What a Below-Quantification Result Tells You
A concentration reported as below the lower limit of quantification (LLOQ) is not a measured concentration of zero. It records an event: the assay did not quantify the sample, and the concentration is below a stated threshold. That event can be informative in independent PBPK validation, particularly when the model predicts concentrations near the limit or when below-limit results cluster at particular times.
The key is to preserve the information actually observed. A censored result does not give an exact concentration, so it cannot be assigned an exact prediction error or fold error. It does, however, constrain the concentration to an interval and can be evaluated against the probability that the model and observation process imply for that interval. The LLOQ, the prediction, the uncertainty model, and the sample time all matter.
For validation, distinguish the model-predicted concentration from the noisy assay result. A PBPK prediction may be treated as the center of a predictive distribution for an observation, provided the distribution and its assumptions have been specified. The predictive spread should reflect the uncertainty relevant to the comparison; it should not be chosen after seeing which values were censored. The earlier material on separating assay noise from model discrepancy is important here: the source and size of the predictive variation should be stated rather than silently conflated.
Representing Censoring in a Predictive Model
One useful working model assumes that, conditional on a model prediction \(C_i\), a positive observed concentration \(Y_i\) has a lognormal predictive distribution. In a simple no-bias form, \(\ln Y_i\) is normal with mean \(\ln C_i\) and standard deviation \(\tau\). Here \(\tau\) is the specified predictive standard deviation on the log scale. This model assumes the prediction is positive, the log-scale distribution is appropriate, the LLOQ is known, and the residual distribution is adequately described by the chosen \(\tau\).
The second expression follows by taking logarithms of \(Y_i<L_i\), then standardizing the normal variable \(\ln Y_i\). It gives the model-implied probability of a below-limit result, not the probability that the PBPK model is true. If a specified log-scale bias \(\delta\) is part of the predictive model, the standardized value becomes \(\{\ln(L_i/C_i)-\delta\}/\tau\). Bias and spread must be set by the validation model, not adjusted to make a particular censored sample look acceptable.
A below-limit observation contributes its event probability to a likelihood. For an uncensored concentration \(y_i\), the model contributes its density; for a censored result, it contributes the probability of the interval \(0 \leq Y_i<L_i\). Thus, for conditionally independent observations with a common predictive distribution \(F_i\), the likelihood contains a factor \(f_i(y_i)\) for each quantified value and \(F_i(L_i)\) for each below-limit value. If the quantified-data density is expressed on the concentration scale, it is the lognormal density; if expressed on the log scale, the corresponding density is for \(\ln Y_i\). The censoring factor is the same event probability in either representation.
The likelihood construction assumes that the censoring rule and limit are correctly recorded and that the event “below \(L_i\)” is determined by the assay process represented in the model. If observations within a subject are correlated, multiplying separate marginal contributions as if they were independent can overstate the information. A joint model or another method that respects that dependence is then needed.
Worked Example: Interpreting One Censored Sample
Suppose a hypothetical validation sample has a model prediction \(C=0.80\) ng/mL and an LLOQ \(L=0.50\) ng/mL. Assume the prespecified predictive model is lognormal with no log-scale bias and \(\tau=0.30\). The assay reports the sample as below quantification. What does the result say, and how probable is it under the model?
The observation establishes \(0 \leq Y<0.50\) ng/mL, not \(Y=0\). Under the stated model,
Therefore \(P(Y<0.50\mid C=0.80)=\Phi(-1.56668)\approx 0.0586\), or about 5.86%. This is a relatively uncommon but possible result under the assumed predictive distribution. It is not, by itself, proof of model failure; the conclusion depends on the prespecified validation criteria and the pattern across observations.
The censored observation cannot be assigned a fold error because its exact \(Y\) is unknown. It does establish that the observed-to-predicted ratio is less than \(0.50/0.80=0.625\), so its log ratio is less than \(\ln(0.625)=-0.4700\). That is a bound, not an exact residual.
Worked Probabilities and the Meaning of the Limit
The LLOQ can lie below or above a prediction. A below-limit observation is unsurprising when the model itself assigns high probability to that event; it is more challenging when the model predicts a concentration well above the limit and assigns a low event probability. The direction of the calculation must be kept clear: \(P(Y<L)\) and \(P(Y>L)\) are complementary probabilities under a continuous distribution.
Worked Example: Checking the Probability Direction
Assume again a no-bias lognormal predictive model with \(\tau=0.30\). First consider \(C=0.50\) ng/mL and \(L=0.30\) ng/mL. The standardized limit is
Thus \(P(Y<0.30)=\Phi(-1.70275)\approx 0.0443\), while \(P(Y>0.30)=1-0.0443\approx0.9557\), or about 0.956. If a hypothetical assay quantifies this sample above 0.30 ng/mL, that outcome is not surprising under the model. The probability 0.956 refers to being above the limit, not below it.
Now use the same \(C=0.50\) ng/mL and \(\tau=0.30\), but consider limits \(L=1.20\) and \(L=0.90\) ng/mL. For \(L=1.20\),
The probability of a below-limit result is \(\Phi(2.91823)\approx0.9982\), and the probability of exceeding the limit is about 0.0018. For \(L=0.90\),
Here \(P(Y<0.90)\approx0.97496\), while \(P(Y>0.90)\approx0.02504\). A concentration below either of these high limits would be expected under this model. The examples show why the reported status alone is insufficient: the same status can be weak or strong evidence against a prediction depending on the ratio \(L/C\) and the predictive spread.
Why Timing Changes the Validation Interpretation
Below-quantification results often occur late in a concentration-time profile, but their timing is not a detail to discard. If predictions decline with time, the model may imply a rising probability of falling below a fixed LLOQ. A sequence of late censored samples may therefore be consistent with the predicted profile. Conversely, a below-limit result at an early time when the prediction is much higher than the LLOQ can be more surprising. The appropriate comparison is time-specific: use each sample’s prediction \(C_i\), limit \(L_i\), and predictive distribution.
Worked Example: A Time-Ordered Censoring Pattern
Consider four hypothetical samples collected at increasing times after dosing. Their predictions are 0.80, 0.30, 0.15, and 0.08 ng/mL. The LLOQ is 0.50 ng/mL for every sample, and the prespecified predictive model has \(\tau=0.30\) with no bias. The results are quantified at the first time and below quantification at the next three times. Compute the model-implied below-limit probabilities.
At the first time, \(z=\ln(0.50/0.80)/0.30=-1.56668\), so \(P(Y<0.50)\approx0.0586\). A quantified result is therefore likely under the model, with probability about \(1-0.0586=0.9414\).
At the second time, \(z=\ln(0.50/0.30)/0.30=1.70275\), so \(P(Y<0.50)\approx0.9557\). At the third time, \(z=\ln(0.50/0.15)/0.30=4.01324\), giving a probability very close to 1 (approximately 0.99997). At the fourth time, \(z=\ln(0.50/0.08)/0.30=6.10860\), also giving a probability essentially 1 to the shown precision.
The three late below-limit results are broadly compatible with the model’s predicted detectability pattern. That does not establish that the full profile is correct: the first result and all quantified concentrations still need evaluation, and repeated measurements may be correlated. But counting three censored samples as three unexplained failures would ignore that the model predicts they should be below the limit. In contrast, if the early 0.80 ng/mL prediction had also produced a below-limit result, its model-implied probability of 0.0586 would make it more diagnostically noteworthy.
Timing also affects what validation can learn. If nearly all late samples are censored, the data may provide little information about the exact shape or magnitude of the terminal decline. They can support the statement that concentrations were below specified thresholds, but cannot distinguish among all possible concentrations below those thresholds. Report the times and limits, and retain the observed-versus-predicted time structure when interpreting the results.
Diagnostics, Alternatives, and Limitations
A practical diagnostic is to compare observed censoring with the probabilities implied at the actual sample times. Summarize how many samples were below limit, where in time they occurred, and whether the model assigned those events low or high probabilities. If many low-probability censoring events occur in a structured region of the profile, that pattern can indicate a predictive mismatch. It should be interpreted alongside quantified residual patterns and prespecified error criteria, rather than as a stand-alone test with an automatic pass/fail threshold.
Check that each sample’s LLOQ is correct and that the censoring status is represented consistently. Inspect whether low predicted probabilities cluster at early or otherwise informative times, and whether the predictive spread was fixed independently of the validation observations. A poor distributional assumption can also produce misleading probabilities: a single lognormal spread may be inadequate if variability changes substantially over time or if residuals have dependence not represented by the model.
| Approach | What it does | Main limitation |
|---|---|---|
| Use the censoring limit in a likelihood | Uses the probability that the observation falls below its actual LLOQ. | Requires a defensible predictive distribution and its uncertainty assumptions. |
| Replace each value by zero | Treats every censored observation as an exact zero. | Invents an exact value and can create artificial bias or extreme errors. |
| Replace each value by LLOQ/2 | Assigns a convenient point estimate below the limit. | Imputes a value not observed; resulting errors depend on the arbitrary substitution. |
| Omit below-limit samples | Evaluates only quantified concentrations. | Discards potentially informative events and can distort conclusions about time regions with frequent censoring. |
Substitution by zero or \(L_i/2\) can be shown as a clearly labelled sensitivity analysis if there is a compelling reason to illustrate how a chosen summary changes. It should not be presented as the measured concentration or as the primary analysis. Similarly, dropping censored samples is not neutral when censoring is related to concentration and time. Neither substitution nor omission preserves the interval information in the observation.
The likelihood approach also has limits. A probability near 1 for one censored sample does not demonstrate overall predictive validity; a probability near 0 does not, by itself, identify the cause of disagreement. Probability calculations inherit the assumptions about assay behavior, predictive variation, and dependence. With a very high LLOQ, even substantially different profiles may all be below the limit, leaving weak discrimination. If limits vary by sample, use the corresponding \(L_i\) rather than a single pooled threshold.
A Practical Interpretation Sequence
For each below-quantification sample, retain its sample time and actual LLOQ. Interpret it as \(0 \leq Y_i<L_i\), not as zero.
Specify how the PBPK prediction relates to the observation, including any prespecified bias, predictive spread, and dependence assumptions.
Calculate \(F_i(L_i)\) under that model, or use the corresponding censoring contribution in a likelihood.
Assess whether low- or high-probability censoring events cluster in parts of the profile that matter for the intended validation domain.
Common Mistakes
- Calling a censored result zero. The assay has established a threshold interval, not an exact zero concentration.
- Reporting an exact fold error. A censored value gives a bound on the ratio, not the ratio itself.
- Confusing the two tail probabilities. \(P(Y<L)\) is \(\Phi\{ \ln(L/C)/\tau \}\); the probability of exceeding the limit is its complement.
- Ignoring when censoring occurs. A late sample with a low predicted concentration can be expected to fall below the limit, while an early censoring event may carry different evidence.
- Treating one probability as a verdict. An event probability is interpreted under a stated model and alongside the full prespecified validation assessment; it is not a probability that the PBPK model is true.
Check Your Understanding
Answer each question using the censoring limit and timing as part of the interpretation. Brief solutions follow.
- A result is reported below 0.20 ng/mL. What concentration interval does this establish, and why is it not an observed zero?
- Under the no-bias lognormal model, \(C=0.40\) ng/mL, \(L=0.40\) ng/mL, and \(\tau=0.25\). What is the model-implied probability of a below-limit result?
- Why is replacing every censored result by half the LLOQ not equivalent to using a censoring likelihood?
- A model predicts low concentrations at late times, and nearly all late observations are below the LLOQ. What should be examined before calling this a validation failure?
Solutions. (1) The result establishes \(0 \leq Y<0.20\) ng/mL; the exact concentration is unknown and may be positive. (2) \(z=\ln(0.40/0.40)/0.25=0\), so \(P(Y<0.40)=\Phi(0)=0.5000\). (3) Half the limit assigns an exact value that was not observed, whereas the likelihood uses the probability of the full interval below the limit. (4) Compare each event with its time-specific predicted probability using its LLOQ and the prespecified predictive distribution, then assess the overall time pattern and quantified observations under the validation criteria.