Concentration Addition Mixture

Predict a concentration-addition mixture response under common-shape Hill curves from normalized component doses; assumes compatible endpoint and modes of action.

Description

Normalize concentrations by component EC50s and predict a Hill response with one shared slope.

Concentration addition expresses component concentrations on a common potency scale and combines their equivalent contributions. This reference is suited to a specified common response model; it differs from combining effect probabilities under independent action.^[1]

This tool uses positive component EC50s, meaning concentrations associated with half of the modeled effect range, and a shared positive Hill slope. It assumes every component follows a compatible zero-to-one response relationship on that same effect scale.

Each component concentration and its EC50 must use compatible units. Their ratios sum to a dimensionless equivalent dose, which is converted to a predicted mixture effect fraction. An EC50 and a slope do not independently validate the concentration-addition assumption.

When to use Concentration Addition Mixture

Use it to inspect a common-slope Hill concentration-addition scenario from externally established potencies, or compare the reference prediction with effect data.

How to use Concentration Addition Mixture and interpret its calculation

Enter aligned, nonempty componentConcentrations and componentEc50Values lists. Concentrations may be zero; every EC50 must be positive.
Enter a positive commonHillSlope appropriate to the shared response model and run the calculation.
Read summedEquivalentDose and predictedMixtureEffectFraction. Assess observed departures and the model assumptions separately.

S = Σ(C_i/EC50_i); predicted effect = S^h/(1 + S^h)

h is the shared Hill slope. S equals one at the model’s half-effect mixture concentration.

For S at least one, the implementation evaluates 1/(1 + S^(−h)); for smaller S, it evaluates the positive-power form. This equivalent branch avoids large positive powers at high normalized concentration. It does not fit separate component slopes.

json

{
  "input": {
    "componentConcentrations": [
      1,
      2
    ],
    "componentEc50Values": [
      2,
      4
    ],
    "commonHillSlope": 2
  },
  "output": {
    "summedEquivalentDose": 1,
    "predictedMixtureEffectFraction": 0.5
  }
}

All concentrations zero give S zero and effect zero. At S one, effect is 0.5 for any accepted slope. A single component produces the corresponding Hill response. A zero EC50 or nonpositive slope is rejected.

Limitations and assumptions

The model assumes compatible effect scales, full normalized response range, and a common slope. Different slopes, partial maximal effects, nonparallel potency relationships, and interactions are not handled.

No measured mixture response, single-agent fitting, uncertainty estimate, mechanism inference, or toxicity classification is performed. A predicted effect fraction must not be relabeled as excess cancer risk or a hazard quotient.

Alternative or Complementary methods

Use independent action when the intended reference combines component response fractions under independence. A two-agent isobole compares effect-matched dose pairs with supplied target doses without predicting a full shared-slope response.

References

Supplementary Guidance for Conducting Health Risk Assessment of Chemical Mixtures (2000) — U.S. Environmental Protection Agency

Similar or alternative tools

Independent Action Mixture
Multiply unaffected fractions to obtain a different reference response.
Isobologram Analysis
Calculate a normalized dose sum and deviation at a specified common effect level.