Deriving Human Equivalent Dose from an Animal NOAEL

Why Convert an Animal NOAEL?

A no-observed-adverse-effect level (NOAEL) is expressed for the tested animal species and study conditions. A dose in mg/kg/day therefore cannot be transferred directly to a human by carrying over the same numerical value: the relationship between body weight and body-surface area differs across species. The human equivalent dose (HED) is a dose expressed in human mg/kg/day, calculated from the animal NOAEL using a specified body-surface-area conversion.

The conversion is a way to put doses on a common body-surface-area basis. It does not establish that the animal dose and HED produce the same drug concentrations, pharmacologic response, or toxicity in humans. The HED is a translation of dose units under a defined scaling assumption, not a direct prediction of human exposure or a clinical starting dose.

Definition: Let $D_a$ be the animal NOAEL in mg/kg/day, $D_{\mathrm{HED}}$ the human equivalent dose in mg/kg/day, $K_{m,a}$ the animal body-surface-area factor, and $K_{m,h}$ the human body-surface-area factor. The body-surface-area conversion is $$ D_{\mathrm{HED}}=D_a\frac{K_{m,a}}{K_{m,h}}. $$ Here $K_m$ is body weight divided by body-surface area, conventionally expressed in kg/m². The ratio of two $K_m$ factors is dimensionless.

Deriving the Conversion

The conversion follows from assuming equal dose per unit body-surface area in the animal and human. This is a scaling convention used to derive an HED; it is not a claim that surface area alone determines drug disposition or safety.

Let $W_a$ and $W_h$ be animal and human body weights in kg, and let $A_a$ and $A_h$ be their body-surface areas in m². If the animal receives $D_a$ mg/kg/day, its total daily amount is $D_aW_a$ mg/day. Dividing by surface area gives its dose per unit area. The corresponding human quantities are defined in the same way. Equating these quantities gives:

\frac{D_aW_a}{A_a}=\frac{D_{\mathrm{HED}}W_h}{A_h}.

Rearrange to solve for the human dose per kilogram:

D_{\mathrm{HED}} =D_a\frac{W_a/A_a}{W_h/A_h} =D_a\frac{K_{m,a}}{K_{m,h}}, \qquad K_m=\frac{W}{A}.

Since $W/A$ has units kg/m², multiplying the animal dose in mg/kg/day by the ratio $K_{m,a}/K_{m,h}$ leaves the result in mg/kg/day. The smaller the animal $K_m$ relative to the human $K_m$, the smaller the HED than the animal dose on a mg/kg basis.

Assumptions and scope: The calculation assumes that the animal NOAEL is correctly identified for the relevant study, that the dose is expressed in compatible units (typically mg/kg/day), and that the chosen species and human factors are appropriate for the intended calculation. It applies the equal-dose-per-surface-area convention. It does not adjust for differences in route, bioavailability, dosing schedule, pharmacokinetics, pharmacodynamics, or sensitivity to toxicity.

Applying the Formula

Use the species-specific factor for the animal and the specified human factor. Keep the ratio in the order animal divided by human. The animal dose is a dose per kilogram, not the total amount administered to an individual animal. If a human total daily amount is also needed for a stated body weight, calculate it only after obtaining the HED by multiplying by that human body weight.

Worked Example: Rat NOAEL to HED

Suppose a hypothetical rat study identifies a NOAEL of 18 mg/kg/day. For this calculation, use $K_{m,a}=6$ for the rat and $K_{m,h}=37$ for the human. These factors are supplied for the example; selecting or justifying factors is a separate task.

Check the input. The NOAEL is a dose rate per unit body weight, in mg/kg/day, so it can be used directly in the conversion.

Substitute into the formula.

D_{\mathrm{HED}} =18\ \text{mg/kg/day}\times\frac{6}{37} =\frac{108}{37}\ \text{mg/kg/day} \approx2.919\ \text{mg/kg/day}.

The ratio is evaluated without first rounding it to 0.1622; using that rounded ratio would introduce a small rounding difference. The result is an HED of approximately 2.919 mg/kg/day under the stated conversion convention.

For a hypothetical 70 kg adult, the corresponding total daily amount would be $2.918918\ldots\ \text{mg/kg/day}\times70\ \text{kg}\approx204.3\ \text{mg/day}$. This multiplication changes the units from mg/kg/day to mg/day; it does not make the amount a recommended human dose.

Worked Example: Dog NOAEL to HED

Suppose a hypothetical dog study has a NOAEL of 12 mg/kg/day. Use $K_{m,a}=20$ for the dog and $K_{m,h}=37$ for the human.

The dose is already expressed in mg/kg/day, so no conversion from total animal amount is needed. Applying the formula:

D_{\mathrm{HED}} =12\ \text{mg/kg/day}\times\frac{20}{37} =\frac{240}{37}\ \text{mg/kg/day} \approx6.486\ \text{mg/kg/day}.

The HED is approximately 6.486 mg/kg/day. It is lower than the animal NOAEL of 12 mg/kg/day because the animal factor is below the human factor, making $20/37<1$. The comparison is only on the body-surface-area-scaled dose basis; it does not show that the human dose would produce the same exposure or tolerability.

Worked Example: Comparing HEDs From Two Species

Consider two hypothetical studies of the same candidate drug. The rat NOAEL is 20 mg/kg/day, and the dog NOAEL is 8 mg/kg/day. For this example, use $K_{m,\mathrm{rat}}=6$, $K_{m,\mathrm{dog}}=20$, and $K_{m,h}=37$. Calculate both HEDs before comparing them.

For the rat:

D_{\mathrm{HED,rat}} =20\ \text{mg/kg/day}\times\frac{6}{37} =\frac{120}{37}\ \text{mg/kg/day} \approx3.243\ \text{mg/kg/day}.

For the dog:

D_{\mathrm{HED,dog}} =8\ \text{mg/kg/day}\times\frac{20}{37} =\frac{160}{37}\ \text{mg/kg/day} \approx4.324\ \text{mg/kg/day}.

Although the rat NOAEL is numerically higher in animal mg/kg/day, its calculated HED is lower: approximately 3.243 versus 4.324 mg/kg/day. The conversion changes the numerical comparison because it uses species-specific factors. It does not, by itself, decide which study is most informative or establish that either HED is an appropriate human dose; those judgments require the study context and other evidence.

Checks, Alternatives, and Limitations

A reliable calculation can be checked in several ways. First, inspect the units: the input should be a dose rate such as mg/kg/day, and the HED should retain those units. Second, inspect the factor ratio. If $K_{m,a}<K_{m,h}$, the conversion must reduce the animal mg/kg dose; if $K_{m,a}>K_{m,h}$, it must increase it. Third, retain enough precision in intermediate arithmetic and round the final result to a sensible number of significant figures given the input data.

Body-weight scaling is a different conversion from body-surface-area scaling. Directly carrying the same mg/kg/day value from animal to human is not the same operation as applying the $K_m$ ratio. The latter can be derived from equal dose per unit area; the former simply preserves the dose per kilogram. Neither rule should be interpreted as a general law that guarantees equivalent exposure. Exposure-based translation uses pharmacokinetic information and asks a different question from the surface-area conversion taught here.

The HED also depends on the NOAEL chosen as its input. A NOAEL is tied to a particular species, study, dose regimen, and observed endpoints. Changing the study or the identified NOAEL changes the resulting HED. Furthermore, an HED does not account for differences in absorption, clearance, target engagement, or species-specific toxicity. It should therefore be reported with its source NOAEL, factors, units, and assumptions, rather than as a context-free dose.

Common pitfall: Reversing the ratio and calculating $D_a(K_{m,h}/K_{m,a})$ is incorrect for this derivation. For example, if $K_{m,a}=6$ and $K_{m,h}=37$, the correct multiplier is $6/37$, not $37/6$. The reversed calculation would increase an 18 mg/kg/day rat NOAEL to 111 mg/kg/day, contrary to the equal-dose-per-area derivation and the expected direction when the animal factor is smaller than the human factor.

Another frequent error is to apply the conversion to an animal's total dose in mg as though it were a dose in mg/kg/day. If only a total administered amount is given, first divide by the animal's body weight in kg and confirm the dosing interval to obtain a compatible mg/kg/day value. Likewise, do not report an HED in mg/day unless a human body weight has been specified and the conversion from mg/kg/day is shown.

Practical Calculation Sequence

Identify the animal NOAEL.
Record the species, dose, units, and dosing interval from the relevant study.

Confirm dose units.
Use a dose rate in mg/kg/day, converting from a total amount only when the animal body weight and regimen are known.

Use the stated factors.
Place the animal factor in the numerator and the human factor in the denominator.

Calculate and interpret.
Report the HED in mg/kg/day, state the factors used, and describe it as a body-surface-area-scaled dose rather than an exposure-equivalent or recommended dose.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A hypothetical animal NOAEL is 30 mg/kg/day, with $K_{m,a}=8$ and $K_{m,h}=40$. What is the HED?
For a species with $K_{m,a}=12$, a human factor of 37, and an animal NOAEL of 5 mg/kg/day, should the HED be greater or less than 5 mg/kg/day? Calculate it.
A researcher calculates an HED by multiplying the animal NOAEL by $K_{m,h}/K_{m,a}$. Identify the error and give the correct ratio.
Why does an HED of 4 mg/kg/day not establish that a human dose of 4 mg/kg/day will produce the same exposure as the animal NOAEL?

Solutions. (1) $30\times(8/40)=6.000$ mg/kg/day. (2) Since $12/37<1$, the HED is lower than the animal dose: $5\times(12/37)=60/37\approx1.622$ mg/kg/day. (3) The factors are reversed. The derivation requires the animal factor divided by the human factor, $K_{m,a}/K_{m,h}$. (4) The HED matches dose per unit body-surface area under the conversion convention; it does not account for species differences in pharmacokinetics, pharmacodynamics, or toxicity, so exposure equivalence is not guaranteed.