Modeling adaptive response profiles in a vaccine clinical trial

Dicle Hasdemir, Robert A van den Berg, Antoine van Kampen, Age K Smilde, Dicle Hasdemir, Robert A van den Berg, Antoine van Kampen, Age K Smilde

Abstract

Background: Vaccine clinical studies typically provide time-resolved data on adaptive response read-outs in response to the administration of that particular vaccine to a cohort of individuals. However, modeling such data is challenged by the properties of these time-resolved profiles such as non-linearity, scarcity of measurement points, scheduling of the vaccine at multiple time points. Linear Mixed Models (LMM) are often used for the analysis of longitudinal data but their use in these time-resolved immunological data is not common yet. Apart from the modeling challenges mentioned earlier, selection of the optimal model by using information-criterion-based measures is far from being straight-forward. The aim of this study is to provide guidelines for the application and selection of LMMs that deal with the challenging characteristics of the typical data sets in the field of vaccine clinical studies.

Methods: We used antibody measurements in response to Hepatitis-B vaccine with five different adjuvant formulations for demonstration purposes. We built piecewise-linear, piecewise-quadratic and cubic models with transformations of the axes with pre-selected or optimized knot locations where time is a numerical variable. We also investigated models where time is categorical and random effects are shared intercepts between different measurement points. We compared all models by using Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Deviance Information Criterion (DIC), variations of conditional AIC and by visual inspection of the model fit in the light of prior biological information.

Results: There are various ways of dealing with the challenges of the data which have their own advantages and disadvantages. We explain these in detail here. Traditional information-criteria-based measures work well for the coarse selection of the model structure and complexity, however are not efficient at fine tuning of the complexity level of the random effects.

Conclusions: We show that common statistical measures for optimal model complexity are not sufficient. Rather, explicitly accounting for model purpose and biological interpretation is needed to arrive at relevant models.

Trial registration: Clinical trial registration number for this study: NCT00805389, date of registration: December 9, 2008 (pro-active registration).

Keywords: AIC; Adjuvant; BIC; Conditional AIC; DIC; Immunology; LMM; Linear mixed model; Model selection; Quantification of individual differences; Random effect selection; Vaccine.

Conflict of interest statement

RvdB is employee of the GSK group of companies. RvdB reports ownership of shares and/or restricted shares of the GSK group of companies.

Figures

Fig. 1
Fig. 1
Time profiles of antibody (Ab) levels. Each Ab-profile (plotted in a different color from the gradient blue color scale) denotes a subject. The y-axis denotes the antibody levels on log10 scale. The x-axis indicates the measurement points: prior to vaccinations (PRE), Day 30 (PI(D30)), Day 44 (PII(D44)), Day 60 (PII(D60)), Day 180 (PII(D180)) and Day 360 (PII(D360)). PI and PII show the measurements after the first and second vaccinations, respectively. The five panels correspond to different adjuvants (AS01B, AS01E, AS03A, AS04 and Alum) and each group consists of different subjects. The bird’s eye view on Ab levels presented here helps to visualise the increased divergence between the individual responses observed especially in the AS04 and Alum groups
Fig. 2
Fig. 2
Population estimates for the 2-segment and 3-segment PW linear models with a-priori fixed knot locations. Black lines show the time profile (interconnected data points) of an individual. Red and blue lines (estimates interconnected at the measurement points) show the population estimates obtained by the 2-segment and 3-segment PW linear models with a-priori fixed knot locations. These 5 individuals are typical examples of their adjuvant group. Therefore, the population estimates obtained by the models (time profile prediction obtained by using only the fixed part of the model) should reflect the example individual time profile
Fig. 3
Fig. 3
Selected fits from the Alum group with optimized knot locations for the 3-segment first order models. Each panel represents an individual from the Alum adjuvant group that was selected for demonstrative purposes. Red lines indicate the model fit and the black lines connect the measured data points. Individuals 160 and 420 are examples whose final line segment, starting from D45, still show an increasing response due to insufficiency of the model structure used. Individual 449 is an example of a good model fit. Overall, the graph shows inter-individual variation in the Alum group
Fig. 4
Fig. 4
Selected cubic model fits of individuals from the Alum group. Each panel represents an individual from the Alum adjuvant group that was selected for demonstrative purposes. Red lines indicate the model fit and the black lines connect the measured data points. The model is inadequate to fit the steep increase in response for individuals 187 and 343. The model is able to correctly fit the prolonged delay in response shown for individuals 386 and 518
Fig. 5
Fig. 5
Selected fits of individuals fits from the AS04 group with quadratic piecewise models. Each panel represents an individual from the AS04 adjuvant group that was selected for demonstrative purposes. Red lines indicate the model fit and the black lines connect the measured data points. Individuals 181, 309 and 658 are examples where both steep increases and slight decreases with different maxima across individuals are handled well by the model. Individual 322 shows a case in which a prolonged delay was handled well
Fig. 6
Fig. 6
Model fit for all groups. The y-axis shows the normalized sum of squared residuals from the fitted data for all groups separately. The value is always normalized to the sum of squared residuals from the first model of each panel. The x-axis shows the models with random point-wise intercepts. For example, model t12+t3+t4+t5 shows a model with no random intercept at t0, a shared random intercept at t1 and t2, and different intercepts at t3, t4 and t5. The first model is always the most flexible model with separate random intercepts for t1, t2, t3, t4 and t5
Fig. 7
Fig. 7
Selected categorical model fits for four individuals from the Alum group. Red lines indicate the model fit and the black lines connect the measured data points. Individual 315 shows a too perfect fit. Similarly, the other individuals still show very good fits, much better than the numerical models

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