Choose a Structure for the Prediction You Need
A PBPK diagram is not a picture of the body for its own sake. It is a choice about which physiological differences the model will represent separately and which it will treat together. The right choice depends on the prediction: a structure adequate for plasma concentration may be inadequate for concentration in a particular organ, even when both models reproduce the same plasma data.
This workshop applies that principle to a specific decision: given a target, measurements, and candidate diagrams, which distinctions should remain explicit? The answer should be justified by the output the model must predict, the information in the measurements, and the physiological rationale for the proposed structure. A more detailed diagram is not automatically better, and a good match to measured plasma concentrations does not by itself validate unmeasured tissue predictions.
A Practical Selection Sequence
Before comparing diagrams, state the prediction in operational terms. Identify the quantity, location, time range, and conditions of interest. For example, “predict drug concentration” is incomplete unless it says whether the target is plasma or a specific tissue, at what times, and under which dosing and physiological conditions. Then check that each candidate can actually produce that output. A model that lumps the target organ into a general tissue group cannot directly predict that organ’s concentration.
Next, ask what the measurements can distinguish. Plasma samples may help assess the systemic concentration-time profile, but they do not necessarily reveal how drug is divided among unmeasured tissues. Organ-specific measurements provide more direct evidence for retaining those organs as separate structures. The available data may therefore support a simple model for one target while leaving a more detailed model necessary—but weakly validated—for another.
A useful comparison keeps the target and conditions fixed while changing structure. Inspect whether candidate predictions differ in a way that matters for the target, and whether measured profiles show patterns that the simpler structure cannot reproduce. Consider physiology as well: differences in blood flow, tissue partitioning, or organ-specific processes can justify a distinction, especially when the prediction depends on it. These considerations support a reasoned choice; they do not make an unsupported prediction certain.
State the output, anatomical location, time window, and scenario. Do not compare models against an unspecified goal.
Confirm that each candidate can produce the target quantity. If a needed tissue is lumped away, the candidate may be unsuitable for a tissue-specific prediction.
Compare model predictions with relevant measured concentrations. Distinguish direct target-tissue evidence from indirect evidence such as plasma-only measurements.
Keep tissues separate when their distinct behavior is needed to represent the target or is supported by data and physiology. Otherwise, test whether pooling preserves the outputs that matter.
Look for systematic profile mismatches and assess whether the decision depends on sparse data, uncertain assumptions, or a narrow prediction scenario.
Why Similar Concentration Curves May Not Justify Pooling
Suppose a measured signal is a weighted sum of concentrations from two tissues, and each tissue follows a single exponential decline. This simple case shows why distinct tissue kinetics can leave a signature in a combined profile. Let the weighted signal be \(C(t)=a_1e^{-k_1t}+a_2e^{-k_2t}\), where \(a_1\) and \(a_2\) are positive weights and \(k_1\) and \(k_2\) are positive rate constants. A single-exponential lumped signal would have the form \(Be^{-kt}\).
For a single exponential, the quantity \(C''C-(C')^2\) equals zero at every time. For the two-component sum, differentiating and simplifying gives the following expression.
If both components contribute and their rates differ, the right-hand side is positive, not zero. Thus the sum cannot be represented exactly by one exponential over the full time course. If the rates are equal, or one component contributes nothing, this particular distinction disappears. In real PBPK models, profiles are generally more complex than exponentials, but the practical lesson holds: similar values at a few sampling times do not establish that two tissues behave identically between those times or under a different scenario.
Worked Example: Retaining a Slow Component for a Late Plasma Prediction
Hypothetical data. The goal is to predict plasma concentration through 8 hours after a dose. Plasma measurements at 1, 4, and 8 hours are 0.95, 0.48, and 0.20 mg/L. Two candidate models have been calibrated using earlier observations, and their predictions at these three comparison times are shown below. The listed measurements are held out from that calibration.
| Time (h) | Measured (mg/L) | One-compartment prediction (mg/L) | Central plus slow-component prediction (mg/L) |
|---|---|---|---|
| 1 | 0.95 | 0.90 | 0.94 |
| 4 | 0.48 | 0.35 | 0.46 |
| 8 | 0.20 | 0.08 | 0.18 |
For a compact comparison, calculate the sum of squared errors (SSE), the sum of the squared differences between measured and predicted concentrations. It is a descriptive comparison here, not a formal test or a proof that either structure is true.
The central-plus-slow-component candidate is closer at every comparison time, including the late 8-hour measurement, and has the smaller SSE. For the stated late plasma prediction, retaining a slow component is better supported by these held-out measurements than using the one-compartment candidate. The data do not identify that component as a particular organ: without tissue-specific measurements, the conclusion is about the usefulness of the structure for plasma prediction, not proof of a specific anatomical interpretation.
Worked Choices for Different Prediction Goals
Worked Example: Keeping a Target Organ Explicit
Hypothetical scenario. The target is liver concentration at 2 hours. Available measurements are plasma concentrations at 0.5, 2, and 6 hours: 1.18, 0.70, and 0.27 mg/L. A liver-resolved model predicts plasma values of 1.20, 0.72, and 0.25 mg/L; a model that pools liver with other tissues predicts 1.19, 0.71, and 0.26 mg/L.
For the liver-resolved model, the plasma SSE is \((1.18-1.20)^2+(0.70-0.72)^2+(0.27-0.25)^2=0.0012\;(\mathrm{mg/L})^2\). For the pooled model, it is \((1.18-1.19)^2+(0.70-0.71)^2+(0.27-0.26)^2=0.0003\;(\mathrm{mg/L})^2\). On these plasma data alone, the pooled model has a smaller SSE.
That comparison does not settle the stated goal. The pooled model does not directly represent liver concentration; a liver-resolved candidate can. Because there are no liver measurements, the available data do not establish that the resolved model predicts liver concentration accurately. The appropriate decision is to retain liver as a distinct compartment if liver concentration is genuinely required, while reporting that the liver prediction needs additional support or validation. The slightly better plasma match of the pooled model is not evidence that it gives a valid liver-specific output.
Worked Example: Choosing the Simpler Structure for Plasma AUC
Hypothetical scenario. The target is systemic plasma exposure over 0–12 hours, summarized by area under the concentration-time curve (AUC). Measured concentrations at 0, 2, 6, and 12 hours are 0, 1.00, 0.55, and 0.20 mg/L. A central-plus-one-peripheral-compartment candidate predicts 0, 0.96, 0.52, and 0.22 mg/L. A candidate that splits the peripheral tissues predicts 0, 1.02, 0.54, and 0.19 mg/L. For this comparison, assume an absolute relative AUC error of at most 5% is acceptable for the stated purpose.
Estimate each AUC with the trapezoidal rule, which approximates the area between consecutive observations by a trapezoid. For the measured profile:
Applying the same calculation to the one-peripheral-compartment predictions gives \(2(0+0.96)/2+4(0.96+0.52)/2+6(0.52+0.22)/2=0.96+2.96+2.22=6.14\ \mathrm{mg\cdot h/L}\). The split-tissue predictions give \(2(0+1.02)/2+4(1.02+0.54)/2+6(0.54+0.19)/2=1.02+3.12+2.19=6.33\ \mathrm{mg\cdot h/L}\).
The relative errors compared with the measured-profile estimate are \(100(6.35-6.14)/6.35=3.31\%\) for the simpler candidate and \(100(6.35-6.33)/6.35=0.315\%\) for the split candidate. Both are within the stated 5% tolerance for this AUC target. The simpler structure is therefore a defensible choice if the decision concerns only this exposure summary and the tolerance is appropriate. This does not establish that it will predict tissue concentrations or other time-dependent targets adequately.
Diagnostics, Alternatives, and Limits
Structural selection is not a contest to minimize one numerical discrepancy in isolation. A model can miss measurements systematically even when its overall error appears modest. For example, errors that are consistently positive early and negative late may indicate a profile-shape mismatch relevant to the time window of interest. Check predictions over the full relevant interval, not just at a single peak or endpoint. If measurements are sparse, a curve can pass near every observed point while behaving poorly between them.
The main alternatives are to use a lumped structure, retain selected organs separately, or construct a more extensively organ-resolved model. A lumped model has fewer distinctions and may be suitable when the target is an aggregate systemic output and pooling preserves it. Selective resolution retains only distinctions important to the target, such as a target organ or tissue with distinct kinetics. A more organ-resolved structure may be needed when several individual organs are prediction targets or when their different behavior materially affects the output. Its extra complexity also needs data or a clear physiological rationale; otherwise the additional distinctions may be poorly supported.
A useful report states both the chosen structure and its scope: which target it was selected for, which measurements informed the decision, which tissues were pooled, and what remains uncertain. Where a target-specific output is not measured, say so. Where a prediction changes substantially across plausible structures, treat that variation as a limitation of the prediction rather than hiding it behind the best-fitting candidate.
Check Your Understanding
Try each question before reading its solution.
- A model pools kidney with other tissues. The target is kidney concentration, but only plasma data are available. What can and cannot be concluded from a good plasma fit?
- A target is plasma concentration at 1 hour, and two candidate structures make nearly identical predictions at that time but differ substantially at 12 hours. Which additional consideration determines whether the difference matters?
- In the exponential-mixture derivation, why does the single-exponential representation fail when both weights are positive and the rate constants differ?
- For a plasma AUC target, a simple and a split-tissue model both meet a pre-specified acceptable error. What is a reasonable selection, and what should not be inferred from it?
Solutions. (1) The plasma fit supports the model’s ability to reproduce the measured plasma profile, but it cannot validate a kidney-specific concentration that the pooled structure does not directly represent. A kidney-resolved structure may be required for that target, with its prediction uncertainty acknowledged because kidney measurements are absent. (2) The prediction horizon and intended use matter: if only the 1-hour value is needed, the 12-hour difference may be irrelevant to that decision; if late exposure or terminal behavior matters, it is consequential and should be evaluated. (3) The derived quantity \(C''C-(C')^2\) is \(a_1a_2(k_1-k_2)^2e^{-(k_1+k_2)t}\), which is positive under those conditions, whereas it must equal zero for a single exponential. (4) Choosing the simpler model is reasonable for the stated AUC target if the tolerance is appropriate. It does not show that the simple model predicts individual tissue concentrations or other outputs adequately.