Why Compartment Count Is Not a Measure of Accuracy
A model with separate compartments for more tissues may look more anatomically detailed. That detail can be useful when it represents differences that matter to the prediction. But adding a compartment also adds assumptions: for example, that a tissue has a distinct kinetic rate, that its contribution to the measured output is known, and that the available evidence can support those distinctions. If those assumptions are not justified, the model may be more elaborate without being more accurate.
Model structure is a choice about which processes and distinctions to represent. Predictive accuracy is a comparison between model predictions and the relevant values for a specified target, such as plasma concentration at a particular time. A compartment count, by itself, does not make that comparison. The earlier material on model structure and prediction targets provides the basis for making the target explicit; here the emphasis is on the risk of interpreting added detail as added predictive value.
Comparing Models at the Prediction Target
A fair structural comparison holds the prediction question fixed. State the output, its units, and when or under what conditions it is to be predicted. Then compare the candidate models on that output. A detailed model could make a different prediction from a simpler model, yet still not be more accurate: without relevant observed evidence or well-supported inputs, the direction and size of that difference do not establish which prediction is closer to reality.
For a simple illustration, consider predictions of a concentration \(C(t)\), in mg/L, after a dose. Suppose a one-rate model predicts exponential decline at rate \(0.4\ \mathrm{h}^{-1}\). A two-rate model assigns most of the initial concentration to that decline and a small fraction to a slower decline at rate \(0.04\ \mathrm{h}^{-1}\). These are hypothetical models, not clinical data. Each assumes that its stated rates and initial contributions describe the system over the times being predicted.
Worked Example: A Small Difference at One Prediction Time
Compare the one-rate prediction \(C_s(t)=100e^{-0.4t}\ \mathrm{mg/L}\) with the two-rate prediction \(C_d(t)=99e^{-0.4t}+e^{-0.04t}\ \mathrm{mg/L}\). Here \(t\) is in hours. Both models predict \(100\ \mathrm{mg/L}\) at \(t=0\). Calculate their predictions at \(t=1\) h and compare them.
Assumptions for this comparison. Treat the initial concentration contributions and rate constants as specified and fixed. Both models use exponential decline; the two-rate model assumes one contribution declines more slowly. This calculation compares the stated predictions. It does not establish that either model is accurate.
For the one-rate model:
For the two-rate model:
The difference is \(67.3225-67.0320=0.2905\ \mathrm{mg/L}\), or about \(100(0.290469/67.0320)=0.4333\%\) of the one-rate prediction. At this time, the two predictions are close in absolute terms. That closeness alone does not show that the two-rate model is better; it shows only that the specified models give similar predictions at \(1\) h.
The target time matters. At \(t=10\) h, the one-rate prediction is \(100e^{-4}=1.8316\ \mathrm{mg/L}\), while the two-rate prediction is \(99e^{-4}+e^{-0.4}=2.4836\ \mathrm{mg/L}\). The difference is \(0.6520\ \mathrm{mg/L}\), about \(35.6\%\) of the one-rate prediction. Thus a small difference at one time does not imply a small difference at every time, and a larger difference later does not, by itself, tell us which prediction is more accurate.
When Extra Compartments Leave the Output Unchanged
An added subdivision need not change the prediction at all. This can happen when the subcompartments have identical kinetics and the output depends only on their total. The following calculation makes that point directly. It uses a simple decline model to illustrate an exact pooling condition, not to prescribe a particular PBPK tissue equation.
Worked Example: Two Identical Subcompartments
Suppose a hypothetical model divides an amount of \(80\ \mathrm{mg}\) into two subcompartments with initial amounts \(30\ \mathrm{mg}\) and \(50\ \mathrm{mg}\). Both amounts decline at the same rate, \(0.3\ \mathrm{h}^{-1}\), and the predicted concentration is their sum divided by a fixed volume of \(20\ \mathrm{L}\). Compare this with one compartment containing the full initial amount and having the same rate.
Assumptions. Each subcompartment follows the same first-order decline, there is no exchange that changes either decline equation, and the prediction uses only the sum of the amounts. The one-compartment comparator has the same total initial amount and rate. Under these assumptions, the output must be the same.
Let \(A_1(t)\) and \(A_2(t)\) denote the subcompartment amounts. Their equations and initial values are:
Each equation has the solution \(A_i(t)=A_i(0)e^{-0.3t}\). Therefore their total is \(30e^{-0.3t}+50e^{-0.3t}=80e^{-0.3t}\ \mathrm{mg}\). The one-compartment model gives that same total, \(A(t)=80e^{-0.3t}\ \mathrm{mg}\). Dividing by \(20\ \mathrm{L}\), both predict:
At \(t=2\) h, both give \(C(2)=4e^{-0.6}=2.195\ \mathrm{mg/L}\). The extra compartment labels have not altered this output. They could still matter for another target that distinguishes the two subcompartments, but this total-concentration prediction provides no evidence that the subdivision improves its accuracy.
Derivation: Why Matching Kinetics Permit Pooling
The pooling result follows by adding the amount equations. Let \(A_i(t)\) be the amount in subcompartment \(i\), with every subcompartment obeying \(dA_i/dt=-kA_i\) for the same fixed rate \(k\). Define the total amount \(A_{\mathrm{tot}}(t)=\sum_i A_i(t)\). Differentiating the sum gives:
The pooled amount therefore obeys the same one-compartment equation, with initial value equal to the sum of the subcompartment initial amounts. If the measured output depends only on this total through the same volume conversion, subdivision cannot change that output under the stated assumptions. If rates differ, or if the target observes subcompartment-specific information, this argument no longer establishes equivalence.
Unsupported Detail and Extrapolated Predictions
A more detailed model may require parameters that are poorly constrained by the evidence available for the intended use. In the two-rate illustration, the slow component has a small initial contribution, but its rate can have a substantial influence later because the faster component has declined. If evidence does not reliably determine the slow contribution or its rate, the resulting later prediction may be fragile. The added compartment has created a way for predictions to differ; it has not automatically supplied evidence to determine how they should differ.
Worked Example: Sensitivity to an Uncertain Slow Contribution
Consider the family of hypothetical predictions \(C_b(t)=(100-b)e^{-0.4t}+be^{-0.04t}\ \mathrm{mg/L}\), where \(0\leq b\leq2\ \mathrm{mg/L}\). The total initial concentration is \(100\ \mathrm{mg/L}\) for every value of \(b\). Suppose the available evidence does not distinguish values of \(b\) within this range. Calculate the range of predictions at \(1\) h and \(10\) h.
Assumptions. The two rates are fixed as shown, the uncertain quantity is the initial slow contribution \(b\), and the range from \(0\) to \(2\ \mathrm{mg/L}\) is an illustrative uncertainty range, not a statistical confidence interval. The component contributions remain nonnegative throughout this range.
Rewrite the prediction as the one-rate prediction plus the effect of \(b\):
At \(1\) h, the coefficient of \(b\) is \(e^{-0.04}-e^{-0.4}=0.290469\). Since \(b\) ranges from \(0\) to \(2\ \mathrm{mg/L}\), the prediction ranges from \(67.0320\) to \(67.0320+2(0.290469)=67.6129\ \mathrm{mg/L}\).
At \(10\) h, the coefficient of \(b\) is \(e^{-0.4}-e^{-4}=0.652004\). The prediction therefore ranges from \(1.8316\) to \(1.8316+2(0.652004)=3.1356\ \mathrm{mg/L}\). A modest range of possible slow contributions has a larger effect on the prediction at \(10\) h than at \(1\) h. This sensitivity calculation identifies a consequence of the assumed structure and parameter range; it does not tell us which value of \(b\) is correct.
Diagnostics, Alternatives, and Limitations
Several checks help distinguish useful structural detail from detail that merely increases complexity. First, define the prediction target precisely: the measured quantity, units, time range, and relevant scenario. Second, compare candidate models on that target, not only on internal states that are not part of the question. Third, check whether added compartments have distinct kinetics or other relevant features supported by evidence. Finally, examine whether plausible changes to their parameters materially change the target prediction, especially over the time range where the model will be used.
Warning signs include a more detailed model producing markedly different predictions while its new parameters are weakly supported; predictions changing sharply under modest parameter changes; or multiple compartments contributing only through an aggregate output that the simpler model already represents. These signs do not prove that the detailed model is wrong. They indicate that its apparent detail should not be mistaken for demonstrated predictive improvement.
| Approach | When it is useful | What it does not establish |
|---|---|---|
| Use a simpler model aligned with the target | When the target output does not require distinctions represented only by extra compartments | That omitted structure is unimportant for every other target or scenario |
| Add compartments with distinct, supported behavior | When the distinction matters to the target and evidence can support the assumptions and parameters | That anatomical detail alone improves accuracy |
| Compare predictions and test parameter sensitivity | When deciding whether added structure materially affects the intended prediction | Which model is more accurate without relevant evidence for comparison |
These checks have limits. Two models can agree for a particular output and time while differing for another output or scenario. Conversely, a difference between predictions does not identify which model is closer to reality. A model comparison is informative only in relation to its stated target and the evidence available to evaluate it. The practical question is not simply whether the model has more compartments, but whether the extra distinctions help answer the intended question reliably.
A Common Mistake
Wrong reasoning: “The model with more anatomical compartments must be more accurate because it represents the body in more detail.”
Correction: Anatomical detail is not evidence of predictive accuracy. An added compartment may introduce assumptions or parameters that are unsupported, may leave the target output unchanged, or may alter that output without evidence showing the alteration is more accurate. Compare predictions for the stated target and assess whether the additional structure is relevant and supported.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two subcompartments have identical first-order rates, and the target is their total amount. Under what assumptions can they be pooled without changing that target?
- In the two-rate example, why does agreement at \(1\) h not guarantee agreement at \(10\) h?
- Does a detailed model that predicts a higher late concentration establish that the real late concentration is higher? Explain.
- What does a sensitivity range of \(1.8316\) to \(3.1356\ \mathrm{mg/L}\) at \(10\) h show in the uncertain-\(b\) example?
Solutions. (1) Both subcompartments must obey the same linear decline rate, and the target must depend only on their sum; their initial amounts add to the pooled initial amount. (2) The slow component decays more gradually, so its contribution becomes more prominent as the fast component declines; the model difference therefore depends on time. (3) No. A prediction difference is not evidence that either prediction is closer to reality; comparison with relevant evidence is needed. (4) It shows how the target prediction varies across the specified illustrative values of \(b\), given the fixed model rates. It is not a confidence interval and does not identify the correct prediction.