- The number of animals in each group
- The exposure concentration or dose of each group
- The number of animals with an adverse response after exposure in each group
- If the exposure duration varies from one group to another, the exposure duration of each group
- If there was only one duration, BMD and BMDL values are calculated for other exposure durations (1, 10, 20, 30, 60, 120, 240, and 240 minutes) if the user specifies the exposure duration of the data.

Data can be input in an online form or by uploading a file (.txt format, tab delimited, points as decimals, one header line, concentration in first column, number of subjects in second column, incidence in third column, and optionally exposure duration in fourth column).

## 4. REFERENCES

**Davis JA, Gift JS, Zhao QJ.** Introduction to benchmark dose methods and U.S. EPA's benchmark dose software (BMDS) version 2.1.1. Toxicology and Applied Pharmacology 2011; 254: 181-191.

**Jarabek A. **Consideration of temporal toxicity challenges current default assumptions. Inhalation toxicology 1995; 7: 927-946.

**Péry ARR, Troise A, Tissot S, Vincent JM.** Comparison of models to analyze mortality data and derive concentration–time response relationship of inhaled chemicals. Regulatory Toxicology and Pharmacology 2010; 57: 124–128.

**ten Berge WF, Zwart A, Appelman LM.** Concentration—time mortality response relationship of irritant and systemically acting vapours and gases. Journal of Hazardous Materials 1986; 13: 301-309.

**US EPA (Environmental Protection Agency).** Benchmark Dose Software (BMDS), 2008.