Handling Below-Quantification-Limit Data in Population PK

01 · The core problem

1. What Does Below-Quantification-Limit Mean?

In pharmacokinetic studies, an assay has a lower limit of quantification (LLOQ). Concentrations below that threshold may still contain information about drug exposure, but the assay cannot reliably report their numerical value.

Such observations are commonly described as below the limit of quantification (BLOQ), below quantification limit (BQL), or BLQ. The terminology varies across laboratories and software, but the statistical issue is the same: the exact concentration is not observed.

Core idea: a BQL observation is not necessarily a concentration of zero. It tells us that the true concentration is below the assay's quantification threshold.

If the LLOQ is $L$, a BQL observation provides the information

\[ C_i < L \]

rather than an exact measured value $C_i=c_i$.

This distinction becomes particularly important in population PK because low concentrations often occur during the terminal phase, where they can contain information about elimination and between-subject variability.

02 · Why it matters

2. Why Are BQL Observations Important in Population PK?

BQL observations are often concentrated in exactly the region of the concentration-time profile where the model is trying to learn about drug disposition at low concentrations.

Situation	What the BQL observation tells us	Why it may matter
Terminal sampling	The concentration is below the assay threshold	Can provide information about the late portion of the PK profile
Long half-life drug	Drug persists, but measured concentrations become difficult to quantify	Can affect estimation of terminal disposition
High assay sensitivity relative to exposure	Few observations are BQL	Often little impact, depending on their timing
Many late BQL observations	Multiple concentrations are known only to be below LLOQ	Ignoring their censoring mechanism can influence model estimates
Early BQL observations	Concentration is below LLOQ before substantial measurable exposure	May contain information about absorption or delayed appearance

The practical importance therefore depends on how many BQL observations occur, where they occur in the sampling schedule, and how low the LLOQ is relative to the predicted concentration distribution.

03 · Statistical interpretation

3. BQL Data Are Censored, Not Missing

The most important conceptual distinction is between missing data and censored data.

If a concentration sample was never collected, there may be no direct information about the concentration at that time. In contrast, a BQL sample was collected and tested, and the assay established that the concentration falls below a known threshold.

A BQL result provides an inequality rather than an exact concentration. The observation is therefore censored at the LLOQ.

Mathematically, if $Y_i$ denotes the latent concentration that would have been observed without the assay limitation, then a BQL result records the event

\[ Y_i < L \]

The statistical model can therefore use the probability of that event rather than pretending that the concentration equals a particular substituted value.

04 · Common approaches

4. What Are the Common Ways to Handle BQL Data?

Several approaches have historically been used in population PK analyses. They differ in how much information from the censored observation is retained.

Approach	How BQL observations are treated	General implication
Omit BQL observations	Remove BQL records from the analysis	Discards information that the concentration is below LLOQ
Set BQL = 0	Replace BQL observations with zero	Assumes an exact value that was not observed
Set BQL = LLOQ/2	Substitute a fixed fraction of LLOQ	Convenient but treats censored observations as exact measurements
Use LLOQ	Replace BQL with the quantification threshold	Also treats a censored observation as an exact concentration
M3 likelihood	Use the probability that the concentration is below LLOQ	Retains the censoring information explicitly
Related likelihood methods	Modify how BQL observations enter the likelihood	Can provide alternative approximations or modeling strategies

Key distinction: substitution methods convert a censored observation into an artificial exact value. Likelihood-based methods instead use the information that is actually known: the concentration lies below the quantification limit.

05 · Substitution

5. Why Can Substitution Be Problematic?

Suppose the LLOQ is $0.5$ mg/L and a sample is reported as BQL. Replacing it with $0.25$ mg/L creates an apparent quantitative observation:

\[ C_{\mathrm{BQL}} \longrightarrow \frac{\mathrm{LLOQ}}{2}=0.25\text{ mg/L} \]

But the assay did not establish that the concentration was $0.25$ mg/L. It established only that the concentration was below $0.5$ mg/L.

This distinction can become important when a large fraction of observations are BQL or when BQL observations occur in an informative part of the concentration-time profile.

5.1 BQL = 0

Setting BQL observations to zero can be especially problematic for a PK model whose predictions are strictly positive. A zero concentration may imply something qualitatively different from a concentration that is merely below the assay threshold.

5.2 BQL = LLOQ/2

The LLOQ/2 rule is simple and reproducible, but its numerical value is arbitrary from the perspective of the individual observation. If the true concentration could be anywhere below the LLOQ, assigning every BQL sample exactly the same value does not represent that uncertainty.

5.3 Omitting BQL observations

Removing BQL records avoids assigning artificial concentrations, but it also removes the information that those samples were tested and found to be below the threshold.

Whether this matters materially depends on the design and the amount and timing of BQL data. It is not appropriate to assume that every dataset will be affected equally.

06 · Likelihood

6. The Likelihood-Based Idea

The central idea behind likelihood-based BQL methods is straightforward: use the probability of obtaining a BQL result under the model.

For a quantifiable observation $y_i$, the likelihood contribution is based on the probability density at the observed value:

\[ L_i = f(y_i\mid \theta) \]

For a BQL observation, the exact value is unknown, but we know that \(y_i

\[ L_i = P(Y_i

where $F$ is the cumulative distribution function implied by the observation model and $\theta$ denotes the model parameters.

Core statistical idea: a quantifiable result contributes a density evaluated at the observed concentration; a BQL result contributes the probability of being below the LLOQ.

07 · M3 method

7. The M3 Method for BQL Data

The M3 method is a likelihood-based approach that explicitly treats BQL observations as censored observations.

For a quantifiable concentration, the likelihood contribution is based on the observation density. For a BQL concentration, the contribution is the cumulative probability below the LLOQ.

Conceptually, the likelihood can be written as

\[ L(\theta)= \prod_{i\in Q} f(y_i\mid\theta) \prod_{j\in B} P(Y_j

where:

$Q$ denotes the set of quantifiable observations.
$B$ denotes the set of BQL observations.
$L_j$ is the relevant quantification limit for BQL observation $j$.
$f(\cdot)$ is the density implied by the residual error model.
\(P(Y_j

The resulting objective function therefore incorporates both the numerical concentrations that can be quantified and the inequality information supplied by BQL samples.

08 · Observation model

8. Why the Residual Error Model Matters

The probability of being below the LLOQ depends on the model for residual unexplained variability.

For example, under a normal observation model on the concentration scale, if the model predicts a typical concentration $F_i$ with residual standard deviation $\sigma_i$, then

\[ Y_i\sim N(F_i,\sigma_i^2) \]

and the probability of a BQL result is

\[ P(Y_i

where $\Phi$ is the standard normal cumulative distribution function.

With a proportional or combined error model, the variance structure changes, and so does the probability associated with the BQL event.

Residual error model	General form	Implication for BQL likelihood
Additive	$Y=F+\epsilon$	Constant absolute residual scale
Proportional	$Y=F(1+\epsilon)$	Residual variation depends on prediction magnitude
Combined	$Y=F+F\epsilon_p+\epsilon_a$	Both proportional and additive components contribute

This is one reason BQL handling should not be separated completely from the rest of model development. The censoring likelihood and residual error model are mathematically connected.

09 · Alternative likelihood

9. What Is the M4 Approach?

The M4 method is another likelihood-based approach for handling BQL observations. Its key distinction is that BQL observations are incorporated using a likelihood contribution while the corresponding censored measurements are not treated as ordinary continuous observations in exactly the same way as fully quantified data.

The important practical point is that M3 and M4 are both attempts to preserve information from censored observations rather than replacing them with arbitrary numerical values.

Feature	M3	M4
BQL observations recognized as censored?	Yes	Yes
Uses probability below LLOQ?	Yes	Yes
Quantifiable observations treated as continuous measurements?	Yes	Yes
Primary concept	Full censored-data likelihood	Likelihood treatment incorporating censored observations

The exact implementation should be matched to the modeling software and estimation framework being used. The terminology M3/M4 is most useful as a conceptual framework for distinguishing likelihood-based censoring from simple data substitution.

10 · Practical workflow

10. A Practical Workflow for BQL Data

A useful BQL workflow starts before model estimation.

Identify the assay limits. Determine the LLOQ for each relevant analyte, matrix, assay, and study period.
Identify BQL records. Distinguish BQL results from missing samples and other types of non-quantifiable observations.
Examine their timing. Plot BQL observations against time and dose to determine where censoring occurs.
Quantify the extent of censoring. Calculate the proportion of observations that are BQL overall and by study, dose group, subject, and time region.
Assess whether censoring is informative for the scientific question. Terminal BQL observations may matter differently from occasional early BQL observations.
Select a BQL strategy. Consider whether a likelihood-based censored-data method is appropriate.
Specify the observation model carefully. The residual error model determines the probability associated with BQL observations.
Estimate the model. Incorporate quantifiable and BQL observations according to the selected likelihood.
Evaluate diagnostics. Examine predictions and residual behavior, including the model's ability to reproduce the observed frequency and timing of BQL results.
Perform sensitivity analyses when appropriate. Compare reasonable alternative treatments when BQL handling could materially affect conclusions.

11 · Diagnostics

11. How Should a BQL Model Be Evaluated?

Standard population PK diagnostics remain important, but BQL data require additional attention because the observations are censored.

11.1 Inspect BQL timing

Plot BQL observations on concentration-time plots. Marking BQL records separately from quantified concentrations makes it easier to see whether censoring occurs primarily in the terminal phase or elsewhere.

11.2 Compare observed and predicted BQL frequency

If a model predicts that concentrations frequently fall below the LLOQ, the observed dataset should contain a correspondingly plausible number of BQL results.

A useful conceptual check is

\[ P(\mathrm{BQL})= P(Y

and comparison of this probability with the observed proportion of BQL observations under comparable conditions.

11.3 Examine predictions near the LLOQ

Model performance around the quantification boundary is especially relevant. A model that systematically predicts concentrations just above the LLOQ when many observations are BQL may not adequately represent the lower tail of the data.

11.4 Examine standard population PK diagnostics

Goodness-of-fit plots, conditional weighted residuals, prediction-corrected diagnostics, visual predictive checks, and parameter uncertainty remain useful, provided their interpretation accounts for the censoring mechanism.

Important: a conventional concentration-vs-prediction plot cannot fully show whether a censored-data model is reproducing BQL observations correctly. The BQL portion of the observation process should be evaluated explicitly.

12 · Visual predictive checks

12. BQL Data and Visual Predictive Checks

Visual predictive checks can be adapted to assess whether the model reproduces the observed distribution of concentrations over time, including the behavior of observations near and below the LLOQ.

For a dataset with substantial BQL information, it can be useful to examine:

the fraction of observations that are BQL at different times;
the predicted probability of being below LLOQ over time;
the distribution of quantified concentrations above LLOQ;
the relationship between BQL frequency and dose;
the terminal portion of the concentration-time profile.

A conventional VPC that simply drops BQL observations may hide systematic discrepancies in the lower tail.

13 · Worked example

13. Worked Example: One BQL Observation

Suppose an assay has an LLOQ of 0.5 mg/L. At 24 hours after dosing, a concentration is reported as BQL.

Step 1: What do we actually know?

We do not know that the concentration is zero, $0.25$ mg/L, or $0.5$ mg/L. We know only that

\[ C(24)<0.5\text{ mg/L} \]

Step 2: Suppose the model predicts 0.30 mg/L

Assume a simplified normal observation model with prediction

\[ F=0.30\text{ mg/L} \] and residual standard deviation $$ \sigma=0.20\text{ mg/L}. $$

Step 3: Calculate the probability of a BQL result

The standardized LLOQ is

\[ z=\frac{L-F}{\sigma} = \frac{0.50-0.30}{0.20} =1.0. \]

Therefore, under this simplified observation model, the probability of a result below the LLOQ is

\[ P(Y<0.50) = \Phi(1.0) \approx0.841. \]

Thus the model assigns approximately an 84.1% probability to obtaining a BQL result at this time point.

Step 4: Compare with substitution

A substitution method might replace the observation with $0.25$ mg/L. The likelihood-based approach does something fundamentally different: it does not pretend to know the exact concentration. It asks how compatible the observed BQL event is with the model.

Interpretation: the model is evaluated against the information actually supplied by the assay: the concentration was below 0.5 mg/L.

14 · Multiple censored observations

14. What Happens When Many Observations Are BQL?

The impact of BQL handling becomes more important as the fraction of censored observations increases.

Fraction of BQL observations	Potential modeling concern
Very low	May have limited influence, depending on timing and design
Moderate	Choice of BQL method may affect parameter estimates and diagnostics
High	The lower tail of the concentration distribution may be strongly informed by censoring
Very high	Identifiability and assay sensitivity should be examined carefully

There is no universal percentage at which BQL observations suddenly become important. The scientific and statistical impact depends on their timing, dose relationship, assay sensitivity, sampling design, structural model, and residual error model.

For example, ten BQL observations concentrated at the very end of a long sampling interval may provide different information from ten BQL observations occurring throughout the absorption and distribution phases.

15 · Dose and censoring

15. BQL Observations Across Dose Levels

Dose information can be particularly useful when evaluating BQL patterns.

If a drug is approximately linear over a dose range, higher doses generally produce higher concentrations. Consequently, the proportion of BQL observations may differ by dose group.

Dose group	Illustrative BQL pattern	Question to investigate
Low dose	Many late BQL observations	Is the assay sufficiently sensitive for the intended terminal sampling?
Medium dose	Fewer late BQL observations	Does the model reproduce the dose-related change?
High dose	Few or no BQL observations	Does the model predict plausible lower-tail behavior?

Unexpectedly similar BQL frequencies across very different dose levels can sometimes signal issues with the assay, data handling, model structure, or assumptions about residual variability. Such patterns should be investigated rather than automatically attributed to one cause.

16 · Special cases

16. Situations Requiring Extra Care

16.1 Different LLOQs

If different samples have different quantification limits, the likelihood contribution should use the appropriate limit for each observation:

\[ P(Y_i

Using one global LLOQ when the actual assay threshold varies can misrepresent the censoring mechanism.

16.2 BLQ caused by a data-processing rule

A record can be unavailable for reasons other than the analytical assay's quantification limit. Such records should not automatically be treated as BQL observations.

16.3 Missing samples

A missing sample is not equivalent to a BQL result. If no sample was collected, there is generally no observed inequality \(C

16.4 Concentrations below LLOQ at baseline

Predose BQL observations may require particular attention because they can relate to residual concentrations from prior dosing, carryover, or assumptions about initial conditions.

16.5 Nonlinear PK

For nonlinear PK, BQL information can interact with the structural model in more complicated ways. A model that predicts the lower tail incorrectly may produce misleading inferences about parameters governing dose-dependent clearance, elimination, or other nonlinear processes.

17 · Model development

17. Where Does BQL Handling Fit in Model Development?

BQL handling should be considered part of the observation model and estimation strategy, not merely as a preprocessing step.

A population PK analysis separates structural PK assumptions from the observation process. BQL handling is part of that observation process.

This perspective helps avoid a common mistake: deciding how to replace BQL records before considering what information those records actually contain.

18 · Sensitivity analysis

18. Should You Perform a Sensitivity Analysis?

When BQL observations are sufficiently numerous or influential, sensitivity analyses can help determine whether the scientific conclusions depend strongly on the selected treatment.

Possible comparisons include:

a likelihood-based censored-data approach;
a carefully justified alternative treatment;
different plausible residual error structures;
different assumptions about records near the LLOQ;
alternative structural models when BQL observations occur in a region that is difficult to characterize.

The purpose of sensitivity analysis is not to choose whichever method produces the preferred parameter estimate. Rather, it is to determine whether reasonable modeling choices materially change the interpretation.

Good practice: if BQL handling materially affects an important PK parameter, document the alternative analyses and explain why the primary approach was selected.

19 · Common mistakes

19. Common Mistakes When Handling BQL Data

Mistake	Why it is problematic	Better question
Assuming BQL means zero	BQL establishes an upper bound, not necessarily zero concentration	What range of concentrations is consistent with the assay result?
Automatically using LLOQ/2	Creates an artificial exact concentration	Can the censoring information be modeled directly?
Deleting all BQL observations	Discards information supplied by the assay	How much information is lost by treating them as absent?
Ignoring the LLOQ	The model may be evaluated against observations it could not actually quantify	What was the relevant quantification threshold for each sample?
Treating missing and BQL as identical	Missing means no observed result; BQL provides an inequality	Was the sample collected and tested?
Ignoring the residual error model	The BQL probability depends on the observation model	How does the specified error model translate predictions into BQL probabilities?
Checking only quantified observations	Diagnostics may miss systematic problems in the lower tail	Does the model reproduce both quantified and censored observations?

20. Key Takeaways

A BQL observation means the concentration is below a known quantification limit; it does not necessarily mean the concentration is zero.
BQL observations are best understood as censored observations, not simply missing values.
Substitution methods such as BQL = 0 or BQL = LLOQ/2 replace censored observations with artificial exact concentrations.
Likelihood-based methods preserve the information that the concentration lies below the LLOQ.
The M3 approach represents BQL observations through their probability of falling below the quantification limit.
The residual error model determines how the model translates a predicted concentration into the probability of a BQL result.
The number and timing of BQL observations determine how important their treatment may be for a particular population PK analysis.
BQL observations occurring in the terminal phase can contain information about the lower tail of the concentration-time distribution and should not automatically be discarded.
Different LLOQs should be represented using the appropriate threshold for each observation when applicable.
Missing samples and BQL samples are different data situations and should not automatically be handled identically.
BQL handling should be evaluated together with the structural PK model, residual error model, and diagnostic strategy.
When BQL treatment could materially affect important conclusions, sensitivity analyses can help assess the robustness of the model.

Next step

Where to Go Next

A natural progression is to study residual unexplained variability models, followed by goodness-of-fit diagnostics in population PK, visual predictive checks, prediction-corrected visual predictive checks, and normalized prediction distribution errors.

These topics build directly on the observation-model concepts introduced here. Understanding how residual error and censoring are represented is essential for interpreting diagnostics and determining whether a population PK model adequately describes both quantified and BQL observations.

References

Beal SL. Ways to fit a PK model with some data below the quantification limit. Journal of Pharmacokinetics and Pharmacodynamics. 2001;28:481–504.
Ahn JE, Karlsson MO, Dunne A, Ludden TM. Likelihood based approaches to handling data below the quantification limit in pharmacokinetic/pharmacodynamic studies. British Journal of Clinical Pharmacology. 2008;65:681–686.
Holford N, Ma SC, Ploeger BA. Clinical pharmacokinetics and pharmacodynamics of population models. General concepts relevant to model-based analysis of censored observations.
European Medicines Agency. Guideline on Reporting the Results of Population Pharmacokinetic Analyses. European Medicines Agency.

Residual error model	General form	Implication for BQL likelihood
Additive	\(Y=F+\epsilon\)	Constant absolute residual scale
Proportional	\(Y=F(1+\epsilon)\)	Residual variation depends on prediction magnitude
Combined	\(Y=F+F\epsilon_p+\epsilon_a\)	Both proportional and additive components contribute

1. What Does Below-Quantification-Limit Mean?

2. Why Are BQL Observations Important in Population PK?

3. BQL Data Are Censored, Not Missing

4. What Are the Common Ways to Handle BQL Data?

5. Why Can Substitution Be Problematic?

5.1 BQL = 0

5.2 BQL = LLOQ/2

5.3 Omitting BQL observations

6. The Likelihood-Based Idea

7. The M3 Method for BQL Data

8. Why the Residual Error Model Matters

9. What Is the M4 Approach?

10. A Practical Workflow for BQL Data

11. How Should a BQL Model Be Evaluated?

11.1 Inspect BQL timing

11.2 Compare observed and predicted BQL frequency

11.3 Examine predictions near the LLOQ

11.4 Examine standard population PK diagnostics

12. BQL Data and Visual Predictive Checks

13. Worked Example: One BQL Observation

Step 1: What do we actually know?

Step 2: Suppose the model predicts 0.30 mg/L

Step 3: Calculate the probability of a BQL result

Step 4: Compare with substitution

14. What Happens When Many Observations Are BQL?

15. BQL Observations Across Dose Levels

16. Situations Requiring Extra Care

16.1 Different LLOQs

16.2 BLQ caused by a data-processing rule

16.3 Missing samples

16.4 Concentrations below LLOQ at baseline

16.5 Nonlinear PK

17. Where Does BQL Handling Fit in Model Development?

18. Should You Perform a Sensitivity Analysis?

19. Common Mistakes When Handling BQL Data

20. Key Takeaways

Where to Go Next

References

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