BQL Data During Repeated Dosing

Why Repeated Dosing Changes the BQL Question

A sample collected after a dose does not reflect only that dose. Drug remaining from earlier doses may still contribute to the concentration, even when the most recent dose was given many hours later. This matters for below-quantification-limit (BQL) data: whether a measurement falls below the assay limit is evaluated against the concentration predicted from the entire relevant dose history, not against the contribution from the latest dose alone.

The introductory material on censored likelihoods establishes the key idea: a BQL record reports an event, not an exact concentration. In a repeated-dose setting, the event probability depends on the predicted concentration at that sample time. Getting that prediction right requires carrying forward residual concentrations from previous doses.

Definition: A BQL observation is a sample whose assay result is reported as below the lower limit of quantification, denoted \(L\). For a continuous assay outcome \(Y\), the recorded information is the event \(Y<L\), rather than an exact value of \(Y\). During repeated dosing, the model prediction used to evaluate that event must include the concentration contributions from all relevant earlier doses.

Constructing the Repeated-Dose Prediction

First consider a one-compartment model with instantaneous intravenous bolus doses and first-order elimination. For subject \(i\), let dose \(D_{ij}\) be administered at time \(t_{ij}\). Let \(V_i\) be the volume of distribution and \(k_i\) the elimination rate constant. For a sample at time \(t\), after the doses included in the model, the predicted concentration is the sum of the contributions from those doses:

C_i(t)=\sum_{j:t_{ij}\le t}\frac{D_{ij}}{V_i}\exp\{-k_i(t-t_{ij})\}.

Each term uses the dose-to-volume ratio for that dose and the elapsed time from that dose to the sample. Its units are concentration: \(D_{ij}/V_i\) has units such as mg/L, and \(k_i(t-t_{ij})\) is dimensionless when \(k_i\) is in inverse time and the time difference is in the matching time unit. The sum is the full predicted profile at the sample time.

This formula is specific to the stated bolus model. In a more general NLME analysis, the structural model may use different dosing inputs or disposition assumptions. The principle remains: use the structural model’s predicted concentration at the actual sample time, based on the recorded dose history and the subject’s model parameters. In a mixed-effects model those parameters may depend on subject-specific random effects, so the prediction can differ between subjects even when their nominal doses are the same.

Assumptions for the bolus calculation: The dose amounts and administration times are known and recorded correctly; the doses are instantaneous intravenous boluses; elimination is first-order with a constant \(k_i\); and the stated one-compartment model adequately describes the concentration contributions being summed. The assay limit and sample time must also be correctly assigned. If these assumptions do not fit the data, use the appropriate structural model rather than this formula.

The sum also shows directly how accumulation occurs. With equal doses separated by a fixed interval, contributions from older doses are successively smaller, but they do not become zero immediately. For example, if the amount added to concentration by each dose is \(A=D/V\), the dosing interval is \(\tau\), and \(r=\exp(-k\tau)\), then a predose sample after \(m\) doses has contributions \(Ar, Ar^2,\ldots,Ar^m\), provided the first dose was given \(m\tau\) before the sample. The finite geometric sum is:

C_{\mathrm{pred}}=A(r+r^2+\cdots+r^m) =A\frac{r(1-r^m)}{1-r}, \qquad 0<r<1.

The formula follows by multiplying the sum \(r+r^2+\cdots+r^m\) by \(1-r\): all intermediate powers cancel, leaving \(r-r^{m+1}=r(1-r^m)\). Dividing by \(1-r\) gives the stated result. For the BQL likelihood, the sum form is often more useful because it makes the individual dose contributions visible.

Worked Example: Three Dose Contributions at One Sample Time

Suppose a hypothetical subject receives 8 mg bolus doses at 0, 12, and 24 hours. Take \(V=1\) L and \(k=0.10\) per hour. Find the predicted concentration at 34 hours and identify how much comes from doses before the latest dose.

The sample is 34 hours after the first dose, 22 hours after the second, and 10 hours after the third. Each dose adds \(8\text{ mg}/1\text{ L}=8\text{ mg/L}\) before elimination. Therefore:

\begin{aligned} C_{\mathrm{pred}}(34) &=8e^{-0.10(34)}+8e^{-0.10(22)}+8e^{-0.10(10)}\\ &\approx 0.266986+0.886425+2.943036\\ &\approx 4.096447\ \mathrm{mg/L}. \end{aligned}

The displayed contributions are rounded to six decimal places; the total is calculated from the unrounded terms. The two earlier doses contribute \(8(e^{-3.4}+e^{-2.2})\approx1.153411\text{ mg/L}\). Thus the latest dose accounts for about \(2.943036\text{ mg/L}\), but it is not the full predicted concentration. Ignoring the earlier residuals would omit about \(1.153411\text{ mg/L}\) at this sample time.

Evaluate the Threshold Against the Full Profile

A BQL record says that the assay outcome crossed a threshold. To see how the predicted profile enters, let \(f(y\mid C_{\mathrm{pred}})\) be the conditional density of the assay outcome \(Y\), given the model-predicted concentration. The BQL likelihood contribution is the probability of all assay outcomes below \(L\):

P(Y<L\mid C_{\mathrm{pred}}) =\int_{-\infty}^{L} f(y\mid C_{\mathrm{pred}})\,dy.

The observation model determines the density and therefore the probability. Whatever observation model is used, its conditioning prediction must be \(C_{\mathrm{pred}}\) from the full dose history. For a quantified result \(y\), the corresponding likelihood contribution is the density evaluated at the observed value, \(f(y\mid C_{\mathrm{pred}})\). For a BQL result, it is the probability over the interval reported by the assay. These are different data contributions: a BQL value should not be replaced by an invented exact concentration to make it look quantified.

In an NLME analysis, the subject’s random effects affect the predicted concentration and hence the BQL probability. Conditional on those effects, the likelihood combines the contributions from that subject’s quantified and BQL observations. The subject-level likelihood then accounts for uncertainty in the random effects by integrating the conditional likelihood over their distribution. The repeated-dose sum is evaluated using the subject’s dose history and parameters at each relevant sample time; it is not reset just because a sample is BQL.

Worked Example: Earlier Doses Change the Threshold Assessment

A hypothetical subject receives 6 mg bolus doses every 12 hours at times 0, 12, 24, and 36 hours. Let \(V=1\) L and \(k=0.10\) per hour. A predose sample at 48 hours has \(L=2.2\text{ mg/L}\). Calculate the full prediction and compare it with the latest-dose-only contribution.

At 48 hours, the four doses have been eliminated for 12, 24, 36, and 48 hours. Set \(r=e^{-0.10(12)}=e^{-1.2}\). The full profile is:

\begin{aligned} C_{\mathrm{pred}}(48) &=6(r+r^2+r^3+r^4)\\ &=6\left(e^{-1.2}+e^{-2.4}+e^{-3.6}+e^{-4.8}\right)\\ &\approx2.564794\ \mathrm{mg/L}. \end{aligned}

This approximation is calculated from the unrounded exponential terms. The latest dose alone contributes \(6e^{-1.2}\approx1.807165\text{ mg/L}\). Since \(1.807165<2.2\), a latest-dose-only calculation would put the prediction below the assay limit. The full prediction is \(2.564794>2.2\text{ mg/L}\), so the complete profile is above the limit. Under a deterministic threshold comparison these predictions fall on opposite sides of \(L\). With a stochastic observation model, a BQL result remains possible on either side; its likelihood contribution is determined by the model’s probability for \(Y<2.2\) at the full predicted concentration.

Worked Example: A BQL Likelihood Contribution From an Irregular Dose History

Consider a hypothetical subject receiving bolus doses whose dose-to-volume contributions are 5, 7, and 4 mg/L at times 0, 10, and 21 hours. Let \(k=0.08\) per hour. A sample at 30 hours is BQL with \(L=4.0\text{ mg/L}\). Suppose the already-specified observation model has residual \(e=Y-C_{\mathrm{pred}}\) with cumulative distribution function \(F_e\), and, for this worked calculation, \(F_e(0.186126\text{ mg/L})=0.58\). Find the predicted concentration and BQL likelihood contribution.

The elapsed times are 30, 20, and 9 hours. The model prediction includes all three doses:

\begin{aligned} C_{\mathrm{pred}}(30) &=5e^{-0.08(30)}+7e^{-0.08(20)}+4e^{-0.08(9)}\\ &\approx 0.453590+1.413276+1.947009\\ &\approx 3.813874\ \mathrm{mg/L}. \end{aligned}

The difference between the limit and prediction is \(4.0-3.813874=0.186126\text{ mg/L}\). Because \(Y=C_{\mathrm{pred}}+e\), the BQL event \(Y<L\) is equivalent to \(e<L-C_{\mathrm{pred}}\). Its likelihood contribution is therefore:

P(Y<4.0\mid C_{\mathrm{pred}}=3.813874) =F_e(4.0-3.813874) =F_e(0.186126) =0.58.

The contribution to the log-likelihood is \(\log(0.58)\approx-0.5447\). The value 0.58 is based on the hypothetical residual distribution specified in the question; it is not inferred from the predicted concentration alone. The example separates two tasks: calculate the full profile from dose history, then use the chosen observation model to evaluate the threshold event.

Practical Checks and Common Pitfalls

Before interpreting a BQL record during repeated dosing, verify that the model prediction is tied to the actual sample time and the complete dose history. A dose given after the sample must not contribute; a prior dose should not be omitted merely because another dose was administered later. Also check that dose amounts, time units, \(k\), \(V\), and the assay limit use compatible units. A time entered in minutes when \(k\) is per hour can change every exponential term.

Check dose records. Confirm dose times and amounts, and account for documented missed, delayed, or changed doses according to the structural model.
Check the prediction at each sample. Recalculate the sum using the elapsed time from each relevant dose to that specific sample.
Check the threshold and observation model. Confirm that the appropriate assay limit applies and that the BQL event probability is computed from the observation model at the full prediction.
Inspect implausible contributions. A term that increases with elapsed time, or a total below a positive dose contribution when all terms should be positive, signals an implementation or unit error in this bolus model.

A common mistake is to reason, “The latest dose predicts less than the LLOQ, so this sample should be BQL.” That reasoning drops residual concentrations from earlier doses. In the 48-hour example, the latest dose contributes \(1.807165\text{ mg/L}\), below the \(2.2\text{ mg/L}\) limit, but the full prediction is \(2.564794\text{ mg/L}\). The correction is to evaluate the event using the sum of the dose contributions, followed by the observation model’s threshold probability.

The main alternatives do not repair an incorrect profile. Deleting BQL records discards their threshold information. Substituting a fixed value such as half the LLOQ or the LLOQ treats an event as if it were an exact measurement and may distort the fitted profile. A censored likelihood retains the BQL event, but it is only as appropriate as its structural prediction, dose history, assay limit, and observation model. During repeated dosing, errors in any of these can affect the likelihood systematically.

This approach also has limits. The bolus sum is not suitable without modification for other administration routes or structural models. Unrecorded doses, incorrect sample times, changing assay limits, or model misspecification can undermine the event calculation. A BQL observation by itself does not identify which dose contributed what; those contributions come from the assumed structural model and its parameters. When many records are BQL, the data may provide limited information about some parameters, even when each event is encoded correctly.

Key takeaway: A BQL record during repeated dosing constrains the assay outcome relative to \(L\), and the event probability is evaluated at the model prediction for that sample. In a repeated-dose profile, that prediction must account for residual concentrations from the full relevant dose history.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A one-compartment bolus model predicts 0.4 mg/L from the latest dose and 0.7 mg/L from all doses combined. The LLOQ is 0.6 mg/L. Which prediction belongs in the BQL likelihood calculation, and why?
In the repeated-dose example with \(r=e^{-1.2}\), what does \(r^2\) represent for the predose sample at 48 hours?
Why is the BQL likelihood contribution not generally the same as checking whether \(C_{\mathrm{pred}}<L\)?
A dose was administered at 40 hours, but the sample was collected at 38 hours. Should that dose appear in the prediction for the sample? Explain.

Solutions. (1) Use \(0.7\text{ mg/L}\), because the prediction must include all relevant dose contributions; the latest-dose-only value omits residual concentration. (2) It represents the relative contribution of a dose that has undergone two dosing intervals of elimination: the corresponding dose contribution is multiplied by \(r^2\). (3) The assay outcome is modeled as a random variable. The BQL likelihood is the probability \(P(Y<L\mid C_{\mathrm{pred}})\) under that observation model, so it need not be zero or one according to whether the prediction lies below the limit. (4) No. The dose occurs after the sample and cannot contribute to the concentration at 38 hours in this model.

BQL Data During Repeated Dosing

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