Residual Patterns From a Misspecified Liver Model

Why Residual Patterns Matter

A liver model is a mathematical description of how hepatic blood flow and intrinsic clearance combine to produce hepatic clearance. Even when its parameter estimates look plausible, the model may systematically miss the observed data. Residuals make that mismatch visible: they show not only how far a prediction is from an observation, but whether the errors change in an organized way with flow, intrinsic clearance, or predicted clearance.

The key distinction is between random measurement error, which fluctuates around the model’s expected value, and systematic residual structure, which tends to have a direction or shape under particular conditions. A pattern is evidence to investigate, not automatic proof that a liver model is wrong: an incorrect error model, a data-processing problem, or an omitted experimental factor can also produce structure.

Definition: For observation \(i\), the raw residual is \(r_i = y_i - \hat{y}_i\), where \(y_i\) is the observed hepatic clearance and \(\hat{y}_i\) is the clearance predicted by the fitted model at that observation’s inputs. A positive residual means the model underpredicts clearance; a negative residual means it overpredicts. The residual has the same units as clearance.

For the well-stirred and parallel-tube models, a common input scale is \(x=f_{u,b}CL_{int}\), the unbound intrinsic clearance expressed on the blood basis, and \(Q_h\), hepatic blood flow. Under the standard steady-state formulations, the predicted clearances are:

\hat{CL}_{h,\mathrm{WS}}=\frac{Q_hx}{Q_h+x}, \qquad \hat{CL}_{h,\mathrm{PT}}=Q_h\left(1-\exp\left(-\frac{x}{Q_h}\right)\right).

These equations describe different assumptions about how blood and drug interact within the liver. The well-stirred model treats the organ as a single, instantly mixed compartment; the parallel-tube model represents blood moving through parallel pathways with extraction along the way. The dispersion model instead represents axial dispersion within the liver. Residual checks are useful for any of these formulations: calculate the residual from the selected model’s prediction, then examine whether residuals vary systematically with relevant inputs.

What a Systematic Pattern Means

A useful starting point is to ask what residuals should look like if the chosen model and error assumptions are adequate. Suppose clearance is measured as \(Y_i=f(X_i;\theta)+\varepsilon_i\), where \(f\) is the chosen liver model, \(X_i\) denotes the inputs, \(\theta\) its parameters, and \(\varepsilon_i\) is measurement error with conditional mean zero. If the model is correct and the error model is appropriate, then before accounting for parameter estimation the expected residual at a given input is approximately zero. Residuals may vary, but they should not repeatedly rise or fall with \(Q_h\), \(x\), or the fitted clearance.

If the true mean relationship is \(g(X_i)\), but the fitted model uses \(f(X_i;\theta)\), then the expected residual is approximately \(g(X_i)-f(X_i;\theta)\). A nonzero difference that changes with the inputs creates a pattern. Fitting parameters can reduce the overall discrepancy, but it cannot generally remove a shape the model cannot represent.

E(r_i\mid X_i)\approx g(X_i)-f(X_i;\theta).

This gives a practical interpretation of common diagnostic plots. Residuals against \(x\) can reveal that a model increasingly underpredicts at higher intrinsic clearance. Residuals against \(Q_h\) can reveal a flow-dependent discrepancy. Residuals against fitted clearance can reveal curvature or a change in spread. A sequence by experiment or time can expose drift or batch effects. The horizontal reference at zero is important: the question is whether residuals are scattered around it without a repeatable shape, not whether every point equals zero.

Residual scale matters. Raw residuals are easy to interpret in clearance units, but if measurement variability increases with clearance, a wider vertical spread at larger predictions may be expected under the measurement model. When the standard deviation for observation \(i\) is \(\sigma_i\), a standardized residual is \(r_i/\sigma_i\). For a multiplicative error model, a log residual, \(\log(y_i)-\log(\hat{y}_i)\), may be more appropriate; it is dimensionless and describes relative rather than absolute discrepancy. Use a scale consistent with the assumed error process, and do not label a fan-shaped raw-residual plot as structural mismatch before checking whether the error variance was expected to change.

Worked Examples: Reading Residual Evidence

Worked Example: A Trend With Intrinsic Clearance

Consider hypothetical clearance data measured at \(Q_h=100\) L/h. The values of \(x\) are 20, 60, 120, and 180 L/h. Suppose the observed values are 18.13, 45.12, 69.88, and 83.47 L/h, respectively. Compare the well-stirred model predictions with these observations.

At \(x=20\), the prediction is \(100(20)/(100+20)=16.67\) L/h, so \(r=18.13-16.67=1.46\) L/h. At \(x=60\), the prediction is \(100(60)/(100+60)=37.50\) L/h, so \(r=45.12-37.50=7.62\) L/h.

At \(x=120\), the prediction is \(100(120)/(100+120)=54.55\) L/h, giving \(r=69.88-54.55=15.33\) L/h. At \(x=180\), the prediction is \(100(180)/(100+180)=64.29\) L/h, giving \(r=83.47-64.29=19.18\) L/h.

All four residuals are positive, and their size generally increases with \(x\). This pattern says that, for these hypothetical data, the well-stirred predictions increasingly underpredict clearance as intrinsic clearance rises. It is more informative than a single average residual near zero would be. The pattern motivates checking an alternative liver formulation and the measurement process; by itself, it does not establish which explanation is correct.

Worked Example: Replicates Separate Scatter From a Repeated Offset

Suppose a model predicts 30, 40, and 50 L/h at three experimental settings. At each setting, three hypothetical replicate measurements produce residuals \((-2,1,1)\), \((-1,0,1)\), and \((7,9,8)\) L/h, respectively. Calculate the mean residual at each setting.

The means are \((-2+1+1)/3=0\) L/h, \((-1+0+1)/3=0\) L/h, and \((7+9+8)/3=8\) L/h. The first two groups fluctuate around zero, while all three residuals in the third group are positive and close to one another.

For the third group, the sample standard deviation is 1 L/h: the deviations from the mean 8 are \(-1,1,0\), and their squared deviations sum to 2, so \(s=\sqrt{2/(3-1)}=1\) L/h. The standard error of its mean residual is \(1/\sqrt{3}=0.577\) L/h. The repeated offset is large relative to the replicate scatter, so random replicate error alone is an incomplete explanation to investigate. This calculation is diagnostic, not a formal test: it does not account for parameter fitting, the number of settings inspected, or other sources of variation.

Worked Example: Checking the Pattern Under a Flow Perturbation

A hypothetical experiment measures clearance at \(Q_h=60\) L/h and \(x=60\) L/h. The observed clearance is 38.40 L/h. Calculate residuals under both the well-stirred and parallel-tube models.

The well-stirred prediction is \(60(60)/(60+60)=30.00\) L/h, so its residual is \(38.40-30.00=8.40\) L/h. The parallel-tube prediction is \(60[1-\exp(-60/60)]=60(1-\exp(-1))\). Since \(\exp(-1)\approx0.3679\), this is \(60(0.6321)=37.93\) L/h, and the residual is \(38.40-37.93=0.47\) L/h.

For this one hypothetical observation, the parallel-tube prediction is closer. A flow-perturbation data set can provide a more informative comparison: if residuals under one model repeatedly change with flow while those under another remain centered around zero, the former model may be missing a flow-dependent feature. One point cannot establish that pattern, and a small residual on one observation is not enough to select a model.

A Practical Diagnostic Sequence

A disciplined sequence reduces the chance of confusing a visible pattern with an explanation. First confirm that observations and predictions use compatible clearance definitions and units, including the intended blood-to-plasma basis. Then inspect raw residuals against the inputs that distinguish the liver models, especially \(x\) and \(Q_h\), as well as against fitted clearance. If the experiment includes flow perturbations, label those conditions directly rather than relying only on an overall plot. Where the error model predicts changing variability, inspect standardized or appropriately transformed residuals as well.

Calculate residuals consistently.
Use observed clearance minus the fitted model’s prediction at the same inputs, with matching clearance units and blood or plasma basis.

Plot against model-relevant inputs.
Check residuals against \(x\), \(Q_h\), fitted clearance, and experimental condition. Look for runs of one sign, curvature, or a change in spread.

Compare the pattern with the error assumptions.
Use raw residuals for roughly constant absolute error, or a suitable standardized or log-scale residual when the assumed variability is proportional or otherwise changes across observations.

Check replication and alternatives.
Ask whether replicate residuals fluctuate around a common mean, then compare the pattern under plausible liver models and, where possible, conditions not used to fit them.

The main alternatives are not simply “model wrong” and “measurement error.” A trend can arise from an inappropriate liver model, but also from a biased assay, a flow value recorded on the wrong basis, a changing experimental condition, or an unsuitable error distribution. Replicates help estimate within-condition scatter; flow perturbations can reveal whether the discrepancy tracks flow; and fitting a plausible alternative model tests whether its predicted shape better matches the data. A residual plot cannot identify the cause on its own.

Comparisons should use the same observations and a compatible treatment of measurement error. A model with more flexibility can reduce residuals simply by fitting more closely, so a smaller residual sum alone is not proof of a better mechanistic description. Examine whether residual structure is reduced without creating new patterns, and whether the improvement persists in independent or held-out conditions when such data are available. This is a diagnostic comparison, not a substitute for checking model assumptions and biological plausibility.

Common Mistake and Limitations

Mistake: treating a positive mean residual as sufficient evidence of misspecification. A constant offset may instead reflect a calibration bias or a systematic input-scale error. Check units, assay calibration, and residuals across conditions before attributing the offset to liver structure.
Mistake: calling every increase in residual spread a structural trend. If the measurement error is expected to grow with clearance, raw residuals may fan out even when their conditional mean is zero. Check standardized or transformed residuals consistent with the error model.
Mistake: concluding that a model is adequate because residuals sum to zero. Parameter fitting can impose a zero-sum constraint in some settings, while positive residuals at low \(x\) and negative residuals at high \(x\) can still cancel. Examine residuals against inputs, not just their total.
Limitation: residuals depend on the fitted data. Parameter estimation can make in-sample patterns look smaller, and inspection of many plots can make chance patterns seem compelling. Replication, prespecified diagnostics, and validation conditions help, but no single diagnostic proves the correct biological mechanism.

Key takeaway: A residual pattern is a structured discrepancy between observed and predicted clearance. Trends with intrinsic clearance or flow are reasons to question whether the chosen liver formulation captures the data-generating relationship, but interpretation must also account for the error model, data scale, experimental design, and possible measurement or input problems.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

An observed clearance is 52 L/h and the fitted prediction is 47 L/h. What is the residual, including its interpretation and units?
Residuals are mostly negative at low \(x\) and mostly positive at high \(x\). What pattern does this suggest, and what can it not prove by itself?
Why might a fan-shaped plot of raw residuals be compatible with the assumed error model?
At a fixed setting, replicate residuals are \(-1\), \(0\), and \(1\) L/h. What is their mean, and what do they suggest about scatter at that setting?

Solutions. (1) \(r=52-47=5\) L/h; the model underpredicts by 5 L/h for that observation. (2) The discrepancy changes with \(x\), suggesting a systematic relationship the fitted model may not capture; it does not prove structural misspecification because measurement, input, and other experimental problems can also create patterns. (3) The measurement standard deviation may increase with predicted clearance, so larger absolute residuals can be expected at larger predictions even when standardized residuals remain appropriately scattered. (4) The mean is \((-1+0+1)/3=0\) L/h; these replicates are centered on the prediction with modest observed scatter, but three values do not establish the full error distribution.