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An Improved Approach for Confirmatory Phase III Population Pharmacokinetic Analysis

From Pharmacokinetics, Modeling & Simulation, Clinical Pharmacology Sciences, Centocor R&D, Inc, Malvern, Pennsylvania.

Address for reprints: Chuanpu Hu, PhD, Pharmacokinetics, Modeling & Simulation, Clinical Pharmacology Sciences, Centocor R&D, Inc, C-4-5, 200 Great Valley Parkway, Malvern, PA 19355; e-mail: CHu25{at}cntus.jnj.com.

	ABSTRACT

TOP ABSTRACT METHOD RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
In contrast to the traditional extensive exploratory approach, the authors propose a confirmatory approach to phase III population pharmacokinetics for regulatory submissions. In their approach, they recommend a prespecified primary analysis based on phase I/II data and phase III study design. Their approach also incorporates several specific sensitivity analyses to evaluate the robustness of the conclusions. According to statistical rationale, this approach eliminates certain biases that may occur in the estimated covariate effects, thereby precluding potentially unnecessary dosing adjustments and allowing for more accurate assessments of ambiguity in the results. Because exploration is vastly reduced, the analysis time is also substantially shortened. The proposed analyses are relatively easy to implement, although careful and prospective planning is required. Applications of the approach are provided, including 2 phase III analyses for regulatory submissions. Differences between the proposed approach and the commonly used extensive exploratory analyses submitted to regulatory agencies were small and consistent with practical expectations.

Key Words: Nonlinear mixed effect modeling • sparse sampling • drug-drug interaction

It is possible to build a POPPK model at phase III by incorporating data from earlier phases. Although perhaps scientifically desirable, this may complicate the interpretation of results, such as what characteristics can be assumed as similar across populations and trials at different stages. Addressing these concerns is not the purpose of this article. Instead, we focus on an approach in which information from earlier phases may be available, but phase III data will be analyzed separately.

What Can Best Be Identified From Phase III Data?
Typical phase III PK design employs sparse sampling designs, which usually consist of only trough samples and perhaps 1 or 2 additional nontrough samples per subject. Such designs allow the estimation of only relatively simplistic structural models achieved by borrowing information across subjects. This approach provides a means to assess covariate influence on exposure, such as age, sex, race, and concomitant medications,¹ and, accordingly, to recommend potential dosing adjustments for drug product labeling. Because sparse PK sampling at phase III may not allow elaborate models regarding absorption or distribution processes to be identified, these models serve as better guidance for long-term (steady-state) average exposure than for short-term concentration rise and fall. However, the number of subjects included in phase III POPPK data sets is usually large enough to provide statistical power for detecting covariate relationships. Therefore, in the POPPK model, covariate detection and quantitation can be viewed as the main points of interest, serving as the basis for the recommendation of potential dosing adjustments for drug product labeling. Although the base model selection may still influence the covariate model,² it may be considered a set of nuisance parameters in this analysis context.

Exploratory Versus Confirmatory
Common POPPK analyses involve, implicitly or explicitly, extensive searches among a large number of candidate models, with the majority of them being covariate models. Although at times mechanistic information is available, the drive is largely to search for the best fit among all possible models. Because of the potentially large number of covariate model combinations, stepwise model selection is frequently used to limit the search space. The number of models explicitly summarized is at least dozens but usually hundreds or even more. Conclusions are based on the final model, typically irrespective of the model selection process. This is commonly recognized as the "exploratory analysis" approach. Regulatory agencies have issued guidance on the topic of POPPK analysis^3,4 due in part to its subjectivity. The exploratory approach often involves stepwise model building, including covariate model selection using repeated hypothesis testing with associated nominal P values (eg, .005). However, interpretation of the outcomes may sometimes be difficult. Both statistical theory⁵ and POPPK research⁶ have shown that for any given hypothesis (eg, the age effect on clearance), the actual P value can be much larger among the repeatedly tested than the nominal ones. Noting these shortcomings, a full-covariate model approach has been proposed, both in statistical framework⁵ as well as in POPPK and PK/pharmacodynamic (PD) modeling applications.^7,8 In the exploratory context, the nominal P value is used as a tool for model selection instead of hypothesis testing. Although it is possible in principle to account for the influence of model exploration on inference, this is difficult in practice.⁹

As a contrast, typical statistical analysis plans of clinical endpoints (such as efficacy endpoints), especially those of phase III studies, use the "confirmatory" analysis approach. Analysis models are prespecified even though alternative models are possible. For example, the endpoint could be assumed normally or lognormally distributed, or certain covariates may or may not have significant effects. However, these alternatives are generally not considered in confirmatory analyses. Although a few limited circumstances for similar analyses may be planned, they are usually considered "sensitivity analyses."

Given the similar uncertainty on the "best" model in both exploratory and confirmatory approaches, it may be interesting to consider the following: should the confirmatory approach become more exploratory, or does it offer something that traditional POPPK analyses can benefit from? Ribbing et al¹⁰ attempted to address this by applying the LASSO method in covariate selection to shrink the inflated covariate effect estimates due to model selection, which improved accuracy and prediction of covariate effects. This method is appealing in theory and involves the use of cross-validation to select the tuning constant that controls the shrinkage. As cross-validation requires repeated fitting, the implementation may be complex.

Based on the principle of parsimony, the exploratory approach is powerful because of its ability to examine a large number of alternative models and to choose a candidate using available data. Nevertheless, this flexibility comes with a price, especially if one wants to conclude in a confirmatory manner. In principle, ignoring the model selection process results in biased estimates and underestimated estimation variability, thus producing overly optimistic inference. This phenomenon has been well recognized in statistical literature. Faraway¹¹ noted that model selection results in biased inferences if uncertainties in model selection are ignored. In linear regression analysis, model selection is known to lead to inflated effects, and methods of correcting the inflation have been developed.⁵ Nonlinear modeling is much more complex. To prevent selection bias, Ribbing and Jonsson⁶ have suggested that covariate model selection should not be performed in the case of low power. Hu and Dong⁹ made similar suggestions in the context of structural model selection. In addition, it may not be easy to report on the robustness of the results. It is noted that the impact of these drawbacks is relatively limited when applied to phase I/II analyses intended for a "learning" purpose, with an understanding that later there will be a "confirming" stage.¹²

In other words, the exploratory approach selects a "most likely" or "best-fit" model among the possible choices. In contrast, the confirmatory approach may be better suited for making unbiased estimates because it addresses questions regarding alternative assumptions by sensitivity analyses. As soon as the choice of exploratory or confirmatory analysis is made, one has to accept the advantages and disadvantages associated with each approach. Therefore, deciding which approach to take depends on the objective of the analysis (ie, how the modeling results will be used).

Phase III POPPK—Learning or Confirming?
A major purpose of POPPK analysis is to assess covariate influence on long-term exposure for the purpose of guiding dose adjustments or label claims. Although a list of collected covariates is available for POPPK analyses intended for regulatory submissions, there is usually no clearly formulated statistical hypothesis to address which covariates need to be used for dosing adjustments and by how much. Therefore, the objective for POPPK appears to be more learning than confirming. Nevertheless, the results of POPPK used for labeling recommendations are treated as if from confirmatory analyses because usually more data will not be collected to confirm the phase III findings. The inflated covariate effect caused by exploratory approaches translates to upward biases in this context and may lead to unnecessary dosing adjustments. To what extent this happens in actual phase III analyses is difficult to determine, as "the truth" is generally unknown. In this sense, so far the best clues reside in the selection biases shown in simulations by Ribbing and Jonsson.⁶ For the purpose of accurate covariate-based dosing adjustment, finding a parsimonious "final" model, although attractive in principle, may be less relevant. Instead, it is more important to obtain unbiased estimates of the covariate effect on long-term systemic drug exposure, such that dose adjustment recommendations are robust with respect to modeling assumptions. To achieve these goals, confirmatory approaches would be better suited than exploratory ones (eg, the confirmatory approach used in bioequivalence assessment based on sparsely sampled PK data).¹³

The need for aligning model choice and purpose has been recognized (eg, Ette et al¹⁴ discussed "model appropriateness"). More recently, Hu and Dong⁹ commented on the need for basing model selection criteria on the intended use of the model.

	METHOD

TOP ABSTRACT METHOD RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
As noted above, the 2 major goals for POPPK analysis in phase III are that (1) estimates of the covariate effects with respect to dosing recommendations be unbiased and (2) the extent of uncertainty in the conclusions be clearly understood with respect to the underlying assumptions. A typical confirmatory statistical analysis plan for clinical studies may provide a good heuristic. It achieves unbiased estimation by prespecifying the analysis model and examining covariates or subgroup effects with sensitivity analyses.

To decide on the best choice for a prespecified model, it helps to examine what information the data provide. From a PK perspective, the maximum drug concentration (C_max) and area under drug concentration-time curve (AUC) are important parameters defining systemic drug exposure. As sparse sampling designs usually do not allow accurate assessment of C_max, the POPPK analysis could reasonably be focused on steady-state AUC only. This may also be assessed as average concentration or apparent clearance (CL/F). Therefore, the goal of phase III analysis should be on identifying covariates affecting CL/F. Considering the relevance to dosing adjustment, it seems reasonable to estimate covariate effects on only CL/F. Although this approach may seem too restrictive, concerns regarding its effect may be assessed by sensitivity analyses. It is noted that, although estimating covariate effects on other structural PK parameters could potentially improve the estimation of CL/F, it could also easily lead to loss of power. One related distracting issue is whether covariate testing needs to be performed explicitly for absolute bioavailability (F), as it affects systemic exposure as does CL/F. In principle, this should be based on a priori mechanistic considerations. This seems to be less often considered for phase III POPPK, however, and we shall discuss more on this at the end of this article.

Criteria for recommending covariate-based dose adjustments should depend on the therapeutic window of the compound. In the absence of clear criteria, it seems reasonable to exclude those discrete covariates from dose adjustments if their effects are within the usual bioequivalence range of (0.8, 1.25). For continuous covariates, it may be reasonable to require that the ratio between the first and third quartiles (ie, 25th and 75th percentiles) be within the range of (0.8, 1.25). With reasonable sample sizes to ensure the precision of the estimates, this should also ensure that the effects discovered are real and not false positive.

As mentioned earlier, the base model in this analysis context is a set of nuisance parameters but still needs to be determined. Early POPPK models can usually be built from phase I/II data and can provide important insights. However, they often are not directly applicable, as sparser sampling in phase III may not allow the structural model to be as complex as phase I/II data would allow. This is in contrast to covariate modeling, where phase III data will have a wider range of covariate values and allow better identification of covariate relationships. In addition, the random effect model best identifiable for phase III data could potentially also be different from that for phase I/II.

Proposal
Based on these considerations, the following steps for model selection and analysis are proposed for phase III data analysis.

(1) Establish Structural and Random Effect Models Based on Earlier Data
An earlier POPPK model built from phase I/II data (ie, the model that best represents existing data/information) should be used to simulate data under the phase III study design to find out which model may be reliably identified from the phase III study design. This is best achieved using the phase III POPPK data with concentrations removed, either informally or with a formal blinding procedure. The best-fit identifiable model with the simulated data will be prespecified for the phase III data. Note that for this purpose, in a case when only data from extravascular administrations are available from phase I/II while only intravenous administrations are given in phase III, a model with an absorption phase may provide enough information. One should still be able to use the phase I/II extravascular model with the absorption component stripped away to retain only the disposition component and then to simulate and see what models can be identified from phase III data. Every effort should be made to avoid the possible "flip-flop" situation. In the (unlikely) case of unstable estimation in phase III data, reducing the number of between-subject random effects in a predetermined sequence can be attempted first, with the next choice being simplifying the structural model. The model is prespecified in the statistical sense because the dependent variable has not been used to select it.

(2) Establish or Build Covariate Models
Prior to including any covariates, correlation among covariates should be examined together with pharmacological rationale to ensure to the best ability that only relevant covariates are selected. In cases of highly correlated covariates, only the more physiologically/pharmacologically/clinically relevant covariates should be retained.⁶ The total number of covariates should be small enough so that there are at least approximately 20 subjects⁵ for each estimated parameter. These choices may be difficult to make in practice but are important to ensure accuracy and precision of the results.

For continuous covariates, the model form should be prespecified with a standard choice, usually the power model scaled by the population median. That is, for a positive continuous covariate X, the effect of covariate X_i for the ith subject is modeled in the form of

(1)

where ₀ represents the "typical" population parameter and ₁ the magnitude of influence of covariate X, relative to its unit. Appropriate transformation is needed if the covariates are not positive—for example, using (1 + X) when minimum(X) = 0. Sparse sampling designs are unlikely to allow accurate identification among alternative covariate model forms.

For discrete covariates, model forms have no impact because they are all mathematically equivalent. We used the following parameterization: for a binary covariate X taking values 0 and 1, the effect of X_i is modeled in the form of

(2)

where ₀ represents the parameter for population X = 0 and ₁ the relative change from ₀ in population X = 1. Discrete covariates taking more than 2 values will be modeled similarly with more parameters. In this formulation, parameters ₀ represents the covariate effects on percent scale.

We propose to apply the full model, including all remaining covariates, on CL/F only. This is proposed as the primary analysis for phase III POPPK modeling as a means for assessing covariate-based dose adjustments.

To understand and assess the robustness of the primary analysis results, we propose several sensitivity analyses below.

(3) Sensitivity Analyses
In the primary analysis, concerns may arise over the influence of factors such as (a) model misspecification for both the base and covariate models and (b) inaccuracies in dosing and sampling time records. Depending on the situation, some or all of the following sensitivity analyses could be of interest:

Allometric scaling by weight (analysis a). Some literature suggests that clearance (CL) parameters are proportional to weight to the power of 0.75, and volume parameters are proportional to weight.^15-20 Note that this does not require any additional parameters. Although this may be arguable and the actual coefficients could be estimated, if a relationship indeed exists, the data may not have the power to distinguish between the alternatives because of lack of power or confounding effects. Using this as an alternative base model allows assessment of the robustness of the results relative to covariate modeling on other structural model parameters.

Analyzing covariate effect on observed concentrations (analysis b). The following simple model is proposed:

where Y_ij is the jth observed concentration of the ith subject, and X_ni is the nth covariate of the ith subject. The covariate that influences f_n(X_ni) takes the form of equation (1) or (2), depending on whether the covariate is continuous or discrete. In addition, _i is the between-subject random effect, and _ij is the error term, both assumed to be normally distributed. Parameter b0 typically may be fixed at 1, assuming linear PK. In this model, parameter a represents a type of average concentration scaled by dose, and f_n(X_ni) represents the influence of covariate X_n on average observed (log) concentration, which is closely related to CL, especially at steady state. In an approximate sense, the influence on CL is represented by –₁ under equation (1) for continuous covariates and by exp(–₁) under equation (2) for discrete covariates.

After log-transforming both sides, the proposed simple model becomes a simple linear mixed effect model in the log-domain for continuous covariates and can be implemented in standard statistical software such as SAS or S-PLUS. As the model form is linear, the P values associated with covariate effects can be referenced with little controversy,²¹ unlike that of nonlinear mixed effect models.

Base model random effects (if applicable) (analysis c). Phase III data may be more variable than that of phase I/II for many reasons, such as more and/or a larger range of covariates, compliance, dosing accuracy, sample collection accuracy, concomitant medications, and disease conditions. Therefore, increasing the number of random effects in a predefined order of importance may be attempted, especially if the prespecified base model contains the same number of random effects as the previous model. However, this rationale does not seem to apply to the within-subject error model. Therefore, the form of within-subject error model should not be explored.

Notes
Several notes may help to clarify the purposes of these analyses, especially analysis (b). Frequently, concentrations are collected at different time points relative to doses by design (eg, peak/trough). The resulting time effect can usually be represented as a discrete covariate depending on the specific study design. Statistically, this will be a nuisance effect because it cannot be used as a sensible covariate for dose adjustments. The proposed model analyzes a certain type of dose-normalized, time-integrated exposure measure as an endpoint, in part depending on the time covariate to be used. For the ease of data handling, the application examples in this article used each subject's nominal dose as Dose_i. In an intuitive sense, the parameter a represents F/CL up to a multiplicative constant with unit of time (day^–1), and thus the model may be expected to provide an assessment of the covariate effect on CL/F. The model can be improved by representing Dose_i as each subject's dose rate, if dosing schedules differ substantially among subjects (eg, when compliance is an issue). This model could be viewed as a simplistic way to account for the time effect and is less efficient than compartment models because information on dosing times is not fully used. However, such information is often inaccurate in phase III trials, and testing the amount of such information in the data would be of interest. In this regard, this analysis could show whether the results are robust with respect to the structural model assumptions. In other words, if analyses (a) and (b) give similar conclusions, then the findings are confirmed. Differences in results may suggest that results of (a) could be less robust to sample/dose time inaccuracies.

In these analyses, (a) and (b) are prespecified, and (c) is only partially data driven. Additional data-driven explorations could be interesting to address other issues (eg, potential influence of covariates on other structural model parameters suggested by the data). However, attempting this immediately causes selection bias to become an issue, making the uncertainty and robustness of the result difficult to assess. Thus, if conducted at all, such explorations should be limited to, for example, 1 to 2 model runs, instead of intensive exploratory searches. In addition, results from all such efforts should be interpreted as secondary.

Because analyses (a) and (b) are prespecified, either can be used as a primary analysis depending on one's preference in particular situations. It is also possible to use fewer or more (but limited) sensitivity analyses, if desirable.

	RESULTS

TOP ABSTRACT METHOD RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
The following examples demonstrate applications of this proposed approach. The prior phase I/II data are briefly described. Although details of the model building may be of curiosity, they are not described here as model building or its evaluation is not the purpose of this article. The results are compared with submission analyses that used the traditional exploratory approach. Analysis (b) was implemented with S-PLUS (Version 8.0, Insightful, Seattle, Washington) and other analyses with NONMEM (Version VI, Level 1, ICON Development Solutions, Ellicott City, Maryland).

Application I: Therapeutic Protein A
Combining 2 pivotal phase III multicenter studies generated a submission POPPK data set consisting of 1937 subjects and 9948 concentration measurements. According to the proposed approach, the following analysis plan and rationale had been specified before beginning the analysis on observed concentrations.

(1) Structural and Random Effect Models
A prior phase I/II data set was available, consisting of 307 patients and 2132 concentration observations, after single and multiple subcutaneous administrations of up to 52 weeks. A previous POPPK model based on this data set was developed, with a structural model of a 1-compartment model with extravascular absorption. It had between-subject variance on absorption rate constant (Ka), CL/F, and apparent volume of distribution (V/F), as well as an additive-plus-proportional residual error. In addition, CL/F was proportional to weight raised to the power of 0.75 and V/F proportional to weight. This model was used to simulate data with the actual dosing information and sampling times in the phase III studies. The best random effect structure was developed from the simulated data, based on the usual fitting criteria.

(2) Covariate Model
The full model approach was used on CL/F only. That is, no covariate models were used on V/F or Ka. The covariates were classified into the following order, based on a priori considerations of relevance: weight, immune response (positive/nonpositive), age (<65 vs 65), sex, race (Caucasian, Black, Asian, other), geographic region, past use of biologics (yes/no), past use of methotrexate (yes/no), past use of cyclosporine (yes/no), duration of disease, baseline disease score, baseline physician global assessment score, baseline disease quality index, concurrent disease A status (yes/no), concurrent disease B history (yes/no), concurrent disease C history (yes/no), alcohol history (yes/no), smoking status (yes/no), and the following concomitant medicines: D1, D2, D3, D4, D5, D6, D7, D8, D9, and D10. This ordering was intended to specify the covariate exclusion ordering, in case too many covariates were included through overly optimistic power estimation. In this example, the following selection considerations were used for inclusion of covariates in the covariate models: physiologically relevant covariates (eg, weight, age, sex, race), pharmacologically important covariates (eg, immune response status), baseline disease severity and/or status (eg, duration of disease, baseline disease score), geographic location (eg, geographic region), past use of relevant medications (eg, past use of biologics, past use of immunosuppressants), relevant concurrent comorbidities (eg, concurrent disease #1 status), and most commonly used concurrent medications (eg, D1, D2). There are no fixed rules in deciding which covariates should or should not be included for testing using this full-model approach. Nevertheless, several key elements may be worth considering: inherent pharmacokinetic behavior, intended indication and patient population, intended product label, and anticipated important/relevant covariates as learned from similar or related molecules.

(3) Sensitivity (Secondary/Exploratory) Analysis
For analysis (b), upon inspection of sampling times, indicators were created for time 4 weeks and every 4 weeks from weeks 8 to 40 to account for the time effect. In addition, because of the complication of a dosing regimen switch for partial responders, observations beyond 40 weeks were excluded, resulting in excluding <10% of total concentrations. Because the phase III study design was not expected to allow accurate estimation of Ka, analysis (c) was not implemented.

Results
The results are summarized according to the corresponding structure of the proposed approach. The (0.8, 1.25) bioequivalence criterion for the estimated effect on CL/F was used to determine the relevance of a covariate.

(1) Structural and random effect models. The phase I model had a between-subject random effect term on Ka. Base model exploration on simulated data showed that, as expected, this term could not be identified. Influence of weight on CL/F and V/F was deemed estimable. This model was reliably estimated from the observed data.

(2) Covariate model. The full model had weight, concurrent disease #1 status, and immune response as relevant covariates affecting CL/F, with estimated effect being 1.13, 1.22, and 1.37, respectively. As the first and third quartiles for weight in the population are 76.4 kg and 104.7 kg, the weight effect was calculated as (104.7/76.4)^1.13 = 1.428, or 43%.

(3) Sensitivity (secondary/exploratory) analysis. Analysis (a) had weight affecting CL/F by default, with an effect of (104.7/76.4)^0.75 = 1.267, or 27%. Concurrent disease #1 status and immune response were the only other relevant covariates affecting CL/F, with the estimated effect being 1.25 and 1.46, respectively.

Analysis (b) also had weight, concurrent disease #2 status, and immune response as relevant covariates affecting CL/F, with estimated effect being 1.157, 1.294, and 1.36, respectively. In addition, it had a concurrent medication D7 effect of size 1.259. Complete results are presented in Table I.

All analyses consistently suggested that weight, concurrent disease #1 status, and immune response were important covariates, with a 27% to 43%, 22% to 26%, and 30% to 37% effect, respectively. Sensitivity analysis (b) also had concomitant medication D7, with an effect size at borderline importance, with other analyses showing approximately half that. Therefore, concomitant medication D7 was considered unimportant. Overall, the effect estimates appeared robust.

Comparison With the Submission Analysis
The submission analysis had the same base model as (1) above but different covariates affecting CL/F and V/F. However, regarding recommendation of potential dose adjustment, it also had weight, concurrent disease #1 status, and immune response as the only covariates affecting CL/F, with estimated effect sizes of 0.862, 1.282, and 1.358, respectively. Given the difference in the amount of exploration used between the approaches, these results are notably similar.

The submission analysis had sex affecting CL/F, with a 5.9% increase in women. It also had covariates albumin, creatinine clearance (CrCL), and alkaline affecting CL/F, which were excluded from the prespecified analyses based on a priori considerations. Although these covariates were deemed insignificant in the submission analysis, CrCL had a high correlation (= 0.72) with weight, with an effect estimate of 0.181, which at the first and third quartiles gives (150.2/99.8)^0.181 = 1.077, or 7.7%. This may account for a majority of the 12% (= 43%-31%) difference with the primary analysis. Because renal function is unlikely to contribute notably to apparent clearance of therapeutic protein A, the CrCL effect is likely spurious. This would suggest the weight effect as perhaps closer to 43% than to 31%.

Overall, both approaches suggested that weight, concurrent disease #1 status, and immune response were the only important covariates. Our proposed approach achieved this by using fewer than 10 models altogether, including base model exploration using simulated data. In comparison, the submission analysis used more than 100 documented models alone.

Application II: Therapeutic Protein B
A prior phase I/II data set was available, consisting of 197 patients and 2257 concentration observations, after single and multiple subcutaneous administrations of up to 48 weeks. Two phase III studies for therapeutic protein B were conducted in 594 patients with 3422 serum drug serum concentrations included in the POPPK data set. According to our proposed approach, the following analysis plan and rationale were specified before beginning the analysis on observed concentrations:

(1) Structural and Random Effect Models
A previous POPPK model based on phase I/II data was developed, with a structural model of a 2-compartment model with extravascular absorption. It had between-subject variance on all 5 structural model parameters and an additive-plus-proportional residual error. In addition, power models of weight on clearance and volume were identified. This model was used to simulate data with the actual dosing information and sampling times in the phase III studies. The best random effect structure was developed from the simulated data based on the usual fitting criteria.

(2) Covariate Model
The full-model approach was used on clearance only. That is, no covariate models were used on V/F or Ka. The covariates were classified into the following order, based on a priori considerations of relevance: weight, antibody status (yes/no), age, sex, race (Caucasian, Black, Asian, others), use of methotrexate, use of concomitant nonsteroidal anti-inflammatory drugs (NSAIDs), use of concomitant corticosteroids, disease duration, and baseline disease activity (C-reactive protein). To ensure power, an a priori cutoff of at least 20 subjects per covariate category was used.

(3) Sensitivity (Secondary/Exploratory) Analysis
To account for different sample types corresponding to the design, we included a time indicator with term week 4 trough, week >4 trough, early nontrough, or late nontrough in analysis (b). Because the phase III study design was not expected to allow accurate estimation of Ka, analysis (c) was not implemented.

Results
The results are summarized according to the corresponding structure of the proposed approach. The (0.8, 1.25) bioequivalence criterion for the estimated effect on CL/F was used to determine the relevance of a covariate.

(1) Structural and random effect models. Base model exploration based on simulated data showed that only a 1-compartment model with a block random between-subject effect on CL/F and V/F could be identified. The influence of weight on CL/F and V/F was deemed estimable.

(2) Covariate model. The full model had weight as the covariate with the largest effect. The weight effect estimate was 0.707, which at the first and third quartiles gives (82/60)^0.707 = 1.247, or 24.7%.

(3) Sensitivity (secondary/exploratory) analysis. Analysis (a) had weight affecting CL/F by default, with an effect of (82/60)^0.75 = 1.263, or 26.3%. This is similar to the full-model result because the change of weight effect was small. Other effect estimates were similar for the same reason.

Analysis (b) gave a weight effect estimate of 0.585, or (82/60)^0.585 = 1.2, or 20%. Use of methotrexate was an additional covariate, with an effect estimate of (0.712 – 1) = –28.8%. Complete results are presented in Table II.

All analyses consistently suggested weight as a potentially relevant covariate, with a 20% to 26% effect. Sensitivity analysis (b) had use of methotrexate as an additional covariate, with an effect estimate much larger than that of the other analyses. Further inspection of observed concentrations showed that the differences between those with and without methotrexate vary over time. This suggests that there is an interaction between methotrexate and time effects, which is implicitly accounted for in the PK models but not at all in sensitivity analysis (b). Therefore, estimates based on PK models may be more accurate. Overall, the conclusion of weight as the only potentially relevant covariate and its effect estimates appeared robust.

Comparison With the Submission Analysis
The submission analysis identified 1 observation as an "outlier" and removed it from the analysis. The analysis had the same structural model but an additional random effect on Ka. In addition, it used a proportional residual model with a multiplicative between-subject random effect term.¹⁹ It also found a weight effect on V/F. Statistically significant covariates found to affect CL/F included weight, C-reactive protein, and use of methotrexate, with effect estimates of 0.605, 0.0746, and 0.839, respectively. The weight effect was (82/60)^0.605 = 1.208, or 20.8%. The C-reactive protein estimate was similar to the primary analysis at 16% and was considered not relevant. It also found antibody status as a relevant covariate, with effect estimates of 29%. This covariate was excluded from the prespecified analyses because of the small number (n = 16) of subjects with antibody status. The effect estimate is unlikely to be precise.

Other than the issue with immune response, both approaches reached similar conclusions. The proposed approach achieved this by using fewer than 10 models altogether, including base model exploration using simulated data. In comparison, the submission analysis used nearly 50 documented models with many more nondocumented runs and required much deliberation time over which models to adopt at different stages.

	DISCUSSION

TOP ABSTRACT METHOD RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
Compared with common POPPK analysis strategies, our proposed approach better fulfills the objective of accurately identifying covariate-based dosing adjustment factors at phase III. Our approach has 1 primary analysis and 2 to 3 sensitivity analyses. By using previous information and vastly reducing the amount of explicitly and implicitly considered models, our new approach has better statistical properties, including accuracy and robustness, compared with the traditional extensive exploratory approach. In addition, our approach allows more clarity of the uncertainty of the conclusions and is relatively easy to implement with appropriate preplanning before analysis.

The proposed approach was also tested on an unpublished small-molecule phase III submission POPPK data set, which consisted of over 3000 subjects and more than 13 000 concentration measurements, under a "peak/trough" sampling design. A similar conclusion was obtained, although results from sensitivity analysis (b) were closer to the primary analysis than the submission analysis. This is understandable, as the drug was administered orally, and thus the dosing time information was conceivably less accurate. In contrast to <10 models used with the proposed approach, the submission analysis was governed by a diligent analysis plan dictating that more than 2000 models be tested, which may have resulted in a larger selection bias.

One might regard this lightly by arguing that "artful" modelers generally are able to reach a final model with much fewer runs. However, the point here is that the more diligent the work, the more it could hurt—in statistical terms, the more exploration, the more selection bias.⁵ A further advantage of our proposed approach is that it leads to standardized analysis, avoiding the phenomenon that different modelers give different results. This could make the POPPK results more credible and interpretable.

Because sensitivity analyses (a) and (b) are prespecified, either could be used as the primary analysis, depending on prior considerations. Analysis (a) may be preferable when allometric scaling is performed. Analysis (b) may be especially appealing when observation times are imprecise. Analysis (c) is partially data driven and did not prove useful in the examples, but it could have some appeal in future situations.

Differences between our proposed approach and the extensive exploratory analyses used for regulatory submissions were usually small. This is consistent with both theoretical predictions and practical expectations, in the sense that disadvantages with model exploration become small with relatively large sample sizes. Nevertheless, the exploratory approach usually does not deal with the issue of inaccuracies in dosing or sampling time records. Neither does it formally address the issue of model misspecification, for the following reasons. First, there is no guarantee that a stepwise model search will find the best fit. In our therapeutic protein B example, a stepwise search did not initially identify weight as an influential covariate. Second, the best-fit model may not be best for assessing influential covariates.^5,6

As a first attempt on applying the confirmatory approach, certain aspects will need further deliberation in the future. For example, our experience so far suggests that the random effect model in the base model may be prespecified as the earlier phase I/II model, but more evidence may be needed to confirm this. Basing covariate models on F may need further consideration, as mentioned earlier. The step to establish structural and random effect models based on earlier data could be less essential for the following reasons. First, the choice of base model might not have large effects on the estimation of the covariate-CL/F relationship. This can make it more acceptable to use a prespecified model. Second, more post hoc exploratory-type analyses could still be conducted without compromising the statistical properties as long as they are "sensitivity analyses" and their results interpreted as such. Details of implementation can be worth further consideration. The most important steps in our proposed approach are to prespecify the covariate models and to adopt the common statistical analysis approach by prespecifying a primary analysis and sensitivity analyses. This will result in the benefits of confirmatory results that may be more suitable for assessing dosing adjustment needs.

The concept of model-based drug development has received increased attention and has been recognized by the Food and Drug Administration.²² In this context, our approach illustrates the use of a previously developed population model for inferential use in phase III studies. The previously developed model is an important step that makes this approach much more applicable, which brings all the benefits of confirmatory interpretations.

	ACKNOWLEDGEMENTS

TOP ABSTRACT METHOD RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 
The authors are grateful for the excellent editorial assistance of the Medical Affairs writing group at Centocor.

Financial disclosure: This study was funded by Centocor Research and Development, Inc. All authors are employees of Centocor Research and Development, Inc and own stock in of Johnson & Johnson.

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TOP ABSTRACT METHOD RESULTS DISCUSSION ACKNOWLEDGEMENTS REFERENCES

 

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