Below-Quantification Data In A PK Case

Why Below-Quantification Results Need Their Own Likelihood Contribution

A concentration reported as below the lower limit of quantification (LLOQ) is not a measured concentration of zero, nor is it an exact concentration equal to the LLOQ. It tells us that the underlying concentration was not quantified by the assay and, under the usual reporting rule, fell below a known threshold. When a substantial fraction of late PK samples have this result, replacing them with a single number can shift the apparent terminal profile and its derived exposure.

A likelihood analysis uses the information actually recorded. For a quantified result, it uses the probability density of the observed concentration. For a below-quantification result, it uses the probability that the model would produce a value below the LLOQ. This approach retains the distinction between an observed value and an inequality.

The method relies on a specified concentration model, a suitable residual-error distribution, and a correctly recorded quantification threshold. In the examples below, the conditional distribution of concentration is lognormal: the logarithm of concentration is normally distributed around the logarithm of the model prediction. We treat observations as conditionally independent given the model, and take the assay LLOQ as known. If assay-specific thresholds or error models apply, they must be represented rather than assumed away.

Likelihood Contributions for Quantified and Censored Samples

Let \(C_i(\theta)\) be the PK model prediction for sample \(i\), at its dose and sampling time, under parameter vector \(\theta\). The lognormal residual model is

Here \(\sigma\) is the standard deviation on the log scale. For a quantified result \(y_i\), the contribution can be written as the normal density of \(\log y_i\) with mean \(\log C_i(\theta)\) and standard deviation \(\sigma\). Equivalently, on the concentration scale the density includes the lognormal \(1/y_i\) factor. That factor does not depend on \(\theta\), so it does not change the parameter estimates when maximizing the likelihood.

For a below-quantification result, integrate the model density over every concentration that could have produced the reported result. On the log scale, this is the normal probability below \(\log L_i\):

where \(\Phi\) is the standard normal cumulative distribution function. The total likelihood is the product of the quantified density contributions and the censored probability contributions, or, more conveniently, the sum of their log likelihoods. Maximizing it estimates parameters using both types of information without pretending that a censored result has a specific concentration.

The reason this is the right contribution follows directly from what was observed. If the assay reports only \(Y_i<L_i\), the event consistent with the data includes all values below \(L_i\). Its probability is the integral of the density over that region:

Using the probability of the whole compatible region is the likelihood principle applied to a censored observation. Substituting \(L_i/2\), in contrast, uses the density at one chosen point as though that exact concentration had been observed.

A Small Censored PK Case

Consider four hypothetical samples taken at the same late time after dosing, with an LLOQ of 1.00 mg/L. Two concentrations are quantified at approximately 1.221 and 1.492 mg/L; two are reported below the LLOQ. For illustration, assume this group shares one typical concentration \(M\) at that time, with known log-scale residual standard deviation \(\sigma=0.250\). This is a deliberately simple local model for showing the censored likelihood calculation, not a general PK structural model.

The quantified log concentrations are 0.200 and 0.400, since \(\log(1.221)\) and \(\log(1.492)\) are approximately those values. Let \(\theta=\log M\). Each quantified result contributes a normal density on the log scale, and each censored result contributes \(\Phi((0-\theta)/0.250)\), because \(\log(1.00)=0\). Up to constants that do not depend on \(\theta\), the log likelihood is

Differentiating gives the likelihood score. Setting it to zero yields the equation solved to obtain the maximum-likelihood estimate:

where \(\phi\) is the standard normal density. Numerically solving this equation gives \(\hat{\theta}\approx0.061\), so the estimated typical concentration is \(\exp(0.061)\approx1.063\) mg/L. This solution uses the two quantified values and the fact that the other two fall below 1.00 mg/L. The censored results pull the estimate down relative to what the quantified values alone would suggest, but they do not pull it as far as arbitrarily assigning each one a very low fixed concentration.

What the Choice Means for Terminal Exposure

A difference in a late concentration estimate can propagate into terminal exposure. To make that connection explicit, suppose the hypothetical terminal profile after the sampling time follows a mono-exponential decline with a known rate constant \(k=0.100\ \text{h}^{-1}\). If \(M\) is the typical concentration at the start of a 12-hour terminal interval, the model is \(C(t)=M\exp(-kt)\). Its interval AUC is obtained by integrating the concentration over time:

The multiplier is \((1-\exp(-1.2))/0.100\approx6.988\) h. This calculation assumes that the same terminal rate constant applies under each estimation approach and that the typical profile is represented by the estimated \(M\). It is an illustration of the effect on a typical-profile AUC, not an estimate of the population mean AUC when individual concentrations vary lognormally.

Diagnostics, Alternatives, and Limits

A censored likelihood is not automatically reliable simply because it handles below-LLOQ results explicitly. The residual model, structural PK model, and observation process still need scrutiny. Useful checks include:

• Confirm the assay rule. Verify the LLOQ for each sample or assay, and check that “below quantification” is not being mixed with missing, rejected, or otherwise invalid results.
• Inspect predictions and censoring by time. Compare the observed fraction below LLOQ with model-predicted probabilities over the sampling schedule. Systematic disagreement can indicate a poor structural model, an unsuitable residual distribution, or an incorrect threshold.
• Check residual patterns among quantified samples. Look for time- or concentration-dependent spread, skewness, and systematic trends. A constant log-scale \(\sigma\) should not be retained without justification if the data show a different error pattern.
• Assess identifiability. If nearly all terminal observations are censored, the data may provide limited information about the terminal concentration or slope. A converged optimizer does not guarantee that the parameter is well determined; inspect uncertainty and sensitivity to plausible model choices.
• Review numerical behavior. Very small censoring probabilities can cause numerical underflow if calculated directly. Stable log-CDF computations and sensible initial values help distinguish numerical failure from genuine lack of fit.

The main alternative is substitution, such as LLOQ/2 or LLOQ, followed by ordinary analysis. It is simple to implement, but treats invented values as measurements and ignores where within the censored region the model places probability. Another option is to omit censored observations. That discards information and can select a systematically unrepresentative subset, especially when censoring is common at late times. The likelihood method is generally preferable when the censoring threshold and an appropriate observation model are available, but it cannot recover information that the design did not collect. If the model is misspecified or nearly all relevant samples are censored, the resulting estimates may remain uncertain or sensitive to assumptions.