Fractional effective dose

Fractional effective dose (FED) expresses an accumulated exposure as a fraction of the dose associated with a specified effect. Fire engineering uses the concept to estimate when exposure to combustion products or heat may impair or prevent escape. The effect being modelled, the exposure-response relationship and the contributing hazards must be stated.

A value of one represents the nominal endpoint of the selected model. It is not a universal boundary below which every occupant remains unaffected. Human susceptibility varies, response models contain uncertainty, and design criteria can require margins relative to the nominal endpoint. FED is therefore an index for a defined assessment, not a probability of injury or survival.

Within fire toxicity assessment, FED links a concentration history to a physiological model. An instantaneous gas concentration alone does not describe that history. Equally, the same concentration-time product does not have an identical meaning for every gas or exposure duration.

Dose, endpoint and normalisation

The basic construction divides each increment of received dose by a reference dose associated with the selected effect. The fractions are accumulated during exposure. Where a constant reference dose is appropriate, the relationship can be written as follows.

Fractional dose accumulates exposure relative to a specified response endpoint. Based on Purser (2002), pp. 2-88–2-90.

FED(t) = Σ_j Δd_j/d_ref

Here Δd_j is the dose received during interval j, and d_ref is the effective dose for the stated endpoint. Numerator and denominator must use the same dose definition and units. Their ratio is dimensionless. The normalisation does not remove the need to explain what was measured or calculated.

For a gas whose effect follows a constant concentration-time relationship, an increment can be represented by concentration multiplied by exposure duration. The reference dose then has matching concentration-time units. Purser describes this as a practical way to integrate a changing fire exposure until the nominal endpoint is reached.

However, FED is more general than division by one universal concentration-time constant. Where the exposure-response relationship changes with concentration, the appropriate reference or dose-rate expression must also change. Applying an unchanged denominator to every gas would erase the physiological differences that the model is intended to represent.

The endpoint needs equally careful definition. Purser distinguishes a fractional incapacitating dose from a fractional lethal dose. The same word dose does not make those endpoints interchangeable. A model intended to describe loss of escape capability should not be interpreted as a prediction of death at the same numerical value.

The choice also depends on the engineering question. Impairment can delay movement before full incapacitation occurs, while some consequences develop after exposure. A calculation focused on one endpoint does not automatically address all the other ways that smoke can affect an occupant.

Concentration-time relationships and their limits

Haber's rule treats concentration and duration as interchangeable through a constant product for a specified effect. Under that approximation, doubling concentration halves the time required to accumulate the same dose. Purser explains both the usefulness of this concept and the circumstances in which it breaks down.

For carbon monoxide (CO), the physiological quantity of interest includes the accumulated carboxyhaemoglobin concentration in blood. Uptake and removal occur together. During a short exposure to a relatively high inhaled concentration, a nearly linear uptake approximation can be useful. During longer exposure at lower concentrations, equilibrium and removal become more important.

Activity influences the calculation through respiratory minute volume and the demands placed on the person. A model for a resting subject and one for an active subject need not predict the same time to incapacitation. Body size and physiological parameters also affect the relationship between inhaled concentration and internal dose.

Hydrogen cyanide (HCN) provides a contrasting example. Purser describes a strong dependence on uptake rate and concentration, with a marked departure from a constant concentration-time endpoint. A short high-concentration exposure cannot generally be treated as equivalent to a longer low-concentration exposure merely because their products are equal.

Reduced oxygen concentration introduces another response pattern. Following equilibration, the severity of oxygen-deficiency effects is strongly concentration-dependent. High carbon dioxide can also have direct effects and can increase the uptake of other gases by stimulating breathing. The mixture therefore requires more than one interchangeable dose constant.

These distinctions matter when processing time histories. A simple average concentration may preserve a linear concentration-time product, but it need not preserve a nonlinear response calculation. The selected physiological relationship determines whether averaging, segmentation or a more complete uptake model is appropriate.

A specified carbon monoxide dose model

Purser presents a short-exposure CO approximation derived from the Stewart uptake relationship. For the stated light-activity assumptions, the fraction of the nominal incapacitating dose can be calculated over discrete time intervals. The following notation keeps the exposure concentration separate from the internal endpoint.

F_CO = Σ_j[a C_CO,j^1.036 Δt_j/H_lim]

Symbol	Meaning and required convention
FCO	Dimensionless fraction of the model's CO incapacitation endpoint
CCO,j	CO concentration during interval j, entered in parts per million (ppm)
Δtj	Interval duration, entered in minutes
a	Empirical coefficient 8.2925 × 10−4 for the stated units and respiratory minute volume
Hlim	Nominal endpoint of 30 percent carboxyhaemoglobin, entered as 30

The coefficient corresponds to a respiratory minute volume of 25 L/min and the source's light-activity condition. The denominator is expressed as a numerical percentage, rather than the fraction 0.30. Those conventions are part of the empirical equation. Changing units or percentage representation without adjusting the coefficient changes the result.

This is a specific historical model formulation. Its inclusion illustrates how a dose calculation is made auditable through explicit inputs and an explicit endpoint. It does not establish that this equation represents every occupant, every CO exposure pattern or the complete toxicity of a fire atmosphere.

Purser discusses the boundary between this approximation and a more complete uptake calculation. Long exposures near equilibrium, or substantial decreases in inhaled concentration, can require treatment of elimination and changing internal concentration. The simple accumulation used here does not calculate recovery following exposure.

The activity assumption affects more than the label attached to the result. Purser relates both CO uptake and the response endpoint to the subject's activity. Changing from rest to movement can therefore require reconsidering the parameter set. Retaining a fixed uptake coefficient while describing a different activity would leave an inconsistency between the exposure narrative and the numerical model.

The exponent is also part of the model. It is close to, but not exactly, one. The calculation therefore should use the stated power for each exposure segment rather than silently replacing it with a linear concentration-time expression. Numerical convenience does not justify changing the response relationship.

Worked accumulation over two exposure intervals

Consider an illustrative CO exposure of 1000 ppm for two minutes, followed by 2000 ppm for one minute. Use the light-activity parameters above and start the calculation with zero accumulated model dose. These are selected inputs for demonstrating the calculation, not a measured fire or a permissible exposure schedule.

The first interval contributes approximately 0.0709 to F_CO. The second contributes approximately 0.0727, giving a cumulative value of approximately 0.144 after three minutes. The calculation applies the concentration exponent separately to each interval and retains unrounded intermediate values.

The calculated CO fraction accumulates more rapidly during the higher-concentration interval. Based on Purser (2002), pp. 2-102–2-103.

Interval	CO, ppm	Duration, min	Dose increment	Cumulative FCO
First	1000	2	0.0709	0.0709
Second	2000	1	0.0727	0.144

The shorter second interval contributes slightly more than the first because its concentration is higher and the response is not exactly linear in concentration. This demonstrates why a cumulative dose must be calculated from the applicable response relationship, rather than inferred from duration alone.

Splitting either constant-concentration interval into smaller intervals gives the same accumulated result. The accompanying calculation checks one-minute, half-minute and tenth-minute subdivisions. This agreement checks the arithmetic implementation for the selected piecewise-constant history. It does not validate the physiological assumptions or establish accuracy for an unresolved changing exposure.

The result is only the CO component under the stated model. It excludes other gases, oxygen depletion, enhanced breathing caused by carbon dioxide, heat, obscuration and irritant effects. An overall fire assessment cannot assume those contributions are absent merely because the demonstration did not calculate them.

Nor should 0.144 be read as a 14.4% probability of incapacitation. It is a fraction of a specified nominal endpoint. Population variability and the chosen design criterion require separate consideration. A probability model would need its own exposure-response definition and supporting parameters.

Mixtures, model scope and independent hazards

Asphyxiant assessments can combine contributions from several gases according to a defined model. Purser identifies CO, HCN and reduced oxygen as important contributors, with carbon dioxide affecting uptake. The combination rules belong to that model; they are not permission to add every available fire-hazard number without regard to its endpoint.

Model names alone do not reveal identical coverage. The supplied CFAST reference describes a gaseous FED calculation using CO and HCN with a carbon-dioxide breathing factor. The supplied New Zealand method instead specifies that its named FED_CO assessment include CO, carbon dioxide and oxygen. The contributing species must be checked explicitly.

Thermal dose is another application of the fractional-dose concept. Both supplied documents describe thermal assessment involving radiation and convection. Its input quantities and response equations differ from those for inhaled toxicants. A gas-dose value and a thermal-dose value should retain their separate model meanings unless a documented combination is provided.

Different hazards require their own response models and criterion checks. Based on Purser (2002), pp. 2-88–2-91, 2-130–2-132, and Ministry of Business, Innovation and Employment (2025), p. 17.

Fractional effective concentration addresses a different type of relationship. Purser uses a fractional irritant concentration for immediate sensory effects governed by concentration. Such an index can exceed its selected endpoint even when a cumulative asphyxiant dose remains below its criterion.

Visibility introduces another independent limitation. A low toxic dose does not establish that an occupant can see the route, identify an exit or move efficiently. The broader assessment needs the applicable optical, irritant and thermal criteria alongside the gas-dose calculation.

This separation is useful when reporting the controlling hazard. The assessment can identify whether dose, obscuration, irritation or heat first limits escape under the selected scenario. Combining unlike quantities into an unexplained total would make that physical interpretation harder to recover.

Endpoints, criteria and population variability

An FED endpoint is a model normalisation; tenability criteria state how the model is used in an engineering decision. A criterion can be set below the nominal endpoint to account for the purpose of the assessment and uncertainty. The selected value should be identified by hazard and framework.

For example, the supplied ABCB handbook uses 0.3 for its thermal dose criterion at the prescribed assessment height. This should not be transferred automatically to a CO calculation merely because both outputs are called FED. The source treats supplementary CO assessment for smouldering scenarios separately.

Human sensitivity is not uniform. Purser explains that underlying response models draw on limited human observations and experimental evidence, including animal studies. The nominal effect level represents a central response assumption, while susceptible groups can respond differently. That uncertainty cannot be eliminated by reporting more decimal places.

A probit function is one of the other human-exposure modelling approaches identified by the ABCB handbook. Its existence does not make FED itself a probability. The quantity being calculated must remain clear before a result is used to characterise the fraction of a population affected.

The difference between toxic potency and scenario hazard is also relevant. A substance can have a high potency, but the resulting hazard depends on how much is produced and what exposure reaches the occupant. FED assessment therefore needs a scenario-specific concentration history as well as the response model.

The history must be associated with a location or an occupant exposure path. CFAST calculates its tenability quantities at specified target positions. Such a stationary target describes conditions there; interpreting it as the complete history of a moving occupant requires additional exposure assumptions.

Calculation records and interpretation

A reviewable FED calculation records its endpoint, contributing hazards, concentrations, time units, activity assumptions and initial condition. It identifies the exposure interval and the mathematical combination used. These details allow a reviewer to distinguish a source-term error from an exposure-history error or a response-model error.

The concentration data require their own justification. Product yields depend on the combustion conditions, and a material does not necessarily produce the same effluent in a small test and a full-scale fire. Selecting a physiological equation does not resolve uncertainty in the atmosphere supplied to that equation.

Numerical checks should preserve the response model. Constant segments can be subdivided to check accumulation, while changing profiles require sufficient temporal detail to represent the chosen nonlinear expression. A calculation should not reset an accumulated dose at each output time or substitute a peak concentration for the full exposure history.

The final interpretation should state the calculated dose, the criterion used and the hazards covered. It should also identify any exposure assumptions that could materially change the conclusion. This connects the mathematical fraction to the actual fire-engineering question without treating a nominal endpoint as a guarantee for every person.

See also

• Carbon dioxide
• Oxygen depletion
• Smoke inhalation

References

• Australian Building Codes Board. (2020). Fire safety verification method: Handbook (Version 1.1).
• Ministry of Business, Innovation and Employment. (2025). C protection from fire: Verification Method C/VM2 (2nd ed.).
• Peacock, R. D., McGrattan, K. B., Forney, G. P., & Reneke, P. A. (2026). CFAST: Consolidated model of fire growth and smoke transport, version 7. Volume 1: Technical reference guide (NIST Technical Note 1889v1, revision 7.7.7). National Institute of Standards and Technology.
• Purser, D. A. (2002). Toxicity assessment of combustion products. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-83–2-171). National Fire Protection Association.