Reporting Model Equations and Parameterization Transparently

Why Equations Need More Than Symbols

A model equation is useful only if a reader can tell what each symbol means and how the equation was implemented. Writing “a one-compartment model with weight on clearance” leaves important questions open: Was the dose intravenous or absorbed over time? Was weight entered as a ratio to a reference value or as an unscaled number? What were the units of clearance? Was residual error additive, proportional, or combined? Different answers describe different fitted specifications.

Transparent reporting therefore pairs each equation with definitions, units, transformations, reference values, and assumptions. The aim is not to reproduce software syntax line by line. It is to state the mathematical model precisely enough that another analyst can map the reported specification to an implementation and identify where the two differ.

Definition: A reported model specification is a set of equations and accompanying definitions that identifies the prediction model, observation model, parameter meanings and units, transformations, reference values, and assumptions needed to interpret the fitted model. The structural model describes the systematic prediction; the observation model describes how recorded data vary around that prediction.

Use consistent indices when observations are nested within individuals. In the examples below, \(i\) indexes an individual and \(j\) indexes an observation for that individual. Define these indices when they first appear, rather than relying on readers to infer them from subscripts. Also distinguish a typical population parameter from an individual-specific parameter and a prediction from an observed concentration.

State the Structural Model and Its Assumptions

Consider a hypothetical intravenous bolus model with one compartment and first-order elimination. Let \(A_i(t)\) be the amount of drug in the compartment for individual \(i\) at time \(t\), and let \(C_i(t)\) be the corresponding concentration. The model assumes that the dose enters the compartment immediately, that the compartment is well mixed, and that elimination is proportional to the amount present. It does not describe absorption or distribution into additional compartments.

\frac{dA_i(t)}{dt}=-\frac{CL_i}{V_i}A_i(t), \qquad A_i(0)=D_i, \qquad C_i(t)=\frac{A_i(t)}{V_i}.

Here \(D_i\) is the intravenous bolus dose in milligrams, \(CL_i\) is clearance in liters per hour, and \(V_i\) is volume of distribution in liters. Time \(t\) is in hours, \(A_i(t)\) is in milligrams, and \(C_i(t)\) is in milligrams per liter. The ratio \(CL_i/V_i\) has units of inverse hours.

Derivation: Dividing the differential equation by \(A_i(t)\) gives \(dA_i(t)/A_i(t)=-(CL_i/V_i)dt\). Integrating from 0 to \(t\) gives \(\log(A_i(t)/D_i)=-(CL_i/V_i)t\). Therefore \(A_i(t)=D_i\exp[-(CL_i/V_i)t]\), and dividing by \(V_i\) gives the concentration equation below.

C_i(t)=\frac{D_i}{V_i}\exp\left(-\frac{CL_i}{V_i}t\right).

A clear report also says whether the equation gives a concentration at a sampling time measured from the dose, and whether the dose is assumed to be fully available in the compartment at time zero. If the actual fitted model includes a different dosing input, absorption, a lag time, or additional compartments, those elements must appear in the reported equations or be described precisely alongside them.

Worked Example: Checking a One-Compartment Equation

Suppose a hypothetical individual receives an intravenous bolus dose of 100 mg. For illustration, take \(CL_i=5\) L/h and \(V_i=20\) L, with a sample at \(t=4\) h. Calculate the model-predicted concentration and check the units.

The elimination rate constant implied by the clearance and volume parameterization is \(CL_i/V_i=(5\ \mathrm{L/h})/(20\ \mathrm{L})=0.25\ \mathrm{h}^{-1}\). Substituting into the structural equation gives:

C_i(4)=\frac{100\ \mathrm{mg}}{20\ \mathrm{L}} \exp\left[-\frac{5\ \mathrm{L/h}}{20\ \mathrm{L}}(4\ \mathrm{h})\right] =5\ \mathrm{mg/L}\times e^{-1} \approx 1.839\ \mathrm{mg/L}.

The exponent is dimensionless because \((\mathrm{h}^{-1})(\mathrm{h})=1\), and the concentration unit is mg/L. This is a model prediction under the stated assumptions, not an observed concentration and not evidence that these hypothetical parameter values fit any data.

Define Covariates, Transformations, and Reference Values

A covariate relationship should state both its mathematical form and the scale on which the covariate enters. A ratio to a stated reference value makes a power model dimensionless; a logarithm should likewise be applied to a dimensionless quantity. Report the reference value and unit, not just the phrase “normalized weight.”

CL_i=CL_{\mathrm{ref}} \left(\frac{WT_i}{70\ \mathrm{kg}}\right)^{0.75} \exp(\eta_{CL,i}).

In this example, \(WT_i\) is body weight in kilograms, \(70\) kg is the reference weight, and \(CL_{\mathrm{ref}}\) is clearance at that reference weight when \(\eta_{CL,i}=0\). The exponent 0.75 is the specified weight-power coefficient. The individual effect \(\eta_{CL,i}\) is dimensionless, so its exponential is a dimensionless multiplier. Clearance and \(CL_{\mathrm{ref}}\) have the same units, here L/h. These statements define the parameterization; they do not imply that a particular power model is appropriate for every drug or analysis.

Worked Example: Applying a Weight Transformation

Consider a hypothetical individual weighing 56 kg. Use \(CL_{\mathrm{ref}}=5.00\) L/h, a weight exponent of 0.75, and set \(\eta_{CL,i}=0\) to calculate the covariate-adjusted clearance.

First form the dimensionless weight ratio: \(56\ \mathrm{kg}/70\ \mathrm{kg}=0.8000\). Then apply the specified power and multiply by the reference clearance:

CL_i=5.00\ \mathrm{L/h}\times(0.8000)^{0.75}\times\exp(0) =5.00\ \mathrm{L/h}\times0.8459 \approx4.229\ \mathrm{L/h}.

The result retains the clearance units because both transformed factors are dimensionless. Under this equation and with \(\eta_{CL,i}=0\), an individual at 70 kg has \(CL_i=CL_{\mathrm{ref}}\). Reporting only “weight was included on clearance” would not disclose the power, the reference value, or the individual-effect term.

If the fitted equation instead uses \(\log(WT_i/(70\ \mathrm{kg}))\), state that exact transformation, its coefficient, and the reference value. Do not silently treat a log-linear model and a power model as interchangeable: their functional forms and coefficient interpretations differ. Likewise, specify whether a covariate was centered, scaled, categorized, or otherwise transformed, and give the units or cut points needed to reproduce that operation.

Report the Observation Model Separately

A structural equation predicts a concentration; an observation model connects that prediction to a recorded value. For example, let \(Y_{ij}\) be the observed concentration for individual \(i\) at observation \(j\), and let \(C_{ij}\) be the structural prediction at that individual’s sampling time. A combined additive-and-proportional error model can be written:

Y_{ij}=C_{ij}\left(1+\sigma_{\mathrm{prop}}\epsilon_{\mathrm{prop},ij}\right) +\sigma_{\mathrm{add}}\epsilon_{\mathrm{add},ij}, \qquad \epsilon_{\mathrm{prop},ij},\epsilon_{\mathrm{add},ij}\sim N(0,1).

For the usual variance calculation, the two residual random variables are assumed independent and have mean zero and variance one. The proportional-error coefficient \(\sigma_{\mathrm{prop}}\) is dimensionless; \(\sigma_{\mathrm{add}}\) has the same units as concentration. Conditional on the prediction, the observation has mean \(C_{ij}\) and standard deviation \(\sqrt{(C_{ij}\sigma_{\mathrm{prop}})^2+\sigma_{\mathrm{add}}^2}\). State the exact form and assumptions used by the fitted model; naming an error model without an equation can leave ambiguity about how its terms are combined.

Worked Example: Computing the Combined Residual Standard Deviation

Suppose a hypothetical concentration prediction is \(C_{ij}=1.128\) mg/L, with \(\sigma_{\mathrm{add}}=0.0500\) mg/L and \(\sigma_{\mathrm{prop}}=0.150\). Under the independent, unit-variance residual assumptions stated above, calculate the conditional residual standard deviation.

The additive variance is \((0.0500\ \mathrm{mg/L})^2=0.002500\ (\mathrm{mg/L})^2\). The proportional contribution to the standard deviation is \(0.150\times1.128\ \mathrm{mg/L}=0.1692\) mg/L, so its variance is \((0.1692\ \mathrm{mg/L})^2=0.02862864\ (\mathrm{mg/L})^2\). Thus:

\sqrt{0.002500+0.02862864}\ \mathrm{mg/L} =\sqrt{0.03112864}\ \mathrm{mg/L} \approx0.1764\ \mathrm{mg/L}.

Both variance terms have units \((\mathrm{mg/L})^2\), and the square root returns mg/L. This calculation describes the residual spread implied by the specified hypothetical model at that prediction; it does not estimate an error parameter from data.

Make the Parameterization Reconstructable

A parameterization is the choice of quantities used to express the model. For the one-compartment example, one may write the exponential term using clearance and volume, \(CL_i/V_i\), or define an elimination rate constant \(k_i=CL_i/V_i\). The two forms give the same predictions only when the relationship and parameter units are made explicit.

Worked Example: Relating Clearance–Volume and Rate-Constant Forms

Suppose the same hypothetical values are \(CL_i=5\) L/h and \(V_i=20\) L. Then \(k_i=CL_i/V_i=0.25\ \mathrm{h}^{-1}\), and the model can be written either in terms of \(CL_i\) and \(V_i\), or as \(C_i(t)=(D_i/V_i)\exp(-k_i t)\). At \(t=4\) h:

\frac{100\ \mathrm{mg}}{20\ \mathrm{L}}\exp[-(0.25\ \mathrm{h}^{-1})(4\ \mathrm{h})] =5\ \mathrm{mg/L}\times e^{-1} \approx1.839\ \mathrm{mg/L}.

The prediction agrees with the earlier calculation because \(k_i\) was defined as \(CL_i/V_i\). If a report lists only \(k_i\) and \(V_i\), it should not label \(k_i\) as clearance: \(k_i\) has units h\(^{-1}\), whereas clearance has units L/h. A parameter’s name, definition, and unit must refer to the same quantity.

When documenting a fitted model, distinguish fixed-effect parameters, individual random effects, and residual terms. Give each symbol one consistent meaning, identify whether a parameter is typical or individual-specific, and state any defining relationships between parameters. If the model uses an exponential random effect, for example, show where it enters and whether the underlying random effect is on a log scale. Avoid listing symbols in a table without connecting them to the equations.

Write the equations.
Separate the structural prediction from the observation model and show any covariate or random-effect terms explicitly.

Define every symbol.
State indices, parameter meanings, units, and whether a quantity is individual-specific, typical, predicted, or observed.

Specify transformations and references.
Give the transformation, its scale, and any reference value with units; check that ratios and exponential arguments are dimensionless.

Check agreement with implementation.
Compare the reported equations and definitions with the model actually fitted, including the input conditions and residual-error form.

Diagnostics, Alternatives, and Common Mistakes

Dimensional analysis is a fast diagnostic. Every term added to another term must have the same units; an exponent must be dimensionless; and an equation’s output must have the units claimed for it. A mismatch often reveals an omitted reference value, a mistaken parameter label, or a transformation that was described imprecisely. A second diagnostic is to substitute a simple reference condition, such as \(WT_i=70\) kg and \(\eta_{CL,i}=0\), and verify that the equation returns the stated reference parameter.

The main alternative to a fully written equation is a verbal description or a parameter table alone. A verbal summary is concise but can conceal whether weight was entered linearly, as a power, or on a log scale. A table can clarify units and definitions but usually does not show how terms combine. In practice, equations plus a compact parameter-definition table are complementary: equations communicate structure, while the table makes symbols and units easy to inspect.

Omitting the reference value. “Weight was normalized” does not specify the denominator. State the reference, such as 70 kg, and show the ratio.
Mixing parameter meanings. Clearance and the elimination rate constant are related but have different units and are not synonyms. Define any derived parameter explicitly.
Leaving error structure implicit. “Combined error” is insufficient if readers cannot tell whether additive and proportional terms are summed as random contributions or variances. Give the fitted equation and assumptions.
Reporting an equation that differs from the fitted model. A clean theoretical equation is not transparent reporting if it omits transformations or terms actually used in the implementation. Verify the report against the implemented specification.

Key takeaway: A reconstructable model report links each equation to symbol definitions, units, transformations, reference values, and assumptions. Check the algebra and dimensions, then verify that the written specification matches the model that was actually fitted.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the one-compartment IV bolus equation, what are the units of \(CL_i/V_i\), and why must its product with time be dimensionless?
For the weight model, what clearance does the equation assign at 70 kg when \(\eta_{CL,i}=0\)?
If an additive residual standard deviation is reported in mg/L, what are the units of its variance?
Why is it insufficient to report that clearance was “adjusted for weight” without the transformation and reference value?

Solutions. (1) \(CL_i/V_i\) has units \((\mathrm{L/h})/\mathrm{L}=\mathrm{h}^{-1}\); multiplying by time in hours gives a dimensionless exponent. (2) The weight ratio is \(70/70=1\), so \(CL_i=CL_{\mathrm{ref}}\times1^{0.75}\times\exp(0)=CL_{\mathrm{ref}}\). (3) The variance is in \((\mathrm{mg/L})^2\), the square of the standard-deviation unit. (4) The phrase does not identify the functional form or scaling; different transformations and reference values define different equations and can yield different predictions.